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> clusters to p then either , Then'HCI" 'Ct" is a closure operator on X and ( contains afundamental Cauchy net \jf if\jf converges to p. 'V such that (Ec) c) = ) is frequently in Va C W Wbb c Star(p,b) eN. c N. cofinal in D with respect to ~, Since is Cauchy, converges to p. (Proof 'V \jf converges if converges to p and N is a neighborhood of p. Then there are a,b E A with a <* band Star(p,b) c N. Also, there is a residual ReD with respect to ~ =:; with or 'V \jf is not eventually in V this implies that if V E J.l with V < U then either and 'V \jf are Cauchy for each V E E V. Pick W E J.l with W** < U. Since both $ there are WI, W 2 E W with eventually in W WI1 and 'V \jf eventually in W 2. Now = 0 so W;nW; W* < U implies W InW 22 = W~nW; = = 0.
80
3. Transfinite Sequences
Proof: The validity of statement (1) follows immediately from the definition of
clustering and statement (2) is easily verified. To establish (3) we note that for each neighborhood U of p and each a < Ythere Y there is some <Pa(x) E U for some x E Ya' But <Pa C {
We are now ready to define the classes of transfinite sequences that were mentioned aboveo above. Let X be a set and C a class of ordered pairs ('V,p) (\jI,p) where 'V \jI is a transfinite sequence in X and p E X. We call C a cluster class on X if C
THEOREM 3.4 Let C be a cluster class on a set X andfor and for each A eX let CI(A) be the set of all x E X "y..Jith
=
A let Ya be a limit ordinal and define 'Va:Ya \jIa:Ya -t X by 'Va(e) \jIa(e) = a for each e E Ya' By statement (1) of Proposition 3.4 (\jIa,a) ('Va,a) belongs to C and hence a E CI(A) for each a E A. Thus A c CI(A). E
To show CI(CI(A» CI(CI(A» = CI(A) suppose a E CI(CI(A» which ilnplies implies the existence of a transfinite sequence 'V:y \jI:Y -t CI(A) with ('V, (\jI, a) E C. For each e E Y, 'V(e) \jI(e) E CI(A) which implies there is a transfinite sequence 'Ve:Ye \jIe:Ye ---7 A such y, that ('Ve,\V(e» (\jIe,\jI(e» E C. But then the sum L of the transfinite sequences {'Ve} {\jIe} and the point a form a pair (L, a) E C by statement (4) of Proposition 3.4. Therefore a E CI(A) since LeA. Consequently CI(CI(A» CI(CI(A» c CI(A) and hence CI(CI(A» = CI(A). To show CI(AuB) = = CI(A)uCI(B) first note that a E CI(A) implies there is \jI c A and hence 'V \jI c AuB so a E CI(AuB). Similarly a a pair (\jI,a) ('V,a) E C with 'V CI(B) implies a E CI(AuB) so Cl(A)uCI(B) CI(A)uCI(B) c CI(AuB). Finally, p E E Cl(B) (\jI, p) EE C with 'V \jI c AuB. Let M = CI(AuB) implies the existence of a pair ('V' YIl\jl-l(A) = yn'V-1 YIl\jl-l(B) \jI:Y-t Y= yn'V-1 (A) and N = (B) where 'V:y -t X. Then Y = MuN and by statement Since'VM \jIN c B (2) of Proposition 3.4, either (\jIM,P) ('tiM, p) or (\jIN,P) ('VN, p) E C. Since '11 M c A and 'VN either P p E Cl(A) or P p E Cl(B). p E Cl(A)uCl(B). CI(B). Hence P CI(A)uCI(B). We conclude that 0
= CI(A)uCI(B). CI(AuB) =
3.3 Transfinite Sequences and Topologies
81
It remains to show that ('V, ('I', p) E C if and only if \V 'I' clusters to a relative to the topology associated with Cl which will now be referred to as 'to First ('I', a) E C but \V 'I' does not cluster to a relative to 'to 'to Then there is an suppose ('V' 'I' is not frequently in U which implies a open neighborhood U of a such that 'V 'I'(R) c X - U where 'I':y --t X. By statement (2) of residual R c y with 'V(R) \V:Y ~ CI(X - U). But this is a Proposition 3.4, ('I'R, ('VR, a) E C which implies a E Cl(X 'I' clusters to a relative to 'to contradiction since U is open. Therefore 'V Conversely, suppose 'V:y 'I':y ~ --t X clusters to a. Then a E CI(M CI(Me) e ) for each e E {'I'(~) Ie :<::; ~). Therefore there is a transfinite sequence 'Ve:Ye 'l'e:')'e ~ ---7 y where Me = {'V(~) ~ ~}. Me with ('I'e, ('lie' a) E C for each e E y. By statement (3) of Proposition 3.4 we must then have ('V, ('I', a) E C.· c.Proposition 3.4 and Theorem 3.5 set up a one-to-one correspondence between the various topologies a set can have and the cluster classes on the set. It is clear from the definition of clustering that if C 1J and C 2 are two cluster 'tJ and't2 are the associated topologies, that C 1J C C C2 classes and 'tl 2 if and only if C 'tJ. Moreover, it is interesting to note that if (C 1J nC 2) denotes the smallest 't2 C'tl. cluster class that is larger than each of C 1J and C 2 then (C 1JnC 2) is the cluster class associated with 't1n't2. 'tJ n'!2' We conclude this chapter with a characterization of continuous functions in tenns terms of transfinite sequences. Various other interesting mappings (e.g., open mappings, closed mappings, quotient mappings, etc.) can be characterized in a similar manner but this will be left for the exercises.
PROPOSITION 3.5 Let f:X --t ~ Y be a function from a space X into a space Y. Thenfis Then f is continuous if and only if for each transfinite sequence {xa) {x a} iffor in X X that clusters to some p E X, X. {f(x a)} a ) I clusters to f(p). a) is a transfinite sequence in X Proof: Assume f is continuous and suppose {x a} that clusters to some point p E X. Let U be a neighborhood of f(P) in Y Y and pick a neighborhood V of p such that f(V) a} frequently in V fey) c U. Then {x a) implies {f(x a)} in f(V) c U, so {f(x a)} to f(P). a )) is frequently inf(V) a )) clusters tof(P).
a) in X that Conversely, assume that for each transfinite sequence {x a} {f(xa)) clusters to some point p in X, {f(x a)} clusters to f(P). Suppose f is not continuous. Then there is apE X and a neighborhood U of f(P) such thatf(V) is not contained in U for each neighborhood V of p. By Theorem 3.4 there y) of P p such that < is exists a well ordered neighborhood base {Val a < y} compatible with the partial ordering of set inclusion. For each a E A pick x a EE V a such that f(x a) is not contained in U. Then {x a} Va a) clusters to p but {f(x a)} a)) does not cluster to f(P) which is a contradiction. We conclude that f must be continuous. •-
82
3. Transfinite Sequences
EXERCISES 1. Show that X is T 1I if and only if for each pair of distinct points in X there are two transfinite sequences clustering to the two points respectively but neither clustering to the other point. 2. A function is said to be open if the image of each open set is again an open set. Show that an onto function f is open if and only if for each transfinite (p) there is rl (P) sequence {y ~} in Y that clusters to some p E Y and for each q E /-1 {xa} u{f-l(Yj3)} } in U{/-I (y~)} that clusters to q. a transfinite sequence {x a 3. A function is said to be closed if the image of each closed set is a closed set. Show that a function f/ is closed if and only if whenever a transfinite sequence {y ~} clusters to some p E Y there is a transfinite sequence {x Ix a} that is a subset of /-1 ( {y ~ }) that clusters to /-1 (p ). ofrl({y~}) tor1(p). 4. A function is said to be a quotient if whenever the inverse image of a set is f is a quotient if and open, then the set itself must be open. Show that a function [is {y~} only if for each transfinite sequence {y ~} clustering to some p E Y there is a /3} under f/ that is transfinite sequence {x a} contained in the inverse image of {y ~} (p). frequently in each open inverse image of an open set in Y that contains /-1 r 1(P). 5. Show that Corollary 3.2 still holds when the assumption of complete regularity is removed from the hypothesis; i.e., show that any space is compact if and only if each transfinite sequence clusters.
Chapter 4 COMPLETENESS, COFINAL COMPLETENESS AND UNIFORM PARACOMPACTNESS
4.1 Introduction In 1915, A paper by E. H. Moore appeared in the Proceedings of the National Academy of Science U.S.A. titled Definition of limit in general integral analysis. This study of unordered summability of sequences led to a theory of convergence by Moore and H. L. Smith titled A general theory of limits which appeared in the American Journal of Mathematics in 1922. In 1937, G. Birkhoff applied the Moore-Smith theory to general topology in an article titled Moore-Smith convergence in general topology, which appeared in the Annals of Mathematics, No. 38, pp. 39-56. In 1940, J. W. Tukey made extensive use of the theory in his monograph titled Convergence and uniformity in topology published in the Annals of Mathematics Studies series. Tukey worked with objects that were generalizations of sequences that he referred to as phalanxes. They were a special case of the objects that are usually called nets today. An equivalent theory of convergence using objects called filters emerged in the thirties from the Bourbaki group in France. The theory of filters is the convergence theory of choice for many topologers. There are many things to recommend it but it is also very awkward to use in certain situations. For instance, in the treatment of hyperspaces (Chapter 5), the objects of the hyperspace are subsets of the original space. Filters, which are themselves collections of subsets of a space, become awkward to construct in the hyperspace because they are collections of collections of subsets. For some proofs, filters carry along too much baggage. In proofs about the convergence completions of uniform spaces it is sometimes desirable to pick a canvergence object from the original space that is arbitrarily close to a convergence object in the completion. Then conclusions are made about the convergence object in the concl~sions are then shown to hold for the convergence original space and these conclusions object in the completion due to its proximity to the object in the original space. A natural way to attempt this with filters is to restrict each subset in the completion filter to the original space, make conclusions about the restricted filter, then take the closures of the members of the restricted filter to get back to the completion. The problem here is that taking the closure picks up too many
84
4. Completeness, Cofinal Completeness and Uniform Compactness
points. Since the original filter in the completion is not always obtained, it may not be possible to draw conclusions about the original filter in the completion.
time, one should be able to find filter type proofs for the Surely with tirne, theorerns theorems we will be presenting in the sequel, but this will not be our viewpoint. Instead, in some of the areas where the filter type proofs are especially nice, the recasting of the results into filter terminology will be suggested as exercises.
4.2 Nets A non-void set 0D is said to be directed by the binary relation the following conditions hold:
~
provided that
(1) if m,n and d E D with m ~ nand n ~ d then m ~ d, (2) d ~ d for each d in D and (3) if m,n E D then there exists a d in D with m ~ d and n ~ d.
Clearly a directed set is a particular type of partially ordered set. Condition (3) in the definition of a directed set is what sets the directed set apart from an ordinary partially ordered set. Also, it should be noticed that well ordered sets are necessarily directed sets. Some useful examples of directed sets that are not necessarily well ordered are given below. Let X be a topological space and let B be a local neighborhood base for a point p E X. Define the relation ~ on B as follows: U ~ V if U,V E B and V c U. Clearly (B, (B,~) ~) satisfies conditions (1) and (2) above so (B, ~) is a partially (B,~) also satisfies (3) assume U,V E B. Then W = UnV ordered set. To show (B,~) belongs to Band W c U and W c V. Thus U ~ Wand V ~ W, so (B,~) satisfies (3). Another directed set that we will employ frequently is the uniformity itself. Let (X, (X. f.!) J.l) be a uniform space. The relations < (refinement) and <* (star refinement) have already been defined for f.! J.! (see Section 2.1). Clearly (J.!, (f.!, <) and (11, (f.l~ <*) satisfy conditions (1) and (2) defined above so that both are partially ordered sets. That < also satisfies condition (3) above follows from condition (1) of Section 2.1. At this point we are careful to remind the reader that we consider these orderings as "proceeding 'proceeding toward the left" in the sense that in proving condition (3) for <* we will show that if U,V E 11 J.! then there exists aWE 11 J.! such that W* < U and W* < V. By (1) of Section 2.1 there exists a Z E 11 J.! such that Z < U Win and Z < V. By (3) there is a W in 11 J.! with W <* Z. Therefore W <* U and W <* V. Consequently both (11, (J.!, <) and (11, (J.!, <*) are directed sets. 4
'F- 0 of a directed set (D, ~) is said to be residual if whenever A subset R -:j:. m,n E D with mER and m ~ n then n E If C c D such that whenever m EE D E R. IfC there is an nEe n E C with m ~ n we say C is cofinal in D. A net is a function 'V:D \jI:D
4.2 Nets
85
-t X from a directed set D into a space X. In the event D is well ordered then 'V '" ex is simply a transfinite sequence with which we are already familiar. For each a D let xX aa = 'V(a). ",(ex). We will often identify a net", ",(D) = {xa {x a I a ex E Diet net 'V with its range 'V(D) ED}. a}. We say a net {x a} is E D). If the set D is understood we simply write {x {xa)' {xa) frequently in U c X if there is a cofinal C ccD D with {x~ II ~ E C) C} CcU. U. We say {x a) a} is eventually in U if there is a residual ReD with {x c U. {x a} (x yy IlYE 'Y E R} R) cU. a) is said to converge to a point p in X if it is eventually in each neighborhood of p and to cluster to p if it is frequently in each neighborhood of p.
PROPOSITION 4.1 A subset U of a space X is open net in X - U converges to a point of U.
if and only if no
Proof: Assume U is open in X and let {x aa)} be a net that converges to some p E U. From the definition of convergence {x aa)} must eventually be in U. But then {x a) cannot lie entirely in X - U. Conversely, assume no net in X - U {xa} converges to a point of U and suppose U is not open. Then there is apE U every neighborhood of which meets X - U. Let D be a local basis for p and let :s; ~ be the partial ordering of set inclusion on D. We have already seen that D is Vn(X - U). directed with respect to this ordering. For each V E D pick Xv E Vn(X {xv) converges to p. Indeed, Then {xv) {xv} is a net in X - U. It is easily shown that {xv} = {V E D IV c W}. W). Then R(W) is residual in D and let WED and put R(W) = V E R(W). But this is a contradiction since {xv} lies in X - U. Xv E W for each V We conclude that U must be open. •PROPOSITION 4.2 A point p of a space X belongs to the closure of E c X X if and only if there is a net in E that converges to p. Proof: Assume p E Cl(E) CI(E) and let D be a local basis for p. Then D is directed by V E D pick Xv EE V V nE. the partial ordering of set inclusion. For each V nEe Then {vv) {vv} is a net in E. An argument similar to the one in Proposition 4.1 shows that {x a} {xv} converges to p. Conversely, assume (x a) is a net in E converging to a point p E X. Let U be an open set containing p. Then {x a )} is eventually in U which implies U nE nE:t Cl(E). •*" 0. Thus p E CI(E). PROPOSITION 4.3 A point p of a space X is a limit point of E c X and only if there is a net in E - {p} {p) that converges to p.
if
Proof: Let p be a limit point of E c X and let D be a local basis for p. For each {xv) is a net in E - {p) V E D pick Vv EE V neE - {p). {p}). Then {xv} {p} converging to p. {p) converging to p E X then for each open Conversely, if {x a} is a net in E - {p} set U containing p, {x a) a} is eventually in U. But then Un(E - {p}) {p) :t *" 0 so Pp is a limit point of E. •-
86
4. Completeness, Cofinal Completeness and Uniform Compactness
PROPOSITION 4.4 A space X is Hausdorff if and only if each net in X converges to at most one point. Proof: Let X be a Hausdorff space and assume p and q are distinct points of X. Since X is HausdorfL Hausdorff, there are disjoint open sets U and V such that p E U and q Y. If {x a} is a net in X that converges to p then {x a} is eventually in U so it E V. cannot eventually be in V. Y. Thus {x a } cannot converge to q. We conclude that a net can converge to at most one point. Conversely, assume each net in X can converge to at most one point and suppose X is not Hausdorff. Then there exists two distinct points p and q such Un V oF 0. Let B(P) and that for each pair of neighborhoods U of p and V of q, UnV"# B(q) be local bases for p and q respectively and put D = = B(P) x B(q). Define $~ = (A, B) and c == (U,V), put a $~ c if on D as follows: for each a,c E D where a = U c A and V c B. Then (D, $) ~) is a directed set. For each c = = (U,V) E D pick Y. Then {xc} is a net in X. To show that {xc} converges to both p and Xc E Un V. $ a}. q let U E B(P) and V E B(q). Put d = (U,V) and let R(d) = {a E Did ~ Then R(d) is residual in D. If a E R(d) then a = = (A, B) for some A E B(P) and B c V. Moreover, XXaa E AnB ccUnY. UnV. B E B(q) such that A c U and BeY. Therefore, for each a E R(d), XXaa E U and XXaa E V, so {x {xa} a } is eventually in both U and V. We conclude {x {xa} a } converges to both p and q which is a contradiction. Hence X must be Hausdorff. •-
A net
~
X if there exists a
(l)
Such a function A is called a cofinal function. It is easily shown that if C is a cofinal subset of D that the identity function i:C ~ D is a cofinal function and hence a net 'II:D \V: D ~ X restricted to a cofinal subset C is a subnet of'll. of \V. However, in the theory of convergence of nets, we will see that such subnets are usually not very useful. Normally Nonnally we will be interested in subnets whose domains are not subdomains of the domain of the original net. It should be noted that a subnet is a more complex object than a subsequence of a transfinite sequence and that a subsequence of a transfinite sequence is a subnet sub net when the transfinite sequence is considered as a net. The principle advantage of nets is that one works with convergence rather than clustering (compare Proposition 4.4 with Proposition 3.3).
PROPOSITION 4.5 If B is a family of subsets of the space X with the property that the intersection of two members of B contains a member of Band if the net {x a} ex} has a subnet that is a) is frequently in each member of B then {x a} eventually in each member of B.
87
4.2 Nets
Proof: Let {xa {xa I a E A} be the net in the hypothesis above. Since the hy intersection of any two members of B contains a member of B, B is directed by set inclusion. Then E = {(a, F)
E
A x B Ix a
E
F}
is directed by the ordering :c; ~ defined as follows: if (aI, F I) 1) and (a2' F 2) E E I) if a2 :c; then (a2, F 2) :c; ~ (aI, F 1) ~ al and FIe F 2. Now define a function A:E ~ A by A(a, F) = = a for each element (a, F) of E. Clearly A A is cofinal so the net lYe leE E} defined by Ye = xA(e) for each e E E is a subnet of {x a}. {Ye a }. {xa} Next let H E B. Since {x aBE A such a } is frequently in H, there exists apE that x~ E H. Hence e' = (P, (~, H) E E. Let e = (a, F) be an arbitrary member of E such that e' :c; P:c;~ a and F c H so ~ e. Then ~ Ye
XA(a,F) = X Xa = xA(a,F) aE
F c H.
Thus {x a} is eventually in H.-
PROPOSITION 4.6 A net clusters to a point if if and only subnet that converges to the point.
if it has a
Proof: Let p be a cluster point of the net {x aa II a E A} and let B be a local basis for p. Then the intersection of any two members of B contains a member of B and {x {xa} a } is frequently in each member of B. Hence by Proposition 4.5, there sub net {y {y~} {xa} ~} of {x a} that is eventually in each member of B. Hence exists a subnet {y ~} converges to p. {y~} Conversely, assume that the net {x aa }} has a subnet {Y~} {y~} that converges to ~} and A the cofinal function from D to A that p. Let D be the domain of {y {Y~} {Y~}. () E A. Since A is cofinal ~ }. Let U he be a neighborhood of p and let SEA. defines {y ~} there is an element E E D 0 such that A(~) A(P) ~ S() for each ~P ~ E. Since {y {y~} converges to p there is an element E' ~ E with Y£' YE' E U. Now let S' ()' = A(E'). Then ()' ~ S () and xXli'(5' = xXA(E') =YYf'£' E U. Hence {x {xa} p.a} clusters to p. we have b' A(€') =
PROPOSITION 4.7 A space is compact convergent subnet.
If if and only if if each net has a
Proof: By Proposition 4.6 it suffices to show that a space is compact if and only = {x a I a EE A} be a net in the compact space if each net clusters. For this let x = p ~ a. Since A is X. For each a E A let M a be set set of all x~ such that B directed by ~, {Mal {M a Ia E A} Al has the finite intersection property so the family {CL(M A}I also has the finite intersection property. Since X is compact, {CI(M a) Ia E A by Proposition 0.35, there is apE X which belongs to CI(M CL(M a) for each a E A. We will show that x clusters to p. For this let U be a neighborhood of p and let a E A. Since p E CI(M CL(M a), it follows that ManU "# =I:- 0. Hence there is a f3 ~ EE A
88
4. Completeness, Cofinal Completeness and Uniform Compactness
with ~ ~ a and x~ EE U. This implies x is frequently in U. We conclude x clusters to p. Conversely, assume every net in X clusters. Let F be a family of closed sets in X with the finite intersection property_ property. Let G denote the family of finite intersections of members of F. Then G also has the finite intersection property. Since F c G, it suffices to show that the intersection of all members of G is nonempty to show that X is compact. Now G is directed by set inclusion c. FE Xp EE F. Then the assignment F -7 XF Xp defines a net x = {xp} For each F E G pick XF = {XF} in X. By hypothesis, x clusters to some p EE X. Let H, KEG KEG such that K K c H. 4.l Then XK EEKe K c H, so x is eventually in the closed set H. By Propositions 4.1 E H for each H and 4.6, p EE H. Hence p E HEE G. G. We conclude that X is compact.
•
PROPOSITION 4.8 A function f:X -7 Y is continuous at p E X if and if for each net {x a} only lffor p, the net (f(x {f(x a)} a } in X converging to p. a )} in Y converges to j(p). f(p)· Proof: Proof' Suppose f:X -7 Y is continuous at p and let V be a neighborhood of f(P) in Y. By definition of continuity at p, there is a neighborhood U of p in X such thatfCU) V. Since {x a} converges f(U) c Vo verges to p, there is a residual R in the domain of that a } con x~~ E E U for each ~ E f(x~) E E feU) {x a} E R. Thus f<x~) f(U) c V for each ~ EE R, a } such that x (f(x a)} converges to f(P). so {f(x Conversely, assume f is not continuous at p. Then there is an open (Y -V). neighborhood V of f(P) in Y such that every neighborhood of p meets f- 11 (Y-V). Let B be a local basis for p. Then B is directed by set inclusion. For each U EE B Unj-l1 (Y - V). The assignment U -7 Xu defines a net {xu} in X that pick Xu in U0f(f(xu)} is a net in Y - V. Since V is open converges to p. Now the composition {f(xu)} and f(P) EE V, {f(xu)} (f(xu)} cannot converge to f(P). •-
r
In the last chapter, cluster classes of transfinite sequences were discussed and it was shown that the various topologies a space can have correspond to the space . A similar situation various cluster classes of transfinite sequences in the space. holds for nets, only now it is possible to use convergence instead of clustering. In order to define the concept of a convergence class, it is first necessary to have a result on iterated limitso limits. Iterated limits are usually first encountered in calculus where the limits are the limits of ordinary sequences. Iterated limits of nets can be defined in an analogous manner as in what follows. If (D, (D,
4.2 Nets
89
e if d a
°
Next, consider the case where D is a directed set and for each bED, E D, E 88 is ~ = {(o,£) E D and ££ E E 8}. Consider a function another directed set. Let L {(b,£) I1o bED o/:L it is possible that 'I':~ ~ X X where X X is a topological space. Then for each bED EDit '1'8 = {'I'(O,£) the net 0/8 {o/(b,£) I1££ E E 8} converges to some point p 8 E X. We also denote p 8 by lima o/(b,a) and say that the limit of 0/8 8. lima'l'(O,a) '1'8 exists and equals p 8. Furthermore, it is possible that the net q><1> = 81 bED} = {p 81o ED} also converges to some lim8Iima'l'(0,a) 'I' point p E X. In this case we say that p is the iterated limit lim8lima o/(b,a) of 'V with respect to b and a. Let P = D x n {E 81 bED} fl{E810 ED}..
°
°
°
THEOREM 4.1 Let D be a directed set and for each bED let E E Diet E88 be Land another directed set. Let ~ and P be defined as above. Then there exists a net 'A:P f:L ~ X A:P ~ L ~ such that for any function f:~ X for which the iterated limit lim8limJ(b,a) © 'A A converges to the iterated limit. lim8Iima!(0.a) exists,f exists,f© Proof: Define 'A:P A:P ~ L ~ as follows: for each (b, (0, d) E P put 'A(b, A(O, d) = = (b, (0, d(b». d(O». Suppose lim8limJ lim8lima! (b,a) (o,a) = = q and U is an open neighborhood of q. We must find (0, d) of P such that if (b, (0, d) < (B, (~, g) thenf© then f © 'A(B, A(~, g) E U. Pick YE yE D a member (b, such that lim~f(B,v) liml3f(~,v) E U for each B ~ following y and then, for each such B ~ pick d(B) E~ such thatf(B'lv) d(~) E EI3 thatf(~.v) E U for all v following d(B) d(~) in E~. EI3' If B ~ is a member d(~) be an arbitrary member of E 13' If (B, (~, g) > of D D which does not follow y let d(B) E~. ~ > y, hence lim~f(B,v) liml3f(~,v) E E U, and since g(~) d(~) we have f © (y, d), then B g(B) > deB) A(~,g) = f(~, g(~» E Uo U. We conclude thatf© 'A 'A(B,g) f(B, g(B» A converges to q.-
u.
Let C be a collection of pairs (q>, (<1>, p) of nets
°
~ and P and 'A A be defined as in Theorem 4.1. directed set. Let Land f:"L ~ X for which lim8limJ Then for each function functionf:~ lim8lima! (b,a) (o,a) exists, (j© A,p) EE C.
THEOREM 4.2 Let C be a convergence classfor class for a set X andfor and for each A c X let CI(A) be the set of all p E HCI" is E X X with ('V' ('I'. p) E E C and 0/ 'I' cA. c A. Then "Cl" X and ('V' a closure operator on X ('I'. p) EE C if and only if'V if 'I' converges to p relative to the topology associated with Ct. C/. The proof of Theorem 4.2 is left as an exercise (see Exercise 2). A net in a space X is said to be a universal net if for each A c X the net is eventually in A
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4. Completeness, Cofinal Completeness and Uniform Compactness
or eventually in X-A. Universal nets have the property that if they are frequently in a set that they must eventually be in the set. Consequently, universal nets converge to each of their cluster points. PROPOSITION 4.9 Every net has a universal subnet.
Proof: Let x = {xa Ix 0.1I a a. ED} be a net in X and let
Let S be the set of all collections of sets in X that contain
THEOREM 4.3 A space is compact if if and only if each universal net nef converges. Proof: Let X be compact and let let x be a universal net in X. By Propositions must also converge. 4.6 and 4.7, x must cluster. But a universal net that clusters tnust Hence x converges. Therefore compactness implies that each universal net converges. Conversely, if every universal net in X converges, then by Proposition 4.9, every net has a universal subnet so every net in X has a convergent subnet. Then by Proposition 4.7, X must be compact. •unifonn space. A net {x a} a.} in X is said to be Cauchy if it is Let (X, f..l.) f.l) be a uniform eventually in some sphere of radius U for each U EE ,..1. f..l.. {x a} a.} is said to be
4.2 Nets
91
E !l. cofinally Cauchy if it is frequently in some sphere of radius U for each U E IJ-. The proofs of the following two propositions are left as exercises (Exercise 3).
uniform space (X,!l) PROPOSITION 4.10 A net {xa} {x a } in a un(form (X, IJ-) is Cauchy if and only if iffor IJ- there is a U E U such that {x a} is eventually in U, } for each U E !l U a a J in (X, !l) PROPOSITION 4.11 A net {x a} J.l) is cofinally Cauchy if and only lffor IJ- there is aU a U E U such that {xa} {x a } isfrequently in iffor each U E !l ill U. U
EXERCISES
OJ be a net in the pseudo-metric space (X, 1. Let {xa {x a I aa. EED} (X~ d). Show that {xa {x a}J X if and only if the net {y a I a ED} ~ defined by Ya = d(p~ converges to p E E {Ya a. E 0 J, d(p, x a) for each aa. EE 0, D~ converges to zero. 2. Prove Theorem 4.2 (see Theorem 3.4). 3. Prove Propositions 4.10 and 4.11.
Let {xa Ia. E D}beanetinR(thereals). O} be a net in R (the reals). Ifa<~inDimpliesxa~X~(xa If a. < ~ in 0 impliesxa ~ XI3 (xa 4. Let{xalaE xl3), then {x ~ x~), {xu} a } is said to be monotone increasing (decreasing). Show that a monotone increasing net that is bounded above, or a monotone decreasing net that is bounded below converges. FILTERS 5. A filter on a set X is a collection F of subsets of X satisfying the following properties: (1) F is closed under finite intersections, (2) the empty set does not belong to F, and (3) every subset of X containing a member of F belongs to F. Show that the following three examples of filters satisfy properties (1) through (3) above. E X. Let F denote the The Neighborhood Filter: Let X be a space and let p E neighborhood base at p.
The Filter Associated with a Net: Let {xa {x a I aa. EO} ED} be a net. Put F = {F c X IxXI3~ E ReD}. E F for each ~ E E R for some residual ReO}. E A} A J be a non-empty family of filters on X. The Intersection Filter: Let {F a I aa. E Then put F = = (IF nF u. a.
6. A filter is said to converge to a point p E X if each neighborhood of p belongs to F. F is said to cluster to p if every neighborhood of p meets every member of F. A filter base is a collection B of subsets of X satisfying
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4. Completeness, Cofinal Completeness and Uniform Compactness
properties (1) and (2) in the definition of a filter above. Clearly, the collection F of all subsets of X containing a member of B is a filter called the filter generated by B. The filter base B is said to converge to p E X if the filter generated by B converges to p and to cluster to p if the filter generated by B clusters to p. Show the following:
(a) U c X is open if and only if no filter base in X - U converges to a point of U. (b) p E Cl(E) if and only if there is a filter base in E that converges to p. (c) p is a limit point of E c X if and only if there is a filter base in E - {p} that converges to p. 7. Show that a space is Hausdorff if and only if each filter canverges converges to at most one point. 8. Filters can be compared in the following way: Let F and G be two filters on X. F is said to be finer than G and G is said to be coarser than F if G c F. In :f:. G then F is said to be strictly finer than G and G is strictly addition if F =Fcoarser than F. Two filters are said to be comparable if one is finer than the other. Show the following: (a) If a filter F clusters to a point p E X then there is a filter G converges to p. finer than F that canverges (b) A space is compact if and only if each filter clusters. (c) A space is compact if and only if each filter is contained in a convergent filter.
4.3 Completeness, Cofinal Completeness and Uniform Paracompactness We have already encountered the concept of completeness in metric spaces (Chapter 1, Section 6). Just as with metric spaces, completeness plays a fundamental role in the theory of uniform spaces. The concept of completeness in metric spaces generalizes in a natural way to uniform spaces. The next three propositions show that Cauchy nets and cofinally Cauchy nets behave in a manner analogous to Cauchy sequences and cofinally Cauchy sequences in metric spaces. Their proofs are also left as exercises (Exercises 1 and 2). PROPOSITION 4.12 A convergent net is Cauchy. PROPOSITION 4.13 A net that clusters is cofinally Cauchy. PROPOSITION 4.14 If a Cauchy net clusters to P p then it also converges to p.
4.3 Completeness, Cofinal Completeness and Uniform Paracompactness
93
The following theorem is the net version of Theorems 3.1, 3.2 and 3.3. It appeared in the 1971 paper titled On Completeness (Pacific Journal of Mathematics, Volume 38, Number 2, pp. 431-440). The proof is similar silnilar to the 3.l, 3.2 and 3.3, so we leave it as an exercise (Exercise 6). proofs of Theorems 3.1,
THEOREM 4.4 (N. Howes, 1971) A space is paracompact (resp. LindelOf or compact) if and only if each net that is cofinally Cauchy with Lindelof respect to the u (resp. e or ~) uniformity clusters. or~) A uniform space is said to be complete if each Cauchy net converges. It is said to be cofinally complete if each cofinally Cauchy net clusters.
COROLLARY 4.1 A co.finally cofinally complete uniform space is conlplete. complete. The Lebesgue property for metric spaces also generalizes in a natural way unifonn spaces. We say (X, 11) f.l) has the Lebesgue property if for each open to uniform J.l such that V can be refined by the covering covering V of X, there is a U E 11 consisting of spheres of radius U. An equivalent characterization of the Lebesgue property is the following:
PROPOSITION f.l) has the Lebesgue property if and only if PROPOS1T10N 4.15 (X, 11) each open covering of X has a refinement in 11 f.l (i.e., 11 J.l is fine and X is paracompact). W Proof: Assume each open covering V of X has a refinement say U in 11. J.l. Let W <* U. Then the spheres of radius W also refine V J.l) has the Lebesgue V so (X, 11) property. Conversely, if X has the Lebesgue property and V is an open J.l such that the spheres of radius U refine V. Clearly covering of X, pick U E 11 then U refines V.· PROPOS1T10N 4.16 A compact Hausdorff uniform space has the PROPOSITION Lebesgue property.
Proof: Let (X, 11) J.l) be a compact Hausdorff uniform unifonn space. Then by Theorem 2.7 ~ is fine. Since X is paracompact, it has the J.l = B and Lemma 3.7 we have 11 4.l5.· Lebesgue property by Proposition 4.15.· PROPOSITION 4.17 A uniform space with the Lebesgue property is cofinally complete. Proof: Let (X, 11) J.!) be a unifonn space with the Lebesgue property. Suppose (X,J.!) is not cofinally complete. Then there exists a cofinally Cauchy net {x (X,ll) (x u} uI that does not cluster. For each p EE X let U(P) be an open set containing p such (X, 11) has X}. Since (X,IJ) that {xu} is eventually in X-U(P) and put U= {U(P)lpE Xl. V Et:: 11. the Lebesgue property, by Proposition 4.15, U has a refinement V f..l. Since
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4. Completeness, Cofinal Completeness and Uniform Compactness
{xu} {x a } is cofinally complete it is frequently in V for some V EE V. But VV c U(p) U(P) for some P EE X so {xu} {x a} cannot be eventually eventuall y in X - U(P) U (P) which is a contradiction. Thus (X, f.!) Il) is cofinally complete. •
eX, f.!) Il) is said to be precompact or totally bounded if A uniform space (X, each uniform covering has a finite subcovering. Clearly each precompact metric space is a precompact uniform space.
THEOREM 4.5 A uniform space is precompact if and only if each net has a Cauchy subnet. Proof: Assume each net in the uniform space (X, Il) Jl) has a Cauchy subnet and Il such that {S(x, U) I x EX} suppose X is not precompact. Then there is a U E f.! p 2 does not belong to has no finite subcovering. Pick PI, P 2 E E X such that P S(p 1, U). Next assume {p 1 . . . Pn} has been defined such that for each pair i,j S(Pi, U). Since u{S(P 1, U) ... with i < j ::::; ~ n we have Pj does not belong to S(Pi' S(pn, U)} i:cf. X we can choose Pn+l E S(p" U) for S(pn' E X with Pn+l not belonging to S(Pi' n. Consequently it is possible (by induction) to construct a each i = = 1 ... . non. sequence {Pn} cC X such that for each pair of positive integers i,j with i ,,jj we Pj does not belong to S(Pi' S(Pi, U). have Pi
ex ED} be such a Cauchy subnet of the sequence {Pn} for Let x = {x ua I a x a = PA(U) PAra) for each aex EE D. Let V VEil some cofinal function A:D ~ N such that Xu E f.l POE X such with V <* U. Since x is Cauchy there is a residual ReD and a POE PAra) EE S(p o,V) o,v) for each a ex EE R. Since A is a cofinal function, A(R) is that PA(U) o,V)o Pick positive integers cofinal in N which implies {Pn} is frequently in S(p S(po,v). thatpkoPm S(Po,V). Then S(Pk,V)US(po,V) S(PkoV)US(PO,V) c S(Pk, S(Pko U). k,m with k < m such that Pk, Pm EE S(po,V). But then Pm EE S(Pk, S(Pko U) which is a contradiction. We conclude (X, f.l) Il) is precompact. Il) is precompact and let x = = {x Ua},}, aex EE D, be a net Conversely, assume (X, f.l) E f.!, in X. If U E Il, then U has a finite subcovering, say {V lUI1 ..• ... V UN}' Ix a} N }. Since {xu} cannot be eventually in X - V Uj for j = 1 ... N, it must be frequently in one of Il) is cofinally Cauchy. By Proposition 4.9, them. Therefore, every net in (X, f.l) {xu} {x a } has a universal subnet {XA(~)} IXA(~)} where A:B ~ D is a cofinal function from a VEil. IXA(~)} is cofinally Cauchy, there exists directed set B into D. Let V E f.!o Since {XA(~)} {XA(~)} is frequently in V. But then {XA(~)} must eventually a V EE V such that {XA(~)} IXA(~)} is Cauchy.· be in V, so {XA(~)} 0
•
0
PROPOSITION 4.18 A closed subspace of a complete uniform space is complete.
PROPOSITION 4.19 A uniform space is compact if and only if it is complete and precompact.
4.3 Completeness, Cofinal Completeness and Uniform Paracompactness
95
The proofs of the two propositions above are essentially the same as the proofs of Propositions 1.23 through 1.25 with sequences being replaced by nets. They are left as an exercise (Exercise 5). In a 1977 paper titled A note on uniform paracompactness that appeared in the Proceedings of the American Mathematical Society (Volume 62, Number 2, pp. 359-362), M. Rice introduced the concept of uniform paracompactness which is defined as the property that every open covering has a uniformly V of a covering U is said to be locally finite open refinement. A refinement V uniformly locally finite if there exists a uniform covering W such that each member of W meets only finitely many members of V.
THEOREM 4.6 A uniform space is cofinally complete if and only if if it is uniformly paracompact. Proof: We will prove that a uniform space (X, f.l) Il) is cofinally complete if and only if each directed open covering of X is uniform and then use the equivalent fonn of uniform unifonn paracompactness given in Exercise 7(b) to finish the argument. To prove the sufficiency, sufficiency. assume each directed open covering is unifonn uniform and (X, f.l) Il) is not cofinally complete. Then there exists a cofinally Cauchy suppose (X" H a = {'V(o) net 'V:D \V:D ~ X that does not cluster. For each a E D put H {o/(D) I 0D ~ a} and let F = CI(H = 0 since'll since \V does not cluster. For each a E D put Faa = et(H a). Then nF a = Ua =X - F Fa. Il. But a. Then U = {U a} is a directed open covering of X, so U E f.l. 0/ is eventually in X - U a which is a contradiction. Therefore, for each a E D, 'V Jl) is cofinally complete. (X, Il) Conversely, assume (X, Jl) Il) is cofinally complete. Let U = {U a II a ED} be a directed open covering of X where (D, <) is a directed set. If X = U a for Faa = X - U a =f. some a E D then U E Il Jl so assume F ::j:. 0 for each a. Put E = Fa. {(F a,x) a~) II a E D and x E Fa} and for each a let D. F 8 = {o/(F Ix E 8 , x) F o}, so V c U o. V < U, so UE U E J.l.• Fo},soVcU U,SO Il.e . Therefore, V<
EXERCISES 1. Prove Propositions 4.12 and 4.13. 2. Prove Proposition 4.14. J.!) be a uniform space and let U E Jl. 3. Let (X, Il) Il. A c X is said to be IJ-small II-small if A c U for some U E U. A is U-Iarge if its complement is U-smalI. U-small. A
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4. Completeness, Cofinal Completeness and Uniform Compactness
~ there is an HE collection H of proper subsets of X is heavy if for each U E J.l HE H that is U-Iarge. U-large. Show that X is complete if and only if each heavy covering has a finite subcovering.
4. A collection U of proper subsets of X is said to be bound to another X. collection V if for each finite subcollection W of U that does not cover X, 7= X in V. A collection that is bound to a neither does Wu {{V} V} for each V V ::j; heavy collection is called heavily bound. Show that a uniform space is cofinally complete if and only if each heavily bound open covering has a finite subcovering. 5. Prove Propositions 4.18 and 4.19. 6. Prove Theorem 4.4.
UNIFORMLY PARACOMPACT SPACES 7. [M. Rice, 1977] The following statements are equivalent: (a) X is uniformly paracompact. (b) Each directed open covering is uniform. (c) If U is an open covering of X, there exists a uniform covering V such that U I V has a finite subcovering for each V E V. 8. [M. Rice, 1977] A locally compact uniform space is uniformly paracompact if and only if it is uniformly locally compact. A uniform space (X, J.l) ~) is said to ~ consisting of compact be uniformly locally compact if there exists a U E f.l sets.
UNIFORML Y PARACOMPACT METRIC SPACES UNIFORMLY 9. [M. Rice, 1977] The collection of points of a uniformly paracompact metric space that admit no compact neighborhood is compact. 10. [A. Hohti, 1981] A necessary and sufficient condition for a metric space to be uniformly paracompact is that there exists a compact K c X with X Star(K,c:) is uniformly locally compact for each c:£ > o. O. Star(K,£) 11. [A. Hohti, 1981] If (X, d) and (Y, 8) are uniformly paracompact metric spaces, then (X x Y, d x 8) is uniformly paracompact if and only if one of the following conditions hold: (a) either (X, d) or (Y, 8) b) is compact or (b) both (X, d) and (Y, 8) b) are locally compact. ex be an infinite ordinal. Put H(a) == 12. [A. Hohti, 1981] Let a = ([0,1] x a)/E Where
4.4 The Completion ofa Uniform Space
97
P I (x) = 0 = PI xEy if and only if x = y or PI PI (y). Define a metric d on H(a) by P I (x) + PI (y) otherwise. d(x,y) = Ip 1I (x) - PI PI (y) I1if P2(X) = P2(Y) or d(x, y) = PI Then (H(a), d) is the hedgehog metric space with a spines. H(a) is a uniformly
paracompact metric space that is not locally compact. It is therefore a counter example to a statement by P. Fletcher and W. Lindgren in 1978 that the product of a uniformly locally compact space with a C-complete (uniformly paracompact) space is C-complete.
4.4 The Completion of a Uniform Space
As seen in Chapter 2 (Section 2.1, Exercise 1), every metric space is a uniform space. Our first proposition below states that a complete metric space is also complete when considered as a uniform space. This may not seem surprising at first glance, but it might if we rephrase it in the following manner: In a metric space, the convergence of all Cauchy sequences forces the convergence of all Cauchy nets (of all cardinalities). The proof of this proposition is left as an exercise (Exercise 1). PROPOSITION 4.20 A complete metric space is also complete when considered as a uniform space.
Our next result will show that Proposition 4.20 can be generalized to uniform spaces; i.e., Le., that the convergence of all Cauchy nets on a certain ordered set forces the convergence of all Cauchy nets. We will then use these specialized nets to construct a completion of the uniform space from equiValence equivalence classes of these specialized nets in a manner similar to the construction of the metric completion in Chapter 1. Our first step will be to J.1) be a uniform space and let v be define these specialized nets. For this let (X, J.l) a cofinal subset of J.1 J.l of least cardinality. Then v is a basis for J.1. J.l. Next, well order v by some ordering < such that the cardinality of each initial interval is less than the cardinality of v. From the proof of the Ordering Lemma, it can be seen that there exists a cofinal subset A of v (with respect to the ordering <*) such that the well ordering < is compatible with <* on A (i.e., U <* V V implies that V V < U).
J.l of least cardinality such that the well Then A is a well ordered basis for J.1 ordering < is compatible with the directed ordering <* and the cardinality of each initial interval (with respect to <) is less than the cardinality of A. We call J.l and a net {xa I1 a E A} C X a fundamental net A a fundamental basis for J...l with respect to A. Let {x a Ia ED} be a net in X and let (E, <) be a directed set. A function ~:E ~ D is called a compatible function if whenever a, ~ E E with {y 13 I ~ E E} c X where a < ~ then ~(~) is not strictly less than ~(a) in D. A net {y~ ~(13) for each ~ E E is said to be contained in the net {x a}. Note that a y 13~ = = x ~(~) cofinal function is a compatible function so a subnet of a net is contained in the 1
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4. Completeness, Cofinal Completeness and Uniform Compactness
net. Also note that a compatible function need not be cofinal so that a net {y ~j3 }} contained in a net Ix {x a} need not span the net {x a} as a subnet would. THEOREM 4.7 Each Cauchy net
\jf) Let
4.4 The Completion ofa Uniform Space
99
(from metric spaces) and some of their topological consequences in uniform spaces as soon as possible. Strictly speaking, the completion of a uniform space can be introduced without introducing the concept of a subspace, but the most useful way of thinking of the completion is, that it is a (perhaps) larger uniform space that contains the original uniform unifonn space as a subspace. Consequently, we now give a simple definition of a uniform subspace, but will revisit the concept in the next chapter in a more formal setting.
f.l) can be given a uniform structure that A subset A of a uniform space (X, 11) is derived from the uniformity 11 f.l in the following way: for each U E E 11 f.l put UA = {VnA IV EE U}. Then let IlA f.lA = {U A IU f.l}. It is easily shown that IlA f.lA is a {UnAIU U EE Ill. uniformity on A. IlA unifonnity f.lA is called the uniformity induced on A by the uniformity 11. f.l. f.lA is also said to be the unifonnity f.l relativized to A. It should be noted IlA uniformity of 11 IlA causes the identity mapping iA:A -7 ~ X to be unifonnly uniformly uniformity J.lA that the unifonnity continuous. Our next task is to construct a completion for (X, 11) J.l) from the fundamental Cauchy nets. For this let L be the set of all fundamental Cauchy nets in X and - 'V \jf if for each U E E A there is a V UE define an equivalence relation - on L by
VA U~
= {$' {' E E x'i $ is eventually in V U for each $ E E $'}. '}.
A
Il~ = = {U {U~ I U EEll} Then f.lA f.l} is a basis for a uniformity for X'. Let 11' J.lP be the uniformity generated by J.lA. J.l') is a uniform space. Note that if x E X Il~. Then (X', 11) there is a unique $' E X' that consists of all fundamental Cauchy nets that <1>' E converge to x. For each x EE X let i(x) denote the unique equivalence class $' <1>' ~ X' is well defined and since X is whose members converge to x. Then i:X -7 Hausdorff, i is one-to-one. Let U Eiland E J.l and U E U. It is easily shown that i(V) i(U) = = UAni(X). f.l. Then for each U' E 11' J.l' U~ni(X). Define i(U) = {i(V) Ii(U) I U E U} for each U E 11. A U~ E J.lA Il~ that refines U' so UAni(X) U~ni(X) = {UAni(X) {U~ni(X) I U E U} = i(U) there is a U refines U'ni(X). But then U refines j-l(Uni(X)) so i-1(U'ni(X)) E 11. J.l. Consequently i(X) is a uniform subspace. It is easily shown that i(X) is dense in X'. <1>' and 'V' \jf' be distinct elements of X'. Then To show X' is Hausdorff let $' there exists $
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4. Completeness, Cofinal Completeness and Uniform Compactness
q>'.. There is aWE Wand residual P,Q cC A A with 8(P), q>(Q) Let 8 E
A A
)
Next let {
A
EVa' Since i(X) is dense in X', for each a E A there is an XXaa E X with i(xaa ) EVa. Then {i(x aa )} )) is a fundamental net in i(X). It is easily shown that {i(x {i(xa)} a )} is Cauchy and consequently converges to some p' E X'. Let N be a neighborhood A A with Star(p', bA) c N. Pick a E A with a <* b. Since {i(x {i(xa)} of p' and b E A a )} A A aCE A A with c <* a and i(x i(xJ converges to p' there is acE c ) E Star(p', a ). Thus p' and A A i(xJ i(xc) {q>~} is eventually in i(x V~ and {
W:
W: W:nV:
fJ.) is called the completion of (X, f.l). fJ.). Since the function i is one-to(X', f.l) one and uniformly continuous we often identify X with the dense uniform subspace i(X) of X'. In this case, i(x) = xX for each xX E X; i.e., the function i is the identity mapping on X. Using this identification, we notice that for each U U' nX = {U'nX {U'nX I U' E U'}. E fJ. J.l there is a U' E fJ.' f.l' such that U = i-I (U'), so U = U'nX Also, if V' E fJ.' f.l' then V' nX E ,.1. fJ.. We record these results as:
THEOREM 4.8 Every uniform space has a completion; i.e., if (X, fJ.) J.l) is ij(X, space, there is a complete uniform space (X', fJ.') a uniform space f.l') and a one-to-one uniformly continuous function i:X ~ X' such that i(X) is a dense uniform X'. subspace of X', 9
Theorem 4.8 corresponds to Theorem 1.15 for metric spaces. Theorems 1.16 and 1.17 l.17 can also be generalized to uniform spaces as follows:
4.4 The Completion of a Uniform Space
101
continuous function on a subset A of THEOREM 4.9 iff Iff is a uniformly continuousfunction a uniform space X into a complete uniform space Y then f has a unique f' to C/(A). uniformly continuous extension extensionj' CI(A).
Proof: Let f.1 be the uniformity on X and v the uniformity on Y. For each x ProoF Let!l ex) that A let {x a} be the constant fundamental Cauchy net (x a = x for each a) converges to x and for each x E CI(A) - A pick a fundamental Cauchy net {x a } c A that converges to x. Let V E v. Since f is uniformly continuous., continuous, there a,b E A with a.,b a,b E U for some U E U., U, then exists a U E 11 Jl such that whenever a.,b f(a),f(b) f(a), f(b) E V for some V E V. But then the net {f(x a)} a )} is Cauchy in Y for each x E CI(A). Since Y is complete, {f(x a )} converges to some x' E Y. j':CI(A) -7 j'(x) = x' for each x E CI(A). Clearly,j'(a) Define f':CI(A) ~ Y by f'(x) Clearly, f'(a) = f(a) for j' is and extension of.f. of f. each a E A so f' E
To show f' j' is well defined, let x E CI(A) and let {Ya} be another fundamental Cauchy net converging to x. Suppose x' "# -j:. y'. Then there exists a S(x',v)nS(y',v) = 0. By the uniform continuity of f, there exists a V E v with S(x',V)nS(y'.,V) U E 11 Jl such that a,b E U E U implies f(a)., f(a), feb) E V for some V E V. Since x, there exists a ~ such that a ex > ~ implies both {x a} and {Ya} converge to x., x a,Ya E U for some U E U which in turn implies f(x a), f(Ya) E V for some V S(x',V) and {f(y {f(Ya)} E V. But this is impossible since {f(x a)} is eventually in S(x'.,V) a)} is S(y',v). Hence x' = y' so f' j' is well defined. eventually in S(y',V). j' is uniformly continuous on CI(A), let W To show f' W E v and pick V E v with V* < W. Since f is uniformly continuous on A? J.l such A, there exists a U E 11 that whenever a,b E A with a,b E U for some U E U, then f(a), f(b) E V for f(a),f(b) Z* < U. Let x.,y X,Y E CI(A) such that x,y X,Y E Z for some V V E V. Let Z E 11 Jl with z* some Z E Z. Since {xa} converges to x., y., {f(x a )} x, {Ya} {Yu} converges to Y, {fey a)} converges to y'., y', it is possible to choose a ~ such converges to x', and {fCY ex > ~ implies x a,x E Z 1, I, and YaS j, Z2 E Z, and x'., x',f(x that a Ya,Y E Z2 for some Z 1, f(x a) a) VI1 and y', y',f(ya) V2 for some VI, Vj, V V2 E V. Then Xa,Ya E Star (Z,Z) c U E V f(ya) E V a), f(Ya) x',y' E for some U E U. Hence f(x a ), flY a) E V for some V E V. But then x',Y' Star(V,V) c W for some W WE Therefore,x,y E W. Therefore, x,y E E Z which impliesj'(x),j'(y) implies f'(x), f'(y) j' is uniformly continuous. E W, so f'
To show f' j' is unique, suppose F is another uniformly continuous extension of ffrom A to CI(A). Let x E CI(A) - A. Then {x a } converges to x. Since both f' f' (x) and {F(x a) } j' and F are continuous, it follows that {f'(x {j'(x a)} {F(xa)} a )} converges to j'(x) {xa} {j'(x a)} )} = = {F(x {F(xa)}' converges to F(x). But {x a} cC A implies {f'(x a) }. Since a )) = {f(x ua)} j'(x) = = F(x), so f' j' is a net in a Hausdorff space has (at most) a unique limit, f'(x) unique. •11') THEOREM 4./0 4.10 For each uniform space (X, 11), IJ), its completion (X', J.l') is unique; i.e., if (X', (X/\, 11') J..L A) is another completion of (X, 11) J.l) then there is a uniform unifornl homeomorphism h:X' ~ -7 X/\ X' that keeps each point of X fixed.
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Proof: Let j:X ~ X X' denote the uniform hOlncolnorphisln homeomorphism that Inaps maps X into the cOlnpletion completion X X'. By Theorem 4.9 there exists a unique unifonnly uniformly continuous extension }':X' j;:X ~ X j" is an extension of j we have Jj = jf © i where i:X X'. Since Since}' ~ X' denotes the unifonn X uniform hOlneolnorphism homeomorphism that Inaps maps X X into its cOlnpletion completion X'. Similarly, there is a unique uniformly continuous extension t:X' iA:X'" ~ X" of i. Then i = (© i'" © j. A i'(j'(x')). Then g' = }' X' is For each x' E X' put g'(x') = i"'(j'(x'». j' © i( so g':X' ~ X X then g'(x) = = i (x) = uniformly continuous. If x E X = {(j'(x» iA(j'(x» = = {(j i"'U (x» = = x so g' is an extension of the identity map i:X ~ X'. But by Theoreln Theorem 4.9, this extension is unique so g' == i' (the identity map on X'). Hence j' © {i = = i' so {i == (j')-l. Hence}' (j'r i . But X' are unifonnly uniformly homeomorphic. •then X' and X'" A
A
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EXERCISES 1. 1, Prove Proposition 4.20.
2. Show that the cOlnpletion completion of a uniform unifonn space is cOlnpact compact if and only if the precompacr. uniformity is precompacL 3. Show that a complete subspace of a Hausdorff uniform space is closed.
4. [K. Morita, 1951] Let 11* ~ * be the uniformity of~. Jl, of!J.X. For each U = {U a} a I in 11, fJX -- Clf1X(X CI~(X - U a) for each u. Jl} is put U' = {U~} {U~ I where U~ = !J.X a. Then {U" {U'II U E III a basis for 11*. Jl *. CAUCHY FILTERS AND WEAKLY CAUCHY FILTERS A filter F in a uniform space (X, 11) Jl) is said to be Cauchy if for each U E 11, Jl, there exists an F E F such that FeU for some U E U. It is said to be weakly Cauchy if for each U E 11, Jj-, there exists a U E U with UnF t= aU of- 0 for each F E F. 5. Show that (X, 11) Jj-) is complete if and only if each Cauchy filter in X converges. complete if and only if each weakly 6. [N. Howes, 1971] (X, 11) f.!) is cofinally cOlnplete Cauchy filter in (X, (X? u) clusters.
7. [H. Corson, 1958] 19581 X is paracompact if and only if each weakly Cauchy filter in (X, u) clusters.
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4.5 The Cofinal Completion or Uniform Paracompactification Since each uniform space has a unique completion. it is natural to define a unifonn space (X~. ~) if (X'. Jl') f..l') to be a cofinal completion of the uniform space (X. f..l) (X'. Jl') f..l') is cofinally complete and (X. Jl) f..l) is uniformly homeomorphic to a dense ~ 'When does (X, uniform subspace of (X', (X'. J.l') f..l') and to ask: "When (X. J.l) f..l) have a cofinal completion. and if a cofinal completion exists, exists. is it unique?" cOlnpletion, unique?" Providing answer~ to these questions is the objective of this section. answers
It turns out these same techniques can be used to provide answers to a variety of other questions asked by K. Morita and H. Tamano. In the late fifties, fifties. '~What is a necessary and sufficient Tamano considered the following question: "What condition for a uniform space to have a paracompact completion?" completion?" Although he did not arrive at a solution, solution. he obtained elegant characterizations of completeness. paracompactness and the structure of the completion by means of cOlnpleteness. a concept called the radical of a uniform space. These results were published in the Journal of the Mathematical Society of Japan in 1960 (Volume 12, No.1, pp. ofJapan 104-117) under the title Some Properties of the Stone-Cech Compactijication. Compactification. We will analyze these results in Chapter 6. In 1970, K. Morita presented a paper titled Topological completions and M-spaces at an international Topology Conference held at the University of Pittsburgh. In Section 7 of that paper, five unsolved problems were listed including a special case of Tamano's question mentioned above and the ~ 'What is a necessary and sufficient condition for a Tychonoff space question: "What to have a Lindelof LindelOf topological cOlnpletion?" completion?" By a topological completion we mean the completion with respect to the finest uniformity. In this section and in later chapters, we provide answers to all these questions. All of the solutions are in terms of the cofinally Cauchy nets and their behavior with respect to various uniformities. Since some of these uniform space has a certain topological questions ask if the completion of a unifonn property, one might be interested in knowing if there are solutions in terms of property. topological properties, or if topological properties exist that are only necessary or only sufficient. sufficiem. Since the literature is rich with characterizations of paracompactness and the Lindelof LindeWf property, one might expect a variety of solutions to these problems. However, at the present time, this area is largely unexplored. We define a uniform space to be preparacompact if each cofinally Cauchy net has a Cauchy subnet. Recall that a uniform space is precompact if every net has a Cauchy subnet. Consequently, preparacompactness is a generalization of precompactness. To continue the parallel. parallel, recall that complete precompact spaces are compact and the completion of a precompact space is compact. We will see that complete preparacompact spaces are paracompact and that the completion of a preparacompact space is paracompact.
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unifonn space will be called countably bounded if each uniform unifonn A uniform Jl) is a uniform unifonn space and (X*, Jl J..1 *) covering has a countable subcovering. If (X, fJ,) unifonnity for x* X* and let v be the its completion, let u* denote the universal uniformity unifonnity is called the uniformity uniformity induced on X by u*. The v uniformity derived from u* or simply the derived uniformity. A directed set 0D is said to be ffi C 0 co directed or countably directed if for each countable {d {d;} D there is a d EE i } c D such that d d;i ~ d for each i. A net {x {xaa I a ED} E D} is ffi co directed if 0D is an co o m directed set. LindelOf if and only if each LEMMA 4.1 A completely regular space is Lindelof m co directed net clusters.
m directed net clusters in the completely regular space Proof: Assume that each co Linde16f. Then there is a covering U of X having no X. Suppose X is not Lindelof. countable subcovering. = u {u IU EE V} sUbcovering. For each countable V c U put G(V) = and F(V) = = X - G(V). Let S be the set of all countable subsets of U and for each ~ Xv defines an ffi VE E S pick Xv E E F(V). The assignment V -7 co directed net {xv IV E E S} in X that clusters to some p E E X where S is directed by set inclusion. Let UE E U. Then {xv} is eventually in F( {U} E U such that p E {U})) and hence cannot be frequently in U which is a contradiction. Consequently X is Lindelof. Linde16f.
LindeWf and let {x a II a ED} be an co Conversely assume X is Lindelof m directed net in X. Let U be a member of the e uniformity unifonnity of X. Then there is a countable = {Vi} {Vd such that VEe and V refines U. Suppose {x a} is not subcovering V = D i = {b frequently in some member of V and put D; {O EE 0D Ix 8Ii EE Vi}' V;}. Then there exists a ()i () E i} would be 0; E E 0 D such that ()0 ~ ()i 0; for each bEDi 0 E D; or else {x {Xli8110 E D D;} V;. Since {x a} is ffi CO directed there is abE a 0 E 0D such that ()i 0; ~ ()0 for frequently in Vi' E V each i. Now X8 Xli E Vjj for some j since V covers X. Hence () 0E E D jj which implies ()0< < ()j' OJ. But ()j OJ ~ ()0 which is a contradiction. Therefore xXaa must frequently be in some member of V. Since V was chosen arbitrarily, {xa} is cofinally Cauchy with respect to e and therefore clusters by Theorem 4.4. • The following theorem appeared in an article titled On completeness in the Pacific Journal of Mathematics in 1971 (Volume 38, pp. 431-440). THEOREM 4.11 (N. Howes, 1970) Let (X, Jl) J.1) be a uniform space and v the derived uniformity. Then: Jl) has a paracompact completion if and only if (X,v) is ((1) 1) (X, fJ,) preparacompact and preparacompactand (2) (X, fJ,) Jl) has a Lindelof LindelOf completion if and only if (X,v) is countably bounded and preparacompact, and (3) (X, fJ,) ,V ) is precompact. Jl) has a compact completion if and only if (X (X,v) Proof of (1): Let (X', J..l) Jl') be the completion of (X, J.l) Jl) and let u' be the universal u'). Assume uniformity unifonnity for X'. Then (X,v) is a dense uniform unifonn subspace of (X', U). ",:D -7 (X,v) is preparacompact and that \V:D ~ X' is a cofinally Cauchy net with
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respect to u'. Since (X'" (X', Il') complete, so is (X', u'). Let E = D x u' and define Jl') is complete" $ on E by (d, U') s $ (e,v') $ e and V' <* U'. For each (d, ~ (e,V') if d s (d" U') EE E put S(d,U,) = a for some a EE X such that a and 'V(d) both belong to some V' U' E U'. 8(d,U') ---+ S(d" Sed, U') defines a net S:E ---+ Then the correspondence (d, U') ~ ~ X. Let U' E u' and pick V' E u' with V' <* U'. Since 'V is cofinally Cauchy there is a cofinal C c D with 'V(C) c V' for some V' E V'. Put = {(d,W')ldE CandW'<*V'}. C and W'<* V'}. A =
(d,W') E A. Then S(d"W') S(d,W') = Then A is cofinal in E. Let (d,W'') = Yy E X such that y and 'V(d) both belong to some W' E W', W'. Since (d"W') (d,W') E A" A, dEC which puts 'V(d) 'tI(d) in V'. Consequently we have:
Y Star(V',W') c Star(Y',V') Star(V',V') c V' YE Star(y',W') U' U' E U'. Therefore SeA) c V' U' which implies S is cofinally Cauchy in for some V' (X', u'). But SeE) c X 8 X implies 8S is cofinally Cauchy in (X,v). Consequently S
net ~. But then ~ converges to some x' E X'. Therefore S has a Cauchy sub subnet~. clusters to x'. It remains to show that 'V clusters to x'. For this let 0 be an open set containing x'. Then there is a U' EE u' such that x' EE Star (x',U') cO c 0 where the members of U' are open sets. Pick V' E u' such that V' <* U'. Let S be S(S) c V' for some V' EE V' containing x'. Put cofinal in E such that 8(S) D(S) = = {dE {d E DI(d,W')E DI (d,W') E SforsomeW'<*V'}. S for some W' <* V'}. Then D(S) is cofinal in D. For each d E D(S), 'V(d) and S(d,W') are contained in some W' EE W' for some (d,W,) (d,W') E S where W' <* V' which implies 'tI(d) 'V(d) and S(d,W,) are contained in some Y'j 8(d,W') V~ E E V' for each (d,W') E S. But (d,W') E S implies 8(d,W') E V'. Hence S(d,W') E 'V(d) E V'uV~ V'uY~ c Star(Y',V') U' 'V(d)E Star(V',V') c V'
U' EE U'. U' . But x' E V' implies x' EE V'. U/. Then V' U' c 0 since Star (x',U') for some V' dEE D(S) which implies 'V clusters to x'. c O. Consequently 'V(d) EE 0 for each d Therefore each cofinally Cauchy net in X' clusters which implies (X', u') is cofinaIly complete. But then X' is paracompact by Theorem 4.4. cofinally Conversely suppose X' is paracompact. By Theorem 4.4 (X', u') is cofinally complete. Let 'V be a cofinally Cauchy net with respect to v. Since u,), we know that'll that 'V is cofinally cofinaIly Cauchy in uniform subspace of (X', u'), (X,v) is a unifonn (X', u'). Also since (X', u') is cofinally complete,\tf complete, \jI clusters to some p EX'. But then 'V has a subnet e that converges to p. Then e is Cauchy in (X', u'). But SeX cofinaIly Cauchy e c X and therefore e is Cauchy in (X,v). Consequently, each cofinally net in (X,v) has a Cauchy subnet so that (X,v) is preparacompact.
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LindelOf. Then X/ X' is paracompact and Proof of (2): Assume first that X' is Lindelof. hence (X/, (X', u/) u') is cofinally complete. But then (X,v) is preparacompact as was shown in part (1). Next let V EE v. Then V has a uniform refinement U consisting of closed sets. For each V E U put V/ {V/ Iu' UJ' UE U' = Clx'(V) Clx'(U) and let U/ U' = {U' E U}. Then U EE u/ E U). u' and hence has a countable subcovering say {V;}. {V;). Then { Vi} p E X then P p E E X/ p EE vj for some V,) covers X. In fact, if P X' which implies P positive integer j. Hence P p E Clx'(Uj)' Clx'(Uj ). Let 0 be open in X such that P p E O. = 0/nX 0/ that is open in X'. Now P p EE 0' 0/ which implies Then 0 = 0' nX for some 0' O/nU p EE Clx'(V 0' nVjj "# :j. 0 since P Clx'(Uj)' 0'nVj. t EE V Vjj implies t EE X j ). Then there is atE O/nVj. (O/nX)nV j "# p E Clx(V so we have (O'nX)nV :j. 0. Hence OnVj "# :j. 0 so that P Clx(Uj) Vj . j ) = Vj. {V j ) covers X there exists some {Vi} {Vj ) C V such that uVij = = X. Since {Vi} Consequently X is countably bounded. Conversely assume (X,v) is countably bounded and preparacompact. (X/,f.!/) (X',j.!') is paracompact by part (1) so that (X/, (X', u/) u') is cofinally coftnally complete by Theorem 4.4. Let U/ U' E u/. u'. Since X/ X' is paracompact there exists a locally finite open refinement ~' == {Vp IB ~ EE B}. B). Since X is norm~ norrn~l we c~n shrink V' to an {W~ IBEE B} such that Clx'(W~) cC VI3 V~ for each BE B. open covering W = {W~IB W/ is also locally finite. Since W/ Then W' W' is an open covering in a paracompact space X', it must be a member of the universal uniformity u/. u'. But then W = {W'nX E W'} belongs to v which implies there is a countable subcovering {W'nX IW' E {W~,) c W. Since W' {W~,) {W ~l} C W/ is locally finite in X', {W ~i} is locally finite in X'. Ui'=l Clx'(W~) = CIX,(Ui'=1 W~) = = since X is dense in X'. Hence Ui=lCl = Clx,(ui=lW~i) = Clx'(X) = X' ,(Wl3i) x {Clx'(W~)} Therefore {Clx'(W ~l)} covers X and
for each positive integer i. Hence {V~i} {V~,} covers X'. But since V' refines U' {V~,) cC U/ U' that covers X'. Therefore there exists a countable subcovering {V~i} (X/,u/) is also countably bounded. We are now in a position to show that X' is (X',u') Lindelof. LindelOf. We will use the fact that (X/, (X', u') is cofinally complete and countably bounded to show that each co-directed net in X' X/ clusters. We then invoke \jf:D ~ X/ X' be an co directed net and Lemma 4.1 to obtain the desired result. Let 'V:D let U' EE u/. U/ has a countable subcovering say {V;}. Put D ij = {d E 0 u'. Then U' DI 'V(d) \jf(d) EE V;} V;) for each i and suppose D ij is not cofinal in D for each i. Then there dE \jf is co directed E D i. exists a dij E D for each i such that d ~ d ij for each d j • Since 'V there is a do EE D such that dij ~ do for each i. Since {V;} covers X', 'V(d \jf(d a o) E vj for some positive integer j which implies do d a EE D jj which in turn implies D j is \jf is frequently in Vj. V j • Hence 'V \jf is cofinal in D since D = ui'=lD Ui=lD ij .• Therefore 'V LindelOf by Lemma cofinally Cauchy and consequently must cluster. Thus X' is Lindelof 4.1. j.!) has a compact Proof of (3): By Problem 3 of Section 4.4, (X, Jl) completion (X/, (X', Jl') j.!') if and only if Jl j.! is precompact. But if v is precompact then J.l j.! j.! is precompact which in turn implies X' is compact. Conversely, if c v implies Jl
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X' is compact then Jl' /-l' = u' which implies Jl /-l = v and Jl /-l is precompact which implies v is precompact. •-
COROLLARY 4.2 Let u be the universal uniformity for a completely regular T 1I space X. Then: (1) ( 1) (X, u) has a paracompact completion if and only if it is preparacompact, preparacompacl, (2) (X, u) has a Lindelof Linde16f completion if and only if it is countably bounded and preparacompact. COROLLARY 4.3 The completion of a preparacompact uniform space is cofinally complete. uniform COROLLARY 4.4 A countably bounded cofinally complete unifornl space is Lindelof. Linde16j. Linde16f if and only if it is COROLLARY 4.5 A paracompact space is Lindelof countably bounded with respect to the universal uniformity. In view of the existence of a unique completion for each uniform space, it is natural to ask when a uniform space has a cofinal completion, and if a cofinal completion exists, when is it unique?
cofinal completion if and only THEOREM 4.12. A uniform space has a cofinal if it is preparacompact. In this case it is unique and identical to the ordinary completion (i.e., the ordinary completion is cofinally complete). /-l) is preparacompact and (X', Jl') /-l') is its Proof: (Sufficiency) Suppose (X, Jl) /-l') is cofinally complete. In the completion. It needs to be shown that (X', Jl') sufficiency part of the proof of Theorem 4.11.( 1), it was shown that if v is the 4.11.(1), /-l rather than the uniformity Jl /-l and if (X,v) is uniformity derived from Jl /-l') is cofinally complete. The proof that (X', Jl) /-l) is preparacompact then (X', Jl') /-l) is preparacompact is cofinally complete based on the assumption that (X, Jl) similar so it will not be included here. /-l) has a cofinal completion (X', J.l'). /-l'). Let {x Ix a I a E (Necessity) Assulne Assume (X, J.!) D} be a cofinally Cauchy net in (X, J.l). {x a} clusters to sorne D) /-l). Then {xa} some p E X' so {x a } has a subnet {y~} that converges to p. But then {y~} is Cauchy in (X',/-l'). (X',Jl'). {xa} Since {y ~} lies in X it is Cauchy in (X, Jl). {Y/3} /-l). Consequently, every cofinally /-l) has a Cauchy subnet so (X, J.l) /-l) is preparacompact. Since Cauchy net in (X, J..l) (X', J..l) /-l') is cofinally complete it is also complete. But then (X', J..l') /-l') is a completion of (X, J..l). /-l). Since the completion of a uniform space is unique this means the cofinal completion (when it exists) is identical to the completion.completion. •
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EXERCISES 1. Show that a completely regular T 1 space is paracompact if and only if it is complete and preparacompact with respect to the universal uniformity.
J.!) is countably bounded if and only if each w (t) 2. Show that a uniform unifonn space (X, J.l) directed net is cofinally Cauchy. J.! be a unifonnity uniformity that generates the topology of X and let U be an open 3. Let J.l J.! if covering of X. We call P p E X a residue point for U with respect to V E J.l (p, V) c U. Let H v be the set of residue there does not exist aU a U E U with Star (P, F v · Then W = points with respect to V. Put F Fvv = Cl(H CI(H v )) and G v = X - Fv· {G v I V E J.!} ~} is called the residue covering derived from U. Show that W is an open covering of X. 4. Show that a completely regular T 1 space is paracompact if and only if each residue open covering with respect to the universal unifonnity uniformity has a finite subcovering. 5. Show that a completely regular T 11 space is paracompact if and only if each heavily bound open covering (Exercise 5, Section 4.3) with respect to the universal uniformity has a finite subcovering. 6. Show that in a completely regular T 1I space the following are equivalent: (a) the Lindelof LindelOf property, (b) each residue open covering with respect to e has a finite subcovering, (c) each heavily bound open covering with respect to e has a finite subcovering. 7. A topological space is said to be entirely normal if the collection of all neighborhoods of the diagonal in X x X forms an entourage uniformity (Exercise 4, Section 2.1). Show that entire normality and almost-2-fully normal nonnal (Exercise 6, Section 3.2) are equivalent.
8. A net in a topological space X will be called cofinally L1 t1 Cauchy if for each open covering U of X there is apE X such that the net is frequently in S(p, U). t1 complete if each cofinally L1 t1 Cauchy net clusters. Show that an X is cofinally ~ t1 complete. entirely normal space is paracompact if and only if it is cofinally L\ ~ complete. 9. Show that a metacompact space is cofinally t1
collection wise normality. 10. Show that entire normality implies collectionwise
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11. Show that a metacompact space is paracompact if and only if it is collection wise normal. nonnal. collectionwise 12. RESEARCH PROBLEM We would like to have a characterization of preparacompactness in terms tenns of coverings, perhaps analogous to the characterization of precompactness in terms tenns of coverings. The following result is in this direction, but something better is needed. A collection ~ of subsets of X is said to be a directed ~, AuB E ~. collection if for each A,B E ~,AuB
THEOREM A uniform space (X, !!) fJ) is preparacompact if and only if each heavily bound collection having no finite subcollection that covers X X is contained in a heavy directed collection. See Exercises 3 and 4 of Section 4.3.
Chapter 5 FUNDAMENTAL CONSTRUCTIONS
5.1 Introduction In this chapter we consider some important constructions of uniform spaces from other uniform spaces. Our first concern will be to consider the so called classical constructions that are studied for most spaces and algebraic structures that arise in the study of mathematics, namely subspaces, sums, products and quotients. Our approach will be to derive these constructions from a few fundamental concepts. These fundamental concepts take the form of limits of collections of unifonnities. uniformities. We will make these concepts precise in the next section. The reason for waiting until now to introduce the fundamental constructions is that many of the interesting results about these constructions involve concepts related to completeness. For example, it can be shown that the product of complete unifonn uniform spaces is a complete uniform space. Without these other concepts, it is difficult to appreciate the utility of these constructions. We opted instead to first develop the theory of completeness in uniform spaces and then see how it applies to these constructions. After the classical constructions, we proceed to some other constructions (some of which also apply to a variety of other structures such as topological spaces). We should remark that we have already seen one such construction~ construction; namely, the completion of a unifonn space. Of great interest to us will be the uniform concepts of the hyperspace of a unifonn space, the inverse limit of a directed uniform uniform spaces, the weak completion of a uniform space and the family of unifonn uniform space. If the hyperspace is complete, the original space spectrum of a unifonn completeness. supercompleteness is said to be supercomplete. Like cofinal completeness, turns out to be a strong form of completeness. In fact, it was recently discovered that supercompleteness is a property that lies between completeness and cofinal completeness. This result will allow us to strengthen several results from Chapter 4. The concept of supercompleteness was introduced by S. Ginsburg and J. Isbell in 1954 in an abstract titled Rings of convergent ,functions functions that appeared in the Bulletin of the American Mathematical Society, Volume 60, page 259. But the definition of supercompleteness in the simple form mentioned above is
5.2 Limit Uniformities
111 III
due to Isbell in a paper that appeared in the Pacific Journal of Mathematics in 1962 titled Supercornplete Supercomplete spaces (V'olume (Volume 12, pp. 287-290). The concept of the inverse limit of a family of uniform spaces leads to the concept of the spectrum of a uniform space, and the spectral analysis of uniform spaces whose spectra exist. The study of uniform spaces by means of their Also, spectra has been pursued by B. Pasynkov (1963) and K. Morita (1970). Also. the concept of the in verse lilni inverse limitt of a directed falnil familyy of unifonn uniform spaces is needed to express the weak completion cOlnpletion of a space with respect to a uniformity introduced by K. Morita in 1970. Finally, we will introduce the locally fine coreflection corefiection and the subfine sUbfme corefiection of a given uniform space. These were introduced by Ginsberg and coreflection Isbell in 1959 and shown to be equi equivalent valent by J. Pelant in 1987. This new construction has a variety of uses. In the last section of this chapter we will introduce the concepts of categories and functors. This material is optional as far as being needed in later chapters. It is included for two reasons. First, much of the literature of uniform spaces after Isbell makes use of the category theoretic vocabulary, and second, category theory, like set theory, is a tool that is often helpful in the study of uniform spaces. The tenn term '''coreflection'' "coreflection" refers to a category theoretic concept, and it will be shown that the locally fine coreflection satisfies this notion. Thereafter, categories and functors will only coreftection be used in the exercises.
5.2 Limit Uniformities Given a collection {'taa }} of topologies for a set X, there exists a finest topology that is coarser than each 'ta urn topology on X with respect to 'ta called the infim infimum {'t {'taa }} and denoted infa't a . Similarly, there exists a coarsest topology that is finer than each 't aa called the supremum topology on X with respect to {'t {'t aa }} and denoted sUPa't supa't a . To see this, note that rY't n'taa is a topology for X that is coarser than each 't'tao If't't is another topology for X that is coarser than each 'taa , then 't't a . If cC (l't n'taa so (l't n'taa is the finest topology for X that is coarser than each 'ta 'ta (I.e., (i.e., infa't a = = (l't n'ta)' a ). Also note that if L is the collection of topologies that are finer L 7:7= 0 since the discrete topology belongs to L. than each 'taa , then L Consequently, (lL nL is a topology for X. For each a, 'ta 'ta C cr for each cr E L. Hence Hence't'taa Cc (lL nL for each a so (lL nL is finer than each 't'tao If't't is another topology a . If finer than each 't'taa ,, then 't't E L which implies (lL nL C 'to 'to Therefore, (lL nL is the coarsest topology that is finer than each 'ta 'ta (i.e., supa't sUPa't aa = = (lL). nL). It can be shown (Exercise 1) that a sub-base for supa't sUPa't a is the set u't aa . The concepts of infimum and supremum uniformities are analogous to the topological concepts. If {J..la} lila} is a collection of unifonnities uniformities for a set X, there is Ila called the infim infimum urn uniformity a finest uniformity that is coarser than each J..la infa/la. There also exists a coarsest uniformity that is finer than and denoted infaJ..lao each Ila J..la called the supremum uniformity and denoted sUPaJ.la. supalla. By Theorem
112
5. Fundamental Constructions
2.1, uflu Jl on X which implies I{(l7=1 IU ul-4x is a sub-basis for a uniformity !-lon ni'=1 U Ua, Ua, a, E ai I 1-4x, for!-l. If!-l' Jla, for some i = 1 ... n} is a basis for Jl. If Jl' is another uniformity for X that is finer than each I-4x flu then !-l' J.l' must contain each (17=1 J.1 C !-l'. Jl'. Hence J.l = ni'=1 U Ua, Hence!-l a , so !-l supa!-la. Similarly, if L = = {O'y 'Y E G} is the collection of all unifonnities uniformities that sUPaJla. {ay IlYE Jla then L "# are coarser than each !-la :F 0 since {X} I X} E L. Put v = supya sUPyO'y. y. If V E v 0'1 ... .. . O'n then by the definition of v, there exists al an ELand Wi E O'i ai for each i = 1 ni'=1 Wi < V. For each a, ex, O'i ... n such that (17=1 ai C I-4x flu for each i = 1 ... n so Wi E Jla for each i = 1 ... n which implies (17=1 Jla. Hence V E I-4x flu so !-l Jl C !-la Jla I-4x ni'=1 Wi E !-la. ex. So VEL and by the definition of v, v is finer than any other for each a. member of L so v is the finest uniformity that is coarser than each !-la. J.la. Therefore v = infaflu. infal-4x. PROPOSITION 5.1 If {J.la} l!-la} is a collection of uniformities for a set X and {'t J.la's, I 'ta }} is the corresponding collection of topologies generated by the !-la's. Supal-4x is Supa't then the topology generated by sUPaJlu supa't a . supa!-la. Since sUPaJ.la supa!-la is finer than Proof: Let 't be the topology generated by sUPaJ.la. each !-la, J.la, 't is finer than each 't'taa so supa't sUPa'taa C 'to If U E 't then for each P E U Supa!-la such that S(p, n7=1 U Ua) there exists a basic covering n7=1 Ua, U a, EE sUPaJ.la a,) C U. Now S(P,U aa) EE supa't for each i = 1 ... n since supa't is finer than 't'ta, sUPa'ta sUPa'taa a , for each i == 1 ... fl. n. But then (17=1 ni'=1 S(P,U a,) E supa't a . If x EE (17=1 ni'=1 S(P,U a,), then X,p X,P E U a, for some U a, EE U for each i = 1 ... n. But then X,p E (17=1 U ai E , Ua, = x,P E ni'=1 a, E a n7=1 ni'=1 U Ua, E S(p, (17=1 ni'=1 Ua,). Therefore a , which implies x E
supa't a . Hence't C supa't sUPa't a which implies 't = = supa't sUPa't a .• .which implies U EE sUPa't infa!-la. Unfortunately, a proposition similar to 5.1 does not hold for infaJ.la. u!-la and u'ta are sub-bases for the unifonnity uniformity sUPaJ.la supa!-la and the Whereas uJ.la topology supa't sUPa't a respectively, (lJ.la n!-la is not necessarily even a uniformity while n't n'taa = infa'ta . The reason for this is explored in the exercises at the end of the section. Letf:X ~ Y be a function from a set X into a topological space Y. Thenf Let f:X -7 Then f determines a coarsest topology 'tf that f is continuous. A basis for the 't/ on X such thatfis X -7 open sets in 'tf 't/ is the collection B = {f{f-I1 (U) I U is open in Y}. If F = {f a: a:X ~ Y a II aex EE A} is a family of functions from X into topological spaces Y a, a , we suPa't/rJ.' define the topology 'tF to be SUPa 'tfa. 'tp is called the projective limit topology for 'tF is the for the collection F. It can be shown (Exercise 3) that a sub-basis for'tF = lra ex} and that 'tF 'tp is the coarsest set S = {f~/l (U a) I U a is open in Y aa for some a} topology on X such that each fa in F is continuous. ~ X is a function from a topological space Y onto a set Similarly, if g:Y -7 continuoUS. A basis for the X, there is a finest topology 't't g on X such that g is continuous.
5.2 Limit Uniformities
113
= {V (U Cc Xlg-I(U) Yl. If G = = open sets in t'tg is the collection B = Xlg-l(V) is open in Y}. Ya onto {g a: Y a ~ X I ex E A} is a family of functions from topological spaces Y X, we define the topology tc 'tG to be inlatg in!a't gexrx.• 'tG tc is called the inductive limit topology for the collection G. It can be shown (Exercise 4) that a basis for tc 'tG is l the set B = {V c X Ig"ix g~1 (V) (U) is open in Y = {U Ya for each aex E A} and that tc 'tG is the finest topology on X such that each g aa EGis continuous. The concepts of projective and inductive limits are also relevant to collections of uniformities. The only difference is that now we are interested in making collections of functions uniformly continuous rather than simply I:X ~ Y is a function from a set X into a uniform space (Y, v), continuous. If !:X J.!t on X then by Exercise 2 of Section 2.2, I! determines a coarsest uniformity f..lf such that that! If-I1(V) II V E v}. I is uniformly continuous. A basis for f..lf J.!t is 1-1l (v) == {/Let F = = {!a:X {Ia: X ~ (Ya,v a )I) I ex E A} be a collection of functions from X into uniform spaces (Ya,v a ). Define f..lF J.!F to be sUPaf..lfex' sUPaJ.!trx. f..lF J.!F is called the projective limit uniformity on X. It can be shown (Exercise 5) that a sub-basis for f..lF J.!F is = {f~} (fal (Va) I Va E Va va for some ex} and that f..lF the set S = J.!F is the coarsest I a in F is uniformly continuous. uniformity on X such that each each!
r
Similarly, if g:Y ~ X is a function from a uniform space (Y,v) into X, by Exercise 3 of Section 2.2, g determines a finest uniformity J.!g f..l g on X such that g is {U I U is a covering of X uniformly continuous. A basis for f..l J.!gg is the set B = {u such that g-1 g-l (U) E v}. Let G = {g a:Y a: Y aa ~ X II ex E A} be a family of functions from uniform spaces (Ya,v a ) onto X. We define the uniformity f..lG J.!c to be in!af..l g rxex and call J.!c f..lG the inductive limit uniformity on X. It can be shown inlaJ.!g (Exercise 5) that a basis for J.!c g~1l (U) E f..lG is the set {U II U is a covering of X and g"ix Va ex} and that f..lG Va for each a} J.!c is the finest uniformity on X such that each g aa in G is uniformly continuous. PROPOSITION 5.2 The topology generated by the projective limit uniformity is the projective limit topology.
The proof follows immediately from Proposition 5.1 and the definitions of projective limit topology and projective limit uniformity.
EXERCISES 1. Let {t {'taa } be a collection of topologies for a set X. Show that the set S = uta u'ta . is a sub-basis for SUPa't SUPata. a 2. Let F = {f a:X a: X ~ Y a II a = If a. E A} be a family of functions from a set X into Y a and let 'tF topological spaces Y tF be the projective limit topology with respect to a.} is a sub-basis F. Show that the set S = {fal {hI (U a) I U a is open in Y aa for some a}
114
for 1F tp and that continuous.
5. Fundamental Constructions tF 1F
is the coarsest topology on X such that each
Ifaa in F is
C = {ga:Ya ~ Xla E A} be a family of functions from topological 3. Let G spaces Y tc be the inductive limit topology with respect to Ya onto a set X and let 1c G. Show that B = {U c X I g~l g~.} (U) is open in Y Ya for each a E A} is a basis for 1C tc and that 1c tc is the finest topology on X such that each g a in G is continuous.
Ifa:X ~ (Ya,v (Y a,v a ) I a E A} be a family of functions from a set X X into 4. Let F = {I /-!F be the projective limit uniformity with uniform spaces (Ya,v a ) and let JlF {fal1 (V (Va) E Va for some a} is a subrespect to F. Show that the set S = {k a ) I Va E basis for /-!F JlF and that /-!F JlF is the coarsest uniformity on X such that each Ifaa in F is uniformly continuous. Y a ~ X I a E A} be a family of functions from uniform spaces 5. Let G = {g a: a:Y (Ya,v a) onto a set X X and let f.!c /-!c be the inductive limit uniformity with respect to C. Show that the set B = {u {U I U is a covering of X and g~l (U) E Va for each a} G. is a basis for f.!c a in G /-!c and that Jlc /-!c is the finest uniformity on X X such that each g a
is uniformly continuous.
5.3 Subspaces, Sums, Products and Quotients In this section, the classical constructions will be examined as special cases of inductive and projective limit uniformities. It will be seen that subspaces and product spaces are special cases of projective limit uniformities, while sum and quotient spaces are examples of inductive limit uniformities. If (X, f.!) /-!) is a uniform space and Y c X, define i:Y ~ X = Y for each y E Y. Let /-!i Jli X by ICy) iCy) = denote the projective limit uniformity on Y Y determined by the single function i. It is left as an exercise (Exercise 1) to show that the uniformity f.!i /-!i is identical to the subspace uniformity introduced in Section 4.4. Given a collection F = {X a II a E A} of uniform spaces with uniformities /-!a for each a, we define the uniform product space (P,1t) to be the Cartesian f.!a I1X a (introduced in the Forward) together with the projective product set P = nx limit uniformity 1t determined by the collection {p a II a E A} of all canonical fA ~ uX a projections p a:P ~ X a' Recall that the members of P are functions I:A f( a) E X a for each a. The mappings P p a are defined by P p a (j) (f) = f( a) such that I(
I1X a, it for each f E P. When dealing with uniform product spaces such as P = nx is customary to refer to the individual uniform spaces X a from which the product space is constructed as the coordinate (uniform) spaces. PROPOSITION 5.3 The topology associated with the uniform product space is the topology of the topological product of the coordinate spaces (considered as topological spaces).
5.3 Subspaces, Sums, Products and Quotients
115
Proof: 'ta be the topology associated with Ila J.la for each a and let 't denote the Prool: Let Let'ta nx a'u. It will suffice to show that for each sub-basic U product topology on P = IlX E 't containing x EE P, there exists a V 1t with S(x,V) S(x,v) c U and for each S(x,W) E V EE 1t where W E 1t, there exists a basic V E 't with x EVe S(x,W). If U E "('t is a sub-basic open set then U = {f f(P) E U ~} for some open U ~ eX B. Since If E pi pi/(p) c X~. x EE U, p~(x) = x(P) EE U~ which implies there exists some V~ EE fJ-~ p~(x) = /..l~ with S(x(P),V~) c U~. Since 1t 1t is the projective limit uniformity determined by the S(x(P),V~) collection {p a} of canonical projection, by Exercise 5 of Section 5.2, p~l P ~l (V~) is Let/E S(X,Pi31(V~))={p~I(V~)lx(P)E S(X'Pi31(V~)) = (Pi31(v~)ix(p) E VeE V~ E V~}. a sub-basic memberof1t. LetfE asub-basicmemberofn. Then fIE E p~l Pi3 1 (V~) for some V ~ E V~ that contains x(P). Hence f(P) I(P) E V ~ C S(x(P),V~) S(x(P),v~) c U ~ so fEU. lEU. Therefore, Sex, PB Pi3 11 (V~) (V~)) c U so "('t is coarser than the uniform product topology on P. Conversely, suppose W is a basic member of 1t. 1t. Since 1t 1t is the projective limit unifonnity uniformity determined by the collection {p a} u} of canonical projections, by Exercise 5 of Section 5.2, there exists a finite collection W W a1 . .. Wan of ••• ~an /..la n respectively such that W = n'!=lPai n7=1Pa~1l (Wa). coverings in 1la J.la l1 ... (W a). For each i = x(a;) EVa,. n7=1 Va, == = 1 ... n choose Va, Val E W U1 E Val. Then x E V = = n'/=1 a, with x(ai) n'!=IP n7=1P ail a~l (V (Va,) of 'to If fI E V then both x and fbelong Ibelong aj ) and V is a basic member of't. l (W aj n7=1Pa~1l (Va,). (Va). But S(x,W) = {fE (IE plx,fE PiX,fE n'!=IPai n7=1Pa~1(W a)) for some W al to n'!=IPai a, E W W a , for each i = 1 ... n}. Hence x EVe S(x,W) Sex,W) so 't is finer than the uniform product topology on P. Therefore, 't is the uniform product topology on P. -
Quotient uniformities are an important example of inductive limit uniformities. If (X, ~) /..l) is a uniform space and R is an equivalence relation on X, let q:X -) ~ X/R be the canonical projection of X onto the quotient set X/R (introduced in Chapter 0). If we then give Q Q == X/R the inductive limit /..lq determined by the single function q, then (Q, J.l /..lq) uniformity J.!q q ) is called the quotient uniform space with respect to R. It is left as an exercise (Exercise 5) to show that the topology associated with /..lq f.l q is the topology of the quotient (topological) space X/R, also introduced in the Introduction.
Q may not be Hausdorff (see A word of caution is in order here. The space Q Exercise 7). To insure that the quotient of a Hausdorff uniform space is again Hausdorff, we need the concept of a uniform quotient (see Exercise 8). It is of interest to observe that every onto mapping !:X I:X -) ~ Y from X to a set Y = I(y). determines an equivalence relation Rff on X by taking (x,y) E Rff if I(x) f(x) = fey). ~ X/Rff determines a one-to-one Moreover, the canonical projection q:X -) function g:X/Rfr -) ~ Y such that If = g © q. g is the function defined by g(x*) = I(x) f(x) where x* is the equivalence class containing x. If Y is also a uniform space with uniformity v, then by Proposition 0.20, If is continuous if and only if g is continuous. It is easily seen that this continuity condition also holds for uniform continuity; i.e., If is uniformly continuous if fact. if g is uniformly continuous, then and only if g is uniformly continuous. In fact, fI must be uniformly continuous since it is the composition of the uniformly
116
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continuous function g and the canonical projection q (which is uniformly continuous by definition). Conversely, if f is uniformly continuous then for each V E E V, J.lq (the quotient uniformity on v, f- 1I (V) EE J.l f.l implies that q [f-I [{~l (V)] EE f.l X/R rr). We will show that g-I(V) g~I(V) = q[f-1(V)] qW1(V)] thereby showing g to be uniiormly continuous. For this let V EE V and suppose x* EE g -1 ~l (V). Then g(x*) uniformly =f(x) EE V. Let y EE f- 1I [f(x)]. f(y) = = f(x) which implies y EE x* so q(y) = [((x)]. Then fey) x*. Thus x* EE q(j-I(V» q(f~I(V» so g-l(V) g~I(V) cC q(j-l(V». q(f~l(V». Next suppose z* E E q(j-I(V». q(f~I(V». I Then Z E E f- 1(V) which implies fez) f(z) E E V. But g(z*) = fez) f(z) so z* EE g -1 ~l (V). q(f~l (V» (V» c g-l g~l (V) which implies g-1 g~l (V) = = q(j-l q(f~l (V». Therefore q(j-1
r
r
r
In Section 5 of Chapter 1, we introduced the concept of the metric space associated with a pseudo-metric space (X, d) by defining the equivalence relation - in X by x - y if d(x,y) = 0 and defining a metric p on X/- by p(x*,y*) = d(p-1(X*), p-1(y*» d(p ~l (x*), P ~l (y*» where p:X --t ~ X/- was the canonical projection and x* and y* were the equivalence classes with respect to - containing x and y respectively. A similar construction can be done for uniform spaces; i.e., if (X,J.l) is a non-Hausdorff uniform space, it is possible to construct a Hausdorff (X,f.l) E S(y,U) for uniform space from X. For this, define R Jl1.1 on X by (x,y) EE R Jl1.1 if x E each U EE J.l. f.l. Then let q:X --t ~ X/R Jl 1.1 be the canonical projection of X onto the quotient set X/R 1.1Jl and let f.l J.l q be thelluotient uniformity determined by q. Then (X/R 1.1' Jl' f.lq) J.lq) is a uniform space called the Hausdorff uniform space associated with (X, J.l). f.l). PROPOSITION 5.4 The Hausdorff uniform space associated with (X,J.l) (X.f.l) determines a Hausdorff topology and the canonical projection q:X --t ~ X/R XIR Jl1.1 is both an open and closed mapping. Proof: Let x* and y* be distinct members of X/R I.l' Jl' Then y does not belong to x* which implies y does not belong to S(x,U) for some U E E J.l. f.l. By Exercise 6 of ~l (V) EE J.l} f.l} is a basis for f.l Section 5.2, 5 .2~ B = {V I V is a covering of X/R 1.1Jl and q -1 J.lq. q. Now q(U) = {q(U) I U EE U} covers X/R I.lJl and U < q-1 E f.l J.lq. q~l [q(U)] so q(U) E q. If E S(x*, q(U» y* E q( U» then both x* and y* EE q(U) for some U EE U which implies x,y E U so Y E E S(x,U) which is a contradiction. Hence y* does not belong to E E f..l S(x*,q(U». Let W WE f.l q q such that W* < q(U). Then S(x*,W)nS(y*,W) = 0 so (X/R >" J.l f.lq) (X/R~, q ) determines a Hausdorff topology.
To show that the canonical projection q is both an open and closed = mapping, first let U be an open set in X and suppose x* EE q(U). Then q(x) = q(y) for some y EE U. Therefore, x EE y*. Let V EE J.l f.l such that S(y,v) S(y,V) ccU. U. Then q(V» c q(U). Hence q(U) is open in X/R Jl1.1 x E S(y,V) which implies x* EE S(y*, q(V» so q is an open mapping. If F is closed in X and if x* does not belong to q(F) f.l such that S(x,W)nStar(F,W) = == then x EE F which implies there exists aWE J.l 0. Let V EE J.l =F- 0. Then there exists f.l with V* < W. Suppose S(x*, q(V»nq(F) q(V»nq(F)"#X/R>,Jl such that x* ,y* EE q(V) for some V EE V and such that y* EE some y* EE X/R q(F). y* EE q(F) implies that there exists some Zz EE F with q(z) = q(y) which in y EE S(z, V). Moreover, x* ,y* EE q(V) implies there are r, s tum implies y EE z*, so Y S(z,V).
5.3 Subspaces, Sums, Products and Quotients
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E V with q(r) = q(x) and q(s) = q(y) so Y E S(s,V) and x E S(r,V). V I, V V2, and V V 3 E V such that x,r E VI, V I, S,y s,y E V V2, and Consequently, there exist VI, y,zE uV 3 y,Z E V 3 •. But then there are WI andW and W 2 E WwithVIuV2CWl. W with V I UV 2 C WI andV and V2 UV cW 2 W2impliesWInW2~0,WI 2 .• NOWX,SE WI ands,zE W 2 impliesW l nW 2 "#0,W I cS(X,W) and W 2 C Star (F. ~ 0 which is a contradiction. (F,. W). Hence S(x,W)nStar(F,W) "# Therefore, S(x*, q(V»nq(F) = 0. Thus q(F) must be closed so q is also a closed mapping. -
°
Our last example of an inductive limit uniformity is the uniform sum of a collection {(X a, J.la)} /-1a)} of uniform spaces. We define a new uniform space LX a by first defining the points of LX a to be the ordered pairs (x, a) where x E Xa' X a' ---7 LX a defined by i a(X) = (x, a) are called the canonical The mappings i a:X a ~ injections. The uniformity cr a of LX a is defined to be the inductive limit uniformity determined by the collection {i a} of canonical injections. The a,a) is called the uniform sum of the uniform spaces Xa. X a' uniform space (LX a,cr) The proof of the following proposition is left as an exercise (Exercise 6). PROPOSITION 5.5 The topology associated with the unIform uniform sum of a collection of uniform spaces is the topology of the disjoint topological sum of these spaces considered as topological spaces.
/-la from the The canonical injections essentially transfer the uniformities J.la n X a onto the disjoint "pieces" a is the "pieces i a(X a) of LX a and since cr spaces Xa a is the finest uniformity on inductive limit uniformity determined by the i a's, cr a 's uniformly continuous. LX a that makes all the i a's
EXERCISES /-1) is a uniform space and Y c X, that the projective limit 1. Show that if (X, J.l) /-1, on Y determined by the function i:Y ~ ---7 X such that i(y) = Y for uniformity J.li each y E Y is identical with the subspace uniformity on Y introduced in Section 4.4. 2. Show that a net in a product space converges to a point p if and only if its projection in each coordinate subspace converges to the projection of p.
3. Show that a net in a product uniform space is Cauchy if and only if the projection of the net in each coordinate subspace is Cauchy. 4. Show that the product complete.
nx a
of complete uniform spaces (X a' a, J.la) /-1a) is
/-1) be a uniform space and let R be an equivalence relation on X. 5. Let (X, J.l) Show that the topology associated with the quotient uniform space with respect
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to R is the topology of the quotient (topological) space X/R introduced in the Introduction. 6. Prove Proposition 5.5.
(X, J.l) /.1) is a non-normal uniform space with the finest uniformity 7. Show that if (X~ u, u~ that there exists an equivalence relation R on X such that X/R is not Hausdorff. UNIFORM QUOTIENTS (X~ J.l) A uniform relation in a uniform space (X, /.1) is an equivalence relation R on X such that for each pair of non-equivalent points x,y E X, X., there exists a sequence {Un} C J.l /.1 satisfying:
yRy', then x' and y' are in no common member of U 1I .• (1) If xRx' and yRy'~ (2) If Un + U E + n 1 nn ++l , there exists a Un E Un such that if p E Unn ++1 and pRq, then Seq, U Un+d n+ l ) C Un. A uniform quotient is a quotient uniform space X/R where the equivalence relation R is a unifonn uniform relation. 8. Show that a uniform quotient of a Hausdorff uniform space is Hausdorff.
9. Show that the canonical projection q:X uniformly continuous.
--7 ~
X/R of a uniform quotient is
UNIFORMLY UNIFORML Y LOCALLY COMPACT SPACES uniform space is said to be uniformly locally compact if it has a A unifonn uniform covering consisting of compact sets. For each countable ordinal a put unifonn X a = a + 1 and let 't'!au be the order topology on X a' X a is compact, so Xu Xa Xu u. Then Xu has a unique unifonnity uniformity J.lu /.1a consisting of all open coverings. Let (L. (L,. J.l) /.1) be the sum of the collection {(X u., a, J.la) /.1a) I a < WI }. Let R be the equivalence relation on L defined by x - y if and only if x and y belong to the same X a' u. Let Y = W1 WI with the order topology. 10. [So Ginsburg and J. Isbell, 1959] Y has a unique uniformity with a basis uniform relation on (}:, (L, J.l). /.1). L/R = consisting of all finite open coverings. R is a unifonn = --7 Y be the canonical projection. There exists a U E Il Y. Let q:L ~ J.l such that q(U) {q(U) q( U) = {q( U) I U E U} is not a uniform covering in Y. L is complete, but q(L) = = compact. Y is not complete. L is uniformly locally cornpact.
5.4 Hyperspaces
119
5.4 Hyperspaces (X, f.l) /.1) be a uniform space and let X' denote the set of all non-void closed Let (X~ E X' with He H c Star (K, (K~ U) and K c Star (H, (H~ U) for some U subsets of X. If H, K E E f.l~ /.1, then Hand K are said to be U-close~ U-close, denoted by IH - K I < U. Note that E this relationship is reflexive, i.e., I H - K I < U implies I K - HI HI < U. u. If U EE f.l/.1 x'ilII F - K KII < U) IFE X'). and F E X', put B(F,U) = {K E X' U} and let U' = {B(F,U) IF EX'}. /.1* = {U' I U EE f.l}. /.1). We want to show that f.l* /.1* is the basis for a Then define f.l* /.1' on X' but first we establish some useful lemmas. uniformity f.l' LEMMA 5.] If u* u* < V.IH-FI U.and IK-FI < U,then U.then IH-KI LEMMAS.] v, IH-FI < U,and
<
v.
Proof: IH - F I < U implies H c Star (F, F c Star (H, U). IK - F I < U (F~ U) and FeStaI' implies K c Star (F, U) and FeStaI' F c Star (K, U). Therefore H c Star (Star (K. (K~ (K~ U),U) c Star (K, V). SiInilarly (H~ V) so I H - K StarCK. Similarly K c Star (H. K!I < V.· V.u* < V and FE X' then S(f",U') S(F.U') c B(F,V) B(FY) c S(F,V'). S(F.V'). LEMMA 5.2 If u* Proof" Let H E S(F.U'). Then H EE B(K~U) B(K.U) for some K E E Proof: E S(F,U'). E X' such that F E B(K,U). Therefore, H c Star Star(K,U), Star(H.U), Fe Star(K,U). B(K~U). (K, U)~ K c Star (H, U), FeStaI' (K, U)~ and K c (F, U), F c Star (H, V) Star (f", U)~ so H c Star (Star (F. (F~ U). U)~ U) c Star (F. (F~ V). Similarly, FeStaI' B(F,V). B(F,V) B(F,v) c S(F,V') which implies H EE B(F,V). Therefore S(F,U') c B(F~V). S(F.V').follows from the definition of S(F~V'). •
/.1* is a basisfor basis for a uniformity f.l' /.1' on X'. PROPOSITION 5.6 f.l* Proof: We need to establish (1) and (3) of the definition of a uniform space. V',W' EE f.l*, We can do this simultaneously by showing that if V'.W' /.1*, there exists a U' E f.l* /.1* with U' <* V'nW'. For this let U EE f.l /.1 such that U* < VnW. Let V' U' c E U'. U' = F E Star (U',U'). Then F E U' . Then V' = B(E,U) for some E EE X'. Let FEStaI' E B(K,U) some K EE X' such that K EE B(E~U) B(E,U) which implies IF - K I < U and IK - E iI for SaIne IF - EI < VSOFE V so F E B(E,V)E B(E,V) E V'. Therefore, Star (U',U') < U. By Lemma 5.1, IF-EI B(E,v) so U' <* V'. Similarly. /.1* is a c B(E,V) Similarly~ U' <* W'so U' <* V'nW'. Thus f.l*
/.1' on X'. •basis for a uniformity f.l' /.1') is called the hyperspace of (X, f.l). /.1). The uniform space (X', f.l') Hyperspaces of topological spaces have also been studied. The original notion of a hyperspace is due to F. Hausdorff (see Mengenlehre, 3rd edition, Springer, Berlin, 1927). Hausdorff defined a metric on the set of non-empty closed and bounded hounded subsets suhsets of a given metric space. L. Vietoris generalized the concept to topological spaces (see Bereiche Zweifel' Zweiter Ordnung, Monatshefte fur 33~ 1923, pp. 49-62). N. Bourbaki introduced Mathematik und Physik, Volume 33, uniform space the uniformity for the hyperspace of non-void closed subsets of a unifonn Paris. Hermann. unifonnity (Topologie General. General~ Paris~ Hermann~ 1940). in terms of an entourage uniformity
5. Fundamental Constructions
120
In this section we translate Bourbaki's approach in tenns terms of covering uniformities. unifonnities.
PROPOSITION 5.7 The hyperspace of a uniform space is Hausdorff. Jl) be a unifonn space and (X', 11') Jl') its hyperspace. Choose H, K Proof: Let (X, 11) E :f. K. Then either H contains a point not in K or K contains a point E X' with H :t not in H. Without loss of generality we may assume there is an x EE H such that x does not belong to K. Theorem 2.6 can be used to show there exists U,V EE 11 Jl S(x,V)nStar(K,v) = = 0. Suppose K EE 8(H,U). with U* < V such that S(x,V)nStar(K,V) B(H,U). Then K c Star (H, U) and H c Star (K, U) which implies x EE Star (K, U) which is a contradiction. Therefore, K does not belong to 8(H,U) B(H,U) so X' is T I. 1. Since X' is regular, it is also Hausdorff. •{Fa} LEMMA 5.3 A net {F a} in X' converges (clusters) to some F and only lffor each V E }.l, B(F,V). 11. {F a} is eventually (frequently) in 8(F,v).
E
X'. if X',
Proof: Let V EE 11 Jl and choose U EE 11 Jl with U* < V. If {F a} converges to F S(F,U').). But by Lemma 5.2, S(F,U' S(F,U')) c then {F a} is eventually (frequently) in S(F,U' B(F,V). Conversely, if for each V E E 11, J..l, {F a} is eventually (frequently) in 8(F,V). {Fa} 8(F,V), S(F,v,) by Lemma 5.2, so {F a} converges to F. F.B(F,V), then it is eventually in S(F,V') • {Fa} LEMMA 5.4 {F a} is Cauchy in X' if and only iffor each V E 11, Jl, there
p, Fa exists a ~ such that for each a ~ ~~.
E
B(F J3,v). ~,V). 8(F
Proof: Let U EE 11 Jl with U* < V. Since {F a} is Cauchy, there exists a ~ such Faa EE 8(H,U) Faa c B(H,U) for each a ~~. Then for each a ~ ~, F that for some HEX', H E X', F Star (H, U) and H c Star (F a,U). Also, F ~J3 c Star (H, U) and H c Star (F J3,U), ~,U). (Star (FJ3'U),U) c Star(F~,V). Star (FJ3,V). Similarly, F~ FJ3 c Therefore, Fa C Star Star(Star(F~,U),U) Faa EE 8(F ~ ~. The converse it obvious since Star (F a, V) so F B(F J3,v) ~,V) for each a ~~. B(F~,V)E 8(F J3'V) E V'.· V'. {Fa} Let {F a} be a net in X'. A point p EE X, each of whose neighborhoods meets cofinally many of the Fa's is said to be a cluster point of {F a} (in X).
PROPOSITION 5.8 The set K of cluster points of a net {F a} in X' is closed. If {F a} converges in X'. X', it converges to K. Proof: Let p be a limit point of K and let U be a neighborhood of p. Then U contains a point k EE K. Since k is a cluster point of {F a}, U frequently meets {F a} which implies p EE K. Hence K is closed. Next assume {F {Fa} ex} converges to F EE X'. Suppose x does not belong to F. Then, as pointed out in Proposition = 0. By Lemma 5.3, {F IF al a} is 5.7, there is a U E E 11 IJ- with S(x,U)nStar(F,U) = Faa EE B(F,U) for each a ex ~ ~ eventually in B(F,U). Therefore, there is a ~ with F Faa c Star (F, U) for each a ~~. Hence x is not a cluster point of which implies F
5.4 Hyperspaces
121
F. Consequently, F c K and there can be no points of K that are not in F, so F =
K.Our first undertaking with hyperspaces will be to characterize supercompleteness (see Section 5.1 for the definition) in terms of the behavior of a certain class of nets in the original space X. In 1988, 1988. B. Burdick informed the author that he had shown that cofinal completeness implies supercompleteness and wondered if the reverse implication held. The author pointed out that real Hilbert space (Chapter 1) can be shown to be a supercomplete metric space that is not cofinally complete (see Exercise 1). Thereafter, Burdick discovered the following characterization of supercompleteness that yields the implication: cofinal completeness => supercompleteness as a corollary. Burdick defines a net {x Ix a}, a E A, in a uniform space (X, J.l) IJ.) to be almost IJ. there exists a collection C of cofinal subsets of A Cauchy if for each U E f.l such that for each K E C, {x Ixyy I'Y E K} c U for some U E U, and such that uC is residual in A. Clearly, Cauchy nets are almost Cauchy and almost Cauchy nets are cofinally Cauchy. His characterization of supercompleteness is that a uniform space is supercomplete if and only if each almost Cauchy net clusters. Burdick's proof depends on a theorem of Isbell on partially convergent functions. Isbell's development of partially convergent functions will be given in a later section. For now we present an alternate constructive proof directly from the definitions of supercompleteness and almost Cauchy nets. This constructive proof has the advantage of making clear just how the clustering of almost Cauchy nets is related to supercompleteness. The proof is rather long, so we decompose it into two lemmas plus the main constructions. We define a net {F IF a} in X' to be proper if {F IF a} is ordered by set inclusion (a ~ B ~ if and only if Fpc {F a}. F 13 c F a) and if F Faa C F E X' for some a then F EElF LEMMA 5.5 A proper net {F IF a} in X' is Cauchy if and only iffor if for each V E IJ. f.l there is a ~ with Fp F 13 c Star(F a,V)for )jar each a.
a.v
IF a} is a proper Cauchy net in X'. Let V E IJ.. IJ. Proof: Assume {F J.l. Pick U E J.l : : : 'y, Y, F Faa E with U** < V. By Lemma 5.4, there is a 'Y such that for each a ~ B(F y,U). Now CI(Star(F y,U» y,U)) = = F 13P for some B ~ since {F IF a} is proper. Let a a:::::~ 'Y. y.U) which implies F y c Star(F Star (F a, U) so Star(F Star (F y,U) c Then Fa E B(F y,U) a,U) Star(F a,U*), which implies F Fpc 13 c Star (F a, V). If a < 'Y then F yy c F Faa so Fpc F 13 c Star (F y' V) c Star (F a, V). Consequently, there is a B ~ with Fpc F 13 c Star (F a, V) for each a.
Conversely, assume that for each V E J.l Fpc IJ. there is a B ~ with F 13 c Star (F a, V) for each fJ.. Ff3 c Star(Fp,V). a. If fJ. a :~: : 13~ then FF~13 c Star(F a,V) and Fa C FI3 Star (FI3'V), Faa E B(F f3'V) ~.V) for each a a:::::~ 13~ so {F IF a} is Cauchy.Therefore F Cauchy. -
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5. Fundamental Constructions
J..l) is supercomplete if and only if each proper Cauchy LEMMA 5.6 (X, J.l) net in X' converges. Proof: Let {F a} be a Cauchy net in X'. For each aa put H a u{F~131I ~ ~ a}. {Fa} a = = u{F Let {S y} denote the collection of elements of X' containing some H a where {S y} is directed by set inclusion (y < 8 b if and only if S8 S8 C Sy). S y). Clearly {S y} is y }. To see this, let V E J.l J..l and pick U E proper. Since {F a} is Cauchy, so is {S y}. J..l with U* < V. By Lemma 5.4, there exists a ~ such that Fa E B(F~,U) B(F 13 ,U) for J.l each a ~~. Therefore, Fa c Star (F 13'U) ~,U) for each a ~ ~ which implies CI(H CI(H~) 13) Star (Fj3,V). Now CI(Hj3) 8, Sy C S8 58 c Star(F~,V). Ct(H~) = S8 for some Sii S8 E {Sy}. For each y~ b, 13, V). Also, for each y ~ 8, which implies S yy c Star (F (F~, b, there exists a A ~ ~ with HAA c Sy which implies FA FA c Sy and F~ Fj3 c Star Star(FA,U) Fj3 c Star (Sy,U) H (FA,U) so F~ which implies S ~,V) for each y ~ 8. b. Therefore, {S y} is a proper Cauchy 5 y E B(F 13'V) net. Then by hypothesis, {S y} converges to its set of cluster points K. Let V E J..l. Pick U E J.l J..l with U*** < V. By Lemma 5.3, {S y} is eventually in B(K,U) so J.l. {Sy} b with S yy E B(K,U) for each y ~ 8. b. Now there exists a ~ with there exists a 8 Ct(H~) 8. Since {F a} is Cauchy, there exists a ~ with F 13) Cc S 58' {Fa} Faa E B(F ~ ,U) for CI(H Ct(H~) each a ~~. Pick A such that A ~ ~ and A ~~. Then Ct(H CI(H A) Cc CI(H 13) c S 8 so CI(H A) = S cr for some (j ~ b. 8. Hence CI(H A) E B(K,U). Since A ~ ~, F Faa E B(F~,U) for each a ~ A which implies Fa C Star(F~,U) and F~ c Star(F a,U) for each a ~ A. Thus CI(H A) Star(F ~ ,U*) and F~ F ~ c Star(Cl(HA),U*). Stare CI(H A)'U*), Hence A) cC Star(F;,U*) Cl(H CI(H A) E B(F ~ ,U*). Consequently, we have
IFa-F~1
J..l) is superTHEOREM 5.1 (B. Burdick, 1991) A uniform space (X, J.l) complete if and only if each almost Cauchy net clusters. Proof: Assume each almost Cauchy net clusters and suppose (X, Jl) J..l) is not supercomplete. Then by Proposition 5.8 and Lemma 5.6, there exists a proper Cauchy net {F a}, a E A, in X' that does not cluster to its set of cluster points K. There are two cases to consider here. First, K might be the empty set, in which case K would not belong to X' and therefore could not be a cluster point of {F a}. On the other hand, K may not be empty. This is the case we will assume first. Later we will see that an argument similar to the one we will use here can be used to show that under the hypothesis of this theorem, K cannot be the empty set.
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5.4 Hyperspaces
If K *~ 0, there exists a U E f.l {F a} is eventually outside /-1 such that {Fa} B(K,U) which implies there exists a ~ with Fa F a not contained in B(K,U) for each a ~~. Let P pEKe p E Star (F y,U) for E K. Then there exists a cofinal C c A with P each yY E C. Pick 8 E C with 8 > ~. Then F'6 p E F Ii c F ~ which implies P Star(F~,U). Therefore, K c Star(F~,U) which implies F~ is not contained in Star (K, U) or else F ~ E B(K,U) which is a contradiction.
Similarly, Fa F a is not contained in Star (K, U) for each a ~~. Suppose y is not greater than or equal to~. Then there is a 8 such that y ~ 8 and ~ p ~ 8 which implies f-' '6 is not contained in Star (K, U) and F F'68 cC F l' y. Hence F y is not F8 contained in Star (K, U). Consequently, Fa F a is not contained in Star (K, U) for = Star (K, U) and for each a E A let H a == Fa each a E A. Put U = F a - U. Then {H a} is a Cauchy net in X' that is directed by set inclusion. Let {G E B, IHa} IGy}, yE y }, Y elements of X', directed by set inclusion, that contain a be the collection of elelnents memher of {H IH a}. Then {G I G y} is a proper Cauchy net in X~. X'. Let D = = {(x, ify~ I (x, y) E X x B 1Ix x E G yy}} and define ~ on D by (x, y) ~ (y, 8) if Y~ \jI(x, y) = = x. Then \jI:D 8. For each (x, y) E D put 'V(x, o/:D ~ X is a net which we will WEE J-l. /-1. Pick U E f.l /-1 with U* < W. Since show is almost Cauchy. For this let W {G } stable, there exists A with G E B(GA,U) for each y ~ A. Pick x E IGy} is a y y Star(GA,U). Then some W E W contains Star(x,U). Let V = {W IW E wlw wlw contains Star(x,U) for some x EGA}. p) E DI E Gd. Let R = {(y, I(y,~) 01 ~ ~ A}. Then R is residual in D. Let (y, ~) E R which implies p (y,~) ~ ~ A which in tum implies G~ c Star(GA,U) so y E Star(GA,U) which implies y E S(x,U) for some x EGA. EGA' \jI(y, ~) = YEW. yEW. Let C w = Then there exists aWE V containing y. Therefore, 'V(Y, {(y,~)E l(y,~)E RI'V(y,~)E RI\jI(y,~)E WforsomeWE V}. Thenu{CwlWE ThenulCwlWE V}=R. So {C I C w I W E V} is a family of subsets suhsets of D whose union is residual in D \jI(C such that 0/( C w) w ) eWE V. It remains to show that C w is cofinal in D for each WE (y,~) W E V. For this let WoE V and (y, ~) E D. Then there exists an Xo E G A with ThenG'6 S(xo,U)cW o. Pick(z,8)E Rwith(xo,A)~(z,8)and(y,p)~(z,8). Rwith(xo,A)~(z,8)and(y,~)~(z,8). ThenG8 eGA which implies Z EGA. EGA' But (z, 8) E R implies 8 ~ A which in turn tum implies G8 B(GA,U), A'U), so G A cC Star (G'6'U). (G 8,U). Therefore,xo Therefore, x E S(s,U) for some SS E G'6 G8 8 E B(G (xo, 8), (y, (y,~) \jI(s, 8) E which implies S E S(xo,U). Sex 0 ,U). Then (x 0, A) ~ (s, (s,8), p) ~ (s, 8), and 'V(s, S(xo,U) cWo. Hence Cwo is cofinal in D so 'V \jI is almost Cauchy. 1
°
By hypothesis, 'V p E X. Let V be a neighborhood of p. \jI clusters to some P Then there exists a cofinal C c D with o/(C) Fa = \jI(C) c V. Let a E A. Then Fa = G ~ for some p ~ E B. Pick x EE G~. Then there exists (y, 8) E C with (x, ~) ~ (y, 8) and o/(Y, 8) EE V. Therefore, ~ ~ b \jI(y, 8 which implies G 8'6 C G ~ and \jI(y, 8) = Y y E G'6 G8 C G~ c Fa F a so SO F anV:t= arN *- 0. Hence p is a cluster point of {F IF a} so P E K. But \jI c X - U which is closed, so p E X - U which is a contradiction.
= 0. In this case, let D in the argument above be Next we have the case K = Define:5; \jI:D the set {(x, y) E X x A Ix E Fa}. Define ~ on D as before and define 'V: D ~X \jI(x, a) = x for each (x, a) E D. Then just as in the argument above, we can by 'V(x, show that 'V \jI is almost Cauchy and hence clusters to some p E X. But then as
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5. Fundamental Constructions
the argument above shows, p E K, so K *' 7: 0. Consequently, (X, f.l) /J.) must be supercomplete. Conversely, assume X is supercomplete. Let {xa)' {x a}, a E A, be an almost {x~ II B a) and F Faa = CI(H a). Then Cauchy net in X. For each a put H a = {xp ~ ~ a} {Fa) /J.andpick UE f.lwith /J.with U*< V. Since {xa} {xa) is {Fa} isanetinX'. Let VE f.landpick y), y Y E B, of cofinal subsets of A and almost Cauchy, there exists a collection {C r}' y) C U such that urC uyC ry is residual in A and {x {x~p I ~BE y} cUr C Uy a collection {U r} E C r} 0:0 E A such that B uyC ry for each ~ B ~ ao. 0:0. Then H ao cC for each y. Pick !X{) ~ E urC uyC r. y' For each y pick V VyrEV E V such that Star (U·r,U) (Uy,U) cC V r. y' We want to show urC . CUrUy. LetyE F . IfYE Huo,clearlyyE UyV Supposey is a limit F Fao CUyU LetYE Fao' Hao,clearlYYE Supposeyisalimit y. y uo ao point of H H uo ao' Let U E U be a neighborhood of Yo y. Then there exists an x p~ E H Hao x~ E U. Now xp x~ E H Hao x~ E U ry for some y so U C Yr. V y. ao with xp ao implies xB ao C UyV y. Hence F uo urVr· 0
ao. Then F al C F ao which implies F al C Star (F ao' V). Pick y Let al a1 E H ao' E B. Let ~BE E C ry such that ~B > ale a1' Then xp x~ E U rand y and xp x~ E F al which implies nF x~ Star(F Star(F x p E VynF Vr al so uV uVry C Star (F al ,V). But F ao C UyV u r V ry so F ao C Star (F al ,V). 8(F ao,v) a1 E H ao so {Fa} Hence F al E R(F ao ,V) for each al {F a} is Cauchy. Since X' is {Fa) complete, {F a} converges to its set of cluster points K. Let p E K. Let W be a anW 7: 0 for neighborhood of p in X. Then there exists a cofinal C C A with F Fan W *' each a E Co y E F anW then y E H a or y is a limit point of H a' a. In either C. If Y case, W contains some xp x~ with B B ~ a. For each a E C pick K(a) ~ a such that x 1CK(a) (a) E F anW. (a) II a E C} c C W is cofinal in A. Hence, {xa} Fan W. Then {x 1CK(a) Hence. {x a} clusters top. -
COROLLARY 5.1 Cofinal completeness implies supercompleteness which in turn implies completeness. (1. 1sbell. paracompact. then it is COROLLARY 5.2 (J. Isbell, 1962) If X is paracompact, supercomplete with respect to u.
Notice that our proof that supercompleteness implies each almost Cauchy net clusters only relies on the fact that there exists a cluster point of {Fa). {F a}. Consequently, the existence of a cluster point for each proper Cauchy net in X' is equivalent to supercompleteness. Furthermore, if p is a cluster point for the {F a} and U E J.l, proper Cauchy net F = = {Fa) /J., then for each U E U containing p, UnF ~ *' 7= 0 for cofinally many F ~~'s 's in F. Let y be any index. Then there exists 7: 0 and since F is directed by inclusion we have UnF UnE' yr *' "* 0. a () 8 ~ y with U(lF UrlF B *' Faa E F. Thus we can pick an Xu E Therefore, S(p, U)nF a*,0 a 7: 0 for each F /J.. Then {x {xu} S(P,U)nF aa for each U EE J.l. u} converges to p which implies p is a limit point of F Faa for each a. Therefore, p EE F Faa for each a since each F F a is closed. Hence nF aa*' 7= 0. Similarly, if nF aa*,0 7= 0 then any p E nF ex is a cluster point of F. We record these observations as
5.4 Hyperspaces
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PROPOSITION 5.9 If (X, J.!) is a uniform space then the .following (X. 11) following statements are equivalent: (J) (X, 11) f.l) is supercomplete (I) (X. supercomplete, XI' has a cluster point and (2) Each proper Cauchy net in X' (3) For each proper Cauchy net F in X'I nF"# 0. X', nF 1
*'
EXERCISES 1. A function f:X -t ~ Y from a uniform space (X, 11) f.l) into a uniform space (Y,v) detennines ~ Y' where X' and Y' are the hyperspaces of X and determines a function f':X' j':X' -t j' is defined by f'(A) j'(A) = Cl E X'. .fj'' is called Y respectively. f' Cly(fIA]) y (f1Al) for each A EX'. the hyperfunction of f. Show that f' j' is uniformly continuous if and only if f is unifonnly uniformly continuous. THE HYPERSPACE OF A METRIC SPACE 2. The Hausdorff distance h between two (closed) sets A and B in a metric space (X, d) is defined as the maximum of sup{d(a, B)la sup I dCa, B) I a E A} A) and sup {d(A,b) I b E B} B) where d(x, S) = =inf{ in!{ d(x,y) lYE S} S) for any subset S. Show (a) h is a metric on HX (the hyperspace of X) that generates the topology of HX so that the hyperspace of a metric space is again a metric space. (b) If (X, d) is complete then (HX, h) is cOlnplete. complete. THE HYPERSPACE OF A COMPACT SPACE 3. Show that the hyperspace of a compact space is compact. 4. Show that a discrete space of the power of the continuum, with the uniformity determined by all countable coverings, is complete but not unifonnity supercomplete. 5. Show that real Hilbert space is supercomplete but not cofinally complete.
{Fa) inf{F a) = = {x E 6. Define the limit inferior of a net {F a} in X' to be the set inf{ Fa} x, {Fa) X Ifor each neighborhood U of X, {F a} eventually meets U}. Similarly, define the limit superior to be the set sup{F = {x suplF a} a) = Ix E X II for each neighborhood U of x, {F a) a} frequently meets U}. If K is the set of cluster points of {F a} then sup{Fa}=K=inf{F (a) sup{F a} = K= inf{F a} a} (b) (X, 11) J.!) is supercomplete if and only if whenever {F IF a} is Cauchy then for each U E I..l J.1 there is a ~ with F Faa C StarCK, U) for each aex ~ ~.
5. Fundamental Constructions
126
paracompact, then it is 7. Use the results of Exercise 8 to show that if X is paracompacL supercomplete with respect to u, without reference to cofinally Cauchy or almost Cauchy nets (i.e., do not appeal to Corollary 5.1).
5.5 Inverse Limits and Spectra We begin our discussion of inverse limits of topological and uniform spaces with a special case that will motivate the concept in general. Let {X nn }) be a sequence of topological spaces where n ranges over the non-negative integers In:Xn and suppose that for each positive integer n, there is a continuous function .in :X n ~ X n- 1 • The sequence of spaces and mappings {X·n,fn} --7 {Xn,fn) is known as an inverse limit sequence. Inverse limit sequences are often represented by diagrams like the one below:
In ~
...
fn-l In-!
f2
~Xn ~Xn-l ~
...
fl
~Xl ~Xo
r::
If m < n, then the composition mapping 1: = = fm+1 fm+l © fm+2 © ... © fn is a lx nn }) such continuous mapping from Xn X n to X mm•. Consider the sequence of points {x X n and XXnn = In+l fn+l (x (X n+l)' that for each n, XXnn E Xn +!). Then {x {xnn }) can be identified with a n;:'=oXnn by means of the function g:J g:l --7 ~ uX nn,, where point of the product space n;=ox 1 J denotes the non-negative integers, defined by g(n) = XXnBy means of this • n identification, the set Y of all such sequences can be considered to be a subset of n;=ox n;:,=oxnn.. Then Y, equipped with the subspace topology, is called the inverse limit space of the sequence {X n, In}. fn)' Y is denoted by limf-X lim.,...Xnn or by X X~. The functions In fn are sometimes called bonding maps. e>o'
LEMMA 5.7 If {X {Xn,fn) n, fn} is an inverse limit sequence with onto bonding n1aps {an} of positive integers maps and for some countable counrable set {an) inlegers there is a set {x an an }} such that xxaann E X Xaann for each n and such that if m < fl, n, then fa~m la~m (x an =x am an )) = am '' then eXlsrs a point in limf-X lim.,...X n whose coordinate coordinare in X Xaann is Xxaann for each n. there exists {an) is infinite. For each positive integer m, there exists a Proof: First assume {an} ~ m. If an = m put X Xmm = x an least positive integer an such that an 2:: an '' otherwise let X = f:nn (x ). Then {x } is a point of limf-X such that x E X Xmm f,:: an IXm) lim.,...X n xaann Xaann for each fl. an )· m Ian) is finite. Then there exists a greatest an, say aj' aj. If m < Next, assume {an} (xa). aj, Xm aj put XXmm = = ;:;. t:/z (x ). For each m ~ aj' assume X has already been defined. m aj Xm+! +!) = XXm • Then by Since fm+l is onto, pick XXm+! (X m+l) m+1 such that fm+l (x m+l E X IXm) Ix mm }} E lim~Xn lim.,...Xn and for induction we can complete the sequence {x m } such that {x each n, xxaann E X On an " U
-
Lemma 5.7 illustrates the fundamental property of inverse limit spaces, Xb all elements of limf-X lim"",Xnn having that coordinate have that for any coordinate Xk,
5.5 Inverse Limits and Spectra
127
all other coordinates XXmm with m < kk determined by the inverse limit sequence, sequcncc~ whereas there may be sOlne some room for choice of the coordinates XXmm with m > k. If the bonding maps are not onto, limf-X n may be the empty set. An Xnn are all countable discrete spaces, say Xn = example of this occurs when the X Xn = {x~} and the bonding Inaps {x;:'} maps fn:Xn ~ X n---1 are of the form fn(x~) fn(x;:') = x~~\. x;:,~il' {Xn> Clearly {X n , fn} is an inverse limit sequence, but if we begin with a point x~, it xi and there is only possible to pick the first m coordinates before we reach xT does not exist an X'J'+1 X'J+l such that fm+l (xj+l) Xl. Consequently~ (x'J'+l) = Xl' Consequently, there do not exist any sequences {x {xnl Xnn E X Xnn and fn(x nn )) = = XXn-l n -1 for each n > 1. n } such that X = 0. Under certain conditions, limf-X n ~ l' Therefore, limf-X nn = conditions~ we can assure that lirn~Xn 0. For instance, instance. THEOREM 5.2 If each space Xn X n in the inverse limit sequence {X·fP.fn} /XnJn} Hausdorff space then limf-X n "# i= 0. is a compact Hausdotff Proof: For each positive integer n let Ynn cC OX DXnn be defined by {Yi} E Yn if for each j < n, Yj-l =h(Yj). Jj(y). Then limf-Xn = nY nYn' n, Ynn n. We will show that for each n~ is closed. Suppose p E E ITX Then for some j < n we have fj+l (Pj+l) "# • DXnn -- Y YnJj+l (Pi+l) i= Pj. Pi' n Since X V j containing p Xjj is Hausdorff, there exists disjoint open sets V Ujj and Vj Pjj and h+1 Vj +1 = Jj+l (Pj+l) respectively. Put Vj+l = r;ll J;ll (V (Vi) Upp be a basic open set in j ) and let V nx + 1 as its j/hth and j+ 1 sf factors DXnn containing p and having V Ujj and V Vj+l Fi j Upp since if q = qj+l respectively. Then no point of Yn lies in V , = {qn} E VUp, then qj +1 E p Vjj + Up. Therefore, Yn is closed. + 1 which implies qj E Vj V j so q does not belong to Vp. DXn, nYnn Since {Ynn }} is a decreasing chain of closed sets in the compact space nx n , nY :j; i= 0 so limf-X nn i-: i= 0. •
There are many applications of inverse limit spaces of inverse limit sequences in topology. What we are interested in here is extending the concept to uniform spaces, and extending it in a more general setting. For this first notice that by changing the definition of an inverse limit sequence {Xn.fn} {X n , fn} so Xnn are now uniform spaces and the bonding maps are uniformly that the X limf-Xnn that is a uniform continuous, we get an inverse limit uniform space iin1f-X n0 This follows from the parallel subspace of the uniform product space OX DXno between product topological spaces and product uniform spaces [see Section 5.3]. more general than this. Let (D,<) But what we have in mind is something Inore be a directed set and suppose that for each a EED, D ~ X a is a topological space. Further suppose that whenever u, a, ~ E D with a :s; ~, there is a continuous ~ B, function fl:X i3~ ~ X a, and that these functions satisfy the following rules: (1)
(2)
.fl;. is the identity function for each a f&.
fr =.f{ =.ff whenever a < ~ < Yo y. fl © fi
Y= {xala E D} and andF= Let Y = {X a I ex ED} F =
UWI t.rg I a, ~ E
E
D
a:S; {Y,F} D with a ~ ~Pl. }. Then the pair {Y, F}
128
5. Fundamental Constructions
is called an inverse limit system. Since the set of non-negative integers forms a directed set, it is clear that an inverse limit sequence is a special case of an inverse limit system. To define the inverse limit space of an inverse limit system, we let {x Ix u} a} denote a net in uX ua such that Xu xa E X X au for each a E 0 and 0 then .!If ~) = X u. a' Then {x a} such that if a < ~ in D fl (x (x~) u} can be identified with a point p of nx au having coordinates p au = = X u. a' The collection of all such nets, under this identification, is a subspace of nx ua that we denote by lim<-X lim~X a and F). call the inverse limit space of the system {Y, F}.
LEMMA 5.8 lim<-X lim~Xu a of the inverse limit system {Y, I Y, F} F) is a closed subspace of the product space nx u. a. The proofs of Lemma 5.8 and of the following Theorem are left as exercises (Exercises 1 and 2).
THEOREM 5.3 The inverse limit space of an inverse limit system of Hausdorff spaces is a compact Hausdoiff Hausdorff space, and if each space of compact Hausdoiff the inverse limit system is non-empty, then the inverse limit space is non-empty. Again, if in the definition of the inverse limit system of {Y, F}, F), we require X ua E Y to be uniform spaces and the bonding maps in F to be uniformly the X a that is a lim~X u continuous functions, we get an inverse limit uniform space lim<-X uniform subspace of the uniform product space nx a' u. We now give an important example of an inverse limit uniform space that we will use later on. Let (X,v) be a uniform space and let {
5.5 Inverse Limits and Spectra
129
that U~ < U~, then we put A< 11. J.l. Then (A, <) is countably directed. To see this let {An} be a countable set of members of A. For each positive integer k let U~ = U}n ... nU} and put cP
X
X X)lil
~
~ -7'
.tJ" ·A0'A
II ~
X XA
X/cP X/
.tJ"
lj.l t)l
iiAA
X
<1>~
X/
where i~ denotes the identity map on X considered as a mapping from X X)lIl onto X A. A' It is left as an exercise [Exercise 3] to show that
veX) is called the weak completion of X with respect to v. A uniform space is said to be weakly complete if each (I)-directed co-directed Cauchy net clusters. We will show that (X,v) is uniformly homeomorphic to a dense uniform subspace of veX) and that veX) is weakly complete. The weak completion of a uniform space with respect to a uniformity was introduced by K. Morita in 1970 in a paper titled Topological completions and M-spaces published in Sci. Rep. Tokyo Kyoiku Daigaku 10, No. 271, pp. 271-288. To see that veX) is weakly complete, let \V:D 'I':D ~ veX) be an (I)-directed co-directed Cauchy net. Then for each A E A, 1t A © \V:D A,1t 'I':D ~ X/
130
5. Fundamental Constructions
belong to CI('VA(R a». u». Let m be the least positive integer such that m S(y,2-m)nStar(CI('V)...(R a», 2- mm)) = 0. Now for each n > m, y E 'VA(R 'V),JR n) which S(y,2)nStar(CI('VA(R u», m m ) C S(y, 2-so 'VA(Rn)n'VA(R u) = 0 which is a contradiction. implies 'VA(R ) 'V".
To see that (X,v) is uniformly homeomorphic to a dense subspace of veX), (x) I A EX} defines a point of vex) [see notice that for any x E X, {
contradiction. Therefore,
5.5 Inverse Limits and Spectra
and only
131
THEOREM 5.4 (K. Morita,
if
Proof: We first show that if X is weakly complete with respect to v then
countable collection of indicies in A. Since (A, <) is countably directed, there exists ayE a 'Y E A such that An < 'yY for each positive integer n. Therefore, for each n
Consequently,