A Categorical Property of the Stone - Cech Compactification

c Allerton Press, Inc., 2007. ISSN 1055-1344, Siberian Advances in Mathematics, 2007, Vol. 17, No. 4, pp. 291–296.  c R. B. Beshimov, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 1, pp. 132–140. Original Russian Text 

ˇ A Categorical Property of the Stone–Cech Compactification R. B. Beshimov1* 1

Institute of Mathematics, Tashkent, 100125 Uzbekistan Received June 19, 2006

Abstract—We study some categorical properties of the functor Oβ of weakly additive functionals acting in the category Tych of the Tychonoff spaces and their continuous mappings. We show that Oβ preserves the weight of infinite-dimensional Tychonoff spaces, the singleton, and the empty set, and that Oβ is monomorphic and continuous in the sense of T. Banakh, takes every perfect mapping to an epimorphism, and preserves intersections of functionally closed sets in a Tychonoff space. DOI: 10.3103/S1055134407040037 Key words: weakly additive functional, functor, weight.

In [6], E. Shchepin defined a normed functor in the category of bicompacta and their continuous mappings. A covariant functor F : Comp → Comp is called normal if F is continuous, monomorphic, and epimorphic and preserves weight, intersections, preimages, the singleton, and the empty set. In [4] A. Ch. Chigogidze constructed an extension of a normal functor F : Comp → Comp to a covariant functor L : Tych → Tych with preservation of normality. We call a functor L : Tych → Tych normal if L is continuous, preserves weight, embeddings, intersections, preimages, the singleton, and the empty set, and takes k-covering mappings to surjections. Recall that a continuous mapping f : X → Y is called k-covering if, for every bicompactum B ⊂ Y , there exists a bicompactum A ⊂ X with f (A) = B. Every perfect mapping is k-covering. In [5] T. Radul defined a covariant functor O : Comp → Comp of weakly additive normed order-preserving functionals on the category of bicompacta. He proved that O satisfies all normality conditions but the condition of preservation of preimages. In this article, we extend the functor O : Comp → Comp to a covariant functor Oβ : Tych → Tych acting from the category of Tychonoff spaces and their continuous mappings into itself. We prove that the functor Oβ satisfies all normality conditions (of course, in the sense appropriate for Tych) with some changes but the condition of preservation of preimages. Some properties of Oβ are studied in [3]. Suppose that X ∈ Comp. Denote by  C(X) the set of all continuous functions ϕ : X → R with  the usual sup-norm ϕ = sup ϕ(x) : x ∈ X . Given c ∈ R, denote by cX the constant function defined by the formula cX (x) = c for all x ∈ X. Let ϕ, ψ ∈ C(X). We say that ϕ ≤ ψ if and only if ϕ(x) ≤ ψ(x) for all x ∈ X. A mapping v : C(X) → R is called a functional. A functional v : C(X) → R is called (1) weakly additive if v(ϕ + cX ) = v(ϕ) + c for all ϕ ∈ C(X) and c ∈ R; *

E-mail: [email protected],[email protected]

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(2) order-preserving if, given ϕ, ψ ∈ C(X), the inequality ϕ ≤ ψ implies the inequality v(ϕ) ≤ v(ψ); (3) normed if v(1X ) = 1. Let X be a bicompactum. Denote by O(X) the set of all weakly additive order-preserving normed functionals. For brevity, call the elements of O(X) weakly additive functionals.   Consider O(X) as a subspace of the space Cp C(X) of all continuous functions on C(X) with the topology of pointwise convergence. The neighborhood base of a weakly additive functional ν ∈ O(X) is formed by the sets of the form     (1) (ν; ϕ1 , . . . , ϕk ; ε) = ν  ∈ O(X) : ν(ϕi ) − ν  (ϕi ) < ε, i = 1, 2, . . . , k , where ϕi ∈ C(X), i = 1, 2, . . . , k, and ε > 0. Suppose that X, Y ∈ Comp and f : X → Y is a continuous mapping. Define a mapping O(f ) : O(X) → O(Y ) by the formula   O(f )(μ) (ϕ) = μ(ϕ ◦ f ), where μ ∈ O(X) and ϕ ∈ C(Y ). T. Radul proved the following in [5]: Theorem 1. Let X be a bicompactum. Then O(X) is a bicompactum too.   Theorem 2. Let X be an infinite bicompactum. Then ω O(X) = ω(X). We say that a functional μ ∈ O(X) is concentrated on a closed subset A of a bicompactum X if μ ∈ O(A). The least closed set A ⊂ X with respect to inclusion for which μ ∈ O(A) is called the support of the functional μ ∈ O(X) and denoted by supp μ; i.e.,   supp μ = A : A ⊂ X, μ ∈ O(A) . Let X and Y be Tychonoff spaces and let f : X → Y be a continuous mapping. Consider the Stone– ˇCech extension of f , i.e., βf : βX → βY . Put   Oβ (X) = μ ∈ O(βX) : supp μ ⊂ X , Oβ (f ) = O(βf ) | Oβ (X). Then the mapping Oβ (f ) : Oβ (X) → Oβ (Y ) acts by the formula   Oβ (f )(μ) (ϕ) = μ(ϕ ◦ f ),

(2)

where μ ∈ Oβ (X) and ϕ ∈ Cb (Y ). Here Cb (Y ) stands for the set of all bounded continuous functions defined on a Tychonoff space Y . If μ ∈ Oβ (X) then Oβ (f )(μ) ∈ Oβ (Y ). Show that Oβ (f ) is continuous. Indeed, assume that μ ∈ Oβ (X) and Oβ (f )(μ) = ν ∈ Oβ (Y ). Consider an arbitrary neighborhood V = O(ν; ϕ1 , . . . , ϕk ; ε) of ν. Put U = O(μ; ϕ1 ◦ f, . . . , ϕk ◦ f ; ε). Take μ ∈ O(μ; ϕ1 ◦ f, . . . , ϕk ◦ f ; ε). Then    μ (ϕi ◦ f ) − μ(ϕi ◦ f ) < ε, i = 1, . . . , k. By (2) this is equivalent to the inequality      Oβ (f )(μ )(ϕi ) − Oβ (f ) μ(ϕi )  = Oβ (f )(μ )(ϕi ) − ν(ϕi ) < ε. Hence, Oβ (f )(μ ) ∈ V . Proposition 1 [3]. The functor Oβ is a covariant functor acting in the category of Tychonoff spaces, i.e., Oβ : Tych → Tych. Proposition 2. Suppose that X and Y are bicompacta, f : X → Y is a continuous mapping, and μ ∈ O(X). Then f (supp μ) ⊃ supp O(f )(μ). SIBERIAN ADVANCES IN MATHEMATICS Vol. 17 No. 4 2007

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Proof. Let L = supp μ, let M = f (L), and let iL : L → X and iM : M → Y be the inclusion mappings. Then f ◦ iL = iM ◦ (f | L). Consequently, O(f ) ◦ O(iL ) = O(iM ) ◦ O(fL ). In particular, O(f ) ◦ O(iL )(μ) = O(iM ) ◦ O(f | L)(μ).

(4)

Since O is a monomorphic functor (see [5]), the mapping O(iL ) : O(L) → O(X) is the identity. By (4) we infer      O(f )(μ) ∈ Im O(iM ) ◦ O(f | L) = O(iM ) Im O(f | L     = O(iM ) O(f | L) = O(iM ) O(M ) = O(M ). Thus, supp O(f )(a) ⊂ M . However, M = f (supp a). ˇ Theorem 3 [2]. Let X be a Tychonoff space, let βX be its Stone–Cech bicompact extension, and let bX be some bicompact extension. Then Oβ (X) and Ob (X) are homeomorphisms, where   Ob (X) = μ ∈ O(bX) : supp μ ⊂ X . Proposition 3 [2]. Let X be an infinite Tychonoff space. Then   ω Oβ (X) = ω(X). Proposition 4. The functor Oβ is monomorphic, i.e., Oβ preserves the injectivity of mappings. Proof. Let μ1 = μ2 ∈ Oβ (X) ⊂ O(βX). Since O is monomorphic, we have O(βf )(μ1 ) = O(βf )(μ2 ). By the definition of Oβ , we have Oβ (f )(μ1 ) = O(βf )(μ1 ) = O(βf )(μ2 ) = Oβ (f )(μ2 ), i.e., Oβ (f )(μ1 ) = Oβ (f )(μ2 ). Proposition 5. The functor Oβ : Tych → Tych preserves (a) the singleton, (b) the empty set. Proof. (a) Take x ∈ X ⊂ βX. Define a functional δx : C(βX) → R by the formula δx (ϕ) = ϕ(x), where ϕ ∈ C(βX). Clearly, δx ∈ Oβ (X), since supp δx = {x} ⊂ X. If y ∈ βX \ X then δy (ϕ) = ϕ(y) / Oβ (X). and supp δy = {y} ⊂ βX \ X. Therefore, δy ∈ Let X = {x} be a singleton. Obviously,   Oβ (X) = O(X) = O {x} = {δx }. (b) Let X = ∅. Then βX = ∅. By [5], we have O(βX) = ∅. From the inclusion Oβ (X) ⊆ O(βX) = ∅ we obtain Oβ (X) = ∅. Proposition 6. Suppose that f : X →Y is a continuous mapping and f (X) is everywhere dense in Y . Then the image Oβ (f ) Oβ (X) is everywhere dense in Oβ (Y ).   Proof. The image Oβ (f ) Oβ (X) includes the set     Oω f (X) = μ ∈ O(βY ) : |supp μ| < ∞ and supp μ ⊂ f (X) everywhere dense in Oβ (Y ) ⊂ O(βY ). Now, consider the question of continuity of the functor Oβ in the sense of T. Banach [1]. Let A be a directed partially ordered set (the directedness means that, for all α, β ∈ A, there exists γ ∈ A such that γ ≥ α and γ ≥ β).   Let Xα , pβα , A be an inverse spectrum (indexed by A) consisting of Tychonoff spaces. Denote by lim Xα the limit of this spectrum; and by pα : lim Xα → Xα , α ∈ A, the limit projections. The inverse ←− ←−   β spectrum Xα , pα , A generates an inverse spectrum   Oβ (Xα ), Oβ (pβα ), A SIBERIAN ADVANCES IN MATHEMATICS

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whose limit is denoted by lim Oβ (Xα ) and whose limit projections are denoted by prα : lim Oβ (Xα ) → ←− ←− Oβ (Xα ). The mapping   Oβ (pα ) : Oβ lim Xα → Oβ (Xα ) ←− yields the mapping   R : Oβ lim Xα → lim Oβ (Xα ). ←− ←− If all the Xα ’s are bicompacta then R is a homeomorphism. This follows from the continuity of the functor O in the category of bicompacta [5].   Proposition 7. The mapping R : Oβ lim Xα → lim Oβ (Xα ) is an embedding. If the limit pro←−  ←−    jections pα : lim Xα → Xα are everywhere dense i.e., pα lim Xα are everywhere dense in Xα ←−   ← −  then the image R Oβ lim Xα is everywhere dense in lim Oβ (Xα ). ←− ←−   β ˇ Proof. Let Xα , pα , A be a spectrum of Tychonoff spaces. Consider the Stone–Cech extension  β   βXα , β pα , A of this spectrum. Note that lim Xα is embedded in lim βXα . Moreover, if the limit ←− ←− projections pα : lim Xα → Xα are everywhere dense then the image of the space lim Xα is everywhere ←− ←− dense in lim βXα . Since the functor O : Comp → Comp is continuous, the corresponding mapping ←−   R : O lim Xα → lim O(βXα ) ←− ←−   is a homeomorphism. Since Oβ preserves embeddings, the mapping R : Oβ lim Xα → lim Oβ (Xα ) is ←− ←− embedded in the homeomorphism R; hence, R is an embedding. Moreover, Oβ preserves mappings with everywhere dense image; therefore,  the limit projections pα are everywhere dense. Then, under the embedding R, the spaces Oβ lim Xα are everywhere dense in lim Oβ (Xα ). ←− ←− Recall that a subset L ⊂ Cb (X) is called an A-subset if 0X ∈ L and ϕ + cX ∈ L for all ϕ ∈ L and c ∈ R. The following lemma may be regarded as an analog of the Hahn–Banach Theorem. Lemma 1 [7]. Let X be a Tychonoff space. Then, to each A-subspace L ⊂ Cb (X) and each weakly additive functional ν : L → R, there is a weakly additive functional ν  : Cb (X) → R such that ν  | L = ν. Lemma 2. Suppose that X is a Tychonoff space, μ ∈ Oβ (X), and A is a bicompact subset of X. The weakly additive functional μ is concentrated on A if and only if μ(ϕ) = 0 for every function ϕ ∈ Cb (X) with ϕ(A) = 0. Proof. Necessity. Let iA : A → X be the identity mapping and let μ ∈ Oβ (iA )(A). Then there exists a functional ν ∈ O(A) such that μ = Oβ (iA )(ν). Since ϕ(A) = 0, we have     μ(ϕ) = Oβ (iA )(ν) (ϕ) = ν(ϕ ◦ iA ) = ν ϕ(A) = ν(0A ) = 0. Sufficiency. Suppose that μ ∈ Oβ (X) and μ(ϕ) = 0 for every function ϕ ∈ Cb (X) such that ϕ(A) = 0. Consider the identity mapping i : A → X. Define a weakly additive functional ν ∈ Oβ (A) as follows: ν(ϕ) = μ(ϕ ), where ϕ : X → R is such that ϕ | A = ϕ. Then Oβ (i)(ν)(ϕ ) = ν(ϕ ◦ i) = ν(ϕ | A) = ν(ϕ | A) = μ(ϕ) = 0. Since μ(ϕ) = 0 for all ϕ ∈ Cb (X) with ϕ(A) = 0, we conclude that μ(ϕ ) = 0. Consequently, Oβ (i)(ν)(ϕ ) = μ(ϕ ) for all ϕ ∈ Cb (X). Hence, μ = Oβ (i)(ν), i.e., supp μ ⊂ A. Theorem 4. The functor Oβ : Tych → Tych takes perfect mappings to epimorphisms. Proof. Let f : X → Y be a perfect mapping between Tychonoff spaces. Suppose also that Oβ (f ) : Oβ (X) → Oβ (Y ), ν ∈ Oβ (Y ), and supp ν = B ⊂ Y . Put E = f −1 (B). Then E is a bicompactum SIBERIAN ADVANCES IN MATHEMATICS Vol. 17 No. 4 2007

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  in X. Consider C = ψ ◦ f : ψ ∈ Cb (Y ) . Obviously, C ⊂ Cb (X). We have 0X ∈ C, since 0Y ◦ f = 0X . Moreover, (ψ + cY ) ◦ f = ψ ◦ f + cY ◦ f = ψ ◦ f + cX for every c ∈ R. Hence, C is an A-subset in Cb (X). Let ν ∈ Oβ (Y ). Define a weakly additive functional μ : C → R by the formula μ (ψ ◦ f ) = ν(ψ), where ψ ∈ Cb (Y ). The weakly additive functional μ can be extended to a weakly additive functional μ : Cb (X) → R by Lemma 1. Moreover, μ | C = μ , i.e., μ(ψ ◦ f ) = ν(ψ) for every ψ ∈ Cb (Y ). Consequently, Oβ (f )(μ)(ψ) = μ(ψ ◦ f ) = ν(ψ), i.e., Oβ (f )(μ) = ν. Demonstrate that supp μ ⊆ E. Take ϕ ∈ Cb (X) with ϕ | E = 0. Show that μ(ϕ) = 0. By construction of μ, there exist functions ψ1 , ψ2 ∈ Cb (Y ) such that ψ1 ◦ f ≤ ϕ and ψ2 ◦ f ≥ ϕ. Indeed, we may assume without loss of generality that |ϕ| ≤ 1. Put

 1 Ai = x ∈ X : ϕ(x) ≥ i , i = 1, 2, 3, . . . . 2 Consider the function ψi2 : Y → [0, 1] such that ψi2 | f (Ai ) =

1 2i−1

,

ψi2 (B) = 0.

1 Clearly, 0 ≤ ψi2 ≤ i−1 . 2 ∞ 2 Putting ψ2 = i=1 ψi , we obtain the desired function. The function ψ1 is constructed similarly:

 1 Ai = x ∈ X : ϕ(x) ≤ − i , i = 1, 2, 3, . . . , 2 1 ψi1 | f (Ai ) = − i−1 , ψi1 (B) = 0. 2 1 1 Clearly, − i−1 ≤ ψi1 ≤ 0. Put ψ1 = ∞ i=1 ψi . Then 2 μ(ψ1 ◦ f ) ≤ μ(ϕ) ≤ μ(ψ2 ◦ f ), or, which is the same, ν(ψ1 ) ≤ μ(ϕ) ≤ ν(ψ2 ). However, supp ν = B. Consequently, ν(ψ) = ν(ψ ◦ iB ) and 0 = ν(ψ1 ◦ iB ) ≤ μ(ϕ) ≤ ν(ψ2 ◦ iB ) = 0. Hence, μ(ϕ) = 0. Recall that a set A ⊂ X is called a functionally closed subset if there exists a nonnegative function f ∈ Cb (X) such that A = f −1 (0). From [5] we have Proposition 8. The functor Oβ preserves intersections of functionally closed subsets of Tychonoff spaces, i.e., Oβ (∩Fα ) = ∩ Oβ (Fα ), where Fα is a functionally closed subset for all α. SIBERIAN ADVANCES IN MATHEMATICS

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Remark. The functor Oβ does not preserve preimages. Give the example taken from [5, 7]. Let X = {x1 , x2 , x3 } and Y = {y1 , y2 } be finite bicompacta (all points x1 , x2 , x3 , y1 , and y2 are distinct). Construct the mapping f : X → Y as follows: f (x1 ) = y1 and f (x2 ) = f (x3 ) = y2 . Consider a functional δy2 ∈ Oβ (Y ) concentrated on {y2 } ⊂ Y . Define the functional μ ∈ Oβ (X) by the formula        μ(ϕ) = max min ϕ(x1 ), ϕ(x2 ) , min ϕ(x1 ), ϕ(x3 ) , min ϕ(x2 ), ϕ(x3 ) . Take an arbitrary function ψ ∈ C(Y ). We have Oβ (f )(μ)(ψ) = μ(ψ ◦ f )    = max min ψ(f (x1 )), ψ(f (x2 )) ,     min ψ(f (x1 )), ψ(f (x3 )) , min ψ(f (x2 )), ψ(f (x3 ))     = max min ψ(y1 ), ψ(y2 }, ψ(y2 ) = ψ(y2 ) = δy2 (ψ),   / Oβ {x2 , x3 } , though f −1 (y2 ) = {x2 , x3 }. Hence, Oβ does i.e., Oβ (f )(μ) = δy2 . On the other hand, μ ∈ not preserve images. REFERENCES 1. T. O. Banakh, “Topology of spaces of probability measures. I. The functors Pτ and P ,” Mat. Stud. 5, 65–87 (1995). 2. R. B. Beshimov, “On weakly additive functionals,” Mat. Stud. 18 (2), 179–186 (2002). 3. R. B. Beshimov, “Some properties of the functor Oβ ,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 313, 131–134 (2004) [J. Math. Sci. (N. Y.) 133 (5), 1599–1601 (2004)]. 4. A. Ch. Chigogidze, “Extension of normal functors,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. (6), 23–26 (1984). 5. T. Radul, “On the functor of order-preserving functionals,” Comment. Math. Univ. Carolin. 39 (3), 609–615 (1998). 6. E. V. Shchepin, “Functors and uncountable powers of compacta,” Uspekhi Mat. Nauk 36 (3 (219)), 3–62 (1981) [Russ. Math. Surv. 36 (3), 1–71 (1981)]. 7. A. A. Zaitov, “On categorical properties of order-preserving functionals,” Methods Funct. Anal. Topology 9 (4), 357–364 (2003).

Translated by Ya. Kopylov

SIBERIAN ADVANCES IN MATHEMATICS Vol. 17 No. 4 2007