Preface
In theoretical and applied areas of mathematics we frequently deal with sets endowed with various structures. However, it may happen that the consideration of a set with a specific structure, say topological, algebraic, order, uniform, convex, et cetera is not sufficient to solve the problem posed and in that case it becomes necessary to introduce an additional structure on the set under consideration. To confirm this idea, it will do to recall the theories of topological groups, linear topological spaces, ordered topological spaces, topological spaces with measure, convex topological structures, and others. This list is not complete without adding the theory of bitopological spaces and also the theory of generalized Boolean algebras connected with certain classes of bitopological spaces. The notion of a bitopological space (X, ~-1,T2), that is, of a set X equipped with two arbitrary topologies T 1 and T2, was first formulated by J. C. Kelly in [151]. Kelly investigated nonsymmetric distance functions, the so-called quasi pseudometrics on X x X, that generate two topologies on X that, in general, are independent of each other. Previously, such nonsymmetric distance functions had been studied in [262] and in [219]. Although [151] is beyond any doubt an original and fundamental work on the theory of bitopological spaces, nevertheless it should be noted that both the notion of a bitopological space and the term itself appeared for the first time in a somewhat narrow sense in [181], [182] as an auxiliary tool used to characterize Baire spaces. For this use, the topologies T1 and ~-2 on a set X, one of which was finer than the other, were connected by certain other relations as well. Mention should be made of the viewpoint of A. A. Ivanov [137], following which a pair (X, T), where X is any set and ~- is any topological structure on X x X, is called a bitopological space in the general sense. Therefore to a bitopological space (X, ~-1,~'2) in the sense of Kelly, there corresponds a bitopological space in the general sense of the type (X, T), where ~- = T1 X 7 2 is the product topology on X x X, which in [137] is called a decomposable bitopological structure (see also [139], [141], and the interesting work [143] together with the bibliography on bitopological spaces [laS], [140], [142]). We shall adhere to Kelly's notion which at present seems to be more flexible for various usage the main objective of this monograph - developing the theory of bitopological spaces with its applications. Distance functions, uniformity, and proximity are the related notions in defining the topology and, naturally, the situation treated in [151] is by no means the
x
Preface
only way leading to a symmetric occurrence of two topologies on the same set; the investigations of quasi uniformity [190], [250] and quasi proximity [203], [124], [243] also lead to an analogous result. These topics are best covered by H. P. A. Kiinzi in [157]. Keeping in mind the symmetric generation of two topologies on a set, along with the above-mentioned cases, we can also consider ordered topological spaces [191], [208], [5], [53], [177], [178], partially ordered sets [10], and hence directed graphs [64], [65], semi-Boolean algebras [212], S-related topologies [252] and so on. On the other hand, there are many examples of nonsymmetric occurrence of two topologies on a set, particularly in general topology, analysis, and potential theory, as well as in topological convex structures (see, for example, [1], [2], [4], [7], [21], [22], [67], [260], [125], [44], [255], [172], [173]). From the above-said it follows that due to the specific properties of the considered structures two topologies are frequently generated on the same set and can be either independent of each other though symmetric by construction or closely interconnected. Certainly, the investigation of a set with two topologies, interconnected by relations of "bitopological" character, makes it possible on some occasions to obtain a combined effect, that is, to get more information than we would aquire if we considered the same set with each topology separately. If we compare all the results available in the theory of bitopological spaces from the general point of view, we shall find that in different cases two topologies on a set are not, generally speaking, interconnected by some common law that takes place for all bitopological spaces. However if, when defining a bitopological notion, the closure and interior operators are successively applied in an arbitrary initial order to the same set, then, in general, these operators will interchange in topologies as well. As a weighty argument in favour of the above reasoning, we can consider the natural bitopological space (R,a~l,aJ2), where R is the real line with the lower wl = {2~,R} U {(a,+oo) : a e R} and upper w2 = {2~,R} U { ( - o c , a) : a e R} topologies [31] playing nearly the same role in the theory of bitopological spaces as R with the natural topology co = { ~ , R } U {(a, b): a, b E R} plays in general topology and analysis as a whole. Indeed, if for an arbitrary subset A c R we take its interior in the topology c01 (respectively, co2), then the smallest closed subset, which contains this interior, is the closure of this interior in the topology co2, but not in wl (respectively, in a;1, but not in w2). Now, if for an arbitrary subset firstly we take its closure in the topology CO1 (respectively, u;2), then the largest open subset, which is contained in this closure, is the interior of this closure in the topology co2, but not in col (respectively, in u;1, but not in w2). This simple example confirms the essence of closure and interior operators, which to each subset A c R put into correspondence respectively the smallest closed set containing A and the largest open set contained in A, on the one hand, and confirms convincingly the above-mentioned interchange principle, on the other hand. In addition to our motives for studying a bitopology, that is, an ordered pair of topologies (T1, T2) on a set X, we have also derived a stimulus from G. C. L. Briimmer [50], where important problems of the same kind are referred to, in particular, hyperspaces and multivalued functions [242], [27]; function spaces [207]; H-closed,
Preface
xi
almost real-compact, nearly compact and k-compact spaces [114], [164], [231], [232]; Wallman compactifications [46], [231]; topological semifields [135]; algebraic geometry and continuous lattices [50]. It should be also noted that at present there are several hundred works dedicated to the investigation of bitopologies; most of them deal with the theory itself, but very few deal with applications. These latter papers have been published after the late 70s (see, for example, [208], [5], [53], [64], [65], [173], [50], [242], [27], [207], [114], [164], [231], [232], [46], [135], [25], [89], [90], [93]-[103], [9]). We should mention [109], [110], where J. Ewert, shows that a separable Banach space with the weak topology and the topology determined by the norm, has interesting bitopological properties, on the one hand, and gives the Baire classification of multivalued functions of topological to bitopological spaces, on the other hand, and [249] since according to its author J. Swart the axiomatic topological characterization of Hilbert spaces is due to a large extent to the bitopological analogue of the notion of an open cover from [113]. In the above context we can also recall [258], where the term "consistent" equivalent to "bitopological Hausdorff" is one of the key notions, and [173], where the "bitopological boundary" is essentially used for establishing the minimum principle for finely hyperharmonic functions. The theory of bitopological spaces and its applications owe much to J. M. Aarts [3], [4]; D. Adnadjevid [5], [6]; S. P. Arya [16]-[19]; B. Banaschewski [23]-[251; T. Birsan [301-[33]; G. C. L. Briimmer [46]-[52]; A. Csgszgr [69], [70]; M. C. Datta [72]-[74]; J. Dei~k [76]-[78]; D. Doitchinov [80], [81]; P. Fletcher [112]-[117]; M. Jelid [144]-[147]; Y. W. Kim [152], [153]; H. P. a. Kiinzi [154]-[159]; E. P. Lane [165][167]; M. Mrgevid [3], [4], [183]-[188]; M. G. Murdeshwar [189]; S. a. Naimpally [189], [194], [195]; C. W. Patty [202]; W. J. Pervin [203]-[205]; H. A. Priestley [208]; I. L. Reilly [213]-[218]; S. Romaguera [221]-[225]; M. J. Saegrove [228]; S. Salbany [229]-[233]; A. R. Singal [235]-[238], [240]; M. K. Singal [235], [237]-[239]; J. Swart [248], [249], and to many other authors, who are not listed here and to whom we offer our apologies. This monograph is a versatile introduction to the theory of bitopological spaces and its applications. It considers the topics of bitopology that were studied perfunctorily or not studied at all and presents original results and examples which, we dare think, will stimulate the reader to further research. The monograph consists of eight chapters, of which Chapters III, IV, V, VI, VII form the core because they contain the basic results related to the abovementioned topics. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, theory of ordered topological spaces, and graph theory. Moreover, a high level of modern knowledge of bitopological spaces theory has made it possible to introduce and study an algebra of new type, the corresponding representation of which brings one to the special class of bitopological spaces.
xii
Preface
To conclude the preface, we would like to note that we firmly believe that from the standpoint of applications the theory of bitopological spaces has no less promising prospects than the theory of topological spaces. The areas of such applications are, in our opinion, the theories of linear topological spaces and topological groups, algebraic and differential topologies, the homotopy theory, not to mention other fundamental areas of modern mathematics such as geometry, analysis, mathematical logic, the potential theory, the probability theory and many other areas, including those of applied nature. In particular, the study of strong and weak topologies in analysis, the initial and the Alexandrov topologies on a manifold in the global Lorentzian geometry, cohomologies of spaces with two topologies, and the theory of foliations seems very promising for future research.
CHAPTER 0
Preliminaries Besides being auxiliary, this chapter also contains the internal characterization of pairwise completely regular bitopological spaces. In Section 0.1, along with the symbols and notations, we give a survey of the basic concepts and results from the theory of bitopological spaces to be used in our further investigation. In particular, we recall various kinds of pairwise separation axioms and their interrelations established by J. C. Kelly [151]; E. P. Lane [166]; M. G. Murdeshwar and S. A. Naimpally [189]; J. Swart [248]; I. L. Reilly [215], [217]; Y. W. Kim [152]; T. Birsan [31]; A. R. Singal [236]; M. K. Singal and a. R. Singal [238]; C. W. Patty [202]; D. N. Misra and K. K. Dube [180]; M. J. Saegrove [228]; W. J. Pervin and H. Anton [205], and others. Since the study of relations between the theory of bitopological spaces and some other branches of mathematics in Chapter VII demands special knowledge of bitopologies, we recall the appropriate notions of pairwise compactness, pairwise local compactness, pairwise paracompactness, pairwise local LindelSf property, and pairwise paraLindelSf property. These notions were formulated for the first time by P. Fletcher, H. B. Hoyle, and C. W. Patty [113]; J. Swart [248]; M. C. Datta [72]; M. Mrgevid [183], [184]; R. A. Stoltenberg [245]; I. L. Reilly [214]; T. G. Raghavan and I. L. Reilly [209]; T. Birsan [30]. We also present bitopological versions of connectedness and similar type properties, the study of which was initiated by W. J. Pervin [204]; H. Dasgupta and B. K. Lahri [71]; J. Swart [248], and C. Amihg~esei [12]. Consideration is given to the bitopological notions of continuous, open, closed, and homeomorphic maps introduced by J. Swart [248] and A. R. Singal [2361. In the topological case the complete regularity in internal terms, that is, without using the notion of a function, was characterized by O. Frink [119], E. F. Steiner [244], and V. I. Zaicev [264]. Their modifications for bitopological spaces were studied by Saegrove, who used the generalization of Steiner's method, and by us with the aid of the generalized method of Frink and Zaicev. 0.1. S y m b o l s a n d N o t a t i o n s . Basic C o n c e p t s of B i t o p o l o g y Throughout the book, along with the generally accepted symbols, we use our own notations and those from [68] and [200]. Sets are usually denoted with italic capitals A, B , . . . and elements of sets with lower case italic a, b, . . . . Sets whose elements are sets are called families of sets, while their elements are called members. Families of sets and classes of functions,
2
0. P r e l i m i n a r i e s
with rare exceptions, are denoted with one or two script letters A, B , . . . , followed by one italic capital in brackets for families of sets signifying spaces, and two italic capitals in brackets, in the case of classes of functions, denoting the corresponding spaces. The empty set is denoted by ;~, while the symbols N, Z, Q and R are respectively used for the sets of all natural numbers (excluding zero), of all integers, of all rational numbers, and of all real numbers. We also use oc to indicate "an infinite number". Other standard symbols are defined for each n E N as follows: n - 1, k means that n E { 1 , 2 , . . . , k } and n - 1, oc means that n E { 1 , 2 , . . . }. The closed (open) real segment joining a and b, denoted by [a, b] ((a, b)), is the set { x c R " a_<x_
0.1. Symbols and Notations. Basic Concepts of Bitopology
3
Their interest was naturally preconditioned by the essence of many topological and bitopological properties, especially by the axioms of separation. Further, the reasons connected either in an obvious or a veiled manner with the investigation of bitopologies (7"1,7"2) on a set X, where 7"1 and 7"2 are either independent of each other or interconnected by the inclusion, S-, C-, and N-relations or by their various combinations (see Chapter II), on the one hand, and, with applications of bitopologies, on the other hand, led us to the following basic objectives regarding two general cases: (1) to establish pairwise properties using the properties of 7"1 and 7"2 (or the properties of one of them) or other pairwise properties (or their combinations); (2) to establish properties of 7"i using the properties of 7"j or pairwise properties (or their combinations). As our study shows, (1) suggests the development of the theory of BS's, while (2) is natural and typical of applications, especially when i = 2, j = 1, 7"1 C 7"2. Finally, please note that all bitopological generalizations in this work are constructed in the commonly accepted manner so that if the topologies coincide, we obtain the original topological notions. Let (X, 7"1,7"2) be any BS and A c X be its any subset. Then 7"i clA and 7-i int A denote, respectively, closures and interiors of A in the topologies 7"i. If A = {As}sos c 2 x is any family, then i-C1A - {7"i cl As }s~s,
i-Int A - co i-C1 co A - { 7"i int As }ses'
(i, j)-C11nt at - { 7"i cl 7"j int As } scs' (i,j)-IntC1A - co(i,j)-ClIntcoA
- {7"~int 7"j clAs}s~s ,
i-~-~ (X) - { A E 2 X" A is an i-Fo-set } and i - G s ( X ) - coi-)r~(X) - {A 6 2 X" A is an
i-a~-set}.
Let (I~,wl,a;2) and (R, cv) be respectively the natural BS and the natural TS. If I - [0, 1] - {x c R " 0 < x < 1} is the closed unit segment, then (I,c~,a;~) and (I,a;') are, respectively, the BsS of (R, czl,a;2) and the TsS of (R,c~). Moreover, ({0, 1}, cJ') is the two-point TsS of (R, cJ) and (R, ~) is the extended natural TS. D e f i n i t i o n 0.1.1. A subset A of a BS (X, 7-1,7-2) is termed p-open in X if A - A1 [J A2, where Ai E 7-i. The complement of a p-open set in X is p-closed in X, that is, a subset B of a BS (X, 7-1,7-2) is p-closed in X if B - B1 N B2, where B~ ~ co ~ [S4]. Thus a subset A c X is p-open (p-closed) in (X, 7-1,7-2) if and only if A 7-1 int AU7-2 int A (A - 71 clAnT"2 clA). The family of all p-open (p-closed) subsets of a BS (X, 7"1,7"2) is denoted by p - O ( X ) ( p - C I ( X ) ) . It is clear that 7"1 [J 7"2 C p - O ( X ) (co 7-1U co7-2 C p - e l ( X ) ) and so in a BS (X, 7-1 < 7"2), we have p - O ( X ) (p-el(x)
-
The notion of a p-open (p-closed) set is equivalent to the notion of a quasi open (quasi closed) set given by D a t t a [72].
4
0. P r e l i m i n a r i e s
Definition 0.1.1 immediately implies that an arbitrary union (intersection) of p-open (p-closed) sets is p-open (p-closed), while the finite intersection (union) of p-open (p-closed) sets need not be p-open (p-closed) ([72, Remark 7]). Recall that for a subset A of a TS (X, T) the derived set of A, that is, the set of all accumulation points of A, is denoted by A d and A i = A \ A d is the discrete set of all isolated points of A. Hence to avoid confusion with our notation, for a BS (X, T1, T2) we shall use the following double indexation: A d - {x c X " x is an/-accumulation point of A} and Aji _ {x c A" x is a j-isolated point of A} so that the lower indices i and j denote the belonging to the topology and, therefore, always i, j E {1, 2}, while the upper indices d and i are fixed as the accumulation and isolation symbols, respectively. Thus for a BS (X, wl, T2), we have:
Aji = A \ Aj,d A is a j-discrete set ~ A is a j-dense in itself set ~
A - Aj,i
A c Ajd
and A isaj-perfectset
~A-Aj.
d
D e f i n i t i o n 0.1.2. Let (X, T1, "/-2) and (Y, Yl, Y2) be BS's. Then a function f : (X, T1, ~-2) --4 (Y, ~/1, ~f2) is said to be/-continuous (/-open,/-closed, an i-homeomorphism) if the induced functions f : (X, 7i) ~ (Y, 7i) are continuous (open, closed, homeomorphisms)[204], [236]. The classes of all/-continuous,/-open,/-closed functions of X to Y are denoted by i-C(X, Y), i-O(X, Y), i-Cl(X, Y), respectively, and the classes of all i-homeomorphisms of X to Y are denoted by i-7-l(X, Y). It is obvious that
d-C(X, Y) - 1-C(X, Y) A 2-C(X, Y), d-O(X, Y) = 1-O(X, Y) N 2-O(X, Y), d-Cl(X, Y) = 1-Cl(X, Y) A 2-CI(X, Y), d-7-l(X, Y) = 1-7-/(X, Y) A 2-7-/(X, Y). Following [31], the bitopological cube is the product
::(x) =
I-[ (I, ::f(x)
where .7"f(X) is the class of all functions t f " (X, 71, T2) --~ (I, Wl, w~), f c d-C(X, I).
Invariants of d-homeomorphisms are called bitopological properties. Thus a bitopological property P belongs to a BS (X, T1, w2) if and only if P belongs to every BS, that is, d-homeomorphic to (X, Wl, T2). Since the object of the theory of BS's is to study bitopological properties, from the bitopological viewpoint two d-homeomorphic BS's can be regarded as one and the same object. In our further discussion we shall abbreviate "lower (upper) semicontinuous"
to 1.(u.)s.c. D e f i n i t i o n 0.1.3. A function f : (X, 7-1,7-2) ---+ (I,O./)is said to be (i,j)-l.u.s.c. if f is i-l.s.c, and j-u.s.c.
0.1. Symbols and Notations. Basic Concepts of Bitopology
Proposition
and only if f :
0.1.4. A function f " (X, rl, r 2 ) ~
!
5
!
(I,~v~,~j) is d-continuous if
( X , T i , T 2 ) --~ ( 1 , ~ ' ) i8 ( i , j ) - l . u . 8. C.
Proof. We shall consider only the case with i = 1 and j = 2 because the other case is proved similarly. Let f " (X, ri, r2) --~ (I, w i, w~) be d-continuous (i.e, 1- and 2-continuous). Then for every 1-open set (a, 1], a c [0, 1), the preimage f - l ( a , 1] c TI and for every 2-open set [0, a), a c (0, 1], the preimage f - l [ 0 , a) E r2. Thus by [79, 12.7.2], f : ( X , T1,T2) ---+ (/,CO')is (1,2)-I.u.s.c. Conversely, let f : (X, rl,r2) ~ (I, co') be (1,2)-l.u.s.c. Then for every set (a, 1], a ~ [0, 1), the preimage f - l ( a , 1] E ~-1, and for every set [0, a), a < (0, 1], the preimage f - l [ 0 , a) c ~-2 since f is (1, 2)-l.u.s.c. Recalling that co~ - {~, I} tO {(a, 1] : a c (0,1)} and co~ = { ~ , I } t 0 { [ 0 , a) : a c (0,1)}, we conclude that f " (X, rl, r2) ~ (I, w~, w~) is d-continuous. [--] Adapting Definition 2.1 from [166] to our terminology, we come to D e f i n i t i o n 0.1.5. Let (X, rl, r2) be a BS and A, B be subsets of X. Then A is (i,j)-completely separated from B if there is an (i,j)-l.u.s.c. function f : (X, rl, r2) -+ (I, co') such that f ( A ) = 0 and f ( B ) = 1. Since f : (X, rl,r2) --+ (I, c0') is (i,j)-l.u.s.c. -: ;- (1 - f ) :
( X , T1, T2) +
-' :-
(I, co t) is (j, i)-I.u.s.c.,
it clearly follows that A is (i, j)-completely separated from B <---> -' ;- B is (j, /)-completely separated from A. We next introduce various properties of BS's which describe how the points a n d / o r / - c l o s e d sets of a BS are separated by means of disjoint/-open and j-open sets. These definitions are necessary to study the influence of pairwise separation axioms on bitopology. Note that in what follows the letter "W" abbreviates the word "weakly". D e f i n i t i o n 0.1.6. Let (X, rl, r2) be a BS. Then (1) (X, rl, r2) is MN-p-T0 (i.e., p-To in the sense of Murdeschwar and Naimpally) if for every pair of distinct points, there exists a 1- or a 2-neighborhood of one point not containing the other [189]. (2) (X, 7-1, r2) is MN-p-R0 if for every set U E 7-1 \ {l~} it follows from x ~ U that r2 cl{x} C U and for every set V E r2 \ {~} it follows from y c V that rl cl{y} C V [189]. (3) (X, r~, r2) is MN-p-T1 if for every pair of distinct points x, y c X, there exists a 1- or a 2-neighborhood of x not containing y [189]. (4) (X, rl, r2) is S-p-T1 (i.e., W-p-T1 in the sense of Swart) if for every pair of distinct points at least one point has a 1-neighborhood not containing the other, while the second point has a 2-neighborhood not containing the first [248]. (5) (X, rl, r2) is R-p-T1 (i.e., p-T1 in the sense of Reilly) if it is d-T1 [215].
6
O. Preliminaries
(6)
(X,T1,T2) is R-p-R1 iffor each pair of points x, y E X such that x E ~-~cl{y} there are an /-open set U and a j-open set V such that x E U, y c V and U N V - 2~ [217]. (7) (X, T1,7-2) is MN-p-R1 if for every pair of points x, y E X, T1 e l { z } r r2 cl{y} implies that x has a 2-neighborhood and y has a 1-neighborhood which are disjoint [189]. (8) (X, T1, ~-2) is W-p-T2 if for every pair of distinct points x, y c X there exist a 1-open set U and a 2-open set V such that either x c U, y E V or x E V, y E U, and U A V - 2~ [152] (see also [31], [248]). (9) (X, ~-~,~'2) is p-T2, (i.e., p-Hausdorff) if for each pair of distinct points x, y E X there exist a l-open set U and a 2-open set V such that x E U, yEV, andUC~V-~ [151]. (10) (X, ~-~,T2) is (i, j)-semiregular if for each point xEX and each/-open set U, x E U, there exists an /-open set V such that x E V C ~-i int ~-j cl V C
u [236]. (11) (X, T1,72) is (i,j)-almost regular if for each point x c X and each (i, j)-closed domain F (see Definition 1.3.3), x E F, there exist a n / - o p e n set U and a j-open set V such that x E U, F C V and U N V - 2~ [236]. (12) (X, T1, r2) is (i, j)-regular if for each point x E X and each/-closed set F, x g F, there exist a n / - o p e n set U and a j-open set V such that x E U, FCVandUAV-o [151]. (13) (X, T1, T2) is (i, j)-almost completely regular if every (i, j)-closed domain F is (i, j)-completely separated from each point x E F [238]. (14) (X, ~-1,T2) is (i, j)-completely regular if every i-closed set F is (i, j)-completely separated from each point x ~ F [166], [189]. (15) (X, T1,7-2) is (i,j)-seminormal if for every j-closed set F and every (i,j)-open domain U (see Definition 1.3.3), F C U, there exists an /-open set V such that F c V c 7-i int ~-j cl V c U [238]. (16) (X, T1, T2) is (i, j)-almost normal if for every pair of disjoint sets A, B in X, where A is j-closed and B is an (i,j)-closed domain, there exist an /-open set U and a j-open set V such that A c U, B c V and U A V [238]. (17) (X, ~-1,~-2) is p-normal if for every pair of disjoint sets A, B in X, where A is 1-closed and B is 2-closed, there exist a 2-open set U and a 1-open set V s u c h t h a t A c U , BcVandUAV-~ [151]. (18) (X, ~-1,T2) is (i,j)-D1 if every j-closed set F has a countable/-base for /-open sets containing F, that is, for every set F E co Tj there exists a countable family {V~}n~=I C ~-i such that for each/-open set U, F c U, there is n E N such that F C Vn c U [180]. Though Definition 0.1.6 provides a wide class of axioms of separation of BS's, it can in fact be expanded further. For the purpose of abbreviating the implications below, obtained on the strength of [189] and Definition 0.1.6, and also the implications in Remark 0.1.9, instead of writing spaces, we shall indicate only the corresponding axioms of separation.
0.1. Symbols and Notations. Basic Concepts of Bitopology
7
p -T4 r (p-norm.AR -p -T~) ::~ p -T389r (p -comp.reg.AR-p-T1) ::~ p -T3 r (p -reg.AR -p -771) :=> R-p-R1
--,
R-p-T1 A MN-p-R1
p-T2 W -p -T~
S-p -T1
d-To
MN-p-T1
MN -p - R0
MN -p-T0 P r o p o s i t i o n 0.1.7. The statements below hold for a BS (X, T1, T2): (1) (X, T1, T2) is (i,j)-almost regular if and only if for each point x c X and each (i,j)-open domain U with x c U, there exists an i-open set V such that x E V C rj clV c U. (2) (X, T1,T2) is (i,j)-regular if and only if for each point x c X and each i-open set U with x c U, there exists an i-open set V such that x c V c 7 j c l V C U. (3) (X, T1,T2) is (i,j)-regular if and only if it is (i,j)-semiregular and (i, j)-almost regular [236]. (4) (X, 7-~,T2) is p-normal if and only if for each 2-closed (1-closed) set F and each 1-open (2-open) set U with F C U, there exists a 1-open (2-open) set V such that F c V c ~-2 el V c U (F c V c 3-1 el V c U). (5) (X, T1, T2) is p-normal if and only if it is p-seminormal and p-almost normal [238]. Moreover, for a BS (X, T1 < T2), we have: (6) (X, 7-1,7-2) is d-reg. ~ ( X , 7-1,72) is 1-reg. ~ ( X , T1, 7-2) i8 (1, 2)-reg.
(x, ~1, ~)
i~ 2 - ~ g . . -
(x, ~, ~) i~ (2,1)-~g.
(7) (X, 7-1, T2) is (1, 2)-completely ((2, 1)-completely) regular if for each point x c X and every 1-closed (2-closed) set F with x-~ F, there exist a 1-1.s.c. a~d 2 - ~ o ~ t i ~ o ~ (1-~.~.~. a~d 2 - ~ o ~ t i ~ o ~ ) / ~ t i o ~ f : (X, ~1, ~2) (I, oa') such that f (x) = 1 and f (F) = O. Therefore (X, rl, T2) is d-compl, reg. ~
(X, rl, T2) is 1-compl. reg. ~
(X, ~-1,72) is 2-compl. reg. ~
(X, ~-1, 72) is (2, 1)-compl. reg.
II
(X, ~-1,T2) is (1, 2)-compl. reg.
C o r o l l a r y 0.1.8. In a p-normal BS (X, T1, T2), for every pair of disjoint sets A , B , where A is l-closed and B is 2-closed, there exist a 2-open set U and a l-open set V such that A c U, B c V and 7-1 C1 U N 7-2 c1 V ~- ~ . Remark
0.1.9. Following [228], W-p-T3 ~
(p-regularity A S-p-T1),
W -p-T3~1 ~, , (p-complete regularity A S-p-T1),
8
0. Preliminaries
W-p-T4 z--> (p-normality A S-p-T1). Moreover, by [205], p-complete regularity implies MN-p-R0 and, hence, for a p-completely regular BS,
W-p-T31 ,z---5, W-p-T3 ,z---5, W-p-T2 ,z----5, S-p-T1 ,z--> MN-p-T0 [205]. D e f i n i t i o n 0.1.10. A BS is said to be hereditarily p-normal if every one of its BsS is p-normal [84]. The class of such spaces will play an essential role in Chapter III dealing with the dimension theory of BS's. D e f i n i t i o n 0.1.11. A BS (X, 7"1,7"2) is said to be (i,j)-perfectly normal if it is p-normal and co 7"i c j-G5(X) [202]. One can easily see that the BS (R, CO1 < CO) is (1, 2)-perfectly normal, but it is not (2, 1)-perfectly normal. By Theorem 2.5 in [202] every p-perfectly normal BS is p-completely normal (see Theorem 0.2.2 below) and so it is hereditarily p-normal. P r o p o s i t i o n 0.1.12. If a BS (X, 7"1,7"2) is p-regular and d-second countable, then it is p-perfectly normal.
Proof. By Lemma 3.2 in [151] and Definition 0.1.11, it suffices to prove that 7-i c j - ~ ( X ) . Let U c 7-i \ {2~} be any set and Bi be a countable/-base. Then for each point x E U there is a set U(x) E 13~ such that x E U(x) c 7-j cl U(x) c U so that U = U 7-j cl U(x). The latter union is countable because all sets U(x) xEU
belong to Bi.
71
C o r o l l a r y 0.1.13. A R~v-T1 and d-second countable BS is p-perfectly normal if and only if it is p-normal.
Proof. Since every R-p-T1 and p-normal BS is p-regular, the proof is an immediate consequence of Proposition 0.1.12 and Theorem 0.2.2 in conjunction with its Corollary 0.2.3. [-1 Using the notion of J. C. Oxtoby [198], we now define (i,j)-quasi regular BS's which play an important role in the formulation and treatment of (i, j)pseudocomplete BS's in Chapter IV. D e f i n i t i o n 0 . 1 . 1 4 . A BS (X, 7l,T2) is (i,j)-quasi regular if for every set U E 7-i \ {2~} there exists a set V E 7-~ \ {~} such that 7-j cl V c U.
It is obvious that every (i,j)-regular BS is (i,j)-quasi regular, but not conversely. Further, if 7-2 C 71 for (X, ~-1,7-2), then (X, 7-1,v2) is (1, 2)-quasi regular if and only if (X, T1,72) is (vl, 7-2)-regular in the sense of [260], in which some interesting examples of such spaces are given. Moreover, for a BS (X, 7-1 < 7-2), we have (X, T1,3-2) is d-quasi reg. ~
(X, Vl, 3-2) is 1-quasi reg. ===, (X, T1,3-2) is (1, 2)-quasi reg.
(X, T1,3-2) is 2-quasi reg. r
(X, 3-1,3-2) is (2, 1)-quasi reg.
0.1. Symbols and Notations. Basic Concepts of Bitopology
9
P r o p o s i t i o n 0.1.15. The conditions below are equivalent for a BS (X, 71,72):
(1) (x,
(i,
(2) For every set UE7~\{2~} there exists a (j,i)-closed domain F such that FcU. (3) The topology 7.~ has a pseudobase (see Definition 2.1.1) consisting of j-closed sets. C o r o l l a r y 0.1.16. The following conditions are satisfied for a BS (X, 71, T2): (1) If (X, 7.1,7-2) is (i,j)-quasi regular and Y c 7-i \ {~} U i-D(X) (see Definition 1.1.9 for j - i ) , then (Y, 7-;, T~) is also (i,j)-quasi regular, and so (2) /f (X, 7.1,7-2) is p-q~tasi regular and Y C ((T 1 if)7-2) \ {~})[-J (1-T)(X)N 2-D(X)), then (Y, 7-{, 7-~) is also p-quasi regular.
Moreover, for a BS (X, 7-1 < 7-2), we have (3) If (X, T1, 7-~.) is p-quasi regular and Y c 7.1 U 2-D(X), then (Y, 7.[, 7.~) is also p-quasi regular. D e f i n i t i o n 0.1.17. Let (X, 7-1,7-2) be a BS. Then (1) (X, T1,T2)is (i,j)-(countably)-paracompact if each (countable)/-open covering of X has a n / - o p e n refinement which is j-locally finite [209]. (2) (X, 7-1,7-2) is (i, j)-locally LindelSf if each point x c X has a j-neighborhood U(x) which is i-LindelSf [210]. (3) (X, 7-1,T2) is (i, j)-paraLindelSf if each/-open covering of X has an/-open countable refinement which is j-locally finite [210]. (4) (X, ~-1,7.2) is p-LindelSf if every proper 1-closed set is 2-LindelSf and every proper 2-closed set is 1-Lindelbf [238]. D e f i n i t i o n 0.1.18. Let (X, 7"1,7-2) be a BS. Then (1) (X, 7-1,7-2) is p-connected if X cannot be expressed as a union of two disjoint sets A and B such that A c 7-1 \ {~}, S C T2 \ {~} [204] (see also [17], [71], [187]). (2) (X, 7-1,7-2) is weakly totally disconnected if for each pair of distinct points there exists a disconnection X = AIB (i.e., X = A U B, where A c 7-1 \ { e } , B C 7-2 \ {~} and A N B = 2~) such that one point belongs to A, the other to B, and the roles of the points need not be interchangeable [248]. (3) (X, 7-1,7-2) is (i, j)-extremally disconnected if rj cl U = 7-i int 7j el U for every set U E Ti or, equivalently, if 7-j el riint A = ri int Tj cl 7-i int A for every subset A c X [12]. It is obvious that for a BS (X, ~-1 < T2) the following implications hold: (X, 71, T2) is d-connected > (X, T1, T2) is 2-connected ~ (X, T1,72) is p-connected ==> (X, 71, T2) is 1-connected.
10
0. P r e l i m i n a r i e s
Using (1) of Lemma 0.2.1 it is not difficult to see that the following three conditions are equivalent: (X, T1,7-2) is (1, 2)-extremally disconnected, (X, 7-1,T2) is (2, 1)-extremally disconnected, and (X, 71,7-2) is p-extremally disconnected. Therefore, later we shall consider only p-extremally disconnected BS's. In contrast to the topological case, the notion of bitopological compactness depending on special coverings of BS's has not been defined uniquely. Many approaches to this notion, in particular the ones given below, are available (see, for example, [113], [248], [72], [88]). D e f i n i t i o n 0.1.19. Let (X, 7-1,7-2) be a BS. Then
(1) (X, 71,7-2) is compact in the sense of Fletcher, Hoyle, and Patty (briefly, FHP-compact) if every p-open covering// = {Us }~cs of X, that is, a f a m i l y / / = {Us}~cs such that // c T~ U7-2, X = U us a n d / / N w ~ s6S
contains a nonempty set, has a finite subcovering [113]. (2) (X, T1,7-2) is quasi compact if the TS (X, sup(7-1,7-2)) is compact [184] (see also [248] and [72]). It is obvious that quite a variety of notions of bitopological compactness gives rise to an even greater variety of notions of bitopological local compactness. How these notions are related to one another is discussed schematically in [184]. Below we recall some of them needed for our discussion. D e f i n i t i o n 0 . 1 . 2 0 . Let (X, Wl, 7-2) be a BS. Then
(1) (X, 7-1,7-2) is (i,j)-locally quasi compact (briefly, (i,j)-lqc) if each point x E Z has an /-neighborhood U(x) such that 7-jcl U(x) is quasi compact [183]. (2) (X, 7-1,7-2) is (i,j)-locally compact in Stoltenberg's sense (briefly, (i,j)-Slc) if each point x E X has an /-neighborhood U(x) such that 7-j cl U(x) is j-compact [245]. (3) (X, 7-1,7-2) is (i,j)-locally compact in Reilly's sense (briefly, (i,j)-Rlc) if each point x E X has an /-neighborhood U(x) such that 7-j cl U(x) is FHP-compact [214]. (4) (X, 7-1,7-2) is (i,j)-locally (countably) compact in Raghavan and Reilly's sense (briefly, (i, j)-RRlc ((i, j)-RRlcc)) if each point x 9 X has a j-neighborhood U(x) which is i-(countably) compact [209]. (5) (X, 7-1,7-2) is (i,j)-locally compact in Birsan's sense (briefly, (i,j)-Blc) if each point x 9 X has an/-neighborhood which is j-compact [30]. Following [184], (i, j)-Rlc .r
(i,j)-lqc --->. (i, j)-Slc ~
(i,j)-Blc ~
(j, i)-RRlc.
In [190] L. Nachbin observed that if the symmetry axiom for a uniform space (X,//) is omitted, the resulting "quasi uniform" space defines, in a natural manner, two topologies 7-1 and 7-2 on the underlying set X (see also D. Tamari [250]), that is, {U(x) : U 9 b/} is the filter of all neighborhoods of a point x E X for the topology 7-1 = 7-(//) and { V - l ( x ) : V -1 9 //-1} is the filter of all neighborhoods of the point x for the topology 7-2 = 7-(//-1), w h e r e / / - 1 = {U-1 : U 9 b/}. Thus for
0.2. I n t e r n a l C h a r a c t e r i z a t i o n of Pairwise C o m p l e t e R e g u l a r i t y
11
every quasi uniform space there is an associated BS. On the other hand, according to [166], a BS (X, T1, ~-2) is quasi uniformizable if there exists a quasi uniformity b / o n X such t h a t T 1 = 7-(~/[) and T2 = ~-(b/-1). In particular, it is shown in [166] t h a t a BS (X, T1, T2) is quasi uniformizable if and only if it is p-completely regular.
0.2. Internal C h a r a c t e r i z a t i o n of Pairwise C o m p l e t e R e g u l a r i t y We begin this section by proving a few elementary facts t h a t will be frequently used in our further investigation. L e m m a 0.2.1. For a BS (X, 7-1, r2), we have (1)
riint F = ri int rj cl r i i n t F for every set F c co 7-j so that ri cl U = ri cl rj int ri cl U for every set U G ~-j
or, equivalently, ri int rj cl A = r i i n t ry cl r~ int ry cl A
so that ri cl rj int A = ri cl rj int ri cl rj int A for every set A c X.
Moreover, for a BS (X, rl < r2), we also have (2) ri cl rj cl A = rl cl A and so ri int rj int A = r l i n t A for every set A c X . Proof. (1) Clearly, it suffices to prove only one of the four equalities, for example, the third one. It is obvious t h a t ri cl rj int ri cl 7-j int A c ri cl rj int A. Conversely, if x c r~ cl rj int A, then U ( x ) • rj int A # ~ for e v e r y / - o p e n neighborhood U(x). Therefore U(x) N rj int ri cl 7-j int A # ~ as rj int A c rj int ri cl Tj int A, and since U(x) E T~ is arbitrary, we obtain x E Ti cl Tj int Ti cl Tj int A. (2) Let us prove the first equality. It is evident t h a t r2 cl rl cl A = rl cl A and rl cl A C rl cl r2 cl A. If x c T1 cl T2 cl A is any point, then for every 1-open neighborhood U(x), we have U(x) NT2 c l d # 2~. Hence V ( x ) ~ d # 2~ because U(x) e 7"1 C 7"2. Since U(x) e T1 is arbitrary, we have x c rl cl A. [2]
T h e o r e m 0.2.2. A BS (X, 71,'/-2) i8 hereditarily p-normal if and only if it is p-completely normal in the sense of Patty [202], that is, if and only if whenever A and B are subsets of X such that (T1 el A N B) U (A N T2 cl B) = 2~, there exist a 2-open set U and a 1-open set V which are disjoint and for which A C U and BcV. Proof. We begin by assuming t h a t (X, "rl, "r2) is hereditarily p-normal and A, B are the subsets of X satisfying the condition (~-~cl A A B) U (A N T2 cl B) = ~. If Y = X \ (T1 cl A ~ T2 cl B), then A C Y, B C Y. By virtue of the hereditary p-normality of (X, T1,7-2) the BsS (Y, T~,7-~) is p-normal and, therefore, for the
12
0. Preliminaries
disjoint 1-closed set A' - 7-1 c1 A N Y and 2-closed set B ' - 7-2 cl B n Y in (Y, 7-~, 7-~) there exist disjoint sets U' c ~-~ and V' c 7-{ such that A' c U' and B ' c V'. Let g l C 7-2, V1 E 7-1 and U1 n Y = U', V1 n Y = V'. Then A c Y n 7-1 cl A = A' c U1,
B C Y n 7-2 cl B = B' c V1.
Since Y -- X \ (T1 c l A n w2 clB) = (X \ T 1 c l A ) u (X \ w2 c l B ) , we have
A c X \ 7.2cl B C 7.2 and B C X \ wl cl A 6 7.1. It is not difficult to verify that U--UIN(X\T2c1B)
CT2 and V = V I N ( X \ 7 - 1 c 1 A )
cT1
are the desired disjoint sets containing A and B respectively. Conversely, let (X, 7.1,7.2) be a p-completely normal BS and (Y, 7~, 7.~) be its any BsS. If ' B E c o T2' and A a B - ~, A c co T1, then (T~ cl A a B) u (A n 7.~ cl B) : ~. Therefore (~-1 cl A n B) U (A N 7.2 cl B) = and, by condition, there exist a 2-0pen set U and l-open set V in (X, 7.1, T2) which are disjoint and for w h i c h A c U a n d B c V. Hence the sets U' = U N Y and V' - V N Y are respectively 2-0pen and 1-0pen in (Y, 7.~, 7.~) which are disjoint, A c U' and B c V'. Thus (Y, 7.~, v~) is p-normal and, consequently, (X, T1,7"2) is hereditarily p-normal. D C o r o l l a r y 0.2.3. Every hereditarily p-normal BS is p-normal. The remainder of this section deals with an internal characterization of p-completely regular BS's. The main result (Theorem 0.2.5) was originally formulated in [83]. A double family, t h a t is, a pair of families Z = {Z1,Z2}, where Zi is an /-closed base of a BS (X, 7.1,7.2), is called a d-closed base and co Z = {co Zi, co Z2 }, where co Zi is a n / - o p e n base, conjugate to Zi, is called a d-open base, conjugate to Z = { Z 1 , Z 2 } . D e f i n i t i o n 0.2.4. A d-closed base Z = {Z1, Z2 } of a BS (X, 7.1,7.2) is said to be p-normal if the following conditions are satisfied: (1) For every point x c X and its any neighborhood U(x) E co ~1 (U(x) E co Z2), there exists a set A c Z2 (A E Z1) such that x c A c U(x). (2) If A E Z1, B E Z2 and A N B = z , then there exist U c co Z2, V E co Z1 such that A c U, B C V and U N V = z . T h e o r e m 0.2.5. A BS (X, 71, T2) is p-completely regular if and only if it possesses at least one p-normal base. To prove this theorem, we have to formulate a few auxiliary statements.
0.2. Internal C h a r a c t e r i z a t i o n of Pairwise Complete Regularity
13
D e f i n i t i o n 0.2.6. Let (X, T1, T2) be a BS. Then the double family A = {A1,A~}, where Ai = {(F, U ( F ) ) : F c coT~ and U(F) is its j-neighborhood}, is said to be p-dense if for every pair (F, U(F)) c A~ there exist a j-neighborhood V(F) and an/-closed set (I) satisfying the following conditions:
(1)
d v(F) c 9 c
(2) (F, V(F)), (q~, U(F)) e A~. L e m m a 0.2.7. Let Z - { Z I , Z 2 } be a p-normal base of a BS (X, T1, T2). Then the double family , 4 - {Al,fl~2}, w h e r e A i - {(A, U(A)) " A c Z~, U(A) c co Zj }, is p-dense.
Proof. Let (A, U(A)) c A~. Consider the set B = X \ U(A) c Zy. It is obvious that A N B = o and by (2) of Definition 0.2.4, there exist disjoint neighborhoods V(A) e c o z y , V ( B ) e cozy. Assume that q~ = X \ V ( B ) e Zi. The equality V(A) N V ( B ) = ~ implies V(A) c 9 and so ~-~cl V(A) c q). On the other hand, (X \ U(A)) ~ q~ - ( X \ U(A)) c~ ( X \ V ( B ) ) - B n ( X \ V ( B ) ) - ~, that is (I) c U(A). Hence AC'qclV(A)
c(FcU(A)
and V(A) c coZj,
q~ c Zi D
imply (A, V(A)), (~o, U(A)) E A~.
L e m m a 0.2.8. Let A and B be disjoint, respectively, i-closed and j-closed subsets of a BS (X, T1, %). If for the set A there exists a family of j-neighborhoods {U~}, enumerated by all dyadic rational numbers {r " 0 < r < 1} in a manner such that U1 - X \ B and rl < r2 implies ~-~cl UF1 C Vr2 , then A is (i,j)-completely separated from B.
Proof. We shall use a generalization of Urysohn's procedure. Let lit = ~ if t < 0, and Ut = X if t > 1. Besides, we shall assume t h a t to each t c [0, 1], which is not a dyadic rational number, there corresponds the j-open set lit = U u~. It ~
r
Hence by condition, ~-i el Utl C Ti cl ST,1 C Sf2 C gt2. Next, let z C X be any point. Then the point z together with the family {Ut : t c R} defines the section of the set R as follows: a real number t belongs to the lower class A x of this section if z-c Ut and t belongs to the upper class B x if z c Ut. Clearly, this section defines a real number ~x. If f : (X, rl, r2) ~ (I, co') is defined as f ( x ) = ~x, then from the definition of this function it immediately follows that f ( A ) = 0 and Z(B) = 1. Thus it remains to prove t h a t f is (i, j)-l.u.s.c. Indeed, assume t h a t z E X is any point and t c R is any real number. If f (x) > t, then t c A x so t h a t z-~ Ut. Since f (x) > I(x)+t 2 = t', where ~'i cl Ut C Ut, and x-4 Ut,, we obtain z E X \ ri cl lit = U(x). On the other hand, if f ( z ) < t,
14
0. P r e l i m i n a r i e s
then t c B z so that x c Ut - V(x). Thus f is (i,j)-l.u.s.c. since U(x) 9 ~-i and V(x) c D L e m m a 0.2.9. Let (A,U(A)) 9 Ai, where A {A1,A2} is any p-dense family of a BS (X, T1, T2). Then A is (i,j)-completely separated from X \ U(A).
Proof. Let (A, U(A)) c Ai. Clearly, it suffices to establish for A the existence of a family of j-neighborhoods {U~} enumerated by all dyadic rational numbers {r 9 0 < r < 1}, which satisfies the conditions of Lemma 0.2.8. Let U1 - U(A). Since A - {A1,A2} is p-dense, there exist a j-open set U0 and an/-closed set (I)0 such that A c U0 c T~ clU0 c ~0 c U~. By the same reasoning as above, there are a j-open set U89 and an/-closed set ~ such that ~o c U!2 c w~clU!2 c ~12 c U1. Thus for each dyadic rational number r - ~ , where 0 _< r _< 1, we can find by induction a j-open set U ~2 n and an/-closed set ~__~ such that 2n (I) _~7
C
C
U 2p-l- 1 2n+ 1
7"i
cl U 2p+ 1 C (~ 2p+ 1 C 2"n+ 1
2'n+ 1
U
2 P+nl ~ "
~-]
Note that following [166] any /-zero set in a BS (X, TI,T2) is of the form f ( x ) - O, f is (i,j)-l.u.s.c. and f _> 0}. This definition immediately implies that a n / - z e r o set is/-closed. Now we can proceed to prove our main result, that is, Theorem 0.2.5. Necessity. Let (X, TI, ~'2) be a p-completely regular BS in the sense of (14) of Definition 0.1.6. Also, let Z - {Z1,Z2}, where Zi is the family of all i-sero sets. By [166, Proposition 2.9], Z - {Zx, ~-2} is a d-closed base of (X, T1, T2). Let us show that conditions (1) and (2) of Definition 0.2.4 are satisfied. (1) Let x 9 X be any point and g(x) 9 co Zj be its any neighborhood. Since (X, Wl, T2) is p-completely regular, the point x is (i,j)-completely separated from the j-closed set X \ U(x), that is, there exists an (i, j)-l.u.s.c, function
{x C X "
f " (X, TI,~'2) --, (I,a/) such that f ( x ) -
0 and ( f ( X \ U ( x ) ) - 1.
Assume that A - f - l ( 0 ) . Clearly, A 9 Z~ and x 9 A c U(x). (2) Let A 9 Z1, B 9 Z2 and A n B - ~. Then there exist a (1,2)-l.u.s.c. function r _> 0 and a (2,1)-l.u.s.c. function ~ _> 0 such that A - r B - ~-1(0). By [166, Proposition 2.8], there exists a (1,2)-l.u.s.c. function h" (X, T1, w2) ~ (I, co') such that h(A) - 0 and h(B) - 1. Let 0 hi(x)
--
3
1 {h(x)-~}
if 0 _< h(x) <_ -~, 1 if < h(x)<_1
and 0 gl(x)
-
-
1 ~ _< h(x)<_ 1, 1 if O < h ( x ) < if
/ 1 \ 3(-z - h(x)) \j
/
0.2. Internal C h a r a c t e r i z a t i o n of Pairwise Complete Regularity
15
It is obvious t h a t 0 < hi(x) < 1, 0 < gl(x) <_ 1 for each point x E X, hi is (1, 2)-l.u.s.c. and gl is (2, 1)-l.u.s.c. Let Ai - h11(0), B1 - g~-l(0). Then A1 e Z1, - U(B) e B1 E Z2. Moreover, 0 < h(A1) < 1 and h ( B ) - 1 g i v e B c X \ A 1 co Z1, while ~1 <_ h(B1) <_ 1 and h ( d ) - 0 give A c X \ B1 - V ( d ) e co Z2 It is easy to see t h a t A1UB1 - h11(0)Ug11(0) - X so t h a t U ( A ) N U ( B ) - ~. Sufficiency. Let Z - {Z1, Z2} be any p-normal base of a BS (X, ~-1, T2). Consider the double family . 4 - {A1,A2}, where A~ - {(A, U ( A ) ) " A e Z~, U(A) e c o z y } . Then by L e m m a 0.2.7, A - {A1,A2} is p-dense. If x c X, F E coT~ and x ~ F , then there exists a set A E Zi such t h a t F c A and x ~ A since Zi is the /-closed base. It is obvious t h a t U(x) - X \ A c co Zi. Since Z - {Z1, Z2} is the p-normal base, there exists a set B c Zj such t h a t x E B c U(x). Hence A N B and by L e m m a 0.2.9 there exists an (i,j)-l.u.s.c. function f 9 (X, T1,T2) ~ (I,w') suchthat f(A)-Oandf(B)1. B u t x e B a n d F C d . Thus f ( x ) land I(F) -0. Further, we have
Proposition 0.2.10. A d-closed base Z = {Z1, Z2} of a BS (X, p-normal if and only if the following conditions are satisfied:
71,72)
is
(1) For every point x E X and every set A c Z1 (A c Z2), x-~ A, there exist U c c o Z 1 , V c co Z2 (U c co Z2, V E co Z1) such that x c U, A c V, andUNV=~. (2) I r A c Z1, B c Z2 and A N B = ~, then there exist U c coZ2, V c coZ1 such that A c U, B c V, and U n V = ~.
CHAPTER I
Different Families of Sets in Bitopological Spaces
We begin the study of different families of subsets of a BS (X, T1,7"2), using the bitopological modifications of the fundamental topological notions of a dense set; a boundary (also called codense) set; a derived set; a dense in itself set; a perfect set and a scattered set belonging to G. Cantor [54], [55], [57], a nowhere dense set defined by P. du Bois-Reymond [34], first and second category sets introduced by R. Baire [20], and the boundary of a set studied by G. Cantor [56] and C. Jordan [148]. The obtained modifications - - the (i, j)-boundaries and (i, j)category n o t i o n s - play an essential role, the former in constructing the dimension theory for BS's in Chapter III and the latter in establishing the principal Baire-like properties of BS's in Chapter IV. This chapter also deals with the bitopological analogues of open and closed domains of K. Kuratowski [160], [162], and semiopen and semiclosed sets of N. Levine [170]. Moreover, the notions of (i, j)-open domains and that of (j, /)-closed domains make it possible to define the families of (i, j)-semiopen domains which equal the families of (j, i)-semiclosed domains. Their corresponding topological families are new for topology. Since the obtained families, including their two topological versions, are not comparable by inclusion, which means that none of them is contained in the other even when one of the topologies is finer than the other, we introduce the "quotient"-families leading to the solution of the problem of comparability. The p-dense in itself, (i,j)-perfect and p-scattered sets are also investigated [95] and a bitopological analogue of the well-known Cantor-Bendixson's theorem is proved. Furthermore, we introduce and study three special operators on 2 X, which give an exact determination of degrees of nearness of the four boundaries of any subset of a BS (X, 7-1 < 72). Based on our earlier studies, we end the chapter by defining and investigating the so-called relative properties, namely the properties of subsets of BsS's of BS's. Among the obtained results the most important one shows that for a BS (X, T1 < 72) every non-empty 1-open set has the same (1, 2)-category both in itself and in the whole space, while for a non-empty 2-open set the corresponding (2, 1)-categories may not coincide. The chapter provides a few simple and illustrative examples.
16
1.1. (i,j)-Nowhere Dense Sets and ( i , j ) - C a t e g o r y Notions
17
1.1. (i, j)-Nowhere Dense Sets and (i, j)-Category Notions The notion of an (i, j)-nowhere dense set is one of the frequently used notions in this work. It will underlie the definitions and investigation of the Baire-like properties for BS's some of which are defined in [173].
Definition 1.1.1. A subset A of a BS (X, r l,r2) is termed (i,j)-nowhere dense (or (i,j)-rare) in X if rj cl A contains no n o n e m p t y / - o p e n set, that is, to say if r i i n t rj cl A = ~. The families of all (i,j)-nowhere dense subsets of X (i,j)-N'D(X). A simple application of the foregoing definition yields
are denoted
by
Proposition 1.1.2. Let (X, rl, r2) be a BS and A c X . Then A e (i,j)-JVD(X) ~
rj clA c r~ cl(X \ rj clA).
Proof. The implication from left to right is obvious. If rj cl A c ri cl(X \ Tj cl A),
then rj cl A N r~ int rj cl A - r, int rj cl A - ~.
Theorem 1.1.3. Let (X, evcfy set U C (T1 n 7"2) \ {~} V n A = ~. Conversely, if for such that V c U and V C~A =
[21
rl,r2) be a BS and A E ( i , j ) - N ' D ( X ) . Then for there is a set V c rj \ {2~} such that V c U and every set U e ri \ {2~}, there is a set V e rj \ {2~} ;~, then A c ( i , j ) - A f D ( X ) .
Proof. First assume that A E ( i , j ) - A f D ( X ) and U c (rl At2) \ {2~} is an arbitrary set. Then U \ rj cl A r 2~ since the contrary means that
2~ - U C~ri el rj int(X \ A) - U c~ (X \ ri int rj el A) - U, which is impossible. Therefore V = U \ rj cl A is the desired set. On the other hand, if A g (i, j ) - N ' D ( X ) , then every set V E rj \ {~}, satisfying the inclusion V c r i i n t rj cl A, is contained in rj cl A, that is, V N rj cl A = V -r 2~. Hence V N A r 2~. [B
Corollary 1.1.4. For a BS (X, T 1 < 7"2) a set A c (1, 2)-2VD(X) if and only if for every set U c 7-1 \ {~}, there is a set V c 7"2 \ {~} 8ugh that V c U and VNA=~. Corollary 1.1.5. The inclusions
(2,1)-ND(x) c 2-:VD(x) n n 1-AfD(X) c (1,2)-AfD(X) are satisfied for a BS (X, rl < r2).
Corollary 1.1.6. For a BS (X, r l < w2) the condition A c (2,1)-AfD(X) implies that for every set U c ~-2 such that rl int U 7~ 2~, there is a set V E rl \ {2~ }
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20
I. Different Families of Sets in Bitopological Spaces
Proof. Let 72 int 7-1 cl A - ~ - Te int T1 cl B. Then 71 cl A E 2-Bd(X) and by (2) of L e m m a 0.2.1, 72 int we el T1 el B -- 7"2 int T1 el B -so t h a t 71 c l B C 2-N'Z)(X). For the topological case, Proposition 1.1.12 gives T1 C1A U T1 cl B E 2-Bd(X). Since T1 cl A U T1 cl B - T1 cl(A U B) c co T1,
we obtain ~-2 int 7-1 cl(A U B) - z ,
t h a t is A U B E (2, 1)-N'~D(X).
V]
D e f i n i t i o n 1.1.16. A family A - {As}sEs of subsets of a BS (X, T1, ~-2) is t e r m e d / - l o c a l l y finite at a j-dense set of points of X if for every set U E ~-i \ {2~} there e x i s t s a s e t V e T j \ { ~ } s u c h t h a t VC Uand {so S" VnAs r Z}is finite. Clearly, if A - {As}sES is/-locally finite at a j-dense set of points of X, then the family j - C 1 A - {~-j cl As}sES is also /-locally finite at a j-dense set of points of X. T h e o r e m 1.1.17. If a family A - {As}sES C (2, 1)-AfZ)(X) is 2-locally finite at a 1-dense set of points of a BS (X, T1% 7"2), then U As c (2, 1)-N'~D(X). sES
Proof. Let A - {As}sES be 2-locally finite at a l-dense set of points of X. Then I-C1A - {~-1clAs}sES must also be 2-locally finite at a l-dense set of points of X so t h a t for every set U E ~-2 \ { z } , there exists a set V c T1 \ { ~ } such t h a t V C U and {sk E S " V N T l c l A s k ~ 2~} is finite. Let {~-lclA~k}~=l be the corresponding finite family. Since 7-1 clAsk C (2, 1)-AfZ?(X) for every k - 1,n, X \ T1 el Ask E T1 N 2-~)(X) for every k - 1, n. One can easily verify t h a t W = n V A ( N ( X \ T1 el Ask)) ~ ~ . Contrary, let W - 2~ so that k=l n n
c
\(
U k=l
k=l
In t h a t case n
n
n
~ T1 int U 7"1 c1 Ask - - T 1 int T1 c1 U Ask C T2 int T1 C1 U Ask' k=l k=l k=l n
which is impossible since, by Corollary 1.1.15,
U Ask c (2, 1)-N'D(X).
Hence
k=l
W c T1\{o}. On the other hand, VC~71 clAs - ~ for s ~ sk a n d s o V c X\T1 clAs for every s r sk. Therefore the set W c V satisfies the inclusion WC
N (X\rlClAs)-X\ sES
UT-lclAs" sES
Hence W E ~-1 \ {2~} implies W C TliIlt ( X \ U T l a l A s ) ~ X \T1c1 U TlClAs C X \T1C1 U As.
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22
I. Different Families of Sets in Bitopological Spaces
Pro@ (1) Using Theorem 2 from [161, p. 78], we obtain {x E X :
T2 el A-~ 1-Bd(X, x) } = 7"1 c17"1 int 7"2 el A.
Thus, it remains to use Proposition 1.1.19. (2) It is clear that { X C A"
A c (1, 9~)-.~:)(X, x) } - A \ 7"1 cl 7"1 int 7"2 cl A c 7"2 cl A \ 7"1 int 7"2 cl A
and 7"1 int(T2 el A \ T1 int 7"2cl A) = n i n t 7"2cl A A (X \ 7"1 el 7"1 int 7"2cl A) = ;~. Therefore 7"2clA \ 7"1 int T2 clA E c07"2 A 1-Bd(X) C (1, 2)-A/D(X).
K]
In particular, if A is (1, 2)-nowhere dense at each of its points, then A = A \ 7"1 cl 7"1 int 7"2cl A so that 7"1 int 7"2 clA = 2~ <---> A e (1, 2)-AfD(X). D e f i n i t i o n 1.1.21. A subset A of a BS (X, 7"1,7"2) is termed (i, j)-somewhere dense in X if A-~ (i, j)-AfD(X), that is, if T~ int 7"j cl A ~ ~. The families of all (i,j)-somewhere dense subsets of X are denoted by Clearly, (i, j ) - S D ( X ) = 2 X \ (i,j)-N'D(X) and hence for the natural BS (R, aJl,W2), we have aJi \ {~} c i-D(R) = 2 R \ i-AfD(R) = i - $ D ( R ) . It is easy to ascertain that the inclusions
(i, j ) - S D ( X ) .
(1,2)-SD(X) c 2-SD(X) A A 1 - $ D ( X ) C (2, 1)-SD(X) hold for a BS (X, T1 < 7"2). In order to use category notions to study bitopological concepts of Baire spaces, we must relate them to the bitopology on a set. D e f i n i t i o n 1.1.22.
A subset A of a BS (X, T1,72) is of (i,j)-first category oo
(also called (i,j)-meager, (i,j)-exhaustible) in X if A -
[.J An, where An e n=l
(i,j)-AfD(X) for every n = 1, oc and A is of (i,j)-second category (also called (i, j)-nonmeager, (i, j)-inexhaustible) in X if it is not of (i, j)-first category in X. A subset A of X is of (i, j)-first category if A is of (i, j)-first category in itself and A is of (i, j)-second category if it is of (i, j)-second category in itself. The families of all sets of (i, j)-first ((i, j)-second) categories in X are denoted by (i, j)- Catg~ (X) ((i, j)- Gatgii (X)), while the statement X c (i, j)- Catg I (X) (X ~ (i, j)-Catgii (X)) is abbreviated to X is of (i, j)-Catg I (X is of (i, j)-Catg II). It is clear that
(i, j)-AfD(X) c (i, j)-datg~ (X) = 2 X \ (i, j)-Catgii (X). R e m a r k 1.1.23. Let T1 = ~ and T2 be respectively the natural and the discrete topology on R. Then n e N implies {n} e (1,2)-AfD(R) and hence N e ( 1 , 2 ) - d a t g i ( R ). On the other hand, if 7~ and T~ are the corresponding
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24
I. Different Families of Sets in Bitopological Spaces oo
n -- 1, oo. Hence F -
U F~ E (i, j)- Cat& (X), and since X \ A
c F, by (1)
n=l
above X \ A E (i, j)- Catg~(X). Thus A E (i, j)- Catgii (X) since by virtue of (1) the contrary means that X is of (i, j)-gatg I. (6) First let {Un}n~176 be a sequence in X, where Un E 7-j a i-D(X) for every oo
n--l,
oo
cxDand n Un--~. T h e n X - X \ n=l
oo
n Un-- U (X\Un),where n=l
n=l
X \ Un c co 7-j n i-13d(X) C (i, j)-Afg)(X) for every n - 1, oc. Therefore X is of (i, j)-Catg I. Conversely, i f X is of (i, j)-gatg I, then by (3), X E j-J~(X)n(i,j)-Cat&(X), oo
i.e. X -
U Fn, where Fn E co 7-j N i-13d(X) for every n - 1, oo. It is clear that n--1
(3o
N(X\Fn)-z,
where {X\Fn }~--1 is a sequence of/-dense j-open subsets of X. IN
n--1
C o r o l l a r y 1.1.25. If (X, 7-1 < 7-2) is of 1-CatgII, then 1-Ga(X) N 2-D(X) C (2, 1)-Catg~i (X).
Proof. By virtue of Proposition 1.5 from [133], X being of 1 - C a t g I I implies 1-Ga(X) n 1-D(X) C 1-Catg~ (X). Hence since 7-1 C 7-2, w e obtain 1-Ga(X) n 2 - D ( X ) C 1 - G a ( X ) n 1 - D ( X ) C 1-Catgi,(X ) C (2,1)-Catgii(X).
D
Further we shall prove some statements connected with Corollary 1.1.25. For the most general result among bitopological results of the type of Proposition 1.5 from [133] reads as follows: if X is of (2, 1)-datg II, then 7-1 C 7-2
2-Gb(X) n 1-D(X) C (1, 2)-Catgii (X) (see (2) of Corollary 2.1.13 below). Nevertheless without resorting to additional assumptions, we have the following weaker statement. Proposition
1.1.26. If (X, 7.1 < 7.2)is of (1,2)-CatgII, then 2-Ga(X) a 1-D(X) C (1, 2)-Cat9~i (X). oo
Proof. Assume that A E 2 - G a ( X ) n 1-/)(X), that is, A -
n An, where An E n=l
r2 n 1-D(X) for every n = 1, oo. Then oo
X \ A - U ( x \ An) E (1, 2)-Catg I (X) n--1
since X \ An E COT2 O 1-Bd(X) C (1, 2)-AfD(X) for every n = 1, oc. If A E (1, 2)-Catgi (X), then by (1) of Theorem 1.1.24 the union A U (X \ A) = X is of (1, 2)-Catg I, which is impossible. D C o r o l l a r y 1.1.27. If a BS (X, Wl < 7.2) is of (1, 2)-datg II, then
1.2. (i, j)-Locally Closed Sets
25
1-Gs(X) N 2-7?(X) C 2-Gs(X) N 2-D(X) N G 1-{75(X) N 1-79(X) C 2 - G a ( X ) n 1-D(X) C (1, 2)-CatgIi (X). Finally, we establish the conditions for sets of (2, 1)-first category in the whole space to become 2-boundaries. Theorem
1.1.28. /f a BS (X, T 1 < 7"2) is 1-compact and (2.1)-quasi regular,
then (2, 1)-Cat&(X) c 2-Bd(X). O<3
Proof. Let A c (2, 1 ) - C a t g i ( X ) be any set, that is, A -
O An, where An n=l
(2, 1)-AfD(X) for each n = 1, oc. Then we must show that V N (X \ A) : / o for each set V c r2 \ {0}. It is evident t h a t if V c r2 \ {0} is an arbitrary set, then V \ rl cl A1 ~ r2 \ { 0 } since V ~ r2 and r2 int rl cl d l = 0. Hence, by condition, there exists a set U1 ~ r2 \ { 0 } such that rl cl U1 C V and 7-1 cl U1 N A1 = 0. Again, there exists a set U2 c r2 \ {0} such that U2 C rl cl U2 C U1 c rl cl U1 and rl cl U2 N A2 = 0 . Following the above reasoning, we can construct a sequence {Un}n~176of non-empty 2-open sets such that rl cl U~ C Un-1 and rl cl UnnAn -- 0 oo
for each n -
1, o0.
But (X, rl < r2) is 1-compact and thus
n r l c l U n =~ 0. n--1
Therefore there exists a point x c X such t h a t x c rl cl Un and x g An for each oo
n-1,
co. T h u s x e V \
U An-V\AandsoVN(X\A)=/=o.
[-1
n=l
1.2.
(i,j)-Locally Closed Sets
D e f i n i t i o n 1.2.1. A subset A of a BS (X, rl, r2) is said to be (i,j)-locally closed at its point x if there exists a set U E ri such t h a t x E U and U O A -
U nrjclA. For every point x E X the families of all sets t h a t are (i,j)-locally closed at the their common point x are denoted by (i, j ) - s x). A subset A of a BS (X, rl, r2) is (i, j)-locally closed if it is (i, j)-locally closed at each of its points. The families of all such subsets of X are denoted by (i, j)-s For a BS (X, rl < r2) the following inclusions hold for every point x c X : 1-/2C(X,x) C (2, 1)-/2C(X,x) n N
< 2-cc(x,
)
and, therefore, 1-s N
c (2, 1)-s N c
Theorem
1.2.2. Let A be a subset of a BS (X, 7-1 ~ 7"2). Then the set
{x c A" A is not (2, 1)-locally closed at a point x} coincides with the set A N r2 el(r1 clA \ A).
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28
I. Different Families of Sets in Bitopological Spaces
so that U n A is j-closed in U. Hence U n A - U n Tj cl A so that A is (i, j)-locally closed at the point x. D Note that the requirement that the BS (X, 7-1,7-2) in Theorem 1.2.5 be (i,j)-regular, is essential. Indeed, if (X, 7-1,7-2) is not (i,j)-regular, then there are a point x 9 X and a neighborhood U(x) 9 7-4 such that 7-j el V ( x ) n (X \ U(x)) r 25 for every/-open neighborhood V(x). From Corollary 1.2.4 it follows that U(x) 9 ( i , j ) - E d ( X ) and thus U(x) 9 (i, j ) - s x). If the condition of Theorem 1.2.5 is fulfilled, then there exists an/-neighborhood W ( x ) such that
n U(x) -
cl ( W ( x ) n U(x))
Let H - 7-4int W(x). Then H n U(x) - E(x) 9 Ti and 7-j clE(x) - 7-j cl(H N U(x)) = 7-j cl (7-4int W ( x ) n U(x)) C 7-j cl (W(x) n U(x)) C U(x), which contradicts the assumption. 1.3. (i, j ) - B o u n d a r i e s . (i, j ) - O p e n D o m a i n s a n d ( i , j ) - C l o s e d D o m a i n s . (i, j ) - S e m i o p e n Sets a n d (i, j ) - S e m i c l o s e d Sets. (i, j ) - S e m i o p e n Domains and (i,j)-Semiclosed Domains Of considerable importance later on is D e f i n i t i o n 1.3.1. For any subset A of a BS (X, 7"1,7"2), the (i, j)-boundaries (i.e., bitopological boundaries) of A are the p-closed sets (i, j)-Fr A - 7-4clA n 7-j cl(X \ A) [84]. The most important properties of these operators are listed in T h e o r e m 1.3.2. For a BS (X, as follows:
7"1,7"2) the
(i,j)-boundaries have the properties
( i , j ) - F r A = B1U B2, where B1 9 j - B d ( X ) and B2 9 i-Bd(X). 7-i cl A = 7j int A U (i, j)- Fr A. Tj int A = A \ (i, j)- Fr A. (i, j ) - F r A = (j, i)-Fr(X \ A). X = Tj int d U (i, j ) - F r A U 7i int(X \ A). (i, j)- Fr T4 cl A U (i, j)- Fr Tj int A c (i, j)- Fr A. A 9 Tj ~ ( i , j ) - F r A = T4clA \ A. A 9 co T4 ~ (i, j)- Fr A = A \ Tj int A. A 9 Tj n co Ti ~ (i, j ) - F r A = ~. (i,j)-Fr A U ( i , j ) - F r B = ( i , j ) - F r ( A U B ) U ( i , j ) - F r ( A N B ) U ( ( i , j ) - F r AN ( j , i ) - F r B ) d ( ( i , j ) - F r B n (j,i)-FrA). (11) If (X1, T} 1) , 7-2(1)) and (X2, w}2) , 7(2)) are BS's with their Catresian product (X, 71, w2) so that X - X1 • X2, 74 - @1) x 7-(2), then the formula
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
(i, j)- Fr(A1 • A2) - ((i,j)-FrA1 • T~2) clA2)t2 (7-(1) clA1 • (i,j)-FrA2) holds for every subset A1 • A2 of X , where Ai c Xi.
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34
I. D i f f e r e n t F a m i l i e s of S e t s in B i t o p o l o g i c a l S p a c e s
so that A c (1,2)-OD(X). Hence, taking into account the inclusions
1-OD(X) c (1,2)-OD(X) c
T1
and (2.1)-OD(X) C 2-OD(X),
we find that 7-1 (]
(2, 1)-OD(X) -- (1, 2)-OD(X) N (2, 1)-OD(X) -
= 1-OD(X)fq (2, 1)-OD(X) c 1 - O D ( X ) n 2 - O D ( X ) C
7"1 (-] 2 - O ~ ) ( X ) -
2)-or(x) n 2-Or(X).
:
The rest is obvious since A E (i,j)-OD(X) ~
X \ A E (i,j)-CD(X).
[3
Here note that such a simple argument as T1 c ~-2 is helpful in establishing the inclusions 1-OD(X) C (1,2)-OD(X) and (2, 1)-OD(X) c 2-OD(X). Indeed, A c I-OD(X) implies A
-
T1
int
T1
el A and so A
-
T1
int A.
Hence A G T1
int T2 cl A c
T1
int
T1 c l A -
that is A
A,
-
T1
int ~'2 cl A.
If A c (2, 1)-OD(X), then A - ~-2int T1 clA. Thus A c ~'2 int T2 cl A c ~-2int
T1
el A
--
A so that A c 2-OD(X).
The reverse inclusions in the foregoing corollary are not, generally speaking, correct for a BS (X, T1 < T2). E x a m p l e 1.3.8. Let X and
T1 be as
in Example 1.3.5, 7-2 - {~, {a},{c},{a, c},
{a, b},{a, b, c}, X}. Then {a} E 1 - O D ( X ) c (1,2)-OD(X),
{a} -@2-OD(X)
so that {a} ~ (2, 1)-OD(X). Also, it is clear that {a, b} c (2, 1)-OD(X)
<
2-OD(X),
but {a, b} ~ (1, 2)-OD(X) and, therefore, {a, b} g 1-OZ)(X). E x a m p l e 1.3.9. Let X be as in Example 1.3.5, T 1 - - {~, {a}, {a, b, c},X}, T2 - {Z, {a}, {d}, {a, d}, {b, c}, {a, b, c}, {b, c, d}, X}. Then {a} E (1,2)-OD(X) \ 1-OD(X), {a,d} c 2-OD(X) \ (2, 1)-OD(X). By Example 1.3.9, T1
cl{a} \ {a} = X \ {a} = {b, c, d} -4 2-13d(X)
so that, generally speaking, the difference Tj cl A \ A need not be a n / - b o u n d a r y set for a subset A c ~-i of a BS (X, T1, T2). However, we have P r o p o s i t i o n 1.3.10. The following conditions are satisfied for a BS (X, T1, T2): (1) If A E (i,j)-OD(X), then Tj clA \ A e i-Bd(X).
1.3. (i, j)-Boundaries. (i, j)-Open Domains and ...
(2) If A1, A2 ~ ( i , j ) - 9
A1 c A2 ~
35
then rj clA1 c rj clA2
so that if A1, A2 e (i, j ) - d D ( X ) , then A1 c A2 ~
rj int A1 c rj int A2.
For a BS (X, rl < 72), we also have: (3) U c r2 ~ (U = V \ r 2 c l A , where V c ( 2 , 1 ) - O : D ( X ) a n d A c (1, 2)-N'T)(X)) so that F c coT2 ~ ( F = B U T 2 c l A , where B c ( 2 , 1 ) - r i D ( X ) and A c (1, 2)-AfT)(X)).
Proof. (1) ~-~int(Tj cl A \ A) = Ti int Tj cl d n ~-i int(X \ A) = d A ~-i int(X \ A) = ~. (2) It is obvious that A1 c A2 implies ~-j cl A1 c ~-j cl A2. On the other hand, if ~-j cl A1 c Tj cl A2 (i.e., Tj cl A1 N Tj int(X \ A2) = ~), then
A1 nT-j i n t ( X \ d 2 ) = 2~ (i.e., A1 NTiclTj i n t ( X \ A 2 ) = 2~) since A1 e(i,j)-OTP(X). Hence
A2 c (i,j)-O~D(X)
i m p l i e s n l ffl ( X \ A2) -- z &lid so A1 c A2.
The second part is obvious. (3) Let U c ~-2 and V = 7~ int 7-1 cl U. The difference 7-1 CI V \ U C co 7-2 n 1-gd(X) c (1, 2)-JVZ)(X)
and hence A = V \ U e (1, 2)-AfD(X). Clearly, U = V \ A and r2 cl A = r2 cl(V \ U) c r~ cl V N r2 cl(X \ U) = r~ cl V \ U. Thus
U= VNU=
V \ ( r 2 c l V \ V) c V \ r 2 c l A
c V\A=U,
that is, U = V \ r2 cl A. The reverse implication is obvious. If F c co r~, then X \ F = V \ r 2 c l d , where V c (2, 1)-OD(X) and d c (1, 2)-N"T~(X). Hence F = (X \ V) U w~ clA = B U ~-2 clA with B e (2, 1)-UT?(X).
D
D e f i n i t i o n 1.3.11. A subset A of a BS (X, T1,T2) is called (i,j)-semiopen in X if there exists an /-open set U C X such that U C A c TjclU. The complement in X to an (i, j)-semiopen set is (i, j)- semiclosed in X, that is, a subset B of X is (i,j)-semiclosed in X if there exists an/-closed set F C X such that ~-j int F c B c F [861, [1751. The families of all (i, j)-semiopen ((i, j)-semiclosed) subsets of X are denoted by ( i , j ) - S O ( X ) ( ( i , j ) - $ C ( X ) ) . The theorem below is an immediate consequense of Definition 1.3.11. Theorem
1.3.12. Let (X, T1, ~-2) be a BS. Then
A E ( i , j ) - $ O ( X ) <--> A c Tj cl ~-~int A
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~
"~
~
~
U
38
I. Different Families of Sets in Bitopological Spaces
(4) if ~-j int Bt c C c Bt for some t E T, then C E 13. Then ( i , j ) - S C ( X ) C 13 and, therefore, ( i , j ) - $ C ( X ) is the smallest family of subsets of X satisfying (3) and (4).
Proof. It is clear that by (1) of Corollary 1.3.13 and the first condition in Proposition 1.3.16, the family ( i , j ) - S O ( X ) satisfies (1) and (2). If A E ( i , j ) - S O ( X ) , then by Definition 1.3.11, there is a set U c ~-~ such that U c A c Tj cl U. Since U E ~-i C A, by (2) above A c A. Thus (i, j ) - S O ( X ) c A and (i, j ) - S O ( X ) is the smallest family of subsets of X, satisfying (1) and (2). The rest is obvious. [-1 P r o p o s i t i o n 1.3.19. The following equalities hold for a BS (X, T1, T2) : ~-i = i-Int(i, j ) - S O ( X ) = i-Int(i-SO(X) ).
Proof. If U c T~, then U c ( i , j ) - S O ( X ) . But U = T~intU so that U c i-Int(i,j)-SO(X) and thus ri C i-Int(i,j)-SO(X). The reverse inclusion is obvious by virtue of the definition of the family i-Int(i, j ) - S O ( X ) . Lemma 2 from [170] gives the second equality. D R e m a r k 1.3.20. For a BS (X, 71, r2) any union (intersection) of (i, j)-semiopen ((i, j)-semiclosed) sets is (i,j)-semiopen ((i,j)-semiclosed)[86], [175]. But the intersection (union) of two (i,j)-semiopen ((i,j)-semiclosed) sets may not be (i, j)-semiopen ((i, j)-semiclosed) [175, Example 3]. However, there holds P r o p o s i t i o n 1.3.21. For a BS (X, T1 < T2), we have
(U E 7-1 and A E ( i , j ) - S O ( X ) ) ~
U n A E (i,j)-$O(X)
so that ( F E COT1 and A E ( i , j ) - S C ( X ) ) ---5, F U A E ( i , j ) - S C ( X ) .
Proof. Let U E rl, A E (i, j ) - S O ( X ) and U n A r 2~. For the set A, there is a set Vcrisuchthat VcAcrjclV. Clearly, U N A C ~ i m p l i e s U n r j c l V C ~ s o that U n V r ~ as U E T1 C 7"2. Hence UNV C UNA c UnrjclV
C 7-jcl(UNV),
where U N V E Ti. The second implication follows from the first one. Remark 1.3.20 leads to
[-1
D e f i n i t i o n 1.3.22. Let (X, T1,7"2) be a BS and A c X be any subset. Then
(i,j)-sintA - U {As
As E ( i , j ) - S O ( X ) , As c A for every s c S}
sES
and (i, j ) - s c l A - n sES
Is6], [ 75]
{As
As E (i, j ) - S C ( X ) , A c As for every s E S}
1.3. (i, j ) - B o u n d a r i e s .
(i, j ) - O p e n
Domains
and . . .
39
Thus ( i , j ) - s i n t A are the largest (i,j)-semiopen sets, contained in A and (i, j)- scl A are the smallest (i, j)-semiclosed sets, containing A. Hence
A c ( i , j ) - $ O ( X ) <--> A = ( i , j ) - s i n t A ,
A c (i,j)-Sg(X)
<--> A = (i,j)- sclA
and (i, j)- sint A = X \ (i, j)- scl(X \ A). Moreover, by (1) and ( 2 ) o f Corollary 1.a.la, T~ int A c (i, j)- sint A n i- sint A and
(i,j)- sclA U i- sclA c ~-i clA for every subset A c X. R e m a r k 1.3.23. Following (1) of Corollary 1.3.15, for every subset A of a BS (X, rl < r2), we have: r l i n t A c (1, 2)- sint A c 1- sint A N O O T2intAc 2-sintA c (2,1)-sintA so that (2,1)- sclA C 2- sclA c r2clA N N N 1- sclA C (1,2)- sclA c T1 clA. T h e o r e m 1.3.24. subset A c X "
For a BS (X,
7-1,7-2)
the equivalences below hold for any
(1) A E ( i , j ) - A f D ( X ) ~ (i,j)-sint(j,i)-sclAz. (2) A e i - D ( X ) ~ i-sclA- X ~ (i, j ) - s c l A X.
Proof. (1) Let A c (i,j)-Af~P(X), that is, r i i n t r j c l A - z . Then ( i , j ) - s i n t T j c l A = z . Contrary: (i, j)- sint rj c l d # ;~ and so, there is a set B e (i, j ) - $ O ( X ) \ {;~} such t h a t B C rj clA. By Definition 1.3.11, there is a set V E ri \ { z } such t h a t V C B c r j c l V , and, therefore, V c B c r j c l A . Hence ~ - i i n t r j c l A # ~, which contradicts A c (i,j)-Af~P(X). Since (j,i)- scl A c rj clA, we find t h a t (i, j)- sint(j, i)- scl A - z . Conversely, let (i, j)- slut(j, i)- scl A - ~. Then T~ int(j, i)- scl A - ;~. Assume t h a t F - (j,i)- sclA c ( j , i ) - $ d ( X ) . By Definition 1.3.11, there exists a j-closed set B such that r i i n t B C F c B and, consequently, A c B so t h a t rj cl A c B. Moreover, r i i n t B C r i i n t F - Ti int(j, i)- sclA - ~ and, therefore, Ti int B - z , which eventually gives r i i n t rj cl A - ~. (2) Since (i, j)- scl A U i- scl A C ~-~clA, it is evident t h a t (i, j)- sclA - X or i- scl A - X implies A c i - D ( X ) . Thus it remains to prove t h a t Ti clA - X implies i- scl A - X since the implication ri cl A - X > (i, j)- scl A - X can be proved in a similar manner. Let ri clA - X and i- sclA # X. Then there exists a set F c i - $ d ( X ) such that A c F # X, and, therefore, B - X \ F c i - 8 0 ( X ) \ { ~ } . Clearly, B N A ~.
40
I. Different Families of Sets in Bitopological Spaces
By Definition 1.3.11 for j - i, there is a set V E 7-~\ {~ } such that V c B c 7-~cl V and V n A - ~, which contradicts ~-i cl A - X. [-] The condition (1) of Theorem 1.3.24 and Theorem 1.11 in [68] allow us to define the category notions from Chapter I, the Baire and Baire-like properties from Chapter IV also by means of (i, j)-semi-interiors and (i, j)-semi-closures for BS's and, consequently, to define the category notions and Baire property for TS's too. Proposition (XtT1 < T2) "
1.3.25.
The following statements are satisfied for a BS
(1) If A E (1, 2 ) - $ O ( X ) , UNB-~. (2) If A E (2, 1 ) - S O ( X ) , andUnB-~. (3) If A - U U B, where A c (1, 2 ) - $ O ( X ) . (4) If A - U u B, where A E (2, 1 ) - S O ( X ) .
then A - U U B, where U E 7"1, B E 2-J~f~)(X) and then A -
U U B, where U E T2, B E (1, 2 ) - A f t ( X )
A is 2-connected, U ~ T1 \ {2;} and Bd2 -- ~, then A is 1-connected, U c ~-2 \ {~} and Bdl -- ~, then
Proof. (1) A E (1, 2)-8(9(X) implies that U c A c ~-2 cl V for some V E T1. Hence A - V U (A \ U) and A \ U c T2 cl V \ U c co T2 N 2-Bd(X) c 2 - A f t ( X ) so that A - U U B, where B - A \ U. The proof of (2) is straightforward, proceed as in (1). (3) Clearly, it suffices to show that B c w2 cl U. Contrary: B n (X \ T2 cl U) D#~. ThenB-DUE, whereE-BNT2clU. Hence A-(UUE)
UD,
UUE#~
and U U E c T 2 c l U
so that w2clUNA-UUE
EcoT~
in (A, T[, T~). Since D d C B d, that is, D d - o, it is likewise clear that D E co ~-~. Thus U U E and D constitute the 2-separation for A, which is impossible. Assertion (4) can be proved by analogy with (3). E] Of the above statements on subsets of a BS (X, T1 < 7"2) the first two give the necessary conditions and the other t w o - the sufficient conditions for the belonging of subsets to the families (i, j ) - S O ( X ) . We close this section by introducing new families of subsets of a BS (X, T1,T2) whose elements belong to the meet ( i , j ) - S O ( X ) N (j, i)- S C ( X ) . It is important to note the fact that their corresponding topological family is also new, although the elements of the latter family were casually mentioned as semiclopen (i.e., both semiopen and semiclosed) sets [226]. D e f i n i t i o n 1.3.26. A subset A of a BS (X, T1, T2) is called an (i, j)-semiopen domain if there exists an (i, j)-open domain U c X such that U C A c Tj cl U.
1.3. (i, j ) - B o u n d a r i e s .
(i, j ) - O p e n D o m a i n s a n d . . .
41
The complement in X to an (i,j)-semiopen domain is an (i,j)-semiclosed domain, that is, a subset B of X is an (i, j)-semiclosed domain if there exists an (i, j)-closed domain F c X such that Tj int F C B C F. Hence a subset A of a TS (X, T) is a semiopen domain (semiclosed domain) if there exists an open domain U c X (closed domain F c X) such that U C A c clU ( i n t F C A c F). The families of all (i,j)-semiopen domains ((i,j)-semiclosed domains) of X are denoted by (i,j)-$O:D(X) ((i,j)-$CZ)(X)). Theorem
1.3.27. The following conditions are satisfied for a BS (X,
7"1,7"2)
"
-
and thus for a TS (X, T), we have
s o D ( x ) - sc (x). (2) If A c ( j , i ) - S d ( X ) (A c ( i , j ) - $ O ( X ) ) and
A c TjclTiintTjclA
(TiintTjclTiintA C A),
then A e (i, j)-SOlP(X) - ( j , i)-$ClP(X) and thus for a TS (X, T), we have if A c $C(X) (A E S O ( X ) ) and A c c l i n t c l A ( i n t c l i n t A c A), then A c SOTs(X)Proof. (1) Let A c (i,j)-SOlP(X). that U c A c T j c l U . But
$CZ)(X).
Then there is a set U c (i,j)-OlP(X) such
U E (i, j ) - O D ( X ) ,z---->,U - ~-i int Tj cl U and by Remark 1.3.4, Tj cl U C (j, i)-CD(X),
that is ~-j cl U - Tj cl T~ int ~-j cl U.
Thus for a set A there is a set Tj cl U -- F E (j, i ) - r i D ( X ) such that T, int F C A C F and, therefore, we obtain A c (j, i)-$CD(X). In a similar manner it can be shown that
(j, i)-SCD(X) c (i, j)-SOIP(X) so that (i, j ) - S O D ( X ) - (j, i)-SCD(X). (2) Suppose A c (j,i)- $ C ( Z ) and A c TjclTiintTyclA. By virtue of Theorem 1.3.12, 7~intTjclA c A as A c (j, i)- SC(X). Assume that U TiintTjclA. Following Remark 1.3.4, V c ( i , j ) - O D ( X ) and V c A. Thus U C A c 7j cl U and, taking into account Definition 1.3.26 and (1) above, we find that A e ( i , j ) - S O Z ) ( X ) = (j, i)-8CT?(X). The rest is proved similarly. F1 C o r o l l a r y 1.3.28. Let (X, ~-1,~-2) be a BS. Then
(i, j ) - O D ( X ) U (j, i ) - C V ( X ) C (i, j ) - S O D ( X ) = (j, i)-scz
(x) - (i, j ) - s o ( x )
n (j, i ) - s c ( x )
+
9
"++
m+
II
.
~-~
9
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,..,_.+
+If +-3 .
9
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t-r"J
.+
~
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44
I. Different Families of Sets in Bitopological Spaces
and so Tjint(X\U) cX\AcX\U. Hence X \ A 9 (i, j)-SCD(X)/co AThe proof of the inverse inclusion is similar since the equality co(co A) = A gives that if co A = B, then co/3 = A. The rest of the proof is an immediate consequence of (2) of Corollary 1.3.7 and (2) of Corollary 1.3.15. [] Proposition 1.3.34 allows us to compare the families i-SOD(X) = i-SCD(X) and (i, j ) - S O D ( X ) = (j, i)-$dD(X) by the set-theoretic operation inclusion for a BS (X, T1 <72). 1.4. Sets Pairwise D e n s e in T h e m s e l v e s . ( i , j ) - P e r f e c t Sets and Pairwise Scattered Sets
This section is devoted to the study of other new families of subsets of BS's. Definition 1.4.1. itself if A~ c A].
A subset A of a BS (X, T1,T2) is termed (i,j)-dense in
The families of all subsets of X, (i,j)-dense in themselves, are denoted by (i, j ) - D I ( X ) and the families of all/-discrete subsets of X are denoted by i-Z(X). P r o p o s i t i o n 1.4.2. Let (X, T1, ~-2) be a BS. Then
(1)
1-D27(X) U 2-D27(X) c (1, 2)-DYT(X) = (2, 1)-D27(X)
and for a BS (X, 71 < Ts), we also have (2) 2-D2"(X) c 1 - D I ( X ) = ( 1 , 2 ) - D I ( X ) = ( 2 , 1)-D2:(X), 1-Z(X)c2-2:(X). Proof. It is clear that the inclusion in (1) follows from the equality in (1) and thus, it suffices to prove only the equality. Let A 9 (i, j ) - D I ( X ) . Then A~ c A j so that X \ A j c X \ A~. Hence d i d A n (X \ Aj) c A n (X \ A~), that is A \ Ajd c A \ A~i and Aji c A \ A ii c A~.
Thus A 9 (j, i)-DI(X). (2) The inclusion 71 C T2 implies that A d c A d and A] c A~ for every subset A c X, that is, 2-DI(X) c 1-DZ(X) and 1-I(X) c 2-I(X). By Definition 1.4.1,
A 9 (1,2)-D:r(Z)~
dil c g d.
Hence d~ c d l d implies A~ - ~. Thus A c d l d, that is, A 9 1-DI(X). From this proposition we conclude that if for a BS (X, has at least one l-isolated point, then
T1 <
[~
T2), & set A C X
A ~ 1-D2:(X) = (1,2)-DI(X) = (2, 1)-D2:(X). By the first equality in (1) of Proposition 1.4.2, it suffices to consider only the family p-D:T(X), where p - D I ( X ) = (1, 2)-D2:(X) = (2, 1)-D2:(X).
1.4. Sets Pairwise Dense In Themselves. (i, j)-Perfect Sets and . . .
45
E x a m p l e 1.4.3. Suppose we are given the BS (R, 7-1,7-2), where 7-1 = w is the 1 1 1 natural topology on R, 7-2 is the discrete topology on R and A - {1, 2, 3, 4 "'" } c R. Then AId - {0} and, therefore, A~ - A \ A1d - A, A~ - A and A d - 2~. Hence A gp-TPZ(R). The reverse of the first inclusion in (2) of Proposition 1.4.2 does not hold in general. E x a m p l e 1.4.4. Suppose 7-1 is the antidiscrete topology on R, 7-2 = w is the natural topology on R and a subset A c R is the same as in Example 1.4.3. Then A~ - 2~, A1d - R, A~ - A, and d d - {0}. Hence A~ - ~ implies that A c 1 - ~ P Z ( R ) - p-~PZ(R), but A ~ 2-9 since A~ r ~. P r o p o s i t i o n 1.4.5. For a BS (X, 7-1,7"2) any union of sets p-dense in themselves is p-dense in itself.
Proof. Let {As }s~s be any family of sets p-dense in themselves so t h a t (As)~ (As \ (As)ld) c (As) d for every s E S. It is well known t h a t U (As) d c ( U As) d sCS
sCS
and hence
\( U
U
sES
U
sCS
sES
\ sU CS
U
sCS
(U
s6S
\
2"
l--i
sES
If there exists a set A E i-TP(X) N i-Bd(X), then by virtue of Theorem 4 from [161, p. 83], X c i-T~I(X) C p-T)Z(X). Based on this fact, our next proposition gives examples of new members of the family p-7?:r(X). Proposition 7-1
1.4.6. For every subset A of a BS (X, 7-1 < 7-2), We have
int (1, 2)- Fr A, A n 7-1 int (1, 2)- Fr A c p -T):Z-(X).
Pro@ As is well-known, if U E 7-1 and A c X is any set, then 7-1 cl (U N 7-1 cl A) - 7-1 c l ( g n A). Therefore 7-1 c I ( U n A ) -
7-1 cl (U N 7-2 el A).
Let U -- T1 int(1, 2)-Fr A - 7-1 int(2, 1 ) - F r ( X \ A). Then it obviously follows t h a t T1 cl U - T1 cl(U n T1 cl A) - T1 cl(U N A) and T, C1V -
T 1 C1 ( V n T 2 c l ( X
\ A)) -
T1 c l ( U n ( X \ A)) -
that is Wl cl T1 int(1, 2)- Fr A -
T1 cl(U \ A),
46
I. Different Families of Sets in Bitopological Spaces
=71cl(AnT1int(1,2)-FrA) -71cl(7.1int(1,2)-FrA\A).
(.)
The first equality in (.) shows that the set A N 7.1 int(1, 2)-Fr A is 1-dense in 2)- Fr A. Moreover, (.) also implies that
7"1 int(1,
7.1 int(1, C 7.1
2)- Fr A c 7.1 el (7.1int(1, 2)- Fr A \
el (7-1 int(1, 2)- Fr A \ ( A n
A) c 7.1 int(1, 2)- Fr n))
so that A n 7.1 int(1, 2)-Fr A is the 1-boundary set in 7.1 int(1, 2) Fr A [161, p. 76]. Hence by Proposition 1.4.2 and by virtue of the remark preceding this proposition, we obtain 7.1 int(1, 2)- Fr A c 1-DZ(X) = p-DZ(X). Since A n 7.1 int(1, 2)- Fr A is 1-dense in 7.1 int(1, 2)-Fr A, by virtue of Theorem 3 from [161, p. 83], we obtain A n 7.1 int(1, 2)- Fr A c 1-DZ(X) = p-DZ(X).
D
D e f i n i t i o n 1.4.7. A subset A of a BS (X, 7.1,7.2) is said to be (i,j)-perfect if
Af
A
The families of all (i, j)-perfect subsets of X are denoted by (i, j)-P(X). Hence
(i, j)-~(x)
=
co ~
p-Z~z(x).
n
P r o p o s i t i o n 1.4.8. For a BS (X, 7.1,7.2), we also have i-7)(X) c (i,j)-7)(X), and for a BS (X, 7.1 < 7.2), we have (1, 2 ) - 7 ) ( X ) = 1 - P ( X ) c (2, 1)-7~(X) D 2 - P ( X ) C 1-DZ(X).
Proof. By (1) of Proposition 1.4.2, i-7)(X)-covi n i-DZ(X) C COTi N p - D Z ( X ) - (i,j)-7)(X). Furthermore, if T1 C T2, then by (2) of Proposition 1.4.2, (1, 2)-7~(X) -- co7.1 n p - ~ ) J Z ( X ) - COT1 n 1-DZ(X) -= 1-7~(X) C co 7.2 N p - D : Y ( X ) - (2, 1)-7)(X). Finally, 2 - 7 ) ( X ) - co7.2 N 2 - D Z ( X ) C 2-DZ(X) C 1-D:Z(X) and 2-~(x)
-
c o ~-~ n 2-Z~:Z(X) c c o r
o
p-Z~Z(X) - (2, 1 ) - V ( X ) .
a
The inverse inclusions in the latter proposition are not, generally speaking, correct. E x a m p l e 1.4.9.
Let (]I~, T1, "/-2) be the BS from the previous example. If then A~ - 2~, A d - {0} and hence A~ c A d c A so that A E (2, 1)-7)(R). But A ~ 2 - P ( R ) as A / : A d. It is likewise easy to observe that for the set Z, we have Z~I - 2~, Z d - R, Z~ - Z and Z d - 2~. Hence Z c (2, 1)-7)(R), but Z ~(1, 2)-7)(R) since R - Z1d is not contained in Z. A -
{1, 2,1 3,1 4 , " 1 . , 0 } ,
P r o p o s i t i o n 1.4.10. For a j-T1 BS (X, 71,72), we have j - C I ( p - D Z ( X ) ) c (j,~)-~(x).
Proof. We shall show that A~ c Ajd implies (Tj clA)~ C (Tj clA) ] c zj clA
1.4. S e t s P a i r w i s e D e n s e I n T h e m s e l v e s .
(i, j ) - P e r f e c t
Sets and ...
47
Clearly,
(rj cl A)jd c rj cl(Tj cl A) - rj cl A. Further,
(rj cl m){ - rj cl d \ (rj cl A) d - (d U A d) \ (A U gd) d = = (A U A d) \ (A d U (Ad) d) - ((A O Aj) \ A d) n ((A U __ ((A \ Ad) u (Ad \ Ad)) N ((A \ (Aj)i) Ud
AJ) \/AJ)f)
(Ad \ (Aj)~))dd .
Since A d \ A d c Aad and A \ A d - A~ C A d, we conclude that the union in the first brackets above is contained in A d. Hence the set (rj cl A){, being the intersection, is completely contained in Aad. But A d - ( r j cl A) d and, therefore, (rj clA)~ c (rj clA) d c rj clA.
[-1
Using the equalities (1, 2)-T)27(X)= (2, 1 ) - D Z ( X ) = p - D Z ( X ) , we introduce D e f i n i t i o n 1.4.11. A subset A of a BS (X, rl, r2) is termed p-scattered if A is nonempty and contains no nonempty p-dense in itself subset. The families of all p-scattered subsets of X are denoted by p - S T ( X ) . The latter definition and Proposition 1.4.2 readily yield P r o p o s i t i o n 1.4.12. The following conditions are satisfied for a BS (X, rl, r2):
(1) p - s ~ - ( x ) c ~ - s r ( x ) n 2-S~-(X). (2) A c p - S T ( X ) and B c A, B ~ ~, imply B c p - S T ( X ) . For a BS (X, rl < r2), we also have (3) 1-ST(X) = p-ST-(X) C 2-87"(X). R e m a r k 1.4.13. Let (IR, rl, r2) be the BS from Example 1.4.4, where
A-
1, 2' 3 ' 4 "
E 1-DZ(N) -p-DZ(N).
If ~ r B c A is any subset, then B c
1-Z~Z(R) = p-Z~Z(R) and so _4 g 1-8~r(R) = p - S ~ r ( R ) .
But B c A implies B~ C A~ - {0} and, therefore, B N B ~ -- 2~. B ~ 2-DZ(R), and since B C A is arbitrary, we obtain A c 2-ST(R).
Hence
Also note that following Example 1.4.4 and the above remark, we have
A-
{111
1,2, a,4, - -
} ~
~
~
c 2-I(R)
~
Thus, generally speaking, the family i-Z(X) is not contained in p - S T ( R ) since A g p - S T ( R ) . However, we have P r o p o s i t i o n 1.4.14. If for a BS (X, rl, r2) a set A c i-Z(X) and for every subset B c A we have B-g j-Di[(X), then A ~ p - S T ( X ) .
Proof. Assume that A c i - l ( X ) and B C A is an arbitrary subset. Then B c i-Z(X), and B - g j - D Z ( X ) implies that Bji # ~. Since B N B d - ~ and Bji c B, we
48
I. Different Families of Sets in Bitopological Spaces
obtain Bji n B d - Z so that B - ~ p - D Z ( X ) . Therefore A E p - S T ( X ) is arbitrary.
since B c A D
P r o p o s i t i o n 1.4.15. Let (X, 7-1,7-2) be a Rep-T1 BS. Then X = A U ( X \ A), where A c (1, 2 ) - 7 ) ( X ) n (2, 1)-7)(X) and X \ A c p - S T ( X ) (it stands to reason that one of these subsets may turn out empty). Proof. Suppose A = U{E : E E p - D Z ( X ) } . Then by Proposition 1.4.5, A c p - D Z ( X ) . It follows from Definition 1.4.7 and Proposition 1.4.10 that 7-j clA E (j, i ) - P ( X ) c p - D r ( X ) . Recalling that A is maximal, we obtain A c co 7-1 n CO 7"2 and, therefore, A
=
7-1
clA = 7-2 clA E (1, 2 ) - 7 ) ( X ) n (2, 1)-7)(X).
It is likewise clear that X \ A contains no nonempty p-dense in itself subset and thus X \ A E p - S T ( X ) . D The latter proposition shows that the maximal p-dense in itself subset of a BS (X, 7-1,7-2) is both 1- and 2-closed. P r o p o s i t i o n 1.4.16. For a BS (X, have (1, 2)-Fr A E (1, 2)-AfD(X).
7-1 < 7-2)
and every set A c p - S T ( X ) ,
we
Proof. By Proposition 1.4.6, the set ANT-1 int(1, 2)- F r A c p - D Z ( X ) and, hence, is empty in view of the fact that A E p - S T ( X ) . Moreover since ANTI int(1, 2)- Fr is 1-dense in 7-1int(1, 2)- Fr A, the latter set is also empty, that is, ( 1 , 2 ) - F r A co 7-2 N 1-Bd(X) C (1, 2)-AfD(X).
it A c D
Note that due to Propositions 1.4.2 and 1.4.12 for a BS (X, 7 1 ( 7 - 2 ) all results that hold for the families 1-DZ(X) and 1 - S T ( X ) also hold for the families p - D r ( X ) and p - S T ( X ) , respectively. We finish the discussion of (i,j)-perfect sets by proving a bitopological modification of Cantor-Bendixson's theorem. In this context recall that for a BS ( X , 7-1 < 7-2) a set A c p - D Z ( X ) if and only if U c 7-1 and U N A =fi 2~ implies that U n A is infinite, and a point x E X is an/-condensation point of a set A c X if e a c h / - o p e n neighborhood U(x) meets A in an uncountable set. The set of all /-condensation points of A is denoted by A ~ T h e o r e m 1.4.17. Let (X, 7.1 < 7.2) be a l-T1 and 1-second countable BS. Then any uncountable 2-closed set F contains a set A E co7.2 n p - D r ( X ) = (2, 1)-P(X). Moreover, F ~ = F ~ = T1 clA. Proof. Let F E cot2 be an uncountable set and U1, U2,... be a countable base of 1-open sets. Suppose that V1, V2,... be those of the sets U1, U2,... that meet F in a countable set. Clearly, the sequence V1, V2,... may be finite or infinite. If (X)
A-
F \ U vk, then rl c r2 implies A c co r2 and k=l OO
OO
u
(u
k=l
k=l
(N:)
(U k=l
1.4. Sets P a i r w i s e D e n s e In T h e m s e l v e s .
(i, j ) - P e r f e c t Sets a n d . . .
49
oo
oo
Since F is uncountable and U vk a F is countable, we obtain t h a t A - F \ U vk k=l
k=l
is uncountable and, therefore, non-empty. It remains only to prove t h a t A c p-7?Z(X). But we would rather prove t h a t A c A ~ that is, each point z c A is a 1-condensation point of A so t h a t U(z) E T1 gives I N ( z ) N A[ > R0. Indeed, let U(x) c T1 be any neighborhood. Then there is a set Un from the 1-countable base 0"1, 0"2,... such t h a t z c Un c U(x). It is obvious t h a t Un /: Vk for each k since the contrary implies z g F \ U vk = A. k=l
This means that IF a U~] > R0. Since A differs from F merely in a countable set, it follows that [ANU~ I > R0. But A N U ~ c A N N ( x ) so that I A N U ( x ) [ > b~0 and so A c A ~ Therefore A c AId and thus A c co ~-2 n p-7927(X) - (2, 1)-7)(X). Now, we have A c A ~ c A d so t h a t T1 C1 A C T1 C1 A~ -
Therefore A ~
-
7-1
A 0 C T1
cl A d - A d c 71 cl A .
cl A. Finally, (x)
(3o
-1~
1
k=l
k=l
1
C o r o l l a r y 1.4.18. Under the hypotheses of Theorem 1.4.17, for any uncountable set B C X there is an uncountable subset A C B such that A c p-T)Z(X).
Proof. Let B c X be any uncountable set. Then the result follows directly from the proof of Theorem 1.4.17 omitting the remark that A is 2-closed. D At the end of this section we shall consider three operators on 2 x , which are used to characterize degrees of nearness of the four boundaries of a set, the S-, C- and N-relations in Chapter II and interrelations of dimension functions in Chapter III. D e f i n i t i o n 1.4.19. For a BS (X, T1 < ~-2) the indicators of nearness of the boundaries are the following three operators: n l , n2, n : 2 X --~ (2, 1)-12C(X), defined as follows: nl
(A) = T1 cl A \ 7-2 cl A, n2(A) = ~-2 int A \ T1 int A
and n(A) - n l ( A ) u n2(A) for each set A c 2 X. It is obvious t h a t n~(A) - n j ( X \ A) so t h a t n(A) - n ( X \ A) for each set A c 2 X, the restrictions nllCOWl
-- n2171
--
n
71neoY 1 z
and, therefore, nlw I - n l ,
n[co~-i - n 2 .
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1.5. Relative Properties
51
If A 9 (72 \ 7"1)n CO 7-1, then n ( A ) -- n 2 ( A ) - A N T1 c l ( X \ A) - T1 c1A AT1 c l ( X \ A) - 1 - F r A -
=T2clANTlcl(X\A)-(2,1)-F'rA, If A
r (~-2 \ 71) N (co 7-2 \ co 7-1),
(1, 2)- Fr A - 2- F r A - 2~.
then
n l ( A ) - 71 c I A n ( X \ A) - 7-1 c l A n
T2 c l ( X \ A) - (1, 2)- E r A ,
n 2 ( A ) - A n T1 c l ( X \ A) - T2 C1A n T1 c l ( X \ A) - (2, 1)- Fr A,
n(A)-(1,2)-FrA
U (2,1)-FrA-(1,2)-FrAA(2,1)-FrA-I-FrA,
2-FrA-2~.
(3) The proof consists of elementary calculations taking into account the fact that the equality nl(A) - n2(X \ A) is fulfilled for each set A 9 2 x. (4) If A 9 p-TPZ(X), then A~ c A d - (A1d \ A d) U A f so that A; c (A d \ A d) - (A U A1d) \ (A-U A d) - nl (A). Conversely, A~ c nl(A) implies A~ c A1d \ A~ c A1d and so A 9 p-7:)Z(X).
[]
1.5. R e l a t i v e P r o p e r t i e s Relative properties, that is, to say, such properties of subsets of subspaces of TS's that are preserved from spaces to subspaces, from subspaces to spaces or in both directions, were investigated in various published works on general topology. Naturally, there arises a question how widespread relative properties are in the theory of BS's because these properties will play an important role in our further investigations. D e f i n i t i o n 1.5.1. Let (X, T1,7-2) be a BS and (Y, r~, r~) be a BsS of X. If A c Y, then (1) A c (i, j)-D(Y) if one of the following equivalent conditions is satisfied:
Y-T~clTjclANY
~
Y c r~ clw] clA ~
Y - r~'clw] clA.
(2) A c (i, j)-Bd(Y) if one of the following equivalent conditions is satisfied: Y - 7-~cl(Y \ ~-~int A) N Y <--> Y C 7-i cl(Y \ 7-] int A) < :, < :- Y - ~-: cl(Y \ T] int A) ~
~-~int r~ int A - 2~.
(3) A c (i, j)-A/'TP(Y) if r] clA c i-13d(Y)so that if one of the following equivalent conditions is satisfied: Y - Ti cl(Y \ ~-] el A) N Y +---> Y C ~-i cl(Y \ ~-] el A) < > Y - r~ cl(Y \ 7-~ cl A) <--->, T~ int r] cl A - 2~. (4) A c (i, j)-SlP(Y) if one of the following equivalent conditions is satisfied: Y r ~-i cl(Y \ Tj cl A) n Y <-->, Y \ T~cl(Y \ T] cl A) / 2~ ,' < ;-Y # T/cl(Y \ T ] c l A ) ~
T~intT~clA =/= 2~.
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53
C o r o l l a r y 1.5.4. Let (X, 7.1 ( 7 " 2 ) be a BS, (7.2clY, 7.~,7.~)and (Y, 7.~',7.~') be BsS's of X , and A c 7.2 cl Y. Then the following conditions are satisfied: A c (2, 1)4VD(~2 d Y) --*, A n Y c (2, 1)-ArD(Y) so that
A n Y c (2, 1)-$/P(Y) ---> A c (2, 1)-S:D(% cl Y) and thus A E (2,1)-Catgi(7.2clY) ~
A N Y c (2,1)-Catgi(Y),
A N Y E (2,1)-Catgii(Y)
A c (2,1)-Catgii(7.2clY).
~
Proof. It suffices to show only the validity of the first implication. By Proposition 1.5.3, Ac(2,1)-AfZ~(7.2clY),z--->7.; c l A c % el (%elY \ 7.~ c l A ) - % el ( % e l y \
7.1
clA),
while (2) of Lemma 0.2.1 gives 7.; cl A c 7.2 cl (7.2 cl Y \ % el 7.1 el A) C 7.2 cl(Y \
7.1
cl A).
Hence 7.; cl(A n Y) c % cl(Y \ 7.; cl A) c % cl (Y \ 7.; cl(A n Y)). But A n Y c Y c T2 cl Y ~
T 1" c l ( A n Y ) - -
T 1' c l ( A N Y )
NY
and thus T;' cl(A N Y) C 7.2 cl (Y \ T; cl(A N Y)) - T2 cl (Y \ T;' cl(A N Y)). Hence, again applying Proposition 1.5.3, we find that A n Y c (2, 1)-A/T)(Y).
D
By analogy with the second part of Theorem 1.1.3, we obtain the sufficient conditions for the relative (i, j)-nowhere densities. P r o p o s i t i o n 1.5.5. Let (X, 7-1,72) be a BS and (Y, 7.~, 7.~) be a BsS of X, where Y c j-7?(X) and A c Y . If for every set V c 7.~ \ {~} there exists a set V e vj \ {~} such that V n A = 2J, then A e (i,j)-A/Z)(Y). For every set U' c T~\{2~} there exists a set U E Ti\{2~} such that U N Y = U'. Hence, by assumption, there is a set V ~ rj \ {2~} such that V c U and V N A = ~. But r j c l Y = X implies that V N Y = V' C r j \ { ~ } and so it remains for us to use the second part of Theorem 1.1.3. D Pro@
7-1,7-2) be a BS and (Y, 7.~, 7.~) be a BsS of X , where Then the following statements hold:
T h e o r e m 1.5.6. Let (X, Y c 7.i and A c X .
A ~ ( i , j ) - A f Z ) ( X ) --->, A n Y E (i,j)-Af:D(Y) so that A n Y c ( i , j ) - S Z ) ( Y ) ----> A E ( i , j ) - S T P ( X ) and thus A c ( i , j ) - d a t g i ( X ) --->, A N Y C (i,j)-Catg~(Y), A n Y C (i,j)-Catg~i(Y) ~
A c (i,j)-Catgii(X).
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C H A P T E R II
D i f f e r e n t R e l a t i o n s B e t w e e n T w o T o p o l o g i e s on a Set a n d B i t o p o l o g i c a l I n s e r t i o n s In Sections 2.1-2.3 we consider the relations on a set, reducing emphasis on points and focusing attention on families of sets, namely on topologies. At first glance one might think that our reasoning goes beyond the framework of this monograph, but in the concluding section we recapitulate the general principles so as to characterize the C- and N-relations through bitopological insertions. The S-relation was introduced by A . . R Todd as an equivalence relation and it expresses a close relationship between two topologies on a set [252]. The coupling of topologies, that is, the C-relation was defined by J. D. Weston to generalize some well-known theorems on topological groups and linear TS's and to connect the same properties of the coupled topologies [258]. The so-called nearness of topologies, that is, the N-relation is defined for the first time in [95]. Along with the above-mentioned relations and their equivalent characterizations, we also consider their combinations with the set-theoretic operation inclusion. In other words, we consider the relations < s , < c , and < x . In contrast to the S-relation, the coupling and nearness of topologies are not symmetric relations, though, topologies on the same set are partially ordered by the relations < c and
The interdependence of the above three relations and the relative properties as well as the relationship between various families of subsets of a given BS are investigated in the case of S-related, <s-related, coupled,
63
64
II. Different Relations Between Two Topologies . . .
2.1. T h e S - R e l a t i o n D e f i n i t i o n 2.1.1. A family B of subsets of a TS (X, 7) is a pseudobase of the topology ~- if the following conditions are satisfied: (1) B c B ----5, int B ~= 2~. (2) For every subset U c 7- \ {~}, there exists a set B E B such t h a t B c U.
By [252] this definition leads to the equivalence relation on the family of all topologies on a set X. D e f i n i t i o n 2.1.2. Topologies 71 and ~-2 on a set X are S-related (briefly, TIST2) if T1 \ {~} is a pseudobase for 72. A set X together with the S-related topologies T1 and ~-2 is denoted by ( X , T1 S T 2 ) .
R e m a r k 2.1.3. The S-relation between two topologies on a set is especially i m p o r t a n t by Proposition 3.4 from [252], following which if one member of an S-equivalence class is Baire, then all members of this class are also Baire. E x a m p l e 2.1.4. Let ( R , s , v ) be a BS, where s is the half-open interval topology, that is, the Sorgenfrey topology on R and so the basic open sets for s are of the form [a, b), and r is the topology with basic open sets of the form (a, b]. It is clear that neither topology is finer than the other, the greatest lower bound inf(s, r ) = s n r = ~ of the topologies s and r is the natural topology on R, and the least upper bound sup(s, T) of the same topologies is the discrete topology on R. Moreover, s S r , s S i n f ( s , r ) and v S i n f ( s , r ) . Hence, by Remark 2.1.3, the Sorgenfrey line, that is, R with the Sorgenfrey topology is a Baire space since s S i n f ( s , r ) and the natural topology is completely metrizable and, therefore, Baire [252]. Theorem
2.1.5. The conditions below are equivalent for a B S (X, T1, T2):
(1) T1 is S-related to T2.
(2)
1- d(X) = 2 - B d ( X ) = (1,
(2,
so that
= 2-z,(x)=
(8)
(1, 2 ) - z , ( x ) =
T l i n t A C TlClT2 int A A T2 int A C T 2 c l T I I n t A
SO that T1
int ~-2 cl A c T1 cl A A w2 int
T1
cl A c 7-2 cl A
for every subset A c X . (4)
71 C (2, 1 ) - S O ( X ) A 7-2 C (1, 2 ) - $ 0 ( X ) ,
so that COWl C (2, 1)-,SC(X)A cow2 C (1, 2)-,S'C(X).
2.1. The S-Relation
65
Proof. It is clear that =
-' > (7-1int A ~ 2~ ~
,,.
;.
7-2 int A ~= 2~ for every subset A c X ) .
Thus, by Proposition 1.1.11, for the equivalence (1) ,z--> (2) it is enough to show that (1) ,z--->, (7-1 int A r ~ ~ 7-2 int A ~ ~ for every subset A c X). We assume that 7-1S7-2, that is, 7-1 \ {2~} is a pseudobase for 7-2, and A c X is any subset. If 7-1 int A ~= z , then by (1) of Definition 2.1.1, 7-2 int 7-1 int A # ~, and hence 7-2 int A # ~. When 7-2 int A # z , by (2) of Definition 2.1.1, there exists a set V E 7-1 \ {Z } such that V c 7-2 int A. Thus 7-1 int 7-2 int A # 2~ so that 7-1 int A # ~. On the other hand, let U e 7-1 \ {2~}. Then 7-2 int U :/= 2~. If V e 7-2 \ {2~}, we have ~ r V = 7-1 int U c U and, consequently, 7-1 \ {2~} is a pseudobase for 7-2. Therefore (1) .z--> (2). The implication (1) ~ (3) is exactly Proposition a.a from [252], (2) is obvious. (3) ,z----5, (4) is an immediate consequence of Theorem 1.3.12. D It is likewise easy to see that if 7-1 = a~ is the natural topology on R and 7-2 is the discrete topology on R, then 7-1 is not S-related to 7-2. In the sequel it will be assumed that (7-1S7-2 A 7-1 C 7-2) ~ 7-1 < s 7-2 and the corresponding BS will be denoted by (X, 7-1 < s 7-2). Furthermore, Example 2.1.4 shows that for a BS (X, 7-1 < s 7-2) the equality 7-1 = 7-2 does not hold in general. C o r o l l a r y 2.1.6. The following conditions are satisfied for a BS (X, 7-1S7-2): 9
(1) X[ - X~ and thus if either of the topologies
7-1 and 7-2 is discrete or antidiscrete, then 7-1 = 7-2. (2) (X, 7-1,7-2) is d-quasi regular if and only if (X, rl, 7-2) is p-quasi regular. (3) If (Y, 7-~,7-~) is a BsS of X , where Z 9 (7-1 N 7-2)[-J 1 - ~ ) ( X ) 2-~:)(X), then 7-~S7-~.
Proof. The proof of (1) is straightforward. (2) We begin by assuming that (X, 7-1,7-2) is a 1-quasi regular and 2-quasi regular BS and U c 7-~\ {~}. By t h e / - q u a s i regularity of (X, 7-1,7-2), there is a set V E 7-~\ {~} such that 7-~cl V C U. But 7-1S7-2 implies that 7-j int V = W r ~ and by the j-quasi regularity of (X, 7-1,7-2), there exists a set E c rj \ {~} such that rj c l E C W. Let O = 7-~int E. Then O r ~ and 7-j cl O c U so that (X, rl, 7-2) is (i, j)-quasi regular. Conversely, if (X, 7-1,7-2) is (1,2)-quasi regular and (2, 1)-quasi regular, then for a set U c r~ \ {~}, there is a set V c 7-~ \ {~} such that 7-j el V c U. For the set r j i n t V = W r ~, there is a set E c rj \ {~} such that r ~ c l E c W. Let O = 7-i int E. Then O r ~ and ri cl O c U so that (X, 7-1,7-2) is/-quasi regular. (3) First we assume that Y E 7-1 N 7-2 and U E 7-~ in (Y, 7-~,7-~). Then U E 7i and by (4) of Theorem 2.1.5, U c (j, i ) - $ O ( X ) . Hence, according to (1) of Proposition 1.5.26, U c (j, i ) - S O ( Y ) so that 7-~ c (j, i ) - S O ( Y ) and, again applying (4) of Theorem 2.1.5, we obtain 7-~$7-~. On the other hand, let Y c 1-Z)(X) = 2-79(X) and A c Y, r~ int A/= ~. Then, there is a set U c 7-i such that U ~ Y = 7-~int A. But 7-1S7"2 implies 7-j int U = V r 2~
66
II. Different Relations Between Two Topologies . . .
and, therefore, V Cq Y = V' # ;~ as Y 9 1-D(X) = 2-D(X). Thus Tj int A # since V' C r" int A. D C o r o l l a r y 2.1.7. The conditions below are equivalent for a BS (X, rl, r2): (1) rl < s v2. (2) 1-13d(X) c 2-13d(X) so that 1-D(X) C 2-D(X). (3) r2 int r2 cl A = re int T1 cl A so that r2 cl r2 int A = r9 cl r l i n t A
for every subset A c X . (4) r2 c (1,2)-SO(X) so that cot2 c (1,2)-SC(X). (5) n(A) 9 2-Bd(X) for each set A 9 2x.
Furthermore, the next statement holds: (6) If (Y, 7-;, r~) is a BsS of X and Y 9 r2, then T 1 S T 2 -----5, T 1
Proof. The equivalences (1) <---> (2) ,e--->, (3) <--> (4) are immediate consequences of the corresponding equivalences of Theorem 2.1.5. (1) ------5, (5). If 7"1S7"2, then by (b) of 4.A.2 in [173], nl(A) 9 2-A/'D(X) and n2(A) 9 2-A/'D(X) for each set A c 2 x. Therefore n l ( A ) c 2-Bd(X) and n2(A) E 2-Bd(X) for each set A E 2x, and thus n(A) c 2-Bd(X) for each set A E 2x. (5) ---> (4). Clearly, if n(A) 9 2-Bd(X) for each set A 9 2x, then n2(A) 9 2-Bd(X) for each set A 9 2 x. Hence ~-2 int(r2 int A \ r l i n t A) = ;~ for each set A c 2 x so that r2 int A c r2 cl r l i n t A for each set A c 2X, that is ~-2 C (1, 2)-$(.9(X). (6) Let A c Y and r ~ i n t A ~= ~. Then r 2 i n t A # ~ and, by (2) above, r l i n t A ~: 2~. But r l i n t A c r~ int A and so r; Sr~. [5] Further, following (1) of Corollary 1.3.15 and (4) of Theorem 2.1.5, for a BS have
( X , T1, T2), w e
rl < s r2 ~
(rl c r2 c (1,2)-,,gO(X)c ( 2 , 1 ) - $ 0 ( X ) ) .
Note that in contrast to (3) of Corollary 2.1.6 and (6) of Corollary 2.1.7 the S-relation is not, generally speaking, hereditary with respect to/-closed subsets. E x a m p l e 2.1.8. Let X = {a,b,c,d}, 71 = {Z, {a, b}, {a, b, c}, X } and 72 = { z , { a , b } , { a , b , c } , { a , b , d } , X } . Clearly, rl < s r2, but if F = {c,d} e cot1 c co 72, then r~ is not S-related to v~ for the BS (F, v~, v~). This fact leads to the following notion to be used in Chapter IV.
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68
II. D i f f e r e n t R e l a t i o n s B e t w e e n T w o T o p o l o g i e s . . .
Proof. To establish (1)-(4) observe t h a t it suffices to prove only the corresponding first conditions. (1) If A c i-N'D(X) and A-~(i,j)-AfD(X), then by (3) of T h e o r e m 2.1.5, # 7.~ int 7.j cl A c 7.~ el A so t h a t 7.~ int 7"~el A # ~, which is impossible. Conversely, if A E (i, j)-N'D(X) and A-~ i-N'D(X), then (3) of T h e o r e m 2.1.5, used for the set 7"i el A, yields the inclusion 7"i int 7"i cl A c 7"i cl 7"j int 7"i cl A and, therefore, 7"i int 7"i cl 7"j int 7"i cl A # 2~. Hence (2) of T h e o r e m 2.1.5 implies t h a t 7"j int 7"i cl 7"j int 7"i cl A # o and by L e m m a 0.2.1, 7.j int 7.~ cl 7.j int 7.~ cl A = 7.j int 7.i cl A # ~. But 7"j int 7"i cl A = 7"j int 7"j int 7"i cl A # 2~ and by (2) of T h e o r e m 2.1.5, we conclude t h a t 7"~int 7"j int 7"~cl A # ~. Thus 7"~int 7"j cl A # ~, t h a t is, A --~(i, j)-N'D(X) since by (3) of T h e o r e m 2.1.5, 7"j int 7"i clA c 7"j clA. The obtained contradiction shows, firstly, t h a t 7"j int 7"i cl A = 2~, t h a t is, A c (j, i)-N'D(X) and, secondly, t h a t 7"~int 7"~cl A = ~ so t h a t A E i-N'D(X). (2) If A c 2-SO(X), then from T h e o r e m 1.3.12 for i = j = 2 and (3) of T h e o r e m 2.1.5, we infer t h a t A c 7.2 clT.2intA c 7"2 c17.1 i n t A and so A c (1, 2 ) - S O ( X ) . If A E 1-80(X), then from T h e o r e m 1.3.12 for i = j = 1 and (3) of Theorem 2.1.5, we infer t h a t A c T1 cl T1 int A C T1 el 7.2 int A and so A E (2, 1 ) - S O ( X ) . (3) T h e first equality and the inclusion follow immediately from (2) above and (1) of Corollary 1.3.15. If A c (2, 1)-SO(X), then from T h e o r e m 1.3.12, (3) of T h e o r e m 2.1.5, and (2) of L e m m a 0.2.1, we have A c 7"1 cl 7"2 int A c 7"1 cl 7"2 cl 7"1 int A = 7"1 cl 7"1 int A and so A c 1-SO(X). Thus, by (2) above (or, by ( 1 ) o f Corollary 1.3.15) (2, 1 ) - 8 0 ( X ) = (4) Let A c X be any subset. Then by (3) of Corollary 2.1.7, A = 7"2 int 7"2 cl A ~
1-80(X).
A = 7"2 int 7"1 cl A.
Hence, by Definition 1.3.3, 2 - O D ( X ) = (2, 1)-OTP(X). Furthermore, 7"1 el 7"2 el 7"2 int A = 7"1 cl 7"2 cl 7"1 int A and by (2) of L e m m a 0.2.1 T1 cl 7"2 int A = 7"1 c17"1 int A so t h a t A = 7.1 cl T2 int A ,z-->, A = 7.1 cl 7"1 int A and so (1,2)-CD(X) = 1 - C D ( X ) or, equivalently, (1,2)-OD(X) = 1-OD(X).
a
E x a m p l e 2 . 1 . 1 1 . Let (X, T1,7.2) be the BS from E x a m p l e 1.3.8. T h e n T1 <S T2, {b,c} C 1 - S O ( X ) = (2, 1 ) - S O ( X ) , but { b , c } c 2 - $ O ( X ) = (1, 2 ) - $ O ( X ) , {a} ~ 1-OD(X) : ( 1 , 2 ) - O D ( X ) , and { a } ~ 2 - O D ( X ) = (2, 1)-OD(X). Also, {a,b} E (2, 1)-OD(X)= 2-OD(X), but {a,b} -4 (1,2)-OD(X) = 1-OD(X). C o r o l l a r y 2 . 1 . 1 2 . Let (Y, 7.~,7.~) be a BsS of a BS (X, T1 <S T2) and A c Y. If Y E 7-2 U 2 - D ( X ) , then
9
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70
II. D i f f e r e n t R e l a t i o n s B e t w e e n T w o T o p o l o g i e s . . .
Proof. Note that, following (3) of Corollary 2.1.6 and (6) of Corollary 2.1.7, in all cases (1)-(8), the <s-relation is hereditary with respect to the corresponding BsS (Y, ~-~ < w~). Hence the coincidence of the families of subsets of a BsS (Y, ~-~ < s r~) in (1)-(6) are immediate consequences of (1)-(4) in Theorem 2.1.10. For the rest,
we have: (1) According to the well-known topological fact, Y c r2 U 2-D(X) implies that A E 2-A/'D(Y) <---> A c 2-A/'D(X). (2) By (2)of Corollary 1.3.7, T1
N
(2, 1)-OD(X) = (1, 2)-OD(X)N (2, 1)-OD(X).
Hence Corollary 1.5.22 gives A e (i,j)-OD(Y) <---> A e (i,j)-OD(X). Assertions of (3) are immediate consequences of Proposition 1.5.25. (4) The inclusions in this condition follow directly from (3) of Theorem 2.1.10 and the equivalences are evident by Corollary 1.5.27. Assertions of (5) are obvious by Corollary 1.5.ao and (3) of Theorem 2.1.10. The conditions of (6) follow directly from (2) of Corollary 1.a.r and Corollary 1.5.32. Finally, by well-known topological facts, Y c r2 implies the equivalence A c 2-Bd(Y) e---->,A c2-13d(X) and Y c 2-D(X) implies the equivalence A E 2-D(Y) e > A E 2-D(X). Thus (7) and (8) are immediate consequences of (2) of Theorem 2.1.5. It is obvious that by (2) of Theorem 2.1.5, in (1) and (8) we can consider r2U1-D(X) and ruN1-D(X) instead of r2U2-D(X) and T2N2-D(X), respectively. E] C o r o l l a r y 2.1.13. Let (X, rlSr2) be a BS and A E rl U COrl U r2 U cot2. Then
( i , j ) - F r A c 1 - H D ( X ) = 2-HZ)(X)= (1, 2)-AYD(X) = (2, 1)-HD(X).
In addition, for a BS (X, rl < s T2), we also have: (1) If A c X is any subset, then n ( d ) = n l ( A ) U n2(A) e 1 - A f D ( X ) = 2-A/'D(X) = (1, 2)-A/'D(X)= (2, 1)-A/'D(X). (2) X is of 1-datg II <---> X is of 2-datg II e---> X is of (1, 2)-Catg II e---> X is of (2, 1)- CatgII ---> 1- G~(X) N 2-D(X) = 1- Ga(X) N 1-D(X) c 2-Ga(X)N1-D(X) = 2 - G a ( X ) N 2 - D ( X ) C 2-Catgut(X)= 1-Catg~i(X ) = (1,2)-Catgix(X) = (2, 1)-CatgI~(X ).
Proof. First assume that A c X is any subset. Then by (3) of Theorem 2.1.5, we have ~-i int rj cl(i, j)- Fr d = ~-i int rj cl (ri cl A a rj cl(X \ d)) c c T~int rj cl ~-~cl A a r~ int rj cl(X \ A) c ~-~cl A N T~cl(X \ A) = i- Fr A. Similarly, riint rj cl(j, i)- Fr A c i- Fr A. If A E T~ U co r~, then by Theorem 3 from [161, p. 74], i-FrA E i-A/'D(X). Hence by (1) of Theorem 2.1.10 and the inclusions above, riint rj cl(k, 1)- Fr A e 1-HD(X) = 2-A/'D(X) = (1, 2)-HD(X) = (2, 1)-A/'D(X)
2.1. The S-Relation
71
and so 7-i int 7-j cl 7-i int 7-j cl(k, l)- Fr A = 2~, where k, l E {1, 2}, k ~ 1. Thus by (1) of Lemma 0.2.1,
(i,j)-Fr A c 1-AF/?(X) = 2-AF/9(X) = (1, 2)-A/'/9(X) = (2, 1)-AF/9(X) for each set A c 7-1 d co 7-1 U 7-2 U co 7-2. (1) It suffices to show that n l ( A ) E 1-Af~D(X) since n2(A) = n l ( X \ A). Obviously, T 1 int 7-1 cl
(7-1 cl A \
7-2 cl
A) c
and, by (3) of Theorem 2.1.5 for
7-1 int 7-1 cl
T 1 C 7"2,
7-1 int
A c~ (X \
T1
cl 7-1 int 7-2 cl A)
we have
rl cl A C 7-2 cl A.
Hence T1
int
7-1 cl
A c 7-1 cl 7"1 int 7-2 cl A
so that T1 int T1 cl A c~ (X \ q cl 7-1 int r2 cl A) = 2~ and thus n 1 ( A ) --- 7-1 cl
A \ 7-2 cl A ~ 1-A/'Z)(X).
The proof of (2) follows directly from Proposition 1.1.26, (2) of Theorem 2.1.5, and (1) of Theorem 2.1.10. [-] R e m a r k 2.1.14. According to [252] let us treat a TS (Y,'7) as the image of a TS (X, r2) under a continuous function f with the property: for every set U c 7-2 \ {2~}, there exists a set V c 7 \ {2~} such that f - l ( v ) C U. Clearly, the family 7-1 = f - l ( ~ ) = { f - l ( p ) : V E ~ } is a t o p o l o g y o n X c o a r s e r t h a n 7-2. I f w e denote this relation between the topologies 7-1 and 7-2 on X by 7-1f72, then 7-1fr2 is a stronger connection between the topologies than the <s-relation. Hence all results obtained for the <s-relation also hold for the f-relation. Furthermore, following [201], a condensation is a one-to-one and continuous function f : (X, 7-2) ~ (Y, "y) such that I ( X ) = Y. It is clear that in this case f : (X, 7-1) --+ (Y, 3') is a homeomorphism and, conversely, for any BS (X, 7-1 < ~-2) the identity function f : (X, 7-2) ---' (X, 7-1) is a condensation. Theorem
2.1.15. If f : (X, 7-2) ---' (Y, 7) is a condensation and 7-1 = f - l ( . y ) ,
then 7-1 <s 7-2 ,z----5, TlfT-2 ~
f is feebly open
(see Definition 5.1.37). Proof. By Remark 2.1.14 for the first equivalence, it suffices to prove that T1 < s T2 implies TlfT2. Let U c T2 \ {2~} be any set. Then T1 int U c 7-1 \ {2~} and, hence, there is a set V c 7 \ {2~} such that f - l ( V ) = 7-1 int U C U, that is, 7-1f7-2. Now, let 7-1f7-2 and U ~ 7-2 \ {2~} be any set. Then there is a set V E 3' \ {2~} such that f - l ( V ) c U. Therefore f(f-l(v))
= V c f(U),
that is Tint f ( U ) r Z
72
II. Different Relations Between Two Topologies ...
and thus f is feebly open. Conversely, let f be feebly open and U c 7"2 \ {~} be any set. Then the set Tint f(U) - V e T \ {2~} satisfies the inclusion f - l ( v ) C U and so 7"1f7"2. E] C o r o l l a r y 2.1.16. If f " (X, T2) ~ (Y, T) is a feebly open condensation, (Y, T) is quasi regular, 7.1 - f - l ( T ) and (X, 7.1,7.2) is 1-quasi regular, then (X, 7.1,7"2) is d-quasi regular and p-quasi regular.
Proof. It is obvious that 7"1f7"2 and by Proposition 3.5 from [252], (X, T1,7-2) is 2-quasi regular. Therefore (X, 7.1,7"2) is d-quasi regular and by (2) of Corollary 2.1.6, (X, T1,7"2) is p-quasi regular since T1 < S 7"2. [-] By the above reasoning, if for a BS (X, 7"1 < 7"2) the identity function (i.e., the inclusion function) j ' ( X , 7.2) ~ (X, 7 - 1 ) i s feebly open and (X, 7.1,7.2)is 1-quasi regular, then (X, 71, ~-2) is d-quasi regular and p-quasi regular. 2.2. T h e C - R e l a t i o n Let (X, 7.1,7.2) be a BS. Then it is obvious that z2 c 7"1 ~ 7"1 cl A c 7"2cl A for every subset A c X. In [258] a topology 7"1 is chosen among different subfamilies of 2 x whose elements satisfy inclusions of the type given on the right-hand side of the above equivalence. This section continues the study begun in [258], while the Section 2.3 will deal with the same inclusion for a topology 7"2. D e f i n i t i o n 2.2.1. A topology T1 is coupled to a topology 7.2 on a set X (briefly, 7.1C7.2) if 7.1 cl U C T2 cl U for every set U c 7.1 [258]. From this definition we immediately find that if 7.1 - co 7"1, then 7"1 is coupled to every topology on X so that the antidiscrete topology on X as well as the discrete topology on X is coupled to every topology on X. R e m a r k 2.2.2. By [258], if T1 is coupled to z2 on X, then T1 is coupled to every topology on X smaller than 7.2. But it is possible for a topology to be coupled to a strictly larger topology and in that case the coupling is mutual. For example, the antidiscrete topology is mutually coupled to every topology on the same set. The possibility for a topology to be coupled with a strictly larger topology leads the author of [258] to the notion of partial order. In [258] more interest is shown in the coupling of topologies than in the situation T1 el U C 7"2 el U for every set U E 7.2 (in our terms - the N-relation). One might think that this preference is conditioned by the following reasoning: the C-relation defines the partial order _< (in our notation < c ) on the family of all topologies on X by virtue of the equivalence T1%C 7"2 ~ (T1CT2 A T1 C T2), and in the subsequent investigations the author considers the cases where 7"1%C 7"2 and (X, 7.2) satisfies the conditions for which it is regular. If instead of the partial order < c we shall consider the relation
2.2. T h e C - R e l a t i o n
73
in [258], the conditions rl < x r2 and (X, r2) is regular (where the regularity of (X, r2) is not superfluous) imply that rl = r2. As distinct from the above situation, we have E x a m p l e 2.2.3. Let X = {a,b,c}, T 1 = {fg, {a}, {b, c } , X } , and r2 be the discrete topology on X. Then rlCr2 since 7"1 = COT1, T1 C T2, that is, rl < c r2 and (X, r2) is regular. However T 1 r T 2. The example below is given in the context of the above reasoning. E x a m p l e 2.2.4. Let X = {a, b, c, d, e}, T 1 = {f~, {a}, {b, c, d, e}, X}, and r2 = {;g, {a}, {a,c}, {a,b,c}, {a,c,d}, { a , b , c , d } , X } . Then rlCr2 and (X,~q,r2) is (2, 1)-quasi regular. However, rl and r2 are not comparable. The coincidence of the topologies for rl < c r2 demands a stronger requirement on (X, rl, r2), which is given below in Corollary 2.2.9. Note that following Definition 2.2.1, T 1 < C 7-2 ~ ( T I C T 2 /N T 2 C T 1 ) , but the inverse implication is not correct. E x a m p l e 2.2.5. Let X : {a, b, c, d}, T1 - - {~, {a}, {b, c, d}, X}, and r2 = {~, {b}, {a, c, d } , X } . Then T I C T 2 and T2CT1, but T1 and 7-2 a r e not comparable by inclusion. T h e o r e m 2.2.6. The following conditions are equivalent for a BS (X, T1,7-2): (1) 7"1 is coupled to r2. (2) 71 cl 71 int A c r2 cl T 1 int A so that r2 int
T1
cl A c
T1
int
T1
cl A
for every subset A C X . (3) vl cl r l i n t A c 72 cl A so that r2 int A c 7-1 int rl cl A
for every subset A c X . (4)
T1
sup(7-1~7"2)"
(5) For every point x E X the 1-closure of any 2-neighborhood of x is a 1-neighborhood of x.
P oof.
quiv l nr
(1) r
(4)
(5)
p ov d in Theorems 2
5 of [258]. (1) ~ (2) is obvious by Definition 2.2.1. If (3) is satisfied, then r l c l d c T2clm for every 1-open set d so that (3) ~ (1), where (1) ~ (2). Since the implication (2) ~ (3) is evident, we obtain (2) ~ (3). 77 A set X together with topologies rlCr2 (rl < c r2) is denoted by (X, rlCr2) ( ( X , T1 < C
T2)).
C o r o l l a r y 2.2.7. The conditions below are equivalent for a BS (X, rl,r2): (1)
T1
7"2"
(2)
T 1 C1 T 1
int A
= T2
cl T 1 int A so that rl int T1 cl A = r2 int
for every subset A c X .
T1 c l A
74
II. Different Relations Between Two Topologies . . . (3)
nl(U)
~ for
-
each set U ~ 7.1 8o that
n 2 ( F ) - 2~ for each set F c co7.1. (4) l-C17.1
--
2-C1 T1 80 that l-Int co 7.1 -- 2-Int co 7.1.
Proof. The equivalences are immediate consequences of Definitions 1.4.19 and 2.2.1, and (2) of Theorem 2.2.6, taking into account the inclusion T 1 C T 2. [-7 C o r o l l a r y 2.2.8. The conditions below are satisfied for a BS (X, 71C7.2)" (1) If (2,7-1,7-2) i8 (1,2)-regular ((1,2)-quasi regular), then (X, 7.1,T2) i8 1-regular (1-quasi regular), and if (X, 7-1, "/-2) i8 1-connected, then (X, T1, T2) is p-connected. (2) If (Y, 7.~, 7.~) is a BsS of X and Y c T1 U T2, then 7.~C7~. Moreover, for a BS (X,
T1 < C
7"2), we have"
(3) (X, 7.1,7.2) is (1, 2)-regular ((1, 2)-quasi regular) .z---->. (X, 7.1,7.2) is 1-regular (1-quasi regular) and (X, 7.1,7.2) is 1-connected .z--->. (X, 7.1, 7.2) is p-connected. (4) If (X, 7.1,7.2) is p-normal, then (X, 7.1,7.2) is 1-normal. (5) If (X, 7.1,7.2)is a l-T1 BS, then X ~ - X~ and, in this case, if 7.2 is the discrete topology, then 7"1 --7.2. (6) 7.1 n 1-D(X) - 7.1 n 2-D(X) so that 9
CO 7.1 n 1 - ~ d ( X )
(7)
1 - I n t e l 7.1 -
-
c o 7.1 (~ 2 - ~ d ( X ) .
2-Int C17.1 - (1, 2)-Int C171
--
(2,
1)-Int C1T1,
so that 1-C11nt co 7-1 - - 2-C11nt co 7.1 - -
(1, 2)
-C11nt co 7.1 - (2, 1) -C11nt co 7.1-
(X, 7.1,7.2) is p-extremally disconnected ~ (X, 7.1,7.2) is 1-extremally disconnected. (9) A E (2, 1)-N'D(X) if and only if for every set U E 7.2 such that 7.1 int U r ~, there exists a set V c 7.1 \ { ;2~} together with V c U and V N A - ;g. (10) If (Y, 7.~, 7.~)is a BsS of X and Y E 2-79(X) then T~
Proof. (1) The first part is an immediate consequence of (2) of Proposition 0.1.7 and Definition 0.1.14, taking into account Definition 2.2.1. Now, suppose that (X, 7.1,7.2) is not p-connected. Then there is a set U E 7.1 f? co 7.2 such that 2~ :/: U r X. Then U c 7.1 and 7.1C7.2 imply that 7.1 C1U C 7.2 cl U - U. Hence U E T1 ('l CO 7.1 SO that (X, 7.1,7.2) is not l-connected. (2) Begin by assuming that Y c 7.1 and U c 7.~ is any set. Then U E 7.1 and by Definition 2.2.1, 7.1 cl U c 7.2 cl U. Hence 7.1 cl U n Y C 7.2 cl U 5 Y so that 7.1' c l U C 7.2' cl U and so 7.1'C7.~ 9 If Y c T2 and A C Y is any set, then by (3) of Theorem 2.2.6, T2 int A C
T1
int
7-1 cl
A
2.2. The C-Relation
75
and so z~ int A = 7.2 int A c T1 int 7-1 cl A • Y c T~ i n t (7.1 cl A A Y) = 7.~ int 7.~ cl A. Thus 7.1'Cry. (3) The first part follows directly from the first part of (1), taking into account (3) of Corollary 2.2.7. The second part is an immediate consequence of the second part of (1) and the implications after Definition 0.1.18. (4) Let F c co7.1, U c T1 and F C U. Then F c co7.2 and by (4) of Proposition 0.1.7, there is a set V c 7.1 such that F C V c 7.2 cl V C U. Hence, by (3) of Corollary 2.2.7, we have F c V c T1 cl V C U and so (X, 7.1,7.2) is 1-normal. (5) The first part follows immediately from (3) of Corollary 2.2.7. The rest is obvious. Assertions of (6) are also consequences of (3) of Corollary 2.2.7. (7) If U c TI~ then by (3) of Corollary 2.2.7, we have T 1 C1U -- 7.2 c 1 U . Hence T1
int 7.1 cl U = 7.1 int 7.2 cl U,
7.1 int 7.1 cl U = 72 int 7.1 C1U -- 7.2 int 7.2 cl U
and thus 7"1 i n t 7.1 cl U -- 7.2
int 7.2 cl U = 7.1 int r2 cl U = r2 int rl cl U.
(8) By ( 3 ) o f Corollary 2.2.7 and (7) above, we have (7.1 cl 7.2
int 7"1 C1A ~- 7-2 int 7.1 C1 A ) ~
(7.1 c1 7.1
int 7.1 cl A - 7.1 int 7"1 C1 A )
for every subset A c X. Hence it remains to use (3) of Definition 0.1.18 and the equivalences after Definition 0.1.18. (9) The necessity follows directly from Corollary 1.1.6. On the other hand, if 7.2 int 7.1 cl A r 2~, then by (2) of Lemma 0.2.1 and (2) of Corollary 2.2.7, we have 7.1 int 7.2 int 7.1 cl A = T1 int 7.1 cl A = 7.2 int 7.1 cl A r 2~. If V c 7.1 \ { ~ } and V c 72 int 7-1 el A , then V c 7.1 cl A so t h a t V ~ 7.1 cl A = V r and thus V C~A r ~. (10) Finally, let Y c 2-~D(X) and g ' c 7.{ be any set. Then, there is a set U c 7.1 such t h a t U ~ = U A Y . By (3) of Corollary 2.2.7, 7.1clU = 7.2clU and Y c 2-Z)(X) c 1-Z)(X) implies t h a t T1
cl U = T1 cl(U c~ Y) = 7.2 c1 g = 7.2 c l ( g ~ Y ) .
Hence 7.~ cl U ~ - 7.~ cl U t and it remains to apply once more (3) of Corollary 2.2.7. F-1 C o r o l l a r y 2.2.9. If (X, TICT2) i8 a (2,1)-regular BS, then 7.2 c 7.1 and, therefore, for a (2, 1)-regular BS (X, 7.1 < c 7.2), we have 7.1 = T2.
Proof. Let U E 7-2 be any set (2,1)-regularity of X, there exists But, according to (5) of Theorem there exists a set W E T1 together x r U being arbitrary implies t h a t
and x c U be an arbitrary point. By the a set V r 7.2 such that x c V c 7"1 cl V C U. 2.2.6, 7"1 C1V is a l-neighborhood of x so that, with x ~ W c T1 C1 V . Hence x E W c U and U c 7"1. [--]
Example 2.2.4 shows that the requirement that (X, 7.1,7-2) be (2,1)-regular in the last corollary is not superfluous.
76
II. Different Relations Between Two Topologies ...
C o r o l l a r y 2.2.10. If (X, 7"1C7"2) is a (2, 1)-quasi regular BS, then for every set U c 7"2 \ {~}, there is a set V c 7"1 \ {~} such that 7"1clV c U.
Proof. It is clear that for every set U c 7"2\{2~} there is a set W c 7"2\{2~} such that 7"1cl W c U. Let x E W be any point. Then by (5) of Theorem 2.2.6, 7"1 cl W is a 1-neighborhood of x and, therefore, there is a set V c 7"1 \ { ~ } s u c h that x c V c 7"1 cl W. Thus 7"1 cl V c U. [3 D e f i n i t i o n 2.2.11. A topology 7"1 is No-coupled to a topology 7"2 on a set X (briefly, 7"1C(R0)7"2) if 7"1 is coupled to 7"2 and 17"1int 7"1cl A \ 7"2int A I < No for every subset A c X. Clearly, by (2) of Corollary 2.2.8, the C(N0)-relation between topologies rl and r2 on a set X is hereditary with respect to any set A c rl tOr2 and, moreover, by (10) of the same corollary, the
It is obvious that if IX[ _< N0, then rlCr2 ,<---->tiC(No)r2. However, we have E x a m p l e 2.2.12. Let X be an uncountable set, ~ r A c X, [A I < N0, A, X \ A, X}, r2 = {~, X}. Then T1CT2, but T1 is not R0-coupled to r2 since for the set X \ A, we have
T 1 -- { ~ ,
Irl int rl cl(X \ A) \ r2 int(X \ A)[ - IX \ A I > N0. The C-relation is not hereditary with respect to/-closed subsets in general. E x a m p l e 2.2.13. Let (X, 7-1,7-2) be the BS from Example 2.1.8. Then T2. Let us consider the set F = {c, d} c co T1 C CO 7-2. Then for the subset {d} of a BsS (Y, r~, r~), we have r~ int r~ cl{d}=2~ and r~ int{d}={d}, that is, r~ is not coupled to r~.
T1 ~ C
D e f i n i t i o n 2.2.14. A topology 7-1 is/-strongly coupled to a topology 7-2 on a set X (briefly, rlC(i)7-2) if the C-relation is hereditary with respect to/-closed subsets of X. E x a m p l e 2.2.15. Let X = {a, b, c, d}, rl = {2~, {a, b, c}, X } and 7-2 = {2~, {a, b}, {a, b, c}, X}. Then T1
is a
1-T1 BS, then
co r2 • 1-DZ(X) = 2-P(X).
Proof. Let F c co r2 Cq 1-DZ(X). Following the condition, r~ < c r~ for (F, r;, r~) and by (5) of Corollary 2.2.8, F~ = F~ = 2~ since F c 1-DZ(X). Hence F c F~, that is, F E 2-DZ(X) and so F c co
n 2-DZ(X) = 2-P(X).
2.2. The C-Relation
77
Thus co T2 N 1-/)Z(X) C 2-7)(X). [-1
The reverse inclusion follows from (2) of Proposition 1.4.2. Note that
(T1C(i)T2 A T1C(}{O)T2) ~
TIC(i, Ro)T2 ~
T;C(Ro)T;
and (T1 "~C(i)7"2 A T1C(~o)T2) ~
T1
T~
for every/-closed BsS (F, T[, T~) of a BS (X, 71,72). Using the p-quasi regularity argument, we obtain the following important statement, interconnecting the C- and S-relations. T h e o r e m 2.2.17. Let (X, T1, T2) be a p-quasi regular BS. Then (7-1CT2 A T2CT1) ~
T1ST2.
Proof. By virtue of (2) of Theorem 2.1.5, it suffices to show that the equivalence T1 int A -r ~ ~ 72 int A -r 2~ holds for any subset A c X. Begin by assuming that T1 int A =/= 2~. Then by the (1,2)-quasi regularity of (X, 71, r2), there exists a set V c T1 \ { ~ } such that 72 cl V c 7-1 int A. If x c V is any point, then by (5) of Theorem 2.2.6, there is a 2-open neighborhood U(x) such that U(x) c r2 el V as T2C~-I. Hence U(x) c T1 int A so that 72 int T1 int A -r 2~ and thus T2 int A =/= ~. Using the (2,1)-quasi regularity of (X, 71, T2) and TICT2 one can prove similarly that T2 int A -r 2~ implies T1 int A =/= 2~. [El C o r o l l a r y 2.2.18. Let (X, 7-1,72) be a p-quasi regular BS, where T1CT-2 f T2CT1. Then (X, T1,72) is a 1-Baire space if and only if (X, 7-1, T2) is a 2-Baire space.
Proof. Under the hypotheses, we have T1ST2 and it remains to use Remark 2.1.3. [-1 C o r o l l a r y 2.2.19. If (X, T1
Pro@ Indeed, it is clear that 7"1
9-1CT2):
(1) I-N'Z?(X) c (2, 1)-Af~(X) so that (2, 1)-SZ)(X) c 1-S~D(X) and thus
1-Ca~gi(X ) ~ (2, 1)-Cai~gi(X), (2, 1)-Catgii(X ) c 1-CaI~gii(X ).
~
-
o
~
II
_
~
~,.
Y
~
.
~
~
il
Rg
~
~-
~
~
~
~
~
~
~
~
~
o
~
~
~
0
~
~
,_,~
9
,
0
~
~
oo
~
~"
.~
~-'~
i-,o
~
~
8 ~
~." ~"
o
~ ~ ~. . ~
~
~
c~
~
~
q
n
~
,~
" T ' ~
n
--
~
~
_~
~
"T"
~
~
~~
~
~
~
C~
II
-.
6~ ~~ ~~
II
~
~
II
,-'
~
~
,~ ~
~
~
.-.
ii
~ ~
~
~
>~
II
~
~
~
~ ~~ ~
~
'
,~
"r"
~
~
~
~
~3
~
~
r~
:b_,~
II
~-.
~
~
'~
-"
~
"~ - - ~
,
9
~.
c"'b
2.2. T h e C - R e l a t i o n
79
and so
~-eD(x)
= (~, 2 ) - e D ( x )
c 2-eD(x).
Furthermore, (2) of Corollary 1.3.7 also gives
~-~ n 2 - e ~ ( x )
=
(1,
2)-eD(x) n 2-oD(x)
= (1,
2)-eD(x)
and, therefore, T1 ["] 2 - O ~ ) ( X )
----- 1-OD(X)
= (2, 1 ) - O D ( X ) - (1,2)-OD(X).
The second condition in (4) is equivalent to the first one. Finally, according to Definition 1.3.26, (1) of T h e o r e m 1.3.27 and the first condition in (4),
A c 1-SOZ)(x)=
1-scz)(x)<
;
<---> (there exists a set U E 1-OD(X) such t h a t U c A c r~ cl U) < ;e, > (there exists a set U c (2, 1)-OD(X) such t h a t U c A c rl cl U) e, > > A ~ (2, 1)-SOD(X) = (1,2)-,5CD(X).
e
Moreover, Definition 1.3.26, (1) of T h e o r e m 1.3.27, the first condition of (4) above and (3) of Corollary 2.2.7 imply t h a t
A c (1,2)-soz)(x)=
( 2 , 1 ) - s c z ) ( x ) ,,, ;
,,' > (there exists a set U c (1, 2)-OD(X) such t h a t U < A < r2 cl U) -,' ;< ;, (there exists a set U c 1-OD(X) such t h a t U c A c r2 cl U = rl cl U) e, ,,
e ;, A < 1-,SOD(X)= 1-SCD(X). Thus, we have
~-soz)(x)
= ~ - s c z ) ( x ) : (~, 2 ) - s o y ( x ) = - (2,1)-soz)(x)
(2,1)-acz)(x) :
= (~, 9 ) - s c z ) ( x ) .
It is also clear t h a t A ~ (1,2)-Soz)(x)=
(2,1)-scz)(x)
~, ,,,
e, ;, (there exists a set U c (1, 2)-O79(X) such t h a t U c A c r2 cl U) < > e, > (there exists a set U E T1 ("12-079(X) such t h a t U C A c r2 cl U) < ?-
e, ,, A < 2-,...qO~)(X)/T1, where the last equivalence follows from Definition 1.3.33. The rest is obvious by Proposition 1.3.34.
D
Note t h a t for the reverse inclusion 2-OD(X) c 1-OD(X), the requirement in (4) of T h e o r e m 2.2.20 for 2-open domains to be 1-open is not superfluous. Example
2 . 2 . 2 1 . Let (X, rl, r2) be the BS from E x a m p l e 2.2.3. T h e n {a, b} c
2-o7)(x), but {a, b}-g 1-OD(X) : (2, 1)-OD(X) = (1, 2)-OD(X) since { a, b} g rl. Corollary
2.2.22.
The following implication holds for a BS (X, rl < c r2):
m
II ~ m
~ -. ~
7-.
& ~ &
II
~
"7"
.,
,
,~, .. ~
7-.
,-~
.,
,~ ,";-" c ~ ~
~
~ ~
7_.
~
m
, ~ ,
m
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2.2. The C-Relation
81
(5) A c (1, 2 ) - $ O ( Y ) -
1-SO(Y) ~
A c 2-80(Y) ~
A c (1, 2 ) - S O ( X ) -
1-SO(X) ~
A c 2 - S O ( X ) ==~ A c (2, 1)-SO(X).
I f Y C T1 n CO 7"1,
A c (2, 1)-SO(Y)
then
(6) A c ( 1 , 2 ) - S C ( Y ) - 1 - S C ( Y ) ~ A E (1, 2 ) - S C ( X ) -
1-SC(X) ~
A c 2 - S C ( Y ) -~. A c (2, 1 ) - S C ( Y )
A c 2-SC(X) ~
A ~ (2, 1)-SC(X).
If Y e (1, 2 ) - O D ( X ) , then
(7) A E (1, 2 ) - O D ( Y ) - 1 - O D ( Y ) -
(2, 1 ) - O D ( Y ) - 2 - O D ( Y ) N 7-;
A E (1, 2 ) - O D ( X ) - 1 - O D ( X ) - (2, 1 ) - O D ( X ) - 2 - O D ( X ) n T1, and thus A c (1,2)-SOD(Y)
- ( 2 , 1)-,SCD(Y)- 1 - S O D ( Y ) - 1-,SCD(Y) - ( 2 , 1 ) - S O D ( Y ) - (1,2)-SCD(Y) - 2-SOD(Y)/'r; = -
A c (1,2)-,SOD(X)
If Y c
2-scz
(Y)/co
- ( 2 , 1 ) - S C D ( X ) - 1 - S O D ( X ) - 1-$CD(X) - ( 2 , 1)-SOD(X) - ( 1 , 2)-$CD(X) - 2 - S O D ( X ) / T 1 = 2-SCTP(X) / c o "q.
--
n (1, 2 ) - c D ( x ) ,
(8) d c (1, 2)-CD(Y) - 1-CD(Y) - (2, 1)-riD(Y) - 2-CD(Y)N co'r~ A c ( 1 , 2 ) - C D ( X ) = 1 - C D ( X ) = (2, 1 ) - C D ( X ) = 2 - C D ( X ) N co T1. Proof. Note that by (2) and (10) of Corollary 2.2.8, in all cases (1)-(8) the C-relation, as well as the
82
II. Different Relations Between Two Topologies ...
(6) It is obvious that (5) of Theorem 2.2.20 also gives the horizontal equalities and implications in this condition. The vertical equivalences follow directly from Corollary 1.5.30 and its topological version. (7) The horizontal equalities of the upper condition are given by (4) of Theorem 2.2.20. Corollary 1.5.22 implies the vertical equivalence. The horizontal equalities of the lower condition are also given by (4) of Theorem 2.2.20 and the vertical equivalence follows from Corollary 1.5.32. (8) By (4) of Theorem 2.2.20 the horizontal equalities are obvious. According to Proposition 1.5.23 if A E (1,2)-CD(X), then A E (1, 2)- CD(Y). On the other hand, Y E (1,2)-CD(X)= (2, 1)- CD(X) and Proposition 1.5.25 imply that (2, 1)-CD(Y) = (1, 2)-CD(Y) C (2, 1)-CD(X) = (1, 2)-CD(X). Thus the vertical equivalence holds.
D
P r o p o s i t i o n 2.2.24. The following conditions hold for a BS (X, 7"1 < c 7"2): (1) X is of 1-CatglI(
II --5, 1-~5(X) N 2-D(X) c 2-~;5(X) N 2-D(X)
N
N
1-Ga(X) ~ 1 - D ( X ) c 2-G5(X)A 1- D ( X ) c
C (1,2)-CatgII(X) C 2-CatglI(X) C 1 - C a t g I I ( X ) = (2,1)-Catgxi(X).
Proof. Assertions of (1) follow directly from the inclusion 2-D(X) c 1-D(X), taking into account (3) of Theorem 2.2.20. The proof of (2) follows immediatelly from Corollary 1.1.27 and (3) of Theorem 2.2.20. 2.3. T h e N - R e l a t i o n In Remark 2.2.2 we have mentioned that the relation existing between two topologies 7-1 and 7"2 on a set X and defined as 7"1 C1 U C 72 c1 U for every set U c 7"2 is not considered in [258] because of the coincidence of topologies 7"1 and 7"2 for a 2-regular BS (X, T 1 < N 7"2). Nevertheless, based on Corollary 2.2.9, we have a full right to consider the relation 7"1
7"1N7"2) if 7"1 c1 U c 72 c1 U for every set U r 72. E x a m p l e 2.3.2. Let (]t~,~l,Cd2) be the natural BS. S-related to Cdj n o r coupled to c~j; however, wi is near a~j.
Then ca~ is neither
From Definition 2.3.1, we readily derive the following simple properties of nearness. P r o p o s i t i o n 2.3.3. The following conditions are satisfied for a BS (X, 7"1,7"2): (1) (7"1N7"2 A 7" C 7"2) : = ~ 7"1N7". (2) (7"1N7"2 A 71 C 7") ~ 7"N7"2.
2.3. The N-Relation (3) (7"2 C 7"1 V 7"2 C CO 7"1) ~
83
7"1NT"2-
Proof. (1) It is clear t h a t for each set U c 7" c 7"2, we have 7"1 C1U C 7"2 CI U C 7" CI U so t h a t 7-1N7.. (2) and (3) are obvious. [~ C o r o l l a r y 2.3.4. The discrete topology on X is near any topology on X and every topology on X is near the antidiscrete topology on X . In contrast to the S-relation, the N-relation, like the C-relation, is not symmetric. E x a m p l e 2.3.5. Let X = {a, b, c}, 7"1 = { ~ , {a, b}, X } , &nd 7"2 = { ~ , {a}, {b}, {a, b}, {b, c}, X } . T h e n 7"2N7"1 since 7"1 C 7"2, but 7"1 is not near 7"2 since for the set {b, c} E 7"2, we h&ve 7"1 cl{b, c} -~ X &nd 7"2 cl{b, c} -- {b, c}. It is obvious t h a t 7"1 ~ N 7"2 ~ Example
X\{a,b}
2.3.6.
:/: ~.
IfT"1
(7"1N7"2 A 7"1 C 7"2).
Let X be any set such t h a t a, b c X, a -~ b, implies t h a t =
{~,{ai, X\{a},X},
7"2 = { ~ , { a } , X \ { a } , { b } , { a , b } , X } ,
then 7"1 < N 7"2Theorem
2.3.7.
The following conditions are equivalent for a BS (X, 7.1,7.2):
(1) 7-1 is near T2. (2) 7-1 cl 7"2 int A C 7"2 cl 7"2 int A so that 7"2 int 7"2 cl A c 7"1 int 7"2 cl A
for every subset A c X . (3) 7"1 C1 7"2 int A C 7"2 cl A so that 7"2 int A c 7"1 int 7"2 cl A
for every subset A c X . (4) For every point x c X l-neighborhood of x.
the 2-closure of any 2-neighborhood of x is a
Proof. It is obvious t h a t (1) ~ (2) ==~ (3). If (3) is satisfied, then 7"1CI U C 7"2 cl U for every set U c 7"2 and so (3) ==~ (1). (1) ==~ (4) Let x c X be any point and U(x) be its any 2-neighborhood. W i t h o u t loss of generality it can be assumed t h a t U(x) E 7"2. T h e n V = X \ ~2 cl U(x) c ~2
and by (1), 7"1 cl V c 7"2 cl V. If V1 = X \ 7"1 C1 V, then Pl -- X \ 7"1 cl ( X \ 7"2 cl U ( x ) ) - 7"1 int 7"2 cl U(x) C 7"2 c1U(x).
x c
cl U(x)
\
cl
U(x) =
7"2 int ( X \ U(x)) N U(x) - ~, we obtain t h a t 7"2 cl 7"2 int ( X \ U(x)) n U(x) - ~.
int(X \
Sinc
84
II. Different Relations Between Two Topologies ...
Hence x ~ v2 cl (X \ ~-2 cl U (x)) - T2 cl V and T1 cl V c T2 cl V implies x ~ T1 cl V so t h a t x E V1 - X \ T1 cl V. Thus V1 is a 1-open neighborhood of x such t h a t V1 C v2 cl U(x). (4) ~ (1) Suppose t h a t U E T2 is any set and x E T1 cl U is an arbitrary point. If U(x) is any 2-open neighborhood of x, then by (4), there exists a set V c TI such t h a t x E V c ~-2 cl U(x). It is clear t h a t V N U --~ ~ so t h a t v2 cl U ( x ) N U ~ ~. Hence U(x) A U ~ ~ and U(x) c 7-2 being arbitrary implies t h a t x E ~-2 cl U. D Remark
2.3.8. By analogy with (4) of Theorem 2.2.6, we have (T1 < N sup(T1,7-2)) ~
TINT2.
Indeed, if T1
Corollary 2.3.9.
The relation
topologies on a set X . Pro@ Clearly, it suffices to show only t h a t (rl < N 3-2 A 3-2 < N 7-3) ~ T1 < N 7-3. Suppose t h a t x c X is any point and U(x) c r3 is its any neighborhood. Since r2 < g ra, following (4) of T h e o r e m 2.3.7, there exists a neighborhood W ( x ) c r2 such t h a t W ( x ) c 7-3 cl U(x). The condition rl
[--]
Examples 2.2.5 and 2.3.2 show t h a t the notions of coupling and nearness of topologies are independent of each other. However, we have the following obvious
Corollary 2.3.10. The implication 7-1
~ N 7-2 ~
7-1 ~ C 7-2 is correct for a
BS (X, 7-1,72).
E x a m p l e 2.3.11. Let X - {a,b,c}, T 1 be the antidiscrete topology on X, 7-2 -- { ~ , {a}, {b, c } , X } . T h e n T1 ~ C T2, b u t T1 ~ N T2 is n o t correct.
Corollary 2.3.12. The conditions below are equivalent for a BS (X, 7"1, T2)" (]) T1 ~ N 7-2. (2) T1 C17-2 int A - ~-2 cl ~-2 int A so that 7"1 int 7-2 cl A - ~-2 int T2 cl A
for every subset A c X . (3) n l(U) - ~ for each set U E T2 so that n 2 ( F ) - ~ for each set F c coT2.
2.3. T h e N - R e l a t i o n
85
(4) l-C1 r2 = 2-C1 r2 so that 1-Int co r2 = 2-Int co r2.
Proof. T h e p r o o f is b a s e d on Definitions 1.4.19 a n d 2.3.1 in c o n j u n c t i o n w i t h (2) of T h e o r e m 2.3.7 a n d t h e inclusion T1 C 7-2. [--] Corollary
2.3.13.
The following conditions are satisfied for a BS (X, ~-INT2):
(1) If (X, rl, r2) is 2-regular (2-quasi regular), then (X, rl, r2) is (2, 1)-regular
2-connected. (2) If (Y, r~, ~;) is a BsS of X and Y c ~ 0 2-Z~(X), then rr
Moreover, for a BS (X, rl < N r2), we have: (a) ( x , (1, (1, 2)-q a i
2-q a i
regular), and (X,rl,r2) is I-connected <--> (X, rl, r2) is p-connected ~' ". (X, rl, r2) is 2-connected .e----->.(X, rl, r2) is d-connected. (4) If (X, rl, r2) is 2-normal, then (X, rl, r2) is p-normal and so (X, rl, r2) is 1-normal. (5) r2 A 1 - D ( X ) = r2 A 2 - D ( X ) so that
co (6)
= co
1-Int C1 r2 - 2-Int C1 r2 - (1, 2)-Int C1 r2 - (2, 1)-Int C1 r2
so that 1-C11nt co r2 - 2-C11nt co 72 - (1, 2)-C1 Int co r2 - (2, 1) -C11nt co r2. (7) (X, r l , 72) is p-extremally disconnected e----->, (X, 71~ 7-2) i8 2-extremally disconnected <---> (X, 71, r2) is 1-extremally disconnected e----> (X, r l , r2)
is d-eztremally disconnected. Proof. (1) T h e first p a r t is an i m m e d i a t e c o n s e q u e n c e of (2) of P r o p o s i t i o n 0.1.7 a n d Definition 0.1.14 in c o n j u n c t i o n w i t h Definition 2.3.1. If (X, rl, r2) is not 2-connected, t h e n t h e r e is a set U c r2 N co r2, ~ :/: U ~- X a n d rlNr2 implies t h a t T1 cl U c ~-2 cl U - U, a n d so U c r2 c~ co ~-1. Similarly, one can prove t h a t X \ U c r2 ca COrl. H e n c e U c rl N c o r l so t h a t (X, r l , r2) is n o t 1-connected. (2) A s s u m e t h a t Y c r2 a n d A c Y is any subset. T h e n by (3) of T h e o r e m 2.3.7, r2 int A c rl int r2 cl A so t h a t r 2' int A - r2 int A c rl int 72 cl A cq Y c !
T 1'
int(r2 cl A Cq Y) -
T 1'
int r~ cl A
!
a n d so T1 N % . O n t h e o t h e r h a n d , let Y c 2 - D ( X ) a n d A c Y be a n y set. Clearly, t h e r e exists a 2-open set U such t h a t U cq Y - r~ int A. B y (3) of T h e o r e m 2.3.7, U c r l i n t r2 cl U a n d hence %' int A C rl int r2 cl U N Y c T1' int(r2 cl U cq Y) - r~ int (r2 cl(U cq Y) N Y) -
-
T 1' i n t
(r2 cl rs int A Cl Y) c r~ int r~ cl A
since Y E 2 - D ( X ) , a n d so r ~ N r ~ .
86
II. Different Relations Between Two Topologies ...
(3) The first part follows directly from (3) of Corollary 2.2.8, the first part of (1) above, and (3) of Corollary 2.3.12, taking into account Corollary 2.3.10. The second part is an immediate consequence of the second part of (1) and the implications following Definition 0.1.18. (4) By (4) of Corollary 2.2.8 and Corollary 2.3.10, it suffices to show only the correctness of the first implication. Let F c co7-1, U c 7-2 and F C U. Since F c co7-1 C co7-2 and (X, 7-1,7-2) is 2-normal, there is a set V c 7-2 such that F c V c 7-2 cl V c U. Hence, by (3) of Corollary 2.3.12, F C V C 7-1 cl V C U and it remains to use (4) of Proposition 0.1.7. Assertions of (5) follow immediately fl'om (3) of Corollary 2.3.12. (6) If U E 7-2, then this proof is identical to that of (7) of Corollary 2.2.8, taking into account (3) of Corollary 2.3.12. (7) The proof of the first equivalence is similar to the proof of (8) of Corollary 2.2.8, taking into account (6) above and (3) of Corollary 2.3.12. The rest follows from (8) of Corollary 2.2.8 in conjunction with Corollary 2.3.10. K] Thus, by (1) of Corollary 2.3.13, the conjunction (7-1N7-2 A 7-2N7-1) implies the equivalence (X, 7-1,72) is 1-connected ,+-->, (X, 7-1,7-2) is 2-connected.
D e f i n i t i o n 2.3.14. A topology 7-1 is R0-near a topology ~-2 on a set X (briefly, near 7-2 and 17-1int 7-2 clA \ 7-2 int A I < Ro for every set A c X.
TIN(Ro)T2) if 7-1 is
It is obvious that if IXI < Ro, then 7-1N7-2 ~ 7-1N(Ro)7-2. However, ~iN~j, but ~i is not Ro-near ~j in the natural BS (R, 021, ~22). Clearly, by (2) of Corollary 2.3.13, the N(R0)-relation as well as the
(71N(~0)7-2/~ 71 C 7-2).
T1
As for the C-relation, Example 2.1.8 also shows that the N-relation is not, in general, hereditary with respect to/-closed subsets. D e f i n i t i o n 2.3.15. A topology 7-1 is/-strongly near a topology 72 on a set X (briefly, 71N(i)7-2) if the N-relation is hereditary with respect to/-closed subsets of X. It is obvious that in Example 2.2.15 we also have 7-1
,
,
'X(
o)4
and (T1
71
T;
for every/-closed BsS (F, 7-~,7-~) of a BS (X, T1, T2). Proceeding with our investigation, note that by Example 2.3.2, the N- and S-relations are independent of each other. Moreover, we have
2.3. The N-Relation
87
E x a m p l e 2.3.16. Let X = {a,b,c,d}, 771 = {25, {a, bI, X } , and 772 = {2~, {b}, {a,b}, { b , c , d } , X } . Then 771
2.3.17. The statements below hold for a BS (X, 711 772):
(1) If (X, 7711772) is d-quasi regular, then (771 N772/k 7-2NT1) ===~71 S772.
(2) If (X, 7711772) is p-extremally disconnected, then 7-1 $772 -----N, (771N7-2/k 772ST1).
Proof. (1) By virtue of (2) of Theorem 2.1.5, it suffices to show t h a t rl int A =/= ~ <---> r2 int A =/= ~ for every subset A c X. If rl int A r 2~, then by the 1-quasi regularity of (X, rl, r2), there is a set V c rl \ {~} such t h a t T1 C1 V C rl int A. Let x c V be any point. Then, by (4) of Theorem 2.3.7, there exists a 2-open neighborhood U(x) such that U(x) c 7-1 C1V since 772N771. Thus U(x) c rl int A and so 2~ :fi 72 int T 1 int A C r2 int A. The implication T2 int A -r 2~ ----5. 771int A -r 2~ can be proved in a similar way using the 2-quasi regularity of (X, 771,772) and TINr2. (2) Following (3) of Theorem 2.1.5, T1 ST2 ~
(T1 int A C T1 C1T2 int A A T2 int A c 72 el T1 int A)
for every subset A C X. Since (X, 71, r2) is p-extremally disconnected, by (3) of Definition 0.1.18, 7-2 CI T 1 int A = 771int 772cl T1 int A A T1 C1 7-2 int A = 72 int T 1 C1 7-2 int A
for every subset A c X. Hence 72 int A c 771int T2 cl T 1 int A c 771int 772el A and T1 int A c 772int T1 C17"2 int A c 72 int 771cl A
for every subset A c X. Thus it remains to use (3) of Theorem 2.3.7.
[-7
C o r o l l a r y 2.3.18. The following conditions are satisfied for a BS (X, 771,T2):
(1) If ( X, 771 < x 772) is d-quasi regular, then 771 < s 772. (2) If (X, 771,7-2) is d-quasi regular, where TINTT2 A 7"2NTT1, then (X, 771,7"2) i8 1-Baire if and only if ( X 1771172) i8 2-Baire. Moreover~ ( X 17711772) i8 p-quasi regular. (3) If (X, T1 < s 72) is p-extremally disconnected, then 771
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2.3. T h e N - R e l a t i o n
(9) A ~ (1, 2)-YD(Y) = 2 - H ~ t Y ) ~ so that A c (1, 2 ) - S T ) ( Y ) = 2-$T)(Y) ~
91
A ~ 2-MD(X) = (1, 2)-HD(X) A c 2 - $ T ) ( X ) = (1, 2)-$T)(X)
and thus A c (1, 2)-Catg I ( Y ) - 2-Catg I (Y) ~
A c 2-Catg I ( X ) - (1, 2)-Catg I (X),
so that A c (1,2)-Catg~i(Y)=2-Catgix(Y) ,z--~, A c 2-Catgii(X)=(1,2)-Catgxi(X).
(10) A c 2 - S O ( Y ) = (2, 1 ) - s o ( Y ) ~ I f Y C 7-1 n
c o T2,
A c (2, 1 ) - S O ( X ) =
2-SO(X).
then
(11) A E 2-SC(Y) = (2, 1)-SC(Y) ,e--->, A c 2 - S C ( X ) = (2, 1)-SO(X). Proof. In common with Corollary 2.2.23 we begin by observing that by (2) of Corollary 2.a.la in all cases (1)-(11) both the N-relation and the
, X is of 2-gary II) implies that
N
n
1-0a(X) N 1-D(X) C 2-0a(X)Cl 1-D(X) C (1, 2)-Catgii (X) 2-Catgli (X) c (2, 1)-Catgii (X) = 1-Catg~i (X).
----
Proof. The proof follows directly from (2) of Proposition 2.2.24, taking into account (3) of Theorem 2.3.19. D Now, we give the appropriate example from [14, p. 159, Exercise 230]. E x a m p l e 2.3.23. Let (X, rl < r2) be a 1-compact and 1-first countable BS, where IUI > R0 for each U E r l \ { ~ } and r2 = { U \ A : U c rl, A c X, IAI < R0}. Then rl
92
II. Different
Relations
Between
Two
Topologies
...
At the end of the section we introduce an equivalence relation D on a special proper subfamily B(T) C A ( r ) , where A ( r ) is the family of all topologies on a given set X, and we prove the following important T h e o r e m 2.3.24. Let X be any set and B(r) be any family of topologies on X , satisfying the following conditions: (1) B(T) is closed under finite intersections. (2) The antidiscrete topology ~-A - { ~ , X } ~ B ( ~ - ) .
Then there exists an equivalence relation D on B(T) such that if one member of an equivalence class is connected, then all members of the class are also connected. Proof. It is not difficult to see that a binary relation ~ on B(T), defined as follows: ~-1 ~ ~-2 if there is a topology ~- c B(T) such t h a t 7"1 C1U - - 7"2 cl U for each U c T, is an equivalence relation on B(T). Further, if we suppose that D is a binary relation on B(T), defined in a manner as follows" 71D72 ~ (71N72 A 72N71), then D is an equivalence relation in the above sense. Indeed, by Definition 2.3.1, TIDT2 ~ ((T1 cl U C_ T2cl U for each U c ~-2) A (T2 cl U c T1 cl U for each U c T1)), and since by (1), B(T) is closed under finite intersections, we obtain T1 CI V - - T 2 CI V for each V c ~ - - T1 N T2. Finally, it follows immediately from (1) of Corollary 2.3.13 t h a t if one member of a D-equivalence class is connected, then all members of this class are also connected. [3 2.4. B i t o p o l o g i c a l Insertions The notion of insertion property of a topology ~- on a set X was introduced in [173] to describe some classes of functions and spaces. In this section we shall define and investigate some bitopological modifications of this notion with an aim to apply them in characterizing the relations between two topologies as well as the small and large p-inductive dimension functions in the next chapter.
D e f i n i t i o n 2.4.1. We shall say that a bitopology (~-1,~-2) on a set X has the (i, j)-A-insertion properties, where A c 2 x is any family, if either of the following two equivalent conditions is satisfied: (1) For every subset A c X there exists a set G c A such that Ti i n t A C G c Tj clA. (2) For every pair of sets (U, F), where U c ri, F c co rj and U c F, there exists a set G c A such that U C G c F. It is obvious t h a t if (T~, r2) on X has the (i,j)-A-insertion properties, then ~, X E A. It is likewise obvious that the antidiscrete topology on X possesses the A-insertion property for any family .4 c 2 x .
2.4. Bitopological Insertions
93
R e m a r k 2.4.2. The following implications hold for a BS (X, 71 < 72) and any family A c 2x: ('rl,r2)
has the 2-A-insertion property ~ ( r l , r 2 )
has the (1,2)-A-insertion property
(rl,r2) has the (2,1)-A-insertion property ==>(rl,r2) has the 1-A-insertion property
so that (T1,T2) has not the l-A-insertion property ===~(Tl,r2) has not the (1,2)-A-insertion property (T1,T2) has not the (2,1)-A-insertion property ==*(~-l,T2) has not the 2-A-insertion property.
Important information on the (i, j)-A-insertion properties is given by the next obvious statement. P r o p o s i t i o n 2.4.3. The following conditions are satisfied for a BS (X, T1, 7-2)2 (1) I r A c 2 x is any family, then (71,72) has the (i,j)-A-insertion properties if and only if (71,72) has the (j, i)-coA-insertion properties. (2) If A1 c A2 c 2 x are any families and (71,72) has the (i,j)-Al-insertion properties, then (71,72) has the (i,j)-A2-insertion properties. (3) If A c 2 X is any family and (71,72) has the (i,j)-A-insertion properties, then (71,72) has the (i,j)-Ad-insertion properties, where Ad c {i-Int A, j-C1 A, (i, j)-Int C1 A, (j, i)-C11nt A }. R e m a r k 2.4.4. Naturally, much importance is attached to the nontrivial aspects of the (i, j)-A-insertion properties, in particular, to cases where a bitopology (rl, r2) on X has the ( i , j ) - ( A \ A4)-insertion properties or (i,j)-B-insertion properties, where B c A4 c A and the bitopology (71,72) on X always has the (i, j)-A4-insertion properties. For example, the nontrivial Ga(X)-insertion property, that is, the (Ga(X) \ r)-insertion property (the insertion of ~a-sets method in the author's terms) was used in [1731 to prove that certain functions belong to the Baire class one. It is clear that there are also nontrivial aspects of bitopological (and hence topological) insertions that are distinct from those mentioned above, when, in particular, the distinctions are conditioned only by the fact that topologies 71 and 72 on X are not comparable. Some of these aspects can be helpful in characterizing the r and N-relations. Theorem
2.4.5. The conditions below are equivalent for a BS (X, 71,72):
(I) 7-i is coupled to ~2. (2) (T1,7"2) has the (2, 1)-l-OD(X)-insertion property (,~-->, (T1,7-2) has the (1, 2)-l-CZ)(X)-insertion property). (3) (71,72) has the (2,1)-rl-insertion property (<---->, (71,72) has the (1, 2)-co rl-insertion property). (4) For every set U E r2 there exists a set V E 7-1 such that U c V and T1 c 1 U :
T1 c 1 V .
Proof. Clearly, by virtue of (1) of Proposition 2.4.3 for the equivalences in (2) and (3) it suffices to consider only the corresponding left sides.
94
II. Different R e l a t i o n s Between T w o Topologies . . .
(1) = ~ (2). Let 7-1 be coupled to 7-2 and A C X be any subset. T h e n by (2) of T h e o r e m 2.2.6, r2 int 7-1 el A C rl int 7-1 C1A and if V - 7-1 int 7-1 el A, then 7-2 int A c V c 7-1 clA. It is evident t h a t V E I - O D ( X ) and by (1) of Definition 2.4.1, (7-1,7-2) has the (2, 1)-l-OZ)(X)-insertion property. On the other hand, by (2) of Proposition 2.4.3, the inclusion 1 - O D ( X ) C 7-1 gives t h a t ( 2 ) ~ (3). (3) ~ (1). If A c X is any subset, then there exists a set V E 7-1 such t h a t 7-2 int A c V c T 1 el A. Hence 7-2 int A c 7-1 int 7-1 c1A and by (3) of T h e o r e m 2.2.6, 7-1 is coupled to 7-2. (3) ----5, (4). If U E 7-2 is any set, then by (2) of Definition 2.4.1 there exists a set V E 7-1 such t h a t U c V c 7-1 cl U. It is obvious t h a t 7-1 cl V - T 1 cl U. (4) ~ (3). Let U c 7-2, F c co 7-1 and U c F . Then there exists a set V c 7-1 such t h a t U C V C 7-1 el g C F and so U C V C F. Hence it remains to use (2) of Definition 2.4.1. D C o r o l l a r y 2.4.6. The following conditions are equivalent for a BS (X, 7-1,7-2)" (1)
T1
(2)
T1
7"2"
and T 2 are weakly connected in the sense of [182].
Proof. The proof follows directly from Definition 1 in [182] and (4) of Theorem 2.4.5. D Theorem
2.4.7. The following conditions are equivalent for a BS (X, 7-1,7-2):
(1) 7-1 i8 near 7-2.
(2) (7-1,7-2) has the 2-(1,2)-OZ)(X)-insertion property (z---->. (7-1,T2) has the 2-(1, 2)-CZ)(X)-insertion property). (3) (7-1,7-2) has the 2-7-1-insertion property (r (7-1,7-2) has the 2-coT-l-insertion property). (4) For every set U E 7-2 there exists a set V E 7-1 such that U C V and 7-2 cl U = 7-2 cl V.
Proof. We consider only the left sides of the equivalences in (2) and (3). (1) ==~ (2). Let T 1 be near r2 and A c X be any subset. Then by (3) of Theorem 2.a.7, r2 int A c 7-1 int 7-2 cl A and if V = 7-1 int 7-2 cl A, then V E (1, 2 ) - O D ( X ) , 7-2 i n t A c V c 7-2clA and so (7-1,7-2) has the 2-(1,2)-OD(X)-insertion property. F u r t h e r m o r e since (1, 2 ) - O D ( X ) c rl, by (2) of Proposition 2.4.3, (2) ~ (a). (3) ~ (1). If A c X is any subset, then there exists a set V ~ 7-1 such t h a t 7-2 int A c V c 7-2 clA. Hence 7-2 int A c 7-1 int 7-2 clA and by (3) of T h e o r e m 2.3.7, 7-1 is near 7-2. (3) ----5, (4). If U E 7-2 is any set, then by (2) of Definition 2.4.1 for i = j = 2 there exists a set V c 7-1 such t h a t U C V C 7-2 el U. Clearly, 7-2 cl U = 7-2 cl V. (4) ~ (3). Let U ~ r2, F E co r2 and U c F. Then there exists a set V E rl such t h a t U c V and 7-2 el U = 7-2 el V so t h a t U c V C F and thus (3) is satisfied. 89 We conclude the discussion of this section by introducing some new notions of bitopological insertions which will be used in C h a p t e r III to describe small and large inductive (i, j ) - d i m e n s i o n functions.
2.4. Bitopological Insertions
Definition
2.4.8.
95
We shall say that a bitopology (7-1,7-2) on X has the
(i,j)-(M,A)-insertion properties, where A4, A c 2 x are any families, if either of the following two equivalent conditions is satisfied: (1) For every set A c Ad and its a n y / - o p e n neighbohood U(A) there exists a set G c A such that U(A) c G c rj cl U(A). (2) For every set A E M and any sets U E 7-i, F E co rj together with A c U c F there exists a set G c A such that U c G c F. R e m a r k 2.4.9. It is clear that if (7-1,7-2) Oil X has the (i, j ) - ( M , A)-insertion properties, then X E A. Also 2~ c Ad implies ~ c A, but in contrast to Definition 2.4.1 it is not necessary t h a t the condition ~ c A be fulfilled. The following implications are correct for any families A/I, A of subsets of a BS (X, T 1 < 7-2):
(rl,r2) has the 2-(2M,.A)-insertion property
==> (rl,r2) has the (1,2)-(2M,A)-insertion property
(rl,r2) has the (2,1)-(2td,A)-insertion property ==> (rl,r2) has the 1-(M,A)-insertion property. P r o p o s i t i o n 2.4.10. Let A , M and 13 be any families of subsets of a BS (X, 7-1,7-2) and ;g c M . Then the following three conditions are equivalent: (1) (T1,7.2) has the
(i,j)-A-insertion properties.
(2) (7-1,7-2) has the (i,j)-(2X,A)-insertion properties. (3) (7-1,7-2) has the (i,j)-(M,A)-insertion properties.
Moreover, each of the conditions (1)-(3) implies (4) (7-1,7-2) has the (i, j ) - ( ~ , j4)-insGrtioft properties.
Proof. It is obvious that (1) = ~ (2) ==~ (3). On the other hand since ~ c M and 2~ c A for every subset A c X so that ~ c U c F, where U c 7-~, F c co7-j, (2) of Definition 2.4.8 implies (2) of Definition 2.4.1 and so (3) = ~ (1). The implication (2) ---> (4) is also obvious, but the inverse implication does not hold. D Hence it is clear that if 2~ c ,4, then (7-1,7-2)on X has the (i,j)-A-insertion properties e---->, (7-1,7-2) on X has the (i, j)-(A, A)-insertion properties. E x a m p l e 2.4.11. Let X = { a, b, c, d, e, f }, 7- = { Z , { a , b , c } , { a , b , c , d } , X } , A = {{a, b, c, d, e}, X}, and M = {{a}, {a, b}}. Then 2~gA/I, 7- on X has the ( M , A)-insertion property and does not have the A-insertion property so t h a t 7does not have the (2x, A)-insertion property since for the set {f} together with int{f} = ~ and cl{f} = { e , f } , there does not exist a set G E A such t h a t G C {e, f}. C o r o l l a r y 2.4.12.
The following conditions are satisfied for a BS (X, 7-1,7-2):
(1) If M and A are any families of subsets of X and ;g c M , then (7-1,7-2) has the (i,j)-A-insertion properties ,e-> (7-1,7-2) has the (i,j)-coA-insertion properties e----->, (7-1,7-2) has the (i, j ) - ( M , A)-insertion properties.
96
II. D i f f e r e n t R e l a t i o n s
Between Two Topologies ...
(2) If A1,A2 and M are any families of subsets of X, where .41 c A2 and (7-1,72) has the (i,j)-(A/t, A1)-insertion properties, then (7-1,7-2) has the (i, j)-(Ad, A2 )-insertion properties. (3) If M I,A/I2 and A are any families of subsets of X, where .All C .Ad2 and (T1,T2) has the (i,j)-(A/12, A)-insertion properties, then (T1,T2) has the (i, j ) - ( M 1, A)-insertion properties.
Pro@ The proof follows from Definition 2.4.8, Propositions 2.4.3 and 2.4.10.
D
C H A P T E R III
Dimension of B itopological Spaces The notion of a zero-dimensional BS was introduced by I. L. Reilly [216] on the basis of the idea of bitopological disconnectedness as examined by J. Swart [248]. A systematic study of bitopological dimension functions was undertaken independent of one another by M. Jelid [144], [145]; D. (~irid [66], and us [84], [86], [87], [101], [102]. As distinct from [66], [144], [145], the ideas set forth in [84], [86], [87], [101], [102] were essentially based on the notion of bitopological boundaries. The nine functions corresponding to the small inductive dimension [107], [179], [254], the large inductive dimension [45], [82], and the covering dimension [168], [192], [11], [111] of a TS are defined for each integer n > - 1 . Each definition is followed by stating and proving the respective properties of these functions. By analogy with [11] the p-small and p-large inductive dimensions are formulated in terms of both bitopological partitions and neighborhoods in a manner such that for n - 0 p-small inductive dimension leads to the notion of Reilly. In the aboveindicated succession of dimension functions, monotonicity with respect to arbitrary BsS's is proved for the first three functions, while monotonicity with respect to the p-closed subsets is established for the remaining six functions. Further, interrelations of the p-inductive dimensions and their topological versions are considered when topologies are comparable by inclusion or are coupled,
Using different methods, we give two equivalent definitions of the p-small inductive dimension p - i n d X for every nonnegative integer n. D e f i n i t i o n 3.1.1. Let (x,A) be a pair in a BS (X, T1,7-2) such that A c coTi and x g A. Then we say that a p-closed set T is a partition, corresponding to the
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3.1. Pairwise Small Inductive Dimension
99
Proof. (1) Let ( i , j ) - i n d X - n < oc. I f x c X, F c cori a n d x - g F , then by ( 2 ) o f Definition 3.1.3, there exists a partition T, that is, X \ T - HI U H2, H~ c Ti \ {2~}, x c Hi, A c Hj and Hi N Hj - ;g. Therefore (X, rl, r2) is (i,j)-regular; for the rest it remains to use Corollary 2.2.9 and Theorem 1 in [258], respectively. (2) It sumces to prove that ( i , j ) - i n d X - k implies ( i , j ) - i n d Y < k. For k - - 1 , k - co, the statement is correct. Let it be correct for k_< n - 1 . We shall prove the statement for k - n. If x r Y, A' c co r'i, and x ~ A', then there exists a set A c co ri such that A' - A n Y. Since (i, j ) - i n d X - n, to the pair (x, A) there corresponds a partition T such that (i, j ) - i n d T <_ n - 1. It is obvious that T ' - T n Y is the partition, corresponding to ( x , A ' ) i n (Y, r{, r~). Hence, by induction, (i, j ) - i n d T' _< n - 1 since T' C T. D C o r o l l a r y 3.1.5. The statements below hold for a BS (X, rl, r2)" (1) /f p - i n d X is finite, then (X, rl, r2) is p-regular. (2) / f ( K r { , r ~ ) i s any BsS of (X, r~, r,e), then p - ind Y < _ p - i n d X . C o r o l l a r y 3.1.6. Let (X, rl, r2) be a BS and n denote a nonnegative integer. Then:
(1) ( i , j ) - i n d X <_ n if and only if for every point x c X and any i-neighborhood U(x) there exists an i-open neighborhood V ( x ) such that rj cl V ( x ) c U(x) and ( i , j ) - i n d ( j , i ) - F r V ( x ) < n - 1 or, equivalently, (X, r1,72) has an i-base gi such that (i, j ) - i n d ( j , i ) - F r V _< n - 1 for every V c gi. (2) If (X, 7-1,7-2) i8 i-second countable, then ( i , j ) - i n d X < n if and only if (X, rl, r2) has a countable i-base Bi such that (i, j ) - i n d ( j , i)- Fr V _< n - 1 for every V c 13i. Proof. It suffices to prove only the first equivalence in (1). Let (i, j ) - i n d X <_ n, x c X and U(x) be any /-neighborhood. It can be assumed without loss of generality that U(x) c 7i. Then for the pair ( x , A - X \ U(x)), there exists a partition T such that X \
-
u H2,
and (i, j ) - i n d T < n -
, A c Hh,
xcH
H t n H2 -
1. It is evident that
rj cl Hi CqHj - ~ and so rj cl Hi c X \ A - U(x). Let Hi - V(x). Then x c
c
cl
c
and by (2) of Remark 3.1.2 and (2) of Proposition 3.1.4, (i, j ) - i n d ( j , i)- Fr V ( x ) < n - 1. Conversely, let the condition in the right-hand part of the equivalence be satisfied and (x, A), where A c co 7i, x c A, be any pair. Then U(x) - X \ A is an /-open neighborhood of x and, by condition, there exists a n / - o p e n neighborhood V ( x ) such that rj c l V ( x ) C U(x) and ( i , j ) - i n d ( j , i ) - F r V ( x ) < n -
1.
100
III. Dimension of Bitopological Spaces
But following (1) of Remark 3.1.2, (j, i)-Fr V ( x ) i s the partition corresponding to (x, A) and thus it remains to apply (2) of Definition 3.1.3. D Hence by (1) of Corollary 3.1.6, we have p - i n d X - 0 ,z--v, (7-1 has a base consisting of 2-closed sets and ~-2 has a base consisting of 1-closed sets) [216]. C o r o l l a r y 3.1.7. The Cartesian product O<3
O<3
(X3
t=l
t=l
t=l
of a ~o~tabl~ fa.~ly {(X.. ~. ~)}7=1 of BS'~ ~ ( ~ . j ) - ~ o d ~ n ~ o n a l (a~d. of ~ o ~ . p-z~o d~.~~o~al) if a~d o~ly if all BS'~ (X,. ~{. ~) ~ (~.j)-z~o d~mensional (p-zero dimensional). Proof. If X # ;~, then each BS (Xt, 7-~,7-~) is d-homeomorphic to a BsS of (X, T1, ~-2) so that if (X, 71,7-2) is (i, j)-zero dimensional, then by (2) of Proposition 3.1.4 and Remark 3.3.7 (following which (i, j ) - i n d X are bitopological properties), all BS's (Xt, T{, ~-~) are (i, j)-zero dimensional. To prove the reverse implication, it is sufficient to consider, for each t - 1, oc, a n / - b a s e B~ for the BS (Xt, 7-~, 7-~), consisting o f / - o p e n and j-closed sets and to recall that sets of the form O<3
k=mq-1
where U~ c B~ for t < m and m - 1, oc, constitute an /-base for (X, rl, 7-2) and a r e / - o p e n and j-closed in (X, 7-1,7-2)" [-7 Proposition
3.1.8. I f f o r a BS (X,~,r2), we have ( i , j ) - i n d X - n, n >_ 1, 1, the BS (X, 7-1,7-2) contains a p-closed subset Y such
then for each k - O, n that (i, j)- ind Y - k.
Pro@ It is enough to show that X contains a p-closed subset Y such that ( i , j ) - i n d Y - n - 1 since it is not difficult to see that p-Cl(Y) c p-Cl(X) for every set Y c p-Cl(X). Since (i, j ) - i n d X > n - 1, there exist a point x c X and a n / - n e i g h b o r h o o d U(x) such that for e v e r y / - o p e n neighborhood V(x), satisfying the condition Tj cl V(x) c U(x), we have (i, j ) - i n d ( j , i)- Fr V(x) > n - 2. On the other hand since ( i , j ) - i n d X that
<_ n, among V(x) there exists V'(x) such
7-jclV'(x) c U(x) and ( i , j ) - i n d ( j , i ) - F r V ' ( x ) <_ n Therefore for the p-closed subset Y (i, j ) - i n d Y - n - 1.
1.
(j, i)- Fr V'(x) of (X, 7-1,7-2), we have K]
C o r o l l a r y 3.1.9. If for a BS (X, 7-1 < 7-9.), we have ( i , j ) - i n d X - n, n >_ 1, 1, the BS (X, T1 < ~-2) contains a 2-closed subset Y such
then for each k - O, n that (i, j)- ind Y - k.
3.1. Pairwise Small Inductive Dimension
Proposition
3.1.10.
101
The conditions below are satisfied for a BS
(X, 7"1,7"2):
(1) If (X, 7"1,7"2) is (i,j)-regular and p-extremally disconnected, then (i,j)-indX =0. (2) If p - i n d X = 0, then (X, 7"1,7"2) is p-completely regular and, thus, it is
also p-almost completely regular. Proof. (1) Let x c X and U(x) be any /-neighborhood. T h e n by (2) of Proposition 0.1.7, there exists an /-open neighborhood V(x) such t h a t 7"j c l V ( x ) C U(x). But by (3) of Definition 0.1.18, 7"j cl V(x) is a l s o / - o p e n . Therefore W(x) = 7"j cl V(x) r 7"i O co T"j and, following (9) of T h e o r e m 1.3.2, (j, i ) - F r W(x) = ~ so t h a t (i, j) ind X = 0. (2) Let p - i n d X = 0. T h e n 7"1 has a base consisting of 2-closed sets and 7"2 has a base consisting of 1-closed sets. If Zi = {A : A c co 7"i N 7"j}, then it is not diMcult to see t h a t Z = {Z1, Z2} is a p-normal base of the BS (X, T~, T2) and, by T h e o r e m 0.2.5, (X, 7"1,7"2) is p-completely regular. The rest follows from (13) of Definition 0.1.6. [--1
7"1,7"2) is
C o r o l l a r y 3 . 1 . 1 1 . If (X, BS, then p - ind X - 0.
a p-regular and p-extremally disconnected
Note also here t h a t a BS (X, 7"1,7"2) is called hereditarily p-disconnected if X does not contain p-connected subsets of cardinality larger t h a n one. Proposition 3.1.12. Every R-p-T1 p- ind X - 0, is hereditarily p-disconnected.
(i.e., d-T~) BS (X, 7"1,7"2), where
Pro@ Let A C X be any subset and ]A I > 1. If Xl, X2 ~ A, Xl ~ X2, then p- i n d X - 0 implies t h a t , there is a set U c 7"2 Nco7.1 such t h a t Xl ~ U c X \ {x2} since X \ {x2} c 7"1 n CO7"2. Therefore !
I
A O U e (~-2 N co 7"1) \ { •}
i
l
and A \ U e (7-1 n CO "/-2)\ { e }
for the BsS (A, 7"~, 7"~) so t h a t A is not p-connected.
Q
R e m a r k 3 . 1 . 1 3 . If for a given fixed point x E X and any pair (x, A), where A c co7"i, x-~A, there exists a partition T such t h a t ( i , j ) - i n d T < n - 1, t h e n we write (i, j)- indx X < n. The meaning of (i, j)- indx X - n or (i, j)- indx X - oc is clear and, we have ( i , j ) - i n d X - s u p { ( i , j ) - i n d x X " x c X } . E x a m p l e 3 . 1 . 1 4 . Let (X, 7"1) be a TS and B c X be a fixed proper subset of X. It is clear t h a t if 7"2 is the B - t o p o l o g y on X (i.e, 7"2 - 7"(B) - {X} U {A c X " n E 2 B} [14, p. 63]) and if (X, 7"1,72) is l-T1, then (2, 1)-indx X - 0 for each point x c B. L e m m a 3 . 1 . 1 5 . Let x be a fixed point of a subset Y c X , where (X, 7"1,7-2) is a hereditarily p-normal BS. Then (i,j)-indx Y < n if and only if for every i-neighborhood U(x), there exists an i-open neighborhood V(x) such that
V(x) c U(x) and (i, j)-ind (Y N (j, i ) - F r V(x)) < n - 1.
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3.1. Pairwise Small Inductive Dimension
103
Conversely, assume t h a t the condition of L e m m a 3.1.15 is satisfied. We shall show t h a t (i, j)-indx Y <_ n. Since a n y / - n e i g h b o r h o o d U(x) contains a n / - o p e n neighborhood V(x) such t h a t Tj cl V(x) c U(x), it is clear t h a t a n y / - n e i g h b o r h o o d U'(x) - U(x) N Y contains a neighborhood V'(x) - V(x) N Y C 7-'i, and
cl
c
cl
n Y, z n ( X \ V ( x ) ) - Z \
so t h a t (j, i)- Fr~ V'(x) c Y a (j, i)- Fr V(x). Hence, by monotonicity,
( i , j ) - i n d ( j , i ) - F r , V'(x) < ( i , j ) - i n d ( Y N ( j , i ) - F r V ( x ) ) < n -
1.
Thus (i, j)- ind~ Y < n.
V~
C o r o l l a r y 3 . 1 . 1 6 . Let ~ r Y c X , where a BS (X, T1,T2) is hereditarily p-normal. Then ( i , j ) - i n d Y = 0 if and only if for each point x c Y and every i-neighborhood U(x), there exists an i-neighborhood V(x) such that V(x) c U(x) and Y O (j, i ) - F r V(x) = ~. Moreover, if (X, ~-1,T2) is i-second countable, then ( i , j ) - i n d Y = 0 if and only if (X, T1,Tv) has a countable i-base Bi such that Y N (j, i ) - F r V = 2~ for every V c Bi.
Proof. The first part is obvious. The second part follows directly from the first part in conjunction with (2) of Corollary 3.1.6. 89 Theorem 3.1.17 (A Generalized F o r m u l a ) . The inequality
Version
of the
Menger-Urysohn
( i , j ) - i n d ( M U N) < ( i , j ) - i n d M + ( i , j ) - i n d N + 1 holds for any subsets M and N of a hereditarily p-normal BS (X, T1,7-2). Proof. If (i, j ) - i n d M - oc or (i, j)- ind N - oc, then the inequality is obvious. It can therefore be assumed t h a t (i, j ) - i n d M - m, (i, j)- ind N - n, where - 1 < m < oc, - 1 <_ n < oc, are the whole numbers. We shall prove by an inductive a s s u m p t i o n with respect to the n u m b e r m + n t h a t ( i , j ) - i n d ( M U N) <_ m + n + 1.
(,)
Inequality (,) holds for m + n - - 2 . Let us assume t h a t (,) holds for m + n < k - 1 and prove it for m + n - k. Let x c M U N be any point and U(x) be any /-neighborhood from ( M U N, r~, r~). It can be assumed w i t h o u t loss of generality t h a t x c M. Since (i, j ) - i n d M - m, we have (i, j)-ind~ M <_ m and, following L e m m a 3.1.15, there exists a neighborhood V(x) c ri such t h a t
V(x) C U(x) and (i, j ) - i n d ( M N (j, i)-FrA,uN V(x)) <_ m On the other hand, (i, j ) - i n d ( N N (j, i)- FrMu N V(x)) < (i, j ) - i n d N < n. Let M 1 - ~ : / n (j, i)- FI'~luN V ( x ) ,
N1 -- N N (j, i)- FraiuN V(x).
Since
(i,j)-indM1 <_m-l,
( i , j ) - i n d N l <_ n
and
( i , j ) - i n d M 1 + ( i , j ) - i n d N 1 <_ m -
1 + n <_ k - 1,
1.
104
III. Dimension of Bitopological Spaces
from inequality (,) it follows that (i, j)-ind(M1 U N1) _< (m - 1) + n + 1 - m + n. But M1 U N1 - ( M N (j, i)-FrMu N V ( x ) ) U ( N N (j, i)-FrMu N V(x)) -= (j,
and so (i, j)- ind(j, i)- FrM.~ V ( x ) <_ m + n. Since V ( x ) is any small/-open neighborhood of an arbitrary point x c M U N (i.e., V ( x ) is contained in any given /-neighborhood U(x)), we ascertain that (i, j ) - i n d ( M U N) _< m + n + 1. C o r o l l a r y 3.1.18. The following statements hold for a hereditarily p-normal BS (X, 7-1,7-2)" (1) If M and N are any subsets of X , then
p - i n d ( M U N) _< p - i n d M + p - i n d N
+ 1.
(2) If Mo, M 1 , . . . ,M~ are any subsets of X , then (i, j ) - i n d (M0 U M1 U . . . U M~) _< <_ ( i , j ) - i n d M o + ( i , j ) - i n d M 1 + . . . + ( i , j ) - i n d M n + n and thus
p - i n d (M0 U M 1 U . . . U Mn) < p - i n d M0 + p - i n d M1 + . . . + p - i n d Mn + n. n
(3) If X
-
U Mk, where ( i , j ) - i n d M k
-
0 for each k -
O, n, then
k--0 n
(i, j)- ind X <_ n and thus, we have p - i n d X <_ n for X -
U Mk, where k=0
p - ind Mk - - 0 for each k -
0, n.
T h e o r e m 3.1.19. If a R-p-T1, d-second countable, and p-normal BS (X, 7-1,7-2) can be represented as a union of two BsS's Y and Z such that ( i , j ) - i n d Y < n and ( i , j ) - i n d Z < O, then ( i , j ) - i n d X < n + 1. Proof. Let x c X be any point and U(x) c 7-i be its any neighborhood. Then by Theorem 3.2.15 below, there exist disjoint sets U E 7-i and V E 7-j such that x E U, X \ U(x) c V and (X \ (U U V ) ) N Z - ;~. Clearly, x c U c V(x). But
(j, i)- Fr U-7-j cl U n (X \ U) C (X \ V) N (X \ U ) - X
\ (V U U) c X \ Z c Y
and by (2) of Proposition 3.1.4, (i, j)- ind(j, i)- Fr U _< n. n+l.
Thus (i, j)- ind X _< K]
C o r o l l a r y 3.1.20. If a R-p-T1, d-second countable, and p-normal BS (X, 7-1,7-2) can be represented as a union of two BsS's Y and Z such that p - ind Y _< n and p - ind Z <_ O, then p - ind X _< n + 1. T h e o r e m 3.1.21. If a 1-T1, d-second countable, and p-normal BS (X, 7-1 < 7-2) can be represented as a union of two BsS's Y and Z, where p - i n d Y p - ind Z - 0 and one of them is l-closed, then p - ind X - 0.
3.1. Pairwise Small Inductive Dimension
105
Proof. Let, for example, Y E COT1 C p - e l ( X ) . S i n c e X \ Y c Z, by ( 2 ) o f Corollary 3.1.5, p - i n d ( X \ Y) = 0, where X \ Y c ~1. By Corollary 0.1.13 the BS (X, T1 < T2) is p-perfectly normal and hence X \ Y E 2-~co(X), t h a t is, (X5
X \ Y -
[_J Fk, where Fk C co ~-2 C p-Cl(X) and p- ind Fk - 0 for every k - 1, oc. k=l
On the other hand, Y c p - e l ( X ) and, therefore, oo
x-ru(x\r)-ru
[_J /~=1
Thus it remains to use Corollary 3.2.28 below since (X, T1 < ~-2) is l-T1 implies ( X , T1 < 7-2) is R-p-T1. K] It is not difficult to see t h a t Theorem 3.1.21 remains valid if one of the sets Y and Z is l-open. T h e o r e m 3.1.22. If a R-p-T1, d-second countable, and p-normal BS (X, T1, T2) can be represented as a union of a sequence F1, F2,... of p-closed BsS's such that (i, j ) - i n d Fk < n for each k - 1, oc, then (i, j)- ind X < n.
Proof. We shall apply induction to the number n. For n - 0 the theorem is already proved in Corollary 3.2.28 below. Assume that the theorem holds for n 1 and oo
prove it for n. Let X -
[.J Fk, where Fk
-
T1 el f k n 7-2 el f k
and (i, j)- ind Fk <_ n
k=l
for each k - 1, oc. By (2) of Corollary 3.1.6 choose for k - 1, oc a countable/-base B) for the BsS Fk such t h a t (i, j ) - i n d ( j , i)-Fr U < n - 1 for every U c B~. By the inductive assumption, for the BsS (N2)
Y-
{U(j,i)-FrU'Uc
UB~} k=l
of the BS (X, T1,72), we have the inequality (i, j)- ind Y <_ n - 1. Now, by the second part of Corollary 3.1.16, for each k - 1, oc, the BsS Zk - Fk \ Y of a BS (Fk, ~-~,~-]) has ( i , j ) - i n d Z k <_ O. Therefore, by Corollary 3.2.28 below, the BsS oo
Z -
U zk - x \ Y of ( x , 7-1, ~-2) satisfies the inequality (i, j ) - i n d Z <_ 0 because k=l
from the equalities
Z k - F k \ Y - - F k N Z - ( T 1 clFk N 72 clFk) N Z--(T1 clFk N Z ) N (72 clFk N Z) it follows t h a t all the Zk's are p-closed in Z. Thus, by virtue of Theorem 3.1.19, we have (i, j ) - i n d X <_ n. [-1 C o r o l l a r y 3.1.23. If a R-p-T1, d-second countable, and p-normal BS (X, T1,7-2) can be represented as a union of a sequence F1, F2,... of p-closed BsS's such that p - ind Fk < n for each k - 1, oc, then p - ind X < n. C o r o l l a r y 3.1.24. If a 1-T1, d-second countable, and p-normalBS (X, T 1 < 7-2) can be represented as a union of a sequence F1,F2,... of i-closed sets, where p - ind Fk < n for each k - 1, oc, then p - ind X < n.
Pro@ The proof follows directly from the inclusions co ~-1 c co ~-2 C p-Cl(X).
E]
106
III. Dimension of Bitopological Spaces
Corollary 3.1.25. If a R-p-T1, d-second countable, and p-normal BS (X, ~-1,7~) can be represented as a union of a sequence F1, F 2 , . . . , where every O0
Fk is a countable union of p-closed sets, that is, Fk -- U F ~ and p - ind Fk _< n m=l
for each k - 1, oo, then p - ind X < n. Proof. By (2) of P r o p o s i t i o n 3.1.4, p - i n d F ~ OO
< n for every k - 1, oo, m -
1, oc
CxD
U
o
k=l m=l
Corollary 3.1.26. If a l-T1, d-second countable, and p-normal BS (X, T1 < T2) can be represented as a union of two B s S ' s Y and Z, where p - ind Y _< n, p - ind Z <_n and one of them is 1-closed or 1-open, then p - ind X <_ n.
Pro@
Let Y c T1.
Then, by Corollary 0.1.13, Y c 2 - ~ ( X ) s o
that Y -
OO
U Fk, where Fk E coT2 C p - C l ( X ) for each k -
1, oc.
Since X \ Y
C Z, we
k=l
have p - i n d ( X \ Y) < n. Clearly, p - i n d Fk < n for each k p - ind X _< n because
1, oc and, therefore,
OO
X - Y U ( X \ Y ) - U Fk U ( X \ Y). k=l
T h e p r o o f for a 1-closed set is similar.
[-]
Theorem 3.1.27. For a R-p-T1, d-second countable, and p-normal BS (X, rl,r2), we have: ( i , j ) - i n d X < n if and only if X can be represented as a union of two B s S ' s Y and Z such that (i, j ) - i n d Y < n - 1 and (i, j ) - i n d Z < 0.
Proof. Let ( i , j ) - i n d X <_ n. T h e n , by (2) of Corollary 3.1.6, X has a c o u n t a b l e /-base Bi such t h a t ( i , j ) - i n d ( j , i ) - F r V < n - 1 for every U c Bi. By Theorem 3.1.22 the BsS Y - U { ( j , i ) - F r U " U e B~} has ( i , j ) - i n d Y <_ n - 1 and by the second p a r t of Corollary 3.1.16, (i, j ) - i n d Z < 0, where Z - X \ Y. To complete the proof, it suffices to apply T h e o r e m 3.1.19. [] Corollary 3.1.28. For a R-p-T1, d-second countable, and p-normal BS We have p - ind X <_ n if and only if X can be represented as the union of two B s S ' s Y and Z such that p - ind Y < n - 1 and p - ind Z < 0.
( X , 7-1, T2) ,
T h e o r e m 3 . 1 . 2 9 . Let (X, 7"1,7"2) be a R-p-Tz, d-second countable, and p-normal BS such that p - ind X < n. Then for every pair A E co ~-1, B E co ~-2 with A • B - 2~, there exists a partition T between A and B such that p - ind T < n - 1.
Proof. By T h e o r e m 3.1.27, X - Y U Z, where p - i n d Y < n - 1 and p - i n d Z < 0. Following T h e o r e m 3.2.15 below for A and B, t h e r e exists a p a r t i t i o n T such t h a t T~Z-2~. SinceTcX\ZcY, wehavep-indT_
3.1. Pairwise Small Inductive Dimension
107
Pro@ Since p- ind Y _< n, by Corollary 3.1.28, Y - P u Q , where p- ind P <_ n - 1, p - i n d Q _< 0. Let A c coT1, B c coT2 and A ~ B - 2~. Then A N Y c co~-~, B A Y ~ coT~ in (Y, ~-~,7-~) and hence, by Theorem 3.2.15 below, there exists a partition T I between A n Y and B N Y such t h a t T' C P so that p - i n d T ~ <_ n - 1. Moreover, by the second part of L e m m a 3.2.14 below, there exists a partition T between A and B such that T CqY C T'. Thus p - i n d ( T N Y) < n - 1. 9 T h e o r e m 3.1.31. For a d-second countable and p-normal BS (X, 7 1 , 7 2 ) , we have: ( i , j ) - i n d X <_ n if and only if X has a countable i-network Af~ such that (i, j ) - ind(j, i)- Ft" N _< n - 1 for each N c Aft.
Pro@ Since a n y / - b a s e is an /-network, by (2) of Corollary 3.1.6, it suffices to prove only t h a t if an i-second countable BS has a n / - n e t w o r k iV - {Nk }k~__l such that (i, j)- ind(j, i)- Fr Nk < n - 1 for every k - 1, oc, then (i, j)- ind X _< n. O(9
Let Y -
[_J (j, i)-Fr Nk and Z - X \ Y. It follows from Theorem 3.1.22 that k=l
(i, j ) - i n d Y _< n - 1. We shall show t h a t (i, j ) - i n d Z <_ 0. For an arbitrary point x E Z and a n / - n e i g h b o r h o o d U(x), there is Nk C Af~ such t h a t x c Nk C U(x). Since
x C X \ Y C X \ ( j , i ) - F r N k - X \ (rj clNk A r~ cl(X \ Nk)) --
-
rj int(X \ Nk) U ri int Nk,
we have x c Vk(x) -- ~-i int Nk C U(x). But (j, i)- Fr Vk(x) c (j, i)- Fr Nk and hence Z N (j, i)-Fr Vk(x) - 2~. Therefore, by the second part of Corollary 3.1.16, (i, j ) - i n d Z <_ 0. Thus it remains to use Theorem 3.1.19. 89 C o r o l l a r y 3.1.32. For a R-p-T1, d-second countable and p-normal BS (X, rl,T2), we have: p - i n d X <_ n if and only if X has a countable l-network A/'I and a countable 2-network A/'2 such that (1, 2)-ind(2, 1 ) - F r N < n - 1 for each
N o H 1 ~nd (2, 1)-i~d(1, 2 ) - F r M _< n - 1 for each M o H 2 Now we are going to give a new characterization of dimensions (i, j ) - i n d X in terms of the bitopological insertions introduced by Definition 2.4.8. D e f i n i t i o n 3.1.33. We say that a bitopology (T1,7"2) on X has the (i, j ) - l o c A-insertion properties if (rl, 7-2) has the (i, j)-(A//, A)-insertion properties for Ad - {{x}" x c X}. T h e o r e m 3.1.34. Let (X, T1,7"2) be an (i,j)-regular BS, let n denote a nonnegative integer, and let (rl, r2) has on X possess the ( i , j ) - l o c A - i n s e r t i o n properties, where A - { V c r~ " (i, j ) - ind(j, i)- Fr V _< n - 1}. Then (i, j ) - ind X _< n.
Pro@ Let x c X be any point and U(x) c ~-~ be its arbitrary neighborhood. By the (i,j)-regularity of (X, T1, ~-2) there exists a neighborhood V ( x ) c 7-~ such that Tj el V ( x ) c U(x). Following the condition, there is also a set V E ~-~ together with V ( x ) c V c Tj cl V ( x ) so t h a t rj cl V - Tj cl V(x). Since V is an /-open neighborhood of x and ( i , j ) - i n d ( j , i ) - F r V _< n 1 by (1) of Corollary 3.1.6, we obtain (i, j)- ind X < n. D
108
III. Dimension of Bitopological Spaces
C o r o l l a r y 3 . 1 . 3 5 . Let (X, 7-1,7-2) ((X, T)) be a p-regular BS and (7-1,7-2) has on X the (1, 2)-1oc(71 ~ co T2)-insertion and (2, 1)-lot(T2 n co n )-insertion properties. Then p - ind X - 0. It should be noted here that by Example 1 from [216], p - i n d X is, in general, different from 1-ind X and 2-ind X. Hence for a BS (X, T1, ~-2) it is natural to establish an interrelation between the dimension functions (i, j ) - i n d X and i-ind X. Recall that a point x0 of a BS (X, rl, 72) is a point of tangency of the topologies T1 and ~-2 if T1 and T2 coincide at x0, that is, if for each 1-open neighborhood U(xo) and each 2-open neighborhood V(xo), there exist a 2-open neighborhood V'(xo) and a 1-open neighborhood U'(xo) such that V'(xo) c U(xo) and U'(xo) c
V(xo) [as] 3.1.36. The conditions below are satisfied for a BS (X, T1, T2)" (1) If T1 c 7-2, then
Theorem
(1,2)-indX<
1-indX
and 2 - i n d X <
(2,1)-indX.
(2) If TICT2, then 1- ind X < (1, 2)- ind X. (3) IfTINT2, then ( 2 , 1 ) - i n d X _< 2 - i n d X . (4) If rl c 72, then x o - c n l (A) for each A E 2 x if and only if Xo is a point of tangency of 71 and T2; moreover, in this case 2-indz(, X <_ (1, 2)-indxo X <_ 1-indx(, X
and 2-indx(, X _< (2, 1)-indxo X _< 1-indx(, X.
(5) If T1
and 2 - i n d X < ( 2 , 1 ) - i n d X .
(6) / f T 1 < N T2, then 1-indX -(1,2)-indX
and 2 - i n d X - ( 2 , 1 ) - i n d X .
Proof. (1) It is clear that the inequality (1, 2)- ind X _< 1- ind X holds for 1- ind X = ec. Thus, assuming that 1- ind X - k < oc, we shall show that (1, 2)- ind X _< k. For k - - 1 the required inequality is obvious. Let us assume that this inequality is also correct for k _< n - 1 and prove it for k - n. Since 1 - i n d X - n, for every point x c X and its any 1-open neighborhood U(x), there exists a 1-open neighborhood V ( z ) s u c h that T1 clV(x) c U(x) and 1 - i n d ( 1 - F r V ( x ) ) _< n - 1. Clearly, T1 C T2 implies that r 2 cl V(x) C T1 cl V(x) and so (2, 1)- Fr V(x) c 1-Fr V(x). Hence, by the monotonicity of the small 1-inductive dimension function, we find that 1-ind(2, 1)-Fr V(x) <_ n - 1 and by the inductive assumption (1,2)-ind(2, 1)-Fr V(x) _< n - 1. Thus for every point x c X and its any i-open neighborhood U(x), there exists a 1-open neighborhood V(x) such that T2 cl V(x) C U(x) and (1, 2)-ind(2, 1)- Fr V(x) <_ n - 1. Hence (1, 2)- ind X _< n. Furthermore, by analogy with the above reasoning we prove that 2-ind X _< (2, 1)-ind X. Let us assume that the inequality holds for (2, 1)-ind X - k _< n - 1 and prove it for k - n. Since (2, 1 ) - i n d X - n, for every point x E X and its
3.1. Pairwise Small Inductive Dimension
109
any 2-open neighborhood U(x), there exists a 2-open neighborhood V(x) such that 7-1 cl V(x) C U(x) a n d (2, 1)-ind(1, 2)- Fr V(x) _< n - 1. But 7-1 C 7-2 implies that 7-2 cl V(x) c 7-1cl V(x) so that 2- Fr V(x) c (1, 2)- Fr V(x). Hence, by the monotonicity of the small (2, 1)-inductive dimension function, we find that (2, 1)-ind(2- Fr V(x)) <_ n - 1 and by the inductive assumption 2-ind(2- Fr V(x)) <_ n - 1. Thus for every point x E X and its any 2-open neighborhood U(x), there exists a 2-open neighborhood V(x) such that 7-2 cl V(x) C U(x) and 2-ind (2-Fr V(x)) < n - 1. Therefore 2-ind X < n. (2) By analogy with (1) let us suppose that the required inequality is correct for ( 1 , 2 ) - i n d X - k <_ n - 1 and prove it for k - n . If ( 1 , 2 ) - i n d X - n, then for every point x c X and its any 1-open neighborhood U(x), there exists a 1-open neighborhood V(x) such that 7-2 cl V(x) c U(x) and (1, 2)-ind(2, 1)- Fr V(x) <_ n - 1. But by Definition 2.2.1, 7-1 cl V(x) C 7-2cl V(x) and, therefore, 1-Fr V(x) C (2, 1)- Fr V(x). Hence, by the monotonicity of the small (1, 2)-inductive dimension function, we have (1, 2)-ind(1-Fr V(x) _< n - 1 and by the inductive assumption 1-ind(1-FrV(x) _< n - 1. Thus for every point x C X and its any 1-open neighborhood U(x), there exists a I-open neighborhood V(x) such that T1 C1 V(x)
C U(x)
and
1-ind
(1-Fr V(x)) < n - 1.
Therefore 1 - i n d X < n. (3) Let us assume that the required inequality holds for 2-ind X - k _
1.
Thus for every point x c X and its any 2-open neighborhood U(x), there exists a 2-open neighborhood V(x) such that 7-, cl V(x) C U(x) and (2, 1)-ind(1, 2)- Fr V(x) < n - 1. Therefore (2, 1)- ind X <_ n. (4) First let us prove the equivalence. If x0 c X is a tangency point of T1 and T2, then x0 c T1 clA implies x0 c T2 clA for each A c 2 x so that x0 c nl(A) for each A E 2 X. Conversely, let x 0 c n l ( A ) for each A E 2 X. If x0 is not a tangency point of 7-1 and 7-2, then there is a neighborhood U(xo) c 7-2 such that for each neighborhood V(xo) c 7-1, we have V(xo)A (X \ U(xo)) r 2~. Therefore X0 E 7 - 1 c l ( X \
U(xo) ).
gilt
Xo-~(X \ g(xo) )
-
7-2cl(X \
g(xo)))
and
so x 0 E
n l ( X \ U(xo)), which is impossible. Now, let Xo E X be a tangency point of 7-1 and 7-2. Since 7-1 C 7-2, by (1) above, it suffices to prove only that 2- indx,, X _< (1, 2)- indxl, X and (2, 1)- indx,, X <_ 1- indx,, X.
110
III. Dimension of Bitopological Spaces
It is obvious t h a t the inequality 2 - i n d x , , X < ( 1 , 2 ) - i n d x o X holds for (1, 2)-indx,, X - oc. Thus assuming that (1, 2)-indx,, X - k < oc, we shall show that 2-indx,, X < k. For k - - 1 this inequality is obviously fulfilled. Let us suppose that it also holds for k 5 n - 1 and prove it for k - n. Let U(xo) c r2 be any neighborhood. Since x0 is a tangency point of rl and r2, there is a neighborhood W ( x o ) c rl such that W ( x o ) c V(xo) and since (1,2)-indx,, X - n, there is a neighborhood V(xo) c rl such that
r 2 c l V ( x o ) C W ( x o ) and (1, 2)- ind(2, 1 ) - F r V ( x 0 ) _< n -
1.
By (2) of Proposition 3.1.4, we have (1, 2)-ind2- Fr V(x0) _< (1, 2)-ind(2, 1)-Fr V(x0) _< n and, by induction, 2 - i n d 2 - F r V(xo) _< n V ( x o ) E T 1 C T 2 such that
1
1. Thus for each U(xo) c r2, there is
r2 cl V(xo) C U(xo) and 2- ind 2- Fr V(xo) < n - 1. Hence 2-indxo X < n, that is, 2-indx0 X < (1, 2)-indxo X. Finally, let us again suppose that the considered inequality is correct for 1- indx,, X - k < n - 1 and prove it for k - n. Let U(xo) c r2 be any neighborhood. Since x0 is a tangency point of rl and r2, there is a neighborhood W ( x o ) c rl such t h a t W ( x o ) c U(xo). But 1-indxo X - n and, therefore, there is a neighborhood V(x0) c rl such t h a t T1
cl V(xo) c W ( x o ) and 1-ind 1- Fr V(xo) <_ n - 1.
By monotonicity, we have 1-ind(1, 2)- Fr V(xo) _< 1-ind 1- Fr V(xo) <_ n - 1. On the other hand, by induction, (2, 1)-ind(1, 2)- Fr V(xo) _< n - 1. Thus for each g ( x o ) E r2, there is V(xo) E rl C T2 such that T1 C1 V ( x o )
C U(xo) and
(2, 1)-ind(1, 2)- Fr V(xo) < n - 1.
Hence (2, 1)- indxo X _< n, that is, (2, 1)- indxo X _< 1- indxo X. The rest is obvious.
D
3.2. Pairwise Large Inductive D i m e n s i o n D e f i n i t i o n 3.2.1. Let ( A , B ) be a pair of subsets of a BS (X, r l , r 2 ) such t h a t A E co rj, B E co ri and A N B = ;~. Then we say that a p-closed set T is a partition, corresponding to (A, B), if X \ T = H1 U/-/2, Hi c ri \ { 2~}, H1 O/-/2 = 2~ and A c Hi, B c Hj. R e m a r k 3.2.2. In the sequel, without loss of generality, we shall sometimes consider a p a i r ( A , B ) where A E CUrl, B c cur2 and A N B - ~. As in Remark 3.1.2, it is easy to verify that in a BS (X, rl, r2) for such a pair (A, B) the following conditions are satisfied:
3.2. Pairwise Large Inductive Dimension
111
(1) If there exists a 2-open neighborhood U(A) (1-open neighborhood U(B)) such t h a t 7-1CI U(A) C X \ B
(7-2cl U(B) c X \ A ),
then (1, 2)- Fr U(A) ((2, 1)- Fr U(B)) is the partition, corresponding to (A, B) in the sense of Definition 3.2.1. (2) If T is a partition, corresponding to (A, B) in the sense of Definition 3.2.1, then (j, i)- Fr H~ C T. D e f i n i t i o n 3.2.3. Let (X, 7-1,7-2) be a BS and n denote a nonnegative integer. We say t h a t (1) ( i , j ) - I n d X - - 1 <---> X - ~. (2) (i, j ) - I n d X < n if to every pair (A, B), where A 9 co rj, B 9 co 7-i and A N B - ~, there corresponds a partition T such t h a t (i, j)- Ind T < n - 1. (3) (i, j)- Ind X - n if (i, j)- Ind X < n and the inequality (i, j)- Ind X < n - 1 does not hold. (4) (i, j)- Ind X - ~ if the inequality (i, j)- Ind X _< n does not hold for any n. As usual, p - I n d X < n ,e---->, ((1, 2 ) - I n d X _< n A (2, 1)-Ind X <_ n). L e m m a 3 . 2 . 4 . Let T be a partition in a BS (X, 7-1,7-2) which corresponds to a pair of disjoint sets (A, B) where A 9 co 7-1, B 9 co 7-2. If Y c X is a p-closed set such that A H Y # ~ # B N Y, then the set T' - T fq Y is the partition in the
Proof. By assumption, X \ T - H l t 0 H 2 , where Hi 9 7-~ \{2~}, H I N H 2 A c H2, and B c H1. Hence T - X \ (H1 O H2). It is obvious t h a t the set T' - T N Y is the required partition since Y \ T'-
- 25,
Y \ ( X \ (H1 U H2)) - (Y \ (X \ H1)) U (Y \ (X \ H2)) (YNH1) U (YAH2)-
H~ UH~,
where L e m m a 3 . 2 . 5 . If (A', B') is a pair of disjoint sets of a p-closed BsS (Y, w~, w~) of a BS (X, Wl, r2) such that A' c co w~ and B' c co 7~, then there exists a pair of disjoint sets ( A , B ) in (X, 7-1,7-2) such that A c co7-1, B c co7-2, A N Y = A', and BNY=B'.
Proof. The fact t h a t Y is p-closed in (X, 7-1,7-2) implies Y = Wl cl Y N w2 cl Y. Let / A ' 9 CO7-1,
B
~ co
l I 7-2/ in (Y, rl,r2) and
A I
N B l --~.
T h e n there are d 9 co
" %" ) and B 9 co r~" in (r2 cl Y, 7-1'" , %' " ') 7-1" in (7-1 clY, 7-1,
such t h a t A N Y = A' and B H Y = B'. It is evident t h a t A 9 co 7-1, B 9 co 7-2 and AHB=~. 89 Corollary
3 . 2 . 6 . A p-closed BsS of a p-normal BS is also p-normal.
112
III. Dimension of Bitopological Spaces
P r o p o s i t i o n 3.2.7. The statements below hold for a BS (X, 7"1,7"2)" (1) If ( 1 , 2 ) - I n d X or (2, 1 ) - I n d X is finite, then (X, 7"1,7"2) i8 p-normal. (2) For every p-closed BsS (Y, 7"~, 7"~) of ( X , 7"1,7"2), we have
( i , j ) - I n d Y <_ ( i , j ) - I n d X . (3) /f (X, 7"1,7"2) is a j-T1 BS, then ( i , j ) - i n d X <_ ( i , j ) - I n d X .
Proof. Assertion (1) is obvious. (2) It suffices to prove that (i, j ) - I n d X = k implies (i, j ) - I n d Y _< k. This inequality is correct for k - - 1 , k - oc. Let us assume that it is correct for k <_ n - 1 and prove it for k - n. If (A', B') is a pair of disjoint sets such that A' c co 7"~ and B' c co 7"~, then by Lemma 3.2.5, there exists a pair of disjoint sets (A, B) where Acco7"j,
B c c o 7 " i and A N Y - A '
BAY-B'.
Since (i, j)- Ind X - n, there exists a partition T for (A, B) such that (i, j)- Ind T _< n - 1. But by Lemma 3.2.4, T' - r A Y is the partition in (Y, 7"~,7"~), corresponding to the pair (A', B'). Hence, by the inductive assumption, (i, j)- Ind T' <_ n - 1 since T' is p-closed in T and so (i, j ) - I n d Y <_ n. The condition (3) is clear. [3 C o r o l l a r y 3.2.8. We have p - I n d Y < p - I n d X for every p-closed BsS (Y, 7"~,7"~) of a BS (X, 7"1,7"2) and, therefore, if Y E co 7"1 U CO 7"2, then p - Ind Y < p - I n d X . Moreover, if (X, 7.1,7.2) is R-p-T1, then p - i n d X _< p - I n d X . C o r o l l a r y 3.2.9. Let (X, T1, ";-2) be a BS and n denote a nonnegative integer. Then ( i , j ) - I n d X <_ n if and only if for any j-closed set F and any i-neighborhood U(F) there exists an i-open neighborhood V ( F ) such that 7.j cl V ( F ) C U(F) and (i, j)- Ind(j, i)- Fr V ( F ) < n - 1. The proof of this corollary repeats in the main that of (1) of Corollary 3.1.6, taking into account (2) of Remark 3.2.2 and (2) of Proposition 3.2.7. C o r o l l a r y 3.2.10. If for a BS (X, 7.1,7.2), we have
(i,j)-IndX-n
(p-IndX-n),
n>_l,
then for each k - O, n - 1 the BS (X, T1,7"2) contains a p-closed BsS (Y, 7"~,7"~) such that (i, j)- Ind Y - k (p- Ind Y - k). Proof. The proof is similar to the proof of Proposition 3.1.8 with Corollary 3.2.9 taken into account. D P r o p o s i t i o n 3.2.11. The following equivalences hold for every BS (X, 7"1,7"2)" (1, 2 ) - I n d X - 0 ~
(2, 1 ) - I n d X - 0 ~
p - I n d X - 0.
Proof. Let ( 1 , 2 ) - I n d X = 0, F E COT1 be any set and U(F) be its any 2-open neighborhood. Then X \ U(F) c X \ F. Hence, there exists a set V E T 1 A CO T2 such that X \ U(F) C V c X \ F and, therefore, F C X \ V c U(F) where X \ V E 7.2 • co 7.1. Thus (2, 1)- Ind X = 0. The inverse implication can be proved in a similar manner. D
3.2. Pairwise Large Inductive Dimension
113
Hence by (3) of Proposition 3.2.7 for every R-p-T1 BS (X, 71,72), we have (1, 2 ) - i n d X -
0 ,e---->, (2, 1 ) - i n d X ,e----f, p_ ind X -
0.
Theorem 3 . 2 . 1 2 . If for a d-second countable BS (X, 71,72) the equality p - ind X - 0 holds, then for every pair of disjoint sets A c co 71 and B c co 72 the empty set is a partition between them, that is, there exists a set V c 72 N co 7-1
such that A c V, B c X \ V and so p - I n d X - O. Proof. Since p- ind X - 0, for each point x c X , there exists a set U(x) c 71Gco 72 or a set V(x) E r2 n cot1 such t h a t A n u(.)
-
e,
o,. B n v ( . )
-
It is evident t h a t for the p-open covering b/ = {{U(x)}, { V ( x ) } I x e x there is a countable subfamily b/'-
{{U(xk)"
k-l,
oo},{V(xv)"
of X,
p-l,~}}
which is also a > o p e n covering of X. T h e sets Uk = U(xk) \ U V(Xp) are 1-open p
h i " - {{Uk " k -
l, oc}, {Vp " p -
l, e c } }
is also a p-open covering of X. It is obvious t h a t UkNA-2~
for each k - l ,
oc and Vv N B - 2 ~
for each p - l ,
oc.
Let V - U{Vp" A N Vp -fl 2~}. T h e n A C V. Since B N Vp - 2~ for each p - 1, oc, (x)
we have B n ( U vp) - 2~ and, therefore, p=l (x)
Box\
Uv,,cx\U{v,,
9
G
p=l
Corollary 3.2.13. If (X, 71,72) is R-p-T1 and d-second countable, then p - I n d X - 0 <--> p - i n d X - 0.
Proof. By (3) of Proposition 3.2.7, p - I n d X - 0 ----> p - i n d X - 0. T h e inverse implication is an i m m e d i a t e consequence of T h e o r e m 3.2.12 and Definition 3.2.3. [E] L e m m a 3 . 2 . 1 4 . Let (Y, r~, r~) be a BsS of a hereditarilyp-normal BS (X,rl,r2) and A r cot1, B c cot2, with A n B = ~. Then for every partition T' in the BsS (Y, w[, T~) between Y N wl cl V1 and Y AT2 cl V2, where Pl and P 2 age 2-open and l-open subsets of X , respectively, such that A c 171, B C 172 and 71 cl Yl N72 cl V2 = ~ , there exists a partition T in X between A and B which satisfies the inclusion TNYCT'. If (Y, r~, 7~) is a p-closed BsS of a hereditarily p-normal BS (X, 71,72) and A C co 71, B C co 72, with A N B - ;g, then for every partition T' in (Y, 7~, 7~) between A N Y and B N Y, there exists a partition T in X between A and B such that T N Y C T'.
114
III. Dimension of Bitopological Spaces
Proof. By Corollaries 0.2.3 and 0.1.8, there exist V1 E ~-2, 1/2 r T1 such t h a t A c V1, B c V2 and 7"1 cl V1 n 7-2 cl V2 - ~ . O n the other hand since (Y, ~-~, ~-~) is p-normal, there are V1~ c T~, V~ c ~-~ such t h a t
Y N n cl Vl C V[, Y N~2cl V2 c V~ and V/ N VJ = ~. Let T' = Y \ (V{ U V~). It is obvious t h a t A N "r2 cl V~ = 2~ = B AT-1 cl VI'.
(1)
Let us consider the BsS (V1~ U V~, ~ ' , ~-~') where V{ E T~', V~ E r['. It clearly follows t h a t V1~ N Ts cl V~ - 2~ - V~ n rI' clVl t. (2) Therefore, by (1) and (2), we have Tl c l ( A U V;) n T2 c l ( B U VJ) = (A U T1 c l V ; ) n ( B U T2C1V~) =
and by T h e o r e m 0.2.2, there exist HI c T2, /-/2 c T1 such t h a t A U 17[ c HI, B U V~ c /-/2 and H i n / - / 2 z ~ . The set T = X \ (H1 U/-/2) is a partition in X between A and B such t h a t
T N Y = Y \ (H1 UH2) C Y \ (VI~U V~)= T' so t h a t we have proved the first part. N o w let (Y - T1 cl Y O ~-2 el Y, T~, ~-~) be a p-closed BsS of (X, T1, ~-2) and T' - Y \ (V[ U V~) be any partition between A N Y r co ~-~ and B N Y c co 7-~ so that A N Y C V; E T~, B N Y C V~ C T~ and V; N V~ = 2;. Let V1" e ~-~' in (~-2cl Y, ~-;', T~') and V~" r T~" in (T1 C1 Y, T;tt, 7"~tt)such t h a t A n 72 el Y c V~', B N 71 el Y c V~". If V1 E T2, V2 c ~-1 are sets for which V1 n 7-2 cl Y = V1" and 1/2 N f l c1Y = V~", then
A N (T2 c l Y \ V 1 ) = A N (T2 e l Y \ V~') = 2~ and B n (T1 c1Y \ V2) = B n (T1 c1Y \ V~tt) = ~ .
Thus
A c X \ (r2cl Y \ V1) = U c r2, B c X \ (rl cl Y \ V2) = V c rl. Since A N B = 2~, there exist G E 72, H E T1 such t h a t A c G, B c H and G N H = ~. Let M = U O G r 72, N = V O H c 71. T h e n A c M , B c N and by (4) of Proposition 0.1.7, there exist Q E T2, P c T1 such t h a t A c Q c T1 cl Q c U, B c P c T2 cl P c V and rl cl Q N 72 cl P = Z. It is evident t h a t TlclQ o Y C UNY
= Y\
(T2 c l Y \ V 1 ) = V;,
T2 c l P n Y C V n Y = Y \ (T1 c l Y \ V2) = V~ and so T ~ = Y \ (VII U V~) is the partition between T1 cl Q N Y and 7-2 cl P N Y. Therefore, there exists a partition T between A and B in X such t h a t T N Y C T ~. D
3.2. Pairwise Large Inductive Dimension
115
T h e o r e m 3.2.15. Let ( X , 7-1,T2) be a R-p-T1, d-second countable and p-normal BS, and Y C X , p - I n d Y - 0(<--> p - i n d Y - 0). Then for every pair of disjoint sets A c co T1 and B E co 7-2, there exists a partition T between A and B such that T N Y - 2~. Proof. Let us consider V1 c ~-2, I/2 c T1 such that A c V1, B c 1/2 and T1 cl V1 N ~-2 cl V1 - 2~. Then by Theorem 3.2.12 the empty set is a partition in Y between Y N 7-1 cl V1 and Y N T2 cl 1/72. Thus it remains to use Lemma 3.2.14 since by Corollary 0.1.13, (X, T1, T2) is p-perfectly normal and hence, be remark before Proposition 0.1.12, it is hereditarily p-normal. []
Of fundamental importance is
T h e o r e m 3.2.16 (A G e n e r a l i z e d Version of Dowker's A d d i t i o n Theorem). A s s u m e that for a hereditarily p-normal BS (X, T1,7"2) We have" X,~ c T1 N 7-2 f o r every m -
1,oc, X,~+I c
(i,j)-Inld(X,~ \ X,~+I) <_ n for each m -
X~,
X1 -
X
and
oo n x,~
m=l
1, oc, then ( i , j ) - I n d X
-
;g.
If
< n.
Proof. The theorem holds for n - - 1 . Let us assume that it holds for n - 1 and prove it for n. If A c coTj, B c coTi and A N B -- 2~, then by Corollary 0.1.8, there exist U0 c 7-~, V0 c Tj such that A c U0, B c V0 and Ty cl U0 N ~-i cl V0 - 2~. Let Dm - X,~ \ Xm+l for every m - 1, oc. We begin by constructing three mutually disjoint sets U1, C1, V1, where U1 is/-open in X1 - X, V1 is j-open in X1 - X,
C1 C D1 is p-closed in D1,
(i, j ) - I n d C1 < n -
1
and D1 C U 1 U C 1UV1,
Tj clU0 c U1, T~clV0 C 1/1, wjclU1 n ~-~clV1NX2 = ~.
Clearly, the sets rj cl U0 N D1 and r~ cl V0 5 D1 are disjoint in (D1,7-{, T~). Since (i, j ) - I n d D 1 < n, there exists a partition C1 in D1, corresponding to the pair (rj cl U0 N D1, ri cl V0 n D1) such that D1 \ C1 = (71 U H1, where (71 is/-open in D1, H1 is j-open in D1, 7-jclU0 ND1 C (71, TiclV0 N D 1 C H1 and G 1 N H 1 = 2~. Since D1 is simultaneously 1- and 2-closed in X1 = X and C1 is p-closed in D1, we conclude that C1 is also p-closed in X1 = X. On the other hand, D1 \ C1 = (X1 \ C 1 ) n D1 so that D1 \ C1 is simultaneously 1- and 2-closed in X 1 \ C1. But D1 \ C1 = G1 U H1, where G1 n H1 = 2~ and, hence, (71 is simultaneously/-open and j-closed in D1 \ C1, H1 is simultaneously j-open and /-closed in D1 \ C1. Therefore (71 is j-closed in X1 \ C1 and H1 is /-closed in X1 \ C1. Let A1 - (Tj el U0 U (71)n (X 1 \ C1) and B 1 - (T i C1V0 U H 1 ) n (X 1 \ C1).
It is obvious that (A1,B1) is a pair of disjoint respectively j-closed and/-closed subsets of X1 \ C1 which is p-normal because (X, T1, T2) is hereditarily p-normal. Therefore, there exists a n / - o p e n set U 1 and a j-open set V1 in (X 1 \ C1, T~t, T~r
116
III. Dimension of Bitopological Spaces
such t h a t A1 c U1, B1 C V1 and ~_~Icl U1 n 7-:I c1 V1 = ~ so t h a t 7-jclV 1 nT-~clV1 n (X 1 \ C 1) = ~.
Thus Tj cl U1 n ri clV1 n (Xl \ C1) :
and, as a result, Tj cl U1 N 7-~cl V1 C C1. Since C1 c D1 = X1 \ X2, we obtain rj cl U1 N ri cl V1 n X2 = ~. Furthermore, C1 -- D1 \ (G1 U H1) = (D1 \ G 1 ) n (D1 \ HI), where D 1 \ G 1 c co T~, D1 \ H1 C co Tj, and, hence, D1 \ G1 c co Ti, D1 \ H1 E co Tj since D1 E co T1 n co 7-2. Now, we have X1 \ C 1 - - X l
\ ((D1 \ G 1 ) n (D 1 \ H 1 ) ) -
- - ( X l \ (D1 \ G1)) U (X 1 \ (D 1 \ H1) ) - ( ( X 1 \ D1) U (~1) U ((X 1 \ D1) U H1). Since U1 C X1 \ C1 and U1 n H1 = ~, we obtain U1 C (X1 \ D1) U G1. But the fact that U1 is /-open in X1 \ C1 implies that U1 is /-open in X1 = X because (X1 \ D1) U G1 is/-open in X1 = X. In a similar manner it can be proved that V1 is j-open in X1 = X. Hence D1 = G1 U C1 U H1 and G1 C A1 c U1, H1 C B1 c V1
imply D1 c U1 U C1 U V1. Moreover, Tj cl U0 N D1 C G1 implies that Tj cI Uo N CI = K~ a n d s o
7j cl Uo = Tj cI Uo N (XI \ C1) c A1 c U1.
Similarly, ~-i cl 170 c V1. Thus for m = 1, we have constructed three mutually disjoint sets Urn, Cm, V.~ satisfying the following conditions: (l~ (2 ~ (3 ~ (4 ~
(5~ (6 ~
U.~ and V.~ are contained in X.~ and are respectively i- and j-open in X. C.~ is contained in Dm - X.~ \ Xm+l and is p-closed in D.~. c u u Tj cl Um n 7-~cl V,~ n X m + l - ~J.
clum_l a x m c
clvm_l
c
(i, j)- Ind Cm <_ n - 1.
Assume that we have constructed three mutually disjoint sets satisfying the above conditions for 1, 2 , . . . , m - 1 and let us construct them for m. Condition (4~ implies that Tj cl Urn-1 O ~-~cl V.~-I n Xm - ~. Recalling that Dm c X.~, that (Tj cl U.~_I O Din, "q C1V.~-I n D.~) is a pair of disjoint j- and/-closed sets in Din, and following the inequality ( i , j ) - i n d D ~ <_ n, there exists, in D.~, a partition C.~ corresponding to the pair above such that (i, j ) - I n d C.~ _< n - 1. To the partition C.~, there also correspond disjoint i- and j-open sets Gm and H.~ such that D.~ = G m U C.~ U H.~ and Tj cl U,~_ 1 n D.~ c G.~, ~-~cl V.~_ 1 n D.~ c H.~. Since C.~ is p-closed in D.~, and D.~ is simultaneously 1- and 2-closed in X.~, Cm is also p-closed in Xm. The sets G.~ and H.~ are j- and/-closed in D m \ C~, and since D.~ \ C.~ = (X.~ \ C.~) N D.~ and D.~ is simultaneously 1- and 2-closed in X.~, Dm \ C.~ is simultaneously 1-closed and 2-closed in Xm \ C.~. Hence G.~
3.2. Pairwise Large Inductive Dimension
117
is j-closed and Hrn is/-closed in Xrn \ Crn. Furthermore, it is easy to verify that the pair (Am, Brn), where
Am - (Tj cl Um-1U Grn) N (Xm \ C r n ) -
(7-j cl Um-1N (Xm \ Cm)) u am,
Bm - (Ti cl Vm_ 1 U H m ) N (Xrn \ Urn) - (7-/cl Vm_ 1 n ( a m \ Cm)) O H m is the pair of disjoint respectively j- and/-closed sets in X m \ C~. Since Xrn \ C~ is p-normal, there exist, in Xrn \ Cm, disjoint respectively i- and j-open sets Um and V,~ such that
Arn c U.~, B.~ C Vm and "rj cl Urn n "ri cl V.~ N ( X ~ \ Crn) -- ~ so that rj cl Urn N ~-i cl Vrn C Crn. Since C.~ c D.~ = Xm \ Xrn+l, we obtain ~j cl U,~ a r~ cl E~ a Xm+l = ~. The set Xrn \ C.~ is p-open in X.~, and since Xrn is simultaneously 1- and 2-open in X, Xrn \ C.~ is p-open in X so that X.~ \ C.~ = O1 U 02, where Oi c ri. Therefore, we obtain Urn c Oi, V.~ c Oj and thus Urn c ri, Vrn E rj. Furthermore, D.~ = G.~ U C.~ U H.~, and since Grn c Arn c U.~, H.~ c Brn c Vrn, the inclusion D.~ c U.~ U C.~ U Vm is correct. Moreover,
Tj C1 Urn-1 n Drn C Grn implies rj cl Urn_ 1 n Crn and, therefore,
Tj cl Urn-1 n Xrn - Tj cl Urn-1 n (Xrn \ Crn) C Arn c Urn. Similarly, ri cl Vrn-1NXrn C Vm. We have thus constructed three mutually disjoint sets Urn, Cm, Vm, satisfying conditions (l~176 oo
Let us now assume that C -
U Crn and prove that (i, j ) - I n d C < n - 1. Let r~--1
oo
Zrn -
U Ck for each m -
k=m
1, oc; in particular, Z1 - C .
Zrn-bl C Zm,
Also,
Z m \ Z m + l - Cm
so that
(i, j)- Ind(Zm \ Zrn+l ) ~ T t - 1. Since Xm N C - Zm, the set Zrn is simultaneously 1- and 2-open in C for each (2<) m
--
1, oc and n
Zm -- 2~. Thus by the inductive assumption (i, j)- Ind C <_ n - 1.
rn=l
Finally, we are to show that C contains a partition, corresponding to the pair
(A,B).
Let U -
c~
oG
U Urn, V -
U Vm. It is clear that U c ~-~ and V c rj.
m--1
m---1 oo
Condition (5 ~
implies that A c U0 c U, B c 170 c V. Also, X -
U Drn and, rn=l
u s i n g D m C Urn U Cm U Vm, we have X - U U C U V. Let us prove that U N V - 2~. Clearly, it suffices to show that Us N Vt - 2~ Ks N Ut. Indeed, s < t implies
Us n Vt c Us n X t C (Us n X s + l ) n X t
C gs+l n X t
and so Us N Vt C gs+l n l/t. Continuing this reasoning, we obtain Us N Vt C U t n V t - 2 ~ . By analogy, V s N U t - ~ s o t h a t UNV--2~. LetT-X\(UUV).
118
III. Dimension of Bitopological Spaces
It is obvious that T is the partition, corresponding to (A, B). Moreover, T C C is p-closed in C and, therefore, (i, j ) - I n d T < n - 1. [] C o r o l l a r y 3.2.17. Let there exist for a hereditarily p-normal BS (X, 7-1,7-2) an X,~ C 7-1 N ~-2 for every m -- 1, o0, such that X,~+I c X,~, X1 - X , and O0
n
xm - z. Ifp-Ind(Xm
\ Xm+t) <_ n for each m - 1, oo, then p - I n d X _< n.
rn= 1
C o r o l l a r y 3.2.18. Let (X, 7.1,7.2) be a hereditarily p-normal BS and A c co 7.1 N co 7.2. Then the following statements hold: (1) If ( i , j ) - I n d A < n and ( i , j ) - I n d ( X \ A) < n, then ( i , j ) - I n d X (2) /jr p - Ind A < n and p - I n d ( X \ A) < n, then p - Ind X < n. Proof. (1) Write X1 - X, X2 - X \ A , rem 3.2.16. Assertion (2) is obvious.
X3 - X4
....
< n.
0 and apply TheoD
The following fact is equivalent to the one proved above. T h e o r e m 3.2.19 9 Let (X, rl, r2) be a hereditarily p-normal BS and {Din} m~176 = l be a disjoint sequence of sets covering X such that Fs - O D,~ c co 7-1 n CO 7"2 m<s
for each s -
1, oo. Then the following conditions are satisfied:
(1) If ( i , j ) - I n d D , ~ <_ n for each m - 1, oc, then ( i , j ) - I n d X (2) If p - I n d D,,~ <_ n for each m - 1, oo, then p - I n d X < n.
<_ n.
Proof. If in Theorem 3.2.16 it is assumed that D,~ - Xm \ Xm+l, then we obtain Theorem 3.2.19. To prove the converse, it is enough to take X~ - X \ U Din. D m<8
T h e o r e m 3.2.20. If (X, 7-1,7-2) i8 a hereditarily p-normal BS and X - P u Q , where ( i , j ) - I n d P <_ n, ( i , j ) - I n d Q <_ O, then ( i , j ) - I n d X <_ n + 1. Proof. Let B be any j-closed subset of X and U ( B ) be its a n y / - o p e n neighborhood. Then there exist a j-open set V and a n / - o p e n set W such that A - X \ U ( B ) C V, B C W and 7.i cl V N Tj cl W - Z. It is obvious that Q \ (7.i cl V N Q) is a n / - o p e n neighborhood of the j-closed set rj cl W n Q in (Q, r~, r~). Since (i, j)- Ind Q _< 0, there exists a simultaneously/-open and j-closed set U in Q such that
- dwnQ
c u c Q \
d v n Q).
Let us consider the sets B U U and A U (Q \ U) and show that
Since 7j cl(B U U) - B U 7y cl U and 7~ cl (A U (Q \ U)) - A u 7.~cl(Q \ U), the proof of (,) reduces to the following facts"
d u n (Q \ U) (b) B N ( Q \ U ) - 2 ~ . (c) 7j cl U N A - ;g.
0 @
@ ;> q;
q; b.O r,,-i
d o4 /
II
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II
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II ~
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r
II ~
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u~ i =~c~ c
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II 1'~
9-
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120
III. Dimension of Bitopological Spaces
Proof. Let A c co Tj, B E co T~ and A A B - 2~. We shall show that to the pair (A, B) there corresponds an e m p t y partition. Since (i, j)- Ind X1 - 0, there exist a j-closed a n d / - o p e n set A1 and an/-closed and j-open set B1 in X1 such that
A n X~ c A1,
A1 U B1 - X1 and A1 G B1 - 2~.
B DI X 1 c B1,
Since X1 E co7-1ACO72, we have A1 E coTj and B1 E coT/. Hence (AUA1, B U B 1 ) is a pair of disjoint j-closed a n d / - c l o s e d sets in X. Since (X, 7"1,1-2) is p-normal, there exist a n / - o p e n set G1 and a j-open set H1 such that A U A1 C G1,
B LJ B1 C H1
and Tj cl G1 N Ti cl H1 -- ~ .
It is obvious t h a t X 1 C G1 U H1. If in our previous reasoning we replace the pair ( A , B ) by the pair (rj clG1, ri clH1), we can construct an /-open set G2 and a j-open set H2 such that X2 c G2 LJ H2, Tj cl G2 A Ti clH2 -- ~ and Tj cl G1 C G2, Ti clH1 C/-/2. Proceeding with our reasoning in a likewise manner, we obtain two increasing sequences of i- and j-open sets in X: G1 c G2 C " " C Gm C "" ,
Tj cl Grn C Grn+ 1,
H1cH2c...cH~
zi c l H m C Hm+l,
c...,
X 1 U X 2 U . . . U X m C Grn U H,~,
LetG-
oo
(2o
U G,~,H-
U Hm. S i n c e X -
m=l
7j cl Gm • 7-~cl Hm - ;g. (ND
UXm, wehaveX-GUH.
rn=l
On
m=l
the other hand, G,~H,~-2~,
G,~cGm+I
and H m c H , ~ + I
for each m - l ,
oo
imply G N H - o. It is obvious t h a t A C G and B C H. Thus (i, j)- Ind X - 0 .
[-]
C o r o l l a r y 3.2.24. Let (X, T1, w2) be a p-normal BS and {Xm} ~.~=1 be a seoo
quence of subsets of X such that X -
U x m , x m cCOWlnCOT2, a n d p - I n d X m - O m=l
for each m -
1, oc. Then p - Ind X -
0. oo
Note t h a t Theorem 3.2.23 also holds if X,~ -
U Fm where F ~ E co 7-1 ACO n ,
7-2
n=l
and (i, j ) - I n d F ~ - 0
for each m -
1, ec, n -
1, oc. oo
C o r o l l a r y 3.2.25. Let (X, ~-1, ~-2) be a p-normal BS and Y -
U F,~, where rn=l
frn C c o
7-1 N c o 7-2
for each m -
1, oc. Then the following implications hold:
(1) ( i , j ) - I n d X - 0 ~ (i,j)-IndY (2) p - Ind X - 0 ~ p - Ind Y _< 0.
<_ O.
oo
Proof. (1) By assumption Y -
U Fro, where F,~ c co T1Aco 72 for each m - 1, oc. rn=l
Hence every F,~ is p-closed in X and by (2) of Proposition 3.2.7, (i, j)- Ind Fm _< 0 for each m - 1, oc. Therefore, by Theorem 3.2.23, (i, j ) - I n d Y _< 0. D Moreover, we obtain
3.2. Pairwise Large Inductive Dimension
Theorem
121
If a d-second countable and hereditarily p-normal BS of p-closed sets,
3.2.26.
( X , 7-1,72) can be represented as a union of a sequence F1, F 2 , . . .
where (1, 2 ) - I n d F ~ - 0 ( ~
for each n -
(2, 1 ) - I n d F ~ - 0 ~
p-IndF~
- 0)
1, oc, then (1, 2 ) - I n d X - 0 ( ~
(2, 1 ) - I n d X - 0 ~
p-IndX
- 0).
Proof. Let A c co T1, B C co T2 and A N B - ~. We shM1 prove that, there exist G c T2, H C T1 such t h a t AcG,
BcH,
GNH-;g
and G U H - X
(1)
and so the e m p t y set is a partition between A and B. By Corollary 0.1.8, there exist U0 c 72, V0 c 7-1 s u c h t h a t A c U0,
B c V0 and 7-1 cIU0 NT-2clV0 - Z .
(2)
We shall define inductively two sequences of 2-open and 1-open sets U0, U1,. 9 9 and V0, V1,..., respectively, satisfying for each k - 0, oc the following conditions" U k -1 C U k ,
Vk _ l C Vk if k _> 1, and 7-1clUk
N T-2 c l V k
--
;g ,
Fk C Uk U Vk, where F0 - 2~.
(3) (4)
Clearly, the sets U0 and V0, defined above, satisfy both conditions for k - 0. Assume t h a t the sets Uk and Vk, satisfying (3) and (4), are defined for k < p. If Fp -- T 1 c1Fp N 7-2 cl Fp, then the sets
7-2ClVp_1 n F p C c o
7-1 cl Up_ 1 n f p C co 7-1' a n d
7"2'
in (Fp,7-~,7-~) are disjoint. Since ( 1 , 2 ) - I n d F p - 0 by virtue of T h e o r e m 3.2.12, there exists a subset V c 7-~ n co 7-~ such t h a t 7-1cI Up_I N Fp C V
and
7-2 cl Vp_l N rp C Fp \ V.
Since 7-1 cl V c 7-1 CI ~ ,
7-2 cl(Fp \ V) c 7-2 cl Yp and Fp - 7-1 c 1 F p n 7-2 cl Fp,
w e have 7-1 C1V n T2 c l ( F p \ V ) - ~ .
Let C -- 7-1 cl V \ 7-2 cl Vp_l and D - 7-2 cl(Fp \ V) \ 7"1 cl Up_ 1. Then c v D) u (C n
cl D) -
Therefore, by T h e o r e m 0.2.2, there exist U c 7-2, W c 7-1 s u c h t h a t C c U, D c W and U N W - 2 ~ . It is evident t h a t V N W - - 2 J , ( F p \ V ) N U - ~ and, consequently, 71CI V N W - ~ , 7-2 cl(Fp \ V ) N U - ;~. Let (I)l -- 7-1c1Up-1 U ( 7 - 1 c l V \ H/r),
(I) 2 - 7-2c1Up- 1 U (7-2cl(Fp \ V ) \ U ) .
Then ~i c coT-i and ~ 1 N ~ 2 -- 2~. Since (X, 7-1,7-2) is p-normal, there exist Up c 7-2, Up C 7-1 such t h a t ( ~ 1 C Up~ ~2 C Up and 7-1c1Up n T-2 cl Vp - (2~.
122
III. Dimension of Bitopological Spaces
T h e sets {Up}~_ 1 and {Vp}p~__l satisfy (3) and (4) for k - p . T h u s the construction of the sequences U0, U 1 , . . . and V0, V1,... is completed. It follows from (2), (3) and (4) t h a t the unions G -
oo
o(3
U Up and H -
U Vp satisfy (1).
p=l
D
p=l
Corollary 3 . 2 . 2 7 . I f a d-second countable and hereditarily p-normal BS (X, T1 < ~-2) can be represented as a union of a sequence F 1 , F 2 , . . . of i-closed sets, where p - Ind F~ - 0 for every n - 1, oc, then p - Ind X - 0. C]
Proof. T h e proof is obvious because co rl c co 72 c p - d l ( X ) .
Corollary 3.2.28. If a R-p -T1, d-second countable, and p-normal BS ( X , T 1 , T 2 ) can be represented as a union of a sequence F 1 , F 2 , . . . of p-closed sets where p - ind F~ - 0 for every n - 1, oc, then p - ind X - 0. Proof. By Corollary 3.2.13, p - I n d F ~ = 0 for each n = 1, oc. On the other hand, by Corollary 0.1.13, (X, 7"1,7-2) is p-perfectly normal and, hence, it is hereditarily p-normal. Therefore, by T h e o r e m 3.2.26, p - I n d X = 0. It remains to use once more Corollary 3.2.13. [3
Corollary 3.2.29. If a (R-p-T1) d-second countable, and hereditarily p-normal (p-normal) BS (X, T1, T2) can be represented as a union of a sequence F 1 , F 2 , . . . , where every Fk is a countable union of p-closed sets, that is, Fk = O(3
U F~, and p - I n d F k = 0 ( p - i n d F k = 0) for each k = 1, oc, then p - I n d X
= 0
n--1
( p - i n d X = 0).
Pro@
x-u
By Corollary 3.2.8, p - I n d F ) O(3
= 0 for every k = 1, co, n = 1, ec and
OO
ur .
k=l n=l
For p - i n d X it remains to use (2) of Proposition 3.1.4 in conjunction with Corollary 3.2.13. D
Corollary 3.2.30.
If a R - p - T 1 ,
d-second countable,
and p-normal BS
( X , T1 < 7-2) can be represented as a union of two BsS's Y and Z, where p - I n d Y -
p - Ind Z - 0 and one of them is l-open, then p - Ind X - 0.
Proof. Let, for example, Y E rl. T h e n X \ Y r co 7"1 C CO T2 C
p-Cl(X), X \
Y c Z
and, by Corollary 3.2.8, p - I n d ( X \ Y) - 0. Moreover, (X, T1 < 72) is p-perfectly oo
U Fk, where Fk E cot2 C p - e l ( X ) .
n o r m a l and so Y E 2 - 5 ~ ( X ) , t h a t is, Y -
k=l
By (2) of Proposition 3.2.7, p - I n d Fk - - 0 for every k -
1, oc. Therefore
O(3
X-YU(X\Y)-
UFkU(X\Y)' k=l
and it remains to use Corollary 3.2.27.
[3
T h e o r e m 3 . 2 . 3 1 . For every R-p-T1, d-second countable, and p-normal BS (X, T1,7-2) we have p - ind X - p - Ind X .
3.2. Pairwise Large Inductive Dimension
123
Proof. By (3) of Proposition 3.2.7, it suffices to prove only t h a t p - I n d X _< p - i n d X . It is evident t h a t one can assume t h a t p - i n d X < oc. We shall apply induction with respect to p - i n d X . Let p - i n d X - 1. Then, by T h e o r e m 3.1.29, fox" every disjoint pair of sets A c co 7-1 and B c co 72, there exists a partition T between A and B such t h a t p - i n d T _< 0. Hence, by Corollary 3.2.13, p - I n d T _< 0 and so p - I n d X < p - i n d X. Let us assume t h a t this inequality is also correct for k <_ n 1 and prove it for k - n. Let A c cot1, B c cot2 and A N B - 2~. Then by T h e o r e m 3.1.29, there exist a partition T between A and B such t h a t p - i n d T <_ n - 1. It follows from the inductive assumption t h a t p - I n d T < n - 1. Therefore p - I n d X < n and so p - I n d X <_ p - i n d X. [] C o r o l l a r y 3.2.32. If a R-p-T1, d-second countable, and p-normal BS (X, ~-1,T2) can be represented as a union of a sequence F1, F2,... of p-closed BsS's such that p - Ind Fk < n for each k - 1, ~ , then p- Ind X < n.
Proof. It is evident t h a t p - i n d Fk < n for each k - 1, cx~ and by Corollary 3.1.23, p- ind X <_ n. Thus p- Ind X < n. D C o r o l l a r y 3.2.33. If a 1-T1, d-second countable, and p-normal BS (X, T 1 < 7-2) can be represented as a union of a sequence F1,F2,... of i-closed sets, where p - I n d Fk < n for each k - 1, cx~, then p - Ind X < n. C o r o l l a r y 3.2.34. If a R-p-T1, d-second countable, and p-normal BS can be represented as a union of a sequence F1, F2,..., where every
( X , T1,7-2)
(N)
Fk is a countable union of p-closed sets, that is, Fk -- U F~ and p - Ind Fk < n m=l
for each k - 1, oc, then p - Ind X < n. Pro@ Since p - I n d F k < n and F~ c p-Cl(X) for each m - 1, oc, by (2) of Proposition 3.2.7, p - I n d F ~ <_ n for each k - 1, oc and m - 1, oc. Thus it remains OO
to use Corollary 3.2.32 as X -- U
(X)
U Fmk"
[-1
k=lrn=l
C o r o l l a r y 3.2.35. If a 1-2/71, d-second countable, and p-normal BS (X, rl < 72) can be represented as a union of two BsS's Y and Z, where p - I n d Y < n, p - I n d Z < n, and one of them is 1-open, then p - I n d X < n.
Proof. Let Y c
T1.
Then T1 C 2-2~'cr(X) since (X, T1,T2)is p-perfectly normal.
OO
Now Y -
U Fk, where Fk C coT2 C p-Cl(X) for each k -
1, oc and by (2) of
k=l
Proposition 3.2.7, p - I n d Fk _< n for each k - 1, oc. On the other hand, X\YcZ,
X\YEcoT-1cp-Cl(X)
and by the same reasoning, p - I n d ( X \ Y) _< n. But (X)
X - Y U ( X \ Y) - U Fk U ( X \ Y) k=l
and it remains to use Corollary 3.2.32.
D
124
III. Dimension of Bitopological Spaces
T h e o r e m 3.2.36. Let (X, T1,T2) be a p-normal BS, let n denote a nonnegative integer and let (71,72) has on X the (i,j)-(coTj,A)-insertion properties (,~->, (7-1,72) has on X the ( i , j ) - ( 2 x , A ) - i n s e r t i o n properties), where A {V c Ti" (i, j ) - Ind(j, i)- Fr V _< n - 1}. Then (i, j)- Ind X _< n. This theorem can be proved by repeating the scheme of the proof of Theorem 3.1.34, taking into account (4) of Proposition 0.1.7 and Proposition 2.4.10. C o r o l l a r y 3.2.37. Let (X, T1, 7-2) be a p-normal BS and (71,7-2) has on X the (i,j)-(covj,T~ C~ covj)-insertion properties (z----> (T1,T2) has on X the (i, j)-(2 x , Ti N COTj)-insertion properties). Then (i, j)- Ind X - 0. 3.2.38. The following conditions are satisfied for a BS (X,7-1,7-2)" (1) If 7-1
Theorem
Proof. (1) It is clear that the inequality 1- Ind X <_ (1, 2)- Ind X holds for ( 1 , 2 ) - I n d X - oc. Let us assume that ( 1 , 2 ) - I n d X - k < oc and show t h a t 1-Ind X _< k. For k - - 1 the required inequality is obvious. Now let us assume that the inequality is correct for k < n - 1 and prove it for k - n. Since (1, 2 ) - I n d X - n, for every 2-closed set, in particular, for every l-closed set F and its any 1-open neighborhood U ( F ) , there exists a 1-open neighborhood V ( F ) such that 7-2 cl V ( F ) C U(F) and (1, 2)-Ind(2, 1)- Fr V ( F ) < n - 1. Since T1 < C 7"2 and V ( F ) 9 T1, by (3) of Corollary 2.2.7, wi cl V ( F ) - - T1 cl V ( F ) and thus (2, 1)- Fr V ( F ) - 1- Fr V ( F ) . Hence (1, 2)-Ind(1- Fr V ( F ) ) <_ n - 1 and by the inductive assumption 1- Ind(1- Fr V ( F ) ) _< n - 1, t h a t is, for every 1-closed set F and its any 1-open neighborhood U(F) there exists a 1-open neighborhood V ( F ) such that T1 cl V ( F ) C U(F) and 1-Ind (1- Fr V ( F ) ) <_ n - 1. Therefore 1-Ind X < n. (2) The first inequality is obvious by (1) and Corollary 2.3.10. Furthermore, by analogy with the above reasoning, we can prove that (2, 1 ) - I n d X _< 2 - I n d X . Let us assume that the inequality holds for 2 - I n d X - k _< n - 1 and prove it for k - n. Since 2 - I n d X - n, for every 2-closed set, and in particular, for every 1-closed set F and its any 2-open neighborhood U(F) there exists a 2-open neighborhood V ( F ) such t h a t ~-2 el V ( F ) C U(F) and 2-Ind(2- Fr V ( F ) ) <_ n - 1. Since ~-1
1. 89
3.3. Pairwise Covering Dimension
125
3.3. Pairwise Covering D i m e n s i o n As in [106, p. 472], by the order of a family A of subsets of a set X we mean the largest integer n such t h a t the family A contains n + 1 sets with the nonempty intersection or oc if no such integer exists. Thus if the order of the family A - {As}s~s is equal to n, then we have Asl n As~ n . . . N As,~+~ - ;g for any n + 2 members As~, A ~ 2 , . . . , As,~+2 of A. In particular, a family of order - 1 consists only of an e m p t y set, while the family of order 0 consists of pairwise disjoint sets not all of which are empty. The order of a family A is denoted by ordA. D e f i n i t i o n 3.3.1. Let (X, T1,7"2) be a BS and n denote a nonnegative integer. We say t h a t (1) ( i , j ) - d i m X - - 1 ,z-----5, X - ;g. (2) ( i , j ) - d i m X <_ n if for every family of /-open sets {Us}sm=l aIld every family of j-closed sets {Fs } ~ 1 together with Fs C Us for each s - 1, m, there exists a family o f / - o p e n sets {Vs}~=l such t h a t Fs C Vs c Us for each s - 1, m, and ord{ (j, i)-Fr Vs } L 1 -< n. (3) ( i , j ) - d i m X - n if ( i , j ) - d i m X <_ n and the inequality ( i , j ) - d i m X <_ n - 1 does not hold. (4) (i, j ) - d i m X - o c if the inequality (i, j ) - d i m X _, ((1, 2 ) - d i m X < n A (2, 1 ) - d i m X < n).
P r o p o s i t i o n 3.3.2. BsS ( Y , T ~ , 4 )
W e have ( i , j ) - d i m Y <_ ( i , j ) - d i m X of a US ( X , T1,7-2).
f o r every p - c l o s e d
P r o @ It ~umces to show t h a t (i, j ) - d i m X - k implies (i, j ) - d i m Y < k. The inequality is correct for k - - 1 , k - oc. Let us assume that it holds for k _< n - 1 m and prove it for k - n. We assume that {U~ } L 1 and {F~! }s=l are respectively families o f / - o p e n and j-closed subsets of (Y, T~, T~) such that F~ C U~ for each s - 1,m. Consider the families {Us}~__l and {Fs}~=l o f / - o p e n and j-closed sets, respectively, in (X, T1,7-2) such that U~ - U s N Y and F~ - F s N Y for each s - 1, m. Since Y - Y1 n Y2, where Yi c co Ti and U - X \ Y~ C Ti, the families {E s 9 m s --
U s
U U}sm=l and {(I)s 9 q)s - Fs N Yj }srn=l
consist respectively of/-open and j-closed subsets of X such t h a t (I)s C Es for each s-l,m.
By virtue of the equality ( i , j ) - d i m X /-open sets in X such t h a t (I)s c V~ c Es for each s -
-
n, there exists a family {Vs}L1 of
1 , m and o r d { ( j , i ) - F r V s }
Let us consider the family o f / - o p e n sets {V~" V / -
"~ s=l -- n.
Vs N Y}~--1 in Y. Then
E~ n Y - (Us U U) N Y - (Us N Y) U (U N Y) - Us n Y, (I)s N Y - (Fs N Yj) N Y -- (Fs n Yj)N (Y1 N Y2) -Fs N (Yl N Y 2 ) - F s N Y - F '
s
126
III. Dimension of Bitopological Spaces
for each s - 1, m. Therefore F~ C V~ c Us' for each s - 1,m. It remains for us to prove t h a t ord{(j, i)- Fry V~}~__1 _< n so that, it suffices to show t h a t (j, i)- Fry V~C(j, i)- Fr Vs for each s - 1, m. But the latter inclusion is obvious and, therefore, (i, j ) - d i m Y<_ n. [-I C o r o l l a r y 3.3.3. We have p - d i m Y (Y, ~-~,~-~) of a BS (X, T1,T2).
_< p - d i m X
for every p-closed BsS
T h e o r e m 3.3.4. The equalities (i, j ) - d i m X - 0 and (i, j)- Ind X - 0 are equivalent for every BS (X, T1, ~-2) and either of them yields the p-normality of
(X, T1, T2). Pro@ We begin by assuming t h a t ( i , j ) - d i m X - O , A c coT j, B c coT i and A A B - ~. Then U - X \ B is t h e / - o p e n neighborhood of A. Since (i, j ) - d i m X - 0, there exists a n / - o p e n set V such t h a t A c V c U and ord(j, i ) - F r V _< 0 so t h a t (j, i)- Fr V - ~ and, therefore, V E Ti C~co Tj. Hence the set
W--X\VcTjAcoTi,
AcV,
BcW
and V N W - - ; g
so t h a t (X, T1, ~-2) is p-normal. Moreover, V O W - X implies t h a t the e m p t y set is the partition in X, corresponding to the pair (A, B) and thus (i, j ) - I n d X - 0. Conversely, assume t h a t ( i , j ) - I n d X - O. Then by (1) of Proposition a.2.7, (X,~-1,72) is p-normal. Let {Us}s-1 and {Fs}srn=l be respectively families of /-open and j-closed subsets of X such t h a t Fs c Us for each s - 1, m. Since ( i , j ) - I n d X - O, we have a family of simultaneously /-open and j-closed sets { V s } ~ l together with F~ c Vs c Us for each s - 1,m. Thus
(j,i)-FrV~ - ~ g for each s -
1, m a n d s o
ord{(j,i)-FrVs}s~__l- -1.
Hence (i, j ) - d i m X <_ 0, and X # 2~ implies t h a t (i, j ) - d i m X - 0. D Note t h a t due to Proposition 3.2.11 the following equivalences hold for a BS
(X, T1, T2)" (1,2)-dim X - 0 ,z---> (2, 1)-dim X - 0 ,z---->,p_dim X - 0. C o r o l l a r y 3.3.5.
The following conditions are satisfied for a BS (X, ~-1, ~-2):
(1) p -dim X = 0 ~
p - Ind X = 0 and either of them yields the p-normality
of (X, Tl, 72). (2) If ( X, T1, ~-2) is j-T1, then
( i , j ) - d i m X - 0 ( ,z---->, ( i , j ) - I n d X
- O) ---5, ( i , j ) - i n d X - 0
so that, if (X, 7-1, ~-2) is R-p-T1, then p - d i m X - 0 ( ,z---->,p - I n d X
- 0) ----5, p - i n d X
- 0.
Proof. The condition (1) is obvious. Assertions of (2) are immediate consequences of (3) of Proposition 3.2.7. C o r o l l a r y 3.3.6.
The following statements hold for a BS (X, T1, T2)"
D
3.3. Pairwise Covering Dimension
127
(1) I f (X, T1,T2) i8 p-normal and {Xm}m~=l is a sequcTtec of subsets of X such that CO
X -- U Xrn~ Xm C COT1 rlCO7-2 and ( i , j ) - d i m X m - 0 m=l - 0 so that if p - d i m X , ~ - 0 f o r
f o r each m - 1, oc, then ( i , j ) - d i m X each m - 1, oc , then p -dim X - 0. (N:)
(2) If (X~ T1, 7"2) i8 p-nof~Ttal and Y -
U F,~, where Fm C co~-i ~ coT2 f o r m=l
each m - 1, oc , then (i,j)-dimX
- 0 ~
p-dimX
- 0 ~
(i,j)-dimY
<_ 0
and so
p - d i m Y < O. n
(3) I f ( X , 7-1, T2) is hereditarily p - n o r m a l and X -
[_J X . ~ , then rrt=O
((i, j ) - dim X.~ <_ 0 f o r each m - O, n ) ---->. (i, j ) - Ind X < n and so
(p- dim X,~ < 0 f o r each m - O, n) -->, p - Ind X <_ n. Proof. (1) and (2) are immediate consequences of Theorem 3.2.23 and Corollaries 3.2.24 and 3.2.25 in conjunction with Theorem 3.3.4. Assertions of (3) follow directly from Corollary 3.2.22 and Theorem 3.3.4. D
R e m a r k 3.3.7. It is clear t h a t each of the nine dimension functions introduced here is an invariant of a d-homeomorphism (i.e., is a bitopological property). Moreover, their values coincide with integer 1 for the natural BS (R, c~1, c~2).
CHAPTER
IV
Baire-Like Properties of Bitopological Spaces A TS is said to have the Baire property if any first category subset has an empty interior in the space. This property was proved for the first time and independently by W. Osgood for the real line [197] and by R. Baire for spaces R ~ [20]. Today it plays an important role in analysis, topology and mathematical logic (see, for example, [63], [227], [199], [211]). Following N. Bourbaki, a Baire space is a TS such that every nonempty open subset is of second category [38]. Thus a Baire space is a space for which the Baire category theorem is true. A brief but cognitive story of the origin and development of the theory of Baire spaces as well as of its applications can be found in [173]. Most of the results on Baire spaces are stated in [133]. It is well-known that a subset A of a TS (X, w) can be of one category in (X, T) and of another category in itself as a subspace of (X, T), while for open subsets of (X, T) these categories coincide. This observation is the principal factor in defining Baire spaces in different equivalent ways. However, as illustrated by Example 1.5.8 and unlike the topological case, a n o n e m p t y / - o p e n subset of a BS (X, T1, ~-2) can be of one (i, j)-category in (X, Wl, 72) and of another category in itself as a BsS of (X, T1,T2). This argument is closely connected with the definition of (i,j)-Baire spaces [92] and serves as a good introduction to our further discussion. We give a general definition of (i,j)-Baire spaces which is based on the (i,j)-category requirement on nonempty /-open sets, though (1,2)-Baire spaces ( X , T 1 < 7-2) also have other descriptions not based on the (1, 2)-category requirement on nonempty 1-open sets. To illustrate this situation it is sufficient to recall the Slobodnik property [241], which is one of the Baire-like properties from [173] (the Baire-type properties in terms of [100]). However, as compared with the topological version of Bourbaki or our method, the description of (i, j)-Baire spaces in terms of the (i, j)-categories has greater subtlety. In order to investigate all the introduced bitopological modifications of Baire spaces, along with (i,j)-Baire spaces, we define the so-called almost-(i,j)-Baire spaces for which n o n e m p t y / - o p e n sets must be of (i,j)-second category with respect to the whole space. The merit of our definition lies in its versatility, which is clearly seen for a BS (X, T1 < w2) because in this case, as is natural of topology, the notions of a (1, 2)-Baire space and of an almost-(1, 2)-Baire space turn out to be equivalent. Virtually in every case, we consider T1 C 7"2. Sections 4.1 and 4.2 contain a thorough investigation of (1, 2)-Baire spaces and almost-(2, 1)-Baire spaces, respectively. In [61] G. Choquet gives an interesting description of Baire spaces by 128
4.1. (1, 2)-Baire Spaces
129
means of the so-called sifter on a TS. Generalizing the latter notion, we obtain a new characterization of almost (i, j)-Baire spaces. Section 4.3 presents the results on (i,j)-Baire spaces in a strong sense, that is, on BS's whose every nonempty /-closed BsS is of (i, j)-second category. The chapter ends with the definitions and study of other possible modifications of Baire spaces. Namely, the (1, 2)-strict, (2, 1)-weak, 2-weak, and 1-strict Baire spaces are introduced and investigated together with their interrelations and relations to the notions from the preceding sections. We would like to remark that the p-Baire BS's introduced by C. Alegre, J. Ferrer, and V. Gregori in [9] are just the p-Baire BS's from [92], that is, are the almost p-Baire spaces from [95], [100]. An overwhelming majority of the results of [9] were obtained in [92], [95]. Moreover, the notion of a pairwise fine BS in [9] is none other than a BS with S-related topologies studied and used for different purposes in [252] and [95] as well. Among the other results from [9] the interesting characterization of a special class of real normed lattices by means of p-Baire (in our terms, almost p-Baire) spaces should be noted.
4.1. (1, 2)-Baire Spaces Definition 4.1.1. An (i,j)-Baire space (briefly, (i, j)-BrS) is a BS (X, T1,T2) such that every nonempty/-open subset U of X is of (i, j)-second category. This definition immediately implies that if (X, T1, T2) is an (i, j)-BrS, then X is of (i, j)-Catg II. E x a m p l e 4.1.2. The natural BS (R, aJl,CU2) is an (i, j ) - B r S since for every set U c czi \ {2~}, the BsS (U, cz~,cz~) contains no nonempty (i,j)-nowhere dense sets. By (ii) of Theorem 1.1.3 in [laa], it is also clear that (R, CJl,W2) is an i-BrS. Therefore, to counterbalance Remark 2.1.3, the BS's (R,C~l) and (R,a~2) are both BrS's, but c~lScZ2 is not correct.
Theorem 4.1.3. The following conditions are satisfied for a BS (X, T1 < 7"2): (1) (X, T1,T2) is a (1,2)-BrS ~ (X, T1,T2) is a 1-BrS. (2) (X, T1,T2) i8 a 2-BrS > (X, T1,T2) is a ( 2 , 1 ) - B r S .
Moreover, we have for a BS (X, T1 <S 7-2)" (3) (X, T1,T2) is a (1,2)-BrS ~ (X, T1,T2) is a 1-BrS ~ 2-BrS ~ (X, 7-1,T2) is a (2, 1)-BrS.
(X, T1,T2) is a
Pro@ (1) By Corollary 1.5.14, if U c T1 \ {~}, then U is of (1, 2)-Catg II ~
U E (1, 2)-Catgli(X)
and, hence, by (7) of Theorem 1.1.24,
U C 1-Catgli (X) ~
U is of 1-Catg II.
(2) If U c T2 \ {2~}, then by (7) of Theorem 1.1.24, U is of 2-Cat9 II implies that U is of (2, 1)-CatgII since T~ C T~ in (U, T~, T~).
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134
IV. Baire-Like P r o p e r t i e s of Bitopological Spaces c~
1, oc. Hence, by (1)ofTheorem 1.1.24, U ( X \ A n ) 9 (1, 2)-Catgi (X).
for each n -
n=l
Following (4) of Theorem 4.1.6, (X, T1,T2)is also an A-(1,2)-BrS and by (3) of Theorem 4.1.4 in conjunction with Definition 4.1.5, oo
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oo
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and
N Ao - x n=l
so oo
O A~ e 2 - ~ ( x ) n 1-D(x). n--i
(3) The equalities are given by (3) of Theorem 2.2.20. For the inclusion assume that there exists a set A 9 2-Ga(X) N (2, 1)-Catgx(X ) such that T2 int T1 clA ~: ~. Hence, by (3) of Corollary 2.2.7, T1 int T1 cl A ~ ~ so that there exists a set V 9 T1 \ {~} such that V C T1 el A. Furthermore, V = V O T1 cl A c T1 cl(V N A) and thus V N A 9 1-:D(V). Clearly, A 9 2-Ga(X) implies that O(3
V\A-
OO
V \ n An - U ( V \ A n ) n=l
n--1
OO
U (v\ (VNAn)) 9 2-~(V) n--1
since every set V \ (V Cl An) is 2-closed in (17, 7-~,~-~), and ~-; int(V \ A) - r; int (V \ (V N d)) - V \ r; cl(V N A) as V Cl A 9 1-2)(V). Therefore, by (2)of Theorem 1.1.24,
v \ A e 2-7~(v) n 1-~d(V) c (1, 2)-Catg, (V). Following the condition, A 9 (2, 1)-Catgi(X ) and by (7) of Theorem 1.1.24, A 9 (1, 2)- datgx (X). Since V 9 71, by Corollary 1.5.9, A N V 9 (1, 2)- CatgI (V) and thus by (1) of Theorem 1.1.24, V = (V \ A) U (V N A) 9 (1, 2)-Catg x(V) so that V is of (1, 2)- Catg I, which contradicts V 9 T1 \ {;~} and (X, T1,T2) is a (1, 2)- BrS. Kl T h e o r e m 4.1.10. Let (X, rl < r2) be a (1,2)-BrS. If A c X is any subset, then A 9 (1,2)-Catgx(X) if and only if X \ A contains a 1-dense 2-G6-subset of X.
Proof. First, let (x)
A 9 (1, 2)-Catgx (X), that is A -
U An, AN 9 (1,2)-A/'2)(X) n=l
for each n = 1, oo. Hence {X \ 7-2cl An}n~1761 is a countable family of subsets of X such that X \ ~-2clAn 9 72A 1-2)(X) for each n - 1, o0. Using (2) of Theorem 4.1.4 and the first equivalence in (4) of Theorem 4.1.6, we obtain cx)
O (x \ ~ clA~) c 2-~6(x) n 1-D(X). n=l
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136
IV. Baire-Like Properties of Bitopological Spaces
C o r o l l a r y 4.1.12.
The following conditions are satisfied for a (1,2)-BrS
( X , TI
(1) I I A c 2-(~5(X)O 1-/)(X), then (A,~-~,T~) is a (1,2)-BrS. (2) If A c (1-Gs(X)N2-~9(X))U(2-Gs(X)N2-1)(X))O(1-Gs(X)N1-7)(X)), then (A,w{,T~) is a (1,2)-BrS.
Proof. (1) Suppose that A c 2-Ga(X)N1-D(X). Then X \ A c 2 - ~ ( X ) O 1 - B d ( X ) and by (2) of Theorem 1.1.24, X \ A c (1,2)-Catgi(X). Hence it remains to use Theorem 4.1.11. Assertions of (2) follow directly from (1), taking into account the inclusion TI C T2.
D e f i n i t i o n 4.1.13. A family of sets II = {Us }scs c T~ is an (i, j)-pseudo-open covering of a BS (X, T1,7"2) if Tj cl U Us = X. sCS
Note that the disjoint bitopological sum of a family of BS's {(Xs, T~, T~)}sCS is defined as a BS ( E Xs, T1, T2) where ( E Xs, Ti) is a disjoint topological sum sCS
sES
of the family of TS's {(Xs, 7-~)}scs. Later we use the abbreviation (i,j)-BrsS for an (i,j)-Baire subspace. T h e o r e m 4.1.14. The union of any family of l-open (1, 2)-BrsS's of a BS (X, T1 < T2) i8 a (1,2)-BrS and the union of any family of 2-open (1, 2)-BrsS's of a BS ( X , T1 < N 7"2) i8 a (1,2)-BrS.
Proof. First, let t ] = {Us}scS be any family of 1-open (1, 2)-BrsS's of (X, T1 < 72) and let U - U us be no (1,2)-BrS so that there exists a set V c T~ \ { ~ } in sCS
(U,T~,T~) such that V c (1, 2)- Catgi (U). Since V # ~, there is a set Us c i l together with V N Us # ~ so that V N Us c T~ \ {~} in (Us,~-~,T~). Since V C (1, 2)-Catgi (U) and Us c T~ \ { ~ } , by Corollary 1.5.9, we obtain that VNUs C (1, 2)-Catgi (Us) which is impossible because (Us, 7-~,T~)is a (1, 2)-BrS. Secondly, assume that il = {Us}soS is a family of 2-open (1, 2)-BrsS's of (X, T1 < N 7"2) and U = U us is not a (1,2)-BrS so that there exists a set sES
V c ~-~\{2~} in (U, T{, T~)such that V c (1, 2)-Catgi (U). Since V -r 2~, there exists a set Us c tl together with V a Us -r ~, and so V N Us c ~-{ \ {~} in (Us, ~-~,T~). By (2) of Corollary 2.3.13, ~-~
V n Us c (1, 2)-Catg I (U) = 2-Catg I (U). Following the well-known topological fact,
V
N
Us C 2-Catg I(U) ~
V
N
Us ff 2-Catg I(Us)
so that V N U s c 7-~ \ {~} and VNUs E (1,2)-Catgi(Us) , which is impossible since (Us, Tf, r~) is a (1, 2)- BrS. [-] C o r o l l a r y 4.1.15. The following conditions are satisfied for a BS (X, 7-1 < T2)"
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138
IV. B a i r e - L i k e P r o p e r t i e s of B i t o p o l o g i c a l S p a c e s
(3) follows via (1) of Theorem 4.1.3 and Proposition 1.30 in [133]. Assertion (4) is an immediate consequence of (1) of Theorem 4.1.3, Proposition 1.31 in [133] in conjunction with inclusion after Definition 1.1.21, and the inclusion T1 c 7-2. The proof of (5) follows directly from the necessary part of (i) in Theorem 1.24 in [133], taking into account (1) of Theorem 4.1.3 and the inclusion (1, 2 ) - S / ) ( X ) c Assertion (6) is an immediate consequence of (5). (7) Indeed, let A c X \ Y and A E 1-~5(X). Then, by (6), A E 1-H/9(X) and since X \ Y c 1-/)(X), by the well-known topological fact and Corollary 1.1.5, A r 1-N'D(X \ Y) c (1, 2)-N'D(X \ Y). [3 T h e o r e m 4.1.18. Let (X, T 1 < 7"2) be a (1,2)-BrS and (Y, ~-~,~-~) be a BsS of X . If Y c 1-Z)(X) and every set A c 1-81P(X)c~ 2 - G h ( X ) i n t e r s e c t s Y, then (Y, 7-;, 7-~) is also a (1, 2)- BrS. Conversely, if (Y, ~-;, ~-~) is a (1, 2)- BrS and Y E 2-Z)(X), then every set A c 1-$Z)(X)c~ 2-65(X) intersects Y. Proof. First, suppose that in our conditions, there exists a BsS (Y, ~-;, ~-~) of X such that (Y, T{, ~-~) is not (1, 2)-BrS. Then, by virtue of the first equivalence in (4) of Theorem 4.1.6 with (2) of Theorem 4.1.4 taken into account, there is a countable o(3
family {U~}~~
of 1-dense 2-open subsets of (Y, T~, ~-~) such that ('1 U~ g 1-79(Y), n=l
and so there exists a set H c ~-; \ {2~} together with (N3
HA n n=l
OO
U~ - N (H N U~) - ~.
(1)
n=l
Let U~ - g N U~ 9 ~-~' in (H, T[', T~') for each n - 1, oc. Then T 1" c l V ~ -
T 1" c l ( H N V ~ ) -
T 1' c l ( H C ~ U ~ ) N H -
T 1' c l H N H - H
as U~ e 1-~P(Y) for each n = 1, oc. Thus, there exists a countable family {U~}~__I oo
of 1-dense 2-open subsets of the BsS (H,T;',T~') and by (1),
n u~ - 2~. It is n=l
obvious that Un E ~-~' and H c T1' implies U~ e ~-~ in (I<, 7-; , ~-~) for each n - 1 , oc. For every n c N let us consider V/ c ~-2, E c 7"1 such that V~ n Y = Un and EAY=H. IfVn=V~nE, then Vn N Y = V~ N E O Y C E N Y = H a n d s o V~ N Y = U~ for each n - 1, oc. Clearly, Vn is 2-open in (E, v-~", 7-(~") for each n - 1, oc. Let us show that V~ E 1-D(E) for each n = 1, oc. If there exists a set V~,,-~ 1-D(E), then there is a set r c T [ " \ {;~} such that T N V~,, - 2~. In view of E C T1, we have T E T1 \ {2~}. Hence Y c 1-D(X) gives T N Y = T' =/= 2~. But T' C E N Y = H so that T' n (V~,, O Y) = T' V/Un,, = 2~, which is impossible since T' E T [ ' \ {2~} in (H, ~-~',~-~') and Ur~o ~ 1-D(H). Thus V~ r 1-D(E) for each n = 1, oc. Following (1) of Corollary 4.1.7, (E, "r~", v-~") is also a (1, 2)- BrS and by the first equivalence in (x)
(4) of Theorem 4.1.6 in conjunction with (2) of Theorem 4.1.4, we obtain O V~ n=l
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142
IV. Baire-Like Properties of Bitopological Spaces
Therefore, we have: V--
V1 ~ V2 ~ . . . ~ Vn ~ . . . .
It is clear that V~+I c V~ N U~ C V n U~ for each n - 1, oc and, hence, (2O
OO
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N vo N T1clPrC N(v n=l
n=2
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n=l
n=l
Since (X, T1,72) is (2, 1)-Slc, we can assume t h a t T1 C1V2 is 1-compact. Then {T1 cl Vn}~__2 is a decreasing sequence of non-empty 1-closed subsets of 7-1 cl V2 oo
oo
and, hence, n T l c l V n r
oo
n(vaun)r
n=2
ThereforeVa
n=l
n u~r n=l
oo
where V c T1 \ { ~ } is an arbitrary set and thus n u~ c 1-T?(x).
D
rt=l
4.2. Almost-(2, 1)-Baire Spaces First, note that for a BS (X, rl < r2) the condition (2) of Theorem 4.1.4 can be made more precise as follows:
Theorem 4.2.1. A BS (X, rl < r2) is an A - ( 2 , 1 ) - B r S if and only if the intersection of any monotone decreasing sequence of 2-dense 1-open sets is 2-dense in X . Pro@ It suffices to prove t h a t if the intersection of any monotone decreasing sequence of 2-dense 1-open sets is 2-dense in X, then (3) of Theorem 4.1.4 also holds. Let A c ( 2 , 1 ) - C a t g i ( X ) be any set. Then, by analogy with the proof of the implication ( 2 ) ~ ( 3 ) in Theorem 4.1.4, we can assume t h a t A c 1 - ~ ( X ) N (x)
(2, 1 ) - d a t g i ( X ) so t h a t A -
U An where AN C coT1 n (2, 1)-N'D(X) for each n=l oo
n -
1, oc.
Suppose that X \ A ~ 2 - D ( X ) ,
that is,
n (X\AN)E2-D(X).
Let
n=l n
U~ -
n ( x \ Ak) for each n - 1, oc. It is obvious t h a t {U~}n~__l is a monotone k=l
decreasing sequence. Moreover, n
x\u
n
UA,
-x\ k=l
k=l
where Ak C COT 1 n (2,1)-AfD(X) for each k = 1, oc and for each n = 1, oc. According to Corollary 1.1.15, X \ g n C c o 7"1 n (2, 1)-N'D(X) (x)
and so U~ c T1 n 2 - ~ ) ( X ) for each n - 1, oc. Hence, by condition, n u~ E 2-79(x). rt=l
But oo
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4.2. Almost-(2, 1)-Baire Spaces
145
that (X \ A) \ B c 2-Z)(X \ A). Thus for every set B E (2, 1)-Catg I ( x \ A) the complement (X \ A) \ B c 2-/)(X \ A) and once more applying (3) of Theorem 4.1.4 gives that (X \ A, 7-~,7-~) is an A-(2, 1)- BrS. To prove the converse, suppose first that (X, T1, ~-2) contains no nonempty sets of (2,1)-first category. Then every set U c T2 \ {2~} is of (2,1)-second category in X and thus by (1) of Theorem 4.1.4, (X, 7-1,7-2) is an A-(2, 1)- BrS. If there exists (x)
a set A c (2, 1)- Catgi (X) \ {2~}, then A -
U AN, where ~-2intT1 clAN - 2~ for rt--1
each n - 1, oc and AN,, # ~ for some no C N. Clearly, An,, c (2,1)- Catgl(X) and following the condition, (X\A~,,, ~-~,7-~) is an A-(2, 1)- BrS. Since ~-2int A~,, c T2 int Tx cl An o - ~, we obtain X \ A~I, c 2-/)(X). Hence, it s u m c e s to use (~) of Proposition 4.2.2. K1 C o r o l l a r y 4.2.6. Let (X, T1 < 7-2) be an A-(2, 1)-BrS. Then A E 1-Gs(X)O 2-/)(X) implies that (A, ~-~,7-~) is an A-(2, 1)-BrS.
Proof. Indeed, A c 1-~5(X) N 2-/)(X) implies that X \ A c 1-3cr and so X \ A c (2, 1)-Catgi(X ).
N
2-Bd(X)
V]
T h e o r e m 4.2.7. The union of any family of 2-open A-(2, 1)-BrsS's of a BS (X, T1 ~ 7-2) is an A-(2, 1)- BrS.
Proof. Let il = {Us }sEs be a family of 2-open A-(2, 1)-BrsS's of X and U = U Us sES
is not an A-(2, 1)-BrS so that there exists a set V c ~-~ \ {2~} N (2,1)- Catgl (U) in (U, T{, T~). Furthermore, there is a set Us c i l such that V N Us C T~ \ {2~} in (Us,T~,T~). Since V E (2,1)-Catgi(U) and Us E ~-~ \ {2~}, by Theorem 1.5.6, V N U s C (2, 1)-Catgi(Us) , which is impossible since (Us,T~,T~) is an A-(2, 1)- BrS. V-] C o r o l l a r y 4.2.8. The following conditions are satisfied for a BS (X, 7-1 < 7-2)" (1) (X, T1,7-2) i8 an A-(2, 1)-BrS if and only if each point of X has a 2-open neighborhood which is an A-(2, 1)-BrS. (2) If (X, T1, 7-2) has a (2, 1)-pseudo-open covering tl = { Us } sE s each of whose members is an A-(2, 1)-BrS, then (X, TI,T2) is an A-(2, 1)-BrS.
Proof. Assertion (1) is obvious. (2) In the considered case r2 cl U Us - X where U us is an A-(2, 1)-BrS. It sES
sES
remains to use (1) of Proposition 4.2.2. C o r o l l a r y 4.2.9.
D
Let ( ~ Xs, T1 < C T2) be the disjoint bitopological sum sES
of the family { ( X ~ , r f , r ~ ) } ~ s
of A-(2,1)-BrS's.
Then ( ~ X s , TI,T2) i8 an sES
A-(2, 1)- BrS.
Pro@ If U c T1 \ {~} is any set, then U n Xs c w~ \ {2~} for each Xs intersected by U. By (2) of Corollary 2.2.8, 7-~ < c 7-~ for each s c S and, hence, by (2) of Proposition 4.2.2, U N X~ is an A-(2,1)-BrS. Therefore, by Theorem 4.2.7, U - U (UOXs)is an A-(2, 1)-BrS. Since U d T1\{2~} is arbitrary, ( 2 Xs,T1,T2) sCS
is also an A-(2, 1)-BrS.
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m
~'~"
~.
"
~
~>0
"---"
>
148
IV. Baire-Like
Properties
of Bitopological
Spaces
Proof.
(1) L e t {~n}n~176 b e & sequence of pseudobases for r2 verifying the (2,1)-pseudocomplete property of Definition 4.2.12. Since rlSre, by Definition 2.1.2, r2 \ {~} is a pseudobase for T 1 a n d , consequently, {Bn}~~176is a sequence of pseudobases for rl. If B~ c B~ is any set, then by (1) of Definition 2.1.1, T1 intBn # 2~ for each n = 1, oc. Since (X, rl,re) is 1-quasi regular and Bn+l is a pseudobase for rl, by (3) of Proposition 0.1.15 for i = j = 1, there is a set Bn+l C Bn+l such that 7"1 c l g n + l C T1 int Bn. Hence T1 clBn+l C 72 int Bn for oo
each n -
1, oc which implies that
n B~ # z because T1 C T2 and (X, rl, re) n--1
is (2,1)-pseudocomplete. It is obvious that by rl c r2 we also have re clB~+l c r l i n t B~ for each n = 1, oc, and (X, rl, re), being 1-quasi regular, means that oo
(X, TI,~'e) is (1,2)-quasi regular. Since n Bn 7~ ;g, once more applying Definin=l
tion 4.2.12 implies that (X, rl, re) is (1,2)-pseudocomplete. (2) Let {Bn}~~176be a sequence of pseudobases for r2 for which (X, rl,re)is (2,1)-pseudocomplete. Since 7 c re and each member of re \ {~} contains a member of 7 \ {~}, we have 7Sr2. From this fact we find that {B~}~~176 1 is a sequence of pseudobases for 7 and thus "7 int B~ 7~ 2~, where B~ E B~ for each n = 1, oc. Since (X, r l , 7 ) is (2,1)-quasi regular and B~+I is a pseudobase for 7, by (3) of Proposition 0.1.15, there is a set B~+I e Bn+l such that T1 clB~+l C 7 int B~. Moreover, 7 c re implies that rl cl Bn+l C T2 int Bn for each n oo
1, oc, and (X, rl, re) is (2,1)-pseudocomplete gives n Bn r 2~. Thus (X, rl, 7) is re=-1 oo
(2,1)-quasi regular and n B~ # ~ whenever B~ c B~ and T l c l B ~ + I
c 7intB~
n--1
for each n = 1, oc so that (X, r l , 7) is (2,1)-pseudocomplete.
[3
T h e o r e m 4.2.14. A (2, 1)-pseudocomplete BS (X, 7-1 < ~-2) is an A-(2, 1)-BrS.
Proof. Let {Un}n~ be a sequence of 1-open 2-dense subsets of X and let {Bn}n~176 be a sequence of pseudobases for ~-2, for which (X, T1, ~-2) is (2,1)-pseudocomplete. oo
It suffices for this to prove that
n u~ c 2-l?(x). Let U c ~-2 \ {~} be any set. n---1
Since U1 c T1Ne-T)(X)C ~-2Ne-T)(X), we have UNU1 c T2\{2~}. But (X, 7-1,7-2)is (2,1)-pseudocomplete so that it is (2,1)-quasi regular and thus, there is a set B1 c B1 such that T1 clB1 C UNU1. Furthermore since 72 intB1 r ~ and U2 E 2-7?(X), we have ~-2 int B1 n 0"2 # 2~. By analogy with the above, there exists a set B2 E B2 such that T1 C1 g 2 C T2 int B1 N U2 C U N U2. Thus in a similar manner for any n > 3, we choose Bn C B~ such that T1 el Bn C 7"2 int B n _ 1 n V n C V n V n and, oo
oo
consequently, n Bn # 2~ since (X, T1,72)is (2,1)-pseudocomplete. But n Bn c n=l oo
n ( u N Un) and U E 7-2 \ {2~} is arbitrary implies that n=l
n=l (9o
n u~ c 2-:D(X).
D
n=l
Finally, we give a new characterization of almost (i, j)-Baire spaces different from that given in Theorem 4.1.4.
4.2. Almost-(2, 1)-Baire Spaces
149
D e f i n i t i o n 4 . 2 . 1 5 . A Nj-sifter on a BS (X, T1,7-2) is a binary relation Kj on the family A o ( X ) - {A - U N V ~ 2~" U c T1, V E T2}, satisfying the following conditions"
(1) (2) (3) (4)
A1 F j A2 ~ A1 C A2. For each A c A0(X), there is U c ~-j \ {~} such that U Kj A. (A~ C A1 Kj A2 C A~)---5, A~ Ej A~. For each sequence { A s } ~ - i c A o ( X ) such that As+l Kj AN for every (3O
n-l,
oc, wehave n A s r n=l
It is evident that every Nj-sifter on A o ( X ) is a j-sifter on the family of all non-empty j-open sets [611 and the result of Choquet [611 together with (5) of Theorem 4.1.6 give: there is a n2-sifter on (X, 71 < 72) ---5, there is a 2-sifter on (X, rl < 72) (X, 71 < 72) is a 2-BrS ==> (X, 71 < r2) is an A-(2, 1)-BrS. In the general case, we have T h e o r e m 4.2.16. If there exists a n j - s i f t e r on a BS (X, 71,72), then (X, 71,7-2) is an A-(j, i)- BrS.
Proof. By (2) of Theorem 4.1.4, it sumces to prove that if AN r r~Nj-2)(X) for each oo
n - 1, ~ , then n As c j-2)(X). Let U c r o \ {~} be any set and let us prove that s=l oo
U N( n AN) -~ 2~. Clearly for U1 - U, we have 2~ r U1 n A 1 E ~[0(X). By ( 2 ) o f n=l
Definition 4.2.15, there is a set [72 c rj \ {~} such that U2 K o U1 N A1. Therefore, by means of the same condition and the fact that As c j-2)(X) for each n - 1, oo, one can define a sequence of j-open non-empty sets U1, U2,... such that Ux - U and Us+l r-o Us N As for each n - 1, o0. Thus Us+l c Us+l K o Us N As c Us and by (3) of Definition 4.2.15, Us+l Kj Us. Therefore (4) of Definition 4.2.15 (3O
gives that
n u s r 2~. On the other hand, we have U~+I K o Us n As and by (1) n=l
of Definition 4.2.15, U2 C A1, U3 c A2, . . . . Hence (XD
(:X)
n=2
n=l
(XD
n=l
(X)
O0
n=2
n=l
Theorem 4.2.16 together with (4) of Theorem 4.1.6 implies the more general result than one of Choquet. Namely take place the following C o r o l l a r y 4.2.17. For a BS (X, 7-1 ~ 7-2) , the following implications hold:
there exists a N1-sifter on (X, T1 < T2)
(X~T 1 < 7-2) is a n A - ( 1 , 2 ) - B r S
(X, l < there exists a 1-sifter on (X, q < ~-2)
a (1,2)-BrS
( X , 71 ~ 72) i8 a I - B r S .
150
IV. Baire-Like Properties of Bitopological Spaces
Therefore one can conclude that for BS's of the type (X, T 1 < 7-2) which have a VIi-sifter and a A2-sifter, respectively, all results, obtained for (1,2)-BrS's and A-(2, 1)- BrS's are valid. 4.3. Strong Baire-Like Properties
D e f i n i t i o n 4.3.1. A BS (X, 7-1,7-2) is an (i,j)-BrS in the strong sense (also called (i,j)-totally nonmeager, briefly, S - ( i , j ) - B r S ) if every nonempty /-closed subset of X is of (i, j)-second category. T h e o r e m 4.3.2. The conditions below are satisfied for a BS (X, 7.1 < 7.2):
(1) (X, 7.1,7.2) is an S-(1,2)-BrS ~ (X, 7.1,7.2) is an S-1-BrS. (2) (X, 71,72) is an S-2-BrS :-(X, 7.1,7.2) is an S-(2, 1)-BrS. For a BS (X, 7.1
(3) (X, 7.1,7.2) is an S-2-BrS --->. (X, 7.1,7.2) is an S-(2, 1)-BrS (X, T1,T2) is an S-(1,2)-BrS ~
(X, Tl,72) is an S-1-BrS.
For a BS (X, 7.1
(4)
S-2-BrS
S-(2, 1)-BrS
(X, 71,72) is an S-(1,2)-BrS ----5. (X, 71,72) is an S-1-BrS.
(5)
S-I-BrS, (X, Tl,72) is an S-2-BrS ~
(X, 7.1,7.2) is an S-(2, 1)-BrS.
Proof. (1) and (2) are immediate consequences of (7) of Theorem 1.1.24.
(3) It suffices to consider only the vertical implication. Let (F, T~, 7~) be any 1-closed BsS of X. Clearly, F is 2-closed and since (X, T1,7.2) is an S-(2, 1)-BrS, F is of (2, 1)-CatgII. But by Definition 2.2.14, 7-1 ~ C ( 1 ) 7-2 implies 7-~ < c 7-~ and, thus, according to (3) of Theorem 2.2.20, F is of 1-Catg II so that (X, 7-1,7-2) is an S-1- BrS. (4) We shall prove only the first vertical implication since 7-1 ~ N ( 1 ) 7-2 implies 7-1 ~C(1) 7-2. Let (F, 7-~,7-~) be any 1-closed BsS of X so that F is 2-closed. Then F is of 2-CatgII since (X, 7-1,7-2) is an S-2- BrS. Following Definition 2.3.15, 7-~ .F is of (i, j ) - d a t g II and so (X, 7-1,7-2) is an S-i-BrS ~
(X, 7-1,7-2) is an S - ( i , j ) - B r S .
[]
T h e o r e m 4 . 3 . 3 . The following conditions are satisfied for a BS (X, 7-1,7-2):
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152
IV. B a i r e - L i k e P r o p e r t i e s of B i t o p o l o g i c a l S p a c e s
4 cl U has no 2-isolated points in itself as a BsS of X. Hence (7"~cl U, 7"~',7"~') is also 1-T1 implies that 2~ = 7"~'int{x} = r~' int 7"{'cl{x} for each z E 7"{ int 7"~el U \ U and thus the set 7"{ int 7"~cl U \ U c (2, 1)- Catg I (7"~el U). On the other hand, U E 2-:D(7"~ el U) and following (2) of Theorem 1.5.13, U E (2, 1)-Catgi(7"~ el U) (U E (2, 1)-Catgli(7"~ el U)) -', '-' :-U is of (2,1)-datgI
(U is of (2,1)-datg lI ).
If V - 7"1 int 7"~el U, then V c 7"1' c 1- Ga(7"~ cl U). Hence U c 7"~int 7"~cl U - V and U E 2-D(7"~ el U) gives that V c 1- G5(7"~el U) O 2-D(7"~ cl U). Moreover, by condition, 7"~el U is of (2, 1)-CatgII. Since 7"1
V-
UU (7"~int7"~clU\ U) E (2,1)-datgi(7"s
which is impossible. Hence U is of (2, 1)-Cat9 II and so (F, 7"~,7"~) is a (2, 1)-BrS. Conversely, let F E co7"2 \ {2~} be any set and (F, 7"{,7"~) be a (2, 1)-BrS. Then F is of (2, 1)-datg II and it remains to use Definition 4.3.1. oo
(2) If A c 1-Ga(X), then A -
n u~, where U~ E 7"1 \ {~} for each n n=l cxD
1,0c.
Let F ' c c07"~\{2~} be any set.
Then F ' - A O 7 " 2 c l F ' -
n (UsA n=l
7"2cl F'), where U~ a 7"2cl F ' E 7"~" in (7"2cl F', 7"~", 7"~") for each n - 1, oc so that F' E 1-Ga(7"2clF')N2-D(7"2clF'). Following (1), (7"2clF',7"~",7"~") is a (2, 1)-BrS. Hence, by (2) of Theorem 4.1.6, (7"2e l F ' , 7"~", 7"~") is also an A-(2, 1)- BrS and from Corollary 4.2.6, we conclude that (F', r~', r~') is an A-(2, 1)- BrS. [:] C o r o l l a r y 4.3.6. /f (X,w~
(2,7.1,7.2) i8 art S-(2,1)-BrS ----5, (X, 7.1,7"2) is a (2,1)-BrS. Proof. \
The proof follows immediately from (1) of Theorem 4.3.5 since X E
D 4.4. S o m e M o d i f i c a t i o n s of B a i r e - L i k e P r o p e r t i e s
D e f i n i t i o n 4.4.1. A BS (X, 7"1 < 7"2) is a (1,2)-strict Baire space (briefly, (1, 2)- SBrS) if every nonempty 2-open subset of X is of (1,2)-second category in X . T h e o r e m 4.4.2. The following conditions are equivalent for a BS (X, 7"1 < 7"2): (1) (X, 7"1,7"2) is a (1, 2)-SBrS. (2) If {Un}~=l is any countable family of subsets of X where Us c 7"2 n oo
1-Z)(X) for each n -
1, oc, then n un c 2-7?(x). n=l
(3) A c (1,2)-Catgi(X) ~
X \ A E 2-:D(X).
~-q
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154
IV. Baire-Like Properties of Bitopological Spaces
Proof. (1) The vertical implication follows from (7) of Theorem 1.1.24, taking into
account Definition 4.4.1. The rest is obvious by (4) and (5) of Theorem 4.1.6. (2) The last implication in the second horizontal line is obvious by (6) of Theorem 4.1.6. (3) The equivalence is an immediate consequence of (3) of Theorem 2.3.19. The rest is obvious by (7) of Theorem 4.1.6. (4) The first equivalence follows directly from (1) of Theorem 2.1.10, and the rest is given by (8) of Theorem 4.1.6. [:] T h e o r e m 4.4.5. The union of any family of 1-open (1, 2)- SBrsS 's is a (1,2)-SBrS for a BS ( X , T1 < T2), while thc union of any family of 2-open (1, 2)-SBrsS's is a (1,2)-SBrS for a BS (X, T1
Pro@
sES
that U \ A ~ 2-TP(g), that is, A E 2-Bd(g). Assume the opposite: T~ int A r in (U, ~-~,T~). Since A a U, there is a set Us c ~.1 such that ~-~int A N Us r ~. Furthermore, A c (1, 2)-Catg I (g) and A n Us a A imply A N Us C (1, 2)-Catg I (g). The BS (Us, r~, r~) is a (1,2)- SBrS, Us E r~ \ {2~} and following Corollary 1.5.9, ANUs E (1,2)-Catgi(Us). Hence, by (3) of Theorem 4.4.2, ~-~ cl(Us\(ANUs)) = Us, that is, r S int(A n Us) = ~ which is impossible since r~ int A N Us C r S int(A n Us). This contradiction completes the proof of the first part. Let now ~1 = {U~}~cs be a family of 2-open (1, 2)- SBrsS's of X and U O us. By (2) of Corollary 2.3.13, r~
in (U, r~,r~). According to (3) of Theorem 4.4.4, every (Us,r~,r~) is a 2-BrS and, hence, by the well-known topological fact, (U, r~, r~) is also a 2-BrS. Again applying (3) of Theorem 4.4.4, we see that (U, r~, r~) is a (1, 2)- SBrS. [-1 C o r o l l a r y 4.4.6. The following conditions are satisfied for a BS (X, T 1 < 7-2): (1) (X, rl,r2) is a (1,2)-SBrS if and only if each point x c X has a 1-neighborhood which is a (1, 2)-SBrS. (2) /f (X, T1,T2) has a (1,2)-pseudo-open covering t1 = {Us}s~S each of whose members is a (1, 2)- SBrS, then (X, T1,T2) is a (1, 2)- SBrS. For a BS (X, T1
(3) If (X, T1, ~-2) has a 2-pseudo-open covering s = {Us }s~s each of whose members is a (1,2)-SBrS, then (X, T1,T2) is a (1,2)-SBrS. Proof. The proof of (1) is obvious by (1) of Corollary 4.4.3, taking into account the first part of Theorem 4.4.5. In both cases (2) and (3), we have U Us c 2-D(X) where U us is a sCS
(1, 2)- SBrS. It remains to use (2) of Corollary 4.4.3.
sES
[5]
4.4. Some Modifications of Baire-Like Properties
155
D e f i n i t i o n 4.4.7. A BS ( X , T 1 ( 7 - 2 ) is a (2,1)-weak Baire space (briefly, (2, 1)-WBrS) if every nonempty 1-open subset of X is of (2,1)-second category in X. T h e o r e m 4.4.8. The following conditions are equivalent for a BS (X, 7"1 < 7-2): (1) (X~ 7-1~7-2) is a (2, 1)-WBrS.
(2) If {U~}~__I is any monotone decreasing sequence of subsets of X where DO
e
n 2-l)(X)
1,
N
c 1-D(X).
n--1
(3) d e (2, 1)-datg~(X) ~ X \ d e 1-2)(X). (4) If {F~}~_ 1 is any countable family of subsets of X where Fn C coT1 N oo
2-Bd(X) for each n -
1, oc, then U F~ 9 1-Bd(X). n--1
Proof. This theorem can be proved by analogy with the proof of Theorem 4.1.4 taking into account the proof of Theorem 4.2.1. V1
C o r o l l a r y 4.4.9. The following condition is satisfied for a BS (X, T1 < ~-2): (1) If (Y, 7-;, ~-~) is a (2, 1)-WBrS and Y < 2-T~(X), then (X, T1, "/-2) i8 also a (2, 1)-WBrS. Moreover, for a BS (X, 7"1 < C T2), We have: (2) If (X, 71, ~-2) is a (2, 1)-WBrS and Y c TI, then (Y, ~-~,~-~) is also a (2, 1)-WBrS. (3) If A c (1,2)- ST)(X) and (A, ~-~,~-~) is a (2,1)-WBrS, then d A, al o (2, 1)-WBrS. Proof. (1) Suppose that Y c X, Y E 2-~P(X) and (Y, ~-~,7-~) is a (2, 1)-WBrS. It suffices to prove that A c (2, 1)-datg I(X) implies that X \ A c 1-T)(X), that is, A 6 (2,1)-Catgi(X ) implies that TlintA = ~. Let TlintA r Z. Then Y c 2-:D(X) C 1-:D(X) implies that T1 int A N Y r ~ and T1 int A O Y C A gives that 7"1 i n t A n Z c (2,1)-Catgi (X). Since Y c 2-~P(X) by (2)of Theorem 1.5.13, T1 int A N Y C (2, 1)-datg I (Y).
But ~-~int(~-i int A N Y) :fi ~ and thus ~-~cl(Y \ (T1 int A N Y)) -~- Y. This fact contradicts the condition that (Y, ~-~,~-~) is a (2, 1)-WBrS. (2) If Z C X , Z C 7-1 a n d A c (2, 1)-r then by (4) of Corollary 2.2.23, A c (2, 1)-Catgi(X ). Hence, by (3) of Theorem 4.4.8, X \ A c 1-2)(X) and T1 C1Z - T1 C1 ( Y n ( X \ A)) - T1 c l ( Y \ A) so that Y - Y n T1 c1Z -- Y n TICI(Y \ A) - T~ cl(Y \ A) and so Y \ A E 1-2)(Y). (3) Let A c (1, 2)-$2)(X) be any set and (A, T~, 7-~) be a (2, 1)-WBrS. Since A ~ 2-2)(r2clA), (1) above gives that w2clA is also a (2,1)-WBrS. Hence it remains to use (2) since T1 int T2 cl A is I-open in ~-2cl A. [] T h e o r e m 4.4.10. The following implications hold for a BS (X, T 1 < 7-2):
156
IV. Babe-Like Properties of Bitopological Spaces (1) (X, T1,T2) is a (1,2)-SBrS=~(X,~-I,~-2) is a ( 1 , 2 ) - B r S ~ ( X , T1,T2) is a 1-BrS (X, 71, ~-2) is a 2-BrS=a (X, 71,72) is an A-(2, 1)-BrS=a (X, 71,72) is a (2, 1)-WBrS. F o r a BS (X, 7"1 < c 7-2), we have"
(2)
(X, 7"1,7-2) is a 2- BrS ::~ (X, 7-1, T2) is an A-(2, 1)- BrS ~ (X, 71,72) is a 1- BrS (X, T1,T2) is a (1,2)-SBrS=~(X, T1,T2)isa(1,2)-BrS=~(X, T1,T2) isa
(2, 1)-WBrS.
F o r a BS (X, 7-1 < N 7-2), We h a v e :
(3) (X, TI,~-2) is a (1, 2)- SBrS c* (X,~-1,~-2) is a 2-BrS=~(X,~-I,T2) is an A-(2, 1)-BrS (X, T1,T2) is
a ( 1 , 2 ) - B r S ~ ( X , T1,T2) isal-BrSr
r (X, ~-1,~-2) is a (2, 1)-WBrS. For a BS (X, 7"1 <S T2), we have:
(4) (X, 7-1,7-2) is a (1,2)-SBrS <---5, (X, 7-1,7-2) is a 2-BrS ,z----5, (X, 7"1,7"2) i8 a (2, 1 ) - B r S ~ (X, 7"1,7"2) i8 an A-(2, 1)-BrS <--> (X, 7"1,7"2) is a (1,2)-BrS <---> (X, 7"1,7"2) is an A-(1,2)-BrS z---->, (X, 7"1,7"2) is a 1-BrS ~' ;, (X, 7"1,7"2) is a (2, 1)-WBrS. Proof. (1) The second vertical implication follows directly from (7) of Theorem 1.1.24. The last implication in the second horizontal line is obvious by Definitions 4.1.5 and 4.4.7 taking into account the inclusion T1 C T2, and the rest follows from (1) of Theorem 4.4.4. (2) By (1) above and (2) of Theorem 4.4.4, it suffices to prove that if (X, 7-1,r2) is a (2,1)-WBrS, then (X, 7-1,7-2) is a 1-BrS. Let U E 7"1 \ {~}, a n d so U E (2, 1)-Catgut(X). Then, by (3) of Theorem 2.2.20, U E 1-Catgli ( X )
or, equivalently, U is of 1-Catg II.
The proof of (3) follows directly from (3) of Theorem 4.4.4 and (2) above since 7-1
T h e o r e m 4.4.12. For a BS (X, T1 < C 7-2) the union of any family of 1-open (2, 1)-WBrsS's of X is a (2, 1)-WBrS. Proof. Indeed, by (2) of Corollary 2.2.8 the two topologies of each member of this family are also
Corollary ( X , T1 < C T2):
4.4.13.
The following
conditions
are satisfied for
a BS
4.4. S o m e M o d i f i c a t i o n s
of Baire-Like
Properties
157
(1) (X, rl,'c2) is a (2, 1)-WBrS if and only if each point x 9 X has a 1-open neighborhood which is a (2, 1)-WBrS. (2) If ( X, rl , 72 ) has a (1, 2)-pseudo-open eoverin9 s = {Us }sos each of whose members is a (2, 1)-WBrS, then (X, rl, r2) is a (2, 1)-WBrS.
Pro@ Assertion (1) is obvious. (2) It is clear that U us 9 2-D(X) where U Us is a (2, 1)-WBrS. It remains sES
sES
to use (1) of Corollary 4.4.9.
D !
T h e o r e m 4.4.14. Let (X, rl
Proof. According to (10) of Corollary 2.2.8, r~ < c r~. Moreover by (3) of Theorem 2.2.20, (2, 1)-8/)(X) - 1-81D(X), and it remains to use (i) of Theorem 1.24 from [133] in conjunction with (2)of Theorem 4.4.10. D D e f i n i t i o n 4.4.15. An (i,j)-regular open filter base of a BS (X, rl,r2) is a nonempty family of sets 13 c ri \ {2~} satisfying the following conditions: (1) IfU, V c 1 3 a n d U n V r (2) For each U c B, there is V c B such that rj cl V c U.
WCUNV.
This definition is closely associated with D e f i n i t i o n 4.4.16. A BS (X, rl,r2) is (i,j)-countably subcompact if there oo
exists an/-open base 13 for (X, rl, r2) such that n u~ # ~ whenever {Un}~_l c B n=l
is a countable (i, j)-regular open filter base.
T h e o r e m 4.4.17. If (X,
7-1 <
72) is a (1,2)-quasi regular and (1,2)-countably
~bco.~v~ct BS, t h ~ (X, r ~ ) i~ ~ (2, ~)-WBrS. Proof.
Assuming that {U~} r~t = l
is a monotone decreasing sequence of 1-open
oo
2-dense sets, we show that
n u~ c 1-7)(x). If 1/1 c T1 \ {~} is arbitrary, then rt--1
U1 c 2-D(X) implies that 171 n U 1 C 7-1 \ {~}. Further, assume that B = {V~}scs is a 1-open base for X verifying the (1,2)-countably subcompact property of X. Since (X, T1, T2) is (1,2)-quasi regular, there exists a nonempty set V2 c 13 such that r2 cl V2 c V1 N U1. For every n > 1 we can define in this manner a nonempty set V~ c 13 such that r2clVn C V~-I n U~-I. It is clear that {V~}~~176 2 C 13 is a countable (1,2)-regular open filter base for (X, 7-1, T2) and X is (1,2)-countable CX3
CXD
CXD
(X)
subcompact gives n v~ r ;~. But n v~ c v1 a n u~ and so n u~ E 1-/)(x). [] n=2
n=2
n=l
n=l
D e f i n i t i o n 4.4.18. A BS (X, rl < r2) is a 2-weak Baire space (briefly, 2-WBrS) if every nonempty 1-open subset of X is of 2-second category in X (or, equivalently, of 2-second category). T h e o r e m 4.4.19. The following conditions are equivalent for a BS (X, TI
158
IV. Baire-Like Properties of Bitopological Spaces
(1) (X, 7"1,7"2) is a 2-WBrS. (2) If {Un}~__l is any monotone decreasing sequence of subsets of X where oo
U~ E T2 n 2-TP(X) for each n -
1, oc, then n u~ E 1-z)(x). n=l
(3) A C 2 - O a t h I ( X ) ~
X \ A c 1-~)(X).
(4) If {Fn}n~__l is any countable family of subsets of X where Fn C co7"2 N oo
2-Bd(X) for each n -
1, oc, then U Fn e 1-Bd(X). n=l
Proof. The proof is standard and, therefore, omitted.
E]
C o r o l l a r y 4.4.20. The following conditions are satisfied for a BS (X, 7"1 ~ 7"2); (1) If (X, 7"1,7"2) is a 2-WBrS and Y c 7"1, then (Y, 7"~,~-~) is also a 2-WBrS. 2-WBrS. (3) [ f A c ( 1 , 2 ) - 8 D ( X ) and(A, 7"~,7-~) is a 2-WBrS, then(7"1 int 7.2 cl A,7.~',2)r" is also a 2-WBrS. Proof. (1) If Y c X, Y c 7"1 and A c 2-datg~(Y) is any set, then A c 2-datgi(X). Hence, by (3) of Theorem 4.4.19, X \ A E 1-Z)(X) and thus 7 . 1 c 1 Y -- T 1 cl ( Y n ( X \ A ) ) - T l c l ( y
\ A)
so that Y - - Y n 7-1 cl Y - Y n 7"1 cl(Y \ A) - r~ cl(Y \ A), that is Y \ A E 1-7P(Y). It remains to use once more (3) of Theorem 4.4.19. (2) Suppose that Y c X, Y E 2-7?(X), (Y, 7.~,7.~) is a 2-WBrS and A c 2 - C a t g i ( X ). Let us prove that (X \ A) E 1-TP(X), that is, 7"1 intA = ~. If 7"1intA ~ ~, then Y c 2-D(X) C 1-7?(X) gives 7-l i n t A n Y r 2~. Hence 7"1 int A n Y C A implies that 7"1int A n Y C 2-Catg I (X) and thus 7"1int A N Y E 2-Catg I (Y) since Y c 2-TP(X). But 7"~int (7"1int A n Y) =~ O so that ~-~cl (Y \ (7"1int A n Y)) -r Y. This contradicts the condition that (Y, T~, 7"~) is a 2-WBrS. Assertion (3)follows directly from (1)and (2).
D
For the purpose of abbreviating the conditions (1)-(3)of Theorems 4.4.21 and 4.4.28 below, instead of writing spaces, we shall indicate only the corresponding Baire and Baire-like properties. T h e o r e m 4.4.21. The following implications hold for a BS (X, 7"1 < 7"2): (1)
2-WBrS
~
2- BrS
4=== (1,2)- SBrS ==~ (1,2)- BrS ==~
(2,1)-WBrS e== A-(2,1)-BrS
1-BrS
2-WBrS
==~ (2,1)-WBrS
For a BS (X, T1 < C T2), we have: (2)
2-WBrS
e==
(2,1)-WBrS ~
2-BrS
e== (1,2)-SBrS ===~ (1,2)-BrS ==~
A-(2,1)-BrS
For a B S (X, T1 < N 7.2), we have:
~
1-BrS
2-WBrS
~===~(2,1)-WBrS
4.4. Some Modifications of Baire-Like Properties
(3) 2-WBrS
~
(2,1)-WBrS r
2-BrS
~==> (1,2)-SBrS ~
A-(2,1)- BrS
-->
159
(1,2)-BrS ~ 1- BrS
r
2-WBrS (2,1)-WBrS
For a BS (X, rl < s r2), we have:
(4) (X, rl,7-2) is a (l12)-SBrS ,e---> (X, 7-1,rD.) is a 2-BrS <---> (X, 7-117-2) is a (2,1)-BrS <---> (XIT11T2) i8 an A-(2,1)-BrS ~ (XITI,T2) i8 a (1,2)-BrS ,e---->,(X, r l , r 2 ) i s a n A - ( 1 , 2 ) - B r S <---> ( X I T l l T 2 ) i s a 2-WBrS ~' > (X, 7-117-2) is a 1-BrS <---> (X, 7-1,7-2) is a (2, 1)-WBrS. Pro@ (1) By virtue of (1) of Theorem 4.4.10, it suffices to prove that if (X, 7-1,7-2) is a (1, 2)- BrS, then (X, 7-1,7-2) is a 2-WBrS and1 hence, (X, 7-1,7-2) is a (2, 1)-WBrS. Let (X, rl,7-2) be a (1,2)-BrS and U c T1 \ {~} b e any set. Then by Definition 4.1.1, g is of (1,2)-CatDII and by Corollary 1.5.14, U E (1,2)-Catgll(X). Hence, by (7) of Theorem 1.1.24, U c 2-CatgII(X), that is, U is of 2-CatgII and by Definition 4.4.18, (X, 7-1,7-2) is a 2-WBrS. Furthermore, if U c 7-1 \ {~;~} and U C 2-Catgli(X) or, equivalently, U is of 2-Catg II,
then by (7) of Theorem 1.1.24, U E (2, 1)-Catgii(X). It remains to use Definition 4.4.7. (2) follows via (2)of Theorem 4.4.10 and (1). Assertions of (3) are immediate consequences of (3) of Theorem 4.4.10 and (2). (4) Let g c 7-1 \ {~} be any set. Then, by (1) of Theorem 2.1.10, U c (1, 2 ) - ~ a t g i i ( X ) ~
U E 2-~atgii(X )
and so ( X l 7-11 7-2) is & (11 2)- B r S ,g-----5, ( X , 7-117-2) is a 2 - W B r S .
Thus it remains to use (4) of Theorem 4.4.10.
[3
R e m a r k 4.4.22. The BS's (R, co < s s) and (R, co < s r ) from Example 2.1.4 are 2-BrS's, and by (1) of Theorem 4.4.21, they also are 2-WBrS's, nevertheless aa -r s and co r r. T h e o r e m 4.4.23. The union of any family of 2-open 2-WBrsS's of a BS (X, 7-1 < 7-2) is a 2-WBrS. [,J us sES and A E 2-CatgI(U ) be any set. Let us show that U \ A E 1-D(U) so that w~intA - 2~ in (U,r~,r~). Contrary: let w~intA -r ~. Since A c U, there exists a set Us c tl such that r~ int A cq Us r ~. It is obvious that A rq Us c A implies A rq Us c 2-Catgi (U), and since Us E r~, the set A N Us E 2-datgi (Us). But (Us, w~, w~) is a 2-WBrS and thus Us\(Ac~Us) c 1-D(Us) so that w~ int(AnUs) = 2~. Since 2~ r r{ int A N Us c r~ int(A N Us), we come to the contradiction. D1 Proof.
Let 11 = {Us}sEs be a family of 2-open 2-WBrsS's of X, U =
C o r o l l a r y 4.4.24. The following conditions are satisfied for a BS (X, Vl < 7-2): (1) ( X I Tll 7-2) i8 a 2 - W B r S
if altd only if each point x c X ha8 a 2-open neighborhood which is a 2-WBrS.
160
IV. Baire-Like Properties of Bitopological Spaces
(2) If (X, q , r2) has a 2-pseudo-open covering t~ = {Us },es each of whose members is a 2-WBrS, then (X, rl, r2) is a 2-WBrS. Proof. This corollary is proved trivially.
D
D e f i n i t i o n 4.4.25. A BS (X, T1 < 7-2) is a 1-strict Baire space (briefly, 1-SBrS) if every nonempty 2-open subset of X is of 1-second category in X. Theorem
4.4.26.
The following conditions
are equivalent for a BS
( X , 7-1 ~ 7-2):
(1) (X, rl, r2) is a 1-SBrS. (2) If {Un}~c~__1 is any monotone decreasing sequence of subsets of X where CX9
Un E 7"1 N 1-D(X) for each n -
1, oc, then ~ Un E 2-D(X). n=l
(3) A E 1-Catgi (X) ~ X \ A E 2-D(X). (4) If {Fn}n~ is any countable family of subsets of X where Fn E
COT1 A
oo
1-13d(X) for each n -
1, c~, then U Fn e 2-13d(X). n=l
Proof. We can omit the proof like in the case of Theorem 4.4.19.
[Z]
C o r o l l a r y 4.4.27. The following conditions are satisfied for a BS (X, rl < re): (1) If (X, Wl, 72) is a 1-SBrS and Y E 72, then (IT, 7~, w~) is also 1-SBrS. (2) If (Y, v-{, r~) is a 1-SBrS and Y E 2-D(X), then (X, 7-1,7-2) i8 also a I-SBrS. (3) I f A c ( 1 , 2 ) - S D ( X ) a n d ( A , r ~ , r ~ ) i s a 1-SBrS, then(r1 intr2 clA, r~',r~') and (r2 int ~-2cl A, r;", 7~") are also 1-SBrS 's. Proof. (1) If Y C X, Y E ~-2 and A c 1-Catgi(Y) is any set, then A c 1-Catgi(X) and by (3) of Theorem 4.4.26, X \ A c 2-D(X). Furthermore,
cl y -
cl ( y
( x \ A)) -
d ( Y \ A)
so that Y = Y cqr2clY = Y n T2 cl(Y \ A) = r2 cl(Y \ A) and thus Y \ A c 2-D(Y). It remains to use once more (3) of Theorem 4.4.26. (2) Suppose that Y c X, Y e 2-D(X), (Y,r~,r~) is a 1-SBrS and A E 1-Catg I (X). Let us prove that X \ A r 2-D(X). Contrary: let T2 int A -r ~. Then r2 int A N Y -r ~ and r2 int A c~ Y c A implies that r2 int A A Y c 1- datg I (X). Since Y E 2-D(X) c 1-D(X), we have ~-2int A A Y E 1-Catg I (Y). But r~ int(r2 int A ~ Y) -/2~ and so r~ cl (Y \ (r2 int A A Y)) r Y, which is impossible. Assertion (3)follows directly from (1)and (2).
[:]
T h e o r e m 4.4.28. The following implications hold for a BS (X, rl < r2): (1)
2-WBrS
4==
(2,1)-WBrS ~
2- BrS
4== (1,2)- SBrS ==~ (1,2)- BrS ==~
A-(2,1)- BrS ~
We have for a BS (X, T 1 % C 7-2):
1-SBrS
~
1- BrS
~
2-WBrS (2,1)-WBrS
4.4. Some Modifications of Baire-Like Properties
(2)
2-WBrS
r
(2,1)-WBrS ~
2- BrS
~
(1,2)- SBrS ==~ (1,2)- BrS ==~
A-(2,1)-BrS r
1-SBrS
~
1-BrS
r
161
2-WBrS (2,1)-WBrS
We have for a BS (X, T1
(3)
2-WBrS
l~
r
(2,1)-WBrS r
2-BrS
g
r
(1,2)-SBrS ==~ (1,2)-BrS ===~ 2-WBrS
g
A-(2,1)-BrS r
g
1-SBrS
==~
1-BrS
r
(2,1)-WBrS
We have for a BS (X, 7-1 < s T2):
(4) (X, T1,T2) is a (1,2)-SBrS ,z---->, (X, TI,r2) is a 2-BrS z---->, (X, T1,T2) is a (2, 1)-BrS z--->, (X, T1,72) i8 an A-(2, 1)-BrS z---->, (X, T1, "/-2) i8 a (1,2)-BrS z---->, (X, T1,T2)is a n A - ( 1 , 2 ) - B r S ,z--5, (X, T1,T2)is a 2-WBrS ~( :, (X, 71,72) is a 1-BrS ,z---->,(X, T1,7-2) is a (2, 1)-WBrS ~ (X, 7-1,~-2) is a 1-SBrS. Proof. (1) Following (1) of Theorem 4.4.21 , it suffices to show that (X, 71,72) is a 1- BrS
(X, T1,T2) is
a
(1,2)-SBrS
--~
(X, 7-1,7-2) is a 1-SBrS ( X , T1,7-2) is aIl A - ( 2 , 1)-
BrS.
Indeed, if U c T2 \ {~}, then by Definition 4.4.1, U c (1,2)-Catgi~(X) and (7) of Theorem 1.1.24 gives U c 1- Catgli ( X ) so that the horizontal implication holds. If U c T1 \ {;~}, then U c ~-2 \ {~} and by Definition 4.4.25, U c 1 - C a t g ~ ( X ) so that the right-hand upper implication holds too. Finally, if U c 7-2 \ {~}, then by Definition 4.4.25, U c 1-Cat9ii ( X ) , and applying once more (7) of Theorem 1.1.24, we obtain U c (2,1)- Catgli ( X ) . Thus, by Definition 4.1.5 the right-hand lower implication is also correct. (2) Following (2) of Theorem 4.4.21, it suffices to show that (X, T1,7-2) is an A-(2, 1)-BrS implies that (X, 7"1,7"2) is a 1- SBrS. If U c 7-2 \ {2~} is any set, then U E (~,l)-Catgii(X). H e I I c e , b y ( 3 ) o f T h e o r e m 9,.2.20, U E 1-~a~gii(X). It remains to use Definition 4.4.25. Assertions of (3) follow directly from (2) above in conjunction with (3) of Theorem 4.4.21. (4) Let U c ~-2 \ {2~} be any set. Then, by (1) of Theorem 2.1.10, U 6 1-Catgli (X) ~
U 6 2-Catg~i (X)
so that (X, T1,T2) is a 1-SBrS <--> (X, TI,r2) is a 2-BrS and it remains to use (4) of Theorem 4.4.21.
[]
T h e o r e m 4.4.29. The following condition is satisfied for a BS (X, T1 < T2): (1) The union of any family of 1-open 1-SBrsS's is a 1-SBrS. For a BS (X, 7-1 < C 7-2), We have:
(2) The union of any family of 2-open I-SBrsS's is a I-SBrS.
162
IV. Babe-Like Properties of Bitopological Spaces
Proof. (1) Let tl = {Us }sos be a family of 1-open 1-SBrsS's of X, U =
U Us sES
and A c 1-Catgi(U ). We shall show that U \ A c 2-D(U) so that ~-~int A = ~ in (U, 7-~,T~). Assume the opposite: 7-~int A -r ~. Clearly, there exists a set Us c tl such that ~-~int A c~ Us # z and A • Us E 1-CatgI(U ). Since Us c ~-~, the set A f-I Us c 1-Catgi(Us ) in (Us,T~,T~). By (3)of Theorem 4.4.26, ~-~ cl (Us \ (A ~ Us)) - Us so that ~-~ int(A c~ Us) - 2~. But 2~ # ~-~int A A Us c 7-~ int(A c~ Us), which is impossible. (2) Let 11 = {Us}sos be a family of 2-open 1-SBrsS's of X and U = U Us. s6S
Since Us c 7-2 for each s E S, by (2) of Corollary 2.2.8, ~-~ < c ~-~ for each s E S. Hence, following (2) of Theorem 4.4.28, (Us, T[, ~-~) is a 1-SBrS ~
(Us,T~,T~) is an A-(2,1)-BrS
for each s E S and Theorem 4.2.7 gives that (U, ~-~,~-~) is an A-(2, 1)-BrS. Since U c ~-2 \ {2~} and once more applying (2) of Corollary 2.2.8, we find that ~-[ < c 7-~ and by (2) of Theorem 4.4.28, (U, ~-~,~-~) is also a 1- SBrS. E] C o r o l l a r y 4.4.30. The following conditions are satisfied for a BS (X, T1 < 7"2) : (1) (X, 7-~,7-2) is a 1- SBrS if and only if each point x c X has a 1-neighborhood which is a I-SBrS. (2) If (X, "t-1,7-2) has a (1, 2)-pseudo-open covering t2 = {Us }~cs each of whose members is a I-SBrS, then (X, 7-1, T2) is a I-SBrS. For a BS (X, T1 < c ~-2), we have: (3) (X, T1, ~-2) is a 1- SBrS if and only if each p o i n t x c X has a 2-neighborhood which is a I-SBrS. Proof. (1) and (3) follow from (1) and (2) of Theorem 4.4.29, respectively. Assertion (2) is an immediate consequence of (2) of Corollary 4.4.27 in conjunction with (1) of Theorem 4.4.29. [3
CHAPTER V
Dynamics of Bitopological Relations, Baire-Like Properties and Dimensions In studying various bitopological concepts, it is of interest to know what types of functions preserve these or other properties of BS's. Since in order to be a BS, the set should have two different structures of its subsets, it is reasonable to expect the function to possess certain additional properties so that it could be used for comparing BS's. It is therefore natural that we first investigate different classes of mappings of BS's together with their interrelationship and then use them to study how the properties of BS's are preserved both to an image and an inverse image [85], [95I. The studied different families of subsets of BS's (X, T1, ~-2) and (Y, 71,72) are of crucial importance in defining different classes of functions of X to Y, while the inclusion and the relations S, < s , C, < c , N, and
164
V. Dynamics of Bitopological Relations and ...
functions [120], and feeble homeomorphisms [133], we prove the preservation to an image of some Baire-like properties and establish that the S-, C-, and N- relations are bitopological properties. Furthermore, the conditions are formulated, preserving the (i, j)-quasi regularity to inverse image, the (2, 1)-pseudocompleteness to image for 7"1 C 7"2, 71 C 72, and the property of a BS to be an (1,2)-Baire space both to an image and an inverse image for 7-1 C T2, 71 C 72" The transformation of an almost-(2, 1)-Baire space into a (1, 2)-Baire space and vise-versa is also interesting. Finally, we study relations between the pairwise small and large inductive dimensions of the domain and the range of a d-closed and d-continuous function.
5.1. Mappings of Bitopological Spaces D e f i n i t i o n 5.1.1. A function f : (X, T1,7"2) --+ (Y, 71,72) is said to be (i, j)-semicontinuous i f f - l ( 7 i ) C ( i , j ) - S O ( X ) so that f - l ( u ) C Tj el Ti int f - l ( g ) for every set U c 7iThe classes of all (i,j)-semicontinuous functions of X to Y are denoted by (i, j)- 8C(X, Y). Similar descriptions of the classes (i, j)- SC(X, Y) can be found in [85] and [175], respectively. T h e o r e m 5.1.2. The conditions below are equivalent for a function f : (X, T1,7"2) ~ (Y, 71,0/2): (1) f E ( i , j ) - $ C ( X , Y ) . (2) f - 1 ( 7 i intA) C 7"j cl~-i int f - l ( A ) for every subset A c Y. (3) f - l ( c o T i ) c ( i , j ) - S C ( X ) so that Tj int Ticl f - l ( F ) C f - l ( F ) for every set F c coT~. (4) Tj int 7"i cl f - l ( A ) C f - l ( 7 i clA) for every subset A c Y. (5) For each point x e X and each i-neighborhood V(y) of the point y = f (x), there exists an (i,j)-semineighborhood U(x) such that f(U(x)) c V(y). (6) For each point x e X the inverse image f-l(U(y)) of each i-neighborhood U(y) of the point y = f ( x ) is an (i, j)-semineighborhood of x. (7) f ( ( i , j ) - s c l A ) c 7 / c l f ( A ) for every subset A c X . (8) (i,j)-scl f - l ( A ) c f - l ( 7 / c l A ) for every subset A c Y.
Proof. The equivalences (1) ,z--->, (2), (3)z--->, (4), (5) ,z---->,(6) and (7) ,z--> (8) are obvious. Thus, it suffices to show that (1) ,z----5,(3), (1) ,z--> (5) and (3) ,z--> (7). It is clear that f-l(v)
c Tj cl ~-/int f - l ( v )
z, : , T j i n t r / c l f - l ( y \ u )
for every set U c 7/~:
cf-I(y\u)
:"
for every set Y \ U 9
and so (1) ,z----5,(3). (1) ~ (5). First, let x 9 X be any point and U(y) be any/-neighborhood of the point y = f(x). Then there is an/-open set U c Y such that y c U C U(y). By (1), f - l ( u ) c rj cl 7-/int f - l ( u ) and thus, following Theorem 1.3.12, f-l(u) c ( i , j ) - S o ( x ) so that U ( x ) = f - l ( g ) i s the required (i,j)-semineighborhood of x. Conversely, let U 9 7i be any set. If x 9 f - l ( u ) is any point, then by (5), there is an (i,j)-semineighborhood U(x) such that f(U(x)) c U. It can be
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V. Dynamics of Bitopological Relations and ...
(2) ~ (3). Let U c (i,j)-OD(Y) be any set and x E f - l ( u ) be an arbitrary point. Then by (2) there is a n / - o p e n neighborhood U(x) such that f(U(x)) c U so that U(x) c f - l ( U ) . Since x c f - l ( u ) is arbitrary, we obtain f - l ( u ) c ri. Conversely, let x c X be any point and U(y) E (i, j)-OD(Y) be any neighborhood of the point y = f(x). Then, by (3), f - l ( g ( y ) ) = g ( x ) E 7-i is the required/-open neighborhood of the point x. D1 5.1.10. Let (X, rl, r2) and (Y, 71, 72) be BS's. Then for have: (1) (1,2)-AC(X,Y) c 1-AC(X,Y) A2-AC(X,Y) C (2,1)-AC(X,Y). (x,n < (Y, < c haw: (2) (1, 2)-AC(X, Y) = 1-AC(X, Y)
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d-C(X, Y) c d-AC(X, Y) n p-AC(X, Y). But the inverse inclusions do not hold. E x a m p l e 5.1.11. Let us consider the real line IR together with the following four topologies: rl = wc is the co-countable topology on IR, r2 = w is the natural
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178
V. D y n a m i c s of Bitopological R e l a t i o n s and . . .
(3) If (Y, 71,72) is p-almost regular, then
p-We(X, z) = p - A t ( x , Y) = p-oc(x, z). (4) If (x, ~, ~-~) i~ p - ~ i ~ S a ~ ,
th~
p - w e ( x , Y) n p-AO(X, Y) < p-AC(X, Y).
(5) If (Z, ~1, ~ ) i~ p-~r162
thr
p-AC(X,Y) =d-C(X,Y). (6) If (Y, ")/1,")'2) is p-regular, then
d-C(X, z ) = p - A C ( X , z ) = p - w e ( x , z ) = p-OC(X, z). Proof. (1), (4) and (5) are immediate consequences of (1), (4) and (5) of Theorem 5.1.35. (2) By (2) of Theorem 5.1.13, p-AC(X, Y) c p-OC(X, Y) so that
p-AC(X, Y) n p-AO(X, Y) c p-oc(x, Y) n p - A o ( x , Y). On the other hand, (2) of Theorem 5.1.35 gives
p- OC(X, Y) N p-AO(X, Y) c p-AC(X, Y) N p-AO(X, Y). (3) If (Y,')'l,~'2) is p-almost regular, then by (3) of Theorem 5.1.35,
p-WC(X, Y) = p-AC(X, Y). On the other hand, following (2) of Corollary 5.1.14,
p-AC(X, Y) c p- oc(x, Y) c p - w e ( x , Y) and hence
p-AC(X, Y) = p-OC(X, Y) = p-WC(X, Y). (6) By (3) of Proposition 0.1.7, (Y, "Yl,72) is p-semiregular and p-almost regular, and thus it remains to use (3) and (5). Vl Definition 5.1.37. A function f : (X, r l , T 2 ) - + (Y,~/1,72)is said to be /-feebly continuous (/-feebly open) if V c 7i \ {0} (V E Ti \ {2~}) and f - i ( v ) :/: 2~ imply that Ti int f - l ( v ) ~ 2~ (~/i int f(V) ~ o) [120]. The classes of all/-feebly continuous (/-feebly open) functions of X to Y are denoted by i-UC(X, Y) (i-UO(X, Y)). Definition 5.1.38. A bijection f : (X,T~,T2) ~ (Y, " f l , ~ / 2 ) i s /-feeble homeomorphism if f c i-JcC(X, Y) ~ i-UO(X, Y) [133].
said to be an
The classes of all /-feeble homeomorphisms of X to Y are denoted by
i-,~7-t(X, Y). It is obvious that i-7-l(X, Y ) c i-oPT-l(X,Y). P r o p o s i t i o n 5.1.39. Let (X, T1, T2) and (Y, ~/1, ~2) be BS 's. Then the follow-
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V. D y n a m i c s of B i t o p o l o g i c a l R e l a t i o n s a n d . . .
implies that f(X) = Y is of (i,j)-Catg II so that Y is of (i,j)-Catg I implies that X is of (i, j)-Catg I. (3) Let Y b e o f (1,2)-datgI. Since f e d-.P~(X,Y), by (3) of Theorem 5.1.41, oo
f e (1,2)- 50(X, Y). Clearly, Y -
U An, where An e (1,2)-AfD(Y) for each n=l
n = 1, oc. Hence (x)
s
(x)
s l(U
U s 1( o) 9
n--1
n=l
By (1) of Definition 5.1.40, / - l ( A n ) e (1,2)-AfD(X) for each n - 1, oc and so X is of (1,2)-CatgI. On the other hand, f E d-.P~(X, Y) implies that f - 1 c d-5~(Y,X) and so f - 1 E ( 1 , 2 ) - 5 0 ( Y , X ) . Hence, if X is of (1, 2)- Catg I, by analogy with the above we find that Y is of (1, 2)-CatgI. The rest is obvious. (4) Indeed, by ( 4 ) o f Theorem 5.1.33, f e (1,2)-AO(X, Y) and, therefore, it remains to use (4) of Theorem 5.1.41. K] D e f i n i t i o n 5.1.43. A function onto f : (X, ~'1, ~-2) ~ (Y, 71,72) is said to be /-irreducible if f(F) r Y for every proper/-closed subset F c X [13]. The classes of all/-irreducible functions of X onto Y are denoted by i-Zr(X, Y). By (i) of Theorem 4.10 in [133], i-lr(X, Y)N i-Cl(X, Y) c i-~O(X, Y). Finally, in comparison with Definition 5.1.37, consider the special modifications of feebly continuous and feebly open functions. D e f i n i t i o n 5.1.44. A function f : (X, T1,T2) ~ (Y, 7 1 , 7 2 ) i s said to be (i,j)-feebly continuous ((i,j)-feebly open) if V E ~/i \ {Z} (V c Ti \ {z}) and f - l ( v ) ~ z imply that Tj int f - l ( v ) ~ z (~/j int f(V) ~ ;~). The classes of all (i, j)-feebly continuous ((i, j)-feebly open) functions of X to Y are denoted by (i,j)-.PC(X, Y) ((i,j)-~O(X, Y)). It is easy to ascertain that the inclusions (2, 1)-FC(X, Y) C 1-SC(X, Y) N
(2, 1)-FO(X, Y) C 1-FO(X, Y) and
N
2-5C(X, Y) C (1, 2)-FC(X, Y)
N
N
2-~c(.9(X, Y) C (1, 2)-~cO(X, Y)
hold for BS's (X, T1 < ~-2) and (Y, ~/~ < 72). The statements below are immediate consequences of the corresponding definitions. P r o p o s i t i o n 5.1.45. For a function f : (X, T1, ~-2) --* (Y, "71, ~/2) the following
conditions are satisfied: (1) If f is onto, then f E (i,j)-.TO(X,Y)~ (2) f C (i,j)-.FC(X, Y ) ~
f-I(j-:D(Y))C i-T)(X).
f(j-D(X)) C i-D(Y).
C o r o l l a r y 5.1.46. For a function f : (X, T1,7-2)
conditions are satisfied:
--+
(Y, ")/1,")/2) the following
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184
V. Dynamics of Bitopological Relations and ...
Hence all three relations are bitopological properties. T h e o r e m 5.2.3. If f : (X, 7-1,7-2) ----+ (Y, ~1, ~2), f C d- C(X, Y), and f-l(7i)fT-i, then (Y, 71,72) is (i,j)-quasi regular implies that (X, 7-1,7-2) is (i, j) -quasiregular.
Pro@ Indeed, let U 6 7-i \ {~} be any set. Then by Remark 2.1.14, there is a set V E 7i \ {2~} such that f - l ( V ) c U. Since (Y, 71,72) is (i,j)-quasi regular, there exists a set W c 7~ \ {z} together with 7j cl W c V. Furthermore, f c d-d(X, Y) implies f E i-d(X, Y), and since f is onto, we obtain f-l(w)
c 7-i \ { ~ } ,
f-l(w)
C f - l ( 7 j clW) C f - l ( v )
C g.
Also, f c d-d(X, Y)implies f c j-C(X, Y) so that 7-j el f - l ( w )
where f - l ( w )
c 7-i \
{~}.
C f - l ( 7 j el W) C U
Hence (X, 7-1,7-2)is (i,j)-quasi regular.
D
C o r o l l a r y 5.2.4. If for a function f : (X, 7-1,7-2) ----+ ( Y , ' ) ' 1 , ~ 2 ) we have: f c d - C ( X , Y ) , f-1(71)/7-1 and f-l(72)fT-2, then (Y, 7 1 , 7 2 ) i s p-quasi regular implies that (X, 7-1,7-2) is p-quasi regular. T h e o r e m 5.2.5. If for a one-to-one function f : (X, 7-1,7-2)
~
(Y,
~1, ~'2),
We
have: 7-1 = f-1(~/1) C f - l ( ' ) / 2 ) C 7-2, (X, 7-1,7-2) i8 (2, 1)-pseudocomplete, (Y, 71,'y2)
is (2, 1)-quasi regular and f(7-2 \ {z}) is a pseupobase for 72, then (Y, 71,'y2) is (2, 1)-pseudocomplete. Proof. It is obvious that T 1 -- / - - 1 ( ~ 1 ) and f-1(~2) c 7-2 imply f c d-C(X,Y). Also, the fact that f is one-to-one onto gives 0/1 C ")'2. Since (Y, 71, ~2) is (2, 1)-quasi regular, by the proof of Theorem 5.2.3, the BS (X, 7-1, f-l(~/2)) is also (2, 1)-quasi regular. Let U C 7-2 \ {2~} be any set. Then f(7-2 \ {~}) being a pseudobase for 72 implies that V = 72 int f(U) r ~ and since f is a bijection, f - l ( V ) c U, where V c 72 \ {2~}. Thus 7-1 C f-1(72) C 7-2 and each member of 7-2 \ {2~} contains a member of f - l ( ~ 2 \ {2~}). Hence, by (2) of Theorem 4.2.13, (X, 7-1, f-1(72)) is (2, 1)-pseudocomplete as (X, 7-1,7-2) is (2, 1)-pseudocomplete. But (X, 7-1,f-1(72)) is d-homeomorphic to (Y, 71,72) and thus (Y, 71,72) is also (2,1)-pseudocomplete. [3 The objects of our further consideration are category notions, Baire-like properties, and pairwise inductive dimensions in the context of different classes of functions. The preceding results enable us to prove Theorems 5.2.6, 5.2.8, 5.2.10 and 5.2.16 below. T h e o r e m 5.2.6. Let a f u n c t i o n f : (X, 7"1 < 7"2) ~ (Y, ~1 < ")'2) be 1-continuous and d-open, where X has a countable 1-pseudobase and Y is of (1, 2)-Catg II. If there is a subset Z C Y such that Y \ Z e (1,2)-Catg~(Y) a n d / - l ( z ) is of (1,2)-CatglI for each point z E Z, then X is of (1,2)-datgII.
Pro@ Contrary, let X be of (1, 2)-datg I. Then there is a sequence { F n " Fn c
n (1,
- 1,
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5.2.
Mappings
Applied
to Some
Bitopological
Notions
189
implies that (Y, 71,72) i8 a (1, 2)- BrS and (Y, 71,72) i8 a~t A-(2, 1)- BrS implies that (X, 7-1,7-2) is a (1, 2)- BrS. Proof. Let {gn}n~ be a sequence where Un c 72 N 1-l)(Y) for each n - 1, oo. We shall prove that the sets 7-1 int f-l(Un) C 2-l)(X) for each n - 1, oc. Let xo C X, n E N be arbitrarily fixed and U(xo) E 7-2 be any neighborhood. Since f c (2,1)-YO(X, Y), we have 71intf(U(xo)) ~ 2~ so that there exists V c 71 \ {2;~} such that V C f(U(xo)). The set V N Un E 72 \ {2~} since Un E 1-:D(Y) and "Yl C 72. But also f E (2, 1)-oPC(X, Y ) a n d hence 2~ r W - 7-1 int f - I ( V N U n ) . Therefore W C f - l ( v N gn) C f - l ( f ( g ( a ? 0 ) ) _ g(xo). It is clear that W c 7-1 int f-l(gn) and, therefore, xo c 7-2 c17-1 int f - l ( U n ) since U(xo) C 7-2 is an arbitrary neighborhood. Let Bn - 7-1 int f - l ( U n ) . Since x0 C X and n E N are arbitrarily fixed, the set B~ c 7-1 N 2-~)(X) for each n - 1, oc. On oo
the other hand, (X, 7-1,7-2) is an A-(2, 1)- BrS implies that
~ B~ c 2-~D(X) and rt=l
oo
hence, by ( 2 ) o f Proposition 5.1.45, f ( ~ B~) E 1-2)(Y)since n=l
f E (2, 1)-.YC(X, Y ) c
(1,2)-}'C(X, Y).
Furthermore, O<3
s( N n=l
O<3
OO
N
cN
n=l
n=l
oK)
implies that the set n u-~ ~ 1-/9(Y) and thus (Y, 71,72) is a (1,2)-BrS. n=l
Finally, it is evident that
f ~ (2, 1)-7C(X, Y) ~
f -~ c (2, 1 ) - f O ( Y , X),
f C (2, 1)-.YO(X,Y) ~
f - 1 c (2, 1)-.YC(Y,X)
and, therefore, the rest is clear.
K]
Proposition 5.2.11. If f : (X, 7-2) ~ (Y, 7) is a feebly open condensation, (Y, 7) is a BrS and T 1 : f - 1 ( 7 ), then (X, T1,T2) is a 2-BrS and, therefore, it is also a 1-BrS, a (1,2)-BrS, an A-(2,1)-BrS, a 2-WBrS, a (2, 1)-WBrS, a 1-SBrS, and a (1,2)- SBrS. Proof. Under the hypotheses, f is a feeble homeomorphism and by Proposition 4.3 in [133], (X, T1,7-2) is gt 2 - B r S . Thus it remains to use (4) of Theorem 4.4.28 since by Theorem 2.1.15, we have T1 ~ S 7-2. [--7 Since f is a feeble homeomorphism, it is evident that if in Proposition 5.2.11, instead of (Y, 7) is a BrS, we assume that (X, 7-1,7-2) satisfies at least anyone of the Baire-like properties, then we obtain that (1/, 7) is a BrS. P r o p o s i t i o n 5.2.12. For a function f : (X, 7-1,7-2) --~ (Y, 71,72) where f C i - C ( X , Y ) N i - O ( X , Y ) N j - C I ( X , Y ) , we have ( i , j ) - i n d X = 0 > ( i , j ) - i n d Y = O.
Pro@ Let y c Y be any point and let U(y) be its a n y / - o p e n neighborhood. For an arbitrary point x E f - l ( y ) , choose a n / - o p e n neighborhood U(x) in a manner
190
V. D y n a m i c s of Bitopological R e l a t i o n s and . . .
such t h a t f ( U ( x ) ) c U(y). Since ( i , j ) - i n d X = 0, there exists a neighborhood V(x) C T i n CO Tj such t h a t V(x) c U(x). T h e n
f ( V ( x ) ) C U(y) and f ( V ( x ) ) c 7i n co~/j so t h a t (i, j)- ind Y = 0.
D
Corollary 5.2.13. For a function f : (X, ~-1,~-2) ~ (Y, ~/1, ~/2), where f c d-C(X, Y) N d-O(X, Y) n d-Cl(X, Y), we have p - i n d X = 0 ~ p - i n d Y = 0. Corollary 5.2.14. For a function f : (X, T1,7"2) --+ (Y~ ~/1, ~/2), where f E d- C(X, Y) N d - O ( X , Y) N d- CI(X, Y) and (X, ~-1,T2), (Y, ~1, ~/2) are both R-p-T1 and d-second countable, we have p-XndX - 0 ( ~
p-dimX
- 0) --, p-IndY
- 0 ( ~
p-dimY
- 0).
Proof. If p - I n d X
= 0, then by Corollary 3.2.13, we have p - i n d X = 0. Hence Corollary 5.2.13 gives t h a t p - i n d Y = 0 and, once more applying Corollary 3.2.13, we conclude t h a t p - I n d Y = 0. T h e rest follows directly from (1) of Corollary 3.3.5. K] It is clear t h a t if a function f : ( X , T1,7"2) ---+ ( Y , ~ 1 , ")/2) is d-closed, d-continuous, and A c X, A c coT1 n coT2, t h e n the restriction flA is also d-closed and d-continuous. But in contrast to this fact, flA is not, generally speaking, d-closed if A c p - C l ( X ) . E x a m p l e 5 . 2 . 1 5 . Let X = {a, b,c,d, e}, 7"1 = {~, { a } , X } , 7-2 = {~, { b } , X } , Y = {0,1}, ~/1 = {2~,{0},Y} and 72 = { ~ , { 1 } , Y } . If A = {c,d,e} and f : (X, T1,T2) ~ (Y, 71,~/2) is defined as f(a) = f(b) = 0 and f(c) = f(d) f(e) = 1, t h e n f is d-closed, but the restriction ZIA, where A c p - C l ( X ) is not d-closed. T h e o r e m 5 . 2 . 1 6 . Let f " (X, T1, 7-2) --~ (Y, ~1, ~2) be a d-closed and d-continuous function of a R-p-T1, d-second countable, and p-normal BS X to a R-p-T1, d-second countable, and p-normal BS Y such that for every set A E p - C l ( X ) the restriction flA " A ~ f (A) is also d-closed and d-continuous. If there is an integer k > 1 such that I f - l ( y ) l < k for every y c Y, then
(i,j)-indY < (i,j)-indX + k-1,
(i,j)-IndY < (i,j)-IndX + k-1
and, therefore, p-indY
p-IndY
Proof. By T h e o r e m 3.2.31, it suffices to prove only the first inequality. We can suppose t h a t 0 < (i, j)- ind X < oc and apply induction with respect to the n u m b e r n + k where n - (i, j)- ind X. If n + k - 1, t h e n k - 1 because f is onto. Therefore f is a d - h o m e o m o r p h i s m and the t h e o r e m holds. Assume t h a t the t h e o r e m holds when n + k < m where m > 2, and consider a d-closed and d-continuous function f " X + Y such t h a t f ( X ) - Y and n + k - m. Let Bi be a c o u n t a b l e / - b a s e of (X, rl, r2) such t h a t (i, j ) - i n d ( j , i)- Fr U < n - 1 for every U E Bi. If U c B~ is an a r b i t r a r y set, t h e n f E d-gl(X, Y) N d-g(X, Y)
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192
V. Dynamics of Bitopological Relations and ...
k >_ 1 such that ] f - l ( y ) ] <_ k for every y E Y, then p - i n d Y and p - Ind Y _< p - Ind X + k - 1.
_< p - i n d X
+ k-
1
Proof. It suffices to note t h a t if 7-1 C 7-2, t h e n (X, T1,7-2) is 1-Zl ((Y, 71,72) is l-T1) z---->, (X, 7-1,7-2) is R-p-T1 ((Y, 71,72) is R-p-T1), (X, T1, "/-2) is 1-second c o u n t a b l e ((Y, 71, "/2) is i - s e c o n d c o u n t a b l e ) ,z--->, (X, rl, r2) is d-second c o u n t a b l e ((Y,'/1,'/2) is d-second c o u n t a b l e ) , a n d co7-2 = p-Cl(X). K] Corollary 5 . 2 . 1 8 . L e t f " ( X , T 1 < N 72) ~ (Y, 71 < N 72) be a d-closed and d-continuous function from a l-T1, 1-second countable, and 2-normal BS X to a l-T1, 1-second countable, and 2-normal BS Y such that for every set A E co~-2 the restriction f A" A ~ f ( A ) is d-closed and d-continuous. If there is an integer k > 1 such that f - l ( y ) l < k f ~ every y C Y, then
i-indY
- ( i , j ) - i n d Y <_ i - i n d X
- (i,j)-indX + k-
1,
i-IndY
- ( i , j ) - I n d Y <_ i - I n d X
- (i,j)-IndX + k-
1,
and, therefore, d-indY
- p-indY
<_ d - i n d X
- p-indX
+ k-
d-IndY
- p-IndY
< d-IndX
- p-IndX
+ k - 1,
1,
where
d-indX
G n ,z----5, ( 1 - i n d X
G n A 2-indX
G n),
d-IndX
_< n ,z----5, ( 1 - I n d X
G n A 2-IndX
G n).
Proof. T h e p r o o f follows directly from C o r o l l a r y 5.2.17, (4) of C o r o l l a r y 2.3.13, a n d (6) of T h e o r e m 3.1.36. D
C H A P T E R VI
Generalized B o o l e a n Algebras and R e l a t e d Problems. R e p r e s e n t a t i o n T h e o r e m s From the late 19th century the tendencies enabling a scholar to view different mathematical theories from the unified standpoint began to gain prevalence and reached rather a high level of development. This observation is confirmed by many studies, of which, keeping in mind our interest, we would like to mention M. H. Stone's remarkable theory of representations that brought together the plain algebraic structures of Boolean algebras with topological type structures, that is, with a special class of TS's [246], [247]. For quite a long time the proofs of theorems had been based on the Boolean algebra laws. G. Boole was the first to formalize these laws [36], [37], thereby providing a powerful incentive to the development of symbolic logic [28]. Subsequently, Boolean algebras became widely adopted in such areas as probability theory, functional analysis, topology, Stone's representation theory, and others. Chapter VI continues and develops the study begun in [91], [95], [100]. If, presumably, the definition of a Boolean algebra employs the notion of a partially ordered set (briefly, poset), whereas a general definition of a TS is based on the notion of a metric, then by considering a more general situation with quasi metrics one may obtain the notion of a BS. A further development of the theory of BS's made it possible to introduce and study an algebra of a new type that is based on a nonordinary variant of a quasi ordered set and the corresponding representation of which brings one to BS's. We believe that this algebra, which we call a generalized Boolean algebra, is interesting not only as the subject of independent research, but also can be used as an important tool in establishing its various relationships with other areas of mathematics. Moreover, since a quasi ordered set of special type, which is the union of two posets forms the basis for defining both generalized Boolean algebras and double Boolean algebras [134], [261] and the latter algebras have already been used in metamathematical studies, due to this common basis, generalized Boolean algebras can also be used in mathematical logic. In Sections 6.1-6.6 we introduce and study generalized Boolean algebras and some other related important questions. To this end, we first define a generalized ordered set (briefly, goset) in terms of a quasi ordered set which is a union of two posets whose partial orders are induced by the quasi order relation and whose intersection contains only zero and unit elements. Furthermore, to define a generalized lattice, for any subset of a goset we introduce the important notions of i~.-inf and i(;-sup. A different approach to defining a generalized lattice as 193
194
VI. Generalized Boolean Algebra and Related Problems
an algebra, satisfying in particular, the generalized associativity and generalized absorption laws, is also used. Theorem 6.1.15 shows how we can pass from the generalized lattice, defined by means of a goset, to the generalized lattice using an algebra and vice versa. The results obtained make it possible to connect the posers, making up the goset, by the one-to-one correspondence and thus to introduce the basic notion of a generalized Boolean algebra. The one-to-one correspondence between the posers is determined by the generalized identity operator which plays an important role in our further studies because, on the one hand, it determines the interdependence of two structures forming a generalized Boolean algebra and, on the other hand, allows us to characterize a generalized Boolean algebra in a different way. We also discuss some other operators and binary operations and give a few interesting examples of generalized Boolean algebras. Furthermore, we assign the V-formation and the corresponding strong amalgamation to every generalized Boolean algebra, and we define a generalized quasi measure on a generalized Boolean algebra and a bitopological generalized Boolean algebra. In Sections 6.2 and 6.3, we obtain key generalizations of an ideal and its variety such as a prime, that is, a maximal generalized ideal, left and right principal generalized ideals, and generalized ideals generated by different pairs of families of sets. We introduce and investigate the notion of a generalized Stone family of prime generalized ideals as well as the notion of a generalized dual to a generalized ideal, that is, the notion of a generalized filter together with its variety. The relation between generalized components and generalized ideals is also established. Other results obtained in the first half of this chapter concern the important notions of generalized homomorphic and generalized isomorphic maps, (i,j)-atoms, and p-atomic generalized Boolean algebras and generalized Boolean factor algebras. The remaining three sections deal with the bitopological modification of Stone's representation theory, generalized complete generalized Boolean algebras, and generalized Boolean rings. We introduce the notions of a generalized field of sets, its reduced version, and generalized field representation. Based on the results of Section 6.2, it is proved, in particular, that there exists a one-to-one correspondence between the reduced generalized field representations of a generalized Boolean algebra .4 and the generalized Stone families of prime generalized ideals of ,4. Thus every generalized Boolean algebra becomes generalized isomorphic to the reduced generalized field of sets. With every generalized field representation of a generalized Boolean algebra we associate a BS and give the necessary and sufficient conditions under which this BS becomes FHP-compact. Our main result is the generalized version of Stone's representation theorem which states that under special hypotheses there exists a one-to-one correspondence between generalized Boolean algebras and p-zero dimensional, p-Hausdorff and FHP-compact (also called Boolean) BS's. We would like to recall that an attempt at connecting the bitopological structure with a generalized algebraic structure was for the first time made in [91]. Throughout this chapter, the term "generalized" will be denoted for brevity by the symbol "G" and k, 1 c { 1, 2} , k ~r 1.
6.1.
Gosets,
Generalized
Lattices,
...
195
6.1. Gosets, Generalized Lattices, Generalized Boolean Algebras, and the Corresponding Operations Definition 6.1.1. A quasi ordered set (P, 4 ) is said to be a goset if P P1 U P2, where (P~, <) are nonempty posets and the restrictions 4 ]P~ -<_. i
i
It is clear that every poset is a goset, but not conversely.
Example 6.1.2.
(X, T1,T2) be
Let
a nonempty BS.
Then the families
j-OZ)(X) r ;g # (i, j)-OT)(X) are partially ordered by the set-theoretic operation inclusion and the binary relations, defined on the sets
P - j-oz)(x)
o (i, j ) - o z ) ( x )
as follows A1, A2 c P, A1 4 A2 <--> Aa c_ ~-j clA2 are the relations of quasi order on P. Moreover,
(P - j - o z ) ( x )
o
j)-oz)(x), 4 )
are gosets. Indeed, by (2) of Proposition 1.3.10, the restrictions 4 A1 4 A2 ~
I(i,j)-oz)(x) give
A1 C_ Tj clA2 <---> Tj clA1 C_ Tj clA2 <----> A1 C_ A2
(i,j)-oz)(x) =c_=<. Similarly, by the topological correspondence of (2)
and so ~
i
of Proposition 1.3.10, 4
Ij-oz)(x) =c_=<_. J
The quasi order 4 , defined in Example 6.1.2, shows that (P (1, 2)-OD(X), 4 ) in Example 1.3.5 is a goset. For the sets
-
2-or(x)u
{a, b} c 2-OT)(X) and {at c (1, 2)-OT)(X) we have {a,b} 4 a and {a} 4 {a,b}, but {a,b}-r {a}. Thus t h e g o s e t (P, 4 ) is not, generally speaking, a poset. If (P, 4 ) is a goset, then for any pair x, y c P the notation x -~ y means z 4 Y and z -r y. Therefore for z c P1 \ P2, Y c P2 \ P1, we have z -< y <---> x 4 Y, whereas for z, y c Pi, we have z -~ y <---> z < y. i
Therefore in Example 1.3.5, we have {a, b} -~ {a} and {a} -~ {a, b}. e b
f
.. -TPx, --- -'- -'-
a
//
1 1 J
d
f
Diagram 1
In
Diagram
goset (P = {a,b,c,d} and Pe - {c,e,f}, the small circles denote the elements; the circles, corresponding to the elements x, y c Pi, are connected by the solid line while for x c Pi and y c PO, the connection is broken.
{a,b,c,d,e,f},4
1
of
the
), where P1
196
VI. Generalized Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
The quasi order r e l a t i o n x 4 y for x E Pi, y E Pj means that y is over x or x and y lies on the same horizontal line; whereas the restriction 4 IP, - < gives that x < y if y is over x or x - y. i
It clearly follows that for a E P1, f E P2, we have a -< f and f -< a, a ~ f. Note that in the sequel, the indices i~; and Ja satisfy as usual the conditions i a , j a C {1,2}, i~ =fi JG, and, moreover, accentuate a more general character of the corresponding notions as compared with usual structural ones. D e f i n i t i o n 6.1.3. Let (P, 4 ) be a goset and A c_ P be a subset such that A N P1 - A1 ~ ~ ~ A2 - A N P2. Then A is called to be a G.chain (G.convex set) if a 4 b o r b 4 a for each pair a, b E A ( a , b c A a n d a 4 c4 bimplycEA). If A c_ P is a G.chain (G.convex set), then A~ are chains (convex sets) in the Usual sense.
An element xE Pi is an ia-lower (/,-upper) bound of a subset A - A 1 U A 2 c_ P, where ( P - P 1 U P 2 , 4 ) isagoset, ifx <_aforeachac Ai a n d x 4 b f o r e a c h i
b E A j (a <_ x for each a c Ai and b 4 x f o r e a c h b c A j ) . i The sets of all ia-lower (/,-upper) bounds of A are denoted by i a - l ( A ) - { x C Pi " x < a, a c Ai and x 4 b, b c A j } i
(ia-u(A)
- { x E Pi " a <_ x, a c Ai and b 4 x, b E A j } ) . i
Moreover, l~;(A) - I ~ - / ( A ) U 2 , - l ( A ) - { x E P " a ~ x, a E A } (ua(A)
- l a - u ( A ) U 2 a - u ( A ) - {x C P " x 4 a, a C A } ) .
Furthermore, if (P, ~ ) is a goset, then a subset A c_ P is G.normal (or A is G.normally included in P) if a c A, x r P and x 4 a imply x r A. Hence, if A is G.normal, then l~; (A) c A. D e f i n i t i o n 6.1.4. An element x c Pi is called an /a-smallest (/a-largest) element of the goset (P - P1 U P2, 4 ) if x is the smallest (largest) element of the poset (Pi, <) such that x 4 a (a 4 x) for each a c Pj. i
The i~;-smallest element (if it exists) will be denoted by the symbol ic-@ (i~-zero element), while the io-largest element (if it exists) will be denoted by the symbol i~jm (iG-unit element). Since the existence of smallest (largest) elements of posets (P~, <) is the neci
essary condition for the existence of i a-smallest (i~-largest) elements of a goset ( P - P1 UP2, 4 ), the uniqueness of/a-smallest (i(~-largest) elements (if they exist) is clear. In the context of the arguments presented above let us consider the following elementary, but necessary, example.
6.1. G o s e t s , G e n e r a l i z e d
Lattices,
...
197
E x a m p l e 6.1.5. Let the set P - { - 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4 } be linearly ordered (by the usual order) and, therefore, quasi ordered as the set-theoretic union of linearly ordered and thus partially ordered sets P1 - { - 4 , - 3 , - 2 , - 1 } , P2 - {1, 2, 3, 4}. Clearly, ( P - P1 U P2, 4 ), where x 4 y ~ x <_ y for each pair x, 9 c P and for the usual order <, is a goset. The element 1 c P2 is the zero element of P2, but it is not the 2a-zero element of P, while - 4 c P1 is the l a - z e r o element of P. Similarly, 4 c P2 is the 2a-unit element of P, while - 1 c P1 is the unit element of P1, but it is not the 1a-unit element of P. In the sequel we shall consider, in general, gosets ( P - P1 U P2, 4 ) such that (P1, <_) and (P2, <) are posets with common zero and unit elements denoted by G 1
2
and e, respectively; obviously, then O - l a - O - 2a-(~ and e - l a - e - 2a-e. Now we introduce the notion which is highly important for our later considerations. D e f i n i t i o n 6.1.6. Let (P, 4 ) be a goset and A c_ P be a subset such that A1 r 2~ r A2 - A N P2. Then an i(;-infimum of A (briefly, i~;-inf A) (an i , - s u p r e m u m of A (briefly, i , - s u p A ) ) is an element from i(;-l(d) ( i , - u ( A ) ) , satisfying the conditions below: A n P1 -
(1) i(;-inf A < _ inf A~ (sup A~ < / , - s u p A) and i
i
i(~-inf A 4 inf Aj
( sup Aj 4 i , - s u p A ) .
(2) I f x c P~, x <_infA~ (supA~ <_x) a n d x 4 a (a 4 x) for e a c h a c Aj, i
then x _< i , - i n f A (iG-sup A < x). i
i
(3) I f x c Pj, x 4 a (a 4 x) for e a c h a c A i
andx
(supAj < x), 5
then x 4 i(;-inf A ( / , - s u p A 4 x). It is obvious that the existence of inf A1 and inf A2 (sup A1 and sup A2) is the necessary condition for the existence of i , - i n f A ( / , - s u p A). Moreover, the notions of i~;-inf A (i~-sup A) will be meaningful if and only if their uniqueness is proved when they exist. As an example, let us consider the case of i~-sup A. If we assume that x and y are both i~;-sup A, a - sup Ai c Pi and b - sup Aj c Pj, then by (1) of Definition 6.1.6, we have a <_ x, b 4 x and a < y, b 4 y. It is obvious that z 4 x for i
each z E Aj since z <_ b ~
i
z 4 b. Therefore x E Pi, a < x and z 4 x for each
2
i
z c Aj. But y - / ( ; - s u p A and from (2) of Definition 6.1.6 it follows t h a t y <_ x.
i On the strength of the same argument, we conclude t h a t x _< y and thus x - y. i
R e m a r k 6.1.7. Let (P, 4 ) be a goset and A c_ P be a subset such that A n P1 - A1 7s 2~ r A2 - A n P2. First, we shall show t h a t j~;-inf A 4 i(;-inf A and j(~-supA 4 /(;-supA. Our consideration involves only the case of infimum since the case of supremum can be proved similarly. By (1) of Definition 6.1.6, jc;-inf A 4 inf Ai. But inf Ai < x ,e---->, inf A~ 4 x for each x E A~. Therefore i
198
VI. Generalized Boolean A l g e b r a and R e l a t e d P r o b l e m s
jc-inf A 4 x for each x E Ai. Further, by (1) of Definition 6.1.6, jc-inf A < inf Aj.
7
Since jG-inf A c Pj, jc;-inf A 4 x for each x c A~ and jG-inf A < inf Aj, by (3) of
7
Definition 6.1.6, we have j~-inf A 4 ia-inf A. Moreover, for A=A~ c_ Pi, Definition 6.1.6 reads as follows: i~-inf A (i~-sup A) is an element inf A E Pi (sup A c P~) such that x E Pj and x 4 a (a 4 x) for each a E A imply x 4 inf A (sup A 4 x). Without loss of generality, let us consider the case o f / , - s u p A. Since A - Ai, we have Aj c_ Ai. By (1) of Definition 6.1.6, sup A _< i~-sup A. On the other hand, assuming that x c Pi and sup A _< x, we i
i
obtain supAj <_ s u p A since Aj c_ Ai - A c_ Pi. i Besides, sup A < x implies i sup Aj <_ x and so a <_ x for each a E Aj, i
that is a 4 x for each a E Aj. Therefore xCPi,
s u p A _ < x implies a 4 i
x for each a c A j
and, by (2) of Definition 6.1.6, we have ia-sup A _< x. Hence sup A <_ x implies i~;-sup A _< x for each x E Pi i
i
and, therefore, i(~-sup A < sup A
so that i ( ; - s u p A - s u p A.
i
Finally, assuming that x c Pj and a 4 x for each a c A, we obtain a 4
x ~
a < x
for each
a
E
Aj
J
since Aj c_ A and hence sup Aj _< x. Thus J xcPj, a 4 x for each a c A
imply s u p A j _ < x J
(since Aj c_ A - Ai) and, by (3) of Definition 6.1.6, i~;-supA 4 z. But, as we have seen above, i a-sup A - sup A and, therefore, x c Pj, a 4 x for each a c A imply sup A 4 x. Clearly for A c_ Pi, the existence of inf A (sup A) is the necessary condition for iG-inf A (it-sup A) to exist and, therefore, iG-inf A (it-sup) (if the latters exist) are unique by virtue of the uniqueness of inf A (sup A). Nevertheless we want to show by giving a simple example below that the existence of inf A (sup A) is not sufficient for i~j-inf A (iG-sup A) to exist. E x a m p l e 6.1.8. Let X - {a, b, c, d, e}, ~-1- {2~, {a}, {b}, {c}, {d}, {a, b}, {a, c},
{a, d}, {b, c},{b, d},{c, d}, {a, b, c},{a, b, d},{a, c, d},{b, c, dI,{a, b, c, d},{a, b, c, e}, X}, and 7-2 be the discrete topology on X. By Example 6.1.2, ( 1 - O r ( X ) U (2, 1)-OZ)(X), 4 ) is a goset.
6.1. Gosets, Generalized Lattices, ...
199
Let us consider the family of sets ~41 = {{a, d}, {b, d}} C 1-Or(X) and the set {d, e} c (2, 1)-OZ)(X). Then {d, e} = {a, d, e} n {b, d, e} = 7-1 cl{a, d} n 7-1 cl{b, d} so that {d,e} ~ {a,d}, {d,e} ~ {b,d}. But inf .At -- {a, d} n {b, d} = {d} E co 7-1. Therefore {d, e} 4 inf A1 does not hold, that is, 1a-inf.41 does not exist. Of considerable importance is E x a m p l e 6.1.9. Let (X, 7-1,7-2) be a nonempty BS. As is well known, 7-i with the set-theoretic operation inclusion as partial orders are complete distributive lattices with respect to the lattice operations
AUs-7-iintNUs and V Us- Uus sES
sES
sES
sES
for every subfamily N - {Us}s~S c_ 7-i. One can easily satisfy oneself that the maps T~ int 7-j c l ' T i ~ ~-i have the following properties: (1) U C_ 7-i int 7-j cl U for each U c Ti. (2) If U c_ V, then T~ int 7-j cl U c_ 7-i int Tj cl V for each pair U, V c 7-i. (3) 7-~int 7-9 el 7-i int 7-r cl U - 7-i int 7-j el U for each U c 7-i. Thus 7-i int 7-j cl are closure operators on 7-i in the sense of [105, p. 13]. Moreover, the "closed" (in the sense of such operators) elements of 7-i are the (i, j)-open domains of (X, 7-1,7-2), and by Theorem 8.1 from [105], the families (i,j)-(.gz)(X) form complete lattices with the same partial order as that of 7-i, while by Theorems 8.1 and 8.2 from [105] the lattice operations in (i,j)-O~?(X) are written aS
A' Us --7-iint
~') Us
sCS
sES
and V' Us --7-,int7-jcl U Us sES
sES
for every subfamily b / - {Us}scs C (i,j)-O2P(X). Note that the latter equality V-s sES
d. U sES
is actually the essence of Lemma 6.1.2, proved in [2381. As Example 6.1.8 shows, for a BS (X, 7-1 < 7-2) and the goset (1-OZ)(X)U (2, 1)-OZ)(X), 4 ), there does not always exist l c - i n f A, where A c 1-O~P(X) is some subfamily. Furthermore, in that case for the BS (X, 7-1 < 7-2) neither does there always exists l c - i n f A for a subfamily .4 c 1-O79(X)U (2, 1)-OZ)(X) such that A N 1-OZ)(X) - J[1 ~ ~ r ~ 2 - - W ~ A (2~ 1)-OZ)(X). Indeed, if we assume that l a - i n f A exists for any such A and take the BS from Example 6.1.8 as a BS (X, 7-1 < 7"2) and the family {{a, d}, {b, d}} U (2, 1)-O:D(X) as A, then by virtue of the second part of Remark 6.1.7, there must exist l c - i n f A1, where A1 - {{a, d}, {b, d}} c 1-OD(X), which contradicts Example 6.1.8. Therefore ic-inf A, where .4
n 1-OT)(X)
- A1 -~ 2~ r A2 - A
n (2, 1)-OT)(X),
200
VI. Generalized Boolean Algebra and Related Problems
does not always exist for a BS (X, 7"1 < 7"2). However, we shall show below that for a BS (X, 7"1 < 7"2) and the goset (1-O2)(X) U (2, 1)-OD(X), 4 ) there always exist 1a-sup A, 2c-inf A and 2a-sup A (which are unique), where either A c_ 1-OD(X) or A _c (2, 1)-OD(X) although 2c-sup.4 exists even for .4 c 1 - O D ( X ) U (2, 1)-O~D(X) and A
N 1-O2)(X) - A1 7s 2~ 7s .42 -- A N (2, 1)-OT)(X).
Choose any subfamily ,4 c_ 1-OT~(X). Again applying Theorems 8.1 and 8.2 from [105], we find that there always exists sup.4 - V 1 A -- 7"1 int 7"1 el U A. If AEA
AEA
there exists a set B E (2, 1)-(.9Z)(X) such that A 4 B, that is, A c_ 7"1c l B for each A c .4, then 7"1 int 7"1cl U A c_ 7"1cl B so that sup A 4 B. Therefore sup A ACA
is l a - s u p A. Next, we shall consider a subfamily .4 c (2, 1)-OZ)(X) and show that
A2A-7"2int N A .~d
infA--
ACA
su A- V2A-7"2intT"lCl U
AC.A
AC.A
A
AEA
always exist. It is assumed that B c 1-OD(X) is a set such that B 4 A, that is, B C 7"1 el A for each A E A. We are to prove that B 4 inf.4 so that B c_ 7"1 cl 7"2 int n A. If we assume the contrary, that is, AEA
B n
(X \7-1C17"2 int N
.,
(X \7"2int N
then B N
AEA
AEA
But B 9 1-O~P(X), A E (2, 1)-OZ)(X) and, therefore, for each A c A the inclusion g C 7"1 clA implies 7"1 int 7"1 cl g C 7"2 int 7"1 cl A, that is, B c_ A for each A c A since T 1 C 7-2. Hence BC_ n
A and thus B C_ 7-lint n
AEA
Ac-7-2int n
ACA
which contradicts the inequality B O (X \ 7-2 int n
A,
AEA
A) r z . Therefore inf A is
AEA
2a-inf A. Now we suppose that B c 1-9 A E A. Then the obvious inclusion
and A 4 B, that is, A c_ 7-1 cl B for each
s u p A - 7"2int T1 cl U A c_ 7-1 cl B AEA
implies that sup A is 2c-sup A. Finally, for a BS (X, T1 < 7"2) let us consider a subfamily A c_ 1-OD(X) u (2, 1)-O:D(X) such that .4 5 1-O2)(X) - A1 7~ ~ 7~ .42 - A n (2, 1)-OZ)(X). We shall show that the set 72 int T 1 cl (72 int T 1 clsup A1 U s u p A2) is 2a-sup A. For this it is sufficient to check that (1)-(3) of Definition 6.1.6 are fulfilled. (1) The inclusions sup A2 C_ 2a-sup A and sup A1 c_ T1 C12a-sup A are obvious.
9
9 ~
.,~
UI
"< r
~
II
~
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~
9-
.,~
~
UI ~
..
uI
"~,
r.~
ro
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@
-a
=
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~
~:,
~
i~ ~
~V
~ <~ II ~ ~
o
-a ku
~
~
u;
-~
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"~ ~
~
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~ ~s
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@ k]) ~_,
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9 ,,,,,4
o
-~=
--
D
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~
Yr
~
VI~
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.-.
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Yr
~
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202
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
Theorem 6.1.11. A goset (L, 4 ) is a G.lattice if and only if for any nonempty finite subset A c L, there exist i(:-inf A and/(;-sup A.
Proof. Clearly, it suffices to prove only the necessary part of the theorem. First we assume t h a t A c Li. If A = {x}, then the equalities i(j-inf A = i~.-sup A = x are obvious. Now let A = {x, y, z}. The proof will be carried out for infimum and s u p r e m u m simultaneously. By (1) of Definition 6.1.10, we have v = ic-inf(x,y ), t = i , - s u p ( x , y) and, therefore, there exist w = i , - i n f ( v , z), r = / ( ; - s u p ( t , z). Let us show t h a t w = i~;-inf A and r = / , - s u p A. We easily find t h a t w = inf A and r = sup A in the usual sense. Assume t h a t a c Lj is an element such t h a t a~
x, a ~
y and a ~
z
(x~
a, y ~
a and z ~
y
a, y 4
and v - i ( ~ - i n f ( x , y )
a).
Since a4
x, a 4
(x4
a)
(t-/(;-sup(x,y)),
by a) of (1) of Definition 6.1.10, a 4 v (t 4 a). By virtue of the same a r g u m e n t a4
v, a 4
z (t4
a, z 4
a)
imply a 4
w (r4
a).
This completes the proof for A - {x, y, z}. If A - { x 0 , x l , . . . , x n - 1 }, where n > 1, then the element i(:-inf((--, i(:-inf(i,-inf(xo, X l ),
X2),...,
Xn-1 )
( / ( : - s u p ( - . - i c : - s u p ( i , - s u p ( x o , X l ), X2 ), . . . , X n - 1 ) ) is i , - i n f A (/(:-sup A), which is proved by induction. Now we suppose t h a t
A N L ~ - A1 r 2~ r A2 - A N L 2 . Clearly, Ai is finite and, as in the first case, there exist x - j , - i n f Aj
(x - j(:-sup Aj ) and y - i~:-inf Ai ( y - / ( : - s u p Ai ).
Since (L, 4 ) is a G.lattice, by (2) of Definition 6.1.10, there exist v - i(j-inf(x, y) and t - i , - s u p ( x , y). Let us prove t h a t v - i , - i n f A and t - / , - s u p A in the sense of Definition 6.1.6. Indeed, by a) of (2) of Definition 6.1.10, v <_ y (y _< t), v 4 x i
i
(x 4 t) so t h a t v <_ inf Ai - i~;-inf Ai ( i(:-sup Ai - sup Ai _< t ) i
i
and v 4 inf Aj - j~;-inf Aj
(j(;-sup Aj - sup Aj 4 t).
Further, let
zcLi,
z4x
( x 4 z), z < y
(y
i
T h e n by b) of (2) of Definition 6.1.10, z < v (t < z). Finally, if i
zcLj,
z<x J
(x
i
and z 4
y ( y 4 z),
2
then by c) of (2) of Definition 6.1.10, z ~ v (t ~ z). Thus the elements v c Li and t E Lj satisfy all the conditions of Definition 6.1.6 so t h a t v - i , - i n f A (t - / ( ; - s u p A). D
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204
VI. Generalized Boolean Algebra and Related Problems
Then
(Xl Vj x2 V j . . . V j Xn) Vi (Yl ViY2 V i " " ViYn) -((x 1 Vj x 2 Vj " " Vj Xn_l) Vj xn) V i ((Yl Vi Y2 V i " " Vi Yn-1) Vi Yn) --
(x n Vj (Xl Vj x2 V j " ' V j
-
-
-
Xn_l) ) V i ((Yl Vi Y2 V i " " Vi Yn-1) Vi Yn) (Xn Vi Yn) Vi ((Xl Vj x2 V j " ' V j Xn_l) Vi (Yl Vi Y2 V i " " Vi Yn-1)) --
-
-
-
-
(Xn Vi Yn) Vi ((Xl Vi Yl) Vi (x2 Vi Y2) V i " " Vi (X,n-1 Vi Y'n-1)) -= (Xl Vi Yl) Vi (x2 Vi Y2) V i " " Vi (xn V~ Yn).
D e f i n i t i o n 6.1.13. Let s = {L = L1 U L2, A1, Vl, A2, V2} be a G.lattice and L~ c L~. Then s "- {L' - Ltl U L~, A1, V1, A2, V2} is said to be a G.sublattice of Z; if a c L~ U L~, b c L~ implies that a A~ b, a V~ b c L~. As we shall see in Sections 6.2 and 6.3, this notion is closely associated with the notions of G.ideal and G.filter. R e m a r k 6.1.14. Note t h a t by Definition 6.1.12, if an element x E Li is on the right-hand side of the operations Ai and Vi, then the definition of the domains of A~ and Vi necessarily implies that x c Li. This fact accounts for the absence of laws, corresponding to the commutativity laws in the well-known sense, t h a t is, the absence of the laws for Ai and Vi, corresponding to L2 when (x, y) c Lj x Li, though as we shall see in the sequel, such G.commutativity laws are available for G.lattices with the G.zero and G.unit elements. Our next important theorem shows the ways how we can pass from the G.lattice, defined by a goset to the G.lattice, defined by an algebra and vice versa. According to this theorem, no m a t t e r in which order we perform such pass-overs initially, they do not change the initial objects. Note that to prove the theorem, we do not need to assume that the G.lattice have the G.zero and G.unit elements. Theorem
6.1.15. The following conditions hold:
(1) Let a goset s = (L, 4 ) be a G.lattice. A s s u m e that x A~ y = i , - i n f ( x , y) and x Vi y = i~;-sup(x, y) for each pair (x, y) e (ni x ni) U (nj x Li). Then the algebra/~a = {L, A1, Vl, A2, V2} is a G.lattice. (2) Let an algebra s = {L, A1, V1, A2, V2} be a G.lattice. A s s u m e that y ~ x if and only if x Ai y = y for each pair (x, y) c (Li x Li) U (Lj x Li). Then 12q = (L, ~ ) is a goset which is a G.lattice. (3) If a goset s = (L, ~ ) is a G.lattice, then (s = s (4) If an algebra 12 = {L, AI~ Vx~ A2~ V2} i8 a G.lattice, then (s __ s
Proof. Conditions (1) and (2) will be proved first for i~-inf, that is, for Ai, and then f o r / , - s u p and so for Vi. Such a proof covers all the cases and we avoid going into a long tiresome discussion. (1) Let a goset Z; = (L, 4 ) be a G.lattice. By Definition 6.1.10 the binary operations Ai, Vi: (Li x L~)U (Lj x L~) ~ Li are defined as x Ai y = i , - i n f ( x , y) and x V~ y = ic~-sup(x , y). Let us show that the algebra s = {L, A1, V1, A2, V2}
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6.1. G o s e t s , G e n e r a l i z e d L a t t i c e s , . . .
207
Therefore, by b) of (2) of Definition 6.1.10, i , - s u p ( y , z) < i , - s u p ( j , - s u p ( x ,
y), z)
i
and thus i ( ; - s u p ( x , i ( ; - s u p ( y , z) ) <_ i G - s u p ( j ( ; - s u p ( x , y), z) i
so t h a t i.-sup(j.-sup(x,
y), z) - / , - s u p ( x , / . - s u p ( y ,
z)).
But this equality means t h a t (x Vj y) V~ z - x V~ (y V~ z). In a similar manner, using Definition 6.1.10, we can prove t h a t (xAjy) A~z--xA~(yAiz)
for x, y C L j ,
z C L~,
or x, z c L~,
y c Lj.
T h u s condition GL3 of Definition 6.1.12 is satisfied. If x c Lj, y c L~, t h e n by a) of (2) of Definition 6.1.10, i(;-inf(x,y) 4 x and, therefore, b) of (2) of the same definition implies j ( x - s u p ( i ( ~ - i n f ( x , y ) , x ) < x. J T h u s j ( j - s u p ( i ( j - i n f ( x , y ) , x ) - x since the inequality x <_ j ( ~ - s u p ( i ( ~ - i n f ( x , y ) , x ) J is also obtained by a) of (2) of Definition 6.1.10 so t h a t (x A~ y) Vj x -- x. F u r t h e r m o r e , by a) of (2) of Definition 6.1.10, y <_ i , - s u p ( x , y ) and, therefore, i
y so t h a t (x Vi y) A~ y -- y. In a similar manner, using
i~:-inf(i(~-sup(x, y), y) -
Definition 6.1.10, we (:an prove the equalities (x V~ y) Aj x - x and (x Ai y) V~ y -- y. Thus condition GL4 is also satisfied and s _ {L, A1, V1, A2, V2} is a G.lattice. (2) Let the algebra s - {L, A1, Vl, A2, V2} be a G.lattice. Assume t h a t y4 Then 4
x .z--->xAiy--y
for each pair (x,y) c ( L i x L i )
U(Lj xLi).
L, --_< are partial orders on Li. Moreover, the binary relation 4
is a
quasi order on L - L 1 U L 2 . Indeed, it is obvious t h a t x 4 x for each x c L. Let us show t h a t x 4 Y and y 4 z imply x 4 z for each elements x, y, z c L. To this end, we shall consider the cases: x, y, z 6 Li;
x 6 Li,
y, z C L j ;
x, y C Li,
z 6 Lj;
and x, z c Li,
T h e case x, y, z c Li is obvious since 4 IL, =<--. i
If x c L~, y, z c L j , t h e n x 4 y ,,+----5, y A ~ x -- x and y 4 z ,z-----~, z A j y -- y.
Therefore x - y Ai x -- (z
t h a t is, x - - z A ~ x ~ x 4 If x, y E L~, z E L j , then
Aj y) Ai x and by GL3, we have z.
x 4 y ,z-~, y A ~ x -- x and y 4 z ,,v----5, z A ~ y -- y.
Therefore x - y Ai x -- (z A~ y) Ai x and by GL3, we have (zAiy) Aix--zAi(yAix)--zAix, t h a t is, x - - z A i x ~ x ~
z.
y c Lj.
208
VI. Generalized Boolean Algebra and Related Problems
Finally, we assume t h a t x, z c Li, y c L j . T h e n x4
y ,,r
y Ai x -- x,
Y4
z ,~-~, z Aj y -- y.
Therefore x - y At x -- (z Ay y) At x and by GL3, we obtain (Z A j y ) A i X - - z Ai
(y At x) - z Ai x
so t h a t x - z At x ~ x 4 z. T h u s the relation 4 is a quasi order on L - L1U L2 such t h a t 4 IL, =_< and s o / : q - (L, 4 ) is a goset. i
Now, assuming t h a t i a - i n f ( x , y) - x A~ y and i(~-sup(x, y) - x Vi y for each pair (x, y) c (Li x L i ) U ( L j x L i ) , let us prove t h a t the g o s e t / : q - (L, 4 ) is a G.lattice in the sense of Definition 6.1.10. First, we show t h a t if x c L j and y E L i , then i~-inf(x, y) - y <--5, i t - s u p ( y , x) - x, t h a t is x Ai y -- y <--> y Vj x -- x. Indeed, by GL4, we have x A i y - - y implies y V j x implies x Ai y -- (y V j X) A i y - - y. Therefore Y4
(xAiy)Vjx-
x and y V j x -
x
x <----~, x A i y = y ,,+--~, y V j x = x.
Now we proceed to considering the fulfilment of (1) and (2) of Definition 6.1.10. Let (x, y) E Li x Li be any pair. Then, according to the a r g u m e n t s t h a t follow Definition 6.1.12, i , - i n f ( x , y) a n d / , - s u p ( x , y) coincide with inf{x, y} and sup{x, y}, respectively, in the usual sense. Moreover, if z c L j and z 4 x, z 4 y, t h e n by GL3, (xA~y) Ajz--xAj(yAjz)--z
and so z 4
xAty~z4
inf{x,y}.
Quite in a similar manner, one can prove t h a t x ~ z Ai y 4 z imply sup{x, y} 4 z. Hence a) is also satisfied and thus (1) of Definition 6.1.10 is completely valid. If ( x , y ) c L j x L~, then by GL4, (x A~ y) V~ y -- y so t h a t x At y _< y and, i
therefore, i~.-inf(x, y) <_ y. Similarly, (x A~ y) Vj x -- x so t h a t x A~ y 4 x, t h a t is, i
i , - i n f ( x , y) 4 x. F u r t h e r m o r e , let z E Li, z _< y and z 4 x. Then, by GL3 and i the definition of 4 , we have z -
y
-
y -
(x
z)
y -
x
A, y)
-
(y
-
y)
so t h a t z < x Ai y and, therefore, z < i~-inf(x, y). Finally, let z E L j and z 4 y, i
i
z < x. Then, by GL3, J (X A i y ) A j z -- x A j ( y A j z) z x A j z z Z a n d s o
z ~
x
At y,
t h a t is, z ~ i~j-inf(x, y). Using a similar reasoning and taking into account the equivalence y ~ x <---> y Vj x = x, we can prove the fulfilment of the rest of the conditions for the s u p r e m u m case in (2) of Definition 6.1.6. Thus the goset L;q = (L, ~ ) is a G.lattice. (3) T h e gosets L; = (L, ~ ) and (L;a) q have the same basic set L = L1 U L2. Hence to prove t h a t L; = (L;a) q, it suffices only to show t h a t the quasi orders on L; and (L;a) q coincide. First, we shall prove t h a t if the goset/2 = (L, ~ ) is a G.lattice, then y ~ x ~ i c - i n f ( x , y ) = y for each pair (x,y) c (Li x L i ) U ( L j x L~).
6.1. Gosets, Generalized Lattices, ...
209
Indeed, if (z, y) c Li x Li, this equivalence is obvious since ~ IL, ----<. Now, let i
(z, Y) C L j x Li and 9 ~ z. By a) of (2) of Definition 6.1.10, i , - i n f ( z , y) <_ 9 since /
s - (L, 4 ) is a G.lattice. On the other hand, y _< y and y 4 x. Hence, by b) of i
(2) of Definition 6.1.10, y < iG-inf(z, Y) so that y -
i~-inf(z, y).
i
Conversely, if the goset s - (L, 4 ) is a G.lattice, then by a) of (2) of Definition 6.1.10, y 4 z. Thus, if a goset s - (L, ~ ) is a G.lattice, then y 4 z implies i~;-inf(z, 9) - Y for each pair (z, 9) c (Li x L i ) U ( L j x Li). Therefore, to prove that (s _ s it remains to apply successively (1) and (2) of Theorem 6.1.15. (4) The algebras s - {L, A1, VI,A2, V2} and (s have the same basic set L -- L 1U L 2. Hence to prove that (s _ s it suffices to show that the operations on s and (s coincide by applying successively (2) and (1) of Theorem 6.1.15 since s {L, A1, V l , A2, V2} is a G.lattice. K] Based on Theorem 6.1.15, we denote by s - { L 1 , A 1 , V l , 4 , L 2 , A2, V2} a G.lattice, defined by means of a goset. It is obvious that s - {L1 - {(9, e},A1,VI, d , L 2 -- {19, e},A2, V2}, where A1 -- A2 -- A and V1 - V2 - V are usual lattice operations, is a G.lattice and s is a G.sublattice of any G.lattice t; - {L1, A1, V1, 4 ,L2, A2, V2} such that L1 n L2 - {19, e}. In the sequel we shall consider only G.lattices with the G.zero and G.unit elements, that is, of the type t ; - {L1,A1,V1,19,4 ,e, L2, A2, V2}, where 19, e c L1K1L2.
Our next theorem is most important for further consideration because it shows that the posets, composing a goset, are in a one-to-one correspondence of special type. Theorem
6.1.16. Let s - {L1, A1, Vl, 19, 4 , e, L2, A2, V2} be a G.lattice and ()~,,X~) be a pair such that X, " Li ~ L j are maps, defined as follows" X, (x) - x Vj 19 f o r each x c L~. T h e n the conditions below are satisfied: -
(1)
(x)
-
9
a ,d
(e)
-
e,
-
(2) X, o X, -- idL, so that X, (X, (x)) - x f o r each x c Li; thus X1, X~_ are bijections and X, - X T, 1 (3) X, o Ai -- A j o (X,, X,) a n d X, o V~ - V j o ( X , , X , ) so t h a t
and
T h u s x <_ y ~ }(, (x) <_ X, (Y) f o r each pair (x, y) c L~ x L~ i j (the i s o t o n i c i t y of the m a p s X,). (4) }(, o Ai -- Aj o (~,,,}(,) and X, o V~ - Vj o ( X , , X , ) so that
and
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y) -
v ; x, (y).
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212
VI. Generalized Boolean Algebra and Related Problems
()~1,)~2)
The pair X -
is called the G.identity operator.
c
b
x. ~. x. .~ . I j -~
d
Let (L = L1 U L2, 4) be the goset in Diagram 2, where the quasi order relation x ~ y is determined as for Diagram 1 by the arrangement of elements x, y E L 1 U L 2 = { a , b , c , n } U { c , d , m , n } at different or the same levels.
n
Diagram 2 If we define the binary operations Ak, Vk : Li x x 4 y <--5, x V k y = y for each pair ( x , y ) c verification of axioms L1 - GL4 shows t h a t s = G.lattice with (9 = n, e = c, X1 (a) = m, ~1 (b) --- d ,
Lk ~ Lk as y A~ x = x ,', ;, Li x Lk, then an elementary { L 1 , A 1 , V1, 4 , L 2 , A2, V2} is a
X2 (~'~) :
a
and X2 (d) = b.
P r o p o s i t i o n 6.1.17. For a G.lattice s = {L1, A1, V1, 0, 4 , e , L2, A2, V2}, the following conditions are satisfied:
(1)
(x /~ y) v~ (x /~ z) = x / ~ (y v~ (~ /~ ~)) ., ,.. .'. :, (z ~ x implies (x Ai y) Vi z -- x Ai (y Vi z)) z. ;.
.'. :. (x v~ y) A~ (x v~ z) - x v~ (y A~ (x v~ z)) ,'. ;. ,'
:. (x ~ z implies (x Vi y) Ai z -- x Vi (y Ai z))
if x, y, z E Li or x C L j , y, z c Li.
(2)
(x Aj y) v~ (~ A~ z) = x A~ (y v~ (~ A~ ~)) ,, ,, .',
,~
(z ~ x implies (x Aj y) Vi z = x Ai (y Vi z))
.'
:.
.' :. (x vj y) A~ (x v, z) = x v~ (y A~ (x v~ z)) .'. '. -(
:.
(x 4 z implies (x Vj y) Ai z = x Vi (y Ai z))
if x, y E L j , z 6 Li or x, z E L i , y E Lj. Proof. Since all these conditions are proved in a similar manner, we can do with proving only (2) for x, y c L j , z 6 Li. First, let (xAjy) Vi(xAiz)--xAi(yVi(xAiz))
and z 4
x.
T h e n z = x A~ z and hence
(x Aj y) v~ ~ = (x A~ y) v~ (x A~ z) = x A~ (y v~ (~ A~ ~)) = ~ A~ (y V~ ~). Conversely, let z ~ x imply (x Aj y) V~ z = x A~ (y V~ z). Then x A~ z ~ x implies that
(x Aj y) v~ (x A~ z) = x A~ (y v~ (x A~ z)) and thus the first equivalence is true.
6.1. Gosets, Generalized Lattices, . . .
213
Using a similar reasoning one can prove that the trird equivalence is also true. By virtue of GL3 and (5) of Theorem 6.1.16, it suffices to prove only that (z 4 x implies (x Aj y) Vi z -- x Ai (y Vi z)) ~' ,<---5, (x 4 z implies (x Vj y) Ai z -- x Vi (y Ai z)). If x 4
z, t h e n x V i z - z a n d
(x v j y) A~ ~ - (x v j y ) / ~ (x v~ ~) - ((x v~ ~) nj (y v5 x)) A~ ~ -
= (((~ v~ ~) Aj y) v~ ~) v~ e - ((z Aj y) v~ ~) v~ e -- (x v j ( z / ~ y)) v~ e - x v~ ((z/~j y) v~ e) - x v~ ( y / ~ ~). Conversely, z ~ x implies x Ai z -- z and
(x A~ y) v~ z - (x Aj y) v~ (x A~ ~) - ((x A~ ~) v~ (y Aj ~)) v~ e -
= (((x A~ ~) vj y) Aj x) v~ e - ((z vj y) A~ x) v~ e - (x Aj (~ vj y)) A~ ~ = x / ~ ((~ vj y) n~ ~) - x / ~ (y vj z).
n
D e f i n i t i o n 6.1.18. A G.lattice s - {L1, A1, Vl, O, 4 , e, L2, A2, V2} is said to be modular (or Dedekind) if it satisfies one of the equivalent conditions both in (1) and (2) of Proposition 6.1.17. This notion is closely associated with D e f i n i t i o n 6.1.19. A G.lattice t; - {L1, butive if the following laws hold:
A1, V l , I~, 4 , e,
L2, A2, V2} is distri-
CDL1 x A~ (y V~ z) -- (x A~ y) V~ (x A~ z) and x Vi (y Ai z) -- (x Vi y) Ai (x Vi z) ifx, y, z c L i o r x 6 L j ,
y, z c L i .
x v~ (y A~ z) - (x vj y)A~ (x V~ ~ ) i f x, y C f j , ~ c f~ or x, z c f~, y c f j . G u n 2 (x v~ V) n~ z - (z A~ z) v~ (v A~ z) and (xAiy) Viz--(xViz)Ai(yViz) ifx, y, z E L i o r x E L j , y, z c L i . (xVjy) A~z--(xAiz)Vi(yAiz) and ( x A j y ) Vi z -- (x Vi z) A~ (y Vi z) if x, y C L j , z ~ L i o r x , z~Li, y~Lj. It is obvious that if s -- {L1,A1,V1,Q),4 ,e, L2, A~,V2} is a distributive G.lattice, then {Li, Ai, Vi, (9, e} are distributive lattices in the usual sense and every G.chain is a distributive G.lattice. GDL1 and GDL2 are called the G.distributivity laws. Let us denote by I GDL1, II GDL1, III GDL1, and IV GDL1 the first, second, third, and fourth equalities in GDL1, respectively, and similarly for GDL2. Theorem have:
6.1.20.
For a G.lattice s
IGDL1 ~
IIGDL2,
-
{L1,A1,VI,~,4 ,e, L2, A~,V~}, we
IIGDL1 ~
IGDL2
and
IIIGDL1 <---->, IVGDL2, so that GDL1 ~
IVGDL1 ~
IIIGDL2
GDL2.
Proof. All the proofs essentially utilize Definition 6.1.12 and Theorem 6.1.16.
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220
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
= y Ai ((a Aj x) Vi (x Ai y)) -- (a Aj x) V~ (x Ai y) --
-- (a Aj x) Vi (a Ai y) Vi (x Ai y) and so x = y Vj @. As for the implication (4) ~ (3), it is an immediate consequence of the notion of a G.neutral element and Theorem 6.1.22. D C o r o l l a r y 6.1.25. For a G.lattice 12 = {L1,A1,V1,Q),4 ,e, L2, A2, V2} the following conditions are satisfied: (1) Every G.neutral element is G.standard and coG.standard. (2) Every G.standard (coG.standard) element is (V,A)-distributive ((A, V)-distributive). (3) Every G.standard and c o G . s t a n d a r d or every G.standard and (A, V)-distributive element is G.neutral.
(1) By (3) and (4) of Theorem 6.1.24, every G.neutral element is (V, A)-distributive, (A, V)-distributive, and
Proof.
aA~x=aA~y,
aV~x=aV~y
aAjx=(aAiy)
Aje,
imply x = y
aVjx=(aViy)
for x, y c L ~ ,
V j O imply x = y V j @
for x C Lj, y E L~. Therefore it remains to use (1) .z---->.(2) in Theorem 6.1.24. Assertion (2) follows directly from the equivalence (1) .z----5. (2) in Theorem 6.1.24. (3) If an element a e Li is G.standard and coG.standard, then by (1) <---5, (2) in Theorem 6.1.24, it is (V, A)-distributive, (A, V)-distributive and a Ai x = a Ai y, a Vi x = a Vi y imply x = y for x, y C Li,
aAjx=(aAiy)
Aje,
aVjx=(aViy)
V j O imply x = y V j O
for x E L j , y E L~. Thus it remains to use (3) z---> (4) in Theorem 6.1.24. If an element a c L~ is G.standard and (A, V)-distributive, then by (1) ~ in Theorem 6.1.24, it is (V, A)-distributive and a Ai
X --- a A i y ,
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a Vi X
Aje,
=
a Vi y
(2)
imply x = y for x, y c Li,
(a Vj x) = (a Vi y) Vj (9 imply x = y V j ( 9
for x C Lj, y E Li. Thus it remains to use (3) ,z---->,(4) in Theorem 6.1.24.
[3
D e f i n i t i o n 6.1.26. A G.lattice s = {L1, A1, V1, 0, ~ , e , L2, A2, V2} is said to be G.complemented if there exists a pair ~ = (pl, p2) such that Pi : Li ---, Lj are maps and x Aj ~ ( x ) = @, x Vj ~ ( x ) = e for each element x c L~. The pair p = (~1, ~2) is called a G.complementation operator. P r o p o s i t i o n 6.1.27. For a distributive G.lattice s = {L1, A1, V1, 0, 4 ,e, L2, A2, V2} the G.complernentation is unique.
6.1. Gosets, Generalized Lattices, ...
221
Proof. Let q p ' - (p~, p~) be another pair, where ~{'L~ -+ Lj are maps such that x Aj p~(x) -- O and x Vj p~(x) - e. Then by GDL2 we obtain ~ ( x ) - e v~ ~ ( x ) - (. Aj ~'~(.)) vo ~ ( . ) = (x vj ~ ( . ) )
Aj ( ~ ( ~ ) vj ~ ( . ) )
- ~ Aj (~'~(.) vj ~ ( . ) )
- ~'~(.) vj ~(~)
so that ~{(x) _< ~i(x). The case ~i(x) <_ ~{(x) is proved in a similar manner and j
J
thus ~ - ~ .
D
Now we are ready to introduce the basic notion. D e f i n i t i o n 6.1.28. A G.Boolean algebra (briefly, GBA) is a distributive and G.complemented G.lattice. In the sequel a GBA will be denoted by A - {A1,A1, V I , ~ I , O , ~ , e , A2, A2, V2, ~2}. It is obvious that if A1 - A2 - A, then A1 -- /~2 -- A and V1 -- V2 -- V are usual lattice operations, (~1 - - ( P 2 - - - - is & complementation operation in the usual sense and, therefore, . 4 - {A, A, V , - , O, e} is a Boolean algebra (briefly, BA) in the usual sense. A G.Boolean subalgebra (briefly, GBSA) of a GBA A is a set B c_ A1 U A2, B • A1 - B1 ~ Z ~ B2 - B A A2, which is closed under the four G.Boolean operations: Ai and ~ , or, V~ and pi. It is obvious that any GBSA contains 0 and e. Take place the following obvious statement.
Proposition
6.1.29.
Yo~ ~ y
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GBSA's of a GBA A, the intersection Bo -
( ~ B~) U ( ~ Bt2) is a GBSA tET
tET
o/A. Therefore for any subset D c A1 U A2, there exists a GBSA of At generated by D, which is the smallest GBSA containing D. Clearly, for a GBA A - { A 1 , A 1 , V I , ( ~ I , ~ , 4 ,e, A2, A2, V 2 , ~ 2 } , the subset {a, ~ l ( a ) , ~=},e} U {X1 (a), (~l(a), {~, e} C n l U n 2 together with the corresponding G.lattice operations is a GBSA. E x a m p l e 6.1.30. Let us consider a nonempty BS (X, the binary operations
A~, v~. ( ( ~ , j ) - o ~ ( x ) •
T1
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7"2)
and define
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as follows" U A~ V -
T~int Tj cl U ~ V and U V~ V - ~-~int rj cl(~-~ int T1 cl U U V).
Simple calculations show that s Vl,%~ ,(2, 1)-O/P(X),Az,V2} is a distributive G.lattice, where the quasi order 4 , defined on (1, 2)-OZ)(X)U (2,1)-O/P(X), is the same as the one defined in Example 6.1.2 and by (4) of Theorem 2.1.10, (1,2)-O/P(X) - 1-O:D(X), 2-O:D(X) - (2, 1)-O/P(X). The quasi order 4 coincides with the quasi order defined by the binary operation A~ as
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6.1. Gosets, Generalized Lattices, . . .
225
Now we assume that (x, y) ~ A j x Ai and x ~ y. Then, by Theorem 6.1.15, L2 and GL3, we have x
~ = 9 no ~ ( ~ ) = (~ n , x) n , ~ ( ~ ) = ~ ( ~ ) n , (~ n , ~) = J(; - ( ~ ( ~ ) n~ ~) no ~ = ~.
On the other hand, by (1) above and (3) of Theorem 6.1.31, we have x - y = (9 <----> x Aj ~ i ( y ) = (9 ~ Jc;
q~j(x) Vi y = e
and hence ( p j ( x ) V ~ y ) A j x = x so that by GDL2, ( ~ j ( x ) A j x ) V j ( y A j x ) = y A j x = x. Thus x 4 y. (6) We shall only consider the conditions without brackets. By condition (3) above
(x,v) e p~(~,) ~
v _< ~,(x) ~ z
~ ( ~ , ( . ) ) _< ~(v).: j
< :, x < ~ i ( y ) ~ J
( y , x ) ~ PA(q~i).
Moreover,
(x, y) e P~(~,) ~
y <_ ~o(x) ~
~o(~(y)) _< ~ ( x ) . :
i
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i
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j
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i
i
by (2) of Theorem 6.1.31, q~j(X Aj z) <_ y Vi v, that is, (x A j z , y V i v ) C P v ( q ~ j ) -
On the other hand, F j ( x ) < y, F j ( z ) <_ v imply F j ( x ) Ai qpj(z) < y Ai v i i i
and thus p j ( x Vj z) <_ y Ai v gives (x Vj z, y Ai v) C Pv(~j).
K]
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Theorems 6.1.16 and 6.1.31 give rise to some interesting statements related not only to generalized, but also to ordinary algebraic structures. C o r o l l a r y 6.1.33. Let A = {A1, A1, V1,991, (~, ~ , e , A2, A2, V2, p2} be a GBA and ~ = (~1, ~2) be a pair such that ~ : Ai ~ A~ are m a p s defined as follows: ga~ = ~ j o Xi so that ~b~(x) = ~ j ( X , ( x ) ) f o r each x r A i . T h e n the following conditions are satisfied: (1) ~ = x, o ~ ~o that ~ ( x ) = X, ( ~ ( ~ ) ) fo~ ~ach 9 e A~. (2) ~&(O) = e and ~b,(e) = 0 . (3) ~b~ o ga~ = idA, so that ~b,(~b~(x)) = x f o r each x c Ai.
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6.1. Gosets, Generalized Lattices, . . .
229
(8) By analogy with the proof of (6) of Corollary 6.1.32, for each pair (x, y) c
Aj x Aj, we have j
J
Note now t h a t ~j o @j - ~i o ~j. Indeed, ~j
O ~)j - - ~ j o ~ i O Xj - - Xj
and ~i o q~j - ~i o ~i o Xj - ;~j.
Moreover since (x, y) e PA(~j) <---> Y <_ ~j(x) and ~j are antitone, we have J
~ j ( ~ j ( ~ ) ) <_ ~j(y) ~
~(~j(~))
i
_< ~j(y) ~
(~j(x), ~j(y)) e P ~ ( ~ ) .
i
Therefore
(~j(x), ~ j ( y ) ) e Finally, (x, y) e PA (~j) ~
P~(~)~
(~j(y), ~ j ( x ) ) 9 P ~ ( ~ ) .
Y < ~j (x) and, by (4) above, J J
(9) We will prove only four implications; the others can be proved similarly. I f ( x , y ) e PA(~j), z 9 Li and z 4 y, then by (7) above, y ~ ~j(x) s o t h a t z < ~ j ( x ) s i n c e ~ A~ --<. Therefore (x,z) 9 PA(~j).
j
J Now, if (x,y) 9 P v ( ~ j ) , z 9 n~ and x 4
z, then we have ~j(x) < y and J
x 4 z. But ~j(x) < y ~ ~ ( y ) < x so t h a t ~ ( y ) ~ z and, by (7) above, we i j have (z,y) 9 Pv(~i). The rest can be proved in a similar manner. Assertion (10) is proved in the same way as (7) of Corollary 6.1.32. (11) By (7) above, for each x 9 A~, the element ~ ( x ) 9 A~ is its complement in the usual sense and so ~i(x) is unique. Hence Xl
-
{A1,
A1, V l , ~)1, O , S , 1
e} and ,.,42 -- {A2, A2, V2, ~2, (~, <_, e} 2
are BA's. To prove t h a t ~, "Ai ~ Aj are isomorphisms, by virtue of (2) and (3) of Theorem 6.1.16, it suffices to prove only t h a t )/, (~i(x)) - ~j(X, (x)) for each x 9 Ai. By (5) above ~ ( x ~ (x)) - ~ j ( x v~ e ) - ~ ( x ) / ~
~(e)
- ~(x)/~
~ - x~ ( ~ ( x ) ) .
Furthermore, let a 9 Ai be an atom of Ai and X~ (a) 9 Aj be not an atom of My. Then there exists an element b 9 Aj \ {O} such that b < X, (a) - a Vj (9. We J
have ;~ (b) < ~ ()~ (a)) - a, where X;~(b) # (9 since X~ are isotonic bijections. The contradiction obtained shows t h a t )/, (a) is an atom of Aj. The reverse implication is obvious since )/j ()/~ (a) - a. Proving the second equivalence is clear. (12) First, let (x,y) 9 A~ x A~. Then x 4 y ~ x < y <---5. x A ~ y - x. Therefore
x - ~ - x a~ ~ ( ~ ) - (x a~ ~) a~ e~(~) - x a~ (~ a~ e~(~)) - x a~ e - o. i
230
VI. G e n e r a l i z e d Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
Conversely, if x - y
- x/~i ~i(Y) - 0 , t h e n by (2), (3), a n d (4) above, we have
i
Oi(xAiOi(y))-
Oi(x)Viy-
e a n d so, by ( 7 ) a b o v e , ( r
Now let (x, y) E Aj x Ai. T h e n x 4 y ~ x - y -
r
-
y) e Pv(r
x < y.
y Aj x -- x. T h e r e f o r e
(y Aj
-
(r
Aj (y Aj x ) ) A,
-
i
= ((r
A, y) Aj
A,
- e.
O n t h e o t h e r hand, if x - y - O, t h e n e i ( x Ai ~i(Y)) -- e j ( x ) Vi y -- e. T h e r e f o r e i
(~j(x), y) c P v ( ~ j ) a n d hence p j ( ~ j ( x ) ) <_ y, t h a t is )/j (x) - x Vi O _< y z----> x 4 y. i
i
(13) We will prove only t h e equivalence in brackets. i
g~i(Y) <_ x Vi (9 ~ ( y , x Vi (9) C P v ( e i ) . i I m m e d i a t e c o n s e q u e n c e of previous r e a s o n i n g is ": :" ~i(Y) 4 x Vi (9 ~
Example
6.1.34.
I-I
C o n t i n u i n g E x a m p l e 6.1.30, let us consider t h e pairs X First, let U e ( 1 , 2 ) - O Z ) ( X ) - 1-OT)(X). T h e n by T h e o r e m 6.1.16 a n d (1) of L e m m a 0.2.1, we have
(X1,X2) a n d ~ - ( ~ 1 , ~ 2 ) .
X1 (U) - U V2 2~ - 7-2 int 7-1 cl (7-2 int 7-1 cl U U 2~) - 7-2 int 7-1 cl U. Now, if U 9 (2, 1)-O~P(X) - 2-OZ)(X), t h e n by (3) of C o r o l l a r y 2.1.7, we have )/2 (U) - U V1 2~ - 7-1 int 7-2 cl (7-1 int 7-1 cl U U 2~) = 7-1 int 7-2 cl 7-1 int 7-1 cl U - 7-1 int 7-2 cl U. Hence ( 1 ) o f L e m m a 0.2.1 a n d ( 3 ) o f C o r o l l a r y 2.1.7 give )(~2(~1 ( U ) ) -- T2 int --
Wl
T1
cl U
V1 ~
-- T1
int r2
(T 1 int 7-1 cl r2 int
cl
T1
cl U U 2~) -
int r2 el rx int 7-1 el r2 int 71 cl U - 7-1 int r2 cl r2 int 71 cl 72 int r l el U -
-- 71 int r2 cl 72 int 7-1 cl U - 71 int r2 cl 71 int 7-1 cl U - 71 int r2 cl U - U so t h a t X~ (XI ( U ) ) )(~1
()(~2(U))
T1
--
U and int r2 cl U V2 2~ - rg, int r l cl (r2 int
= r2 int r2 cl r2 int r2 cl
T1
T1 C1 T1
int r2 cl U U 2~) -
int r2 cl U - r2 int r2 cl r l i n t r2 cl U =
= r2 int r2 cl 72 int r2 cl U - U, a n d so X I ( X 2 ( U ) ) @I(U)-
U. F u r t h e r m o r e , if U 9 (1, 2 ) - ( 9 / ) ( X ) -
X2(~I(U))-
~2(T2
i n t ( X \ U))
-T1
1-O~D(X), t h e n
int 7-2 c172 i n t ( X \ U) -
= r l int r2 cl r2 int rl c l ( X \ U) and, by (3) of C o r o l l a r y 2.1.7, we o b t a i n ~)1 ( g )
- - T1
int 7-2 el T1 int - - 7"1
T1
c l ( X \ U)
- - T1
int ~-2 cl
T1
int ~-2 c l ( X \ U) -
int 7"2 c l ( X \ U) - 7"1 i n t ( X \ U).
6.1. Gosets, Generalized Lattices, . . .
231
Consequently, g A 1 ~)1 ( g )
-
T 1 int r2 cl U ~ T 1 i n t ( X \ U) -
U N T 1 i n t ( X \ U) - 2~,
U Me ~)1 ( g )
-
T 1 int 72 cl (T1 int T1 C1U [_J T1 i n t ( X \ U)) -
= 71 int (72 cl 71 int T 1 cl U U 72 cl 71 i n t ( X \ U)) = 71 int (72 cl U U 72 cl 72 i n t ( X \ U)) = 71 int (72 cl U U 72 cl(X \ 72 cl U)) - X. Now, if U c (2, 1)-OT)(X) - 2-OT)(X), t h e n ~2(U) - X1 (~2(U)) - X1 (71 i n t ( X \ U)) - 72 int 71 cl 71 i n t ( X \ U) = 72 int 71 cl 71 int 72 cl(X \ U) - 72 int 72 cl 72 int 72 cl(X \ U) = 72 int 72 cl(X \ U) - 72 i n t ( X \ U). Therefore U A2 ~2(U) - 72 int 71 C1 g A 7"2 i n t ( X \ U) - U N 72 i n t ( X \ U) - 2~, U V2 ~2(U) - 72 int 7-1 cl (72 int T 1 cl U U 72 i n t ( X \ U)) = 72 int (71 cl ~-2 int 7-1 cl U U T 1 cl 72 i n t ( X \ U)) = 72 int (71 cl U U T 1 c l ( X \ 72 cl U ) ) -
X.
Finally, ~i(~i(U))
- 7i int ( X \ 7i i n t ( X \ U ) ) - 7i int 7~ cl U - U
for each U c ( i , j ) - O D ( X ) -
a
i-07P(X).
c
Let ( A = {A = A l U A 2 = {a, b, c, d, m , n}, 4 } ) b e the goset in D i a g r a m 3 where A1 = { a , b , d , m}, A2 = { b , c , d , n}, and the quasi order relation on A is defined as follows: if x, y c Ai and x is connected with y by the solid line directed upwards from x to y, then x -< y; if x and y are connected by the horizontal broken line, t h e n x -< y and y -< x. Finally, i f x , y c A \ { b , d } are not connected, t h e n x and y are not comparable.
Diagram 3 Let binary operations Ak, V k : A i x Ak ~ Ak be defined as follows: y A i x = x ,z-----+, x ~ y ,z-----~, x V k y = y
for each pair of c o m p a r a b l e elements (x, y) c Ai x Ak and yAix=xAky=d,
yVix=xVky=b
for each pair of n o n - c o m p a r a b l e elements (x, y) c Ai x Ak. T h e n it is not difficult to see t h a t A = { A 1 , A 1 , V l , ~ 1 , 0 , 4 , e , Ae, A2, V 2 , ~ 2 } is a G B A , where (3 = d, e = b, ~1 ( a ) ~-~ ?~, ~1 (T/~) - - C, ~ 2 ( C ) - - T/~, ~2(Tb) = a , ~ ) l ( a ) = ?Tt,
e l ( ? T t ) -- a ,
r
= /t,
e2(/t)
~-- c,
232
vI. Generalized Boolean Algebra and Related Problems
Xl(a)-
c,
xl(m)-n,
x2(n)-m.
X2(c)-a,
Next, for a GBA A - {A1, A1, VI,~I, Q), ~ ,e, A2, A2, V2, ~2}, we shall define two more binary operations >" Ak x Ai ~ Ai in the manner as follows" iG
x-->y-~k(x)
Viy if ( x , y ) ~ A a x A i .
T h e o r e m 6.1.35. For a GBA .4 = { A 1 , A I , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2,~2}, the following statements hold: (1) ~i(x ~
y) = x - y if (x,y) c Ak x Ai.
z(;
(2) ~ - - .
JG
x = ~, e - - - .
iG
x
x = ~, ~ - - .
i(;,
9 = x, ~ - . .
i~
e = ~(~)
iG
>e=eifxEAi. iG
(3) (x ~
y) ~
z~
(x ~
x = x and
kG
ZG
y) ~
~G
y = x V~ y if (x,y) c Ak x Ai.
(4) x ~ y = e ~ x ~
y/f(x,y)~AkxA~.
zG
(5) x ~
y = (~(y) ~
Zc;
x
~ ( x ) ) V~ O if (x, y) e A~ x A~ and
3c;
> Y = W(Y) ~ i(;
V j ( x ) if ( x , y ) E Aj x A~.
zc;
(6) x /~ (x ~
y) = x / ~ y, y / ~ (x ~
ZG
x ~
(x Ai y) = x ~
iG
y if (x,y) c Ak x Ai.
iG
y ~
(x A~ y) = y ~
y ~
(x Ai y) = (y ~
x) Vi (3 if (x,y) E Aj x Ai. ~ ( x ) if (x,y) c A~ x A~ and
~(y) = y ~
3~
x ~
x if (x,y) c A~ x A~ and 3~
zG
(7) x ~
y) = y and
~G
3~
~i(Y) = (Y ~
JG
V j ( x ) ) Vj (3 if (x, y) c A j x Ai.
z c;
(s)
(x ~
y)/~
zo
(x ~
z) = x
---., (~/~ ~)
zG
iG
if x, y, z ~ A~ or x ~ Aj, y, z ~ A~ and
(~ ~
~) ~ (x ~
3c;
z) = ~ ~
ZG
(~ ~ ~)
iG
if x, y c A j , z c Ai or x, z ~ Ai, y ~ A j . (9)
(x
~
~)/~ (~ ~
zo
z) = (~ v~
~) ---.
zG
z
iG
if x, y, z ~ A~ or x ~ Aj, y, z ~ Ai and
(x ~ zG
z)
~ (~ ~
z) = (x v~ ~) ~
zG
z
iG
if x, y ~ A j , z ~ Ai or x, z ~ Ai, y ~ A j .
(10)
x --.~ (y ~ iG
zG
z) = (x Ai y) ---> z = y ~ iG
~G
(x ~ zG
z)
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6.1. G o s e t s , G e n e r a l i z e d
Lattices, ...
237
Once again to emphasize the importance of the G.identity operator, recall that to define a GBA above, we first introduced the notion of a goset, then successively the notions of a G.lattice, a distributive G.lattice, a G.complementation operator and, finally, the notion of a GBA. Now that we have fully covered the (fundamental) details, we can introduce the notion of a GBA more easily by using only the well-known notions of the theory of Boolean algebras and the G.identity operator. T h e o r e m 6.1.37. Let ( L - L1 U L2, 4 ) be a goset, where (L~, A~, V~, (9, e) are lattices and L1 UI L2 - {0, e}. If there exists a pair X - (X1, X2) such that X, " Li --+ Lj are maps and X,(x) ~ x ~ X,(x) for each element x c Li, then X, are isomorphisms. Proof. First, let us prove that X, (X, (x)) - x for each x c Li. Clearly,
x, (x, (~)) v x,, (x) ~ x, (x,, (*)) and, therefore,
so that x _< X, (X, (x)) and X, (X, (x)) < x i
since 4
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X,,(x) and X,(Y) 4 Y 4 X~(Y)
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we obtain that and so because ~ In, --<. J Finally, the case of Vi can be proved similarly and thus X~ are isomorphisms. F1 As in the first case the pair X - (X1, X2), is called the G.identity operator. C o r o l l a r y 6.1.38. Let (L - L1U L2, ~ , X) be a goset, where (L~, A~, V~, 0, e) be lattices, L1 N L2 - {O,e} and X - (X1, X2) be a G.identity operator. Define binary operations
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240
vi. Generalized Boolean Algebra and Related Problems
Conversely, if Z: is a G.lattice, then (Ls, Ai, Vs, O,e) are lattices in the usual sense, satisfying the conditions which correspond to a) and b) of (4). Therefore (Li, Ai, Vi) are Boolean algebras ([26, [29]) and thus, by (c) of Corollary 6.1.38, s is a GBA. D The pair I = (11, 12) is called the G.rejection operator. Our next result is concerned with a V-formation and the corresponding strong a m a l g a m a t i o n in the sense of [127]. Let s = {L1, AI, V1, 4 , L 2 , A2, V2} be a G.lattice. We define the binary relation (c) on L1 U L2 as follows: if x c Li, y c L 1 U L2, then x(c)y ~
y = x or y = x Vj (~.
Let us prove t h a t (c) is the congruence o n L1 U L2. In the first place, note t h a t (c) is the equivalence relation o n L 1 U L 2. (1) Suppose that x, xl, y, yl E L i and x(c)y, Xl(C)yl. Then x = y and XI --- Yl. Hence (x Ai X l ) ( c ) ( y Ai Yl) and (x Vi X l ) ( c ) ( y Vi Yl) since X Ai Xl = y Ai Yl a n d x Vi Xl = y Vi Yl.
(2) If x, Xl E Li, y, Yl E Lj, then y = x Vj I~ -- x Aj e, yl = Xl Vj (~ = Xl Aj e and we have
y Aj yl - (x Aj ~) Aj (Xl Aj ~) - (x A~ x,) Aj ~, y vj yl - (x vr e) vj (x~ vj e) - (x v~ Xl) vr e since/2 is a G.lattice so that
(x Ai Xl)(e)(y Ai Yl) and (x Vi xl)(c)(y Vj Yl). (3) If x, y C Li, Xl, Yl C L j , then x - y, Xl - yl and it follows that
(X Aj Xl)(C)(y Aj Yl), (x Vj Xl)(c)(y Vj Yl). (4) If x, yl 6 Li, X l , y C Lj, then y - x Aj e, Yl -- Xl Ai e and x Aj Xl - (x Aj ~) A~ Xl - y Aj Xl - y Aj ((~1 A~ ~) A~ ~) -
= y A j (Yl Aj e) -- (yA i Yl) Aj e. Similarly, x Vj X 1 -- (y V i Yl) Vj (~1, that is,
(x Aj Xl)(c)(y Aj Yl) and (x Vj Xl)(c)(y Vj Yl). Thus (c) is the congruence on L1 U L2. The congruence class, containing an element x E L1 U L2, is denoted by Ix], while the set of classes [x] - by s Let us define binary operations 9, U ' ( s x s ~ s as follows: [x] N [y] -- [x As y] and [x] U [y] - [x Vi y] for each pair (x,y) E L k x L i . It is not difficult to see that ( s [(9], [e]) is a lattice in the usual sense, obtained from s by sticking each element x c Li \ {(~, e} to the element x Vj (~ and the elements (9, e to themselves. Moreover, the maps
mi " (Ls, Ai, Vi, (9, e) ~ (C/c, n, I I, [0], [el), defined as m s ( x ) -
[x], are isomorphisms.
6.1. Gosets, G e n e r a l i z e d L a t t i c e s , . . .
241
Since the lattice C(2) = {L = {(~, e}, A, V} is a sublattice of any lattice which contains (9 and e, we obtain t h a t if s = { L 1 , A 1 , V I , ( ~ , 4 ,e, L2, A2, V2} is a G.lattice, then the quintuplet (s tl,t2), where t~ : s --, L~ are algebra embeddings, is the V-formation since m l O t l = m 2 o r 2 , t h a t is, the diagram ml
L1
s is commutative. Moreover, (ml, m2, C/c) since
t
this
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t2 > L2 V-formation
is strongly a m a l g a m a t e d
m , ( L 1 ) F1 m2(L2) - m l ( t l ( L ( 2 ) ) - m2t2(L(2))
by
[127].
Our consideration in this section will not be complete unless we make at least a casual mention of the real-valued functions on G B A ' s and bitopological GBA's. For this, we need to introduce a few handful notions. Suppose s {LI,A1,V1,Q},4 ,e, L2, A2, V2} is a G.lattice. Then the elements x E L1 U L2 and y c Li are disjoint, t h a t is, x d , y if x Ai y -- ~ . Furthermore, an element x 6 L1 U L2 is disjoint from a set E C_ L 1 U L 2 , t h a t is, x d c E , if x Ai y -- 0 an for each element y C E~, where Ei - E N Li. D e f i n i t i o n 6 . 1 . 4 0 . Let A = {A1, A 1 , V I , ~ I , Q } , ~ ,e, A2, A2, V2,~2} be a G B A and E = E1 U E2 c A1 U A2. T h e n E is said to be a G.disjoint system of elements of A if xd~((Ei \ { x } ) U Ej), where x c Ei is an arbitrary element, t h a t is, if x Ai y = (3 for each y c E~ \ {x} and x Aj z = (9 for each z c Ej. The notion of a G.disjoint system of elements is used to give the following D e f i n i t i o n 6 . 1 . 4 1 . Let f i [ - {A1, A 1 , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2, q~2} be a GBA. A pair p (#1,#2), w h e r e p i " Ai ~ R are finite maps, is said to be a G.quasi measure on A, if pi(x) >_ 0 for each element x c Ai and
xEE 3
yEEi
xEEj
yCE,
for every finite G.disjoint system E = E1 U E2 C A1 U A2. It is obvious t h a t if p = (pl, p2) is a G.quasi measure, then #~(0) = 0. P r o p o s i t i o n 6 . 1 . 4 2 . If p = (#1,#2) is a G.quasi measure on a G B A A, then x 4 y implies that pk(x) <_ #~(y) for each pair (x, y) ~ Ak x A~.
Pro@ First, let x, y E A~ and x < y. T h e n the elements x and y - x = y Ai ~i(x) i
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6.2. Generalized Ideals and their Variety . . . .
243
i f x 4 Y, then Cli(x) 4 Clj(y) and Inti(x) 4 Inty(y) for each pair (x, y) c Ai x Aj.
(9)
x = Cl~(x) ~
~i(x) = Inti(~i(x)) z--> ~i(x) = I n t j ( p i ( x ) )
and x = Inti(x) ,z--->, ~i(x) = Cli(~i(x)) ,z----5,~i(x) = Clj(~i(x)) for each x c A i. The proof, based on elementary calculations, is omitted. A more detailed development of the questions connected to bitopological GBA's and quasi measures on GBA's, in our view, is of independent interest also.
6.2. Generalized Ideals and their Variety. Stone Family of P r i m e Generalized Ideals All our further constructions are essentially connected with the notion, introduced by
Definition 6.2.1. A G.ideal (briefly, GI) of a G.lattice/2 = {L1, A1, Vl, 1~, 4 , e, L2, A2, V2} is a pair I = (I1,/2), where Ii C_ Li and which satisfies the following conditions: (1) I f x c I l U / 2 andycIi, thenxV, ycIi. (2) If x E I1 U/2, y E L1 U L 2 and y 4 x, then y c I1 U 12. E x a m p l e 6.2.2. Let A = {A1,A1,V1,q21,(~,~,e, A2, A2, V2,~2} be a GBA and p = (p l, p2) be a G.quasi measure on A. Then it is not ditticult to see that I = ( I I , h ) , where I~ = {x c A , : p~(x) = 0}, is a GI. Moreover, note that the pairs ({a,d}, {c, d}) and ({m, d}, {n, d}) in Diagram 3 are GI's. Since for every GBA A = {A1, A1, V1, qP1, (~, 4 ,e, A2, A2, V2, P2}, the system { A 1 , A 1 , V I , ( ~ , 4 ,e, A2, A2, V2} is a G.lattice, in the sequel we shall consider, in general, GBA's.
P r o p o s i t i o n 6.2.3. Let A = {A1, A1,V1,~91,O,~ ,e, A2, A2, V2,~2} be a GBA. Then a pair I = (/1,/2), where I~ c_ A~, is a GI if and only if {I~, A1, V1, 4 , /2, A2, V2} is a G.sublattice of the G.lattice {A1, A1, V1, 0 , 4 , e, A2, A2, V2} and x 6 I1 U 12, y 6 A~ imply x A~ y E Ii. Proof. First, let I=(I1, h ) be a GI and let us prove that {/1, A 1 , V I , ~ , / 2 , A2, V2} is a G.sublattice of the G.lattice {A1,A1, V1,0, 4 ,e, A2, A2, V2}. Indeed, if x c I1 U/2 and y c Ii, then by (1) of Definition 6.2.1, z Vi y c Ii. It is evident that z Ai y 4 z and by (2) of Definition 6.2.1, x Ai y E I1 U/2 ~ x Ai y 6 I/. Thus {/1, A1, V1, 4 ,I2, A2, V2} is a G.sublattice. Now, if x c I1 U 12 and y c A~, then x Ai y 4 z and once more applying (2) of Definition 6.2.1 gives that z Ai y C I 1 U / 2 ,z-----N,x Ai y 6 I~.
244
VI. Generalized Boolean Algebra and Related Problems
Conversely, let {I1, A1, V1, ~ , I2, A2, V2} be a G.sublattice of the G.lattice { A 1 , A 1 , V 1 , O , 4 ,e, A2, A2, V2} and x E I1 U / 2 , y C Ai imply x Ai y C Ii. Let us prove t h a t the conditions (1) and (2) of Definition 6.2.1 are satisfied. Indeed, if x c I1 U / 2 and y E Ii, t h e n x Vi y c Ii since {I1,A1, V1, 4 ,I2, A2, V2} is a G.sublattice, t h a t is, (1) of Definition 6.2.1 is satisfied. Finally, if x c I1 U / 2 , y E A1 U A2 and y ~ x, t h e n if, for example, we consider the case y c A j , we obtain t h a t y - x A j y and, therefore, y c lj c I1U/2, t h a t is, (2) of Definition 6.2.1 is also satisfied. D It is obvious t h a t if I - (I1, I2) is a GI, then Ii are ideals in the usual sense and the pair I - (A1, A2) is a GI. It is likewise obvious t h a t for a GI I - (I1, I2), we have x E I1 U 12, y C Ii ~ x V i y C Ii. M o r e o v e r , I1 7s A1 ~ 12 ~= A2 and, therefore, a GI I - (I1,I2) is said to be proper if I~ ~ A~. Thus, by (2) of Definition 6.2.1, I - (I1, I2) is proper
e---->, e g I1 <-->, e c / 2 .
Hence, using (1) of the latter definition, the a r g u m e n t s between T h e o r e m 6.1.31 and Corollary 6.1.32, and (7) of Corollary 6.1.33, we obtain: I -- (I1,/2) is proper -'
<---5, P v ( ~ l ) N (11 x / 2 ) - 2~ ~' :,
,'- Pv(~2) K~(I2 x I1) - 2~ ~ -'
P v ( ~ l ) K I (I1 • I1) - 2~ ,:
:,
:- Pv(~b2)A (I2 x I2) - 2~.
Note further t h a t I ~ - ({(9}, {@}) is called the zero GI, and for any proper GI (I1, I2), we have 11AI2 - {(9}. It is not difficult to verify how I1 a n d / 2 are i n t e r c o n n e c t e d in any proper GI. Following (6) of T h e o r e m 6.1.16, for each x c Ai, we have x 4 x Vj 1~ and x Vj 1~ ~ X. Now by virtue of (2) of Definition 6.2.1, we obtain x c I~ ,<--->, x Vj (~ E Ij. Therefore any proper GI can be w r i t t e n as I - (I1 - { x } , I 2 - { x V2 (9}), where Ii C Ai so t h a t 1 I I 1 - /21. Moreover, if A1 contains an a t o m a (<---5, A2 contains the a t o m a V2 (~), then it is clear t h a t I - ({a, @}, {a V2 (9, @}) is s t r u c t u r a l l y the simplest type of a GI after the zero GI I ~ Let I - ( I I , h ) and I ' - ( I ~ , I ~ ) be any two GI's o f a G B A A. Then, by virtue of the above reasoning, I
-
I1
--
I~ <-->, I2 -- 1s and I1 C I~ <-->, h C I;.
In the former case we write I - I', while in the l a t t e r - I < I'. Hence I < I ' ,' :, (I < I ' or I - I'). Clearly, the binary relation < is a partial order on the set Z{/t - (I~,/~) " t e T} of all GI's of the G B A A. T h e zero of this poser coincides with the zero GI I ~ and the unity - with the unit GI I e - ( A 1 , A 2 ) . As usual, a subfamily {/t - (I~, I~)" t c To c T} of 2- is a chain i f / t <_/t, o r / t , <_/t for each pair t, t ~ c To. Proposition any intersection
6.2.4. For a GBA A- {A1,A1,VI,~I,0,~ ,e, A2, A2, V2,~2} o f G I ' s a n d the u n i o n o f a n y c h a i n i f G I ' s is a GI.
P r o o f . First, let { I t - ( I f , I t ~ ) " t c To c T} be any family of GI's of a G B A A. We are to prove t h a t I - (I1,/2) - ( ~ I~, ~ I t) is also a GI. Assume t h a t tETo
tETo
6.2. Generalized Ideals and their Variety . . . .
245
x 6 I1 U /2 and y 9 Ii. If x 9 Ii, then x ~ I t for each t ~ To and, by (1) of Definition 6.2.1, x V i y 9 I t for e a c h t 9 To so t h a t x V i y 9 Ii. I f x 9 Ij, then x 9 I t for each t ~ To and, therefore, x V~ y 9 I t for each t 9 To, t h a t is, x V~ y ~ I~. Before we proceed to proving (2) of Definition 6.2.1 note t h a t
( N
( N z:) - N
t E To
t E To
t E To
t E T~~
t E To
t E To
o
Indeed, it is clear that
Hence it can be assumed t h a t x c {@, e} and x c
n (I~ u I~). In that case x c I~ t6To
or 2; C I~ for each t E T O since A1 n A2 -
{E:),e} and, t h e r e f o r e , 2; c
n
I~
tGTo
or x C n
I2t. Hence x c ( n
t 6 To
I1t) u ( n
t 6 To
I2t) 9 Now assume t h a t x c I1 U I2,
t 6 To
y 6 A1 U A2 and y ~ x. W i t h o u t loss of generality let us consider the case where X 9 I1 -- n I~. Then y ~ x and x 9 I~ for each t c To imply x 9 I~ U I2t for each t6Tc) t 9 To and y ~ x. By (2) of Definition 6.2.1, y 9 I~ U I2t for each t 9 To so that y C n (1~ u/2t) -- ( n 1~) u ( n I ~ ) - I l U I 2 . t E Tu
t E Tu
t E Tu
Next, let us assume t h a t {It - (I~,It2) " t 9 To C T} is a chain of GI's a n d I -- (11,22) -- ( U I~, U I~). T h e n x c I x u / 2 a n d y 9 I i i m p l y x c t E To
( U zf)u( U zl) t E To
If x 9
t E To
t E To
.ayc U If. t 6 To
U It, then there exists an index tl 9 To such t h a t x c I tx. Similarly, t6To
there exists an index t2 c To such t h a t y 9 I t2. It is clear t h a t I~ 1 c I~ 2 or I t~ C_ I tl . W i t h o u t loss of generality let I tl C I: 2 . Then x 9 I~ ~ and, therefore, x V~ y 9 I~ 2 C Ii. Now assume t h a t x 9 U I~. Then there exists t l e To such t h a t y 9 I~ ~ tE Tu
sinceyE
U It and there existst2 9 t E To
such t h a t x 9
~ sincex 9
U It. It is t E To
obvious t h a t / t ~ <_/t2 o r / t 2 _
246
VI. Generalized Boolean A l g e b r a and R e l a t e d P r o b l e m s
of a GBA A and
z'-
{I~, - (I~', I~tt ).
t'
c To c T, (B,, B~) _< I~, for ~,~h t' ~ To }
then 27' r 2~ as I c - (A1, A2) E Z'. The smallest (with respect to <_) GI, containing (B1, B2), is said to be generated by the pair (B1,B2). Clearly, the GI, generated by the pair (~, ~), is the zero GI. Also, note that {27, A, V} is a lattice in the usual sense with I ~ and U as the zero and the unit element, respectively, where for I ' - (I[, I~), I " - (I~', I~') E Z, we have
I' A •
n I~', I~ n I~')
and I ' V I " is the GI, generated by the pair (I{ O I{', I~ O I~'). Before proceeding to our next important theorem, let us recall that by (6) of Theorem 6.1.16 if x, y E A~, z E Aj, then X ~ y, i
X ~ Z ,,~,---)', X \ / j (9 ~ y,
X\/j(9
~ Z. j
Hence, taking into account the fact t h a t for each GI I - (I1,I2), we have Ij X, (Ii), the following equivalences are obvious: (For each element x C I1, there exist finite sequences of elements al, a2,. 99 an c A1, b l , b 2 , . . . , bm E A2 such t h a t x <_ al Vl a2 Vl "--Vl an and x 4 bl V2 b2 V2 1 9.. V2 b,~.) .z---> (For each element x E I1, there exist finite sequences of elements al,a2,...,an c A1, b l , b 2 , . . . , b m C A2 such that x V 2 ( 3 <_ bl V 2 b 2 V 2 " " V 2 b m and 2 x V2 O 4 al V1 a2 Vl .-. V1 an.) ~ (For each element x c I1, there exist finite sequences of elements bl, b2, . . . , bm E A2, al, a2, . . . , an C A1 such t h a t x _< (bl Vl 1
(9)vI(b2VI(9)VI" "VI(DrnVI(9) a n d x
4 (al V2(9)V2(a2V2(9)V2. . .V2(anV2(9).) ." "," (For each element x E I1, there exist finite sequences of elements bl, b 2 , . . . , b~ c A2, al, a 2 , . . . , an e A , such that x V~ (3 < (al V2 (9) V2 (a2 V2 (9) V 2 " " V2 (an V9 (9) 2
and x V~ (9 4 (bl V~ (9) Vl (be V~ (9) V l - . . Vl (b,~ Vl (9).) Note t h a t by virtue of GL3 we can easily ascertain by induction that the equalities (a 1 V 2 e ) V 2 (a 2 V 2 1~) V 2 " "
V2 (a n V2 l~) -- (a 1 V 1 a 2 V l . . .
V 1 an) V 2 (~)
and
(hi Vl e) v, (b~ v, e) v , . . . v, (bm V, e) - (hi v~ b~ v~... v~ bin) v, e are valid for finite sequences of elements al, a 2 , . . . , an 6 A1, bl, b 2 , . . . , bm 6 42. T h e o r e m 6.2.5. L e t A {A1, A 1 , V 1 , ~ 1 , 0 , 4 , e , A2, A2, V2,~2} be a G B A and (B1,B2) be a pair such that Bi C A i . T h e n a pair I - (I1,I2) is a GI, w h e n e v e r it satisfies one of the following equivalent conditions: (a) I1 - {x E AI" there exist finite sequences of elements al, a 2 . . . , an C B1 and b l , b 2 , . . . ,bin E B2 such that x < al V1 a2 V 1 . . . V1 an, x ~ bl V2 1
b2 V 2 " " V2 bin}.
6.2. G e n e r a l i z e d Ideals and their Variety . . . .
247
(b) 12 - {y c A2" there exist finite sequences of elements c l , c 2 , . . . , c k and d l , d 2 , . . . ,dz E B2 such that y < dl V2d2 V2 . . . V2dl, y ~ Cl V1 2 9 - . V 1 Ck}. Moreover, this CI I - (I1,/2) is generated by the pair (B1, B2) if and only each element a ~ Bi, there exists a finite sequence of elements b,, b 2 , . . . , bt such that a 4 bl V j be V j . . . V j bt.
c B1 c2 V1
if f o r c Bj
P r o @ First, we shall see t h a t I - (I1, h ) is a GI, t h a t is, (1) and (2) of Definition 6.2.1 are fulfilled. Indeed" (1) Clearly, we have the following four variants: y E I1, x c I1 or x E /2; y C 12, x C I1 o r x C 12. Since all these variants are proved by a similar scheme, we shall consider only the case x c I~, y ~ / 2 . By (a) and (b), there are finite sequences of elements a l , a 2 , . . . , a n ; C l , C 2 , . . . , c a ~ B1, b l , b 2 , . . . , b ~ ; d l , d 2 , . . . , d z ~ B2 such t h a t
x <_ al V1 a2 V 1 - . . V1 an,
x 4 bl V2 b2 V 2 " " V2 bm
y_
y% ClVlC2Vl...VlCk.
,
and 2
Therefore 2
and Let el al,e2 a2,...,en b , , . . . , g,~ - b,~,gm+l - d l , . . . , elements e l , e 2 , . . . , e n - b k C B 1 virtue of which x V2 y c I2. (2) Let x c I1 U I2, y c A1 -
-
-
-
x c I1,
-
y c A,
-
an, en+l Cl,...,en+k ck and gl bl,g2 g,~+z - dz. It is clear t h a t the finite sequences of and g l , g 2 , . . . , g m + l C B2 are those sequences by -
-
-
-
-
-
-
-
U A2 and y d x. Here we also have four variants:
or y c A2;
x c I2,
y c A1 or y c A2.
We will prove only the case where x c I2, y E A1 and y d x; the others can be proved similarly. By (a), there exist finite sequences of elements a l , a 2 , . . . , a , ~ c B1, b l , b 2 , . . . , b m E B2 such t h a t x < bl V2 b2 V2 . . . V2 bm and x d al V1 a2 V1 ..- V1 an. 2
It is obvious t h a t y _~ al V, a2 V1 . . . VlaN and y ~ b, V2 b2 V2 .-. V2 b~ 1
so t h a t y E I1. Thus I = (I1,/2) is a GI. Now let us prove the second part of the theorem. If the GI I = (I1,/2), constructed above, is generated by the pair (B1, B2), then Bi C_ Ii and, therefore, the "only if" part is obvious. Conversely, let there exist for each element a c B~, a finite sequence of elements bl, b 2 , . . . , bt c By such t h a t a ~ bl Vj b2 Vj . . . Vj bt. Then it is clear t h a t a E Ii and, therefore, Bi C_ Ii. Hence, it suffices to prove
248
VI. Generalized Boolean Algebra and Related Problems
that I - ( I i , h ) is a smallest GI such that ( B I , B 2 ) <_ I ,e---->, Bi c_ Ii. Indeed, if I ' - (I~, I~) is any GI for which (B1,B2) <_ I' and x E h is any element, then by assumption, there exists a finite sequence of elements a l, a 2 , . . . , an C Bi such that x <_ al Vi a2 Vi . . . Vi an. Clearly, a l , a 2 , . . . , a n C B i C_ I~ implies x E I~ so i that Ii c_ I ' because x E Ii is an arbitrary element and, therefore, I _< I'. D This theorem gives rise to the following interesting C o r o l l a r y 6.2.6. Let A - {A1, A1, V I , ~ I , 0 , 4 , e , A2, A2, V2, qD2} be a GBA and let I - (11, I2) be the GI generated by a pair (B1,B2), where Bi c Ai. Then the following conditions are equivalent: (1) I - (I1, I2) is nonproper. (2) There is an element a E B1 for which there ezist finite sequences of elements a l , a 2 , . . . , a n E B1, b l , b 2 , . . . , b , ~ E B2 such t h a t p l ( a ) ~ al V1 a2 Vl -.- Vl an and ~ l ( a ) _< bl V2 b2 V2 ... V2 b,~. 2
(3) There is an element b E B2 for which there exist finite sequences of elements C l , C 2 , . . . , c k c B1, d l , d 2 , . . . , d l c B2 such that ~2(b) ~ dl V2 d2 V2 . . . V2 dz and ~2(b) <_ cl Vl c2 V 1 . . . Vl ck. 1
(4) There is an element c E B1 for which there exist finite sequences of elements e l , e 2 , . . . , e t E B1, g l , g 2 , . . . , g r C B2 such that ~bl(C ) 4 gl V2 g2 V2"'" V2 gr a n d ~bl(C ) ~ el V1 e2 V I " " V1 et. 1
(5) There is an element d E B2 for which there exist finite sequences of elements P l , P 2 , . . . , P u c B1, q l , q 2 , . . . , q v E B2 such that r 4 Pl ~/1 P2 V1 . " VlPu and r < ql V2 q2 V2 ... V9 qv. 2
Proof. Applying Theorem 6.2.5, in four cases of the existence of finite sequences of the elements of B1 and B2, we respectively obtain ~ l ( a ) c I2, ~2(b) c 11, r E I l a n d ~ b 2 ( d ) c / 2 . T h e r e f o r e a , c c B1 C_ I1, b, d E B2 C_ /2 and (1) of Definition 6.2.1 imply that a V 2 q ~ l ( a ) - - e e I 2 , bV1 ~ 2 ( b ) - e e I 1 ,
cV1 ~ b l ( c ) - e e I 1 , and d V 2 ~ b 2 ( d ) - e e I 9
so that I - I e - (A1, A2) in each one of the four cases. Conversely, let I - I ~ - (A1,A2). Then e c I1 - A1, e E / 2 - A2. Hence, by virtue of Theorem 6.2.5, there exist finite sequences of elements al, a 2 , . . . , an E B1, bl, b2,... ,bin E B2 such that e - al V1 a2 Vl . . . V1 an -- bl V2 b2 V2 "" V2 bin. Clearly, for each element a E B1 and each element b E B2, we have ~ l ( a ) 4 al V1 a2 V 1 . . . V1 an, ~l(a) < bl V9 b2 V2-.. V2 bm; 2
~2(b) 4 bl V2 b2 V2.-. V2 bin, g)2(b) <_ al V2 a2 V 2 . . . V2 an; 1
~bl(a) 4 b~ V2 52 V 2 . ' - V 2 bin, ~bl(a) _< al V1 a2 V I ' " V1 an; 1
@2(b) 4 al V1 a2 V I " " V1 an, ~2(b) _< bl V2 b2 V2"'" V2 bin. 2
~]
Thus for the GI I - (I~, I2), generated by the pair (B1, B~), to be nonproper it is sufficient that there exists at least one element of the set B1, satisfying the
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252
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
and
{xv =
{y ~ A j "
e.
xe
there exists an element y~ ~ I~ such that y ~ z V~ y~},
is a GI. Moreover, this GI is generated by the GI I' - (I~,I~) and the element z 6 Ai \ I~ (so that I - (I1,I2) is the smallest GI for which I' < I and z E Ii) if and only if for each element y' c I}, there exists an element x E I~ such that y'4x. Pro@ First we are to prove t h a t I - (I1,/2) is a GI. Since in our concrete case z E A~ \ I ' in proving (1) of Definition 6.2.1 we shall consider b o t h y E I~ and ycIj. If y E Ii, z c I1 U/2, t h a t is, if x E I i or x E Ij, then in b o t h cases, we obtain __ z Vi y < z Vi (x' Vi y'), where x' Vi y' E I~ since x', yl E I il and I iI is an ideal in i the usual sense. Therefore, by condition, x V i y E I i . On the other hand, if y E I o, x E I~ or z c Ij, then in a similar manner, we obtain x Vj y 4 z V~ (x' V~ y'), where z' Vi y' E I~, and, therefore, x Vj y c Ij. Let us prove (2) of Definition 6.2.1. For this we assume t h a t x E I1 U / 2 , y 6 A1 U A2 and y 4 x. We shall consider only the cases x E Ii, y c Aj and z E Ij, y E Ai since the other cases are proved quite similarly. For x c Ii and y 6 Aj, we obtain y 4 z Vi z', while for x E Ij, y E Ai, we obtain y _< z Vi z', i
where x' c I[. Therefore y c Ii. Now, let us prove the second part. If a GI I - (I1,/2) is generated by a GI I ' - (I~ , I~) and an element z E Ai \ I '~, then I ' < I and, by the first part, for each element y' c I5 there exists an element y" E I~ such t h a t y' 4 z Vi y". Hence the conditions I~ c Ii and z E Ii imply z V~ y" c Ii, t h a t is, z - z Vi y" is the required element. Conversely, let us assume t h a t I - (I1,I2) is the GI, constructed using the GI I ' - (I~, I~) and the element z E Ai \ I~. It is clear t h a t I~ c Ii and z c Ii. If y' c I}, then the existence of an element x c Ii with the condition y' 4 z implies y' E Ij and, therefore, I~ c Ij since y' is an a r b i t r a r y element of I5. Thus I ' < I and it remains only to prove t h a t if I " - (I{', I~') is any GI such t h a t I ' < I " , and z c I~', then I _< I " . Indeed, for an a r b i t r a r y element x E Ii there is an element x~EI~suchthatx<_zVix ~ ButI ~
x E I~~. Similarly, if y c Ij, then there is an element y' c I~ such t h a t y 4 z Vi y'. Since z Vi y~ C I "i , we obtain y E I~t and so I _< I". D Note that, by virtue the first part of T h e o r e m 6.2.7, for the GI I - (I1,/2), we can write the structure of Ij in more precise terms as follows:
{xv -
e-
xc I,} -
{y E Aj 9 there exists an element y' E I~ such t h a t y 4 z Vi y'} - {y c A j "
there exists an element t' E I} such t h a t y <_ z V j t'}, J
where obviously t ~ - y' Vj 0 .
6.2. Generalized Ideals and their Variety . . . .
253
C o r o l l a r y 6.2.8. Let A = { A 1 , A I , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2,~2} be a GBA, I = (I1, I2) be the GI, generated by a GI I ' = (I~,I~) and an element z ~ Ai \ I~. Then the following equivalences hold:
I = (h, h)
e Ij
9
Proof. Since, by (1) of Corollary 6.1.33, ~ = ~j o X,, (2) of Corollary 6.1.32 and (2) of Theorem 6.1.16 imply X, o X:, o ~ i = X, o ~Pi ~
9~i = X, o ~'i,
and, hence, if x c Ai is any element and I = (I1,/2) is any GI, then c
x,
c 6.
Thus it remains to prove the first of our equivalences. If ~ ( z ) c I}, then I' < I and z r imply that I = I e = (A1, A2). Conversely, let I = I e = (A1,A2). Then there exists an element x I E I~ such that e = z Vi x'. Therefore ~i(e) = ~i(z) Aj ~i(x') = O and by (5) of Corollary 6.1.32, ~ i ( z ) 4 x'. Thus ~ i ( z ) c I~ since x' c I~. rq Now, we have come to the most important variety of GI's which will be essentially used in our further reasoning. A GI I = (I1,I2) of a GBA is said to be maximal if it is proper and has no property to be contained in a proper GI of A so t h a t there does not exist a proper GI I ' - ( I ~ , I~) such t h a t I < I'. D e f i n i t i o n 6.2.9. A GI I = (/1, I2) of a GBA .4 = {A1, A1, V1, ~ 1 , (~, 4 , e, A2, A2, V2, ~2} is said to be prime if it is proper and the following condition is satisfied: if x E A1 u A2, y 6 Ai and x Ai y C Ii, then either x r I1 U/2 or y r Ii (or both x c 11 U 12 and y c / ~ ) . Therefore, if I = (I1,I2) is a prime GI and x c Ai is any element, then x e Ii or pi(x) e Ij(..v---> ~i(x) = X, (pi(x)) e Ii), but not both, because the GI I = (I1,/2) is proper. On the other hand, if there exists an element x c Ai such that x c I~ and V)i(x) c Ij, then x Aj ~ ( x ) = (9 C Ij and to obtain a contradiction, it suffices to apply the condition of Definition 6.2.9. T h e o r e m 6.2.10. Let I = ([1,12) be a GI of a GBA A = {A1, A1, Vl, (/91, (~), ~---~, e, A2, A2, V2, ~2}. Then the conditions below are equivalent: (1) I = (I1, I2) is a prime GI. (2) For every element x c Ai either x E Ii or pi(x) E I j ( < - - > r x. c both
=
Pro@ The implication (1) ----5, (2) is shown in the discussion preceding Theorem 6.2.10. (2) ~ (1). By (2) it is obvious that the GI I = (I1,/2) is proper. Now assume t h a t x, y c A~, x A~ y C/~ and x ~ 5 , YE I~. Then by (2), ~i(x), ~ ( y ) c Ij and by ( 1 ) o f Definition 6.2.1 and ( 2 ) o f Theorem 6.1.31, we have F~(x)Vj F ~ ( y ) = p~(x A~ y) ~ Ij. Therefore by (2), x A~ y r I~, which is impossible. The case x ~ Aj,
254
VI. Generalized Boolean Algebra and Related Problems
y 9 A~ and x A~ y 9 I~ is proved in a similar manner, taking into account (1) of Definition 6.2.1 and (3) of Theorem 6.1.31. (1) - - > (3). Let the prime GI I = (I~,I2) be not maximal. Then there is a proper GI I ' - (I~,I~) such t h a t I < I'. I f x 9 I ~ \ I i is any element, then (1) <---> (2) implies t h a t ~ ( x ) 9 Ij so t h a t ~ ( x ) 9 I5 and, therefore, P v ( ~ ) N (I~ • I5) ~ ;~, which contradicts the condition t h a t I' - ( I { , I ~ ) is a proper GI. Hence I = (I1,I2) is maximal. (3) ---->. (1). Let x 9 A1UA~, y 9 A~ and x A ~ y 9 I~. W i t h o u t loss of generality assume t h a t x 9 A~ \ I~ and consider the GI I ' - (I~, I~), generated by I - (I1, I2) and x-gI~. Clearly, I ' = I ~ = ( A I , A 2 ) since I = (I1,I2) is maximal. Therefore for e 9 Ai, there exists an element z 9 I~ such t h a t e = x V~ z. Hence
sincez 9
andyAiz
If we now assume t h a t x 9 A j \ Ij and I ' - (I[, I~) is generated by I - (I1, I2) and x - g I j , then I' - I ~ - ( A 1 , A 2 ) and thus for e 9 A j , there exists an element z 9 Ij such t h a t e - x Vj Z. Hence, by GDL2, we have
y - (x v j z)
y - (x
y)
(z
y) e
sincez 9 andzAiy4 z. Therefore x g [1 U/2 and x Ai y 9 Ii imply y 9 Ii. Further, assume t h a t x A~ y 9 I~ and y-gI~. First, let x 9 A~. If I ' - (I~,I~) is generated by I = (I1,I2) and y g / ~ , then I ' = I e = (A1,A2) and, by analogy with the above, for e 9 A~ there exists an element z 9 Ii such t h a t e = y Vi z. Therefore Similarly, if x 9 A j , then I ' = I e = (A1, A2), where I ' is generated by I = (I1, I2) and y g Ii. Hence, by the remark between Theorem 6.2.7 and Corollary 6.2.8, there is an element z 9 Ii such t h a t e = y Vj (Z Vj {~). But by the G.commutativity laws, x Ai y 9 Ii .e----->.(x Ai y) Vj ~) = (y Aj x) 9 Ij and, therefore, x -
Aj 9 - (v
sincez 9
z) Aj x - (y Aj x) v j (z Aj x) 9 Ij
zimplyzAjx 9
P r o p o s i t i o n 6.2.11. For a GBA A1 - {A1, A1, V1, ~1, ~, ~, e, A2, A2, V2, p2} any GI I - (I1, I2) is p r i m e if and only if the goset (A1 U A2, ~ ) is a G.chain. Proof. Let any GI of A be prime and, for example, there are elements m E A1, y E A2 such t h a t x ~ y is false and y ~ , is false. Then by Theorem 6.2.12 and Corollary 6.2.13 below there are prime GI's I x - ( I ~ , I ~ ) and Iy - ( I ~ , I ~ ) such t h a t x g I ~ , y c I~ a n d y g I ~ , x E I~. S i n c e x A 2 y ~ x, x A 2 y < y, by (2) of 2
Definition 6.2.1, x A 2 y I x Cq I v. Then x A2 y Let (A1 tO A2, 4 ) xEAltOA2, yEA~.
e I~ and x A 2 y e I y. Let I - ( I i , I u ) - ( I ~ A I ~ , I ~ A I ~ ) -C I2, but x g I1 and y g / 2 , which is impossible. be a G.chain, I (I1,I2) be a GI and x Ai y C Ii, where Thenx4 yory4 x. If y 4 x, t h e n x A ~ y - y E I ~ , a n d i f
6.2. Generalized Ideals and their Variety . . . .
255
x 4 y where, for example, x 9 Aj, then x Ai y 9 Ii implies y Aj x -- (x Ai y) Aj e -x 9 Ij so that in both cases I - (11, I2) is prime. V] Our next theorem underlies many further constructions. T h e o r e m 6.2.12. For a GBA A - {A1, A1, Vl, ~1, (~, ~ , e, A2, A2, V2, ~2} and each element x 9 Aj \ {(9}, there exists a prime GI I* - (I~,I~) such that
Proof. Let A4j - {I - (11,/2)} be a family of all GI's such that x - ~ I j and ~j(x) 9 Ii(~=> ~j(x) 9 Ij), where x 9 Aj \ {(9} is an arbitrarily fixed element. Clearly, A4j ~ ~ since, in particular, the right principal GI (~I(X))I(R)--(II--{ycA
1 " y ~ ~l(X)}, I2-{zcA
2 9 z ~ (~l(X)}) 9 2
and the left principal GI
(992(x))i(L)-(Ii-{yCAl
" y ~
1
992(x)}, I 2 - { z 9
" z 4 ~2(x)}) 9
for x 6 A1 \ {(~} and x 6 A2 \ {(~}, respectively. Following Proposition 6.2.4, every chain in 3dj has an upper bound in A4j since the family Adj is partially ordered by the relation <. Therefore by Zorn's lemma, M j has a maximal element I* - (I~, I~). Let us prove that I* - (Ii~, I:~) is the required prime GI. Clearly, x - c I ] and ~ j ( X ) 9 I[, that is, ~)j(X) 9 I~. By virtue of Theorem 6.2.10, it suffices to prove that I* - (I~, I~) is maximal. Let I' - (I~,I~) be a GI such that I* < I'. It is obvious that ~j(x) 9 I~(~' ",, @(x) 9 Ij). Also, x 9 Ij since if x-E I j, then I' c A4j, which is impossible by the definition of the family 34j and the definition of I* as its maximal element. Therefore P v ( ~ j ) n I} x I~ r ~ so that I ' - I e - ( A 1 , A 2 ) and thus I* - (I~,I~) is a maximal, that is, a prime GI. D C o r o l l a r y 6.2.13. For a GBA A - {A1, A1, V1, 7~1, O, ~ , e, A2, A2, V2, P2} the conditions below are equivalent:
(1) (2)
For each element x c A~ \ {@}, there exists a prime GI such that x-c I~. For each pair (x, y) E (A~ x Aj) \ PA(P~), there exists a prime GI I = (I1,12) such that x -c I~ and y -~ Ij. (3) For each pair (x, y) c Ai x Aj, where x ~ y is false, there exists a prime GI I - (/1, I2) such that x-~ I~ and y E Ij. (4) For each pair (x, y) c (A~ x A~) \ PA(~P~), there exists a prime GI I = (I1, I2) such that x -c Ii and y -c Ii. (s) For each pair (x, y) c A, x A~, where x < y is false, there exists a prime i
GI such that x-E Ii and y c I~.
Pro@ (1) ~ (2). Let (x, y) c (Ai x Aj) \ PA(~i) be any pair. Then by (6) of Corollary 6.1.32, (y,x) c (Aj x Ai) \ PA(~j) and, therefore, y Ai x C Ai \ {@}. By (1), there exists a prime GI I - (I1,I2) such that y Ai x-~Ii and thus, by (2) of Definition 6.2.1, x c Ii and y -c Ij.
256
VI. Generalized Boolean Algebra and Related Problems
(2) ~ (3). Let (x, y) E A~ x Aj be a pair such that x 4 Y is false. Then by (5) of Corollary 6.1.32,
x A~ 9~j(Y) r 0 and (x Ai 9~j(Y), e) 9 (Ai x Aj) \ PA(9~i). Therefore, by (2), there exists a prime GI I - (I1,/2) such that x Ai ~j(y)-EIi so that ~i(x Ai ~j(y)) -- p~(x)Vj y 9 Ij since I - ( I i , h ) is a prime GI. Hence 9~(x), y 9 Ij and x gI~. (3) --->. (4). Let (x,y) 9 (Ai x A i ) \ P~(gi) be any pair. Then, by (7) of Corollary 6.1.33, y ~ 9~(x) is false and, therefore, by (3) above, there exists a prime GI I - (I1,/2) such that y ~ / ~ and 9~(x) 9 Ij so that x c/~. (4) ~ (5). Let (x,y) 9 A~ xA~ b e a p a i r such that x_< y is false. Then, i
by (12) of Corollary 6.1.33, x A~ r r ~ and by (7) of the same corollary, we have (x A~ r e) 9 (A~ x A~) \ PA(~). Thus by (4), there exists a prime GI I - (I1, Is) such that x Ai @i(Y)C Ii so that @i(x A~ t~i(Y)) -- t~i(x) Vi y 9 Ii since I - (I~,/2) is a prime GI. Hence ~bi(x), y 9 I~ and x~I~. (5) ~ (1). Let x 9 Ai \ {O} be any element. Then x _< ~Pi(x) is false and by i
(5), there exists a prime GI I - (I1, Is) such that x-~I~.
[3
C o r o l l a r y 6.2.14. I f { I t - (I~,I~)}tcT is a family of all prime GI's of a GBA then ( ~ I~, r] i ~ ) - ( { e I , { e } ) .
.A-{A,,A1,V1,9~,,O,~,e, A2, A2, V2,9~2},
tET
tCT
Proof. The proof is an immediate consequence of Theorem 6.2.12.
[3
Thus the intersection of all prime GI's of a GBA is the zero GI. However, we shall see that every infinite GBA (i.e., A1 and, therefore, As too is infinite) has families of prime GI's the intersection of which is the zero GI and which do not contain all the prime GI's of ,4. The following definition will be of much use. D e f i n i t i o n 6.2.15. A family S - {I - (I1,I2)} of prime GI's of a GBA A - {A1, A1, V1,9~a, O, 4 , e, A2, Ae, V2, 9~2} is called a Stone family of prime GI's if ( A t 1 , AI2 ) -- ({(~}, ( ( ~ } ) -- I O. IEs
T h e o r e m 6.2.16. Let S be a family ofprime GI's of a GBA A = {A1, A1, V1, 9~1,O, ~ , e, A2, A2, V2,9~2}. Then the conditions below are equivalent:
(1) S is a Stone family. (2) For each element x E A~ \ {(9}, there exists a prime GI I such that x -~ I~. (3) For each pair (x,y) c (Ai x Aj) \ PA(9~i), there exists a (I1, I2) E S such that x -E I~ and y -EIj. (4) For each pair (x, y) c A~ x Aj, where x ~ y is false, there GI I = (I1, I2) c S such that x-~ I~ and y E Ij. (5) For each pair (x, y) E (Ai x Ai) \ PA(~i), there exists a (I1, I2) c S such that x -~ Ii and y -~ Ii.
= (I1, I2) E S
prime GI I = exists a prime prime GI I =
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258
VI. Generalized Boolean Algebra and Related Problems (7) (8) (9) (10)
/ f x = i a - s u p B , then ~ ( x ) = j a - s u p C d a B for any subset B C A1UA2. C d c B is a GI for any subset B c A1 U A2. Cdaax(L) = qPl(a)I(R) if a E A1, and CdaaI(R) = ~2(a)i(L) if a E A2. IflC = {B} is a non-empty family of subsets of A1UA2, then ~ C d a B =
Cd~
U
B61C
B.
B61C
(11) x = / c - s u p B if and only if x E Cd(;(Cdc~B ) and ~ ( x ) ~ C d a B for any
subset B c A1 U A2. Proof. (1) and (2) are immediate consequences of the respective definitions. (3) If x c u c ( B ) N A i , then b <_ x ~ ~ ( x ) A ~ b - O if b E Bi, and
i
b4 x ~ ~ ( x ) Aj b - (~ if b c Bj so that 7)~(x) c CdaB. Conversely, if x E Aj and 7)j(x) E CdaB, then 7)j(x) A~ b - (~ ~ bEB~,andT)j(x)Ajb-(~~b<_xifbEBj. T h u s x C u a ( B ). J (4) By Definition 6.2.18, we have
Cdc;(Cd~B) - {x c A1
U A2
b 4 x if
" xdGCdGB}
and, therefore, if b E Bi is any element, then x Ai b - O ~
b Ak x -- O for each element x E Cd(;B and so, we have
bdGCdaB ~
b E C d a ( C d c B ).
(5) First, let us prove that if B - B1 U B2 has an i c - s u p B , then for each element y c A1 U A2, we have:
yAiia-SUp(BlUB2)--ia-SUp((
U ( y A I a ) ) U ( U (yA2b))). aEB1
Indeed, let y c Aj be any element.
b6Bg,
< y A i i a - s u p B for each i element x c Bi and y Aj z ~ y Ai ia-sup B for each element z E Bj. Therefore
ia-stlp ( (
U (yAla))U
(
a6B1
Then y A i x
U (yA2 b ) ) ) ~ y A i i a - s l l p ( B 1 U B 2 ) . b6B2
Now, let
u
u( u
aEB1
bEB2
be any element. Then for each x E Bi, we have
x <_ x v~ ~,(y) - (y v~ ~j(y)) A~ (x v~ ~,(y)) - (y A~ x) V~ ~j(y) <_ z V~ ~,(y).
i
i
Similarly, for each x E Bj, we obtain x ~ z Vi ~j(y). Therefore
ic-sup(B 1 U B2) _< z Vi r
i
and y Ai iG-sup(B1 U B2) <
i
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260
VI. Generalized Boolean Algebra and Related Problems
(11) Let x = i c - s u p B . By (5), ydcx for each y E CdoB. Therefore y A i x = (9 implies x Ak y = (9 for each y C C d c B and so x E C d o ( C d c B ). Further, by (7), pi(x) = jc;-sup B and, by (5), qpi(x) Ak y = 19 for each y E C d o ( C d o B ) . But by (4), B C C d c ( C d o B ) so that ~i(x)Akz = (9 for each z C B. Hence ~i(x) c C d c B . Conversely, if ~i(x) c CdoB, then by (3), we have x E u o ( B ). If z c u o ( B ) is any element, then applying once more (3) gives that ~k(z) E C d c B , that is, x At ~k(z) = (9 since x C C d o ( C d c B ). Therefore x 4 z and thus x = i t - s u p B since z c u c (B) is an arbitrary element. [-1 D e f i n i t i o n 6.2.20. Let A = { A 1 , A 1 , V I , q O l , ( 9 , ~ ,e, A2, A2, V2,~2} be a GBA and B = B1 O B2 c A1 U A2. Then B is said to be a G.component of A if B = C d o ( C d c B ) , that is, if C d c ( C d ~ B ) c B. Now we can formulate the following important statements. T h e o r e m 6.2.21. For a GBA .4 the conditions below are satisfied:
(1) (2) (3) (4)
=
{A1,A1,VI,~I,O,~
,e, A2, A2, V2, p2},
Every G. component is a GI. Left and right principal GI's are G.components. C d a B is a G.component for every subset B = B1 U B2 c A1 U A2. If = {E} a family of a. ompo t , th n ('1 E al o EC tC
a G.component and thus for every subset B = B1 U B2 c A1 U A2 there exists a smallest G.component A B, containing B. (5) AB = C d a ( C d c B ) for every subset B = g 1 U B 2 C A1 u A2 and, hence, AB is a GI. Proof. Assertion (1) follows directly from ( 8 ) o f Theorem 6.2.19. (2) By (9) of Theorem 6.2.19, CdGa~(c) = pl(a)~(n), and hence Cdc;(Cdc. a l ( L ) ) = C d c ( g p l ( a ) l ( R ) )
= ~92(gPl(a))i(L) = de(L)
if a c A1. Similarly, Cdc(Cdoal(R)
) = Cdc(g)2(a))l(L) = ~l(~2(a))l(R)
= al(R)
ifaEA2. Thus we can conclude that the converse of (1) above is true for the left and right principal GI's. (3) By ( 4 ) o f Theorem 6.2.19, C d o B c C d a ( C d a ( C d o B ) ) . On the other hand since, by (4) of Theorem 6.2.19, B C C d c ( C d c B ) , it follows from (1) of the same theorem that Cd~.(Cd~(Cd~B)) c Cd~B. (4) If E0 = N E, then by (10) of Theorem 6.2.19, EC/C
Zo- [') cd (cd z)ECtC
U CdGZ EC1C
and it remains to use (3). (5) By (4) of Theorem 6.2.19, the set C d a ( C d a B ) is a G.component, containing B, and hence AB C C d a ( C d a B ) . On the other hand, by (1) of Theorem 6.2.19,
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262
VI. Generalized Boolean Algebra and Related Problems
Note t h a t the pairs ({m, b}, {n, b}) and ({a, b), {c, b}) in Diagram 3 are GF's. In the sequel we shall consider GBA's. It is evident t h a t if F = (Fx,F2) is a GF, then Fi are filters in the usual sense and the pair F = (A1,A2) is a GF. It is likewise obvious t h a t for a GF F = (F1,F2), we have x c F1 U F2, y C Fi ~ x A i y E Fi. Furthermore, it is clear t h a t F1 r A1 ,z----5, F2 r A2, and hence a GF F = (F1, F2) is said to be proper if Fi :/: Ai. Therefore, by (2) of Definition 6.3.1, a GF F = (F1, F2) is proper ,z----5, (~ c F1 ,z----5, (9 c F2 and thus, taking into account (1) of Definition 6.3.1 and using the arguments between T h e o r e m 6.1.31 and Corollary 6.2.32, as well as (7) of Corollary 6.1.33 of the same theorem, we find t h a t F - (F1,F2) is a proper *', )" P A ( ~ 2 ) ( ]
GF <--5. P A ( ~ I ) A
(/!72 x F 1 ) - ~ ~ ",
( F 1 x/t72) -- Z ,(
PA(~)I)N (F 1 x F1)-
Z .'
,','.
" PA (~P2) C~ (F2 x F 2 ) - 2~.
Note t h a t the GF F ~ - ({e}, {e}) is called the unit GF, and for any proper G F F - (F1, F2), we have F1 • F2 - {e}. By (2) of Definition 6.3.1, it is clear t h a t every proper GF has the form F - (F1 - {z}, F2 - {x V2 O}), where x r (3, and hence Fll - IF2[. Let F - (F1, F2) and F ' - (F~, F~) be any two G F ' s of a G B A A. T h e n by virtue of the above arguments F1 - F[ ~ F2 - F~ and F1 c F~ z--->, F2 c F~. In the former case we write F - F ~ and in the latter c a s e - F < F ~. Therefore F < F ~ ,z--5, ( F - F ~ or F < F~). It is obvious t h a t the relation _< is a partial order on the set ~ {Ft - (F~,F~) 9 t c T} of all GF's of the G B A A. The zero element of this poset is the unit GF, while the unit element is the zero GF F - F O - (A1,A2). As usual, a family {Ft - (F~, F~)" t E To c T} is a chain if Ft <_ Ft, or Ft, <_ Ft for each pair of indices t, t ~ c To. A G F F - (F1, F2) of a G B A .4 is said to be maximal or a G.ultrafilter if it is proper and has no property to be contained in a proper GF of A so that, there is no proper GF F ' - (F~, F~) of .4 such t h a t F < F'. Definition 6.3.2. A G F F - (F1, F2) of a G B A .4 - {A1, A1, V1, ~1, ~ , 4 , e, A2, A2, V2, ~2} is said to be prime if it is proper and the following condition is satisfied: if x E A1 U A2, y E Ai and x Vi y E Fi, then either x E F1 t2 F2 or y c Fi (or both x c F1 t2 F~ and y c Fi).
By analogy with the reasoning after Definition 6.2.9, the notion of a prime GF immediately implies t h a t if x c A~ is any element, then x c Fi or ~ ( x ) c Fi(< :. ~2~(x) - Xj(~i(x)) E I~), but not both since the GF F - (F1, F2) is proper. An especially useful assertion is T h e o r e m 6.3.3. For a G B A A - {A1,A1, VI, qPl, ~), 4 ,e, A2, A2, V2, ~2} the following condition (the G.duality) holds: I - (I1, h ) is a GI of A if and only if F-(~2(12),~1(/1))
i8 a G F of,A.
Pro@ It suffices to prove only the implication from left to right since the proof in the opposite direction can be carried out by the same scheme, taking into account the equality ~ i ( ~ j ( I j ) ) - Ij.
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264
VI. Generalized Boolean Algebra and Related Problems
(7) Let F = (F1, F2) be the GF generated by a pair (B1, B2), where Bi C Ai. Then the following conditions are equivalent: (a) F = (F1, F2) is nonproper. (b) There is an element a c B1 for which there exist finite sequences of elements al,a2,. .. ,an E B1 and bl,b~, . . . ,bin ~ B2 such that al A1 a2 A1 9 "" A1 a n ~ ~ l ( a ) and bl A2 bz A 2 " " A2 bm < ~ 1 ( a ) 2
(c) There is an element b c B2 for which there exist finite sequences of elements C1~ C2~ 9 9 9 ~ Ck E B1 and d l , d 2 , . . . ,dz E B2 such that dl A2 d2 A2 9". A2 dz ~ ~2(b) and C l A l C 2 A l . . . A l C k <_ ~2(b). 1
(d) There is an element c C B 1 for w h i c h there exist finite sequences of elements e l , e2 ~ 9 9 9 ~ e t E B1 and gl, g2, 999, g~ E B2 such that gl A2 g2 A2 " ' " A2 g r ~ 1~1 (C) and e l A1 e2 A1 " ' " A1 et 1/21(C) 9 1
(e) There is an element d E B2 for which there exist finite sequences of elements P l , P 2 , . . . ,Pu E 81 and ql,q2,... ,qv E B2 such that pl A1 P2 A1 " ' " A1 P u ~ ~ 2 ( d ) and q l A2 q2 A 2 " " A2 qv ~ 1/)2(d).
(8)
2
If a E A1, b E A2 and
F-(F1
-{xEA
1 9 a<_x 1
81
--
{a}, B2 -- {b},
then
andb~ x},F2-{yEA2"
the
pair
b < y a n d a ~ y}) 2
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(FI-
{xcAI"
a<_x}, F 2 - { y c y c A 2 "
aV20
1
2
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there exists a finite sequence of elements a~,a~2,... ,a fk c B1 such that a~ A1 a~ A I " " A1 a~ ~ y}) and if B 1 - - ~ , ~ ~ B 2 C A2, then the GF, generated by the pair (~,B2), has the form: F = (F1 = {x c A1 : there exists a finite sequence of elements b~l,b~,..., b~ E B2 such that b~l A2 b~ A 2 . . . A2 b~m ~ x}, F2 - {y E A2 " there exists a finite sequence of elements bl, b2,..., bz c B2 such that bl A2 b2 A2 ... A2 bz _< y}). 2
(10)
{a}, B2 = z (B1 = z , B2 = {a}), where a E Aa (a E A2) is an arbitrary element, then the CF, generated by the pair ({a}, ~) ((~, {a})), is called the left (right) principal GF and has the form
I f B 1 ~-
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--
{a V 1 x"
(F1 -- {x E AI"
a<x},F2-{yEA2" 1
x 9 A1}, F2 -
{ a V2 y "
a 4 x}, F2 - {y E A2"
a~ y
y})-
e A2})
a < y}) 2
-- (F1 - { a V1 x"
x C A 1 } , F2 - {a V2 y"
y E A 2 } ) ).
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266
VI. Generalized Boolean Algebra and Related Problems
(16) Let F -
(F1,F2) be the GF generated by a GF F ' element z E Ai \ F~. Then F-
(F1,/72)
is nonproper
~
~i(z) c I~ ~
(F~,F~) and an
~i(z) c 1~.
(17) I f .T -
{ F - (F~,F2) 9 F is a GF of a GBA A}, then {U,A,V} is a lattice in the usual sense with F ~ and F ~ as the zero and unit elements, respectively, where f o r F ' - ( F ~ , F ~ ) and F " - ( F ~ ' , F ~ ' ) , we the pair (F~ U F[', F~ U F~').
Proof. Conditions (1)-(17) are immediate consequences of the G.duality and the corresponding statements for GI's. D
T h e o r e m 6.3.5. Let A - {A1, A1, V l , 991, 1~, ~ , e, A2, A2, V2, ~2} be a GBA, ( I i , h ) be a GI, F - (F1,F2) be a GF and I n F - 2J. Then there exists a prime GI I* - (I~, I~ ) such that I <_ I* and I* n F - ;~. I-
Proof. Let 3,t - { I ' - (I~, I ; ) " I < I ' and I ' n F - ~}. It is clear that 34 r ;~ since I E A/t, and Ad is partially ordered by _<. If C - {It - ( I ~ , I t ) " It c All} is any chain in AA, then by Proposition 6.2.4, I " - (UI~, U/t) is a GI. Therefore, by Zorn's lemma, A/l has a maximal element I* - (Ii~, I~). Clearly, I _< I* and I* N F - ;~ so that, it suffices to prove only that I* is prime. Contrary: if I* is not prime, then there exist elements a E (A1 U A2) \ (Ii~ U I~) and b E Ai \ I[ such that a Ai b E I~*. Without loss of generality let us suppose that a c A1 \ I~ and b c A2 \ I~. Since I* is a maximal GI having the property I* n F - ;~, we have a I * n F r 2~ r I*b N F and, therefore, a I[ N Fi r ;g r I[b N Fi. Let, for example, x c aIi~ n F1 and y C I~ b n/7'2. By the definition of aI* and I *b, there exist elements m E I i ~, p c Ii ~ such that x-mVla,
y-pVpb
and m V l a E F 1 ,
pV2bEF2.
Thus t - (m V~ a) A2 (p V2 b) E F2 since F is a GF. By I GDL1 and III GDL2, we have: t
--
=
((?Tt
Vl
a) A2 p)
v)
V2 ((m V1 a) A~ b) p) t,) (a -
t,).
By condition aA2b C I~. Since I* is a GI, it is clear that m A 2 p E I~; also aA2p <_ p 2
andpcIi ~implyaA2pCI~,mA2b4 mandmcIi ~ i m p l y m A 2 b C I ~ . Hence t ~ I~ and thus I* n F ~r ;~, which is impossible, and, consequently, I* - (I~, I~) is the required prime GI. V] C o r o l l a r y 6.3.6. Let I - (I1,/2) be a GI of a GBA A - {A~, A1, V1, Pl, ~, 4 , e, A~, A~, V2, ~ } and a E A~ \ I~. Then there exists a prime GI I* - (I~, I~ ) such that I <_ I* and a-r I [ . P r o @ Without loss of generality let a ~ A2 \ I~ and let us consider the GF aF(R) -- ( F 1 - - { x ~ A l "
a 4 x } , Fg, - {y E A2 " a <_ y } ) .
Then I n a~(~) -- ~ and it remains to use Theorem 6.3.5.
2
6.3. Generalized Filters and their Variety. . . .
267
C o r o l l a r y 6.3.7. Let A - {A1, A1, VI,~I, 1~),~ , e , A ~ , A 2 , V~, ~ } be a GBA, a, b ~ A1 ~ A2 and a r b. Then there exists a prime GI which contains exactly one of the elements a and b. Proof. Let us consider the following cases: (1) a, b ~ A~, a r b. Then for a, b E A1 (a, b c A2) and a <1 b (a < b), it
suffices to consider the GI aZ(L) and the GF bF(c) (respectively, the GI ai(R) and the GF bF(R)) since ai(c) ~ bF(c) -- ~ (ai(R) C~br(R) -- 2~). (2) a, b ~ A~, a r b and a, b are not comparable by 4 IA, = < - If, for example,
i
a, b ~ As, then we also have ai(~) ~ b~(R) - 2~ - a~(n) N hi(n). Indeed, if we consider a <_
a 4
2 and
({:;c C AI" :;c 4 b}, {y c A2" y < b}), 2 then aF(R) A bl(R) r 2~ implies t h a t there exists an element y c A2 such t h a t a _< y _< b and so a < b, which is impossible. The proof for ai(R) and bF(R) is 2 2 2 similar. (3) a E A1, b E As and a -< b. Let us consider bi(R)-
aI(L)-
({X C AI" a: < a}, {g E A2" y ~ a } ) 1
and bF(R) -- ({X ~ n l " b 4 x}, {y ~ n 2 9 b ~ y}).
2
It is clear that aI(L) N bF(R) -- 2~. (4) a E A2, b c A1 and a, b are not comparable by 4 . In this case we also have aI(R)NbF(L) -- ~ -- aF(R)Nbt(L). Indeed, let, for example, aI(R)C~bF(R) r ~. Then there exists y ~ A2 such t h a t b 4 9 < a, t h a t is, b -< a, which is impossible, ffl
2
C o r o l l a r y 6.3.8. A n y GI I - (I1,12) of a GBA A - {A~, A1, V1, ~1, (~), ~ , e, A2, A2, V2, ~2} is the intersection of all prime GI 's, which contain I. Proof. Let M-
{It - (I~,I~) " I <_ It, t c T }
and I ' -
(~I~,~')I2t). tET
tCT
Let us prove t h a t I ' - I. Contrary" I' r I and so I < I'. Then there exists a E I~ \ I1 and by Corollary 6.3.6 there is a prime GI I " such that I < I " and a g I~'. But then a g I~ since I ' <_ I". D T h e o r e m 6.3.9. Let A = { A 1 , A 1 , V I , g ) I , 0 , ~ , e , A2, A2, V2, q~2} be a GBA, I = (I1,/2) be a GI and F = (FI,F2) be a GF. If I N F = (I1 N F1, I2 N F2) r ~, then {I1 to Fj, A1, V1, ~ , / 2 tO F2, A2, V2} is a G.convex G.sublattice of the G.lattice {A1, A1, V1, 0, ~ ,e, A2, A2, V2} and, conversely, any G.convex G.sublattice of {A1, A1, V1, (~, ~, e, A2, A2, V2} can be represented in a unique manner as a nonernpty intersection of a GI and a GF.
268
VI. Generalized Boolean Algebra and Related Problems
Proof. First, let I n F = (11 n F1,/2 N F2) r Q and let us prove that {I1 U F1, A1, V1, 4 , / 2 U F2, A2, Vg.} be a G.sublattice of the G.lattice {A1, A1, V1, ( 9 , 4 , e , Ag~,A2, V~.}. L e t a E ( I 1 N F 1 ) U ( h N F 2 ) a n d b E I i N F i . IfaEIjnFj, then by (1) of Definition 6.2.1, a V~ b c I~. On the other hand, a c Fj and a 4 a V i b . Hence, by (2) of Definition 6.3.1, a V i b E F l U F 2 4---> a V i b E Fi and, therefore, a Vi b E I i N Fi. Furthermore, a E Fj, b E Fi and by (1) of Definition 6.3.1, a A~ b E F~. But we also have a c Ij and a A~ b 4 a. Therefore by (2) of Definition 6.2.1, a Ai b E I~ U/2 e-----F,a Ai b c I~ and thus a A~ b E I~ n F~. Now, let us prove that the set (I1N F1)U (I2 N F2) is G.convex. We will prove only the cases, where aEIinFi,
bcIjnFj,
ccAi
and a < c 4
b;
i
the others can be proved similarly. By (2) of Definition 6.2.1, we have c E I1 U /2 .e-----F. c c Ii and by (2) of Definition 6.3.1, c c F1 U F2 .e----F. c E Fi. Thus c E Ii N Fi implies c C (11 n f l ) U (I2 n/72) so that (I1 n F1) U (I1 n F2)
is G.convex and the first part is proved. By Proposition 6.2.4, any intersection of GI's is a GI. Let us prove that this fact is also true for G.convex GI's. Suppose that {It - (I~, I~)" t c T} is a family of G.convex GI's, that is, I~ U I~ is a G.convex set for each t E T, such that
I-- (/1- n I ~ , / 2 tET
-
N tET
and a, b E I1 U 12, c 6 A1 U A2, a 4 c 4 b. Let us consider only the case, where a c I1, b c / 2 and c c A1. Then a c I~, b c I~ for each t c T and since It - (I~,It) is G.convex, c E (I~ U I t) n A 1 ,g-----5,C E I~ for each t c T. Therefore c E I1 and so c c I1 U/2. Hence the set I1 U I2 is G.convex, that is, I - (I1,12) is G.convex. Suppose that s - {L1,A1, V1, ~ ,L2, A2, V2} is a G.convex G.sublattice of { A 1 , A 1 , V 1 , O , 4 , e , A2, A2, V2}. If I - (I1,I2) and F - ( F 1 , F2) are respectively the GI and the GF, generated by (L1,L2), then Li c_ Ii n Fi. By Theorem 6.2.5 and (6) of Corollary 6.3.4, x c Ii N Fi implies that there exist elements b E Lj, c c Li such that c _< x 4 b. Therefore x E L i because L 1 U L 2 is G.convex and /
thus L i - Ii n F~. Finally, let us prove that this representation is unique. Suppose that there exist I' - (I~, I;) and F ' - (F~, F~) such that L 1 U L 2 - (I~ n f ; ) U (I; n F~). It is clear that Ii _C I ' as Li c_ I ' and I is generated by (L1,L2). Similarly, F~ C_ F'. On the other hand, let a c I ' and b E Lj be any elements. Then a V 3 b c I~ and a Vj b _> b E F~. Following (2) of Definition 6.3.1, a Vj b E F~ and, therefore, J a V j b E I } N F ~ . Furthermore, a 4 a V j b a n d L j - I~NF~ implyaVjb EIj. Therefore, by (2) of Definition 6.2.1, a 4 a Vj b implies a c Ii and thus I - I'. Similarly, one can prove that F - F ' . [2
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274
VI. Generalized Boolean Algebra and Related Problems
The rest can be proved by means of (1) of Theorem 6.1.35 in the manner as follows: if (x, y) E Ai • Ai, then y) -
h
(
j(x h
y)) -
h
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j(x A j
(v) -
h
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h
y) (x)
Similarly, one can prove that
h~(x = ~ y) - hi(x) ~(~
.-ff~' h~(y) for (x, y) c Aj z d~.
~(~,.
(6) This statement follows immediately from Definitions 6.4.1, 6.4.6, and (1)(3) above. (7) If h - (hi, h2)" A ~ A', then by Definition 6.4.1, and (3) above, hi and G h2 are homomorphisms. Therefore the first equivalence is the well established fact. Now assume that (hl(x),h2(y)) e PA(~I) and h - ( h i , h 2 ) " ,4 ~ A' is a G G.isomorphism. Therefore hi and h2 are bijections, and (hl(x),h2(y)) e PA(~I) gives hi(x) A~ ~ ( h 2 ( y ) ) - ~t so that hi(x) A~ hl(~2(y)) - ~t and so hl(x A1 ~2(Y)) - O'. Hence x A1 P2(Y) -- O, that is, (x,y) E PA(~I). Similarly, one can prove that (h2(z), hi(v)) c PA(P~) implies (z, v) E PA(P2). Conversely, let (hl(x),h2(y)) c PA(~/1) implies (x,y) C PA(~I). We are to show t h a t hi and h2 are bijections. Let us consider only the case of h2 since the case of hi can be proved similarly. Assume that a, b c A2 and a J: b. If h2(a) - h2(b), then h2(a) A~I ~;~(h~(b)) - 0 implies h2(a) A~ hi (~2(b)) - O' and so (h2(a),hl(q;2(b)) c P A ( ~ ) . Therefore (a, ~2(b)) c PA(~2) SO that hA1 ~2(b) - ~. Thus by (5) of Corollary 6.1.32, ~ ( b ) < ~2(a) ~ a _< b. In a similar manner 1 2 one can prove that b <_ a, that is, a - b. The contradiction obtained shows that 2 h~ is a bijection. It is likewise easy to see that the implication
(h2(z),hl(v)) ~ P A ( ~ ) ~
(Z,V) ~ PA(~2)
gives t h a t h l and hz are bijections. Hence it remains to show that
((hl(x),hl(y))
~
PA(~i) and (he(z),h2(v)) ~ PA(~Pl)
imply ( x , y ) e PA(~I) and (z,v) e PA(~Pz)) ~ hi and hz are bijections. We shall prove only the first case since the proof of the second one is similar. Let (h~(x),hl(y)) e P A ( ~ ) implies (x,y) e PA(~Pl). We are to show t h a t hi is a bijection. If a, b e A1, a ~ b and h~(a) - h~ (b), then
hi(a) Ai ~ i ( h l ( b ) ) - Q}t implies hi(a) A~ h l ( ~ l ( b ) ) - Q)t and so (hi(a), hi (~1 (b)) ~ PA (1/)~). Hence (a, ~1 (b)) E PA (~1) so that hA1 ~1 ( b ) - ~}. Therefore by (12) of Corollary 6.1.33, a <_ b. Similarly, one can prove that b <_ a, 1 1 t h a t is, a - b. The obtained contradiction shows that hi is a bijection. By a similar reasoning one can prove that h~ is a bijection.
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276
VI. Generalized Boolean Algebra and Related Problems
is a GI of A.
Now let h" A ~ A' and I' - (I{, I;) be a GI of A'. G
(1) Suppose that x 9 h l l ( I ~ ) U h~l(I~) and y 9 h~-l(I~). If x 9 h~-l(I~), then
hi(x) Vri hi(y) 9 I~ and h ~ l ( h i ( x ) Vti hi(y)) - h ~ l ( h i ( x Vi y)) C hi-l(I~). Therefore x Vi y 9 h ~ l ( h i ( x Vi y)) c hi-l(I~). If x 9 h~-l(Ij), then hj(x)V~ hi(y) 9 I~ and, therefore,
h ~ l ( h j ( x ) Vti hi(y)) - h ~ l ( h i ( x Vi y)) c hi-l(I~). Thus x Vi y 9 h ~ l ( h i ( x Vi y)) c hi-l(I~). (2) If x 9 h~-l(I~), y 9 Ai and y <_ x, then hi(x) 9 I~, hi(y) 9 A~ and i
hi(y) <_' hi(x) so that hi(y) 9 I~ C I~ U I~. Hence i
y 9 h~l(hi(y)) C hi-l(I~) C h l l ( I ~ ) U h2-1(Is If x 9 hi-l(/~), y 9 Aj and y 4 x, then y Aj ~i(x)
so that h j ( y ) 4 ' h i ( x ) . y 9 h;l(hj(y))
Since hi(x) 9 I~, we obtain hj(y) 9 Ij c I~ U g and thus
C hjl(z~) c hll(/i)U
C o r o l l a r y 6.4.10. I - (Ii,h)
(~. Therefore
h21(/;).
Let h 9 A ~ A ~ be a G.isomorphism of GBA's. G
[-1
Then
is a GI of A if and only i f h ( I ) - (hl(I1),h2(I2)) is a GI of A'.
C o r o l l a r y 6.4.11. The image of a GF under a G.isomorhism is a G F and the inverse image of a GF under a G.homomorphism is a GF. Thus, if h : A ~ A' is a G.isornorphism, then F = (F1, F2) is a GF of A if G
and only if h(F) = (h~(F1),h2(F2)) is a GF of A'. Proof. The proof is an immediate consequence of the G.duality. 6.5.
(i,j)-Atoms
D
and Pairwise Atomic Generalized Boolean Algebras
Since every pair from PA(~I)[-J PA(~2) is supposed to play the role of zero element of a GBA A, we can extend the notion of an atom to the case of a GBA, that is, to say, we can define the notion of an atom not as an element, but as a pair of elements of a GBA. Moreover, we shall establish a connection between an atom, defined as an element and an atom, defined as a pair. D e f i n i t i o n 6.5.1. An ( i , j ) - a t o m of a GBA A - {A1, A1, Vl,q01, i~, 4 ,e, A2, A2, V2, F2} is a pair ( a , a Vj 0 ) E Ai x Aj, where a r 0 and if (c,d) E Ai x Aj, c _< a, d _< a Vj 0 , then (c, d) - (a, a Vj 0 ) or (c, d) 9 PA (~i). i j It is clear that a 9 Ai \ {0} ~ (a,a Vj O)-~PA(~i), and if a 9 A1, then (a,a V2 O) is a (1, 2)-atom ~ (a V2 O,a) is a (2, 1)-atom.
6.5. (i, j)-Atoms and Pairwise Atomic Generalized Boolean Algebras
277
P r o p o s i t i o n 6.5.2. Let A { A 1 , A 1 , V 1 , ~ 1 , ( 9 , ~ ,e, A2, A2, V2,~2} be a GBA. Then an element a E Ai is an atom of the BA Ai - {Ai, Ai, Vi, !)i, (3, <, e} i
if and only if the pair(a, aVj(3) is an (i,j)-atom of the GBA A (3, 4 , e, A2, A~, V~, q~}.
{A1, A1, V l , ~ l ,
Pro@ Let a c Ai be an atom o f t h e BA Ai. Then by (11) of Corollary 6.1.33, aVj(~ is an atom of the BA Aj. Assume that (c, d) E Ai x Aj is a pair such that c < a and d < a Vj @. Then it is clear that (c, d) - (a, a Vj (9) or (c, d) - ((9, d) c PA (~i) J
or (c,d) - ( c , @) 6 PA(P~) or (c, d) - ((9, (9) 6 PA ( ~ ) . Conversely, let (a,a Vj (9) be an ( i , j ) - a t o m of the GBA A. If a is not an atom of the BA Ai, then there exists an element c E Ai \ {(9} such that c < a. Then the pair (c, c Vj (9) satisfies the conditions:
(c, c Vj (9)-~ PA (p~).
c < a, c Vj (9 < a Vj (9 and i j D
P r o p o s i t i o n 6.5.3. Let c a t - {A1,A1,V1,~l,(9,~ ,e, A2, A2, Vg, p2} be a GBA and (a, a V 9 (3) ~ A~ • Aj. Then (a, a Vj (3) is an (i, j)-atom if and only if for every pair (c, d) E Ai x Aj, we have
a <_ c, a Vj (9 <_ d or (a Ai c, (a Vj (9) Aj d) C PA(9~i)-
i
j
Proof. First, we assume that (a, a Vj (9) is an ( i , j ) - a t o m and (c, d) E Ai x Aj is any pair. Then for the pair (a Ai c, (a Vj (9) Aj d) it is obvious that a Ai c < a, (a Vj (9) Aj d <_ a Vj (9 i
j
and hence
a Ai c -- a, (a Vj (9) Aj d - a Vj (9, that is a
aVj@
or (a Ai c, (a Vj (9) Ajd) 6PA(qDi)
since (a, a Vj (9) is an (i, j)-atom. Conversely, suppose that (a, a Vj O) E A~ x Aj and a pair (c, d) E A~ x Aj satisfies the conditions: c _< a, d <_ a Vj (9. If a <_ c and a Vj O < d, then a - c, i
j
i
j
a V j O - d. If (a Ai c, (a Vj @) Aj d) e PA(~i), then (c,d) c PA(~i) and thus (a, a My (~1) is an (i, j)-atom. D T h e o r e m 6 . 5 . 4 . For a G B A , 4 - {A1, A1, VI, ~I, @, 4 , e, A2, A2, V2, qp2} and an element a E A1, we have the equivalences"
ai(L)--(Ii--{xEAl
" x<_a},I2-{yEA2
" y 6 a}) is a prime GI,'
1
<
;" ( ~ l ( a ) V 1 (9,991(a))
z,: > ~ I ( a ) F ( R ) - ( F I - { x f f A
is a (1, 2)-atom
1 9 ~l(a)~x},f2-{yEA2"
<
',
~l(a)<_y}) 2
is a G. ultrafilter.
>
278
VI. Generalized Boolean Algebra and Related Problems
Pro@ By (3) of Corollary 6.3.4, it suffices to prove only the first equivalence. First, we assume that ai(L) is a left principal GI and (qDl(a) V1 QI, qo1(a)) is a (1, 2)-atom. We have to show that ai(r) is a prime GI, that is, at(c) is a maximal GI so that aI(L) is proper and has no property to be contained in a proper GI of .4. If aI(L) is nonproper, then a - e so that (~l(a) V1 l~, qPl(a)) -- ((~, Q)), which is impossible. Now we assume that there exists a proper GI I' - (I[,I~) such that de(L) < I ~. Then there exists an element b c I[ \ I1, and, therefore, b g I 1 , that is, b _< a is false. But in that case b _< ~)1 (a) and b :/= 2/)1(a) since otherwise 1
1
a, @l(a) E I [ , and so I' - (I~, I;) is nonproper. Thus b < ~)l(a). Consider the pair 1
(b, b V2 (9). It is clear that b < ~1 (a) V 1 1~, bve ~ < ~1 (a) and (b, b V2 O) g PA (~1), 1
2
which contradicts the fact that (~l(a) V1 i~, ~ l ( a ) ) is a (1,2)-atom. Hence aI(L) is a maximal, that is, a prime GI. Conversely, assume that de(L) is a prime GI and (qpl(a) V1 Q), ~ l ( a ) ) is not a (1, 2)-atom. Then there exists a pair (c,d) ~ A1 x A2 such that c < qPl(a)V1 (~), d ~ ~ l ( a ) , 1 2
(c,d) 7~ ( ~ l ( a ) V 1 (~), qDl(a))
and (c, d) c E PA (~1)- Clearly, C < ~ l ( a ) V1 O, 1
d < qpl(a) a n d 2
(c,d) # ( ~ l ( a ) V 1 0 , ~ 9 1 ( a ) )
give C ~ @l(a) V1 O, d < @l(a) or c < @l(a) V l O, d ~ ~ l ( a ) 1
1
2
or both c < ~ l ( a ) V 1 (~) and d < ~l(a). 1
2
Without loss of generality consider the case c <_ ~ l ( a ) V 1 (~, d <2 p l ( a ) . 1
d < ~l(a) ~ 2
a < ~2(d) and so p 2 ( d ) g I 1 . 1
Then
On the other hand, if d E I2, then
d 4 a, that is, d < a V2 (9 - ~(~1(a) so that 2
(~92(~1 (a)) < ~2(d) ~ 1
~)l(a) <_ qD2(d). 1
Hence we conclude that a < p2(d) and ~)l(a) <1 p2(d), that is a V1 ~ l ( a ) -- e -- ~2(d) 1
and hence d - O. But this conclusion contradicts the fact that (c,d)-~ P A ( ~ l ) Thus d g I2 and, therefore, aI(L) is not a prime GI. D T h e o r e m 6.5.5. Let A - { A 1 , A 1 , V l , ~ l , O , 4 ,e, A2,A2, V2,~2} be a GBA and (a, a V j O ) E A~ x Aj. Then (a, a V j O ) is an (i,j)-atom ~ aGF F = (F1,F2), generated by the pair ({a}, {a Vj 0}), is a G.ultrafilter ~ a GI I (~2(F2), pl(F1)), generated by the pair (~i(a) Vi O, ~i(a)), is prime.
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VI. Generalized Boolean Algebra and Related Problems
Thus, as in the first case, a Vj (~ - (~ and so x c Fj or x ~ c Fi (or both x E Fj and x' c Fi). Furthermore, we are to prove that the GF (F1, F2) is proper. Indeed, if there exists an element x c A~ such that x c F~ and ~ ( z ) c Fj, then simultaneously, a _< x and a 4 ~ ( x ) so that a - O, which is impossible since (a, a Vj O) is an i
(i, j)-atom. To prove the converse, assume that F - (F1, F2) is the G.ultrafilter, generated by the pair ({a}, {a Vj 0}) so that Fi - {x c Ai" a <_ x} and Fj - {y c Aj " a Vj I~ ~ y}. i
2
Let us prove that (a, a Vj (~) is an (i,j)-atom. If (c, d) c A~ x Aj is any pair, then c c Fi or ~i(c) E Fj(~=~ r c Fi), but not both since F - (F1,F2) is a G.ultrafilter. Similarly, d E Fj or ~j(d) E Fi(e==~ r E Fj), but not both. Hence there arise the following cases" (1) c E F ~ a n d d E F j . Thena_
i
d) - (a, e) e PA
(a
(3) a Vj 0 <_ p~(c), a Vj 0 _< d, and, therefore, J J
(a
(a Vj e)
d) - (e, a
e) 9
Thus the sufficient condition of Proposition 6.5.3 holds and so (e, a Vj ~) is an (i, j)-atom. D D e f i n i t i o n 6.5.6. A GBA A - {A1, A1, V1, ~91, I~, 4 , e, A2, A2, V2, ~2} is said to be (i,j)-atomic if each pair (c, d ) C P A ( ~ i ) contains an ( i , j ) - a t o m so that for each pair (c, d) g PA ( ~ ) , there exists an (i, j ) - a t o m (a, a Vj (~) such that a <_ c and i
aVj(~<_d. J It is clear that for a GBA A, we have: A is p-atomic
~
A is (1,2)-atomic
~
A is (2, 1)-atomic.
Also, a GBA A - {A1, A1, V1, ~1, t~, 4 , e, A2, A2, V2, ~ 2 } is p-atomic ~ BA's Ai - {Ai, Ai, Vl, ~1, O, <_ e} are atomic.
the
i
Indeed, if A is p-atomic and a E A~ \ {0} is any element, then the pair (a, a Vj (~) g PA ( ~ ) and, therefore, there exists an (i, j ) - a t o m (c, c Vj (~) such that c _< a and c Vj ~ _< a Vj (~. By Proposition 6.5.2, c is an atom in the usual sense i j and thus Ai are atomic. Conversely, let A~ be atomic and (a,b) ~ (Ai x A j ) \ P A ( ~ ) S O that a r (~ r b and b <_ ~i(a) is false. Then b Ai a r ~ and since Ai are atomic, there exists J an a t o m c c A~ such that c_< b A i a . T h e r e f o r e c 4 b, t h a t i s , c V j ( ~ <_ b a n d i j
6.5. (i, j ) - A t o m s and Pairwise Atomic Generalized Boolean Algebras
281
c <_ a. Again applying Proposition 6.5.2, we conclude that (c, cVj@) is the required i
(i, j)-atom. T h e o r e m 6.5.7. A GBA A {A1,AI,VI,~I,O,~ ,e, A2, A2, V2,p2} is p-atomic if and only if( Vi at, Vj bt) E Pv(~i), where {(at, bt)}tET is a family of tET1
tET2
all ( i, j ) -atoms of .4 and T1 U T 2 - T. Proof. First, assume that A is p-atomic and ( Vi at, Vj bt) -g Pv(~i). If V, at - a, t6T1
t6T2
tET1
Vj b t - b, then by ( 6 ) o f Corollary 6.1.32, (Fj(b), F i ( a ) ) C P A ( P i ) and, therefore, t6T2
there exists an ( i , j ) - a t o m (at,,, at,, Vj O) such that at,, < pj(b), at,, Vj 0 <_ r i j since A is (i, j)-atomic. Clearly, to c T - T1 U T2 implies to c T1 or to c T2. Without loss of generality assume that to c T1. Then at(, <_ a. On the other hand, ato Vj 0 <_ q~i(a) <_ ~i(ato) i j j and so (ato, ato Vj @) c PA (P~). This result contradicts the fact that (ato, ato Vj 0 ) is an (i, j)-atom. To prove the converse, let ( Vi at, Vj bt) c Pv(Pi), where T1 U T2 - T. We t6T1
t6T2
shall prove that A is p-atomic. Let us assume the contrary, that is, there exists a pair (c, d ) E PA(P~) such that for each ( i , j ) - a t o m (at, at Vj @), we have: at <_ c is i false or at V j @ <_ d is false. Let J
T1-
{t c T " at _< c is false} and T2 - {t c T " at Vj O <_ d is false}. i j
It is obvious that T1 U T 2 - T. Moreover, by Definition 6.5.1, we have
(at Ai c, at Vj O) 6 PA(~/) for each t C T1 and
(at, (at Vj (9) Aj d) c PA (~i) for each t C T2. Therefore
at Vj 0 ~_ ~i(at) Vj ~i(c) for each t c T1 J and
at <_ ~j(at Vj (9) V~ pj(d) for each t c T2. i
Hence
at Vj 0 _~ qpi(c) for each t c T1 and at <_ pj(d) for each t c T2 j i since
(at Vj O) Aj pi(at) -- O and at Ai ~j(at Vj 0 ) -- 0 for each t c T. Clearly,
V ~ at <_ ~j(d), i
t6T2
~ t6T1
(at Vj @) _< ~i(c) j
282
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
and if V~ at - a, Vj bt - b, t h e n tET2
tET1
a <_ ~ j ( d ) ,
b <_ ~ ( c ) j
and (a, b) 9 Pv(p~)
since T2 t2 T1 - T. Therefore
~ i ( ~ j ( d ) ) <_ ~i(a) <_ b <_ ~i(c), J
J
t h a t is d <_ ~i(c)
J
J
and thus (~j(d), ~i(c)) 9 P v ( ~ i ) . But in t h a t case, by (6) of Corollary 6.1.32, we have (c, d) 9 P~ ( ~ ) , which is impossible. Hence ,4 is p-atomic.
6.6. G e n e r a l i z e d B o o l e a n Factor A l g e b r a s We shall define a factor algebra of a G B A A = {A1, A1, V1, ~1, ~=), ~ , e, A2, A2, V2, p2}. Let I = (I1,/2) be any proper GI of A and F = ( ~ 2 ( 2 2 ) , ~ 1 ( / 1 ) ) b e the corresponding GF. If E~ = A~ U / j t2 ~i(I~), t h e n Ai • / j = { 0 } , Ai • ~i(I~) = {e} since A1 V1A2 = {(~, e}, and Ij A ~i(Ii) = f~J since I = (I1,/2) is a proper a I .
P r o p o s i t i o n 6 . 6 . 1 . Let A = {A1, A1, V1, ~1, (~, 4 , e, A2, A2, V2, ~2} be a G B A and ~ be binary relation on the set Ei, defined in the m a n n e r as follows: i
(1) I f x, y C Ai, then x~y
,z----5, ( x - y ,
i
.,
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ja
y ) v~ e
JG
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; :,
( y :- ~) v~ e 3(;
= y - x e I~). i
j~
= x - y, i
(2) / f x c A~, y c Ij U ~ i ( I i ) , then i ~:
i
iG
:. ( ( ~ - y ) v j e) = ~ - y e I j , ic;
J
JG ( y - ~ ) v~ e Jc;
:
y - x e I~). i
(3) I f x, y 9 Ij U ~i(Ii), then x~y
~.
( x - y, y - x c I~) ~: :.
i
i(;
iG
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J
iC
x 9 Ij). J
Then ~ is an equivalence relation on Ei such that elements f r o m Ij and y)i(Ii) i
belong to different equivalence classes. Moreover, if x, y 9 Ij or x, y 9 ~i(Ii), then x ~ y ,z---5, x ~ y. i j Proof. First, let us prove t h a t ~ is an equivalence relation on Ei. It is obvious i
t h a t x x, x for each x 9 E~ since 11 N / 2 = {(~}. Also, by (2), the implication i
x ~ y ---5, y ~ x is clear for each x, y 9 Ei. Therefore it remains to show x ~ y i
i
i
and y ~ z imply x ~ z. We shall consider the following cases: i
x, y, z 9 Ai;
x 9 A~, y, z 9 Ij U ~i(Ii);
x, y 9 Ai,
z 9 Ij [2 ~i(I~)
6.6. G e n e r a l i z e d B o o l e a n F a c t o r A l g e b r a s
283
and
x, z ~ A~,
y ~ Ij U ~ i ( I i ) .
T h e r e s t r i c t i o n ~ IA, is an e q u i v a l e n c e r e l a t i o n on Ai since Ii is an ideal in t h e usual sense. H e n c e t h e case x, y, z ~ Ai is obvious. Since all t h e o t h e r cases are p r o v e d similarly, we shall consider only t h e case w h e r e x ~ A~ a n d y, z ~ I j U ~ ( I ~ ) . B y (2) a n d (3), we o b t a i n
x,~y i
,,t---5, (x - y ~ Ii, y - x ~ I j ) i(;
j(;
and y~
i
z ,,+---5, ( y - z , i(;
z - y c Ii) i(;
so t h a t
a~ ~j(y) c I~, y Aj ~(~) c Ij, y a~ ~j(z), ~ a~ ~j(y) e I~. T h e r e f o r e G D L 1 , G D L 2 a n d Definition 6.2.1 i m p l y
(x A~ ~j(y)) v~ (y A~ ~j(~))- ((~ A~ ~j(y)) vj y) A~ ((x A~ ~j(y)) v~ ~j(z))(~ v~ y) A~ (~j(y) vj y) A~ (~ v~ ~j(~)) A~ (~j(y) v~ ~j(~)) = (x vj y) A~ (~ v~ ~(~)) A~ ( ~ ( y ) v~ ~j(~)) 9 I~ since I = ( I 1 , / 2 ) is a GI. Clearly, i
Following (2) of Definition 6.2.1, x Ai ~ j ( z ) = x - z e Ii. i(;
In a similar m a n n e r one can show t h a t z - y E Ii a n d y i (;
x E Ij i m p l y
J c;
Z--XCIj.
H e n c e x ~ z and, therefore, ~ is an e q u i v a l e n c e r e l a t i o n on Ei. i
i
F u r t h e r m o r e , let us prove t h a t e l e m e n t s from Ij a n d ~ i ( I i ) b e l o n g to different e q u i v a l e n c e classes. If x c Ij, y c y:i(I~) a n d x ~ y, t h e n t h e r e is an e l e m e n t z E I~ i
such t h a t y = y)i(z) as y c qpi(Ii). Moreover, x ~ y implies x Ai ~ i ( Y ) , Y Ai ~ j ( x ) C Ii, i t h a t is, a n d so z Vj x c ~ ( I ~ ) . O n t h e o t h e r h a n d , z ~ I~, x ~ I o i m p l y z Vj x ~ Ij a n d h e n c e z Vj x ~ Ij N ~ i ( I i ) , which is impossible since I 3 A p i ( I i ) = ~ . Finally, let x, y ~ I o or x, y ~ ~ ( L ) . It is obvious t h a t Ij U ~ ( I ~ ) C E~ C~A j . T h e r e f o r e , on t h e one h a n d , by (3), for a pair x, y ~ Ij U ~ i ( I i ) c Ei, we have x ~ y ~ (x - y, y - x E I~), and, on t h e o t h e r h a n d , by (1), for t h e s a m e pair i
i(,
i(;
x, y ~ Ij U ~ i ( I i ) C Ei, we h a v e x ~ y .z----5. (x - y, y - x ~ I~). T h u s , if x, y ~ Ij j
or x, y C ~ ( I ~ ) , t h e n x ~ y i
i(;
i(;
.z--5. x ~ y. j
Let us d e n o t e t h e set of all ~ - e q u i v a l e n c e classes by E i / I ,
E{/I - {[x] {- x s E{}.
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VI. Generalized Boolean Algebra and Related P r o b l e m s
Thus the algebra . A / I - {El/I, [-]1, l-]1, (1)1, [l~], ~ F, [e], E2/I, R2,112, ~2} is a G B A t h a t we call a G.Boolean factor algebra of the G B A A - {A1,A2, Vl, ~1, (9, 4 , e, A2, A2, V2, ~2}. Moreover, if we consider the pair n - (nl, n2), where the maps ni" Ai ~ E i / I are defined as ni(x) - [x] i, then it is easy to see t h a t n ( r t l , rt2) " fl[ ---+ .A/I. This G . h o m o m o r p h i s m is called a natural G.homomorphism. G
Finally, let us consider any G . h o m o m o r p h i s m h - (hi, h2) 9 .4 ~ A ~. Then, by G
Example 6.4.2, h - 1 ( O ') - (h[l((~'),h~l((~')) is a proper GI of .4. Let A / h - I ( O ') be the G.Boolean factor algebra. If g-(g1, g2), where gi" Ei/h-l((~') ~ A~ are defined as gi([x] i) - hi(x) , then g - (91,92) 9 A/h-l(~)') ~a .4 ~ is a G.isomorphism. Indeed, by (7) of T h e o r e m 6.4.7, it suffices to prove only t h a t gl([X] 1) - (9' and g2([y] 2) - (9' imply [x] 1 - [(9] and [y]2 _ [(9]. But this implication is an immediate consequence of the definition of g - ( g l , 9 2 ) . 6.7. Generalized Fields of Sets and the Generalized Field Representation of a Generalized Boolean Algebra D e f i n i t i o n 6.7.1. Let X be a n o n e m p t y set, Ai(X) be a ring of subsets of the set X , t h a t is, Ai(X) be closed under the finite set-theoretic operations intersection and union, ==<: be a quasi order relation on A I ( X ) u A2(X) such t h a t ==~ IA,(x) = C , A I ( X ) N A 2 ( X ) = {2~, X } and A c Ai(X) implies X \ A c Aj(X). If there exists maps []i: Ai(X) ~ A j ( X ) satisfying the conditions:
(1) A ~ [A]~, [ [A]~]j = A and X \ [A]~ = IX \ A]j for each set A e A~(X). (2) [ ] i o n = No([ ]i, []i) a n d [ ] i o U = Uo([ ]i, []i) s o t h a t [ANB]i = [AIiA[B]i and [A U B]i = [A]i U [B]/ for each pair (A, B) c A i ( X ) x A i ( X ) , then we shall say t h a t Jc(X) = {AI(X)UA2(X),N, u, \, ==<:, []1, []2]} is a G.field of sets (briefly, GFS) for which X is the basic set. Clearly, []1 and []2 are bijections and []i - - [ ] ; 1 It is easy to verify t h a t the equivalence (A =<: [A]i and [ [A]i]j - A) ,z---->,A ==~ [A]i ==<: A holds for each set A E Ai(X). Indeed, A :=<: [A]i and [[A]i]j - A imply [A]i =:<: [[A]i]j - A. On the other hand, if A ==<: [A]i =<: A, then A =:< [A]~ ==< [ [A]/]j =:< [A]~ =:< A and so A ~ [[A]~]j ==<: A. Therefore A c_ [[A]~]j c_ A since ==<: latex) and thus [ [AJ~]j - A. Obviously [2~]i- o and [ X ] i - X. Moreover, for each set A e Ai(X), the set [A]i is a smallest set from A j ( X ) (with respect to the set-theoretic operation inclusion) such t h a t A ==~ [A]~. Indeed, if B c ,Aj(X) is any set such t h a t A ==~ B C [A]i, then [A]i ==<: A ==<: B c [A]i, t h a t is [A]i c_ B c_ [A]~ and thus B -
[A]~.
6.7. G e n e r a l i z e d F i e l d s of Sets a n d . . .
289
For a GFS ~'(X) - { A I ( X ) U . A 2 ( X ) , N , U, \, ==<, []1, []2} let us define binary operations ni, Ui " (A~(X) x A i ( X ) ) u ( A j ( X ) x A i ( x ) ) --+ A i ( X ) as follows: AN~B-ANB,
AU~B-AUB,
if ( A , B ) r ( A i ( X ) • A i ( X ) ) \ {(A, [X \ A]j) " A r A~(X)}, and A Ni [X \ A]j - ~, AN~ B - [A]j N~ B,
A Ui [X \ A]j - X if A c A ~ ( X ) ;
A U~ B - [A]j U~ B if ( A , B ) e A j ( X ) x A~(X)
so that ANiB--
~'
if A - X \ B , if A ~ X \ B ,
( [A]jNB, A U i B - ~X'
if A - X \ B , if A C X \ B .
[ [A]jUB,
Hence for ( A , B ) c A j ( X ) x A~(X), we have A n~ B C_ B,
A o~ B c_ [A]j ~
A and B c_ A U~ B,
A ~
[A]j c_ A U~ B.
From the above arguments, we immediately conclude that [A n~ Bli - [A]i nj [B]i,
[A Ui B]~ - [ A ] i
Uj
[B]i if (A, B) r A i ( X ) x A i ( X )
and [A N, B]~ - [ IA]j N~ B]i - A Nj [B]~, [ A U ~ B ] i - [ [ A ] j U ~ B ] i - A U j I B I i if (A,B) E A j ( X ) x A i ( X ) . Hence it is not difficult to see that every GFS can be treated as a GBA A(~'(X)) with ~ instead of 4 , c instead of <, ~ and X instead of 0 and e, i
respectively, the set-theoretic complementation \ instead of Wi, []~ instead of X , Oi and U~ instead of Ai and Vi, respectively. (The converse of this statement is Theorem 6.7.5.) Indeed, let us first consider the question of coordination of the quasi order - - < and the lattice operations ni, Ui to show that A ==< B .e---->.A Ui B - B for each pair ( A , B ) c ( A i ( X ) x A i ( X ) ) u ( A j ( X ) x A~(X)). If (A, B) c A ~ ( X ) ( x A ~ ( X ) , then A=:< B .<-->. A C_ B ..e->. B - A U B ..~->. B - A U~ B, and if ( A , B ) c A j ( X ) x A i ( X ) , then A ::::< B ~
[A]j c_ B ..v-->. B - [A]j u B ~
B - [A]j u~ B ~
B - A u~ B.
Now the examination of axioms L1, L2, GL3, GL4, GDL1 and GDL2 reduces to simple calculations which readily show that A(.T'(X)) - {AI(X),N1, Ul, \ , ~ , --<:, X, A i ( X ) , N2, U2, \} is a GBA. A GFS 5r(X) - {.41(X) U A2(X), N, U, \, ==<, []1, []2} is said to be reduced if for every pair of distinct points x, y c X there exists a set A c A i ( X ) such that x E A and y - c A .
290
VI. Generalized Boolean Algebra and Related Problems
D e f i n i t i o n 6.7.2. A G.field representation of a GBA .4 - {A1, A1, V l , 991, (~), 4 , e, A2, As, V2,992} is a pair ( f , X ) , where X is the basic set of a GFS Jc(X) considered as a GBA A(Jz(X)) = {A1(X),cll,U1, \,;g, ===~,X,A~(X), Ns, Us, \}, and f 9 A ~ A ( ~ ( X ) ) is a G.isomorphism such that f - (fl, f2), G
where f ~ ' A ~ ~ A~(X) are bijections, and the following conditions hold: (1)
s o Ai -- rli o ( f i , s
and s o Vi -
Ui o ( f i , f i )
so that
f~(~ A~ y) - f~(x) n~ s
~nd f~(~ v~ y) - s
u~ f~(y)
for each pair (x, y) c Ai x Ai. (2)
f i o Ai -- rli o
(fj, fi)
a n d f~ o Vi - Ui o
(fj, fi)
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y) - f j ( x ) n ~
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y)
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n~ f~(y)
and v~
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=
[f;(x)]j u~ f~(y)
for each pair (x, y) E Aj x Ai. (3) \ o fi - fj o 99~ so t h a t X \ fi(x) = fj(99i(x)) for each x 6 A~. Note t h a t here we also identify these (reduced) G.field representations (f, X) and ( I ' , X ' ) of a GBA ,4 = {A1,A1,V1,991, O , ~ ,e, As, As, V2,992} which are equivalent in the sense as follows: there exists a bijection h : X ~ X t and a G.isomorphism h* = (h~,h~) : A ( 5 ( X ) ) ~ A ( ~ ( X ' ) ) such that h~(A) = {h(x) : x c A} for every set A c X i ( X ) . L e m m a 6.7.3. If (f , X ) is a G.field representation of a GBA .4 = {A1, A1, V1, 991,O, ~ ,e, A2, As, V2,992}, then for each x c Aj we have [fj(x)]j = f~(x V~ 0).
Proof. By (6) of Theorem 6.1.16, x c Aj implies x V~ 0 6 x and, therefore, by (1) of Definition 6.7.1 and the G.isotonicity of f = (fl, f2), we have f i ( x Vi (9)==3
[fj(x)]j,
that is fi(x Vi (~) c_
[fj(x)]j.
On the other hand, the same arguments give fj(x)==~ fi(x Vi (9) and, hence, by the definition of the set [fj(x)]j, we have fj(x)==< [fj(x)]j c_ fi(x Vi (9). Thus
[fj(x)]j - f i ( x Vi 0).
[3
We are now ready for the principal result of this section; namely, the next theorem characterizes all the reduced G.field representations of a GBA .4. The existence of a reduced G.field representation will be a consequence of this theorem and of the results of previous sections. T h e o r e m 6.7.4. There exists a one-to-one correspondence between the reduced G.field representations of a GBA A and the Stone families of prime GI's of X.
Pro@ Let S be a Stone family of prime GI's of a GBA A. The reduced G.field representation (f, X) of A can be associated with S in the manner as follows: let
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292
VI. Generalized Boolean Algebra and Related P r o b l e m s
Conversely, if x1 ~ f j ( a ) A~ f~(b) - [fj(a)]j ~ k ( b ) , then
f j ( a ) Ni fi(b) - [fj(a)]j N f~(b) so t h a t x1 ~ [fj(a)]j ~ f~(b) a n d s o x i ~ f ~ ( a V ~ ) ~ f ~ ( b ) . Therefore a V i @ g I ~ z--->, a - g I j and bgI~. Now, Definition 6.2.9 gives aA~b-gIi so t h a t x i e f~(aA~b) and thus f~(aA~b) - f j ( a ) ~ f ~ ( b ) . Furthermore, if x~ ~ f~(a V~ b), where a ~ A j , b ~ Ai, then a V~ b--gI~. By (1) of Definition 6.2.1 the case a ~ Ij, b ~ I~ is impossible. Hence a-g Ij or b g I~ (or both a g Ij and b g/~). We shall consider only the cases a g Ij, b ~ /~ or a g Ij,
bg~. If a-g I j, b ~ I~ and b -
~ j ( a ) , then
f j ( a ) ~ fi(b) - [fj(a)]j ~2~ f~(b) - [fj(a)]j t3~ f i ( ~ j ( a ) ) - [fj(a)]j ~ ( X \ f j ( a ) ) - X so t h a t x i ~ f j ( a ) ~ fi(b). Now we assume t h a t b --/: ~j(a). Then
f j ( a ) tAi fi(b) - [fj(a)]j Ui fi(b) - [fj(a)]j ~ fi(b) and, therefore, g-EIj ~
g V i {~--EI i ,g------5,xI E f i ( g V{ ~ ) -- [fj(g)]j,
t h a t is, x1 ~ f j(a) Ui k ( b ) . I f a - g I j , b-gIi, then, by (2)of Theorem 6.2.10, the case b - ~ j ( a ) is impossible. Hence f j ( a ) U i fi(b) - [fj(a)]j U fi(b) and a - g I j , b-gIi imply a V~ O - g I i , b-gIi so that x1 e k ( a Vi O ) U fi(b) - [fj(a)]j U fi(b) - f j ( a ) Ui fi(b). Conversely, let x , c f j ( a ) Ui fi(b) and b - ~j(a). Then
f j ( a ) t& fi(b) - [fj(a)]j tAi f i ( ~ j ( a ) ) - [fj(a)]j tAi ( X \ f j ( a ) ) - X and, by (2) of Theorem 6.2.10, a-EIj or b - qpj(a)-EIi. It is clear t h a t always a V~ b-~Ii so t h a t x1 c f i ( a Vi b). Now let b 7~ ~j(a). Then f j ( a ) Ui f~(b) [fj(a)]jU fi(b) and so x1 E [fj(a)]j - fi(aVi(~) or x i c fi(b) (or both x i c [fj(a)]j and x i E fi(b)). Clearly, in all cases a Vi b E I i so that x i c f i ( a Vi b) and thus f i ( a Vi b) - f j ( a ) tO~ fi(b). We have shown that the conditions (1)-(3) of Definition 6.7.2 are fulfilled and, therefore, f - (fl, f2) " A - {A1, A1, Vl, q;1, O, 4 , e, A2, A2, V2, P2} -+ f(.A) A ( . T ' ( X ) ) - { A I ( X ) - { f l ( a ) - { x i " a-gI1 ..r a v2 ~ ) g I 2 , I - (11,12) c S } " a c A 1 } , N 1 , U I , \ , , ~ , ===<,X,.A2(X) - {f2(b) - { x I " b-gI2 ~ bV10cIl, I(11,12) E S} 9 b E A2}, C12,tO2, \} is a G.homomorphism of the GBA A onto the GFS ~ ( X ) , considered as the GBA A ( . T ( X ) ) . To prove t h a t f is a G.isomorphism, it remains to show only t h a t fl and f2 are bijections so that by virtue of (7) of Theorem 6.4.7, it suffices to show t h a t if f l ( a ) - f2(b) - ~, then a - b - O. But this implication immediately follows from the definition of f~ and (1) of Theorem 6.2.16. Furthermore, let us prove t h a t ~ ( X ) is reduced. Indeed, if x i , xi, E X and xx ~ x i,, then by the definition of the set X, we have I - (I1, I2) r I ' - (I~, I~). Therefore since I and I ' are prime GI's, there exist elements a, a' E A1, b, b' E Az such t h a t a f t I1,
a-El~,
a'-EI1,
a' E I~, b E I2, b g I ~ ,
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VI. G e n e r a l i z e d Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
and thus x - ~ f j ( b ) U [fj(b)]j, that is, b E I x. Hence we conclude that Ix = (I~, I f ) is a GI. Moreover, the proof of (1) immediately implies that Ix is a proper GI. It remains to prove that Ix = (I~, I~) is a prime GI, that is, the condition of Definition 6.2.9 holds. First, assume that a, b ~ A~ and a A~ b E I x. Then
x-~ f~(a A~ b)U [f~(a A~ b)]~ - (f~(a) n f~(b)) U ([f~(a)]i A [f~(b)]~). If we assume the opposite, that is, a g I x, b g I x, then (I)
x E (fi(a)U [f~(a)]~)N (f~(b)U [f~(b)]~). Since a Ai b E I ,x, (I) immediately implies
x E f~(a)rq (X \ [f~(a)]~) rq (X \ f~(b)) rq [f~(b)]~
(II)
x E (X \ f~(a)) n [f~(a)]~ rq f~(b)0 (X \ [f~(b)]~).
(III)
or
By the definition of maps f~, in the case (II), we have
xE[fi(a)]~ - fj(a Vj e) ,z---->,a Vj e E I f ~
a c Ix ~
x E fi(a),
which is impossible. Similarly,
x--~ f~(b) ~
b E I~ ~
b Vj 0 E I f ~
x-E fj(b Vj 0) - [fi(b)li,
which is also impossible. Using similar arguments, one can prove that (III) is impossible. Thus if a, b E Ai and a A~ b E I~, then
x--~fi(a) U [fi(a)]i or x - c f i ( b ) U [f~(b)]i ( o r both x-E (Si(a)U [S~(a)]i) U (Si(b)U [S~(b)]~) ) and s o a c I x o r b E I x (or b o t h a c I x a n d b E I X ) . Finally, let a E Aj, b E Ai and a Ai b E I x. Then
x-E fi(a Ai b) U [fi(a Ai b)]i - (fj(a) ni fi(b)) U [fj(a) ni fi(b)]i = = ([fj(a)]j ni fi(b)) U ([fj(a)]j Nj [fi(b)]i) -
= ([fj(a)]j N fi(b)) U ( f j ( a ) N [fi(b)]i) and, using the arguments as above, we obtain a E I f or b E I x (or both a E I f and b E I / ) . Thus Ix - (I~, I f ) , where I.~ - {a c A~" x--~f~(a)U [fi(a)]~}, is a prime GI. Now let us prove that S - {Ix - (I~, I f ) " x E X } is a Stone family of prime GI's. Indeed, let a c Ai \ {fg} be any element. Then fi(a) ~ ;g and so, there exists an element x E fi(a). But by the definition of maps f~, we have a g I x and, therefore, (2) of Theorem 6.2.16 implies that S is a Stone family. Thus, we have constructed two maps Ct I and a2, where O~1 associates the reduced G.field representation of A with every Stone family S of prime GI's of .4 and vice versa: a2 associates the Stone family S of prime GI's of A with every reduced G.field representation (f, X) of ,4.
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VI. Generalized Boolean Algebra and Related Problems
(f', X'), then by the definition of (f', X') the points of X ' are in the one-to-one correspondence with the prime GI's of S and, accordingly, with the points of X. Moreover, following the corresponding definition,
f{(a) - { x e X " a-~I f ~
a vj O-~I~, Ix - ( I { , I ~ )
e S}
for each element a 9 Ai and e f'(a)
a-
Ix
a
v; e-cI
x c f
(a)
so that f~(a) = fi(a) for every element a 9 Ai. This completes the proof.
D
By virtue of Corollary 6.2.14, the family S of all prime GI's of a GBA ,4 is a Stone family and on the strength of the preceding theorem, there exists a reduced G.field representation of.4 associated with S. Thus the following principal statement is valid. T h e o r e m 6.7.5. Every GBA is G.isomorphic to a reduced GFS.
Proof. Indeed, the family of all prime GI's of a GBA A is a Stone family. By Theorem 6.7.4, to this Stone family there corresponds the G.field representation ( f , X ) of A and from Definition 6.7.2, it follows that the GFS De(X), considered as the GBA A ( ~ ( X ) ) = {AI(X) = {fl(a) : 9 A 1 } , n l , U l , \ , ~ , ==<,X, A 2 ( X ) = {f2(b) : b 9 A2}, N2, u2, \}, is G.isomorphic to A. [~ C o r o l l a r y 6.7.6. For each real number m > 0 and every non-zero element a 9 Ai there exists a G.quasi measure # = (p1,#2) on a GBA A = {A1, A1, V1, pl, O, ~ ,e, A2, A2, V2, ~2} such that #i(a) = # j ( a V j O) = m .
Pro@ Let m > 0 be any real number and a c A~ \ (2~} be an arbitrary element. Then, by definition, fi(a) -r ;g =/=f j ( a Vj O) and, therefore, there exists an element b 9 f~(a). If we define a pair # = (#1,#2) as
{
?T~, if b e fi(x) if b-~ fi(x) for each x E Ai
-
0,
and
{77~, 'J(Y)
-
0,
if b Vj 0 c f j(y) if b Vj @-E f j (y) for each y c Aj '
then it is not difficult to see that # = (~tl, it2) is the G.quasi measure on ~4 such that #i(a) = # j ( a V j (~) -- m . D The reduced G.field representation of a GBA A, associated with the family of all prime GI's, is called the perfect G.field representation of .4. Theorem 6.7.5 concludes one part of the study of representations of a GBA. In the next section we shall investigate the second part of the modification of Stone's representation theorem.
6.8. Bitopological Representation ...
297
6.8. B i t o p o l o g i c a l R e p r e s e n t a t i o n of a G e n e r a l i z e d B o o l e a n A l g e b r a Let A = {A1,A1,VI,g~I,O, 4 , e , A2, A2, V2, q~2} b e a GBA, S = { I = (11,/2)} be any Stone family of prime GI's of A and (f, X) be the reduced G.field representation of A associated with S in the sense of the first part of Theorem 6.7.4. We can endow the set X with the topologies rl and 72 by taking the rings of sets .41(X) and A 2 ( X ) of the GFS 2r(X) = { A I ( X ) U A 2 ( X ) , N, U, \, ==<, []1, []2} as the corresponding bases of the open sets. Clearly, t h e / - o p e n sets have the base consisting of j-closed sets since A i ( X ) = co A j ( X ) and, therefore, (1) of Corollary 3.1.6 implies that p-ind X = 0. On the other hand, if we assume that x, y E X, x -r y, then/V(X) is reduced implies, in particular, that there exist A, B c A I ( X ) such that x c A, y g A and x g B, y c B. It is clear that
A, B E rl NCOr2, X \ A, X \ B E r2 DICOrl so that
U(x) = A c 7-1, U(y) = X \ A c T2, V(x) = X \ B C ~-2 and V(y) = B E T1 are disjoint neighborhoods of the points x, y E X and thus, by (9) of Definition 0.1.6, (X, T1, T2) is p-Hausdorff.
T h e o r e m 6.8.1. Let S = {I = (I1,/2)} be a Stone family of prime GI's of a GBA A and let (f, X ) be the corresponding reduced G.field representation of A. Then the BS (X, T1,7-2) is FHP-compact if and only if S contains all the prime GI's of .4, that is, if and only if (f, X ) is the perfect G.field representation of A. Proof. First, assume that S = {I = (I1,/2)} is the Stone family of all prime GI's of the GBA A. It suffices to prove that every p-open covering U C { { f l ( a ) 9 a c A1}, {f2(b) " b c A2}} of X such that b/N T1 r ~ r ~ ~'17-2 has a finite subcovering. Assume the opposite, that is, there exists a p-open covering { { f l ( a n ) : rt C N } , {f2(bm) : rrt E M } } of X so that hEN
and
mEM
# ( U Sl/an/)o ( U hEN1
mEM1
for any finite subsets N1 c N and M1 c M. Therefore, alVla2V1...Vlan~e,
bl V2 b2 V2 "" V2 bm C e
and ~1(al V1 ae VI'-" V1 a~) ~ bl Ve be V e " . V2 bm is false for any finite subsets N1 = { 1 , 2 , . . . , n } C N, M1 = { 1 , 2 , . . . , m } C M. Indeed, if a l V1 a2 V1 ... V1 an ~: e, then f l (al V1 a2 V 1 . . . V1 an) 7s f(e) - X
and so
f l ( g l Vl g2 Vl "'" Vl
an)
--
f l ( g l ) Ul f l ( g 2 ) U l " ' " Ul/1(an)
-
-
298
VI. Generalized Boolean Algebra and Related Problems = fl(al) U fl(a2) U... U
since f -
(fl,f2)
al V l a 2 V 1 . . . V l a n
fl(an) 5r X,
is a G.isomorphism. The same is true for bl,b2,...,bm. If - - a , b l V2b2V2---V2bm - b and ~bl(a) 4 b, t h a t is, g91(a) < b, 2
then f - ( f l , f2) is a G.isomorphism implies t h a t q~l(a) _< b ,e---->,f2(~l(a)) C_ f2(b) ,e----->,X \ / l ( a ) c_ f2(b)<--->, fl(a) U f 2 ( b ) - X since 4
2 IA,--~
and =:< A , ( x ) - C _ .
Thus the simultaneous fulfilment of all three cases al Vl a2 V1 "'" V1 an r e, bl V2 b2 V2 "-. V2 bm r e and ~31(al V1 a2 V l " ' " Vl
an) ~ bl V2 b2 V 2 " "
V2 bm
is false enables us to conclude t h a t
X ~ (
U fl(an))U
nCN1
(
U
mCM1
f2(bm))
for any finite subsets N1 C N and M1 C M. Let I - (I1,/2) be the GI generated by the pair of sets (B1 - {an " n E N}, B2 - {bin 9 m e M}). Then I - (I1,12) is proper since the contrary means t h a t e c I1 N I2 and, by Theorem 6.2.5, there exists, in particular, a finite sequence of elements a l , a 2 , . . . ,ak E B1 such t h a t al V l a 2 V 1 - - . V l a k - e, which is impossible. Let us consider a family of all proper GI's ordered by the partial order < defined in the part preceding Proposition 6.2.4. Then, by virtue of the latter proposition, every chain in this family has the upper bound, also defined by this proposition. Therefore Zorn's lemma implies t h a t this family has a maximal element, namely, a prime GI I. - (I~,I~) such t h a t I < I.. Clearly, an c Ii ~ ~ an V2 (9 c I~ for each n E N and bm C I~ ~ bm V1 I~ E Ii ~ for each m c M. Also, I* c S. But ( f , X ) is the reduced G.field representation of the G B A A which corresponds to S and, by (.) in the proof of Theorem 6.7.4, xi. g f l ( a n ) and xI. gf2(bm) for each n c N and m c M. Thus xL g ( U fl(an)) U ( U f2(b~)) - x . This contradiction proves the first part
nCN
mCM
of the theorem. Conversely, let (X, rl, r2) be F H P - c o m p a c t and there exists a prime GI I (I1,/2) such t h a t I g S so t h a t I r Ix for each point x c X. Therefore for each point x E X, the following conditions hold: (1) There exists an element a E A1 such t h a t a c
I1 ,z----F,a V2 0 E / 2
and a-gI~ ,e----->,a V2 ( g g I ~ ,e---->,x c f l ( a )
and there exists an element b c A2 such t h a t b g / 2 <---> b V 1 O g I 1
and b E I~ ,e---->, b V 1 1~ E I~ ,e----->,x-gf2(b).
(2) There exists an element c E A1 such t h a t
c-CI1 ~
c V20g/2
and c c I~ <---> c V2 0 C I~ ,e---->x g fl(c)
6.8. Bitopological Representation ...
299
and there exists an element d c A2 such t h a t d c / 2 ,z---5, dV1 (3 c I1 and d - ~ I ~ ~ Since for each point x e also prime, in both cases (1) d as ~1 (c), that is, b = ~ l ( a ) , condition (1) above. Indeed, for each element that
X the GI Ix = ( I ~ , I ~ ) is prime and I = (I1,I2) is and (2) one can consider the element b as ~ l ( a ) and d = q~l(C). As will be seen below, it suffices to apply x c X, there exists, by (1), an element a x c A1 such
ax V2 O c I2 and ax ~ I~ ~
a~ E I1 ~
dV1 ( ~ g I ~ ,,+----5x c f2(d).
{fl (ax): x c X} is
1-op
ax V2 O ~ I~ <---->.x E fl (ax).
cov i g of X.
b c I2 \ { e }
element. Then f2(b) ~ ;g and it is obvious that {{fl(ax) : x E X}, {f2(b)}} is p-open covering of X so that, there exists its finite subcovering since (X, T1, ~-2) is FHP-compact. If this finite subcovering does not contain the set f2(b), t h a t is, if it has the form { f l ( a x k ) : xk c X, k - 1,n}, then f l ( a x l ) U f l ( a x 2 ) U . . . U f l ( a x , , ) X and so axl V1 ax~ V1 "'" V1 a~,, = a = e since, like in the proof of the first part of this theorem, for a :/: c, we obtain f l (a) ~ X~ that is, f l ( a X l V1 ax 2 V I . . . V1 ax,,) = f l ( a X l ) [-Jl f l ( a x 2 ) U l . . . [-Jl f l ( a z , , ) ~ X ,' :,
;.
> fl(aXl) 9 fl(ax2) U ". U fl(ax,,) r X.
But ax~ c I1 for each k = 1, n and, therefore, by (1) of Definition 6.2.1, a = e c I1, which is impossible since I = ( / 1 , h ) is a prime GI. Thus f l ( a ) r X and hence finite subcovering has the form {{fl(ax~) : xk c X, k = 1,n}, {f2(b)}}. Thus f l ( a x l ) U f l ( a x 2 ) U . . . U fl(ax,~) U f2(b) - X ,', > z, ,, f l ( a ) U f2(b) = X z----->, ~ l ( a ) < b, 2
which is also impossible since a c I1, b c / 2 and I = (I1,/2) is a prime, t h a t is, a proper GI. Therefore I c S so t h a t S contains all prime GI's of the GBA A. D By virtue of the reasoning t h a t precedes Theorem 6.8.1, we see t h a t with every Stone family S of prime GI's of a GBA A one can associate a p-zero-dimensional and p-Hausdorff BS which, by Theorem 6.8.1, is F H P - c o m p a c t if and only if S contains all prime GI's of ,4. The Stone BS of the GBA A is a p-zero-dimensional, p-Hausdorff, and F H P - c o m p a c t BS, which is associated with the Stone family of all prime GI's of S. This BS will be denoted by ( S ( A ) , T1, T2). Theorem
6.8.2. Let A - {A~,A1,VI,~91,1~, ~ ,e,A~,A2, V2,~2} be a GBA, its associated Stone BS, A ( X ) = {A ~ 2 x : A = A1 A A2 or A - A1 U A2, A~ E A ~ ( X ) - 7~ A coTj} and the following conditions hold:
(S(A), T1,7-2) be
(1) the combination of l-closure and 2-interior operators and the l-closure operator as well as the combination of 2-closure and l - i n t e r i o r operators and the 2-closure operator are conjugate over A ( X ) so that ~-2 int T1 C1 A -- T1 C17-2 int T1 cl A and T1 int ~-2 cl A = ~-2 cl T1 int 7-2 cl A f o r each set A c A ( X ) .
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Furthermore, prove t h a t A' is G.isomorphic to .4. Since (S(A),7-1,7-2) is the Stone BS of the GBA A, the pair ( f , S ( A ) ) is the perfect G.field representation of A and, by virtue of Theorem 6.7.5, the GFS .F(S(A)), considered as the GBA A(Jz(S(A))) = {fl(a) : a 6 A~},N1, Ul, \,2~, ==~ ,S(A), {f2(b) : b c A~}, N2, U2, \}, is G.isomorphic to A. Thus it remains only to prove t h a t A(.T(S(A))) is G.isomorphic to A'. First, let us show t h a t {fi(a) : a E A~} = ~-i N coTj. By the definition of ~-i, it is obvious that fi(a) c ~-i n co Tj for each element a c A~. On the other hand, let U c ~-i N co~-j be any set. Since the family {fi(a) : a E A~} is the base for 7-i, we have U = U Ut, where tET
Ut c {fi(a) :
a c A~} for each t c T. Therefore by Lemma 3 from [113], U is /-compact since U c coTj and (S(A),T1,T2) is FHP-compact. Hence the /-open covering {Ut : t c T} has a finite subcovering {Ut~: k = 1, n}, t h a t is, n
U-
U Ut~. But Ut~ c { f i ( a ) " a c A~} for each k c 1, n and since the family
k=l
{fi(a) : a c U c {fi(a) : a the pair h = maps, then it
A~} is closed under the set-theoretic operation union, we obtain c A ~ and hence {fi(a) : a c A~} = ~-i N coTj. Now, if we consider (hi,h2), where hi : {f~(a): a c A~} -+ ~-i N coTj are the identity is obvious t h a t since the GFS's
{{fl(a)'a6
A~Iu{f2(b)'b6A~I,N,U,\,
=~,[]1,[]2}
and
{(TI n COT2)U (T2 n coT1),N,U,\ , ==x(', []i, []~ } coincide, the corresponding GBA's will coincide too. Therefore the GBA's A(~c(S(A))) and A' are the same so t h a t A' is G.isomorphic to A. []
T h e o r e m 6.8.3 (A Generalized Version of Stone's Representation Theorem). Under the hypotheses of Theorem 6.8.2, there exists a one-to-one correspondence (up to a G.isomorphism and a d-homeomorphism) between G B A ' s and p-zero-dimensional, p-Hausdorff and FHR-compact (also called Boolean) BS's such that for every GBA A the GBA
.A(.)~-(S(A)))- {T 1NCOT2, N I , U I , \ , ~ , ==~,S(A),T2 NCOTI,N2, U2,\}, corresponding to the GFS 5(S(A))
\, =<, I11, I
of the associated Boolean BS (S(M), T1, T2), is G.isomorphic to A. Proof. First, assume that A and A' are G.isomorphic. We shall show that the corresponding perfect G.field representations (f, S(A)) and (f', S(A')) are equivalent so that, there exist a bijection h : S(A) + S(A') and a G.isomorphism h* - ( h * I , h ~ ) ' A ( Y ( S ( A ) ) ) ~ A'(gc(S(A'))) such that h~(A) = {h(z) : x c A} for every set By Theorem 6.4.9 the image of a GI under a G.isomorphism preserves prime GI's as well. Ix = ( I ~ , I f ) be the corresponding prime GI.
A c {fi(a) : a c Ai} = ri n c o r j . a G.isomorphism is a GI. Clearly, Let x c S(A) be any point and I f g = (gl,g2) : A + A' is the
G
304
VI. Generalized Boolean Algebra and Related Problems
above-mentioned G.isomorphism, then g ( I x ) = ( g l ( I ~ ) , g 2 ( I ~ ) ) i s the prime GI of A ~. Let the corresponding fixed point from S ( A ~) be the point Yg(lx) and define h: S ( A ) ~ S ( A ' ) in the manner as follows: h(x) = Yg(lx). Clearly, h is a bijection. Let us prove that if B 9 {fi(a) : a 9 Ai}, then
h~(B) - {h(x) " x 9 B } 9 {f[(a') " a' 9 A'i}. By virtue of the corresponding definition, we have S ( A ) = {x~: I = (I1,/2) 9 S}. Since B C S ( A ) and B 9 {f/(a) : a 9 Ai}, there exists an element a 9 Ai such that
B = fi(a) = { x i : Let B' - { h ( x f ) Therefore
9
a9
xI 9 B}. If gi(a)
~aVj -
a' 9 Ai,' then a'-# 9~(Ii) ,,f----~,a ' Vj' e '
hi(B)* - {Y9(I) 9 a'-Egi(Ii) ~
g(I) - ( g l ( i l ) , g 2 ( I 2 ) )
(~-CIj, I = (11,/2) 9 S}. -# 9y ( I j ) .
a' Vj' O ' ~ g j ( / j ) } ,
E S ' - - { h ( x I ) " xI E B }
and so h~(B) e {f/(a') 9 a' c A~}. The pair h* - (h~,h~) is a G.isomorphism since g and f are G.isomorphisms. Indeed, let B' e {f;(a') 9 a' e A~} be any set. Then there exists an element a' c A~ such that B' - f[(a'). Hence, there is an element a E Ai such that gi(a) = a ~ since gi are bijections. Let us consider the set f~(a) = B e {f/(a) : a e Ai}. It is obvious t h a t h'~(B) = B' so that h~ are onto. Now we assume that B1, B2 e {fi(a) : a e Ai} and B1 5r B2. Since f~ are bijections, there exist elements hi, a2 c Ai such that a l r a2 and fi(al) = B1, fi(a2) - B2. Let a~ - gi(ai). Then B~ - f~(al) , B; - f~(a~) imply h*(B1) - B~, h~(B2) - B;, and we have h~(B1) r h~(B2) because gi and f~ are bijections. Thus h~ are also bijections. It is trivial to verify the fulfilment of the following conditions: (1) h~(C1 ni (72) -- h~(C1)n~ h~(C2) and h~(C1 Ui C2) - h~(C1)U~ h~(C2) for each pair (C1, C2) 9 (Ti n co Tj) X (Ti n co Tj). (2) h~(C1 ni C2) - h~(C1) N{ h~(C2) and h~(C1 Ui C2) - h~(C1)U{ h~(C2) for each pair (C1, C2) 9 (rj N cori) x (ri N corj). (3) S ( A ' ) \ h~(C) - h~(S(A) \ C) for each set C 9 r~ N co rj. Therefore h* = (h~,h~) is a G.isomorphism and thus ( f , S ( A ) ) and (f', S ( A ' ) ) are equivalent. Together with this equivalence, the definitions of the topologies r~ and r~ immediately imply that the BS's ( S ( A ) , T I , r 2 ) and (S(A'), r~, r~) are d-homeomorphic. This result means t h a t the G.isomorphy of GBA's implies the d-homeomorphy of the corresponding Boolean BS's. Next assume that (X, 71,72) is any Boolean BS satisfying the conditions of Theorem 6.8.2. W i t h (X, 71,72) we associate the GFS
oT'(X)- {(')'1 n co~'2)U (~2 n co')'1), n,u, \, ~, =:~, []1, []2} and its corresponding GBA
r
{~'1 nco~'2, n l , U l , \ , ==xQ,X,~2 n co~'l, n2, u2,\,}.
The GFS 2 ( X ) is the reduced one since (X, 71,72) is p-zero-dimensional and p-Hausdorff. We want to show that the Stone BS (S(A), r~, r2), associated with
6.8. Bitopological Representation ...
305
the Stone family of all prime GI's of the GBA A(a~(X)), is d-homeomorphic to (X, ~1, ")/2).
It is clear that the GFS Jr(X) -- {(~1 (-I CO ")/2) U (~2 A CO ~I) , ['-I, U, \, ==x(, []1, []2}, considered as the GBA
A(.~'(X))
--
{"71
I"'1
CO ")'2, ["ll, U1, \, 23, ==xQ, X , "/'2 f"l co ~1, A2, U2, \ },
is the reduced G.field representation of the same GBA A(.F(X)) in itself. Hence, by virtue of Theorem 6.7.4, with )c(X), considered as a G.field representation of A ( 5 ( X ) ) , one can associate the Stone family of prime GI's of A(Jz(X)). Therefore, applying the reasoning that precedes Theorem 6.8.1 with this G.field representation, we can associate a p-zero-dimensional and p-Hausdorff BS and, as is easy to see, this BS is d-homeomorphic to (X,'71,'72). But the BS (X,'71, ~72) is FHP-compact and, by Theorem 6.8.1, the corresponding Stone family S contains all prime GI's of A. Hence we conclude that (X, ~/1, ~/2) is d-homeomorphic to the Stone BS (S(A), T1, T2). The rest of the proof follows directly from Theorem 6.8.2. r-] D e f i n i t i o n 6.8.4. A G.lattice Z; = {L1,A1, Vl,@, m--X,e, L2, A2, V2} is said to be G.complete if its every subset has an ic-infimum and an i,~-supremum. Now we say that a GBA A = { A 1 , A 1 , V I , q P l , O , 4 , e , A2, A2, V2, qP2} is G.complete if the corresponding G.lattice {A1, A1, V1, 0, 4 , e, A2, A2, V2} is G.complete. L e m m a 6.8.5. Let f = (fl, f2)
: A
--+
G
A t be a G.isomorphism of GBA's A
and A', where A is G.complete. Then fi ( i , - i n f B) - i , - i n f (fl ( U l ) U f2 (U2) ),
for every subset B c A1 U A2, where B rq A1 = B1 7k 23 7L B2 = B r~ A2, and, therefore, A' is also G.complete. Pro@ It suffices only to prove the equality fi(ic-inf B) = ic-inf(/1 (B1) U f2(B2)) since the proofs of both cases are similar. Since A = {A1,/~l,Vl,~91,l~,4 ,e, A2, A2, V2, qp2} is G.complete and f = (f~, f2) is a G.isomorphism, it is clear that A~ = {A~, A,, V,, ~&, (3, <_, e} are complete BA's and the fact that
f~. A~ - {A~, A,, v,, e,, e, _<, ~} -~ A'~ - { < , A,,l v,,l e,,l e , l ~I , (31} i
are isomorphisms implies that A{ -
i i I I I ~ i {A{, A~, V~, ~,, (3, , e'} are also complete i
BA's. Therefore L(infB~) = infL(B~), and it remains to show that (1)-(3) of Definition 6.1.6 hold for fi(i~-inf Bi) when they hold for ia-inf Bi.
306
VI. Generalized Boolean Algebra and Related Problems
(1) Since A is G.complete, for a subset B C A1U A2, there exists an ic;-inf B. Let b - f~(i~;-infB). Then by (1) of Definition 6.1.6, we have i~-infB <_ inf B~ and ic-inf B 4 inf Bj. Therefore b <_' fi(inf Bi) - inf fi(Bi) and b4'fj(inf By) - inf fj(Bj) i so that (1) holds. (2) Let x E A{, z _<' inffi(Bi) and x4'a for each a E fj(Bj). Since f i (fl, f2) is a G.isomorphism, that is, fl and f2 are isomorphisms, we conclude that f - 1 _ ( f l 1, f~-l) is also a G.isomorphism and, therefore,
f/-l(x) ~ f[-l(inf fi(Bi)) -- infBi, f/-l(x) ~ fj-l(a) i
for each element fj-l(a) c Bj. By (2) of Definition 6.1.6, f / - l ( x ) ~_ ic-infB and i thus x _<' fi(i~-inf B). i (3) Let x c Aj,r Xr 4 a for each a c fi (Bi) and x _<' inf fj (Bj). Then we easily J obtain fj-l(x) _< f 7 1 ( i n f f j ( B j ) ) - i n f B j , fj-l(x) 4 f ( l ( a ) J for each f ( l ( a ) c Bi. Hence, by (3) of Definition 6.1.6, fj-l(x) 4 iG-inf B and thus x4' fi(ia-inf B). Now, Definition 6.1.6 implies that fi(ic-inf B) -iG-inf(fl(B1 ) U f2(B2)). It is clear that if B c_ Ai, then, we have fi(i~;-infB) - i~-inff(B) and fi(ic-su p B) - / c - s u p f(B). [:] Our next theorem converts the property of a GBA .4 being G.complete into the property of the Stone BS (S(N), rl, r2) being p-exremally disconnected. Theorem 6.8.6. Under the hypotheses of Theorem 6.8.2, { A 1 , A1, V1, ~1, 1~, 4 , e, A2, A2, V2, ~2} is G.complete if and only
a GBA
A
-
if (S(A), rl, r2)
is p-extremally disconnected. Proof. First, we assume that A is G.complete. Then, by Theorems 6.8.2 and 6.8.3, the GBA r
{T1 n COT2, N I , U I , \ , { ~ , ==~,S(ce),T2 n COTI,N2, U 2 , \ } ,
which corresponds to the GFS .~'(S(A)) -
{(TI NCOT2) U (T2 NCOTI),N,U,\ , = = x ( , [ ] l , [ ] 2 } ,
is G.isomorphic to A. Hence, by virtue of Lemma 6.8.5, for every family of /-open and j-closed subsets of (S(A),rl, r2), there exists a smallest /-open and j-closed set containing all of them. Let U c r~ \ {~} be an arbitrary set. Since p-ind S(A) - 0, for each point x E U, there exists an/-open and j-closed neighborhood U(x) such that U(x) c U so that U - U U(x). Therefore for the
xEU
families of/-open and j-closed sets {U(x) 9 x E U}, there exists a smallest/-open and j-closed set U* such that U c_ U*. Thus rj cl U - r~ int rj el U - U* and, consequently, (S(A), rl, r2) is p-extremally disconnected.
6.8. Bitopological R e p r e s e n t a t i o n . . .
307
Conversely, let (S(A),T1,T2) be a p - e x t r e m a l l y disconnected BS and B c A 1 0 A2 be any subset. Let us prove t h a t there exist i ( j - i n f B and i c - s u p B . W i t h o u t loss of generality, we consider the case of i~-sup B. We have
B A A1 = B1 ~ ;g ~ B fB A2 = B2. Clearly, f l ( B 1 ) U f2(B2) C S ( A ) since f = ( f l , f 2 )
: A --~ A ( ) c ( S ( A ) ) ) is
a
G
G . i s o m o r p h i s m and u
-
9
a
u
9
where { f i ( a ) 9 a ~ Bi} c ~-i ~ c o T j
and { f j ( b ) "
b ~ By} c rj ~ c o T i .
It is obvious t h a t
U fi(a)-UEz-i, U fj(b)-VETj aEBi
bEBj
and
Tj cl U = T~ int Tj cl U c T~ N COTj , T i c l V = v j i n t T ~ c l V c T j A c o T i since ( S ( A ) , T1, T2) is p - e x t r e m a l l y disconnected. Therefore, there exist elements a c Ai and b c Aj such t h a t f~(a) = Tj cl U, fj(b) = ~-~cl V and 7-i cl V Ui Tj cl U = [Ti cl V]~ Ui Tj cl U = 7-i int Tj cl Ti cl V Ui Tj cl U = 7i int Tj cl (~-~int Tj cl 7-i cl V U Tj int 7-i cl Tj cl U) = ~-i int (Tj cl 7-i int Tj cl 7-i cl V U Tj cl Tj int ~-i cl Tj cl U) c ~-i A co Tj since Tj cl ~-i int Tj cl Ti cl V U Tj cl Tj int 7-i cl Tj cl U C co Tj and ( S ( A ) , T1, T2) is p-extremally disconnected. Therefore T~ cl V U~ rj cl U c ~'i ~ co Tj is the smallest set containing fi (Bi) U f j (Bj) and so Ti cl V U Tj cl U = / ( ; - s u p ( f i ( B i ) U f j ( B j ) ) . By L e m m a 6.8.5, there exists i(.-sup B - f / - l ( i a - s u p ( f i ( B i ) U f j ( B j ) ) ) since f - 1 = (f~-l, f~-l) is also a G . i s o m o r p h i s m and hence ,4 is G.complete.
D
T h e o r e m 6 . 8 . 7 . For any G.component E of a G.complete G B A A = {A1, A 1 , ,e, A2, A2, V2, p2}, we have ai(L) = E = hi(R), where a = 1 , - s u p E and b = 2,:-sup E, that is, a = b V1 (~ and b = a V2 0 .
VI,~I,(:~),~
Pro@ Let E = E1 U E2 C A1 U A2 be a G . c o m p o n e n t , t h a t is, E = C d , (Cd(;E). Clearly, there exist a = 1 , - s u p E and b = 2(~-sup E. Then, on the one hand, by (5) of T h e o r e m 6.2.21, we have A z = C d , ( C d ( ; E ) and on the other hand, by (1) and (2) of T h e o r e m 6.2.22, we have aI(L) = A z = Cd~ (Cd(:E) = E = hi(R). T h e rest is obvious.
If]
308
VI. Generalized Boolean Algebra and Related Problems
R e m a r k 6.8.8. Let .4 = {A1, A1, Vl, (~91, t~, ~--~,e, A2, A2, V2, ~2} be a G.complete GBA, B = B~ U B2 C A1 U A2 and ~ ( g ) -- ~ I ( B 1 ) U ~ 2 ( B 2 ) ,
~)(B) = ~ ) l ( g l ) U @2(B2),
where i~,-infB-(
Aj x) Ai( Ai y) and / a - s u p B - ( x6 B 3
y 6 B~
Vj x) Vi( Ai y). xE B j
y ~ B.~
Then it is not difficult to see that
~( a~ x ) - v3 ~(x), ~( v~ x ) - a~ ~(x), x 6 B,,
x ~ B.i
~( /~ x ) x 6 B~
x ~ B,
v~ V~(x), r x 6 B~
x 6 B~
v~ x ) x 6 B,i
/~ V~(x) x 6 B.,
and, therefore, ic-infB-(
Aj x) Ai ( Ai y ) - x6B 3
= ~j((
y6Bi
Vi pj(x)) Vj ( Vj Pi(Y))) - ~ j ( j a - s u p ~ ( B ) ) x 6 B:i
y6 B i
so t h a t / a - s u p B = pj(jG-inf ~(B)). Similarly,
i~;-inf~ - (
A~ x)A~ ( A~ ~ ) xEBj
=r
Vj ~ j ( x ) ) V i ( Vi r xCBj
yCBi
~i(ie~-sup~(B))
y6Bi
so t h a t / a - s u p B = ~i(iG-inf ~(B)). The proof of Corollary 6.8.12 below will be facilitated by Remark 6.8.8. D e f i n i t i o n 6.8.9. A GBA A = {A1, A1, V1, Pl, (9, 4 , e, A2, A2, V2, ~2} is said to satisfy the countable G.chain condition if every G.disjoint system of non-zero elements of A1 U A2 is countable. T h e o r e m 6.8.10. A GBA .4 = {A1,A1,V1,q~)1,0,4,e, A2, A2, V2,~2} satisfies the countable G. chain condition if and only if every set E = E1 U E2 c A1 U A2 has a countable subset D such that u a (D) = % (E).
Proof. First, let B = B1 U B2 C A1 U A2 be a G.disjoint set (system) of non-zero elements of A1 U A2, D c B, IDI <_ b~0 and u c (D) = u G(B). If, for example, there is an element x c (B \ D) N A~, then d A~ x = (9 ~ d 4 ~ ( x ) for each d E D and hence pi(x) c % ( D ) = u~ (B), which is impossible since x 4 pi(x) is false. Thus B = D. To prove the converse, let the countable G.chain condition be satisfied and E = E1U E2 c A1U A2 be an arbitrary subset. If I = (I1, I2) is the GI generated by E, then u o ( E ) = u , ( I 1 U h ) . Indeed, always u~,(I1 U h ) c % ( E ) and if, for example, x c u ~ ( E ) N Ai, then y 4 x for each y c E. Let z c Ii be an arbitrary element. Then, according to Theorem 6.2.5, there are sequences al, a 2 , . . . , an C Ei and bl, b2,..., b,~ c Ej
6.8. B i t o p o l o g i c a l R e p r e s e n t a t i o n . . .
309
such t h a t z < al Vi a2 V~... V~ a~ and z ~ bl Vj b2
Vj ...
b,~.
Vj
Clearly, ak <_ x for each k - 1, n and bz ~ x for each 1 - 1, m so t h a t alvla2v1.-.vla~<_x,
bl Vj b2 Vj . . . Vj b,~ 4 x i
and, therefore, z <_ x. But z c Ii is an a r b i t r a r y element and hence x c uc;(IltJI2), i
t h a t is, uc; (E) - uc; ((11 CJ/2). Furthermore, suppose t h a t 13 - { B ~ - B ~ U B ~ } ~ c T is a family of all G.disjoint sets consisting of non-zero elements such t h a t Ba C 11 [J/2 for each a E T. It is evident t h a t the family B is partially ordered by inclusion, t h a t is, Ba < BZ ~ B~ c B~ and if/C c 13 is linearly ordered, then ]C is b o u n d e d from above by the set B [.J Ba. It is not difficult to see t h a t B is a G.disjoint set B~6/E
of non-zero elements. Therefore, by Zorn's lemma, 13 contains a m a x i m a l element, say B0 - B ~ U B ~ t h a t is, a maximal G.disjoint set of non-zero elements. Since B1~ ~ C I1 U / 2 , we have u c ; ( I i U I 2 ) c uc;(Bo). On the other hand, let, for example, z C u ~ ; ( B o ) A A j , t h a t is, x ~ z ~ xAi~j(z) -- 0 for each x c B ~ ~ If z-Eu(;(I1 U I2), then, for example, there is an element y c I~ such t h a t y ~ z is false ~ ao - y A ~ j ( z ) ~ @. T h e n x A i a o -- @ since ao < ~ j ( z ) , where i
x C B ~ U B ~ is an a r b i t r a r y element.
Moreover, ao < y implies ao c I~ since i
I - (I1,I2) is a GI. Hence, we have found the element ao c Ii \ {@} such t h a t x Ai ao -- @ for each x c B1~ U B ~ This contradicts the m a x i m a l i t y of Bo and thus uc;(I1 U 12) - uc;(Bo), where Bo < Ro since the countable G.chain condition is satisfied. Clearly, E U Bo c I1 U/2. But we would like to find a set D c E such t h a t ]D[ < Ro and u~ (D) - u , (E). If x c Bo N A~ is an a r b i t r a r y element, t h a t is, if x ~ I~, then there are elements al,a2,...,a~
~ B~
bl,b2,...,bm
~ B~
such t h a t x < al Vi a2 V i - . . V~ a~ and x ~ bl Vj b2
Vj
bin.
".. Vj
i
Suppose t h a t D x --
{{al,a2,...
, an; bl, b2, . . .
,bin}
o
x <_ al Vi a2 Vi
o
o
i
Vi an
i
and x ~ bl Vj b2 Vj . . . Vj bm }. Then [Bo[ _< Ro implies [D] _< Ro, where D -
U
Dx and u c ( D ) - uc~(B0 ) - uG(/1
U
12) - u(;(E).
[-1
x E B oi U B 2o
D e f i n i t i o n 6 . 8 . 1 1 . A G B A A = {A1,A1,VI,(~I,(~,4 ,e, A2, A2, V2,(P2} is said to be a G.Boolean a-algebra if its every countable subset has an ic-infimum (and, therefore, of course, and i , - s u p r e m u m ) .
310
VI. Generalized Boolean Algebra and Related Problems
It is clear that the notion of a G.Boolean a-algebra is an intermediate concept between that of a GBA and a G.complete GBA. C o r o l l a r y 6.8.12. A G.Boolean a-algebra .4 = {A1, A1, V1, ~ 1 , (~, 4 , e, A2, A2, V2, ~2} which satisfies the countable G.chain condition, is G.complete.
Pro@ Indeed, let B = B1 U B2 c A1 U A2 be any subset. Then according to Theorem 6.8.10, there exists a subset E = E1 U E2 c B such t h a t [E I < R0 and u(:(E) = u , ( B ) . Thus i(~-supE = / , - s u p B and by Remark 6.8.8, i(~-infE =
i~-inf B.
[-1 6.9. G e n e r a l i z e d B o o l e a n R i n g s
The notion of a G.Boolean ring is a special G.ring version of the notion of a GBA. D e f i n i t i o n 6.9.1. A G.ring is a non-empty set R - R1UR2 together with four binary operations | @i : (Rk x Ri) -~ Ri called G.multiplication and G.addition, respectively, which satisfy the conditions below: (1) There exists a unique element O E R1 N R2, called the zero element, such that x @~ (9 = (9 @i x = x for each x c Ri. (2) To each x E R~ there corresponds a unique element c~(x) c Rj such that x |
c~(x) = e .
(3) G.addition is G.commutative: xO~y=yOix
if (x,y) E R i x R i
(x@iy)-(y@jx)@iO
and
if (x,y) E Rj x Ri.
(4) G.addition is G.associative:
(x |
y) |
z = x e~ (y |
~)
if x, y, z E Ri or x E Rj, y, z E Ri and
(x e~ y)|
~ = x e~ (y e~ ~)
if x , y c Rj~ Z C R i o r x, z C R i , y E Rj. (5) G.multiplication is G.associative:
(x |
y) |
~- x|
(y |
z)
if x, y, z E Ri or x E Rj, y, z E Ri and
(~ |
y)|
z = x |
(y |
z)
if x, y E Rj, z E Ri or x, z E Ri, y E Rj. (6) G.multiplication is G.distributive with respect to G.addition:
x |
(y |
z) = (~ |
y) |
(x |
z)
y) |
(~ |
~)
(y |
z)
if x, y, z E R~ or x E Rj, y, z E Ri,
x | if
x, y
c Rj,
(y e~ z) = (x |
Z C R i o r x, z C R i , y
(x |
y) |
~ = (x |
c Rj, ~) |
~~ .
~.
~"~,.
t~
~.
II
II
II
~c~
~.
II
9
(i)
~..
~.
~
-~. '~
k"%
@
~ .
~
"
~.
9 ~.
9
~=.
~
,-.
~'~.
~
~
~
II
.< ~ ~~
9
|
~ .~: ~
I::~
~.~.
~.
~
~.
| 1 7 4
~
,q.,~.0
r,
~
~
O ~.
~
O
~
~
~
~ p ~
ob
9
~'=
~
'~'
~
9
~
"
II
r~
9
9
o
~
~0
~" o
9 ~
. . . .
~ .
~
~
~
("b
|
~.
~.
~.
~
---q-- |
ii
|
~.
~11
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6.9. G e n e r a l i z e d B o o l e a n Rings
317
(2) hi o Oi - ED~ o (hk, hi) so that hi(x Qi Y) - hk(x) O'i hi(y) for each pair (x, y) e Rk x R~. (3) c', o hi - hj o ci so that c{(hi(x)) - hj(ci(x)) for each x 9 Ri. A G.homomorphism h = (hi,h2) of GBR's is a G.isomorphism if hi and h2 are bijections. It is not difficult to ascertain that the following statement is true. T h e o r e m 6.9.7. Let A be a GBA, 7g(A) be the GBR, constructed from A by the rules stated in Theorem 6.9.4, and let A(Tg(A)) be the GBA constructed from 7g(A) by the rules stated in the same theorem. Then A is G.isomorphic to A(Tg(A)) in the sense of Definition 6.4.6. Similarly, ifTg is a GBR, A(Tg) is the GBA, constructed f f o m 7g, and 7~(A(7~)) is the GBR, constructed from A(Tg), then 7g is G.isomorphic to 7g(A(Tg)) in the sense of Definition 6.9.6.
C H A P T E R VII
Applications of Bitopologies The investigation of problems connected with the application of the theory of BS's has begun only recently and each fact, elementary as it might seem, deserves consideration. In analysis, potential theory, and general topology, there are many approaches to sets equipped with two topologies of which one may occasionally be finer than the other. So it seems quite natural that the purpose of the discussion in this chapter is to reveal the role that bitopological concepts play in the investigation of such situations. In 1977 J. Lukeg formulated certain new methods to be used in discussing fine topologies, especially in analysis and potential theory [172], which were largely developed by him and his colleagues J. Mal) and L. Zajf~ek in the famous monograph [173]. Since an abstract fine topology ~-2 on a TS (X, T1) is any topology on X, finer than T 1 and, as is frequently the case, fine topologies are not normal, it was natural to consider the nonstandard separation axiom as the property which could take the place of normality. It was expected of this axiom to have bitopological character and, indeed, the so called Luzin-Menchoff property of a fine topology, that is, the p-normality of a BS (X, T1 < 72), proved to be the best suitable tool for subsequent studies [172]. This fact acted as a further stimulus in studying the connections of various problems of analysis and potential theory with BS's. We recall some background [42] and investigate some noteworthy applications of the bitopological concepts introduced earlier. M. Brelot compared the notion of a regular point (in particular, of a boundary point in the sense of the Dirichlet problem) of a set with that of a stable point of a compact set for an analogous Dirichlet problem and thus arrived at a general notion of thinness in classical potential theory [39]. H. Cartan observed that complements of sets A c R ~, A N x0, thin at x0, form neighborhoods of x0 in the topology which is the coarsest topology making all superharmonic functions on R ~ continuous [58]. This new topology was termed a fine topology and the corresponding fine topological notions became the commonly accepted tools in subsequent investigations. The chapter consists of three sections, the first two of which dwell upon many aspects of the above-mentioned topics. Section 7.1 begins by showing the bitopological essence of the Baire-like properties from [173], thereby enlarging information both on these properties and on fine topology varieties such as the density topology on R [126] and the crosswise topology on R 2 [173] in the context of the results from Chapter IV. Sufficient conditions, weaker than those in [173], are established, under which, on the one hand, any 2-G~-set is of 2-second category in (X, wl < 72) and, on the other hand, any 318
7.1. Applications in Analysis
319
BS (X, rl < r2) is an A-(2, 1)-Baire space. Furthermore, one of the bitopological insertion properties discussed in Chapter II makes it possible to find the necessary and sufficient condition for a locally convex linear TS to be barrelled, which implies the results from [258], and the subordination of two locally convex topologies T1 and r2 on a linear space X in the sense of G. Godefroy [125] is formulated by using the p-regularity of (X, rl, r2) and some bitopological insertion. At the end of Section 7.1, the sufficient conditions for the four families of nowhere dense sets to coincide with the four families of first category sets are established for a BS (X, 7"1 < 7-2) by means of a finite measure, which is in agreement with the (1, 2)-category for the special class of Baire BS's. In Section 7.2, we deal with questions of a fine topology in potential theory. Following Brelot, a convex cone (I) of lower semicontinuous functions f, defined on a TS (X, rl), determines a new topology r2 on X, finer than rl, making all lower semicontinuous functions from (I) continuous [40], [44]. (This agrees with Cartan's definition in the case of the classical potential theory.) Here the obtained BS is called a BS in potential theory and is denoted by (X, T1 <(I) 7"2). There naturally arises the question in situation which can a BS (X, rl < r2) be treated as ( X , T1 < ~ T2), to which our answer is as follows: a 1-Tychonoff BS (X, T1 < 7-2) is ( X , 7-1 < ~ 7-2) if and only if it is (2, 1)-completely regular. Clearly, this fact can be regarded as defining the relationship between topological aspects of potential theory and bitopologies. It can also be used as a basis for studying various properties of the fine topology r2 in potential theory either with each property individually or in combination with properties of the initial topology rl. In particular, we establish the topological and bitopological properties of a 1-Tychonoff BS (X, rl <~ r2) and find new sufficient conditions for (X, 7-1 < 7-2) to be (X, 7-1 < ~ 7"2), as well as the sufficient conditions under which the fine topology coincides with the initial one. The latter t h e o r e m - we believe - is related to solving Problem C.10 in [176], concerning the characterization of all harmonic spaces on which the initial and fine topologies coincide. N. Wiener introduced a mathematical interpretation of the physical concept of capacity [259]. A systematized exposition of the theory of capacities was provided by G. Choquet [60] and M. Brelot [41]. In the remainder of Section 7.2 the key notion is that of an abstract capacity on a TS (X, rl) in the sense of B. Fuglede [121]. A BS (X, rl <~ r2) with a fixed capacity C in the sense of Fuglede is denoted by (X, 7"1
320
VII. Applications of Bitopologies
( X , T1
7.1. Applications in Analysis
321
Further, the bitopological solution of one of C. J. Everett and S. M. Ulam's problems [108], [253], [263], which concerns the coincidence of the classes 7-/(X, r) and 7-{(X, 7) of all homeomorphisms of TS's (X, r) and (X, ~/) onto themselves, is also given. The importance of the theory of BS's is fully confirmed by its natural relationship with the theory of ordered TS's, that is, of sets simultaneously having a partial order and a topology [191]. Using the parallels drawn by M. J. Canfell [53] and T. McCallion [177] between the theory of BS's and that of ordered TS's, we construct the dimension theory for ordered TS's and formulate and study the Baire-like properties of the latter spaces, thereby filling in the gap of the monograph [191]. Based on these parallels, the relations between the separation axioms of ordered TS's and the separation axioms of the corresponding BS's are established. Other results include a simple bitopological characterization of continuous mappings of TS's. Finally, in addition to the results belonging to J. Chvalina [641, [65], some simple properties of BS's associated with directed graphs are considered.
7.1. Applications in Analysis To begin with, let us recall some notions which are essentially used in [173]. As usual, if X is a set, (Y, 7) is a TS and 5 is a family of functions from X to Y, then a weak topology induced on X by ~ is the smallest one on X making each function from ~ continuous. Further, if X is any set, then a mapping b : 2 X --+ 2 X is said to be a base operator or, briefly, a base if b(~) = ~ and b ( A t J B ) = b(A)tOb(B) for any sets A, B c 2 x. The pair (X, b) is called the base operator space, while the b-topology based on b is defined as follows: a set F c X is b-closed e--->, b(F) c F. The b-topology is the finest topology w on X, satisfying the inclusion b(A) c w cl A for each set A c 2 x. Obviously, any topology is defined by some base operator which is not defined uniquely. For example, both maps A -+ el et and A -+ A a on a TS have the properties of a base operator and define one and the same original topology. The density topology d on R and the crosswise topology c on R 2, defined by special-type base operators, are given in [173]. Since an (abstract) fine topology w2 on a TS (X, rl) is any topology on X, finer than Wl, the density topology d and the crosswise topology c are the fine topologies on (R, co)and (R2,co2), respectively. The conditions establishing principal relations between the Baire-like properties defined in [173, 4.A] and different kinds of Baire spaces, discussed in Chapter IV, are formulated in T h e o r e m 7.1.1. Let 72 be a fine topology on a TS (X, T1). Then:
(1) (X, 7-2) is a ws Baifc space with respect to T1 ~ ( X , T1,72) is an A-(2, 1)-BrS ,z----5, every subset U e 7-: \ {2~} is of (2, 1)-second category in X . (2) 7-2 on a TS (X, T1) has the Slobodnik property ~
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at go y
322
VII. Applications of Bitopologies
(3) Tu on a T S (X, T1) has the property.hd,z---->,(X, ml,T2) is a (1, 2)-SBrS<,' every subset U c ~-2 \ {2~} is of (1,2)-second category in X . Pro@ (1) By the definition from [173, p. 135], (X, ~-2) is a weak Baire space with respect to ~-1 if the intersection of each countable family of 2-dense and 1-open sets in X is 2-dense. Hence the first equivalence follows from Definition 4.1.5 for i = 2 and j = 1. Now, by (1) of Theorem 4.1.4 for i = 2 and j = 1, and Definition 4.1.5, the second equivalence is also correct. (2) Following the definition from [173, p. 136], a fine topology T2 on a TS (X, 7"1) has the Slobodnik property if the intersection of each countable family of 1-dense and 2-open sets in X is 1-dense. Therefore, the first equivalence is an immediate consequence of Definition 4.1.5 and (2) of Theorem 4.1.4 for i = 1 and j = 2, taking into account the first equivalence in (4) of Theorem 4.1.6. For the second equivalence it suffices to recall Definition 4.1.1. (3) By the definition from [173, p. 1371, a fine topology ~-2 on a TS (X, T1) has the property Ad if the intersection of each countable family of 1-dense and 2-open sets in X is 2-dense. Hence the first equivalence follows from (1) and (2) of Theorem 4.4.2, and the second equivalence from Definition 4.4.1. [5
One easily sees that this theorem is helpful for connecting the results of [173] with those discussed in Chapter IV. In particular, we have the following corollaries: C o r o l l a r y 7.1.2. The statements below are satisfied for the BS (R, aJ < d): (1) (R, co, d) an A-(2, (2) (R, co, d) (3) (R,a~, d)
is a hereditarily 2- BrS and, thus, (R, co, d) is also a 2- BrS, 1)-BrS, a 2-WBrS, and a (2, 1)-WBrS. is not 2-normal. is hereditarily p-normal and, hence, it is p-normal.
Proof. (1) Following (e) of Theorem 6.9 in [173], the TS (R, d) is a hereditarily BrS and it remains to use (1) of Theorem 4.4.28. Assertion (2) is exactly (c) of Theorem 6.9 in [173]. (3) By virtue of (g) of Theorem 6.9 in [173], the density topology d has the complete Luzin-Menshoff property, that is, (R, co, d) is p-completely normal. Then, by Theorem 0.2.2, (R, co, d) is hereditarily p-normal and it remains to use Corollary 0.2.3.
Corollary
7.1.3.
The following
conditions
are satisfied for the BS
(]l~2,& 2 < C):
(1) (R 2,co2,c) is a an A-(2,1)-BrS, a (1, 2)- BrS. (2) (R 2, w 2, c) is not (3) co2 and c are not
(1,2)-SBrS and thus (R 2,co 2,c) is also a 2-BrS, a (2,1)-WBrS, a 2-WBrS, a 1-SBrS, a 1-BrS, and 2-regular. S-related.
However, if cr is the weak topology on R 2 induced by the class of all separately continuous functions, then for the BS (R 2, w2, or), we have:
7.1. Applications in Analysis
323
(4) (R2,w2, a ) i s a (1,2)-SBrS or, equivalently, (R2,w2,cr)is also a 2-BrS, an A-(2,1)-BrS, a (2,1)-WBrS, a 2-WBrS, a 1-SBrS, a 1-BrS, and a (1, 2)-BrS. (5) (R 2, cz2, c) is 2-completely regular and, thus, it is 2-regular.
(6) ~2So. Proof. (1) By (a) of 4.A.8 in [173], the topology c on (R2, aJ2) has the property A4; hence, by (3) of Theorem 7.1.1, (R2,a~2, e) is a (1,2)-SBrS and it remains to use (1) of Theorem 4.4.28. Assertions (2) and (3) are exactly (c) and (b) of 4.A.8 in [173], respectively. (4) If cr is the weak topology on R 2 induced by the class of all separately continuous functions, then by 2.B.3 in [173], the strict inclusions cz2 c cr C c hold. Therefore, by (b) of 4.A.9 in [173], the BS (R2,cz2,a) is also a (1,2)-SBrS and, hence, it remains to use (4) of Theorem 4.4.28 in conjunction with (6) of this corollary. (5) By virtue of Theorem 2.3 in [173], the BS (R2, cJ2, or) is 2-completely regular and, therefore, it is 2-regular. (6) Since (R2,w2,a) is a 2-regular and (1, 2)- SBrS, it remains to use (3) of Theorem 7.1.1 in conjunction with ( d ) o f 4.A.6 in [173]. [--]
C o r o l l a r y 7.1.4. Let a 2-reDular BS (X, 7-1 < 7-2) be an S -1- BrS and (7-1,7-2) has the 2 - ( 1 - 6 5 ( X ) ) - i n s e r t i o n property. Then (X, T1,7-2) is an A-(2,1)-BrS and thus it is also a (2, 1)-WBrS. Proof. By Theorem 4.6 in [173], (X, 7-2) is a weak Baire space with respect to 7-1. Hence, by (1) of Theorem 7.1.1, (X, 7-1,7-2) is an A-(2, 1)-BrS and it remains to use (1) of Theorem 4.4.28. [5]
C o r o l l a r y 7.1.5. Let ( X , 7-1 < 7-2) be a (1,2)-BrS. Then any 2-Baire one function (or, more generally, any 2-Gs-measurable function) f on X is I-continuous at all points of X except a 1-first category set. Proof. By (2) of Theorem 7.1.1, 7-2 on the TS (X, 7-1) has the Slobodnik property and it remains to use 4.A.5 from [173]. E3
Note that when r2 is a fine topology on a TS (X, 7-1), it is obvious that the class of all 2-Baire one functions is broader than that of all 1-Baire one functions. Corollary 7.1.5 is therefore a stronger result than the theorem on Baire one functions (which is a part of the well-known Baire theorem) following which any Baire one function on R is continuous at all points of R except a first category set [199, Theorem 7.3]. D e f i n i t i o n 7.1.6. An (i,j)-completely regular BS (X, 7-1,7-2)is (i,j)-(~echcomplete if there is a sequence {b/~ }~~ 1 of j-open coverings of X such that any family of/-closed sets {F~}~s, having the finite intersection property and the property that for each n c N there exists Fs contained in some U c L/~, has a nonempty intersection. By virtue of implications after Proposition 0.1.7, it is clear that the following implications hold for a BS (X, rl < r2):
324
VII. Applications of Bitopologies
(X, T1,72) is 1-(~ech-complete ~ (X, T1,72) is (1, 2)-(~ech-complete, (X, T1, w2) is (2, 1)-(~ech-complete ~ (X, T1, T2) is 2-(~ech-complete. Furthermore, by Theorem a.9.3 from [106], if A E 2-Catg~(X) in a 2-(~echcomplete BS (X, TI,T2), then X \ A c 2-D(X). Hence, (iii) of Theorem 1.13 from [133] gives that a 2-(~ech-complete BS (X, Wl, ~-2) is a 2-BrS, and taking into account (1) of Theorem 4.4.28, we have the following implications: (X, T1, T2) is (2, 1)-(~ech-complete .~ (X, Wl, T2) is 2-(~ech-complete --~, (X, T1,T2) is a 2-BrS = ~ (X, TI,T2) is an A-(2,1)-BrS ==~ (X, T1,T2) is a (2, 1)-WBrS. Now, applying the arguments from [173], we will prove Theorems 4.2 and 4.6 from [173] under weaker hypotheses by using (1, 2)-(~ech-completeness, instead of 1-r and 2-quasi regularity instead of 2-regularity. T h e o r e m 7.1.7. If (X,T 1 < 7"2) is (1,2)-Cech-complete and (2,1)-regular, then any 2-~5-set is of 2-datg II and, hence, it is of (2, 1)-Cat9 II. oo
Proof. Assume the opposite: let V e 2-Gh(X) \ {2~}, that is, V -
n v~, where n=l oo
V~ c 72 for each n -
1, oc and V is of 2 - d a t g I .
Then V -
(.J A~, where n=l
A~ e 2-AfZ)(V)in (V, T{, 7-~) for each n - 1, ec. It is obvious that A1-~2-I)(V) since the contrary means that 7~ int T~ cl A1 - V ~ O, which is impossible. Let Xl E V \ ~-2clA1 be any point and {/4~}~__1 be a sequence of 2-open coverings of X whose existence is conditioned by the (1,2)-(2ech-completeness of (X, 71, ~-2). Then there is a set U1 E /41 such that Xl E U1. Since Xl E V c V~ for each n - 1, oc, we have Xl c U I N V 1 . On the other hand, x 1 c T 2 c l A 1 and by the (2, 1)-regularity of (X, T1,T2), there exist U(Xl) e T2 and U(T2clA1) 9 T1 such t h a t U(Xl) N U(T2 cl A1) - ~. Hence T1 C1 U ( X l ) n U(T2 clA1) - Z.
(1)
Applying the (2, 1)-regularity once more, we find that for the 2-open neighborhood U1 n V1 of the point Xl, there exists a 2-open neighborhood Vl(Xl) such that VI(Xl) C T1 cl V I ( X l ) C Vl (~ V1.
(2)
Let F1 - T 1 cl(VI(Xl)N U(Xl)). Then, by (1) and (2), we have F1 N A1 - 2~ and F1 c U1 N V1. Clearly, V N w2 int F1 r 2~ because x 1 E V n 7-2 int F1. Let us show that (VAT2 int F1) \w2 el A2 r ~. Indeed, if (VN~-2 int F1) \~-2 el A2 - ~, then V n w2 int F1 C w2 el A2 and since V n 72 int F1 C ~-~ \ { ~ }, we obtain T~int(T2 clA2 fl V) - T2' int T2' cl A2 r ~, which contradicts A2 E 2-N'I)(V). Assume that x2 E (V n ~-2 int F1) \ 7-2 clA2 is any point. As in the first case, there is a set U2 c L/2 such that x2 c U2 and there is a set F2 E co T1 such that x2 ~ F2, F2 C~A2 - 2~ and F2 C U2 C~V~ C~F1. Thus, applying an inductive assumption, we find the sequences {F~}~__I and {Un}nc~=l of
7.1. Applications ira Analysis
325
respectively 1-closed and 2-open sets such that F~cU~nV~,
F~nA~-~
foreach n - l ,
oo, and F I D F 2 D F a . . . .
It is obvious that the sequence {Fn}n~__l has the finite intersection property and by O(3
O(3
the (1, 2)-(~ech completeness of (X, 7-1,7-2), we have n Fn -r 2~. Let x c n Fn be n:l (N9
(X)
(20
any point. Since n Fn c V and n F n N ( U A n ) n=l
n=l
(30
n=l
2J, we o b t a i n x c V \ U An,
n=l
n=l
O<3
which contradicts the assumption V -
U An. n--1
The rest follows from (7) of Theorem 1.1.24.
[:]
C o r o l l a r y 7.1.8. Under the hypotheses of Theorem 7.1.7 a BS (X, 7-1 < 7-2) is a 2-BrS and, therefore, it is also an A-(2, 1)-BrS, a 2-WBrS, and a (2, 1)-WBrS. T h e o r e m 7.1.9. If (X, 7.1 < 7"2) is a 2-quasi regular S-1-BrS and (7.1,T2) has the 2-(1-Gs(X))-insertion property, then (X, 7.1,7.2) is an A-(2, 1)-BrS and, therefore, it is also a (2, 1)-WBrS. Proof. Let {Un}~__l be a sequence of subsets of X, where Un c 7-1 n 2 - T ) ( X ) for each n = 1, oc. By (2) of Theorem 4.1.4 and Definition 4.1.5, it suffices to O<3
show that U -
n un c 2-z)(x). Assume the contrary: X \ 7-2 cl U 5r ~. Since n=l
(X, 7-1,7-2) is 2-quasi regular and (7-1,7-2) has the 2-(1- Gs(X))-insertion property, O<3
there are sets V c 7-2 and A -
n An, where An c 7-1 for each n - 1, oc such that n=l
V c A c 7-2 clV c X \ 7-2 cl U.
(,)
c 2 - D ( x ) for = 1, we n V e 2-D(V). Un N V c 1-D(V) and thus Un N V E 1-D(7-1 cl V) for each n = 1, oc. By (.),
Un N V = Un N V N T-l cl V C Un N An N T-l cl V for each n - 1, oc. Since (X, 7-1,7-2) is a S-1- BrS, the BsS (7-1cl V, 7-~, 7-~) is a 1-BrS. But {Un N An N 7-1 cl V}n~ is the sequence of l-open l-dense subsets of O<3
(7-1 C1V~ 7-~ 7-~) a n d so
n (Vn n A n n 7-1 cl V) C 1-D(7-1 cl P ) . O n the other hand, n--1
(20
(x)
(N:)
(20
n=l
n=l
n=l
n=l
This result contradicts the fact U n A = ~.
77
Below we give two theorems illustrating the naturality and fitness of bitopological insertions for the theory of linear TS's (briefly, LTS's). For the relevant notions one can refer to [2201, [1501 . T h e o r e m 7.1.10. A locally convex LTS (X, 7-1) is barrelled if and only if the bitopology (7-1,7-2) has the (2, 1)-7-1-insertion property for every locally convex topology 7-2 that X admits.
326
VII. A p p l i c a t i o n s of Bitopologies
Proof. By (1) and (3) of Theorem 2.4.5, the arguments from [258, p. 351]. C o r o l l a r y 7.1.11. relled.
7"1
is coupled to T2 and it remains to use [-1
Every locally convex LTS of the second category is bar-
Proof. Let (X, 7.1) be any locally convex LTS of Catg II. If 7.~ is any locally convex topology on X, then the conditions of Theorem 3 from [258] are satisfied and, hence, 7"1 is coupled to 7"2. T h u s it remains to use (1) and (3) of Theorem 2.4.5, and Theorem 7.1.10. [3 C o r o l l a r y 7.1.12. A barrelled LTS (X, 7.1) is of CatglI (or metrizable) if there is a locally convex topology T2 on X , larger than 7.1, such that (X, 7"2) is of Catg II (or metrizable).
Pro@ Let (X, 7"1) be a barrelled LTS and T2 be a locally convex topology on X, larger than T1. Then by Theorem 7.1.10, the bitopology (7.1,7.9) has the (2, 1)-7"1insertion property and so 7"1 < c 7"2. Therefore, it remains to use Theorem 6 (or Theorem 7) from [258]. D It should however be noted that by an example from [257], (X, 7"1) can be a barrelled LTS of the first category although (X, 7"2) is a LTS of the second category with 7"1 c 7"2. D e f i n i t i o n 7.1.13. Let T1 and 7"2 be two locally convex topologies on a linear space X. Then the topology 7"2 is subordinate to the topology T1 if 7"2 is finer than 7"1 and there exists a base of 2-neighborhoods of 0 (zero element) consisting of 1-closed convex circled sets [125]. T h e o r e m 7.1.14. Let 7"1 and T2 be two locally convex topologies on a linear space X . Then 7"2 is subordinate to T1 i f and only if the BS (X, T1,7"2) is p-regular and (7"1, T2) has the (1, 2)-T2-insertion property.
Proof. First, we assume that 72 is subordinate to T1. Then T1 C 7.2 and, therefore, 7"1 CI U for each U E 7"2 so that 7"2 is coupled to 7"1. Hence, by (3) of Theorem 2.4.5, (7"1,7"2) has the (1, 2)-7"2-insertion property. Moreover, (X, 7"2) has a local base consisting of 1-closed sets and so (X, 7"1,7"2) is (2, 1)-regular. On the other hand, (X, Vl) is also a locally convex LTS having a local base, consisting of 1-closed sets [150, 6.5] so that (X, 7.1) is regular. It is clear that (X, 7.1,7.2) is (1, 2)-regular since T1 c T2 and thus (X, 7.1, T2) is p-regular. To prove the converse, assume that a BS (X, T1, T2), where 7"1 and v2 are two locally convex topologies on a linear space X, is p-regular and (7.1,7.2) has the (1, 2)-7.2-insertion property. Hence, by (1) and (3) of Theorem 2.4.5, 7"2 is coupled t o 7"1 and, by Corollary 2.2.9, we have vl c v2. For every neighborhood V(0) E 7.2 choose a neighborhood U(0) E T2 such that 7"1 clU(0) c V(0). Since (X, 7"2) is locally convex, by [150, 6.5], there exists a set F E COT2 with the property 0 c F c U(0). Therefore T l c l F C TlclU(0) C V(0), where the set 7.1elF is convex circled because F is convex circled [220, Proposition 4]. Thus for every neighborhood V(0) c 7.2, there exists a convex circled set (F = T1 e l F such that 0 E 9 C V(0) so that 7"2 is subordinate to 7"1. D 7"2 CI U C
7.1. Applications in Analysis
327
In 4.A.6 of [173] the sufficient conditions are found, for which 7.1S7.2 in a BS (X, 7.1 < 7"2). Below we prove the same result under weaker conditions. T h e o r e m 7.1.15. If (X, 7"1 < 7"2) is a 2-quasi regular and (1, 2)- SBrS, then 7-1S7.2, and, therefore, (X, 7"1,7"2) is a 1-Blumberg space implies that (X, 7"1,7"2) i8 a 2-Blumberg space, where, by [173, p. 140], (X, 7"1,7"2) is an i-Blumberg space if for any i-real function f on X there is an i-dense subset D c X such that the restriction f [D is i-continuous.
Proof. By (3) of Theorem 7.1.1, (X, 7"1 < 7"2) is & (1,2)-SBrS ~
7"2 on (2,7"1) has the property M .
T h ~ , by (c) of 4.A.O in [173], w, h~v, ~-~ \ { ~ } c (1, 2)- SD(X).
Now, 1,t U c
7"2 \ {2~} be any set. Since (X, T1,7"2) is 2-quasi regular, there is a set V E 7"2 \ {2~} such that r2cl V c U and, therefore, 2~ r 7.1 int 7"2cl V C U. Hence, by (2) of Corollary 2.1.7, we have 7"1S7"2. The rest is given by 4.B.4 of [173]. D C o r o l l a r y 7.1.16. If f : (X, 7"2) ~ (Y, 7) is a feebly open condensation and 7"1 ---- f - l ( ~ ) , then (X, 7"1 ~ 7"2) i8 a 1-Blumberg space implies that (X, 7"1 < 7"2) is a 2-Blumberg space.
Proof. By Theorem 2.1.5, we have 7"1 ~ S 7"2.
[--]
The section concludes with establishing the conditions under which the families i-A[7)(X), (i, j)-A[D(X), i- Catgi (X), and (i, j)- Catgi (X) coincide. For this purpose we need to study questions which are connected with finite measures on BS's. D e f i n i t i o n 7.1.17. A subset A of a BS (X, 7"1,7"2) has the (i, j)-Baire property if it can be represented as A = U A C , where U c 7"j and C c (i, j)-Catg I (X). The families of all subsets of (X, 7"1,7"2), having the (i, j)-Baire property, are denoted by (i,j)-13(X). Hence
(i,j)-~(X) - {A C X"
A-
U/kC, U 6 7"j, C 6 ( i , j ) - C a t g l ( X ) } .
It follows immediately from Definition 7.1.17 that 7"j U (i, j ) - C a t g i ( X ) c (~,j)-~(x). It is evident that for a BS (X, 7"1 < 7"2) and a set U c 7"~, the set T1 c 1 g \
g C (1, 2 ) - . / ~ ) ( X ) C
(1, 2)-Catgi (X)
and so 7.1 cl U = U 0 (7.1 cl U \ U) = UA(7.1 cl U \ U) E (1, 2)-B(X). It is also evident that for a BS (X, 7-1 < 7-2) the following inclusions hold: (2, 1)-B(X) c 2-B(X) N N 1-B(X) C (1,2)-B(X). Theorem
7.1.18. For a BS (X, 7"1 < 7-2) the following conditions are satisfied:
(1) A c (1, 2)-B(X) ~ - ~ (A = F A D , where F c co7.2, D c (1,2)-datgi(X)) and, hence, X \ A c (1, 2)-B(X). (2) (1, 2)-B(x) i~ ~ ~-~lg~b~a, g ~ ~ t ~ d by th~ ~ i o ~ ~ u (1,2)-Card(X).
9
C
II
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..
q"b q'~
II
0
II
II
D
~
w
~I~
II
C~
~
II
D
I>
II ~
D
~C~
II
II
,
~
~
q
r
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~
~= IIC~
~
~
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~ . ~ ~
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o
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7.1. Applications in Analysis
329
(4) Let A = U A C , U 9 r2 and C 9 (1, 2)-C, a t g i ( X ). Then by (3) of Proposition 1.3.10, U = V \ r2 c l B , where V = T2 int 7-1 cl U 9 (2, 1)-OT)(X) and B = V \ U 9 (1,2)-Afg)(X). Therefore,
A = UAC = (VA(V \ U))AC = VA(BAC)
= VAM,
where M c (1,2)-Catgx(X). Thus it remains to prove only t h a t for a (1,2)-SBrS this representation is unique. Indeed, let A = V A M = W A N , where V c ( 2 , 1 ) - O D ( X ) , W E r2, M, N 9 (1, 2)-Catg x(X). Then
W\r2clV
c W\ V c WAV=MAN
9 (1,2)-Catgx(X).
Since W \ 7-2 cl V 9 7-2 ffl ( 1 , 2 ) - C a t g I ( X )
and (X, rl so t h a t W A = VAM, (2, 1)-open
< r2) is a (1, 2)- SBrS, we have W \ r 2 c l V = ~, t h a t is, W c r 2 c l V c r2 int r2 cl V c r2 int rl cl V = V. Therefore in the representation the 2-open set V c (2, 1 ) - O D ( X ) is maximal and if V and W are both domains, then V C W and W C V, that is, V = W and M = N. [2]
D e f i n i t i o n 7.1.19. Let (X, T1 < 7-2) be a BS and It be a finite measure on the a-algebra (1, 2)-B(X). Then It is said to be in agreement with the (1, 2)-category, if It(A) = 0 is equivalent of A 9 (1, 2)-Catgi (X). T h e o r e m 7.1.20. Let (X, 7"1 < 7"2) be a (2, 1)-regular (1,2)-SBrS and It be a finite measure on (1, 2)-B(X) which is in agreement with the (1, 2)-category. Then for each 2-open set G and each c > 0 there exists a l-closed set F such that F c G, It(F) > I t ( G ) - c, and for each 2-closed set F, there exists a 1-open set G such that F c G, It(G) < It(F) + s.
Proof. Let b/ = {U} be a maximal family of non-void disjoint 2-open sets such t h a t r l c l U c G for each U c U. Since (X, rl < r2) is a (1, 2)- SBrS, U c b/ implies U c (1, 2)-Catgii (X) so t h a t It(U) > 0 and so the family H is not greater co
than countable, t h a t is, b/ -
{U~}n~176. Then V -
U u~ c G.
Let us prove
n--1
t h a t G c T2cl V. Indeed, if G N (X \ r2cl V) ~ 2~, then by the (2, 1)-regularity of (X, rl < r2), there is a set H 9 r2 \ {2~} such that 71 c l H c G N (X \ r2 clV). Moreover, H N Un = 2~ for each n = 1, oc and so H 9 b/, which contradicts the maximality of N. Hence G \ V c ~ r
\ V 9 >HD(X)
c (1, 2 ) - H D ( X ) c (1, 2)-Catgi(X ) co
and, therefore, It(G \ V) - 0, t h a t is, It(G) - It( U un).
For each U c r2, we
n--1
have T1 C1 V \ V 9 (1, 2)-AfT)(X) c (1, 2) Catg I ( X ) .
~9
~
9
~ ~ ~
II
II
~
~
"~
~
C
II
~~
~ -~ ~
c
~
~
~
8
~~
~
~'~
A
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7.2. Relations with Potential Theory
331
(4) From (2) above we immediately obtain (2, 1 ) - B ( X ) = 1-B(X) c 2 - B ( X ) = (1, 2)-B(X). Thus it remains to prove only that 2-B(X) C 1-B(X). Let A c 2-B(X), that is, A = U A C , where U E 7-2, C c 2 - C a t g i ( X ) = 1 - C a t g i ( X ). But U E r2 \ {~} and 7-1 < S 7-2 imply 7-1 int U -/- 2~. Therefore A - U A C - (7-1int U U (U \ 7-1 int U ) ) A C - (7-1 int U A ( U \ 7-1 int U ) ) A C = = 7-1 int UA ((U \ 7-1 int U) A C ) - V A D ,
where V E 7-1, and D c 1-Catg I ( X ) since C c 1 - C a t g i ( X ) and U \ 7-1 int U = n2(U) c (1, 2)-datg I ( X ) = 1-Catg~ ( X ) .
[]
7.2. R e l a t i o n s w i t h P o t e n t i a l T h e o r y Let us recall one of the methods of constructing fine topologies which is frequently used in potential theory [44]. Assume that (I) is a convex cone of nonnegative 1.s.c. functions on a TS (X, 7-1) such that the following conditions are satisfied" (1) If p E (I), then ~p E (I) for each real ~ _> 0. (2) If ~1, ~2 C (I), then ~1 -Jr ~2 C (I). (3) +oc c (I) together with 0. (oc) - 0. The fine topology 7-2 on X, defined by the cone (I), is the coarsest topology on X, finer than the initial topology 7-1, which makes all functions from q) continuous. It is clear that the fine topology 7-2 is generated by the family - - T 1 U { {X C X "
f (x) < a, f C ~, a C R t } .
The obtained BS is called a BS in potential theory and is denoted by (X, T1 < ~ T2). E x a m p l e 7.2.1. Let (X, 7-1 < 7-2) be a l-T1 and p-normal BS. Then by [172, Remarks, (2)] (X, 7-1 < 7-2)is (X, T1
sup
(
inf
U(x())ET1 xEANU(xo)
W(x)).
It is obvious that if x0 c T1 cl A, then the set A is always thin at the point x0 since it is sufficient to take ~ - 0 and in that case lira inf - +oc. A ~ x---+x(~
By virtue of Cartan's theorem [44, Theorem 1.3], fine neighborhoods of the point x0 c X are complements to those sets A c X which are thin at x0 and do not contain this point. Some useful properties of a BS in potential theory are collected in T h e o r e m 7.2.2. The following conditions are satisfied for a BS (X, 7-1
332
VII. Applications of Bitopologies (2) If (X, rl,T2) is not 1-T2, but for each point x c X
and its every 1-neighborhood U(x), there is a 1-closed 1-neighborhood F(x) such that F(x) c U(x), then (X, rl, re) is (2, 1)-regular. (3) If (X, rl, re) is 1-regular, then it is also 2-regular. (4) If (X, rl, r2) is I-locally compact, then it is a 2-BrS and, therefore, it is also an A-(2, 1)-BrS, a 2-WBrS and a (2, 1)-WBrS. (5) If(X, rl, r2) is 1-uniformizable (<---> 1-Tychonoff), then (X, rx, r2) is also 2-uniformizable (<--> 2-Tychonoff). Proof. The proof of conditions (1)-(5), though without the additions, we have introduced in (1) and (4), is outlined in [44, Theorem 1.4]; here we will give a more detailed proof of (2) and (3). (2) Let x0 E X be any point and U(xo) be any 2-neighborhood of x0. Then U(xo) = X \ A, where A is thin at x0 and, therefore, there exist a function c (I) and a 1-neighborhood V(xo) such that ~(x0) < inf ~(y). Let A yCAnV(xo) be any real number satisfying the inequality ~o(x0) < A < inf ~(y). It is yEAnV(xo) clear by [79, 12.7.2] t h a t the set E = {x c X : ~(x) > A} is 1-open so t h a t the set F = {x c X : ~o(x) <_ A} is 1-closed for ~o c (I) and so ~o is 1-1.s.c. Furthermore, p(xo) < A implies x0 E F. Let us show that E is thin at xo. Indeed, ~(x0) < A < ~(x) for each point x c E and thus p(x0) < inf p(y), y6EnV(x()) where ~ and V(xo) are the same as above, t h a t is, E is thin at x0. Hence we conclude t h a t the set E is a 1-closed 2-neighborhood of x0. On the other hand since A < inf ~(y), we see that ~(y) > A for each point y E A n V(xo) and, yEAnV(xo) therefore, A n V(xo) C E so t h a t
(x \ u(~0)) n V(xo) c E. Hence
x \ U(xo) c E u ( x \ V(xo)) and so
x \ (E u ( x \ V(xo))) c U(~o), t h a t is
(X \ E ) N V(xo) C U(xo) ~
F N V(xo) C U(xo).
Since V(xo) is a 1-neighborhood, there is, by assumption, a 1-open neighborhood W(xo) such t h a t rl cl W ( x o ) c V(xo) and, therefore, F n T1 C1W(xo) C f n V(xo) C U(xo). Let us prove that F N W(xo) is a 2-neighborhood of x0. By Cartan's theorem , it suffices to show t h a t X \ ( F N W(xo)) is thin at x0. Indeed,
x \ (F ~ W(x0)) - (x \ F)u (X \ W(xo)), where X \ F = E is thin at x0 and X \ W(xo) is also thin at x0 because XO-CX \ W(x0) -- T1 C1 (X \ W(x0)).
7.2. Relations with Potential Theory
Hence, by [44, p. 11, (ii)], the union (X \ F)U (X F A W(xo) is the 2-neighborhood of x0. Moreover,
\ W(xo)) is
333
thin at x0 so that
T1C1 (F A W(xo)) C T1C1F A T1 c1W(x0) - F N 7 1Cl W(xO) C U(xo)
and thus (X, T1,7-2)is (2,1)-regular. (3) If (2,7-1,7"2) is 1-regular, then, by (2), (X, 7"1,T2)is (2,1)-regular and, hence, it remains to use (6) of Proposition 0.1.7. K] C o r o l l a r y 7.2.3. /f (X, 7"1 <02 7"2) is 1-Tychonoff, then it is (2,1)-completely regular.
Proof. By (5) of Theorem 7.2.2, (X, 7"1,7"2) is also 2-Tychonoff so that if F E co 7"2, x g f , then there exists a 2-continuous function f ' ( X , 7"1,7"2) ~ (I, cz') such that f ( x ) - 0 and f ( F ) - 1. Moreover, using the function f constructed in proving (c) of Theorem 1.4 in [44], we find that f is also 1-1.s.c. since the basic function u from [44] belongs to 9 so that u is 1-1.s.c. and 2-continuous since (X, 7"1, r2) is (X, 7-1 <02 7-2). Therefore ~ - 1 - f is a 1-u.s.c. and 2-continuous function such that F ( x ) - 1 and q ~ ( F ) - 0 and, hence, (X, rl, 7-2) is (2,1)-completely regular by (7) of Proposition 0.1.7. [-] C o r o l l a r y 7.2.4. If a 1-TychnoffBS (2,7" 1 <02 7"2) is a (1, 2)- SBrS, then (X, 7-1 <02 7-2) is a 1-Blumberg space implies that (X, 7-1 <02 72) is a 2-Blumberg
space. Proof. By Corollary 7.2.3, (X, rl <02 7-2) is (2,1)-completely regular and hence 2-quasi regular. Thus it remains to use Theorem 7.1.15. D Theorem (X, 7"1 <02 T2).
7.2.5.
If (X, T1 < 7-2) i8 (2,1)-completely regular, then it is
Proof. Let (I) = {f : f >_ 0, f be a 1-1.s.c. and 2-continuous function on X}. It is easy to see that r is a convex cone on X. Since (X, rl, T2) is (2, 1)-completely regular, the remainder of the proof is similar to the proof in [172, Remarks, (2)]. As a result we find that 7-2 is the coarsest topology 7-02 on X, finer than 7-1 and making all functions from (I) continuous. D C o r o l l a r y 7.2.6. A 1-TychonoffBS (X, T 1 if it is (2, 1)-completely regular.
<
7-2) is ( X , T 1 <02 7"2) if and only
Proof. The proof follows directly from Theorem 7.2.5 and Corollary 7.2.3.
D
In a certain sense Corollary 7.2.6 can be regarded as an analogue of Proposition 10 from [208], drawing a parallel between the ordered TS's and BS's (see Section 7.3). It is of no wonder that this corollary leads to many interesting consequences.
Theorem
7.2.7.
The statements below are satisfied for a 1-Tychonoff BS
(X, T1 <02 7-2)"
(1) (X, 7"1,T2) i8 p-completely regular ( ~ quasi uniformizable). (2) (X, rl,7-2) has at least one p-normal base.
334
VII. A p p l i c a t i o n s of Bitopologies
(3) If (X,n,7-9) is (1,2)-Cech-complete, then any 2-Gb-set is of 2 - C a t g I I and, thus, (X, r l , T 2 ) i s a 2-BrS, an A-(2,1)-BrS, a 2-WBrS, and a (2, 1)-WBrS. Therefore, if (X, 7-1,7-2) is I-locally compact or 1-completely metrizable, then any 2-Gs-set is o f 2 - C a t g I I and, thus, (X, 7-1,7-2) is a 2-BrS, an A-(2, 1)-BrS, a 2-WBrS, and a (2, 1)-WBrS.
(4) For each point x E X and its any neighborhood U(x) E ( i , j ) - O I ) ( X ) , there exists an i-zero set F which is a j-neighborhood of x contained in
u(~). (5) If A c X , IAI < No, F E ( i , j ) - C I ) ( X ) and A N F = ~, then there is an i-zero set B such that F c B and A N B = ;g.
(6) Every i-compact subset of X is j-closed. (7) If A is an i-compact subset of X , then for every i-closed subset B c X \ A , there exist an i-open set U and a j-open set V such that A c U, B C V and U A V = ;g. (8) If A is an i-compact subset of X , then for every i-closed subset B c X \ A , there exists an i-continuous function f : (X, 7-1,7-2) ~ (I,a/) such that f (A) = 0 and f (B) = 1.
(9) If (x. n . ~ ) i~ 1 - ~ t ~o.~tabl~. t h ~ ~ y 2-clo~d a~d 2-co~tably compact subset of X is 1-closed. (10) If (X, 7-1,7-2) is 2-second countable, then every 2-countably compact subset of X is 1-closed.
(11) /f 1-G~(X) = 7-1, then every 2-Lindelbf subset of X is 1-closed. Proof. (1)(X, 7-1,7-2)is 1-Tychonoff ~ (X, n , T2)is 1-completely regular, and by (7) of Proposition 0.1.7, (X, T1,72) is also (1,2)-completely regular. Hence, by Corollary 7.2.3, (X, 7-1,T2) is p-completely regular and thus, by Theorem 4.2 from [166], (X, 71,7-2) is quasi uniformizable. Assertion (2) follows directly from (1) and Theorem 0.2.5. (3) For the first part note that by (1) above and the implications after Definition 0.1.6, (X, 7-1,7-2) is (2,1)-regular and it remains to use Theorem 7.1.7 in conjunction with Corollary 7.1.8. Furthermore, according to [106, p. 252] and the implications preceding Theorem 7.1.7, if (X, T1, T2) is I-locally compact, then (X, T1,7-2) is 1-(~ech-complete and hence (X, 7-1,7-2) is (1, 2)-(~ech-complete. Finally, if (X, 7-1,T2) is 1-completely metrizable, then by Theorem 4.3.26 from [106], (X, n , ~2) is 1(~ech-complete metrizable and thus it is (1, 2)-(~ech-complete. Therefore, in both cases, it remains to use the first part. (4) By (13) of Definition 0.1.6 and (1) above, (X, 7-1,7-2) is p-almost completely regular. It remains to use Theorem 3.3 from [238]. (5) By analogy with (4) above, this condition is an immediate consequence of Theorem 3.5 from [238]. (6) Obviously, a BS (X, 7-1,T2) is 1-T2 and the case with i = 1 and j = 2 is an immediate consequence of the well-known topological fact and the inclusion 7"1 C 7-2. Now assume that A c X is 2-compact. Since, by (1), (X, T1,7-2) is
7.2. Relations with Potential Theory
335
p-regular, Proposition 2 from [217] gives that (X, rl, r2) is R-p-R1. Hence, by Proposition 6 from [217], the set [_J T1 CI{x} is 1 - c l o s e d and thus A is I-closed because xCA ( X , T1,7-2) is l - T 1 .
Assertion (7) follows directly from Lemma 3.4 in [73] since (X, T1,T2) is p-regular. The condition (8) is an immediate consequence of Theorem 3.1.7 from [106] since, on the one hand, (X, T1, T2) is 1-Tychonoff and thus it is 1-regular and on the other hand, it is (2, 1)-regular and thus it is 2-regular. (9) Since (X,T1,T2) is p-completely regular, by Remark 0.1.9, it is also MN-p-R0. Therefore, 7"1 ( 7-2 implies that 9 U(
)c
that is (X, T1, T2) is MN-2-R0. The rest follows directly from Lemma 3 in [210] since (X, T1, T2) is p-regular. (10) Let A c X be any 2-countably compact subset. Since (X, T1,T2) is l-T2, by (1) of Theorem 7.2.2 it is also p-T2 and so for each pair of points y ~ A and x c A, there exist a 2-open neighborhood U(x) and a 1-open neighborhood UX(y) such that U(x) NUX(y) - 25. The f a m i l y b / - {U(x) 9 x c A} is a 2-open covering of A, that is, A c [.J U(x). But the weight 2 ~ ( X ) < b~0 V(x)~U and, by Theorem 1.1.14 from [106], there exists a countable subset { X l , X 2 , . . . } c OO
[.J U(x) - [_J U(xn). Hence/~' - {U(xn)}n~ 1 is a countable U(x)El~ n=l 2-open covering of A. Since A is 2-countably compact, bff has a finite subcovering
A such that
l/lit-- { U ( X k ) } ~ = l .
Let U -
~ UXk(y). Then U E T1, y E U and U n A -
k=l
2~ so
that y g T1 cl A. Thus ~-1 cl A - A since y g A is arbitrary. (11) Let A c X be any 2-Lindelhf subset. If y g A is any point, then like in the proof of (10), for each point x c A there exist a 2-open neighborhood U(x) and a 1-open neighborhood UX(y) such that U(x)O UX(y) - ~. The family/// {U(x) 9 x c A} is a 2-open covering of A. Since A is 2-Lindelhf,/~ has a countable OO
subcovering b / ' -
{V(xn)}~~ 1. Let U -
n UX'(y) 9 Then U e 1-Gh(X) - T 1 , n=l
y E U and U C~A - ~ so that y g T 1 cl A. Thus T 1 cl A - A.
Z]
R e m a r k 7.2.8. Obviously, the statements given by (3) of Theorem 7.2.7 are stronger results than (4) in Theorem 7.2.2 and Proposition 2 from [260], respectively. T h e o r e m 7.2.9. A BS (X, T1 < 7-2) i8 ( X , 7" <02 7-2) if anyone of the following conditions is satisfied: (1) p - i n d X = 0. (2) For each point x E X and any 1-closed set F, x-cF, there exists a 1-1.s.c. and 2-continuous function f : (X, T1, T2)--~ ({0, 1}, W") such that f ( x ) = 1, f (F) = O, and for each point y c X and any 2-closed set E, y-E E, there exists a 1-u.s.c. and 2-continuous function p : (X, T1,T2) ~ ({0, 1},a/') such that ~(y) = 1 and p(E) = O.
336
VII. Applications of Bitopologies (3) (4) (5) (6) (7) (8) (9)
(X, (X, (X, (X, (X, (X, (x,
T1,7"2) i8 R-p-I~ 1 aTbd (2, 1)-Rlc (or, (2, 1)-lqc). T1,T2) iS l-T1 and (1,2)-IndX or (2,1)-IndX is finite. T1,7-2)is l-T1, (1,2)-regular or (2,1)-regular, and FHP-compact. ~-~,T2) is l-T1, R-p-R1 and FHP-compact. 71,7-2) is 1-T1, p-almost normal and p-seminormal. T1,7-2) is l-T1, p-regular and p-LindelSf. ~,~:) i~ 1-T1, d-~co~d co~tabZ~ a~d p-~g~Za~.
Proof. By Theorem 7.2.5 , it suffices to prove in each of the cases (1)-(9) that (X, Wl < T2) is (2, 1)-completely regular. (1) Following (2)of Proposition 3.1.10, (X, T1 < T2) is p-completely regular and, hence, it is (2, 1)-completely regular. (2) By (i) and (ii)of Theorem 6 from [33], p - i n d X = 0 and, hence, it is p-completely regular. (3) Theorem 2 and the diagram in [184, p. 189] imply that in both cases (X, 7-1 < 7-2)is (2, 1)-completely regular. (4) By (1) of Proposition 3.2.7, (X, 7-1 "< 7-2) is p-normal and since (X, T1 < 7"2) is l-T1, it is also d-T1. Hence, by [166, p. 242], (X, 71 < 72) is p-completely regular. Assertion (5)follows directly from (4)since, by Theorem 13 in [113], (X, T1 < 7"2) is p-normal. The condition (6) is an immediate consequence of (4) since, by Theorem 2 in [217], (X, T1 < 7-2) is p-normal. (7) and (8) are immediate consequences of (4) since, by Theorems 1.2 and 2.10 in [238], (X, T1 < T2)is p-normal. (9) By Lemma 3.2 in [151], (X, T1 < 7-2) is p-normal and it remains to use (4).
D
The arguments used for a BS (X, T1 <~ 7"2) lead to the formulation of some sufficient conditions for which the fine topology 72 coincides with the initial topology T1. T h e o r e m 7.2.10. Each of the conditions (1)-(10) is sufficient for the fine topology 72 to coincide with the initial topology T1 for a 2-Ro [75] and p-regular BS (X, T1 <(~ T2): (1) (X, T1,T2) is (2,1)-aRlc. (2) (X, T1,7"2) i8 (2, 1)-paracompact. (3) (X, TI, T2) is 2-COmpact. (4) (X, T1, "c2) is I-first countable and (2, 1)-RRlcc. (5) (X, T1, T2) is 1-first countable and (2, 1)-countably paracompact. (6) (X, ~'1, T2) is 1-first countable and 2-countably compact. (7) (X, T1,7"2) is (2, 1)-locally LindelSf and 1-G,(X) = T1. (8) (X, T1, T2) is (2, 1)-paraLindelSf and 1-G6(X) = T1. (9) (x, ~1, ~ ) i~ 2-Li~e~z6I and 1 - ~ ( X ) = ~ . (10) EveI'y 2-opelt COVeFiTbgof (X, T1,72) has a 1-open refinement. Furthermore, the fine topology 7-2 coincides with the initial topology Vl for a 1-T2 BS (X, T1 <~ T2) if each of the conditions below hold: (11) (X, T1, T2) is 2-second countable and 2-countably compact.
7.2. Relations with Potential T h e o r y
337
(12) (X, T1,7-2) i8 (1,2)-Sic. (13) (X, T1,72) i8 d-compact. (14) (X, rl, r2) is FHP-compact. Proof. (1)-(9) are immediate consequences of Propositions 1-6 and Lemmas 1-4 in [210] since under each of the conditions (1)-(9), (X, rl, r2) is MN-2-R0 and hence T2 C T1.
Assertion (10) follows directly from Theorem 8 in [113]. (11) Since T1 C T2, by (1) of Theorem 7.2.2, (X, T1,T2) is also 2-T2 and p-T2. Let F c cot2 be any set. Then, by Theorem 3.10.4 from [106], F is also 2-countably compact. Now, from the proof of (10) in Theorem 7.2.7, it follows that F E co 7-1. The equality T1 : 7-2 in each of the cases (12)-(14) is an immediate consequence of Theorem 2.2 from [245] and Theorems 10 and 11 from [113], taking into account that (X, T1,7-2) is d-T2 and p-T2. 7-1 The remainder of this section will be concerned with the questions connected with the compatibility of a fine topology with a quasi topology in the sense of [121]. Also, consideration will be given to the so-called almost notions. In particular, we are going to supplement the results from [90] and [93]. To this end we will use the well-known notions from [121]. D e f i n i t i o n 7.2.11. A capacity on a set X is a set function C : 2 X + R+ satisfying the following conditions: (1) If A1 c A2, then C(A1) < C(A2) for any sets A1,A2 E 2X (increasability). oo
o<9
(2) C( [,J An) < ~ C(An) for every countable family {An}nC~=l C 2 x (COn=l
n=l
untable subadditivity). (3) C(2~) = 0. In our further discussion C will denote a fixed capacity on X in the above sense, while by (X, T 1
C(A) = inf {C(U) : U c
T1,
A c U} ( C ( A ) = C(r2 clA) )
for every set A E 2 x. Following [121], a property 7)(x) involving the generic point x c X is said to hold quasi everywhere in X if C({x E X : non 7)(x)}) = 0. For sets A1, A2 E 2 X,
A1 -< A2 ,e--v, C(A1 \ A2) = 0 (i.e., A1 is quasi contained in A2 or A2 quasi contains A1), and the associated equivalence relation (C-equivalence) on 2 x is denoted by ~ so that
A1 ~ A2 ,.e-->C(A1AA2) = 0 (i.e., A1 is equivalent to A2).
338
VII. Applications of Bitopologies
It is not difficult to verify that if A1 -~ A2 (A1 ~ A2) and A is any set, then AINA--z, A2NA,
AIUA--
(AINA,.,A2NA,
(A1 - ~ A 2 A A I ~ A )
AIUA~A2UA),
~A-~A2,
and hence (A1 -< A2 A A1 ~
A] AA2
~
A;)
~
Ai
-< A;.
Clearly, A -< B <---~, X \ B -< X \ A
sinced\B Also,
= (X \ B) \ (X \ A) and, therefore, A ~ B ~
(A1 -< A A A2 -< A) ~
(X\d)
A1 U A2 -< A and (A -< A1 A A -< A2) ~
~ (X\
B).
A -< A~ n A2.
A point x c X is a polar point of X if C({x}) = 0 and
x0 = {x c x :
c ( { x } ) = 0}.
We have thus arrived at the notion of thinness of a set at its point, namely, a set A c X is thin at a point x0 c A if A \ {x0} is thin at x0 and x0 c X0. The set of all points of X, at which A is thin, is denoted by e(A). D e f i n i t i o n 7.2.12. The base b(A) of a subset A of a BS (X, 7-1 <~ 7-2,C) is the set of all points of 7-2cl A which are not both polar and 2-isolated in A, that is, b(A) - (A d N X0) U (7-2clAN (X \ X0)). Hence, by Cartan's theorem, we have b(A) = X \ e(A) for any subset A of ( X , 7-1 < dp 7-2, C ) .
T h e o r e m 7.2.13. If (X, 7-1 <~ 7-2,C) is a 1-completely metrizable and (1,2)-D1 BS, and for each set A E 2-7)(X) there is a set A c 2 - g h ( X ) such that B E 2-:D(A), then (X, 7-1,7-2) is an S - 2 - B r S and, thus, it is a 2-BrS, an A-(2, 1)- BrS, a 2-WBrS, and a (2, 1)-WBrS. Proof. It is obvious that (X, 7-1,7-2) is a 1-Tychonoff BS and by (1) of Theorem 7.2.7 it is (2,1)-regular, that is, 7-2 is cometrizable with respect to (X, 7-1) in terms of [173, p. 133]. For any set A E 2-P(X) let us consider its base b(A). Following Definition 7.2.12, b(A) - (A d N Xo) U (7-2cl A N (X \ Xo)) - (A d O Xo) U (A d N (X \ Xo)) - A d.
Hence A - A d - b(A) - T2clA. On the other hand since (X, T1,T2) is (1,2)-D1, we have b(A) c 1-gh(X) c 2-65(X) [16, Corollary of Theorem 4]. Thus for the set A c 2-7)(X) the set b(A) c 2 - 6 0 ( X ) n 2-:D(A) and hence, by Corollary 4.5 from [173, p. 135], (X, 7-1,7-2) is an S-2- BrS. Since every S-2-BrS is a 2-BrS, the rest follows from (1) of Theorem 4.4.28. [-7 D e f i n i t i o n 7.2.14. A subset A of a BS (X, 7"1 <(I) 7"2, C ) is said to be quasi closed (quasi open) if inf { C ( A A F ) : F E co 7-1} = 0 (inf { C ( A A E ) : E C 7-1} = O) [121, Definition 2.1].
7.2. Relations with Potential Theory
339
The family of all quasi closed (quasi open) subsets of (X, T1 <~ T2, C) is denoted by QC(X) ( Q O ( X ) ) , where the family Q O ( X ) is said to be the quasi topology on X. Clearly, we have A c QC(X) ~ X \ A E Q O ( X ) and, by Lemma 2.3 from [121], (A1, A2 E QC(X)) ~
A 1 U A 2 c QC(X), oo
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1, oc)==:> U An 9 QO(x). n:l
In addition, A 9 QC(X) (A 9 QO(X)) and A ~ B imply B 9 QC(X) (B 9 QO(X)) so that the notion of a quasi closed (quasi open) set depends only on the equivalence class of the set A. D e f i n i t i o n 7.2.15. Let (X, T1 < ~ T2, O) be a BS in potential theory with a capacity C. Then the fine topology T2, determined by the cone (I), is compatible with the quasi topology Q O ( X ) determined by the capacity C if the following conditions are satisfied: (1) A 9 QC(X) ----> A ~ f for some f 9 coT2 (i.e., every quasi closed set is equivalent to some finely closed set). (2) (C(A) = O) ===>(b(A) = 2~) (i.e., every set of capacity 0 is everywhere thin). (3) (A F1b(A) = Z) ~ (C(A) = 0) (i.e., every set, thin at each of its points, has capacity 0). (4) CoT2 C QC(X) (i.e., every finely closed set is quasi closed). In the case of compatibility the corresponding BS in potential theory with a capacity C will be denoted by (X, 7-1 <m(c) r2). R e m a r k 7.2.16. One can easily verify that (X, TI<~T2, C) is (X, 71
7-2)
(1) A 9 QC(X) ,,e---->, A ~ F for some F 9 coT2 (or, equivalently, A 9 Q O ( X ) ,,e---->,A ~ U for some g 9 T2). (2) (AN b(A) = 2~) ==~ (C(A) = O) ==> (b(A) = 2~) for each set A c X [121, p. 143]. Below, on the basis of Definition 7.2.12, we give elementary proofs of some statements from [121], [122]. Theorem 7.2.17. The following condition is satisfied for a BS (x, ~ <~ 7-2,C): (1) b(A U B) = b(A) u b(B) for any sets A, B C X and, therefore, A c B implies b(A) c b(B). Moreover, for a BS (X, 7-1 <~(c) 7-2) and each set A c X , we also have: (2) b(A) = b(T2 cl A) = T2 cl b(A).
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VII. Applications of Bitopologies
If U(x) \ {x}) C A, then U(x) c A and, hence, x E ~-2cl(X \ A) so that x ~ 2- Fr A, which is impossible. Therefore
(U(x)\{x})
cX\A,
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and so x E A ~ X o .
In a similar manner one can show that x E X \ A implies x E (X \ A)~ ;3 Xo and hence (2-Fr A)~ N Xo c (A~ n Xo) u ( ( X \ A)2i N X o ) .
To prove the reverse inclusion let x E (A~ n X0) U ((X \ A)~ ;3 X0). Clearly, it suffices to consider the case x c A~ ;3 X0 because the proof of the case x c (X \ A)~ A X0 is similar. Since x E A~ implies that x E (~-2cl A)~, there exists a 2-domain U(x) such that U(x) ;3 ~-2el A - {x}. If U(x) N 2- Fr A - ;~, then U(x) N ~-2cl(X \ A) - z since, by virtue of the fact that U(x) is 2-connected, the converse of the latter equality means that U(x) A 2- Fr A # z. Therefore U(x) - {x} E X~, which is impossible. Hence U(x) A 2- Fr A - {x} so that x E (2- Fr A)~ r3 X0. This result completes the proof of the first equality. On the other hand since 2- Fr A is 2-closed, we have b(2- Fr A)C2- Fr A and thus b(2- Fr A) - 2- Fr A \ i(2- Fr A) -
= (T2clAn~-2cl(X\A))n ((X\i(A))N (X\i(X\A)))
-
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and so 7-2 cl(X \ A) \ i(X \ A) - b(X \ A). Thus 6(2- Fr A) - b(A) ~ b(X \ A).
D
C o r o l l a r y 7.2.21. Under the hypotheses of Theorem 7.2.20, for a BS (X, T1 <e(C)7-2), we have:
i(2- Fr F) - 2- Fr i(Y) - i(F), b(2- Fr F) - 2-Fr b(F) b(2- Fr b(F)) b(F) n 2- Fr F -
-
for any finely closed set F c X and i(2- Fr U) - i(X \ U), b(2- Fr U) - b(X \ U) ;3 2-Fr U
for any finely open set U c X.
Moreover~ y - b(Y) ~ for any finely closed set F c X.
2-Fr Y - b(2- Fr F)
Proof. First, let us show that U E 72 implies U C b(U) and, therefore, b(U) = T2 cl U. Indeed since X~ - 0, every 2-open set U c X is 2-dense in itself so that U C Ud. Hence U n b(V) - (V n U~ n Z o ) U (U n ~-~ cl V n ( X \ X o ) ) = ( u n x0) u ( u ~ ( x \ x0)) - u.
Therefore U c b(U) and T2 cl U = U U b(V)= b(U). Now, for any F E co T2, we have i(2- Fr F) - i(F) U i(X \ F) - i(F) U ((X \ F) \ b(X \ F)) - i(F)
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7.2. R e l a t i o n s w i t h P o t e n t i a l
Theory
347
and thus 2- Fr U = 2- Fr V U i(X \ U). From the latter equality it also follows that 2-FRY = 2-Fr U \ i(X \ U ) = 2 - F r U \ i(2-Fr U ) = b(2- Fr U) as 2 - F r V A i(X \ U) = ;g. (2) 2-ext A = e(7-1 cl A)<-->, 7-2 int(X \ A ) = X \ b(7-1 clA),,v-->,7-2clA=b(7-1 clA) and so the first equivalence is satisfied. Furthermore, 7-2 clA = b(7-1 clA) ,z----5,X \ 7-2 int(X \ A) = b(X \ 7-1 int(X \ A)) so that r2 int(X \ A) ---- T1
-
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7-1
7"1
int(X \ A))) -
int(X \ A)).
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-
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7-1
int(X \ A)) \ b(X \
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int(X \ A)U (7-1clA \ b(7-1cl A)).
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=
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int(X \ A)U (T1 C1A \ 7-2 cl A).
It is obvious that b(T1 el A) C T1 cl A, r2 cl A c and thus b(71 cl A) - T2 cl A.
T1
cl A E]
From our viewpoint, under the hypotheses of Theorem 7.2.20 and the assumption that the fine topology is compatible with the quasi topology, the equality 7-2 el A = b(7-1 cl A) is very important since, on the one hand, it shows that the set b(7-1 cl A) of all points z c X, at which T1 c l A is not thin, coincides with the set r2 cl A and the set e(rl cl A) of all points x E X, at which T 1 el A is thin, is a union of the I-open set 1-ext A and the (2, 1)-locally closed set nl(A) = T1 clA\b(T 1 c1A), and, on the other hand, it connects the family r l with the family r2. This connection also shows that every initially open set can be complemented to a regular finely open set by adding to it the (2, 1)-locally closed set of a capacity zero. Thus since every initially open set is equivalent to itself so that condition (1) of Remark 7.2.16 for the subfamily rl c QO(X) is always satisfied, the notion of compatibility makes it possible not only to bypass this trivial equivalence, but also to connect the subfamily QO(X) \ 7-1 with the fine topology r2 as well. Furthermore, as an important supplement to the results from [121, pp. 130132], let us introduce the notions of a quasi boundary and the corresponding quasi closure (interior) of any set.
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7.2. Relations with Potential Theory
353
U( qcl(A U B) n qcl (X \ (A U B))) qFr(A n/3) U (qFr A O qFr B) U qFr(A U/3) so that qFr A -< qFr(A N B) U (qFr A N qFr B) U qFr(A U B). Similarly, qFr B -< qFr(A n B) U (qFr A N qFr B) U qFr(A U B) and, therefore, qFr A U qFr B -< qFr(A U B) U qFr(A N B) U (qFr A N qFr B). Thus qFr A U qFr B ~ qFr(A U B) U qFr(A N B) U (qFr A n qFr B).
D
As proved in [62], in the classical potential theory for a subset A c X the set of points from X, at which A is not thin, can be included in the open set U in a manner such that U N A has any small capacity. The fundamental importance of this result was the fact that the thinness of the set A was required only at the points of the complement X \ A, which naturally implies a close connection of the result from [621 with Cartan's notion of a fine topology. As we already noted before Definition 7.2.23, a fine topology has been connected to a quasi topology by means of compatibility. On the other hand, along with formulating the quasi topological notions, it became possible not only to modify them, but also to obtain their various generalizations. D e f i n i t i o n 7.2.25. Let .4 = {As}ses be any family of subsets of a BS (X, 7"1 <~ 7"2,C). A set E c X is said to be C-near the family A from the outside (from the inside) if to every c > 0 there corresponds a set As E A such that AscEandC(E\As)<e(EcAs andC(As\E)<e). P r o p o s i t i o n 7.2.26. The following statements hold for a BS (X, 7"1 ~(I) 7.2,C)' (1) A set A c X is C-near a family A = {As}sos from the outside if and only if the set X \ A is C-near the family co A = {X \ As}ses from the inside. (2) If C is an outer capacity, then a set E c X is C-near co 7-1(7-1) from the outside (inside) if and only if E c QC(X) (E c QO(X)).
Proof. Assertion (1) is straightforward. (2) It suffices to note that by Lemma in 2.2 from [121], if C is an outer capacity, then a set E c QC(X)(E E QO(X)) ~
inf { C ( E \ F)" F c COT1, F C E } -- 0
( inf {C(U \ E)" U c 7"1, S C U} - 0 ) .
[-]
D e f i n i t i o n 7.2.27. A subset A of a BS (X, 7"1 < ~ 7"2, C ) is said to be almost o p e n (almost closed) if it is C-near 7"1(CO7"1) f r o m t h e o u t s i d e (inside). The family of all almost open (almost closed) subsets of (X, 7"1 < e 7"2,C) is denoted by A O ( X ) (,,4C(X)).
354
VII. Applications of Bitopologies
P r o p o s i t i o n 7.2.28. For a BS (X, T 1 < ~ T2, C), we have" E e A C ( X ) (E c N O ( X ) ) if and only if C(T1 cl E \ E) - C(T2 cl E \ E) - 0
( C ( E \ T1 int E) -- C ( E \ T2 int E) - 0 ).
Pro@ We will prove only the case of quasi closed sets since the case of quasi open sets is proved quite similarly. By Definitions 7.2.25 and 7.2.27, E E At(X) -- C { ( n f
,e---->,i n f { C ( F \ E ) "
F E coT1, E c F } -
\ E)" F c COT1, E C F } - C(T1 c l E \ E) - 0.
Since C is increasing and nonnegative, the inclusion T2 el E C T1 cl E implies that C(T2cl E \ E) - 0. D C o r o l l a r y 7.2.29. The statements below hold for a BS (X, T1 <~ T2, C)" (1) E C A t ( X ) (E C A O ( X ) ) if and only ifC(T1 clE) -- C(T2 clE) - C(E) (C(rl int E) - C(z2 int E) - C(E)). (2) If E E A C ( X ) (E c N O ( X ) ) , then E - F \ A where F E COT1 and (3)
C ( A ) -- 0 ( E - g U B wheI~e g c T1 and C ( B ) - 0). A function f " ( X , T 1 < ~ T2, C ) --+ (N,-~) is q~tasi c o n t i n u o u s (quasi
1.s.c., quasi u.s.c.) in the sense of [121, Definition 3.1], if either of the two conditions below holds" a) For each e > O, there exists a set E c A C ( X ) such that C(E) < e and the restriction flx\~l c l E (0/" f xX~clE) is 1-continuous (1-1.s.c., 1-u.s.c.). b) For each e > 0, there exists a set E ~ A O ( X ) such that C(T1 int E) < e (or C(T2 intE) < e) and the restriction f l x \ E is 1-continuous (1-1.s.c., 1-u.s.c.). Pro@ We will consider only the cases of almost closed sets since the proofs for almost open sets are similar. (1) Since C is increasing, it suffices to prove that E c ,AC(X) ~
C(T1 cl E) - C(E).
By (2) of Definition 7.2.11, C(T1 clE) _< C(T1 c l E \ E) + C(E) and, hence, Proposition 7,2,28 gives that E c A C ( X ) e--->, C(TI el E \ E) - 0 <----> C(T1 cl E) - C(E). (2) Obviously, E - T1 cl E \ (T1 cl/~ \ ET), where C(~-1 cl E \ E) - 0 since E c Ac(x). a) Suppose that for every e > 0 there is a set E c A C ( X ) such that C(E) < e. Then by (1), C(T1 clE) - C(T2 clE) - C(E) < e and if the r e s t r i c t i o n fix\w1 clE (or flx\~2clE)is 1-continuous (1-1.s.c., 1-u.s.c.), then by Definition 3.1 from [121], f is quasi continuous (quasi 1.s.c., quasi u.s.c.). D Here we would like to note that condition (3) in Corollary 7.2.29 compared to Definition 3.1 from [121], on the one hand, narrows the family 2 x and, on the other hand, in a) it diminishes the sets with respect to which the 1-continuity (1-1.s.continuity, 1-u.s.continuity) of f is demanded, and in b) it diminishes the sets of any small capacity.
7.2. Relations with Potential Theory
355 m
D e f i n i t i o n 7.2.30. A function f " (X, T1 <(p T2, C) --~ (]t~,~) is said to be C-near a class of functions 9c - {f~ 9 (X, T1 <~ T2, C) ~ ( R , ~ ) } s e s with respect to some property 7) if the assumption that Z) is valid for every function f~ c 9c implies that for each c > 0, there exists a set E c X such that C(E) < c and the restriction f X\E has the property 7). R e m a r k 7.2.31. From Definition 7.2.30 one immediately concludes that if a function f 9 (X, T1 <~ T2, C) --~ (R,~) is C-near a class of functions $r = { f s " (X, T1 ~ T2~ C) ~ (R, ~ ) } s c S with respect to the 1-continuity (1-1.s.continuity, l-u.s.continuity), then f is quasi continuous (quasi 1.s.c., quasi u.s.c.). T h e o r e m 7.2.32. For a BS (X, T1
Proof. As we will see below, it suffices to prove only the outside case. Let E C X be any set near a family A = {As}sos from the outside. Then for each c > 0, there exists a set As c A such that As c E and C ( E \ As) < c. Since X\(E\As)=(X\E)UAs
and X \ E c X c A s ,
we have
X,,: X\(E\A~) = XA~ IX\(E\A~)" Hence, if )a~ has a property 7), then by the condition, the restriction Xa~ IX\(E\A~) also has this property so that X,,: Ix\(~,\A~) has the property P. Thus X~ is C-near the class {X~a~}ses with respect to the property 7). For the inside case note that
x,... Ix\(a~\~) becauseX\(As\E)=(X\As)UEandX\As
= x~,,
Ix\(x~\.) cX\E.
D
C o r o l l a r y 7.2.33. The following statements hold for a BS (X, T1 < e T2, C): (1) If E C A C ( X ) U QC(X) (E c A O ( X ) u Q O ( X ) ) , then X , is quasi u.s.c. (quasi l.s. c) . (2) If C is an outer capacity, then AC(X) c QC(X) ( A 9 c QO(X)).
Pro@ (1) The case of quasi closed and quasi open sets is considered in [44, (a) of Proposition IV.2]. If E E AC(X) (E c A O ( X ) ) , then by Definition 7.2.27, E is C-near co 71 (T1) from the inside (outside). Since the class of characteristic functions X = {),,,}FeCO~l (X = {X,J}~Ze~l) has the 1-upper semicontinuity (I-lower semicontinuity) property which is hereditary with respect to restrictions on any subsets of X, by Theorem 7.2.32, ) ~ is C-near the class X = {Xi,,}FCco~l (X = {)r }ge-1). Thus, by Remark 7.2.31, X;~: is quasi u.s.c. (quasi 1.s.c.). (2) Let E c A d ( X ) (E E A O ( X ) ) . Then by (1), X~: is quasi u.s.c. (quasi 1.s.c.) and, therefore, it remains to use (b) of Proposition IV.2 from [44]. EJ
356
VII. Applications of Bitopologies
Each of the equivalent conditions below is the Choquet property, that is, the property (c~): co7-2 c QC(X) (or, equivalently, r2 c QO(X)) ,e---->,to any set A c X and each e > 0 there corresponds a 1-open set U such that 2-ext A c U and C(U n A) < e. Clearly, (c~) is the compatibility condition (4) of Definition 7.2.15 so that the Choquet property is always satisfied for a BS (X, rl
7.2.34.
The following conditions are equivalent for a BS
(X, 71 <~ 72, C): (1) cur2 c AC(X) (or, equivalently, r2 c N O ( X ) ) . (2) To any set A c X and each e > 0 there corresponds a 1-closed set F such that A c F and C(F n 2-ext A < e).
Proof. (1) ==> (2). Let A c X be any set. Then r2clA E A g ( X ) and, hence, for each e > 0 there exists a set F c co 7"1 such that r2 el A C F and C ( F \ 7.9 el A) < e. It is clear that C ( F n 2-ext A) = C ( F \ r2 cl A) < c. (2) ----5, (1). Let U E r2 be any set. Then there exists a 1-closed set F such that X \ U C F and C ( F a 2-ext(X \ U)) - C ( F n U) < c. Let V = X \ F. Then V E
7-1, V C
U and
c ( u \ v) = c ( u \ ( x \ F)) = C(U n F) < so that U c A O ( X ) .
V1
If any of the equivalent conditions in Proposition 7.2.34 is satisfied, then the corresponding capacity is said to be of the type (5). C o r o l l a r y 7.2.35. If for a BS (X, rl <~ r2, C) the corresponding capacity C is of the type (5), then the following statements hold: (1) F E COT2 (U C 7.2) implies F = E \ A, where E E co 7-1, C(A) ( u = v u B, V c C ( B ) = 0). (2) C is of the type (/3).
--=
0
Proof. (1) By (1) of Proposition 7.2.34, F c AC(X) (U c A O ( X ) ) and thus it remains to use (2) of Corollary 7.2.29. (2) Let f be any real-valued 2-u.s.c. function with the value 0 or 1. If A = {x c X : f ( x ) = 1}, then f = Xa and, therefore, A c cur2. Hence, by ( 1 ) o f Proposition 7.2.34, A E A(7(X) and it sumces to use (1) of Corollary 7.2.33. D
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358
VII. Applications of Bitopologies
f E 1-C(X, Y) implies that the set U(c) - f - l ( V ( c ) )
E 7"1 and A c U(c). Clearly,
U(e) is the required 1-open neighborhood of A. Thus inf {C2(V) 9 S(A) c V, V c ")'1} -- inf {C2(f(U))" A c U, U c 7"1} -= inf {CI(U)" A c U, U c 7"1} and, hence, C I ( A ) - inf { C I ( U ) 9 A C U, U E 7"1}(3) We will consider only a capacity of the type (c~) since for a capacity of the type (5) the proof is similar. Let A E 7"2. Then f c 2-O(X, Y) implies that f ( A ) E 72 and thus f ( A ) E QO(Y). Hence for every e > 0 there exists a set V C ")'1 s u c h that f ( A ) c U and C2(U \ f ( A ) ) < c. It is obvious that f - l ( u ) c 7"1 since f C 1-C(X, Y) and A c f - l ( U ) . Thus, CI(f-I(u)
\ A) - c 2 ( S ( S - I ( u ) \ A)) -
= C 2 ( f ( f - l ( u ) n (X \ A))) < C2(U n (Y \ f(A))) - C2(U \ S(A)) < c.
[2
C o r o l l a r y 7.2.37. Let f 9 (X, 7"1 < ~ 7"2) --+ (Y, ")'1 < ~ ~/2, C2), and let a capacity C1 on X be defined as C1 - C2 o f, where f c 1-C(X, Y). Then f-l(Ad(y)) I-I(Qc(y))
c AC(X), C QC(X),
f-I( AO(Y)) c AO(X), f - l ( Q o ( y ) ) c QO(X).
Proof. The proof is similar to the proof of (3) of Proposition 7.2.36.
D
D e f i n i t i o n 7.2.38. A BS (X, 7"1 < r 7"2, C ) is said to be (q, 2)-normal if for every quasi closed set A c X and its every quasi open neighborhood U(A), there exists a quasi open neighborhood V(A) such that C(7"2clV(A)\ A) <
C(U(A)) - C ( A ) . T h e o r e m 7.2.39. Let a BS (X, 7-1 <~ 7"2, C) be (q, 2)-normal and 1-second countable. If C is an outer capacity of the type (c~) (or, equivalently, of the type (/3)) CX:)
and C is sequentially order continuous from above in the sense that C( ~ AN) -n=l
inf{C(A~)}~__l for any decreasing sequence of sets {A~}~_ 1 c 2 x [121], then a
set A c X is quasi closed if and only if it is equivalent to some 2-closed set. Proof. First, let A E QC(X) be any set. By condition, we have C(A) = inf{C(U) : U E 7"1, A c U}, and since (X, 7"1 ~(P 7"2, C ) has a countable base of l-open sets, for each n = 1, ~c there exists a set U~ E 7"1 such that A C U~ and C(A) >_ C(U~) - 1/n. It is cleat that U~ E QO(X) since C is of the type (a) and 7"1 C 7"2. But (X, 7"1 <~ 7"2,C) is (q, 2)-normal and, therefore, for each n = 1, ~ there is a quasi open set V~ such that A C V~ and C(7"2 cl V~ \ A) <_ C(Un) - C(A) <_ 1/n. Let us consider the decreasing sequence {F~ }~__1 of finely closed sets, where F~ n
(7 7"2cl Vk and assume that F k=l
(x)
n Fn. It is obvious that C(A \ F) - O. On the n=l
7.3. A p p l i c a t i o n s
in G e n e r a l
Topology
and ...
359
other hand, oo
cx3
c(F \ A)- c( N F. \ A) - c( N (to \ A)) n=l
-- i n f { C ( F n \ A ) } n~= l
n=l
< inf{1/n} n~= l - 0 --
since C is sequentially order continuous from above. Thus
C(AAF) = C((A \ F) U (F \ A)) <_C(A \ F) + C(F \ A) = O and so A ~ F, where F r co 7-2. Conversely, let A ~ F, where F r co T2. Since C is of the type (a), we have F e QC(X) so that A is equivalent to some quasi closed set. Hence A e QC(X) because the notion of a quasi closed set depends only on the equivalence class of the set F. []
Corollary 7.2.40. Under the hypotheses of Theorem 7.2.39, (X, T1
(A N b(A)= ;g) --> (C(A)= O) ~
a BS
(b(A)= 25)
hold for each A c X. Proof. The proof immediately follows from Remark 7.2.16.
D
Note that Theorem 7.2.39 remains valid for a capacity of the type (5). To summarize the section, for the further development of the fine potential theory, as well as of "quasi topological" concepts, we consider essential expanding the range of questions, which will often study combined notions, analogical to the compatibility of the fine topology T2 on a TS (X, 7-1) determined by the cone ~, with the quasi topology QO(X), determined by the capacity C, and the (q, 2)-normality.
7.3. Applications in General Topology and Theory of Ordered Topological S p a c e s The naturality of relations between principal topological and bitopological conceps is a decisive factor largely underlying this study. The concept of a maximal connected space, that is to say a connected TS (X, ~-1) such that no strictly finer topology T2 on X is connected, was introduced by J. P. Thomas in [2511. Later, such spaces have been studied, for example, in [128][130]. Obviously, this problem is closely connected with establishing conditions, under which a connected TS (X, rl) is not maximal connected, that is, when there exists a finer connected topology ~-2 on X. In this direction the important results were obtained in particular in [173] and [234]. The elementary examples of nonmaximal connected spaces are (R, w~) and (R,w) since we c w c d and the density topology d is connected [173, 6.9h]. Below, based on our results from the previous chapters, we continue the research started in [173], where the following most important topological lemma
360
VII. Applications of Bitopologies
is proved: if 7.2 is a fine topology on a strong Baire connected TS (X, 7"1), r2 n c o r 2 C 1-Ga(X) and r e c l U 9 1-D(1-FrU) for each set U 9 rl, then (X, re) is connected [173, 5.1, p. 142]. It is not difficult to see that if in the t o p o l o g i c a l l e m m a we replace the inclusion re O cot2 c 1- Ga(X) by a weaker inclusion 72 O COrl C 1-Gs(X), then the obtained BS (X, rl, re) becomes p-connected. Note here, that the N-relation is a stronger requirement on (X, 7"1 < 7"2) than the inclusion re O core c 1-Ga(X) together with the condition re elU 9 1-D(1-FrU) for each set U 9 rl. Indeed, By (3) of Theorem 2.4.7, t i N t 2 .: ".. (rl, re) has the 2-rl-insertion property and implies that (rl, re) has the 2-(1- Ga(X))insertion property. Hence, if A 9 r2 5 core is any set, then there is a set B 9 1-Ga(X) such that r2intA-
A c B C reclA-
A
so that A = B and, therefore, r2 n cot2 C 1-Ga(X); also, by (3) of Corollary 2.3.12, we have t i N t 2 ~ rl clU = re clU for each set U E re, and thus r2 el U 9 1-79(1-Fr U) for each set U r rl. As the following elementary example illustrates, the inclusion re N core C 1-Ga(X) does not imply the condition that (rl, re) has the 2-(1-Ga(X))-insertion property. E x a m p l e r.3.1. Let X {a,b,c,d}, T 1 - - { ~ , { b } , { a , c , d } , X } - 1- ~5(X) and r2 - {z, {a}, {b}, {a, b}, {a, d}, {a, b, d}, {a, c, d}, X}. Then r2 n co r2 -- rl = 1-Ga(X) and if A - {c, d}, then r2 int A - z , r2 cl A - A and there does not exist B E 1 - 6 a ( X ) - r~ such that ~ c B c {c, d}. Another simple confirmation of the above reasoning is the second part of the condition (1) of Corollary 2.3.13. If instead of being an S-1-BrS, a BS (X, T1 ~ 7"2) is an S-(1, 2)-BrS (which is a stronger requirement by (1) of Theorem 4.3.2), then the condition T2 O co r2 C I-Gs(X) can be eliminated and the obtained result, as well as other theorems of this kind proved below is fully placed within the framework of (2) of the scheme given on the third page. T h e o r e m 7.3.2. Let (X, rl < re) be a 1-connected and S-(1, 2)-BrS such that (F,r~ < r~) is a (1,2)-BrS for each set F c cot1 \ {~}. If r e c l U E 1-D(1-Fr U) for each set U E rl, then (X, rl, r2) is 2-connected.
Proof. Assume X = A U B , Denote
where A , B r ( r 2 O c o r 2 ) \ { O }
and A N B
= 0.
M = rl clA O rl clB = 1 - F r A = 1- F r B . Since (X, 7.1 < 7.2) is 1-connected, M # ~. Let us prove that one of the sets A, B cannot be 1-dense in M. By Definition 4.3.1 the set M is of (1,2)-second category; hence, if A o M, B O M r 1-D(M) and, consequently, A O M, B O M r 1 - D ( M ) N 2 - ~ a ( M ) , then by Proposition 1.1.26, A N M , B N M e (1,2)-Catgli(M). On the other hand (M, 7.~ < r~)is a (1, 2)-BrS, M = ( A N M ) U ( B N M ) and since
AnM-M\(BnM),
BnM-M\(AnM),
7.3. Applications in General Topology and ...
361
A N M, B n M c 1-7)(M) N 2-Gs(M), by T h e o r e m 4.1.10, A N M, B N M E (1,2)-Catgi(M). This contradiction shows t h a t one of the sets A, B (say A) is not l-dense in M. Therefore, there exists a set V c 771 \ {2~} such t h a t
V N M r 2~, (V N M ) N (A C~M ) = 2~ and, hence, V n M C B n M C B. We have
X \ M = 771 int A U 771int B,
V cqd = V Cq ( ( d A M ) U
( d f q X \ M ) ) = V N771intd
and V [~ 771 C1A - V N 771 C1 ((A \ 771 int A) U T1 int A) -- V ('1 (771 cl(A \ 771 int A) U 771 cl 771 int A).
But so t h a t V n T1 cl A = V N T1 cl 77~int A. Therefore, V A M = V n 1 - P r A = V N771 clAN771cl(X \ A) - V n 771cl 771int A n 771cl(X \ 771int A) = V n 1-Pr 771int A. Finally, by the condition, 772cl 771int A c 1-l)(1- Pr 771int A) and hence r 772c177~i n t A n 1- Pr 771i n t A n V c 772c l A n (M n V) c A N B
= ~,
which is a contradiction.
[]
C o r o l l a r y 7.3.3. is a 1-connected S - ( 1 , 2 ) - B r S 772cl U c 1-1P(1-Fr U) for each set U c T1, then (X, T 1 < 7"2) is 2-connected.
and
Proof. The proof follows directly from (1) of T h e o r e m 4.3.3.
[Z]
We are now going to study sufficient conditions for a l-connected BS 7"2) t o be p-connected and for a p-connected BS (X, T1 ~ T2) t o be 2-connected. ( X , T1 ~
T h e o r e m 7.3.4. If (X, 771 < 772) is 1-connected, 772 A co771 C 1 - G s ( X ) and 772cl U E 1-1P(1-Fr U) for each set U c 771, then (X, 77-1 ~ 772) i8 p-connected.
Proof. Suppose X = A U B, where A C (771 f)c0772) \ { ~ } ,
B c (772 n c 0 7 7 1 ) \ { ~ }
and A N B = 2~.
(2<3
Then B -
n n=l
u~ , where U~ c rx for each n -
1 , oc. Besides, if X r U r {U~}~176 n=l,
then X \ U c X \ B = A. Since (X, 771,772) is 1-connected, we have U g co771, 1-Fr U :/: 2~ -r 1-Fr A and hence
/ M - 1-F U u 1-FrA - 1- Fr(U U a ) u 1- F (u n a ) u But 1-Fr(U U A ) = Moreover,
(1- Fr V
1-
A).
1 - F r X = ~ = 1 - F r U n 1 - F r A and hence M = 1-Fr(U N A).
A N M = A N ( 1 - F r U U 1 - F r A ) = A N 1 - F r U = AN771 c l U N ( X \ U )
= 1 - F r U E co771
362
VII. Applications of Bitopologies
and so A O M ~ 1-/)(M). Therefore, there is a set V 9 T 1 \ { 2~ } such that V n M 9 7{ \ {~} in the BsS (M, 7~ < 7~), ( A n M ) U (V N M ) - ~ and M - ( A n M ) U (B N M ) imply that V N M C B N M C B. By condition T2 cl(U N A) C 1-/)(1- Fr(U n A)) since U N A 9 T1 and, hence, r 72 cl(U n A) n 1- Fr(U n A) N V = T2 cl(U H A ) n ( M N V ) C T 2 c l A N B
- ANB
-- ~ ,
which is a contradiction. D As in the case of N-relation, in a BS (X, 7"1 < "/-2) the coupling of topologies is a stronger relation than the inclusion T~ n coT1 C 1-G~(X) together with T2clU 9 1-T?(1-FrU) for each set U c Vl. Indeed, by (3) of Theorem 2.4.5, T1CT2 <---> (T1, w2) has the (2, 1)-Tl-insertion property and implies that (71, ~-2) has the (2, 1)-l-g;~(X)-insertion property. Hence, if A 9 ~-2 n coT~, then there is a set B c 1-6~(X) such that A -
and so A -
~-2 i n t A c
B c
T1 c1
A
-
A
B. Moreover, by (3) of Corollary 2.2.7, we have TICT2 ~
T1 CI U -- 7-2 el U
for each set U 9 ~-1 and, thus, ~-2 clU 9 1-TP(1-FrU) for each set U 9 T1. In general, the converse is not true as the following elementary example shows. E x a m p l e 7.3.5. Let X - {a, b, c, d}, T1 -- { ~ , {a}, {a, b}, {c, d}, {a~ c~ d}, X } 1- Ga(X) and 72 - {~, {a}, {c}, {a, b}, {a, c}, {c, d}, {a, c, d}, {a, b, c}, X}. Then ~-2 Aco71 -- { { a , b } , { c , d } } C T1 - 1- Ga(X) and if A - {b}, then T2intA -- ~, 7-1 cl A - {b}, but {b} 9 ~-1 - 1-Ga(X). T h e o r e m 7.3.6. Let (X, 7-1 < "I-2) be a p - c o n n e c t e d BS, T2 N coT2 C 1-65(X) and E E ~-2 n coT2 implies that there is a set U c {Un},~__l, U r X , where (x)
E --
n Vn, Vn C 7-1 f o r each n n--1 1-D((1, 2)- r v) ach V
1, oc such that ~-1 c l E c U.
(X,
I f ~-2 cl V E
2
Proof. Assume X = A U B, where A, B c (~-2N co ~-2) \ {~ } and A n B = 2~. Then oo
by the condition there is a set U E {Un}n~__l, where E -
n U~ E 1-Gh(X) such n=l
that ~-1 cl B c U. It is obvious that X\UcA,
(1,2)-FrA=TlclA\Ar
as (X, ~-1, T2) is p-connected and 1 - F r U = (1, 2)-Fr U = T1 c 1 U \ U ~ as (X, T1, T2)is 1-connected. Hence, by ( 1 0 ) o f Theorem 1.3.2, =/= M = (1, 2 ) - F r A U (1, 2)- Fr U = (1, 2)- Fr(A U U) U
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364
VII. Applications of Bitopologies
x0 E 7-~'int T(7-1 c l F ) is any point, then Xo E T(7-1 e l F ) and since 7-1' and 7-~' coincide at xo, it follows that xo c 7-2 cl F = F. Hence xo is a point of tangency of topologies 7-~ and 7-~, and so xo E T(F). Since xo c 7-~'int T(7-1 c l F ) - V is an arbitrary point, we have V c T(F). But F c 7-1 c l F , where V c 7-~' and V C T(F) C F. Therefore V C 7-~ and thus 7-~ int T(F) r ;g. E] The next theorem, unlike the t o p o l o g i c a l l e m m a and Theorem 7.3.6, gives sufficient conditions under which a connected TS is not maximal connected without using the strong Baire and Baire-like properties. T h e o r e m 7.3.9. Let (X, 7-1 < 7-2) be a 1-connected BS, let 7-1 be WS-tangent to 7-2 and let T(F) be 2-connected for each set F c co7-1 \ {~}. If 7-2clU c 1-7P(1-Fr U) for each set U c 7-1, then (X, 7-1,7-2) is 2-connected.
Proof. Assume X = A U B, where A, B c (7-2 n co 7-2) \ {2~} and A N B = 2~. Then the set M = 7-1 c l A n 7.1 c l B = I - F r A = I - F r B / : 2~ since (X, 7-1,7-2)is 1-connected. Moreover, V' - 7-~ i n t T ( M ) # 2~ in the BsS (M, 7-~ < ~-~) since 7-1 is WS-tangent to 7-2. Furthermore, T ( M ) is 2-connected implies t h a t T ( M ) c A or T ( M ) c B. W i t h o u t loss of generality, let T ( M ) c A. Then V' N ( M n B) = ~, and, therefore, there exists a set V c 7-1 \ {2~} such t h a t
VNM
= V', V' C A N M
c A
as M - (A N M) U (B n M). It is clear that X \ M
-
T1 int A U 7-1 int B
and V N B - V N ((B N M ) U (B N (X \ M ) ) - V N 7-1 int B.
Moreover, V n7-1 c l B - V N7-1 cl ((B \ 7-1 intB)UT-1 i n t B ) - V n7-1 C17-1 i n t B
since V n 7-1 cl(B \ 7.1 int B) C 7.1 cl (V n (B \ 7.1 int B)) - 2~.
Hence V N M - V n 1 - F r B - V AT-1 c l B AT-1 cl(X \ B) = V n 7.1 cl 7.1 int B N 7"1 cl(X \ 7.1 int B) - V n 1- Fr 7.1 int B. But T1 int B c T1 and by the condition, 7.2 cl 7.1 int B c 1-:D(1- Fr 7.1 int B) so t h a t r
7-2 c1 T1 int B n 1-Fr T 1 int B N V - r2 el T1 int B N ( M n V) c
c ~2 cl B n ( M n V) -- B n ( M n V) -- B n V ' - - ~ ,
which is a contradiction.
D
C o r o l l a r y 7.3.10. Let (X, T 1 < 7"2) be a 1-connected BS, let T1 be strongly tangent to ~-~ a~d l~t T(F) b~ 2 - ~ o ~ t ~ d fo~ ~ach ~ t F c r ~1 \ { ~ } . ~f ~2 cl U c 1-~P(1-Fr U) for each set U E 7.1, then (X, 7.1,7.2) is 2-connected.
7.3. Applications in General Topology and ...
365
Proof. The proof is obvious since T1 is strongly tangent to T2 implies that 7-1 is WS-tangent to 7-2. D C o r o l l a r y 7.3.11. Let ( X , T1 < 7-2) be a 1-connected BS, let T1 be WStangent to T2 and let T(F) be p -connected for each set F c co 7"1 \ {O}. I f 7"2 cl U C 1-79(1-Fr U) for each set U c T1, then (X, T1, T2) is p-connected.
Proof. Following [204, Theorem E], if T(F) is p-connected, T ( F ) C A U B = X , where A E (T1A COT2) \ {~},
/~ E (7-2 ACOT1) \ { ~ }
and A n B = z , then T(F) c A or T(F) c B. The rest of the proof is similar to the proof of Theorem 7.3.9. [2] We continue our discussion by considering the well-known notion of relative compactness [21], [22], which completely fits in our scheme of bitopological applications mentioned in Section 0.1. D e f i n i t i o n 7.3.12. For a BS (X, T1,T2) the topology Ti is compact with respect to the topology Tj if for every/-open covering/A of X and for each point x E X there is a j-neighborhood of x covered by a finite subfamily of/A [22]. Now from Definition 0.1.20 we readily obtain Theorem
7.3.13. The following implications hold for a BS (X, T1,7-2):
(X, T1,T2) is (j,i)-lqc ----->, (X, TI,T2)iS (j,i)-Slc ~ (X, T1,T2)is (j, i)- Blc(r (X, TI,T2) is (i, j)- RRlc) ==~ Ti is compact with respect to Tj.
Proof. By virtue of the implications after Definition 0.1.20, it suffices to prove only the last implication. Let/A = {Us }s~s be a n y / - o p e n covering of X and let z E X be an arbitrary point. Then by (4) of Definition 0.1.20, there exists a j-neighborhood U(z) which is/-compact. Clearly,/A = {Us}sEs is also a n / - o p e n covering of U(x) and hence 6/ has a finite subfamily/A' = {Us~}~=l such that n U(x) c U usk. Thus ri is compact with respect to rj. D k=l It is clear that using the relative compactness argument and the appropriate notions from [14], the bitopological assertion of Theorem 7.3.13 gives many interesting results from [21] and [22]. Now we want to establish conditions under which the reverse of the last implication in Theorem 7.3.13 is true. D e f i n i t i o n 7.3.14. A subset A of a BS (X, T1, T2) is said to be/-extendable if for every/-open covering/At of A, there is a n / - o p e n covering 6 / o f X such that /At C/A and U • A = 2~ for each U c/A \/At. In that case/A is said to b e / - e x t e n d e d from A. It is obvious that every /-closed set F, 2~ r F r X, is /-extendable, while there are n o / - d e n s e / - e x t e n d a b l e subsets of X.
366
VII. Applications of Bitopologies
If for a BS (X, 7-1,7-2) the topology 7-i is compact with respect to the topology 7-j, then for an arbitrary point x c X and every /-open covering g/ of X the j-neighborhood of x, mentioned in Definition 7.3.12, is denoted by Uu(x). D e f i n i t i o n 7.3.15. For a BS (X, 7"1,7"2) the topology 7-i is strongly compact with respect to the topology 7-j if 7-i is compact with respect to 7-j and for every point x c X there is a j-open/-extendable neighborhood U(x) such that U(x) c Uu(x) for every/-open covering b / o f X , / - e x t e n d e d from U(x). T h e o r e m 7.3.16. If for a BS (X, T1,T2) the topology 7-~ is strongly compact with respect to the topology 7-j, then (X, 7-1,7-2) is (i, j)-RRlc, or equivalently, (j, i)- Blc.
Proof. Let x E X be any point and U(x) be a n / - e x t e n d a b l e j-neighborhood of x whose existence is guaranteed by the hypotheses. We will prove that U(x) is the required/-compact j-neighborhood of x. If L/' is a n y / - o p e n covering of U(x), then there exists a n / - o p e n covering b/of X , / - e x t e n d e d from U(x) and containing b/' as a subfamily. Since 7-i is strongly compact with respect to 7-j, there is a j-neighborhood Uu(x) such that U(x) c Uu(x) and Uu(x) is covered by a finite subfamily g#' c b/. It is obvious that U(x) is also covered by b/" and b/" c b/' since U E b / \ b/' implies that U • U(x) = 2~. [3 C o r o l l a r y 7.3.17. Let for any point x of a BS (X, 7-1,7-2), there exists an i-extendable j-neighborhood U(x) such that V(x) c Uu(x) for every i-open covering bl of X , i-extended from U(x). Then (X, r1,7-2) is an ( i , j ) - R R l c (,: ', (j, i)-Blc) if and only if 7-i is compact with respect to 7-j.
Proof. The proof follows directly from Theorems 7.3.13 and 7.3.16.
D
C o r o l l a r y 7.3.18. If for a p-T2 BS (X, T1,7"2) the topology 7-i is strongly compact with respect to the topology 7-j, then 7-i c 7-j.
Proof. It suffices to use Proposition 10 from [30] since by Theorem 7.3.16, (X, T1, 7-2) is (j, i)- Blc. D The arguments used at the beginning of this section are also essential for discussing cotopologies. Indeed, quoting [1], "cotopology can be roughly defined as the part of topology in which cospaces of a space X are used to study the properties of X". In the context of the above and the case (2) on page 3 it would righful to say that bitopology can be roughly defined as the part of topology in which BS's can be also used to study the properties of the corresponding TS's. D e f i n i t i o n 7.3.19. Let (X, r2) be a TS. A topology rl on X is called a cotopology of r2 and (X, rl) is a cospace of (X, r2) if the following conditions are satisfied:
(1) 7-1 is weaker than 7-2. (2) For each point x E X and any 2-closed neighborhood M ( x ) there is a I-closed neighborhood N(x) such that N ( x ) C M(x) [1, the fundamental definition].
7.3. Applications in General Topology and ...
367
It is not difficult to verify that if a BS (X, 71,72) is 2-regular, then the above condition (2) can be replaced by the following equivalent condition: (2') Each point x E X has a 2-neighborhood base whose elements are 1-closed. We have thus obtained the following simple, but important result. T h e o r e m 7.3.20. For a BS (X, 7-1 < T2) the topology 7"1 is a cotopology of the regular topology 72 if and only if (X, 71,72) is (2, 1)-regular. C o r o l l a r y 7.3.21. If for a BS ( X , T 1 < 7"2) t h e dimension ( 2 , 1 ) - i n d X is finite, then T 1 is a eotopology of the regular topology 72. Pro@ The proof follows directly from (1) of Proposition 3.1.4.
89
C o r o l l a r y 7.3.22. If (X, 71,72) is a p-regular BS and (71,72) has the (1, 2)-72-insertion property, then 71 is a cotopology of the regular topology 72. Proof. Indeed, if (71,72) has the (1, 2)-72-insertion property, then by (1) and (3) of Theorem 2.4.5, 72 is coupled to 71. Hence, by Corollary 2.2.9, 71 c 72 since (X, T1,7"2) is (1,2)-regular. Thus it remains to use Theorem 7.3.20. [3
The next two corollaries connect the notion of a cotopology with LTS's. C o r o l l a r y 7.3.23. Let T 1 and 72 be two locally convex topologies on a linear space X . Then 72 is subordinate to T 1 i f and only if 71 is a cotopology of the regular topology 72. Pro@ Let the locally convex topology T2 be subordinate to the locally convex topology 71 on the linear space X. Clearly, T1 and 72 are both regular. By Theorem 7.1.14, (X, 71,72) is p-regular and (71,72) has the (1, 2)-72-insertion property. Hence, by Corollary 7.3.22, the topology T1 is a cotopology of the regular topology 72. On the other hand, let 71 and 72 be two locally convex topologies on the linear space X and 7-1 be a cotopology of the (always regular) topology 72. Since (X, 71,72)is always I-regular and T1 C T2, (X, T1,T2)is also (1,2)-regular. Further since (X, T1,72) is 2-regular, by Theorem 7.3.20, it is (2,1)-regular. Moreover, by Definition 2.2.1, T1 C 7"2 implies 72C71 and, following (1) and (3) of Theorem 2.4.5, (71,72) has the (1, 2)-72-insertion property. Thus it remains to apply Theorem 7.1.14. [-1
C o r o l l a r y 7.3.24. Let T 1 and 72 be two locally convex topologies on a linear space X . Then the following conditions are equivalent: (1) 72 is subordinate to 71. (2) (71,72) satisfies the closed neighborhoods condition in the sense of [67]. (3) 7-1 i s a cotopology of 72. Proof. It suffices to recall that (71,72) satisfies the closed neighborhoods condition if T1 C 7-2 and 72 has a base of zero element consisting of l-closed convex sets and to use condition 6.5 from [150, p. 47]. [3
C o r o l l a r y 7.3.25. Let (X, 71 < 72) be a (2, 1)-regular BS. Then the following statements hold:
368
VII. Applications of Bitopologies
(1) If (X, T1, 7-2) i8 2 - T 2 and 2-locally compact, then (X, 7"1,7"2) i8 1-compact. (2) If (X, 7"1,7"2) is I-compact, then (X, 7"1,7"2) is a 2-BrS and thus an A-(2, 1)- BrS, a 2-WBrS, and a (2, 1)-WBrS. Proof. Following Theorem 7.3.20, 7"1 is a cotopology of the regular topology 7"2. Hence (1) follows from (a) of Example 3.3 in [2]. Now, taking into account (1) of Theorem 4.4.28, (2) follows from Theorem 2.9 in [133] since (X, 7"1 < 7"2) is 2-quasi regular. V] The notion of a bitopology can also be used with success to characterize continuous mappings of TS's and the bitopological solution of one of Everett and Ulam's problem. Let us consider a function f : X ~ Y of a set X to a set Y and let C A = f - l ( f ( A ) ) for any set A c 2 x. It is not difficult to ascertain that if C is the closure operator and if f is a bijection, then C generates the discrete topology on X, while if f is constant, then C generates the antidiscrete topology on X. If I is any interior operator, then the operators I and C are conjugate over a subfamily .4 c 2 X provided that IA = CIA, CA = ICA for any set A c .4. T h e o r e m 7.3.26. Let f : X ~ in the above sense, I be any interior I. Then f : (X, 7") ~ (Y, 7), where 7 if and only if I and C are conjugate
Y be any function, C be a closure operator operator and 7" be the topology, generated by is an arbitrary topology on Y, is continuous over the family { / - I ( u ) : U E 7}.
Proof. First, let f : (X, 7") ~ (Y, 7) be continuous. Then U 9 7 implies f - l ( v ) and so I f - l ( u ) = f - l ( U ) for every set U 9 "7- Therefore Clf-l(v)
C f-
= cf-l(u)
= If-l(v),
ICI-I(u)
= If-l(v)
9 7"
= cf-l(u).
Conversely, i f / a n d C are conjugate over { f - l ( U ) : V e 7}, t h e n I C f - l ( U ) 1 ( U ) , where
C f-I(u) Thus I f - l ( g )
=/-l(g)
= f-l(f(f-l(U)))
=
- /-I(u).
so that f is continuous.
[3
Given a TS (X, 7"), let 7-/(X, 7") be the class of all homeomorphisms of (X, 7") onto itself. In 1948 C. J. Everett and S. M. Ulam [108], [253], posed the following problem: when and how can a new topology 7 be constructed on X such that 7-/(X, 7") = 7-/(X, 7)? Among the (partial or complete) answers to this problem we will use the result of Yu-Lee Lee [263], according to which on a locally compact space (X, 7") there exists a coarse topology 7 such that 7-/(X, 7") = 7-/(X, 7). This result is based on the following simple, but important lemmas. L e m m a 7.3.27. Let (X, T) be a TS and let P ( V ) be the topological property which certain subsets V of X have. If ~ = { V : P ( V ) } is a topology on X , then 7-/(X, 7") C 7-/(X, 7) [263, Lemma 1]. L e m m a 7.3.28. Let 7. and "7 be two topologies for X such that U E 7. ~< U U V c ~/for all nonempty V in 7. Then H ( X , 0/) c ~ ( Z , 7.) [263, Lemma 2].
7.3. Applications in General Topology and ...
369
As we will see below, certain bitopological conditions are imposed on the considered BS, which ensures a satisfactory solution of the problem. It is worth noting here t h a t the requirement t h a t
7-1 \ { ~ } --772 \ { ~ } (-] 2-Catgli(X) is, in general, stronger than the one t h a t (X, 771 < 772) be a 2-WBrS. Indeed, 771 \ {2;~} -- 7-2 \ { ~ } n 2 - C a t g l i ( X ) implies (X, 771 < 772) is a 2-WBrS, but by R e m a r k 4.4.22, the BS's (R,a~ < s s) and (R,a~ < s r ) are 2-WBrS's for which
Cd \ {2~} C 8 \ {2~} -- (8 \ {2~}) r-] 2-Catgli(]~ ) A u.; \ {2~} C "r \ {2~} -= ( r \ {2~})r] 2-datgii(R) since (R, s) and (R, T) are BrS's. Theorem
7.3.29. If (X, T1 < 7-2) is a BS such that
7-1 \ { ~ } = (7-2 \ { ~ } ) ( - / 2 - ~ a t g i i ( X )
and for each point x E X there is a neighborhood U(x) E 772 \ 1-D(X), then ~-~(X, 71) -- ~-~(X,"/-2) and (X, 71 < 7-2) is a 2 - W g r S .
P r o 4 Let P(V) mean t h a t V c 72 and V c 2-Catgll(X)(,,(-----~, V is of 2-CatglI). Then the family {V : V = ~ or P ( V ) } is exactly the topology 7-1. Hence, by L e m m a 7.3.27, ~ ( X , 772) c 7-[(X, 771). If (X, 771 < 772) is a 2-BrS, then T1 = 7-2. Therefore we may assume t h a t (X, 771 < 772) is not a 2-BrS. Now let U r 772 and V c 771 \ { ~ } . Then U U V c 772 and by (4) of Theorem 1.1.24, U U V C 2-Catgli ( X ) ( ~
U U V is of 2-Catg I I ) .
Therefore U U V E 771. Furthermore, let U c772 and x r U \ 772int U. By the condition, there is a 2-open neighborhood V(x) such that 7-1 cl V ( x ) ~ X . It is evident t h a t x c 772 c1(771 cl V(x)~U) since the contrary means that there is a neighborhood W ( x ) c 772 such that
W(x) n (9-1 cl V(x) \ U) = ~, t h a t is, W(x) n V(x) C U and so x c 772int U. Let
E : X \ (T 1 C1V(X) \ U) -- ( X \ T1 c1V(x)) U U. Then x E E, but Ec772 since x E 772c1(771 cl V(x) \ U). Therefore for U~772, there is V = X \ 71 cl V(x) E 71 such t h a t U U V c 772 and hence U U V g 771 since 771 c 772. Thus L e m m a 7.3.28 gives that H ( X , 771) c 7-/(X, 772) and, consequently, =
The rest follows from Definition 4.4.18 since T1 \ { ~ } C 2-Catgii(X). Corollary
7.3.30.
If (X, 7-1
<
D
7"2) is a 2-WBrS such that
((7-2 \ 7-1)U {X}) n 2-Catgii(X) - {X}
and for each point x c X there is a neighborhood U(x) E 772 \ 1-D(X), then ~-~(X, T1) -- ~-l~(X, 7"2).
Proof. The proof follows directly from Definition 4.4.18.
E]
370
VII. Applications of Bitopologies
The importance of the theory of BS's also depends on its natural relation to the theory of ordered TS's (briefly, OTS's) whose systematic study is given in [191]. An OTS is a set X having a topology ~- and a partial order _<. We will denote such a space by (X, T, <_). In [191] L. Nachbin generalizes different topological concepts to the topology-order case in a manner such that one obtains the classical definitions for the discrete order (i.e., when the partial order is defined as x _< y ~ x - y for x, y E X). Obviously, for the natural OTS (R, CO,_<) the binary relation _< is the natural order on R. Following [191], each subset A c X determines in a unique fashion an increasing set i(A) (a decreasing set d(A)) which is the smallest one among increasing (decreasing) sets, containing A. A set A C X is said to be convex if A - i(A) N d(A). The smallest closed increasing set I(A) (the smallest closed decreasing set D(A)), containing A, is also defined in a unique fashion. Hence the set I(A) n D(A) is the smallest closed convex set containing A. In [53] M. J. Canfell draw a parallel between the theories of OTS's and BS's in the following manner: to each OTS (X, ~- _<) there corresponds the BS (X, T1, T2), where ~-1 - {U c r " U - i(U)} and 7-2 - {V c T" V - d(V)} are respectively the upper and the lower topology with respect to the partial order <_ in terms of [53]. On the other hand, Canfell leaves open the question in which cases a BS can be treated as an OTS, that is, if (X, r l , 7-2) is a BS and ~- - sUp(T1, r2), and which conditions are required for the existence of a partial order <_ on X such that T1 coincides with the upper topology and ~-2 coincides with the lower topology of (X, r, _<). By virtue of [208, Proposition 10] the answer to this question is as follows: if (X, T1, ~-2) is a BS and ~- - sup(rl, ~-2) is compact, then there exists a partial order <_ with a closed graph on X such that T1 and ~-2 are respectively the upper and the lower topology of (X, T, _<) if and only if
(1) (X, T1,72) is l-T0 (or 2-T0). (2) (X, T1, ";-2) is p-regular. It is clear that for the natural OTS (R, co, _<) the corresponding BS in the sense of Canfell is the natural BS (R, COl,CO2). The above duality seems essential for discussing different mutually beneficial relations between these two theories. For example, we can give a fact which directly follows from this duality, (3) of Definition 0.1.18 and [208, p. 521]: (X, T1, T2) is p-extremally disconnected ~ (X, T, _<) is extremally order disconnected in T1 @ (X, 7-, <) is extremally order disconnected in 7-2. In the context of the above-said we will consider the axioms of separation of OTS's, taking into account the axioms of separation of the corresponding BS's. We will also introduce and investigate dimension functions and Baire-like properties for OTS's. This program will fill the gap of [191]. Note that owing to the duality, the results constructed here are of a quite simple character. D e f i n i t i o n 7.3.31. An OTS (X, r, _<) is said to be upper (lower) Tl-ordered if for each pair of elements x, y c X, x ~ y, there exists a neighborhood U ( y ) -
7.3. Applications in General Topology and ...
371
d(U(y)) ( U ( x ) - i ( U ( x ) ) ) such that x E U ( y ) ( y g U ( x ) ) , and (X,T, <) is said to be Tl-ordered if it is both upper and lower Tl-ordered [1781. The concept of Tl-order coincides with those of semicontinuous partial order [256] and semiclosed partial order [191]. D e f i n i t i o n 7.3.32. An OTS (X, r, _<) is said to be T2-ordered if for each pair of elements x, y c X, x :~ y, there exist disjoint neighborhoods U(x) - i(U(/)) and U(y) - d(U(y)) [1781. This concept coincides with those of continuous partial order and closed partial order in [256] and [191], respectively. D e f i n i t i o n 7.3.33. An OTS (X, T, <) is said to be upper (lower) regularly ordered if for each set F - I(F) C X (F - D((F) c X) and each element x - g F , there exist disjoint neighborhoods U ( F ) - i(U(F)) and U ( x ) - d(U(x)) ( U ( F ) - d(U(F)) and ( U ( x ) - i(U(x))). (X, 7-, _<) is said to be regularly ordered if it is both upper and lower regularly ordered. (X, r, _<) is upper (lower) T3-ordered if (X, T, <_) is both upper (lower) Tl-ordered and upper (lower) regularly ordered. An OTS (X, r, _<) is T3-ordered if it is both Tl-ordered and regularly ordered [178]. D e f i n i t i o n 7.3.34. An OTS (X, r, <) is said to be normally ordered if for each pair of disjoint sets F1 - I(F1), F2 - D(F2), there exist disjoint neighborhoods U(F1) - i(U(F1)) and U(F2) - d(U(F2)); and (X,r, _<) is said to be perfectly normally ordered if it is normally ordered and each increasing (decreasing) closed oo
set F is Gs-increasing (Gs-decreasing), that is, F -
~ U~, where Un are open rz--1
increasing (decreasing) sets for each n - 1, oc [191], [178], [59]. (X, r, _<) is said to be T4-ordered if it is both Tl-ordered and normally ordered [178]. The work [178] also contains the definitions of the strong Tk-order separation axioms for k - 1, 4 obtained from the Tk-order separation axioms by using the term "open neighborhood" instead of "neighborhood". We denote the Tk-order (strong Tk-order) separation axioms by Tk(0) (STk(o)) for k - 1, 4. It is obvious that the following implications hold: ST4(o) ~ T4(0)
~
ST3(o) ~ T3(0)
ST2(o) ~
-->, T2(o)
ST~(o)
==~ TI(0).
The inverse implications are not valid in general. Of the basic separation axioms only the axiom T31(0) will be recalled. D e f i n i t i o n 7.3.35. An OTS ( X , r , <) is said to be completely regularly ordered if the following conditions are satisfied: (1) For each point x E X and its every neighborhood U ( x ) , there are two continuous real-valued functions f and g on X, where f is order preserving
372
VII. Applications of Bitopologies and g is order reversing such that 0 <_ f _< 1, 0 _< g <_ 1, f ( x ) - 1 - g(x) and inf(/(y), g(Y)) - 0 if y c X \ U(x). (2) If x, y c X, x ;~ y, then there exists an order preserving continuous real-valued function f such that f ( x ) > f ( y ) [191].
(X, T, _<) is said to be T31(0)-ordered if it is both Tl-ordered and completely regularly ordered. Let (X, T, _<) be an OTS. Then by [177] a set A c X is called a decreasing (increasing) zero set in (X, "1-,<_) if there is an order preserving (order reversing) continuous function f : (X, T, _<) ~ (R,~, _<) such that A = {x c X : f ( x ) < 0}. The family of all decreasing (increasing) zero sets of (X, T, <_) is denoted by A1 (A2). If f : (X, T, _<) -+ (R, co, _<) is continuous and order preserving (order reversing), then by Proposition 1.1 from [177], A c A~ (A c A2) defines a continuous order preserving (order reversing) function f : (X, 7, _<) ~ (R, ~, _<) such that A = {x c X : f ( x ) = O, f >_ 0}. Clearly, in both cases A is closed in (X, T, _<) and, therefore, A = D(A) for A E A1 and A = I(A) for A c A2. R e m a r k 7.3.36. Following T. McCallion [177], the family AI(A2) is a base of closed sets of a topology 7-A1 (7-r on X. Moreover, such topologies are characteristic of completely regularly ordered TS's [177, Theorem 1.2]. D e f i n i t i o n 7.3.37. Let (X,r,_<) be an OTS and z~1 (z~2) be a family of decreasing (increasing) closed sets of X. Then Z1 U Z2 is called a normally ordered subbase for (X, r, _<) if the following conditions are satisfied:
(1)
z~1 (z~2) is a base for closed sets of the topology rZl (7z2) on X such that sup(Tzl, Tz2) - 7- and (TZl, ~-z2) is an order defining pair, that is, x e cl{y} . - . x _< y y c cl{x}. (2) If x C X , F c COTzl (F C co7z2) and x ~ F , then there is a set A c Z2 (A E Z1) such that x E A and A A F - ~. (3) If A1 c Z1, A2 c Z2 and A1 N A2 - ~, then there are sets All E Z1, A~ E Z2 such that A~ c_ A~, A2 c_ A~, A1 c~ A~ - ~ - A~ N A2 and d~ U d~ - X [177, p. 466].
We conclude the dicussion of the axioms of separation of OTS's by investigating their relations to the axioms of separation of the corresponding BS's, where the correspondence is of one of the two types mentioned above. T h e o r e m 7.3.38. Let (X, T, <_) be an OTS, (X, 71, T2) and (X, ~-~[1,7j42) be the corresponding BS's in the sense of Canfell and McCallion, respectively. Then the following statements are valid:
If (X~ TI~ 7"2) is R -p-T1, then (X~ T~ ~) i8 ~TI(o). If (X, T1, ~-2) is p-T2, then (X, T, <_) is ST2(o). (X, T1,7"2) i8 p-regular if and only if (X, T, <_) is strong regularly ordered. If (X, T, <_) is completely regularly ordered, then (X, 7-~[1,7A2) is p-completely regular. (5) (X, 71, T2) is p-normal if and only if (X, T, <) is strong normally ordered.
(1) (2) (3) (4)
7.3. Applications in General Topology and . . .
373
(6) (X, 7-1,~-2) is p-perfectly normal if and only if (X, T, <) is perfectly normally ordered.
Proof. (1)-(3), (5) and (6) are immediate consequences of the corresponding definitions. Hence it remains to prove only (4). We begin by assuming that (X, T, <) is completely regularly ordered. Then by Theorem 1.3 from [177], A1 u A2 is a normally ordered subbase for (X, T, <). Therefore on account of Definition 0.2.4, A - {A1,A2} is a p-normal base for the BS (X, TAI,rA~) and it remains to use Theorem 0.2.5. [5] P r o p o s i t i o n 7.3.39. If (X, T ~ I ~ TA2 ) is R -p T 1 and ~- -- sup(TA1, rA~ ), then (X, r, <_), where <_ is the discrete order on X , is completely regularly ordered.
Proof. Since (X, TA1, ~-A~) is R - p T1 (i.e., d-T1) and _< is the discrete order, we have X E T A 1 el{y} ~ y ~ TA2 el{x} ~ x -- y ~ x < y. Hence by (1) of Definition 7.3.37, the bitopology (TA1, ~-A2) is an order defining pair and so, by Theorem 1.2 from [127], (X, T, <) is completely regularly ordered. [U C o r o l l a r y 7.3.40. Let (X, T, <_) be an OTS, where <_ is the discrete order on X and let (X, TA1, TA2) be R-p-T1. If T - SLlp(TA1 , 7"A2 ), then (X, TA1 , TA2 ) is P-T3~-~ if and only if (X, T, <_) is T389 )
Proof. The proof follows directly from (1) and (4) of Theorem 7.3.38, and Proposition 7.3.39. C] Let us define new operators which are also natural and necessary for further investigations. Every subset A of (X, w, <_) determines uniquely the largest decreasing set dl(A) (largest increasing set il(A)), contained in A and the largest open decreasing set DI(A) (largest open increasing set II(A)), contained in A. It is obvious that dl - X \ i(X \ A)
( il (A) - X \ d(X \ A)),
DI(A)-X\I(X\A)
(II(A)-X\D(X\A)).
Moreover, if A is closed and decreasing (increasing), then A - d(A) - D(A) - dl(A)
(A - i(A) - I(A) - il(A)),
and if A is open and decreasing (increasing), then A - dl(A) - DI(A) - d(A)
(A - il(A) - II(A) - i ( A ) ) .
If (X, T, __) is an OTS and (X, rl, T2) is the corresponding BS in the sense of Canfell, then it is clear that D(A)
-
7-1 cl A
I(A) - r2 cl A
( 11 ( A ) -
7-1 int
A ),
( D1 (A) - r2 int A )
and thus A is closed and convex in (X, T, _<) (i.e., A - D(A)A I(A)) ~ A is p-closed in (X, T1, ~-2) (i.e., A - ~-1cl A N ~-2cl A). Due to these relations, we have D(A U B) - D(A) U D(B), I ( d U B) - I(A)U I(B),
D(D(A)) - D(A), I(I(A)) - I(A),
.
~
d:)
IA
~ ."~'o
o
~
= 8
d)
r
~ =
~ ~
,-~~
0
~
~
~.~;b.~
~
~-" ;S="~'a 9 - ~ . ~
tm
=
-
~ ~,
~
~"
oq
,
~
D
i
c ~
,
D
i
II
#~
c
C
'
,
m.--"
,~
~
, 9
=~ ~.
fb
i
~
~
N
N
""~
~ o
D~
~
e
~
~
9
~
il
o~
~
m
~
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~
~
~
~
o
~.
<-
7.3. A p p l i c a t i o n s in G e n e r a l T o p o l o g y a n d . . .
375
In the sequel we will consider only order subspaces of an OTS (X, r, _<). D e f i n i t i o n 7.3.44. An OTS (X, r, <_) is said to be hereditarily strong normally ordered if its any order subspace is strong normally ordered. To characterize such spaces we will make use of the following notions. D e f i n i t i o n 7.3.45. If A and B are subsets of an OTS (X, r, _<), then we write A <' B to indicate that (A N I ( B ) ) U (D(A)N B ) = ~. In addition, if A < ' B and there exist disjoint neighborhoods U ( A ) = DI(U(A)) and U ( B ) = II(U(B)), then we write A <<' B.
Proposition
7.3.46.
The following conditions are satisfied for an OTS
(x,~,<_). (1) (X,T, <_) is hereditarily strong normally ordered if and only if A <' B implies A <<' B for every pair of subsets A c X , B c X . (2) If (X, T, <_) is strong regularly ordered and A c X , then (A, T', <_') is also strong regularly ordered. (3) If (X, T, <_) is strong normally ordered and A = D ( A ) N I(A) C X , then (A, 7-', <') is also strong normally ordered.
Proof. Using (3) and (5) of Theorem 7.3.38, and Remark 7.3.43, the proof immediately follows from Theorem 0.2.2, Corollary 3.2.6 and the obvious fact that the p-regularity is hereditary with respect to any subsets. D C o r o l l a r y 7.3.47. Every hereditarily strong normally ordered space is strong normally ordered.
Proof. The proof follows immediately from (1) of Proposition 7.3.46. Definition 7.3.48. integer. We say that
D
Let (X, r, _<) be an OTS and n denote a nonnegative
(1)1 (u, 1)-indX - - 1 ,e--->, X - 2~. (2)1 (u, l ) - i n d X <_ n if for every point x E X and any neighborhood U(x) 11(U(x)), there exists a neighborhood V ( x ) - I 1 (V(x)) such that I ( V ( x ) ) c U(X) and (u, 1)-ind((/, u)- Fr V(x)) <_ n - 1. (3)1 (u, 1)- ind X - n if (u, 1)- ind X <_ n and the inequality (u, 1)- ind X <_ n - 1 does not hold. (4)1 (u, 1)-indX - oc if the inequality (u, 1)-indX <_ n does not hold for any n. Similarly, (1)2 (1, u ) - i n d X - - 1 ,e--->, X - 2~. (2)2 (1, u ) - i n d X _< n if for every point x E X and any neighborhood U(x) DI(U(x)), there exists a neighborhood V(x) - DI(V(x)) such that D(V(x)) C U(X) and (1, u)- ind((u, 1)- Fr V(x)) <_ n - 1. (3)2 (1, u ) - i n d X - n if (1, u ) - i n d X _< n and the inequality (1, u ) - i n d X _< n - 1 does not hold.
376
VII. Applications of Bitopologies
(4)2 (1, u ) - i n d X - oc if the inequality (1, u ) - i n d X <_ n does not hold for any n. O - i n d X <_ n ~ ((u, 1 ) - i n d X <_ n A (1, u ) - i n d X <_ n). Definition 7.3.49. integer. We say t h a t
Let (X,T, <) be an OTS and n denote a nonnegative
(1)1 ( u , / ) - I n d X - - 1 ~X-~. (2)1 (u, 1)-Ind X <_ n if for every set F - I ( F ) and any neighborhood U ( F ) I I ( U ( F ) ) , there exists a neighborhood V ( F ) - I I ( V ( F ) ) s u c h t h a t I(V(F)) c U(F)
and (u, 1)- Ind((/, u)- Fr V ( F ) ) <_ n - 1.
(3)1 (u, 1)- Ind X - n if (u, 1)- Ind X <_ n and the inequality (u, 1)- Ind X _< n - 1 does not hold. (4)1 ( u , / ) - I n d X - oc if the inequality ( u , / ) - I n d X <_ n does not hold for any n. Similarly, (1)2 ( / , u ) - I n d X - - 1 ~ X - 2~. (2)2 (1, u)- Ind X <_ n if for every set F - D ( F ) and any neighborhood U ( F ) D I ( U ( F ) ) , there exists a neighborhood V ( F ) - D I ( V ( F ) ) s u c h t h a t D(V(F)) c U(F)
and (1, u ) - I n d ( ( u , 1)-Fr V ( F ) ) <_ n -
1.
(2)3 ( / , u ) - I n d X - n if (1, u ) - I n d X <_ n and the inequality ( / , u ) - I n d X _< n - 1 does not hold. (2)4 (1, u ) - I n d X - oc if the inequality (1, u ) - I n d X <_ n does not hold for any n. O-IndX
<_ n .z--->. ((u, 1 ) - I n d X <_ n A ( l , u ) - I n d X
Definition 7.3.50. integer. We say t h a t
<_ n).
Let (X, T, <) be an OTS and n denote a nonnegative
(1)1 (u, 1 ) - d i m X - - 1 ~ X - ~g. (2)1 ( u , / ) - d i m X < n if for all families of sets {Us - I1 (Us) 9 s - 1, k} and {Fs - I(Fs) 9 s - 1, k}, where Fs c Us for each s - 1, k, there exists a family of sets {Vs - Ii(Vs) " s - 1, k} such t h a t Fs c Vs c Us for each s - 1, k and o r d { ( l , u ) - F r V s " s - 1, k} < n. (3)1 (u, 1 ) - d i m X - n if (u, 1 ) - d i m X < n and the inequality (u, 1 ) - d i m X < n - 1 does not hold. (4)1 ( u , / ) - d i m X = o c if the inequality (u, l ) - d i m X < n does not hold for any n. Similarly, (1)2 (1, u ) - d i m X - - 1 ,z---5, X - ~. (2)2 (1, u ) - d i m X < n if for all families of sets {Us - DI(Us) " s - 1, k} and {Fs - D(Fs) 9 s - 1, k}, where Fs c Us for each s - 1, k, there exists a family of sets {V~ - DI(Vs) " s - 1, k} such t h a t Fs c V~ c Us for each s - l , k a n d o r d { ( u , 1)-FrVs" s - l , k } < _ n .
7.3. A p p l i c a t i o n s in G e n e r a l T o p o l o g y a n d . . .
377
(3)2 (1, u ) - d i m X - n if (1, u ) - d i m X _< n and the inequality (1, u ) - d i m X _< n - 1 does not hold. (4)2 (1, u)-dim X = o c if the inequality (1, u)-dim X<_n does not hold for any n. O-dim X _< n ~
((u,/)-dim X _< n A (1, u)-dim X _< n).
R e m a r k 7.3.51. If (X, T, <_) is an OTS and (X, BS in the sense of Canfell, then it is obvious that
T1,7"2)
is the corresponding
(u, 1)-ind(X, 7, <_) = (1, 2)-ind(X, T1, T2), (1, u)-ind(X, T, _<) = (2, 1)-ind(X, T1, ~-2), O-ind(X, 7, _<) = p - i n d ( X , ~-1,T2);
(u, 1)- Ind(X, r, <_) = (1, 2)- Ind(X, T1, T2), (1, u)- Ind(X, T, _<) = (2, 1)-Ind(X, T1, T2), O- Ind(X, T, <_) = p - I n d ( X , 71, T2); (u,/)-dim (X, T, _<) = ( 1, 2)-dim (X, T1, T2), (1, u)-dim (X, 7, _<) = (2, 1 )-dim (X, T1, ~-2), O-dim (X, T, <_) = p -dim (X, T1, ~-2). Thus, by Remark 3.3.7, for the OTS (R,w, _<), the values of nine ordered dimension functions coincide with integer 1. All results presented below and concerning the ordered dimension functions are immediate corollaries of the results discussed in Chapter III in conjunction with Remark 7.3.43. One can easily verify that if two OTS's (X, T, <_) and (Y, 7, <-') are both homeomorphic and order isomorphic in the sense of [177], then O- ind X - O- ind Y, O- Ind X - O- Ind Y and O-dim X - O-dim Y. For the sake of simplicity all results are formulated for the dimension functions O- ind X, O- Ind X and O-dim X. T h e o r e m 7.3.52. The following conditions are satisfied f o r an OTS (X, w, <)" (1) I f O - i n d X
( ( u , l ) - I n d X o r ( l , u ) - I n d X ) i s f i n i t e , then ( X , T, <_)is strong regularly ordered (strong normally ordered). (2) /f (A, T',
quence of subsets in X such that X -
[.J X m , X,~ - D(X,~) - I(Xm) m=l
and O- Ind X,~ - 0 (or, equivalently, O-dim Xm - 0) f o r each m - 1, oc, then O- Ind X - 0 (or, equivalently, O-dim X - 0). (N3
It is clear that (4) remains valid if X,~ -
[.J F ~ ,
where F ~
-
D(F~) -
n=l
I ( F ~ ) and O - I n d F ~ - 0 (or, equivalently, O-dim F ~ - 0) f o r each m - 1, oc, n-
l, oc.
T h e o r e m 7.3.53. The s t a t e m e n t s below are satisfied f o r a hereditarily strong normally OTS (X, T, <)"
378
VII. Applications of Bitopologies
(1) If M0, M 1 , . . . , Mn are any subsets of X , then O-ind(M0 U M 1 U . . . U Mn) _< O-ind M0 + O-ind M1 + . . . + O-ind Mn + n n
and thus if X -
U Mk, where O - i n d M k - O for each k - O,n, then k=0
O - i n d X < n. (2)
f o r each m - 1, ec, Xm+l C Xm, X1 - X and ~ = 1 X m - 0 . If O-Ind(Xm \ Xm+l) < n for each m - 1, oc, then O - I n d X < n. L e t X m - D1 ( X m ) - I I ( X m )
Therefore if A D(A) - I(A), then O - I n d A <_ n and O - I n d ( X \ A) < n imply that O- Ind X <_ n.
(3) Let {Dm}~=l be a disjoint sequence of sets covering X such that Fs = [.J D,~ - D(Fs) - I(F~) for each s - 1, ec. If O-Ind Dm <_ n for each m~8
m - 1, oc, then O- Ind X _< n. (4) If X - P U Q, where O- Ind P _< n, O- Ind Q <_ o, then O- Ind X < n + 1. n
Thus, if X -
U x m , where O- Ind X m <_ 0 (or, equivalently, O-dim Xm < 0) rn=0
for each m - O, n, then O-Ind X _< n.
It is clear that many other results from Chapter III, which we did not include in Theorems 7.3.52 and 7.3.53 because the objective was to demonstrate the duality in the sense of Canfell for dimension functions, remain valid. D e f i n i t i o n 7.3.54. A subset A of an OTS (X, T, <) is u-nowhere dense (also called u-rare) in X if II(I(A)) - ~, and A is/-nowhere dense (also called/-rare) in X if DI(D(A)) - Z . The family of all u-nowhere dense (/-nowhere dense) subsets of X is denoted by u-A/'~P(X) (1-A/'lP(X)). P r o p o s i t i o n 7.3.55. Let (X, T, <) be an OTS. Then the following conditions are satisfied: (1) A 9 u-A/'~P(X) ((A 9 1-A/'~(X)) ~ I(A) c D(X \ I(A) (D(A) c I(X \ D(A)) for any subset A c X . (2) If (Y, ~-', <_') is an order subspace and A c Y, then A 9 u-A/'~)(Y) ((A 9 1-A/'~)(Y)) ~ I'(A) c D(Y \ I'(A) (D'(A) c I(Y \ D'(A)). Proof. The proof immediately follows from Propositions 1.1.2 and 1.5.3, Remark 7.3.43 and using the duality in the sense of Canfell. D
D e f i n i t i o n 7.3.56. A subset A of an OTS (X,T, <) is of u-first (/-first) category (also called u-meager (/-meager), u-exhaustible (/-exhaustible)) in X if oo
A-
U An, where An 9 u-Af~P(X) (An 9 1-A/'~)(X)) for every n -
1, oc and A is
n=l
of u-second (/-second) category (also called u-nonmeager (l-nonmeager), u-inexhaustible (/-inexhaustible)) in X if it is not of u-first (/-first) category in X.
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7.3. Applications in General Topology and ...
381
D e f i n i t i o n 7.3.61. An almost upper (lower) Baire space or, briefly, an A-u- BrS (A4- BrS) is an OTS (X, T, <_) such that U-
II(U) # ~
(U-
DI(U)
~ ;2~) ==~ U E ~t-Catgii(X)
(U E 1-Catgii(X)).
It is obvious that if any U r T1 \ {/~f} (U E 7"2 \ {l~}) is an order subspace of the OTS (X, r, _<), then (X, r, _<) is an u-BrS (1-BrS) implies that (X, r, 5) is an A-u-BrS (A-l-BrS) so that if (X, r, _<) is an I-space, then (X, r, <_) is an u-BrS (1- BrS) implies that (X, r, <_) is an A-u- BrS (A-l- BrS). T h e o r e m 7.3.62. The conditions below are equivalent for an OTS (X, r, <_)" (1) ( X , ' r , < ) is an A-u-BrS (A-1-BrS). (2) If {Un}n~ is any countable family of subsets of X , where Un - DI(Un) 6 (2O 1-I)(X) (Un - I I ( U n ) 6 u-E)(X)) for each n 1, oc, then n U~ c n--1 oo
1-D(X) ( N
n=l
(3) A c u - C a t g x ( X ) ( A
r 1-Catgi(X))~
X \ A C 1-Z)(X) (X \ A r
(4) If {F~}~__I is any countable family of subsets of X , where F n - I ( F ~ ) E oo u-Bd(X) (F~ - D(F~) c l-gd(X)) for each n - 1, oc, then U F~ c n=l oo
( U
rt=l
c
The proof of this theorem repeats that of Theorem 4.1.4, taking into account (3) of Theorem 7.3.57 and L e m m a 7.3.63. The following equivalences are correct for an OTS (X, ~-, <_)" A 6 1-~)(X) (A 6 u-~)(X)) ~ (every set U - I I ( U ) 7s ( U - D I ( U ) 5r 2~) intersects A ) ~ X \ A c u-Bd(X) (X \ A r 1-Bd(X)). Finally, to add to the results of [65] we would like to state some properties of BS's associated with digraphs. To this end, we have to recall briefly the notions and notations from this work. A digraph (i.e., directed graph) is an ordered pair (X, F), where X is a set and F a binary relation on X, that is, F c_ X x X. All graphs are 1-graphs (simple graphs), which means that for every pair of vertices x, y, there exists at most one arc with the initial endpoint x and the terminal endpoint y. A circuit is a chain (Xl,... ,x~) such that no arc appears twice in the sequence, two endpoints of the chain coincide and for all i < n the terminal endpoint of xi is the initial endpoint of xi+ 1. Let vr and T+ be respectively the left and the right associated topology with the digraph (X, F). Then Tr and T+ have the property of a completely additive closure, that is, the intersection of any system of TF--open sets is ~-r-open and the intersection of any system of T+-open sets is T+-open I8], [171].
382
VII. Applications of Bitopologies
The topology ~-+ is called the dual topology to ~-b- and vice versa so that for every set A c X the set T+ cl A is the least Tr--open set containing A, and Tr cl A is the least T+-open set containing A. Following [196], if (X, F) is an ordered set, then the considered topologies T+ and Tr coincide with the left and right topologies on X with all the corresponding properties. Note that the statements below are also valid for the digraphs considered in [64]. The next two simple propositions are closely related to the above reasoning. To retain the notation of this book, (X, T1, ~-2) will denote the BS (X, r +, Tr). P r o p o s i t i o n 7.3.64. Let (X,F) be a digraph. (X, T1, ~-2) has the following properties: (1)
A ~ 7-~ .z--->A E COTj ~
Then the associated BS
(j,i)-FrA-
25
and, therefore, ~
-
co ~j -
(~,
j)-oz)(x)
= (j, ~ ) - s c ( x )
- (~,j)-soz)(x)
= (~,j)-cc(x)
(2)
- (j, ~ ) - c z ) ( x )
< p-cl(x)
-
-(j, 1-z:c(x)
- (~, j ) - s o ( x )
-
~)-scz)(x)-
2-z:c(x).
( i , j ) - A f D ( X ) - {z}, that is, ( i , j ) - S D ( X )
- 2 X \ {2~}
and, thus, (i, j ) - C a t g I ( X ) - {2~},
that is, (i, j ) - C a t g l i ( X ) - 2 X \ {~}.
(3) /f U E v~ \ {2~}, then V is of ( i , j ) - C a t g I I
or, equivalently,
U c (i,j)-~atgxx(X)
and, hence,
(X, T1,T2) is an ( i , j ) - B r S
or, equivalently,
(X, T1,T2) is an A ~ i , j ) - B r S .
(4) ~-i A j - D ( X ) - {X}, that is, coT i N j - B d ( X ) - {~}. (5) ~-i is neither S-related to Tj nor coupled to Tj, but nevertheless 7-i is near Tj.
(6) (X, T1,7"2) i8 Trot p-co~tTtected. (7) (X, 7-1~7"2) is p-extremally disconnected. (8) p - i n d X - p - I n d X - p - d i m X - 0. (9) For each point x c X and its any i-open neighborhood U(x), there exists a function / 9 (X, T1,T2) ~ ({0, 1},w") such that f is (i,j)-l.u.s.c. and f (x) -- 1, f ( X \ U(x) - O. (10) ( X , T1, T2 ) is M S -p-Ro . (11) (X, TI, T2) is hereditarily p-normal. (12) If (X, F) does not contain any circuit, then (X, TI, T2) is d-homeomorphic to a BsS of the bitopological cube I J:~(X). Pro@ The proof of (1) is an immediate consequence of determination of the topologies T1 and ~-2 in conjunction with the definitions of the corresponding families of sets from Sections 0.1, 1.2 and 1.3.
7.3. Applications in General Topology and . . .
(2)
383
By Definition 1.1.1 and (1), we have A c (i,j)-HI)(X)
<---> 7-~int 7-j clA = 2~ <---> A = ~.
Therefore, Definition 1.1.21 implies t h a t A E (i,j)-8~P(X)
~
7-~int 7-j clA r 2~ ~
A r ~.
The rest is obvious. The proof of (3) follows directly from (2) and Definition 4.1.5. (4) Let A c X, A r X be any subset. If A c 7-~ n j - ~ P ( X ) , then 7-j cl A = X, which is impossible since by (1), A c 7-i ,z----5,A c co 7-j so that 7-j cl A = A r X. (5) Let A c X be any set. If we assume t h a t 7-1S7-2, then by (3) of Theorem 2.1.5, T] int A c 7-] cl 7-2 int A A 7-2 int A c 7-2 cl 7-] int A. But by (1), T1 cl 7-2 int A = 7-2 int A A 7-2 cl 7-] int A = 7-1 int A
so that r ] i n t A c r2 int A A 7-2 int A c 7-] int A, t h a t is r l i n t A = r2 int A. Since A c X is an arbitrary set, we obtain that 7-] --- 7-2, which is impossible. Now, if we suppose that 7-1C7-2, then by (2) of Theorem 2.2.6, 7-1 cl 7-] int A c 7-2 cl 7-1 int A
for any subset A c X. Hence (1) gives t h a t 7-1 cl 7-1 int A c 7-1 int A,
t h a t is, 7-1 int A = T1 cl 7-1 int A.
Therefore 7-1 = co 7-1, which is impossible. Finally, if A c X is an arbitrary set, then it is clear t h a t 7-1 cl 7-2 int A = 7-2 int A c 7-2 cl 7-2 int A
and hence it remains to use (2) of Theorem 2.3.7. (6) Following (c) of Theorem A from [204], (X, 7-1,7-2) is p-connected in the sense of (1) of Definition 0.1.18 if and only if X contains no nonempty proper subset which is both 1-open and 2-closed (hence none which is 1-closed and 2-open). Therefore it remains to use (1). Assertion (7) follows directly from (3) of Definition 0.1.18 and (1). (8) By virtue of (1), we have p - i n d X = p - I n d X = 0 and, hence, by Corollary 3.3.5, p - d i m X = 0. (9) By Theorem 6 from [33], this condition is equivalent of the equality p - i n d X = 0 so t h a t it remains to use (8). (10) By Theorem 4.3 in [65], (X, T1,7"2) is p-completely regular and, therefore, it remains to apply Theorem 1 from [205]. ( 1 1 ) B y C o r o l l a r y 4.2 in [65], (X~ T1,7-2) is p-completely normal and it remains to apply Theorem 0.2.2. (12) By Theorem 3.2 in [65], (X, T1,72) is W - p - T 2 and it remains to use the theorem from [31] since (X, 7-1,7-2) is p-completely regular. Q
384
VII. Applications of Bitopologies
P r o p o s i t i o n 7.3.65. Let (X,F) be a digraph. If (X, q , r2) is the associated BS, then the conditions (1)-(7) below are equivalent: (1) (X, F) does not contain any circuit. (2) (X, T1,7"2) i8 MN -p -To. (3) (X, 7"1,7"2) is MN -p -T1. (4) (X, 7"1,7"2) i8 ~ - p - T 1. (5) (X, T1, "/-2) i8 W - p - Z 2 .
(6) (x,
w-p- 3.
(7) (X, 7-1,7-2) is W - p - T 3 1 . Moreover, if anyone of the equivalent conditions (1)-(7) is satisfied, then (8) (X, rl, r2) is weakly totally disconnected. (9) The p-components (i.e., maximal p-connected subsets) of X are its points which are p-closed. Pro@ By virtue of Theorems 3.2 and 4.3 from [65], (1) ~ (2) ~ (5) and (X, T1, T2) is p-completely regular. Hence for the equivalences (1) ~ (4) ,' '(6) <--5, (7) it remains to use Remark 0.1.9. Finally, following the implications before Proposition 0.1.7, we have the implications:
(X, T1,72) is S-p-T1 ---5, (X, T1, ~-2) is MN-p-T1 ~
(X, T1, ~-2) is MN-p-T0,
and since (2) <---5, (4), we obtain (2)<---5, (3). (8) Let (X, T1,7-2) be MN-p-T0. Since by (8) of Proposition 7.3.64, p - i n d X = 0, Theorem 2 from [216] gives that the BS (X, rl, T2) is weakly totally disconnected. (9) By Theorem 2.4 in [248], the components of a weakly totally disconnected BS are its points. Therefore, by Theorem F from [204] for each point x c X, we have x = T1 el{x} c~ T2 cl{x} so that {x} 9 p - C I ( X ) . [:]
385
Bibliography
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19] [2o]
Aarts J. M., de Groot J., McDowell R. H., Cotopology for metric spaces. Duke Math. J. 37(1970), 291-295. Aarts J. M., de Groot J., McDowell R. H., Cocompactness. Nederl. Akad. Wetensh. Proc. Ser. A 74 Indag. Math. 32(1970), 9-21. Aarts J. M., Mrgevid M., Pairwise complete regularity as a separation axiom. J. Austral. Math. Soc. ser. A 48(1990), No. 2, 235-245. Aarts J. M., Mr~evid M., A bitopological view on cocompact extensions. Topology Appl. 42(1991), No. 1, 1-16. Adnadjevid D., Ordered spaces and bitopology. Glasnik Mat. Ser. I l l 10(30)(1975), No. 2, 337-340. Adnadjevid D., Some weak forms of continuity in bitopological spaces. Mat. Vesnik 38(1986), No. 1, 17-23. Agashe P., Levine N., Adjacent topologies. J. Math. Tokushima Univ. 7(1973), 21-35. Ahlborn T. J., Bhargava T. N., On topological spaces associated with digraphs. Acta Math. Acad. Sci. Hungar. 19(1968), 47-52. Alegre C., Ferrer J., Gregori V., On pairwise Baire bitopological spaces. Publ. Math. Debrecen 55(1999), No. 1-2, 3-15. Alexandrov P. S., On the notion of space in topology. (Russian) Uspekhi Mat. Nauk 2(1947), No. 1, 5-57. Alexandrov P. S., Pasynkov B. A., Introduction to dimension theory. (Russian) Nauka, Moscow, 1973. Amih~esei C., Sur les bi-espaces extr6mement discontinus. An. ~ti. Univ. "A1. L Cuza" Ia~i Sect. I a Mat. (N.S.) 19(1973), 19-25. Arhangel'skiY A. V., On hereditary properties. General Topology and Appl. 3(1973), 39-46. Arhangel'ski~ A. V., Ponomarev V. I., Fundamentals of General Topology Through Problems and Exercises. (Russian) Nauka, Moscow, 1974. Arkhangel'skiY A. V., Bokalo B. M., Tangency of topologies and tangential properties of topological spaces. (Russian) Trudy Moskov. Mat. Obshch. 54(1992), 160-185, 278-279; English transl.: Trans. Moscow Math. Soc., 1993, 139-163. Arya S. P., Singhal A., A note on pairwise D1 spaces. Glasnik Mat. Ser. I l l 14(34)(1979), No. 1, 147-150. Arya S. P., Nour T. M., Pairwise locally semiconnected spaces. Indian J. Pure Appl. Math. 18(1987), No. 5, 402-409. Arya S. P., Nour T. M., Quasi semi-components in bitopological spaces. Indian J. Pure Appl. Math. 21(1990), No. 7, 649-652. Arya S. P., Bhamini M. P., Minimal pairwise P-spaces. Glasnik Mat. Ser. III 23(43)(1988), No. 2, 365-372. Baire R., Sur les fonctions de variable r6elles. Thdse. Ann. di Math. 3(1899).
386
[21] [22] [23] [24] [25] [26] [27] [28]
[29] [30] [31] [32] [33] [34] [35]
[36] [37] [38] [39] [40] [41]
Balogh Z., Relative compactness and recent common generalizations of metric and locally compact spaces. Fund. Math. 100(1978), No. 2, 165-177. Balogh Z., Metrization theorems concerning relative compactness. General Topology and Appl. 10(1979), No. 2, 107-119. Banaschewski B., Briimmer G. C. L., Strong zero-dimensionality of biframes and bispaces. Quaestiones Math. 13(1970), No. 3-4, 273-290. Banaschewski B., Briimmer G. C. L., Hardie K. A., Biframes and bispaces. Proceedings of the Symposium on Categorical Algebra and Topology (Cape Town, 1981). Quaestiones Math. 6(1983), No. 1-3, 13-25. Banaschewski B., Brummer G. C. L. Stably continuous frames. Math. Proc. Cambridge Philos. Soc. 104 (1988), No. 1, 7-19. Berstein B. A., Simplification of the set of four postulates for Boolean algebras in terms of rejection. Bull. Amer. Math. Soc. 39(1933), 783787. Berthiaume G., On quasi-uniformities in hyperspaces. Proc. Amer. Math. Soc. 66(1977), No. 2, 335-343. Birkhoff G., Mathematics and psychology. SIAM Review 11(1969), 429469. Birkhoff G., Lattice Theory. (Russian) Nauka, Moscow, 1984. Birsan T., Compacit~ dans les espaces bitopologiques. An. ~ti. Univ. "A1. L Cuza" Ia~i Sect. I a Mat. (N.S.) 15(1969), 317-328. Birsan T., Sur les espaces bitopologiques complStement r6guliers. An. ~ti. Univ. "Al. I. Cuza" Ia~i Sect. I a Mat. (N.S.) 16(1970), 29-34. Birsan T., Contribution h l'~tude des groupes bitopologiques. An. ~ti. Univ. "A1. I Cuza" Ia~i Sect. I a Mat. (N.S.) 19(1973), No. 2, 297-310. Birsan T., Transitive quasi-uniformities and zero dimensional bitopological spaces. An. ~ti. Univ. "A1. L Cuza" Ia~i Sect. I a Mat. (N.S.) 20(1974), No. 2, 317-322. du Bois-Reymond P., Die Allgemeine Funktionentheorie. 1. Tiibingen, 1882. Bokalo B. M., The tangency of topologies and scatterness-type properties. (Russian) Cand. of Science Thesis, Moscow State Univ., 1989. Boole G., The mathematical analysis of logic. Cambridge, 1847. Boole G., An investigation into the laws of thought. London, 1854 (Reprinted by Open Court Publishing Co., Chicago, 1940). Bourbaki N., t~lSments de mathematique. Topologie GSn~rale, Chapter 9. Actualitds Sci. Ind., No. 1045, Hermann et Cie., Paris, 1948. Brelot M., Points irr~guliers et transformations continues en thSorie du potentiel. J. Math. Pures Appl. (9) 19(1940), 319-337. Brelot M., Introduction axiomatique de l'effilement. Ann. Mat. Pura Appl. (4) 57 (1962), 77-96. Brelot M., Lectures on potential theory. Tata Institute of Fundamental Research, Bombay, 1967.
387
[42] Brelot M., La topologie fine en th(~orie du potentiel. Symposium on Probability Methods in Analysis (Loutraki, 1966), pp. 36-47, Springer, Berlin, 1967. [43] Brelot M., Recherches axiomatiques sur un th6orhme de Choquet concernant l'etYilement. Nagoya Math. J. 30(1967), 39-46. [44] Brelot M., Topologies and boundaries in potential theory. (Russian) Mir, Moscow, 1974. [45] Brouwer L. E. J., Uber den nat/irlichen Dimensionsbegriff. J. Reine Angew. Math. 142(1913), 146-152. [46] Briimmer G. C. L., Note on a compactification due to Nielsen and Sloyer. Math. Ann. 195(1972), 167. [47] Briimmer G. C. L., On certain factorizations of functors into the category of quasi-uniform spaces. Proceedings of the Second Symposium on Categorical Topology (Univ. Cape Town, Cape Town, 1976), Quaestiones Math. 2(1977/78), No. 1-3, 59-84. [48] Briimmer G. C. L., Two procedures in bitopology. Categorical topology. (Proc. Internat. Conf., Free Univ. Berlin, Berlin, 1978), pp. 35-43, Lecture Notes in Mathematics, 719, Springer, Berlin, 1979. [49] Briimmer G. C. L., On some bitopologically induced monads in Top. Structure of Topological Categories. (Proc. Conf. Bremen, 1978), pp. 13-30a, Math.-Arbeitspapiere, 18, Univ. Bremen, Bremen, 1979. [50] Briimmer G. C. L., On the nonunique extension of topological to bitopological properties. Categorical Aspects of Topology and Analysis (Proc. Conf. Carleton, 1980), pp. 50-67, Lecture Notes in Mathematics, 915, Springer, Berlin-New York, 1982. [51] Briimmer G. C. L., Functorial transitive quasi-uniformities. Categorical topology. (Proc. Conf. Toledo, Ohio, 1983), 163-184, Sigma Set. Pure Math., 5, Heldermann, Berlin, 1984. [52] Briimmer G. C. L., Hager A. W., Functorial uniformization of topological spaces. Proceedings of the 8th international conference on categorical topology (L'Aquila, 1986). Topology Appl. 27(1987), No. 2, 113-127. [53] Canfell M. J., Semi-algebras and rings of continuous functions. Thesis, University Edinburgh, 1968. [54] Cantor G., Math. Ann. 5(1872). [55] Cantor G., Math. Ann. 15(1879). [56] Cantor G., GSttinger Nachr., 1879. [57] Cantor G., Math. Ann. 23(1884). [58] Cartan H., Th6orie g6n6rale du balayage en potentiel newtonien. Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.) 22(1946), 221-280. [59] Choe T. H., Partially ordered topological spaces. An. Acad. Brasil. Cidnc. 51(1979), No. 1, 53-63. [60] Choquet G., Theory of capacities. Ann. Inst. Fourier, Grenoble 5(195354), 131-295. [61] Choquet G., Une classe reguli~re d'espace de Baire. C. R. Acad. Sci. Paris 246(1958), 218-220.
388
[62]
[63] [64]
[65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81]
Choquet G., Sur les points d'effilement d'un ensemble. Application l'Stude de la capacit4. Ann. Inst. Fourier, Grenoble 9(1959), 91-101. Choquet G., Lectures on Analysis, vol. I. Benjamin, New York, 1969. Chvalina J., On certain topological state spaces of X-automata. Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978), pp. 287-299, Colloq. Math. Soc. Jdnos Bolyai, 23, North-Holland, Amsterdam-New York, 1980. Chvalina J., Separation properties of topologies associated with digraphs. Scripta Fac. Sci. Natur. UYEP Brunensis-Math. 10(1980), No. 8, 399410. (~irid M. D., Dimension of bitopological spaces. Papers presented at the Fifth Balkan Mathematical Congress (Belgrade, 1974). Math. Balkanica 4(1974), 99-105. Cook M. C. H., Sur deux problhmes de sous-espaces ayant une topologie mixte. C.R. Acad. Sci. Paris Sdr. A - B 277(1973), A1095-A1097. Crossley S. G., Hildebrand S. K., Semi-topological properties. Fund. Math. 74(1972), N. 3, 233-254. Cs~sz~r A., Double compactification d'espaces syntopogenes. Ann. Univ. Sci. Budapest. EStvSs Sect. Math. 7(1964), 3-11. Cs~sz~r A., D-complete extensions of quasi-uniform spaces. Acta Math. Hungar. 64(1994), No. 1, 41-54. Dasgupta H., Lahri B. K., Local connectedness in bitopological spaces. Y. Indian Inst. Sci. 64 (1983), No. 2, 37-40. Datta M. C., Projective bitopological spaces. Y. Austral. Math. Soc. 13(1972), 327-334. Datta M. C., Projective bitopological spaces, II. J. Austral. Math. Soc. 14(1972), 119-128. Datta M. C., Paracompactness in bitopological spaces and an application to quasi-metric spaces. Indian Y. Pure Appl. Math. 8(1977), No. 6, 685690. Davis A. S. Indexed systems of neighborhoods for general topological spaces. Amer. Math. Monthly 68(1961), 886-893. De~k J., Extensions of quasi-uniformities for prescribed bitopologies. I, II. Studia Sci. Math. Hungar. 25(1990), No. 1-2, 45-67, 69-91. Deak J., A survey of compatible extensions (presenting 77 unsolved problems). Topology. Theory and applications, II (Pecs, 1989), 127-175, Colloq. Math. Soc. Yanos Bolyai, 55, North-Holland, Amsterdam, 1993. Deak J., A bitopological view of quasi-uniform completeness. I, II, III. Studia Sci. Math. Hungar. 30(1995), No. 3-4, 389-409, 411-431; 31(1996), No. 4, 385-404. Dieudonn~ J., t~l~ments d'Analyse, Tome 2. Gauthier-Villars, Paris, 1972. Doitchinov D., On completeness in quasi-metric spaces. Topology Appl. 30(1988), No. 2, 127-148. Doitchinov D., A concept of completeness of quasi-uniform spaces. Topology Appl. 38(1991), No. 3, 205-217.
389
Is2] Is31 [84] [ss] Is6] Is7] [ss] [sgl [9ol [91] [92] [93] [941 [951 [96] [971 [gsl [99] [lOO]
Dowker C. H., Inductive dimension of completely normal spaces. ./. Math., Ozford, Set. (2) 4(1953), 267-281. Dvalishvili B. P., Separation in bitopological spaces. (Russian) Soobshch. Akad. Nauk Georgian SSR 73(1974), 285-288. Dvalishvili B. P., The dimension of bitopological spaces. (Russian) Soobshch. Akad. Nauk Georgian SSR 76(1974), 49-52. Dvalishvili B. P., Mappings of bitopological spaces. (Russian) Soobshch. Akad. Nauk Georgian SSR 80(1975), No. 3, 553-556. Dvalishvili B. P., On dimension and some other problems of the theory of bitopological spaces. (Russian)Trudy Tbiliss. Mat. Inst. Razmadze 56(1977), 15-51. Dvalishvili B. P., Some principal questions of the theory of bitopological spaces. (Russian) Cand. of Science Thesis, Tbilisi State Univ., 1978. Dvalishvili B. P., Bicompact extension of bitopological spaces. (Russian) International Topology Conference (Moscow State Univ., Moscow, 1979). Uspekhi Mat. Nauk 35(1980), No. 3(213), 171-174. Dvalishvili B. P., Some applications of the theory of bitopological spaces in the theory of ordered topological spaces. (Russian) Trudy Tbiliss. Univ. 225(1981), 35-50. Dvalishvili B. P., On the notions of nearness for sets and functions. (Russian) L Vekua Inst. Appl. Math., Abstr. Seminar on Topological Aspects of Function Theory, Tbilisi, 1985, 4-5. Dvalishvili B. P., Generalized Boolean algebras and their representations. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 78(1986), 104-124. Dvalishvili B. P., Bitopology and the Baire category theorem. Abstr. Tartu Conf. Problems of Pure Appl. Math., 1990, 90-93. Dvalishvili B. P., On some bitopological applications. Mat. Vesnik 42 (1990), No. 3-4, 155-165. Dvalishvili B. P., The (i,j)-G-insertion property in bitopology. Abstr. Seventh Prague Topological Symp., 1991, p. 27. Dvalishvili B. P., Investigations of bitopologies and their applications. Doctoral Dissertation, Tbilisi State Univ., 1994. Dvalishvili B. P., A note on (i,j)-pseudocomplete bitopological spaces. Abstr. 9th Congress of Mathematicians of Yugoslavia, 1995, 2 p. Dvalishvili B. P., On a characterization of a stronger topology by means of a weaker one. Abstr. Kiev Inter. Conf. Topol. Appl., 1995, 15-17. Dvalishvili B. P., Relative compactness from the bitopological point of view. Abstr. ICTA 2000, Inter. Conf. Topol. Appl., Ohrid (Macedonia), 2000, 2 p. Dvalishvili B.P., Bitopological essence of a cotopology and the convenient insertion. Abstr. The First Turkish Inter. Conf. Topol. Appl., 2000, Atlas Document No. caeh-64. Dvalishvili B. P., Bitopological and algebraic structures in the context of Baire-type properties and generalized Boolean algebras, g. Math. Sci. (New York) 111(2002), No. 1, 3227-3338.
390
[101] Dvalishvili B. P., The strengthening and weakening instrument: comparability of topologies spaces. Math. Moray. 6(2002), 21-64. [102] Dvalishvili B. P., The lower and upper topologies as a bitopology Mat. Vesnik 55(2003), No. 1-2, 37-52. [103] Dvalishvili B. P., Relative compactness, cotopology and some other notions from the bitopological point of view. Topology Appl. 140(2004), No. 1, 37-55. [104] Dvalishvili B. P., Generalized lattices, to appear: Nova Science Publishers, Inc. [105] Dwinger Ph., Introduction to Boolean Algebras. Physica-Verlag, Wiltzburg, 1961. [106] Engelking R., General topology. Translated from the Polish by the author. Monografie Matematyczne, Tom 60. P WN-Polish Scientific Publishers, Warsaw, 1977. [107] Engelking R., Dimension Theory. Translated from the Polish and revised by the author. North-Holland Mathematical Library, 19. North-Holland Publishing Co., Amsterdam-Ozford-New York; P WN-Polish Scientific Publishers, Warsaw, 1978. [108] Everett C. J., Ulam S. M., On the problem of determination of mathematical structures by their endomorphisms. Bull. Amer. Math. Soc. 54(1948), abstract 285 t. [109] Ewert J., Przeksztalcenia wielowartogciowe i przestrzenie bitopologiczne. Slupsk, 1985. [110] Ewert J., Baire classification and multivalued maps. Comment. Math. Prace Mat. 27(1987), No. 1, 59-65. [111] Filippov V. V., Feathered paracompacta. (Russian) Dokl. Akad. Nauk SSSR 178(1968), 555-558. [112] Fletcher P., Pairwise uniform spaces. Doctoral Thesis, Univ. North Carolina at Chapel Hill, 1966. [113] Fletcher P., Hoyle H. B., Patty C. W., The comparison of topologies. Duke Math. Y. 36(1969), 325-331. [114] Fletcher P., Naimpally S. A., On almost complete and almost precompact quasi-uniform spaces. Czechoslovak Math. J. 21 (96)(1971), 383-390. [115] Fletcher P., Liu P., Topologies compatible with homeomorphism groups. Pacific J. Math. 56(1975), No. 1, 77-86. [116] Fletcher P., Lingren W. F., Quasi-uniform spaces. Lectures Notes Pure Appl. Math., 77, Marcel Dekker, Inc., New York, 1982. [117] Fletcher P., Frith J., Hunsaker W., Schauerte A., The Nachbin quasiuniformity of a bi-Stonian space. J. Austral. Math. Soc. Set. A 61(1996), No. 3, 322-326. [118] Fomin S. V., Extensions of topological spaces. Ann. of Math. (2) 44(1943), 471-480. [119] Frink O., Compactifications and semi-normal spaces. Amer. J. Math. 86(1964), 602-607. [120] Frolik Z., Remarks concerning the invariance of Baire spaces under mappings. Czechoslovak Math. J. 11 (86)(1961), 381-385.
391
[121] Fuglede B., The quasi topology associated with a countably subadditive set function. Ann. Inst. Fourier, Grenoble 21(1971), fasc. 1, 123-169. [122] Fuglede B., Connexion en topologie fine et balayage des mesures. Ann. Inst. Fourier, Grenoble 21(1971), No. 3, 227-244. [123] Fuglede B., Finely harmonic functions. Lecture Notes in Mathematics, Vol. 289, Springer-Verlag, Berlin-New York, 1972. [124] Gastl G. L., Bitopological spaces from quasi proximities. Portugal. Math. 33(1974), 213-218. [125] Godefroy G., Topologies subordonn6es. Sdm. Choquet (Initiation d l'analyse), 15e annde, 1975/76, Comm. C10. [126] Goffman C., Waterman D., Approximately continuous transformations. Proc. Amer. Math. Soc. 12(1961), 116-121. [127] Gr/itzer G., General lattice theory. (Russian) Mir, Moscow, 1982. [128] Guthrie J. A., Reynolds D. F., Stone H. E., Connected expansions of topologies. Bull. Austral. Math. Soc. 9(1973), 259-265. [129] Guthrie J. A., Stone H. E., Spaces whose connected expansions preserve connected subsets. Fund. Math. 80(1973), N. 1, 91-100. [130] Guthrie J. A., Stone H. E., Wage M. L., Maximal connected expansions of the reals. Proc. Amer. Math. Soc. 69(1978), No. 1, 159-165. [131] GutiStez A., A note on a paper by S. Lal. J. Austral. Math. Soc. 35(1983), 197-199. [132] Hamlett T. R., The property of being a Baire space is semi-topological. Math. Chronicle 5(1976/77), No. 3, 166-167. [133] Haworth R. C., McCoy R. A., Baire spaces. Dissertationes Math. (Rozprawy Mat.) 141(1977), 73 pp. [134] Herrman C., Luksch P., Skorsky M., Wille R., Algebras of Semiconcepts and Double Boolean Algebras. Technische Universitgt Darmstadt. Preprint Nr. 2086, Mai, 2000. [135] Hicks T. L., Satterwhite R. E., Quasi-pseudometrics over Tychonov semifields. Math. Japon 22(1977), No. 3, 315-321. [136] Hodel R. E., Extensions of metrization theorems to higher cardinality. Fund. Math. 87(1975), N. 3, 219-229. [137] Ivanov A. A., Problems of the theory of bitopological spaces. (Russian) Zap. Nauchn. Sere. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167(1988), Issled. Topol. 6, 5-62, 190; English transl.: J. Soviet Math. 52(1990), No. 1, 2759-2790. [138] Ivanov A. A., Bibliography on bitopological spaces. (Russian) Zap. Nauchn. Sere. Leningrad. Otdel. Mat. Inst. Steklov. (LOM1)167(1988), Issled. Topol. 6, 63-78; English transl.: J. Soviet Math. 52(1990), No. 1, 2790-2799. [139] Ivanov A. A., Problems of the theory of bitopological spaces, II. (Russian) Zap. Nauchn. Sere. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POM1) 208(1993), Issled. po Topol. 7, 5-67; english transl.: J. Math. Sci. 81(1996), No. 2, 2465-2496. [140] Ivanov A. A., Bibliography on bitopological spaces, II. (Russian) Zap. Nauchn. Sere. S.-Peterburg. Otdel. Mat. Inst. Steklov. ( POMi)
392
208(1993), Issled. po Topol. 7, 68-81,; English transl.: J. Math. Sci. 81(1996), No. 2, 2497-2505. [141] Ivanov A. A., Problems of the theory of bitopological spaces, III. (Russian) Zap. Nauchn. Sere. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 231(1995), Issled. po Topol. 8, 9-54,; English transl.: J. Math. s i. Yo k) 9 (1998), No. 6, 3339-3364. [142] Ivanov A. A., Bibliography on bitopological spaces, III. (Russian) Zap. Nauchn. Sere. S.-Peterburg. Otdel. Mat. Inst. Steklov. ( P OMI) 231(1995), Issled. po Topol. 8, 55-61; English transl.: J. Math. Sci. (New York) 91(1998), No. 6, 3365-3369. [143] Ivanov A. A., Bitopological spaces. (Russian) Zap. Nauchn. Sere. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 242(1997), Bitopol. Prostran. Issled. po Topol. 9, 7-216; English transl.: J. Math. Sci. (New York) 98(2000), No. 5, 509-616. [144] Jelid M., Certain properties of dimension functions in bitopological spaces. (Russian) Math. Balkanica 4(1974), 309-311. [145] Jelid M., Certain dimension functions in bitopological spaces. Mat. llesnik 12(26)(1974), 38-42. Jelid M., s function of bitopological spaces. Mat. Vesnik 39 [146] (1987), No. 4, 393-398. [147] Jell5 M., Feebly p-continuous mappings. Rend. Circ. Mat. Palermo (2) Suppl. No. 24, (1990), 387-395. [148] Jordan C., Y. de Math. (4)8(1892). [149] Juhs I., A generalization of nets and bases. Period. Math. Hungar. 7(1976), No. 2, 183-192. [150] Kelley J. L., Namioka I., Linear topological spaces. Graduate Texts in Mathematics, No. 36, Springer-Verlag, New York-Heidelberg, 1976. [151] Kelly J. C., Bitopological spaces. Proc. London Math. Soc. (3) 13(1963), 71-89. [152] Kim Y. W., Pseudo quasi metric spaces. Proc. Japan Acad. 44(1968), 1009-1012. [153] Kim Y. W., Pairwise compactness. Publ. Math. Debrecen 15(1968), 87-90. [154] Kiinzi H.-P. A., A note on uniform ordered spaces. J. Austral. Math. Soc. Set. A 42(1987), No. 3, 349-352. [ 55] Kiinzi H. P. A., Strongly zero-dimensional bispaces. J. Austral. Math. Soc. Ser. A 53(1992), No. 3, 327-337. [156] Kiinzi H. P. A., Nonsymmetric topology. Topology with applications (Szekszard, 1993), 303-338, Bolyai Soc. Math. Stud., 4, Yanos Bolyai Math. Soc., Budapest, 1995. [157] Kiinzi H. P. A., Nonsymmetric Distances and their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology. Preliminary version, Topology Atlas Preprint, No. 318. [158] Kiinzi H. P. A., Briimmer G. C. L., Sobrification and bicompletion of totally bounded quasi-uniform spaces. Math. Proc. Cambridge Philos. Soc. 101(1987),No. 2, 237-247.
393
[159] Kfinzi H. P. A., Romaguera S., Salbany S., Topological spaces that admit bicomplete quasi-pseudometrics. Ann. Univ. Sci. Budapest. EStvSs Sect. Math. 37(1994), 185-195. [160] Kuratowski K., Sur l'opdration A de l'Analysis Situs. Fund. Math. 3 (1922), 182-199. [161] Kuratowski K., Topology. Vol. 1. (Russain) Mir, Moscow, 1966. [162] Kuratowski K., Mostowski A., Set theory. (Russian) Mir, Moscow, 1970. [163] Lal S., Pairwise concepts in bitopological spaces. J. Austral. Math. Soc. Set. A 26(1978), No. 2, 241-250. [164] Lambrinos P. Th., Quasi-uniform characterizations of (weak) boundedness and (weak)compactness. Ann. Soc. Sci. Bruxelles Sdr. I90(1976), No. 4, 307-316. [165] Lane E. P., Concernig extensions of semicontinuous functions in bitopological spaces. Notices Amer. Math. Soc. 12(1965), No. 1, 128. [166] Lane E. P., Bitopological spaces and quasi-unform spaces. Proc. London Math. Soc. (3) 17(1967), 241-256. [167] Lane E. P., Quasi-proximities and bitopological spaces. Portugal. Math. 28(1969), 151-159. [168] Lebesgue H., Sur la non applicabilit6 de deux domains appartenant deux espaces de n e t n + p dimensions. Math. Ann. 70(1911), 166-168. [169] Levine N., A decomposition of continuity in topological spaces. Amer. Math. Monthly 68(1961), 44-46. [170] Levine N., Semi-open sets and semi-continuity in topological spaces. Amer. Math. Monthly 70(1963), 36-41. [171] Lorrain F., Notes on topological spaces with minimum neighbourhoods. Amer. Math. Monthly 76(1969), 616-627. [17s] Luke~ J., The Lusin-Menchoff property of fine topologies. Comment. Math. Univ. Carolinae 18(1977), No. 3, 515-530. [173] Lukeg J., Mal3) J., ZajfSek L., Fine Topology Methods in Real Analysis and Potential Theory. Lecture Notes in Mathematics, 1189, SpringerVerlag, Berlin, 1986. [174] Maheshvari S. R., Prasad R., On pairwise irresolute functions. Mathematica (Cluj) 18(41)(1976), No. 2, 169-172. [175] Maheshvari S. R., Prasad R., Semi-open sets and semicontinuous functions in bitopological spaces. Math. Notae 26(1977/78), 29-37. [176] Matematick~ Ustav CSAV, Collection of Problems. II, Potential Theory, Praha, 1987. [177] McCallion T., Compactifications of ordered topological spaces. Proc. Cambridge Phil. Soc. 71(1972), 463-473. [178] McCartan S. D., Separation axioms for topological ordered spaces. Proc. Cambridge Phil. Soc. 64(1968), 965-973. [179] Meager K., Uber die Dimensionalit~t von Punktmengen, I. Monatsh. Math. Phys. 23(1923), 148-160. [1801 Misra M. D., Dube K. K., Pairwise R0 spaces. Ann. Soc. Sci. Bruxelles Sdr. I87(1973), 3-15.
394
[181] Motchane L., Sur la notion d'espace bitopologique et sur les espaces de Baire. C. R. Acad. Sci. Paris 224(1957), 3121-3124. [182] Motchane L., Sur la caract~rization des espaces de Baire. C. R. Acad. Sci. Paris 246(1958), 215-218. [183] Mrgevid M., Local compactness in bitopological spaces. Mat. Vesnik 2(15)(30)(1978), No. 3, 265-272. [184] Mrgevid M., On bitopological local compactness. Mat. Vesnik 3(16)(31) (1979), No. 2, 187-196. [185] Mrgevid M., On some properties of bitopological spaces and their hyperspaces. Report Ser., 187, Auckland, New Zealand, 1982, 1-22. [186] Mrgevid M., On bitopological separation axioms. Mat. Vesnik 36(1986), No. 3, 313-318. [187] Mrgevid M., Reilly I. L., A note on bitopological connectedness. Ranchi. Univ. Math. J. 18(1988), 1-3. [1881 Mrgevid M., Reilly I. L., Separation properties of a topological space and its associated topology of c~-subsets. Kyungpook Math. J. 33(1993), No. 1, 75-86. [189] Murdeshwar M. G., Naimpally S. A., Quasi-uniform topological spaces. P. Noordhoff Ltd., Groningen, 1966. [190] Nachbin L., Sur les espaces uniformes ordonnSs. C. R. Acad. Sci. Paris 226(1948), 774-775. [191] Nachbin L., Topology and order. D. Van Nostrand Co., Inc., Princeton, N. J.- Toronto, Ont.-London, 1965 [192] Nagata J., Modern dimension theory. Interscience Publishers John Wiley Sons, Inc., New York, 1965. [193] Nagata J., A note on Filippov's theorem. Proc. Japan Acad. 45(1969), 30-33. [194] Naimpally S. A., Function spaces of quasi-uniform spaces. Nederl. Akad. Wetensch. Proc. Ser. A 68 Indag. Math. 27(1966), 768-771. [195] Naimpally S. A., Separation axioms in quasi-uniform spaces. Amer. Math. Monthly 74(1967), 283-284. [196] Novotn37 M., Skula L., Uber gewisse Topologien auf geordneten Mengen. Fund. Math. 56(1964/1965), 313-324. [197] Osgood W., Non-uniform convergence and the integration of series term by term. Amer. J. Math. 19(1897), 155-190. [198] Oxtoby J. C., Cartesian products of Baire spaces. Fund. Math. 49 (1960/1961), 157-166. [199] Oxtoby J. C., Measure and category. (Russian) Mir, Moscow, 1974. [200] Page W., Topological Uniform Structures. Pure and Applied Mathematics. Wiley-Interscience John Wiley & Sons, New York, 1978. [201] Parhomenko A. S., On condensations into compact spaces. (Russian) Bull. Acad. Sci. URSS. Ser. Math. 5(1941), 225-232. [202] Patty C. W., Bitopological spaces. Duke Math. J. 34(1967), 387-392. [203] Pervin W. J., Quasi-proximities for topological spaces. Math. Ann. 150 (1963), 325-326.
395
[204]
Pervin W. J., Connectedness in bitopological spaces. Nederl. Akad Wetensch. Proc. Set. A 70 Inday. Math. 29(1967), 369-372. [205] Pervin W. J., Anton H., On the separation axioms for bitopological spaces. Ann. Soc. Sci. Bruzelles Set. I91(1977), No. 4, 195-199. [206] Ponomarev V. I., The metrizability of a finely compact p-space with a point-countable base. (Russian) Dokl. Akad. Nauk SSSR 174(1967), 1274-1277. [207] Predoi M., Sur la convergence quasi-uniforme. Period. Math. Hungar. 10(1979), No. 1, 31-49. [208] Priestley H. A., Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. (3)24(1972), 507-530. [209] Raghavan T. G., Reilly I. L., Metrizability of quasi-metric spaces. J. London Math. Soc. (2) 15(1977), No. 1, 169-172. [210] Raghavan T. G., Reilly I. L., Uniformization of quasi-uniform spaces. Bull. Austral. Math. Soc. 23(1981), No. 3, 413-422. [2111 Rasiova M., Sikorski R., A proof of the completeness theorem of GSdel. Fund. Math. 37(1950), 193-200. [sls] Rauszer C., Semi-Boolean algebras and their application to intuitionistic logic with dual operations. Fund. Math. 83(1973/74), No. 3, 219-249. [213] Reilly I. L., Quasi-gauges, quasi-uniformities and bitopological spaces. Unpublished Ph. D. Thesis, Urbana-Champaign, Illinois Library, Univ. Illinois, 1970. [214] Reilly I. L., Bitopological local compactness. Nederl. Akad. Wetensch. Proc. Set. A 75 Indag. Math. 34(1972), 407-411. [215] Reilly I. L., On bitopological separation properties. Nanta Math. 5(1972), No. 2, 14-25. [216] Reilly I. L., Zero-dimensional bitopological spaces. Nederl. Akad. Wetensch. Proc. Set. A 76 Indag. Math. 35(1973), 127-131. [2 7] Reilly I. L., On essentially pairwise Hausdorff spaces. Rend. Circ. Mat. Palermo (2) 25(1976), No. 1-2, 47-52. [2181 Reilly I. L., Some properties of fine topologies. Abstr. Seventh Prague Topological Syrup., 1991, 93. [219] Ribeiro H., Sur les espaces g m4trique faible. Portugaliae Math. 4(1943), 21-40, 65-68. [220] Robertson A. P., Robertson W., Topological Vector Spaces. (Russian) Mir, Moscow, 1967. [221] Romaguera S., Two characterizations of quasi-pseudometrizable bitopological spaces. Or. Australian Math. Soc. Set. A 35(1983), No. 3, 327333. [222] Romaguera S., On bitopological quasi-pseudometrization. J. Australian Math. Soc. Set. A 36(1984), No. 1, 126-129. [223] Romaguera S., Salbany S., Quasi-metrizable spaces with a bicomplete structure. Eztracta Math. 7(1992), No. 2-3, 99-102. [224] Romaguera S., Salbany S., On bicomplete quasi-pseudometrizability. Topology Appl. 50(1993), No. 3, 283-289.
396
[225] [226] [2271
[228] [2291
[230]
[231]
[232] [233]
[234]
[2351 [236]
[237]
[238] [239]
[240] [241] [242]
Romaguera S., Ruiz-Gdmez M., Bitopologies and quasi-uniformities on spaces of continuous functions. I, II. Publ. Math. Debrecen 47(1995), No. 1-2, 81-93; 50(1997), No. 1-2, 1-15. Rose D. A., Hamlett T. R., Ideally equivalent topologies and semitopological properties. Math. Chronicle 20(1991), 149-156. Rudin W., Real and Complex Analysis. McGraw-Hill Book Co., New York- Toronto, Ont.-London, 1966. Saegrove M. J., Pairwise complete regularity and compactification in bitopological spaces. J. London Math. Soc. (2) 7(1973), 286-290. Salbany S., Bitopological spaces, compactifications and completions. Thesis, Univ. Cape Town, 1970; Mathematical Monographs of the University of Cape Town, No. 1. Department of Mathematics, University of Cape Town, Cape Town, 1974. Salbany S., Reflective subcategories and closure operators. Categorical Topology. (Proc. Conf. Mannheim, 1975), pp. 548-565. Lecture Notes in Mathematics, Vol. 540, Springer, Berlin, 1976. Salbany S., An embedding characterization of compact spaces. Categorical topology (Proc. Internat. Conf., Free Univ. Berlin, Berlin, 1978), pp. 316-325, Lecture Notes in Math., 719, Springer, Berlin, 1979. Salbany S., An embedding theorem for k-compact spaces. Math. Colloq. Univ. Cape Town 12(1978/79), 95-106. Salbany S., A bitopological view of topology and order. Categorical topology (Toledo, Ohio, 1983), 481-504, Sigma Set. Pure Math., 5, Heldermann, Berlin, 1984. Shakhmatov D. B., Tka~enko M. G., Tkachuk V. V., Watson S., Wilson R. G., Neither first countable nor (~ech-complete spaces are maximal Tychonoff connected. Proc. Amer. Math. Soc. 126(1998), No. 1, 279287. Singal M. K., Singal A. R., Almost continuous functions. Yokohama Math. Y. 16(1968), 63-79. Singal A. R., Remarks on separation axioms. General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968) pp. 265-296, Academia, Prague, 1971. Singal M. K., Singal A. R., Bitopological ordered spaces. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student 39(1971), 440-447. Singal M. K., Singal A. R., On some pairwise normality conditions in bitopological spaces. Publ. Math. Debrecen 21(1974), 71-81. Singal M. K., Jain S. C., Separation axions weaker than pairwise Hausdorff. Glasnik Mat. Set. III 16(36)(1981), No. 1, 121-129. Singal A. R., Lal S., A Wallman type compactification for a bitopological space. Bull. Inst. Math. Acad. Sinica 11(1983), No. 1, 23-29. Slobodnik S. G., Expanding system of linearly closed sets. (Russian) Mat. Zametki 19(1976), No. 1, 67-84. Smithson R. E., Multifunctions and bitopological spaces. Y. Natural Sci. and Math. 11(1971), 191-198.
397
[243] Steiner E. F., The relation between quasi-proximities and topological spaces. Math. Ann. 155(1964), 194-195. [244] Steiner E. F., Normal families and completely regular spaces. Duke Math. J. 33(1966), 743-746. [245] Stoltenberg R. A., On quasi-metric spaces. Duke Math. J. 36(1969), 65-71. [246] Stone M. H., The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40(1936), No. 1, 37-111. [247] Stone M. H., Application of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41(1937), No. 3, 375-481. [248] Swart J., Total disconnectedness in bitopological spaces and product bitopological spaces. Nederl. Akad. Wetensh. Proc. Set. A 74 Indag. Math. 33(1971), 135-145. [249] Swart J., An axiomatic topological characterization of Hilbert spaces. Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur. (8)52(1972), 166-174. [250] Tamari D., On a generalization of uniform structures and spaces. Bull. Res. Council. Izrael 3(1954), 417-428. [251] Thomas J. P., Maximal connected topologies. J. Austral. Math. Soc. 8(1968), 700-705. [252] Todd A. R., Quasiregular, pseudocomplete, and Baire spaces. Pacific J. Math. 95(1981), No. 1, 233-250. [253] Ulam S. M., A Collection of mathematical problems. Interscience Publishers, New York-London, 1960. [254] Urysohn P. S., Les multiplicit~s Cantoriennes. C. R. Acad. Sci. Paris 175(1922), 440-442. [255] van de Vel M., Abstract, topological and uniform convex structures. Facul. Wisk. en Infor., De Boolelaan 1081, H V Amsterdam, rapportnr. WS-353, mei 1989. [256] Ward L. E., Partially ordered topological spaces. Proc. Amer. Math. Soc. 5(1954), 144-161. [257] Weston J. D., The principle of equicontinuity for topological vector spaces. Proc. Univ. Durham Philos. Soc. Set. A 13(1957), 1-5. [258] Weston J. D., On the comparison of topologies. J. London Math. Soc. 32(1957), 342-354. [259] Wiener N., Certain notions in potential theory. J. Math. Massachussetts 3(1924), 25-51. [260] de Wilde M., The Baire property in spaces of functions and product spaces. Bull. Soc. Roy. Sc. Li@e 46(1977), No. 11-12, 397-400. [261] Wille R., Boolean Concept Logic. Technische Universitgt Darmstadt. Preprint Nr. 2089, Mai, 2000. [262] Wilson W. A., On quasi-metric spaces. Amer. J. Math. 53(1931), 675684. [263] Yu-Lee Lee, Topologies with the same class of homeomorphisms. Pacific J. Math. 20(1967), 77-83.
398
[264] Za~cev V. I., On the theory of Tychonov spaces. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Meh. 22(1967), No. 3, 48-57.
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Index
A
G.antitonicity (of the pair "~' = (~bl, ~'2)), 226 arc, 381 G.associativity laws, 202 atom, 227, 229, 276, 277 (i,j)-atom, 194, 276-281 (i,j)-atomic GBA, 280 p-atomic GBA, 194, 280, 282 /-atomic BA, 280
absolute bitopological analogue, 2 G.absorption laws, 202 abstract fine topology (on a TS), 321 /-accumulation point, 4, 44-48, 341, 343, 344, 348, 349 G.addition, 310-316 agreement (of a finite measure with a category), 329, 330 Alexandrov topology, xii a-algebra, 310, 327, 328 almost i-Baire space, 131 almost (i,j)-Baire space, 128, 131, 142-150, 153, 156, 158-162, 164, 186, 188, 189, 321-323, 325, 330, 332, 347, 382 almost p-Baire space, 129 almost closed function, 163 (i,j)-almost closed function, 174-178 p-almost closed function, 175, 177, 178 almost closed set, 354-356, 358 (i, j)-almost completely regular BS, 6 p-almost completely regular BS, 334 almost continuous function, 163 /-almost continuous function, 168 d-almost continuous function, 168 (i,j)-almost continuous function, 167, 168, 170, 176, 177 p-almost continuous function, 168-171,177, 178 almost lower Baire space, 381 (i, j)-almost normal BS, 6 p-almost normal BS, 7, 336 almost open function, 163 (i,j)-almost open function, 174-177, 179, 180, 186, 187 p-almost open function, 175, 177, 178 almost open set, 354-356, 358 almost real compact space, xi (i,j)-almost regular BS, 6, 7, 175, 176, 181 p-almost regular BS, 178 almost upper Baire space, 381 antidiscrete topology, 45-47, 65, 72, 76, 83, 84, 92, 171, 175, 368 antitonicity (of the map ~i), 224 G.antitonicity (of the pair
B
Baire category theorem, 128 Baire classification (of multivalued functions), xi Baire-like properties, 128, 164, 321 Baire-type properties, 128 Baire space, 128, 133 i-Baire one function, 323 i-Baire property, 327, 330, 331 (i,j)-Baire property, 327-331 i-Baire BS, 77, 87, 129-132, 137, 146, 149, 153, 154, 156, 158-161, 324, 338 (i,j)-Baire space, 128-141, 149, 151-153, 156, 158-161, 164, 185, 186, 188, 189, 321-323, 360, 382 p-Baire BS, 129 barreled linear TS, 325, 326 base (of a set), 338-349, 351 base operator, 321, 340 base operator space, 321 bijection, 163, 172, 173, 178, 184, 186-188, 209, 210, 223, 229, 237, 272, 274, 288, 290, 292, 295, 303, 304, 317, 368 Binary relation @ F, 285, 288 bitopological cube, 4, 382 bitopological disconnectedness, 97 bitopological GBA, 194, 242, 243 bitopological Hausdorff (i.e., p-Hausdorff), xi, 6, 297, 299, 303305, 320, 331, 332, 335, 337, 372 bitopological insertion, 92, 319 bitopological property, 4, 97, 100, 127, 164, 184 bitopological space, ix bitopological space of (i, j)-first category, 163 bitopological space in the general sense, ix
= (~1, ~2)), 224
antitonicity (of the map ~p~), 226 406
407 bitopological space in potential theory, 319, 331, 333, 335, 336, 338, 339, 341, 343, 344, 346, 348, 353-359 bitopological space of (i, j)-second category, 163 bitopological space in the sense of Canfell, 370, 372, 373, 377 bitopological space in the sense of McCallion, 372, 373 bitopology, x, 3, 92-95, 325, 326, 360, 362, 367, 372, 373 i-Blumberg space, 327, 333 Boolean algebra, 193, 194, 221, 222, 227, 229, 237, 238, 240 G.Boolean algebra, 221, 225, 232, 235, 239, 241-246, 248, 251, 253-257, 260-263, 266, 267, 269, 272, 275-278, 280-282, 284, 285, 288-290, 292, 293, 295-297, 299, 303-310, 312, 315, 317 Boolean BS, 303, 304 G.Boolean factor algebra, 288 G.Boolean a-algebra, 309, 310 G.Boolean operations, 221 G.Boolean ring, 311 G.Boolean subalgebra, 221, 222, 257, 275 Boundary point, 318 boundary set, 16 /-boundary set, 18-20, 22-25, 28-31, 34, 35, 46, 51, 52, 57, 64-66, 69, 74, 85, 382 (i,j)-boundary set, 18, 19, 52, 57, 69 /-boundary set (at a point), 21, 22 (i,j)-boundary set (at a point), 21 /-boundary (of a set), 29, 33, 50, 51, 108110, 124, 360-365 (i, j)-boundary (of a set), 28-33, 50, 51, 98110, 112, 119, 124, 362, 363 (l, u)-boundary (of a set), 374-377 (u, /)-boundary (of a set), 374-377 Brelot's notion (of a fine topology), 319, 331 C Canfell's duality, 370, 372, 373, 378-380 Cantor-Bendixon's theorem, 48 capacity (on a set), 319, 337-344, 346, 348351, 353-359 capacity of the type (c~) ((~)), 356-359 capacity of the type (8), 356, 357, 359 cardinality (of a set), 2, 137, 146, 147 Cartan's notion (of a fine topology), 318, 353 Cartesian product (of BS's), 4, 28, 30, 100 i-(~ech-complete BS, 323, 324, 334 (i,j)-(~ech-complete BS, 323-325, 334 chain (of G.filters), 263
chain (of G.ideals), 244 G.chain, 196, 213, 254, 263 circled set, 326 circuit, 381, 382, 384 d-closed base, 12 closed domain, 36, 37, 42, 43 /-closed domain, 33, 37, 67-69, 78, 81, 88, 90, 93 (i,j)-closed domain, 6, 16, 31-37, 41, 43, 59, 60, 67, 69, 78, 81, 82, 88, 90, 94, 167, 169, 174-176, 382 /-closed function, 4, 174, 182, 186, 188, 189 d-closed function, 4, 164, 175, 190, 191 closed neighborhoods condition, 320, 367 closed partial order, 371 closed real segment, 2-5, 7, 13-15, 166, 372 b-closed set, 321 p-closed set, 3, 28, 97, 100, 105, 106, 110112, 114-117, 121-123, 125, 126, 190, 192, 374, 384 H-closed space, x closure of a set A (in a topology Ti), 3 closure operator, 199 G.closure operator, 242 co-countable topology, 168, 175 cohomologies (of spaces with two topologies), xii combination (of/-closure and j-interior operators), 299, 300 coG.standard element, 217, 218, 220 commutativity laws, 204 G.commutativity laws, 204, 210 compactness (of a topology with respect to another), 365, 366 /-compact set, 303, 334, 365 /-compact BS, 25, 91, 320, 336, 368 d-compact BS, 337 k-compact space, xi FHP-compact BS, 10, 194, 297-299, 303, 305, 336, 337 compatibility (of fine and quasi topologies), 337, 339, 341, 343, 344, 346, 348, 356, 359 T-compatibly ordered subspace, 374 complementation operator (in the usual sense), 221,225-231,248, 253, 255, 256, 262, 264, 266, 272-275, 278, 280, 293, 297, 298, 302, 313-315 G.complemented G.lattice, 220, 221 G.complementation operator, 220-236, 239, 242-244, 248, 253-266, 272-286, 289293, 298, 299, 308, 309, 312-315 complete BA, 305 G.complete GBA, 305-308 G.complete G.lattice, 305
408 G.complete set, 257 complete distributive lattice, 199 complete Luzin-Menshoff property, 322 completely additive closure, 381 completely metrizable topology, 64 /-completely metrizable BS, 334 p-completely normal BS, 11, 12, 322, 383 /-completely regular BS, 322, 334 (i,j)-completely regular BS, 6, 7, 323, 334, 336 p-completely regular BS, 7, 11, 12, 14, 101, 333, 335, 336, 372, 383 completely regularly ordered TS, 372, 373 (i,j)-completely separated (pair of sets), 5, 13, 14 p-component, 384 G.component, 260, 307 condensation (function), 72, 91 /-condensation point, 48, 49 congruence, 240 congruence class, 240 conjugate family (of sets), 2 conjugate operators, 368 conjunction, 2 /-connected BS, 9, 74, 85, 86, 320, 360-362, 364, 365 d-connected BS, 9, 85 p-connected BS, 9, 74, 85, 361, 362, 365, 382, 383 /-connected set, 343 p-connected set, 384 consistent, xi continuous function, 368 /-continuous function, 4, 7, 168-170, 172, 176, 177, 180, 181, 184, 185, 189, 323, 333, 334, 354, 355, 357, 358 d-continuous function, 4, 5, 164, 166, 168, 170, 172, 180, 181, 183, 184, 186, 190, 191 0-continuous function, 163 i- 0-continuous function, 170 (i, j)- O-continuous function, 169, 170, 176, 177 p-0-continuous function, 170, 177, 178 continuous partial order, 371 convex cone (of nonnegative lower semicontinuous functions), 319,331-333 G.convex GI, 268 convex set, 326, 370, 374 G.convex set, 196 G.convex G.sublattice, 267 cospace, 366 cotopology, 320, 366-368 i-countably compact set, 334, 335 i-countably compact BS, 336, 337
countable G.chain condition, 308-310 (i,j)-countable paracompact BS, 9, 336 countable subadditivity (of a set function), 337, 343 countable set, 2 (i,j)-countably subcompact BS, 157 coupling (of topologies), 63, 72-82, 84, 87, 97, 143, 145, 146, 150, 151, 153, 155, 156, 158, 160-162, 164, 165, 168, 170, 171, 183, 362, 382, 383 (i,j)-covering dimension, 125-127 p-covering dimension, 125-127 crosswise topology, 321-323 customs crossing theorem, 343 D
decomposable bitopological structure, ix decreasing set, 370, 371,373 decreasing zero set, 372 degree of nearness (of bitopological boundaries), 16 p-dense family (of sets), 13-15 /-dense set, 18, 19, 23, 24, 61, 62, 69, 74, 75, 80, 82, 85, 89, 91, 130, 132-146, 148, 149, 151-155, 157-160, 178, 182, 183, 186-189, 322, 325, 369, 382 (i,j)-dense set, 18, 19, 69 dense in itself set, 16 /-dense in itself set, 44-48, 76, 77, 344 (i,j)-dense in itself set, 44 p-dense in itself set, 16, 44-51, 61, 62 density topology, 321, 322, 359 derived set, 16 /-derived set, 4, 45-49, 51, 338, 340-342, 349-351 Diagram 1, 195 Diagram 2, 212 Diagram 3, 231 difference (of sets), 2 /-difference operation, 227, 230 it-difference operation, 223-225, 232, 233, 235, 236, 282-285 directed graph (digraph), 321, 381-383 disconnection, 9 disjunction, 2 discrete order, 370, 373 discrete set (of all isolated points of a set), 4, 341-343, 345, 346 /-discrete set, 4, 44-48, 74, 341-343, 345, 346, 349 discrete topology, 19, 21-23, 45, 64, 65, 73, 74, 83, 170, 198, 368 /-discrete BS, 74 disjoint family (of sets), 2
409 disjoint bitopological sum (of a family of BS's), 136, 137, 145 G.disjoint complement (of a set), 257-261 disjoint topological sum (of a family of TS's), 136 G.disjoint system (of elements), 241, 308, 309 (V, A)-distributive element, 217-220 (A, V)-distributive element, 217-220, 271,272 distributive G.lattice, 213, 214, 221 G.distributivity laws, 213 /-domain, 343 double Boolean algebra, 193 double family, 12 double indexation, 4 double property, 2 Dowker's addition theorem, 97 G.duality, 262, 266, 276 duality in the sense of Canfell, 378, 380
empty set, 2 C-equivalence relation, D-equivalence relation, S-equivalence relation, 164, 322, 327, 382, ~.-equivalence relation,
337 92 63-72, 77, 87, 129, 383 282-285
Everett and Ulam's problem, 321, 368 /-extended covering, 365 extended natural TS, 3, 354, 355 /-extendable set, 365 (i, j)-exhaustible set, 22 /-exhaustible set, 378 u-exhaustible set, 378 /-exterior (of a set), 346, 347, 356 i-extremally disconnected BS, 74, 85 d-extremally disconnected BS, 85 (i,j)-extremally disconnected BS, 9 p-extremally disconnected BS, 9, 74, 85, 87, 306, 307, 370, 382 extremally order disconnected TS, 370
p-feebly continuous function, 183 feebly open condensation, 71, 72, 189, 327 feebly open function, 163 /-feebly open function, 178, 180-182, 186, 188 d-feebly open function, 180, 183 (i,j)-feebly open function, 182, 188, 189 p-feebly open function, 183 fibre, 191 G.field representation of a GBA, 290 G.field of sets, 288-290, 292, 295-297, 303305 filter (of all neighborhoods of a point), 10 G.filter, 261-268, 270, 271, 276-280 G.filter generated by a pair (B1, B2), 264 G.filter generated by an another G.filter and an element, 265 fine topology (in potential theory), 318, 319, 331, 336, 337, 339, 347 fine neighborhood (of a point), 318, 331 finely stable capacity, 337, 357 finite measure, 329, 330 first category (LTS), 326 first category set, 16 /-first category set, 23, 52, 67, 69, 78, 80, 88, 90, 133, 136, 158-160, 162, 323, 330, 331 (i, j)-first category set, 22-25, 52-57, 67, 69, 78, 80, 89-91, 130, 132-136, 140-145, 152-155, 159, 160, 173, 327-331 /-first category in itself set, 324 (i,j)-first category in itself set, 22, 24, 56, 57, 132, 133, 152, 173, 180-182, 184, 185, 321 /-first category set, 378-380 u-first category set, 378-380 /-first category in itself set, 379, 380 u-first category in itself set, 379, 380 /-first countable BS, 91, 334, 336 foliation (theory), xii V-formation, 241 function space, x G
feeble homeomorphism, 163 /-feeble homeomorphism, 164, 178-180 d-feeble homeomorphism, 180-183, 188 feebly continuous function, 163 /-feebly continuous function, 178-182, 186188 d-feebly continuous function, 179, 183, 186, 188 (i,j)-feebly continuous function, 182, 188, 189
generalized ordered set (goset), 193, 195, 199-202, 204, 209, 285 global Lorentzian geometry, xii 1-graph (simple graph), 381 greatest lower bound (of topologies), 64 generalized version of Dowker's addition theorem, 115 generalized version of the Menger-Urysohn formula, 103
410 generalized version of Stone's representation theorem, 303 H
half-open interval topology, 64, 159, 369 half-open real segment, 2, 5, 64, 166 hereditarily p-disconnected BS, 101 hereditarily p-normal BS, 8, 11, 12, 101104, 106, 113, 115, 118, 119, 121, 122, 322, 382 hereditarily strong normally ordered TS, 375, 378 Hilbert space, xi homeomorphism, 321,368 i-homeomorphism, 4, 178 d-homeomorphism, 4, 97, 127, 172, 183, 190, 303-305 G.homomorphism (of G.lattices), 269-272 G.homomorphism (of GBA's), 272-276, 288, 292 hyperspace, x I
a-ideal, 23 G.ideal, 243-261, 266-270, 275-278, 282288, 290-299, 303-305, 307 G.ideal generated by a pair (B1, B2), 246248 G.ideal generated by an another G.ideal and an element, 251-253 identity function, 72, 91, 175 G.identity operator, 209-212, 224, 225,227230, 232, 237-239, 272, 273 implication, 2 inclusion, 2 inclusion function, 72 mcreasibility (of a set function), 337, 354 Increasing set, 370, 371,373 increasing zero set, 372 indicators of nearness (of bitopological boundaries), 49-51, 66, 70, 71, 74, 85, 108, 330, 331,347 (i, j)-inexhaustible set, 22 /-inexhaustible set, 378 u-inexhaustible set, 378 ic-infimum (of a set), 197, 305, 306 infinite number, 2, 331 initial topology (in potential theory), 319, 331,336 initial endpoint, 381 (i, j)-A-insertion property, 92 (i, j)-(A \ ~d)-insertion property, 93 (i,j)-(JM, A)-insertion property, 95, 96
(i, j)-(2 X, A)-insertion property, 95 (Gs(X) \ w)-insertion property, 93 2-wl-insertion property, 94, 360 2- co rl-insertion property, 94 2-(1- •5(X))-insertion property, 323, 325, 360 2-(1, 2)-OD(X)-insertion property, 94 2-(1, 2)-CD(X)-insertion property, 94 (2, 1)-rl-insertion property, 93, 326, 362 (2, 1)-l-OD(X)-insertion property, 93 (2, 1)-l-Gs(X)-insertion property, 362 (1, 2)-co wl-insertion property, 93 (1, 2)-l-CD(X)-insertion property, 93 (1, 2)-r2-insertion property, 326, 367 integers, 2 G.interior operator, 242, 243 interior of a set A (in a topology Ti), 3 intersection (of G.filters), 263 intersection (of G.ideals), 244 intersection (of sets), 2 /-irreducible function, 182, 186, 188 irresolute function, 163 (i,j)-irresolute function, 166, 167, 172, 173 p-irresolute function, 172 j-isolated point, 4 isomorphism, 229, 237, 269, 306 G.isomorphism (of GBA's), 272-276, 288, 290, 296, 303-305 isotonicity (of the map X~), 209 G.isotonicity (of the pair X = ()(~1,)(~2)), 210 G.isotonicity (of the pair h = (hi, h2)), 269, 270 italic (letters), 1 L /-large inductive dimension, 124, 192 d-large inductive dimension, 192 (i,j)-large inductive dimension, 111, 112, 115-121, 124, 126, 127, 192 p-large inductive dimension, 111-113, 118120, 122, 123, 126, 127, 192 i(~-largest element, 196 lattice, 201, 240 G.lattice, 201-204, 209, 212-214, 216, 220, 221, 237-240, 243, 261, 267, 269-272, 305 least upper bound (of topologies), 2, 64 left associated topology, 381 left principal G.filter, 264, 265, 267, 271 left principal G.ideal, 249-251,258-261,267, 272, 277, 278, 307 i-LindelSf BS, 336 p-LindelSf BS, 9, 336 i-Lindel6f set, 334, 335
411 linear TS, ix, 63, 319, 326, 367 (i, j)- loc A-insertion property, 107, 108 /-locally closed set, 25, 27, 382 (i,j)-locally closed set, 25-29, 31, 49, 339, 347, 382 /-locally closed set (at a point), 25 (i,j)-locally closed set (at a point), 25, 27, 28 /-locally compat BS, 320, 332, 334, 368 (i,j)-locally compact in Birsan's sense BS, 10, 365, 366 (i,j)-locally (countably) compact in Raghavan and Reilly's sense BS, 10, 336, 365, 366 (i,j)-locally compact in Reilly's sense BS, 10, 336 (i,j)-locally compact in Stoltenberg's sense BS, 10, 141, 337, 365 locally convex linear TS, 325, 326 locally convex (topology), 326, 367 /-locally finite family at a j-dense set, 20 (i,j)-locally Lindel6f BS, 9, 336 (i, j)-locally quasi compact BS, 10, 336, 365 /-locally connected BS, 343 lower Baire space, 380 lower class AX, 13 lower G.homomorphism (of G.lattices), 269, 270 lower topology, x, 370 lower Tl-ordered TS, 370, 371 lower regularly ordered TS, 371 lower semicontinuous function, 4, 333 /-lower semicontinuous function, 4, 14, 15, 331-333, 354, 355 Luzin-Menchoff property (of a fine topology), 318 M
manifold, xii maximal G.filter, 262 maximal G.ideal, 253, 278 maximal connected space, 359 McCallion's duality, 372 (i, j)-meager set, 22 /-meager set, 378 u-meager set, 378 2-Ca-measurable function, 323 Menger-Urysohn formula, 97 metrizable (LTS), 326 modification of Cantor-Bendixson's theorem, 48 modular (Dedekind) G.lattice, 213 G.multiplication, 310-316 multivalued function, x
N
natural BS, x, 3, 19, 31, 127, 129, 370 natural numbers, 2 natural order, 370 natural TS, 3 natural topology, x, 3, 22, 45-47, 64, 65, 159, 168, 175, 369, 370 (i,j)-nearly open function, 174, 180, 181 p-nearly open function, 180, 181 nearness (of topologies), 63, 82-89, 91, 92, 94, 97, 108, 124, 135, 136, 141, 147, 150-154, 156, 158, 160, 161, 164, 165, 168, 170, 171, 179, 183, 360, 382 C-nearness from the outside and from the inside (a family of sets), 353-355 C-nearness (a class of functions), 355, 356 nearly compact space, xi /-network, 107 G.neutral element, 217, 219, 220 (i, j)-nonmeager set, 22 l-nonmeager set, 378 u-nonmeager set, 378 nonsymmetric distance function, ix nontrivial condensation (function), 91 nontrivial Ga(X)-insertion property, 93 nontrivial (i,j)-A-insertion property, 93 non-zero element, 308, 309 G.normal (G.normally included) set, 196, 257 p-normal base, 12-15, 101, 373 /-normal BS, 85, 192, 322 p-normal BS, 6, 7, 12, 85, 104-107, 111, 112, 115, 119, 120, 122-124, 126, 190-192, 318, 336, 373 (q, 2)-normal BS (in potential theory), 358, 359 normally ordered TS, 371 normally ordered subbase, 372 /-nowhere dense set, 17-20, 29-31, 40, 50, 52, 67-71, 77, 78, 80, 88-91, 137, 138, 140, 146, 147, 188, 324, 327, 329, 330 (i,j)-nowhere dense ((i,j)-rare) set, 17-25, 29-31, 35, 36, 39, 40, 48, 50, 52-58, 67-71, 77, 78, 80, 88-90, 130, 131, 133135, 137, 138, 140, 142-144, 146, 147, 173, 174, 180-182, 185, 187, 327-330, 382 /-nowhere dense set (at a point), 21 (i,j)-nowhere dense set (at a point), 21, 58 /-nowhere dense (/-rare) set, 378-380 u-nowhere dense (u-rare) set, 378-380
412
d-open base, 12 /-open covering (of a BS), 303, 335, 336, 365, 366 p-open covering (of a BS), 10, 113, 297, 299 open domain, 36, 37 42, 43 /-open domain, 33, 34, 67-69, 78, 79, 81, 88, 90, 93, 94, 168, 195, 199-201,221, 230, 231 (i,j)-open domain, 6, 16, 31-37, 41-44, 58, 59, 61, 67-70, 78, 79, 81, 88, 90, 94, 167-169, 174-177, 195, 199-201, 221, 222, 230, 231, 316, 343, 382 /-open function, 4, 172, 174, 176, 186, 188, 189, 357 d-open function, 4, 175, 177, 183, 185, 190 g-open function, 163 (i,j)-5-open function, 179-181, 186, 187 open real segment, x, 2 /-open refinement, 336 order defining pair, 372 order of a family (of sets), 125 ordered TS, 320, 370-381 Tl-ordered TS, 370, 371 T2-ordered TS, 371 T3-ordered TS, 371 T31(0)-ordered TS, 372 T4-ordered TS, 371 order preserving real-valued function, 372 order reversing real-valued function, 372 order subspace, 374, 375, 377, 379-381 outer capacity, 337, 355-358 P
(i,j)-paracompact BS, 9, 336 (i,j)-paraLindelSf BS, 9, 336 partial order, 321, 370 partially ordered set (poset), 193-195, 285 partition corresponding to a pair (x, A), 97-99 partition corresponding to a pair (A,B), 110-115, 117, 119, 121, 123, 126 perfect set, 16 /-perfect set, 46, 76, 77, 151, 152, 338 (i,j)-perfect set, 16, 46-49 perfect G.field representation (of a GBA), 296, 297, 303 (i, j )-perfectly normal BS, 8 p-perfectly normal BS, 8, 115, 373 point of tangency (of topologies), 108-110, 363, 364 polar point, 338
presemiopen function, 163 (i,j)-presemiopen function, 171, 172 p-presemiopen function, 171, 172 prime G.filter, 262, 263 prime G.ideal, 253-257, 263, 266, 267, 277, 278, 290, 293-299, 303-305 Problem C.10, 319 proper G.filter, 262 proper G.ideal, 253 /-property, 2 d-property, 2 (i, j)-property, 2 p-property, 2 property l~/~, 321,328 property P, 355 (~-(Choquet)-property, 320, 356-358 ~-property, 356-358 /~-property, 320, 356, 357, 359 pseudobase (of a topology), 9, 64, 148, 184 i-pseudobase, 184 (i,j)-pseudocomplete BS, 147, 148, 164, 184 i-pseudo-open covering (of a BS), 160 (i,j)-pseudo-open covering (of a BS), 136, 145, 154, 157, 160, 162
Q quasi boundary (of a set), 348, 351-353 quasi closed set, 3, 338, 339, 348, 349, 352, 353, 355, 356, 358, 359 quasi closure, 348-353 quasi compact BS, 10 quasi contained (in a set), 337, 348 quasi contains (a set), 337, 348 quasi continuous function, 354 quasi everywhere, 337 quasi 1.s.c. function, 354, 355 quasi u.s.c, function, 354-356 quasi interior, 348-350 G.quasi measure, 241, 242 quasi open set, 3, 338, 339, 348, 349, 352, 353, 355, 356, 358, 359 quasi order, 208 quasi order =:<:, 288-290 quasi ordered set, 193, 195 quasi proximity, x quasi pseudometric, ix /-quasi regular BS, 8, 72, 74, 85, 147, 148, 325, 327, 330, 368 d-quasi regular BS, 8, 65, 72, 77, 87 (i,j)-quasi regular BS, 8, 9, 25, 74, 77, 85, 147, 148, 157, 164, 184 p-quasi regular BS, 9, 65, 72, 77, 184 quasi topology, 319, 337-339, 353, 359 quasi uniformity, x
413 quasi uniformizable BS, 11, 333 quasi uniform space, 10, 11 "quotient" (family of sets), 16 R
rational numbers, 2 real normed lattice, 129 real numbers, 2 reduced G.field representation (of a GBA), 290, 293, 296, 297, 305 /-regular BS, 7, 74, 85, 322, 323, 330, 332, 367, 368 d-regular BS, 7 (i,j)-regular BS, 7, 27, 74, 75, 85, 98, 101, 107, 141, 176, 320, 324, 329, 330, 334, 336, 367, 368 p-regular BS, 7, 99, 101, 178, 319, 326, 335, 336, 338, 367, 370, 372 (i, j)-regular open filter base, 157 regular (2-open set), 340, 346, 347 regular point, 318 G.rejection operator, 239, 240 f-relation, 71, 72, 184
G.ring operations, 310
scattered set, 16 /-scattered set, 47, 48 p-scattered set, 16, 47, 48 script (letters), 2 second category (LTS), 326 second category set, 16 /-second category set, 23, 24, 52, 70, 78, 80, 82, 88-91, 128, 131, 132, 156, 157, 159161, 369 (i,j)-second category set, 22-25, 52-57, 69, 70, 77, 78, 80, 82, 88-91, 129-132, 143, 152, 155, 159, 161, 173, 174, 185, 382 /-second category in itself set, 24, 70, 91, 129, 130, 132, 150, 156, 157, 159, 324, 369 (i,j)-second category in itself set, 22, 24, 54, 56, 70, 82, 91, 129-132, 150-152, 157, 173, 180-182, 184, 321, 324 /-second category set, 379-381 u-second category set, 379-381 /-second category in itself set, 379, 380 u-second category in itself set, 379, 380 /-second countable BS, 48, 191, 192, 334, 358 d-second countable BS, 8,105-107, 113, 115, 121-123, 190 section (of the set IR), 13 semibitopological property, 163, 173, 174 semiclopen domain, 40 i-semiclosure (of a set), 39 (i,j)-semiclosure (of a set), 38, 39, 172, 173 semiclosed domain, 41, 61 i-semiclosed domain, 43, 79, 81, 88, 90 (i,j)-semiclosed domain, 16, 41-44, 78, 79, 81, 88, 90, 382 semiclosed set, 16, 36, 37, 41, 43 i-semiclosed set, 36, 39, 43, 67, 69, 78, 80, 81, 88-91 (i,j)-semiclosed set, 35-43, 50, 60, 61, 64, 66, 67, 69, 78, 80, 81, 88-91, 164, 166, 382 semiclosed partial order, 371 semicontinuous function, 163 i-semicontinuous function, 165, 166, 179 d-semicontinuous function, 166, 179 (i, j)-semicontinuous function, 164-167, 179, 186, 188 p-semicontinuous function, 166, 179, 186, 188 (i,j)-l.u. semicontinuous function, 4, 5, 1315
414 semicontinuous partial order, 371 semihomeomorphism, 163 (i, j)-semihomeomorphism, 172, 173 p-semihomeomorphism, 172, 173 /-semi-interior (of a set), 39 (i, j)-semi-interior (of a set), 38, 39 (i, j)-semineighborhood, 164 (i, j)-seminormal BS, 6 p-seminormal BS, 7, 336 semiopen domain, 41, 61 i-semiopen domain, 43, 79, 81, 88, 90 (i,j)-semiopen domain, 16, 40-44, 78, 79, 81, 88, 90, 382 semiopen function, 163 i-semiopen function, 171 (i,j)-semiopen function, 171, 186 p-semiopen function, 171 semiopen set, 16, 36, 37, 41, 43 i-semiopen set, 36, 39, 43, 67-69, 78, 80, 81, 88-91, 164, 165 (i, j)-semiopen set, 35-43, 50, 60, 64, 66-69, 78, 80, 81, 88-91, 164-167, 171, 172, 187, 188, 382 (i, j)-semiregular BS, 6, 7, 176 p-semiregular BS, 178 semitopological property, 163 sequentially order continuous from above (capacity), 358 separable Banach space, xi separately continuous function, 323 /-separation, 40 set of all points of tangency (of topologies), 363 i-Fa-set, 3, 8, 23, 130, 133, 142, 143, 145, 191, 328 l-F~-set, 379 u-F~-set, 379 i- Gs-set, 3, 8, 23-25, 70, 82, 91, 133, 134, 136-141, 143-147, 151, 152, 323-325, 328, 360-362 l- •5-set, 379 u- Gs-set, 379 Nj-sifter, 149 simple graph (l-graph), 381 Slobodnik property, 321,323 /-small inductive dimension, 108-110, 192 d-small inductive dimension, 192 (i, j)-small inductive dimension, 98-110, 112, 113, 126, 192, 341 p-small inductive dimension, 97-101, 104106, 112, 113, 115, 122, 126, 192 (i, j)-small inductive dimension (at a point), 101-103 smallest G.component, 260, 261
smallest G.filter containing a pair ( B1, B2 ), 263 smallest G.filter containing an another G.filter and an element, 265 smallest G.ideal containing a pair
(B1, B2),
245
smallest G.ideal containing an another G.ideal and an element, 252 /G-smallest element, 196 somewhere dense set, 133 /-somewhere dense set, 22, 52, 67, 69, 78, 80, 88-90, 137-139, 141, 146, 147 (i,j)-somewhere dense set, 22, 52-55, 57, 67, 69, 78, 80, 88-90, 132, 137, 138, 141, 142, 146, 147, 153, 155, 157, 158, 160, 173, 174, 181,382 Sorgenfrey line, 64, Sorgenfrey topology, 64 stable point (of a compact set), 318 G.standard element, 217, 218, 220 Stone BS, 299, 303, 305 Stone family of prime G.ideals, 256, 290299, 304, 305 1-strict Baire space, 160, 162, 189, 322, 330 (1, 2)-strict Baire space, 152-154, 156, 158161, 189, 322, 323, 327-330 strong amalgation, 241 strong Baire space, 360 strong i-Baire space, 150, 323, 325,338, 360 strong (i,j)-Baire space, 150-152, 360, 361 strong compactness (of a topology with respect to another), 366 /-strongly coupled topologies, 76, 150, 151, 361 /-strongly near topologies, 86, 150 /-strongly S-related topologies, 67, 150 strong Ti-order separation axioms (i = 1, 2, 3,4), 371, 372 strong tangency (of topologies), 363-365 strong topology, xii subordination (of two locally convex topologies), 326, 367 G.sublattice, 204, 209, 243, 267, 268 superharmonic function (on IRn), 318 ic-supremum (of a set), 197-202, 257-259 symmetric difference (of sets), 2, 327-329, 331,337, 338, 341, 342, 346, 349, 350
WS-tangency (of topologies), 363-365 terminal endpoint, 381 thinness of a set (at a point), 318, 331-333, 338, 339, 343, 347, 353 9-(B)-topology, 101
415 b-topology, 321 w--topology, 381 w+-topology, 381 topology determined by the norm, ix topological group, 63 topological lemma, 359, 360, 364 topological semifield, xi (i, j)-totally nonmeager BS, 150 two-point set, 3, 382 i-Tychonoff BS, 319, 332-335 U G.ultrafilter, 262, 263, 270, 277, 278, 280 uncountable set 48, 49, 76 i-uniformizable BS, 332 union of a chain (of G.filters), 263 union of a chain (of G.ideals), 244 union (of sets), 2 unit element, 193 /(:-unit element, 196, 197, 204 unit G.filter, 262 unit G.ideal, 244 upper Baire space, 380, 381 upper class B z, 13 upper G.homomorphism (of G.lattices), 269, 270 upper Tl-ordered TS, 370, 371 upper topology, x, 370 upper regularly ordered TS, 371 upper semicontinuous function, 4 /-upper semicontinuous function, 4, 14, 15, 333, 354-356 usual lattice operations, 209, 222 V vertex, 380 W Wallman compactification, xi (2, 1)-weak Baire space, 155-161, 189, 322325, 330, 332, 334 2-weak Baire space, 157-161, 189, 322, 325, 330, 332, 334, 369 weak Baire space (with respect to an initial topology), 321-323 weak tangency (of topologies), 363 weak topology, xi, xii, 321, 366 weakly connected topologies, 94 weakly continuous function, 163 /-weakly continuous function, 170 (i,j)-weakly continuous function, 169, 170, 176, 177
p-weakly continuous function, 170, 177, 178 weakly totally disconnected BS, 9, 384 weight, 335
p-zero dimensional BS in the sense of p-ind, 100, 108, 113, 126, 194, 297, 299, 335, 382, 383 p-zero dimensional BS in the sense of p - I n d , 112, 121, 122, 126, 127, 382, 383 p-zero dimensional BS in the sense of p-dim, 126, 383 zero element, 193, 326, 367 i~;-zero element, 196, 197, 204 zero G.filter, 262 zero G.ideal, 244 /-zero set, 14, 334
101, 104, 303-305,
113, 115,
127, 382,
