Selection Principles in Uniform Spaces

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Note di Matematica 22, n. 2, 2003, 127–139.

Selection principles in uniform spaces

Ljubiˇsa D. R. Koˇcinaci Department of Mathematics, Faculty of Sciences, University of Niˇs, 18000 Niˇs, Serbia [email protected]

Received: 13/01/2003; accepted: 24/09/2003.

Abstract. We begin the investigation of selection principles in uniform spaces in a manner as it was done with selection principles theory for topological spaces. We introduced and characterized uniform versions of classical topological notions of the Menger, Hurewicz and Rothberger properties. The uniform γ-sets are also considered.

Keywords: Uniform space, selection principles, uniform Menger, uniform Hurewicz, uniform Rothberger, uniform γ-set, ω-cover, γ-cover, groupability, weak groupability.

MSC 2000 classiﬁcation: 54E15, 54D20.

Introduction

We first recall the basic facts about selection principles in topological spaces. Let and be collections of open covers of a topological space X. A B The symbol S ( , ) denotes the selection hypothesis that for each sequence 1 A B ( n N) of elements of there exists a sequence (U n N) such that for Un | ∈ A n | ∈ each n, U and U n N [16]. n ∈ Un { n | ∈ } ∈ B The symbol S ( , ) denotes the selection hypothesis that for each se- fin A B quence ( n N) of elements of there is a sequence ( n N) such that Un | ∈ A Vn | ∈ for each n N, is a finite subset of and N is an element of [16]. ∈ Vn Un n∈ Vn B If denotes the collection of all open covers of a space X, then X is said to O S have the Menger property [13], [7], [14], [9] (resp. the Rothberger property [15], [14], [16] if the selection hypothesis S ( , ) (resp. S ( , )) is true for X. fin O O 1 O O Our terminology and notation follow [4]. For a subset A of a space X and a collection of subsets of X, St(A, ) P P denotes the star of A with respect to , that is the set P A P = ; P ∪{ ∈ P | ∩ 6 ∅} for A = x , x X, we write St(x, ) instead of St( x , ). { } ∈ P { } P Let X be a space. If α and β are families of subsets of X we denote by α β ∧ the set A B A α, B β . α < β means that α is a refinement of β, i.e. { ∩ | ∈ ∈ } that for each A α there is a B β with A B. If St(A, α) A α < β we ∈ ∈ ⊂ { | ∈ } iThis work is partially supported by MNTR of Serbia, grant 1233 128 L. D. R. Koˇcinac

write α∗ < β and say that α is a star reﬁnement of β or that α is star inscribed in β. Let us recall now three equivalent deﬁnitions of uniform spaces:

using a family C of covers (see, for example, [17], [3]; also [4]); • using a family U of entourages of the diagonal [4]; • using a family D of pseudometrics [5](see also [4]). • In this paper we shall use the first of these three approaches because it allows us to define uniform selection principles in the standard manner as in general topological case, and to see, after their characterizations, the right nature of them. More precisely, we shall see that uniform selection principles are actually a kind of star selection principles defined and studied in [10].

A uniformity on a nonempty set X is a family C of covers of X which satisﬁes the following conditions:

(C.1) if α C and if β is a cover of X such that α < β, then β C; ∈ ∈ (C.2) if α , α C, then there exists β C such that β∗ < α and β∗ < α . 1 2 ∈ ∈ 1 2 The covers from C are called uniform covers, and the pair (X, C) a uniform space. We assume that all spaces are inﬁnite and consider only Hausdorﬀ unifor- mities satisfying also the condition

(C.3) for any two distinct points x and y in X there is an α C such that no ∈ member of α contains both x and y.

The topology on X generated by C, denoted TC, is deﬁned in such way that St(x, α) x X, α C is a base for TC. { | ∈ ∈ } Let us mention a basic fact regarding uniform spaces. For any uniformity C on X and each α C there is a pseudometric d on X such that B (x, 1) x ∈ { d | ∈ X < St(x, α) x X . } { | ∈ } This fact (or the third approach for the deﬁnition of uniform structures) gives a possibility to consider also measure-like properties in uniform spaces. These properties often give characterizations of selection principles in metric spaces. We do not consider such questions in this paper. Selection principles in uniform spaces 129

1 The uniform Menger property

Let (X, C) be a uniform space. We say that X has the uniform Menger property if for each sequence (α : n N) of elements of C there is a sequence n ∈ (β n N) such that for each n N β is a ﬁnite subset of α and N β is n | ∈ ∈ n n n∈ n a cover of X. S So, in notation above the uniform Menger property is the S (C, ) prop- fin C erty. 1 Theorem. For a uniform space (X, C) the following are equivalent: (a) X has the uniform Menger property;

(b) for each sequence (α n N) C there is a sequence (A n N) of n | ∈ ⊂ n | ∈ finite subsets of X such that X = n∈NSt(An, αn). (c) for each sequence (α n N) CSthere is a sequence (β n N) such n | ∈ ⊂ n | ∈ that for each n β is a finite subset of α and X = NSt( β , α ). n n n∈ ∪ n n Proof. (a) (b): Let (α n N) be a sequence ofScovers from C. By ⇒ n | ∈ (a), for each n choose a finite subset βn of αn such that n∈N βn is a cover of X. For each n and each B β choose an element x(B, n) B and put ∈ n S ∈ A = x(B, n) B β . Then each A is a finite subset of X and n { | ∈ n} n

X = βn St(An, αn), N ∪ ⊂ N n[∈ n[∈ i.e. (b) holds. (b) (c): It is evident. ⇒ (c) (a): Let (α n N) C be a sequence of uniform covers of X. ⇒ n | ∈ ⊂ For each n let γn be a uniform cover of X which is star inscribed in αn. Apply (c) to the sequence (γ n N) and find a sequence (δ n N) such that n | ∈ n | ∈ each δ is a finite subset of γ and NSt( δ , γ ) = X. For each n N n n n∈ ∪ n n ∈ and each D δ pick a member A α such that St(D, γ ) A and let ∈ n D S∈ n n ⊂ D µ = A D δ . Then the sequence (µ n N) witnesses for (α n N) n { D | ∈ n} n | ∈ n | ∈ that X satisfies (a). QED

This theorem shows that the uniform Menger property is actually a kind of star covering properties. In [10] we introduced star-Menger and strongly star- Menger topological spaces just as spaces which satisfy conditions (c) and (b), respectively, from the previous theorem, but with open covers instead of uniform ones. In notation we used there the conditions (b) and (c) of the previous theorem can be written as SS∗ (C, ) and S∗ (C, ), respectively. fin O fin O 130 L. D. R. Koˇcinac

2 Remark. In terms of entourages of the diagonal of X the uniform Menger property is defined in this way: A uniform space (X, U) is uniformly Menger if for each sequence (U n n | ∈ N) of entourages of the diagonal there is a sequence (A n N) of finite n | ∈ subsets of X such that N U [A ] = X, where U [A ] = y X (x, y) n∈ n n n n { ∈ | ∈ U for some x A . n ∈ n} S Recall that a uniform space (X, C) is said to be totally bounded or precompact [4], [3] (resp. pre-Lindelöf or ω-bounded [3], [12]) if each α C has a finite (resp. ∈ countable) subcover, or equivalently, if for each α C there exists a finite (resp. ∈ countable) set A X such that X = St(A, α). ⊂ Evidently that for a uniform space (X, C) we have:

1. If X is totally bounded, then it is uniformly Menger;

2. If X is uniformly Menger, then it is pre-Lindel¨of;

3. If (X, TC) has the Menger property, then (X, C) is uniformly Menger.

3 Note. There is a uniform space (X, C) which is uniformly Menger, but topological space (X, TC) has no the Menger property. Any non-Lindelöf Tychonoff space serves as such an example. To see this we have only to observe that each (Tychonoff) space with the Menger property is Lindelöf and that, by 8.1.19 and 8.3.4 in [4], each Tychonoff space X admits a uniformity C∗ which is totally bounded and thus uniformly Menger, and generates the original topology on X. 4 Note. A regular topological space X has the Menger property if and only if its fine uniformity has the uniform Menger property. Let X be a regular topological space with the Menger property. Then X is Lindelóf and thus paracompact. By a result of A.H. Stone (see 5.4.H.(d) in [4]) it follows that each open cover of X is normal. So, the collection of all open covers of X is the base of a uniformity on X – the universal uniformity on X (see 8.1.C in [4]). 5 Note. Any uniformity C on a Tychonoff space X such that (X, C) is uniformly Menger is coarser or equal to the Shirota uniformity on X ([4]). It follows from the fact that uniformly Menger uniform spaces are pre- Lindelöf and 8.1.I in [4]. Uniform spaces having the uniform Menger property have some properties which are similar to the corresponding properties of totally bounded uniform spaces. Selection principles in uniform spaces 131

6 Theorem. If a uniform space Y is a uniformly continuous image of a uniformly Menger space X, then Y is also uniformly Menger. 7 Theorem. Every subspace of a uniformly Menger uniform space (X, C) is uniformly Menger. Proof. Let (Y, C ) be a subspace of (X, C) and let (µ n N) be a Y n | ∈ sequence of elements of CY . For each n let αn be an element in C such that µ = α Y := U Y U α and let β C be such that β∗ < α . Apply n n ∧ { ∩ | ∈ n} n ∈ n n to the sequence (β n N) the fact that (X, C) is uniformly Menger and find n | ∈ finite sets A X, n N, such that X = NSt(A , β ). For each n put n ⊂ ∈ n∈ n n B = a A y Y withS y St(a, β ) n { ∈ n | ∃ ∈ ∈ n } and for each b B choose an element y Y with y St(b, β ); put ∈ n b ∈ b ∈ n C = y b B . n { b | ∈ n} We claim that the sequence (C n N) witnesses for (µ n N) that (Y, C ) n | ∈ n | ∈ Y is uniformly Menger. Let y Y . There are n N and a A such that y St(a, β ). By definition ∈ ∈ ∈ n ∈ n of B it means that a B and so there is y C satisfying y St(a, β ), n ∈ n a ∈ n a ∈ n hence a St(y , β ). Since y St(a, β ) and β∗ < α it follows y St(y , α ). ∈ a n ∈ n n n ∈ a n This together with y Y gives y St(y , µ ), i.e. Y = NSt(C , µ ). QED ∈ ∈ a n n∈ n n The following theorem describes a property of uniformS spaces which is close to the Menger uniform property. An open cover of a space X is an ω-cover [6] if X does not belong to U U and every finite subset of X is contained in an element of . U An open cover of a space X is said to be weakly groupable [2] if it is a U union of countably many finite, pairwise disjoint subfamilies such that for Un each finite set F X there is an n with F . ⊂ ⊂ ∪Un 8 Theorem. For a uniform space (X, C) the following are equivalent: (1) For each sequence (α n N) C there is a sequence (β n N) of n | ∈ ⊂ n | ∈ finite sets such that for each n, β α and the set St( β , α ) n N n ⊂ n { ∪ n n | ∈ } is an ω-cover of X;

(2) For each sequence (α n N) C there is a sequence (β n N) such n | ∈ ⊂ n | ∈ that for each n β is a ﬁnite subset of α and St( β , α ) n N is n n { ∪ n n | ∈ } a weakly groupable cover of X.

(3) For each sequence (α n N) of uniform covers of X there is a sequence n | ∈ (F n N) of ﬁnite subsets of X such that St(F , α ) n N is an n | ∈ { n n | ∈ } ω-cover of X; 132 L. D. R. Koˇcinac

(4) For each sequence (α n N) of uniform covers of X there is a sequence n | ∈ (F n N) of finite subsets of X such that the set St(F , α ) n N n | ∈ { n n | ∈ } is a weakly groupable cover of X. Proof. (1) (2): It follows from the fact that each ω-cover is weakly ⇒ groupable. (2) (3): Let (α n N) be a sequence of uniform covers of X and let for ⇒ n | ∈ each n N β be a uniform cover of X such that β∗ < α . Apply (2) to the ∈ n n i≤n i sequence (β n N) and choose a sequence (γ n N) such that each γ is n | ∈ n | ∈V n a finite subset of β and the family St( γ , β ) n N is a weakly groupable n { ∪ n n | ∈ } cover of X. Therefore, there is a sequence n < n < < n < of natural 1 2 · · · k · · · numbers such that each finite subset F of X is contained in St( γ , β ) n ∪{ ∪ i i | k ≤ i < n for some k N. For each n and each C γ let A be an element of k+1} ∈ ∈ n C α with St(C, β ) A and let δ = A C γ . For each A in δ pick a n n ⊂ C n { C | ∈ n} C n point x(A ) A , denote by K the set x(A ) C γ and put C ∈ C n { C | ∈ n} M = K i < n , for n < n , n ∪{ i | 1} 1 M = K n i < n , for n n < n . n ∪{ i | k ≤ k+1} k ≤ k+1 Then each M is a finite subset of X. Let us show that the set St(M , α ) n { n n | n N is an ω-cover of X. ∈ } Let F be a finite subset of X. There is some k such that F St( γ , β ) ⊂ ∪{ ∪ i i | n i < n . Then we have k ≤ k+1} F St( γ , β ) n i < n = St(C, β ) ⊂ ∪{ ∪ i i | k ≤ k+1} ∪nk≤i

(i) for each n < n let β = µ ; 1 n i

2 Uniform Hurewicz spaces

In 1925 in [7] (see also [8]), W. Hurewicz introduced a covering property for a topological space X, called now the Hurewicz property, in this way:

For each sequence ( n N) of open covers of X there is a Un | ∈ sequence ( n N) of finite sets such that for each n , Vn | ∈ Vn ⊂ Un and for each x X, for all but finitely many n, x . ∈ ∈ ∪Vn We define now the natural uniform analogue of this property. A uniform space (X, C) is said to have the uniform Hurewicz property if for each sequence (α n N) of uniform covers of X there is a sequence n | ∈ (β n N) such that each β is a finite subset of α and for each x X we n | ∈ n n ∈ have x β for all but finitely many n. ∈ ∪ n The following theorem gives a characterization of the uniform Hurewicz property. 9 Theorem. For a uniform space (X, C) the following statements are equivalent:

(1) X is uniformly Hurewicz;

(2) For each sequence (α n N) of uniform covers of X there is a sequence n | ∈ (F n N) of ﬁnite subsets of X such that each x X belongs to all but n | ∈ ∈ ﬁnitely many sets St(Fn, αn);

(3) For each sequence (α n N) C there is a sequence (β n N) such n | ∈ ⊂ n | ∈ that for each n β is a finite subset of α and each x X is contained n n ∈ in all but finitely many sets St( β , α ). ∪ n n Proof. (1) (2): It is evident. ⇒ (2) (3): Let (α n N) be a sequence of covers from C. By (2) there ⇒ n | ∈ is a sequence (F n N) of finite subsets of X such that each x X is an n | ∈ ∈ element of St(F , α ) for all but finitely many n. For each n and each x F n n ∈ n take an element A(x, n) α containing x and put β = A(x, n) n F . It ∈ n n { | ∈ n} is easy to see that the sequence (β n N) guarantees that (3) is true for X. n | ∈ 134 L. D. R. Koˇcinac

(3) (1): Let (α n N) be a sequence of uniform covers of X. For each ⇒ n | ∈ n N choose a β C with β∗ < α . Applying (3) to the sequence (β n N) ∈ n ∈ n n n | ∈ one may find a sequence (µ n N) such that each µ is a finite subset of n | ∈ n β and each x X is a member of St( µ , β ) for all but finitely many n. For n ∈ ∪ n n each M µ choose an element U(M) α such that St(M, µ ) U(M) and ∈ n ∈ n n ⊂ put ν = U(M) M µ . Then for each n ν is a finite subset of α and it n { | ∈ n} n n is routine to check that the sequence (ν n N) shows for (α n N) that n | ∈ n | ∈ X is uniformly Hurewicz. QED

10 Remark. When studying a uniform spaces using entourages of the diagonal we shall use the condition (2) of the previous theorem as an official definition of the uniform Hurewicz property: A uniform space(X, U) is uniformly Hurewicz if for each sequence (U n n | ∈ N) of entourages of the diagonal there is a sequence (F n N) of finite subsets n | ∈ of X such that each x X belongs to all but finitely many sets U [F ]. ∈ n n Notice the following simple facts. For a uniform space (X, C) we have: 1. If X is totally bounded, then it is uniformly Hurewicz;

2. If X is uniformly Hurewicz, then it is uniformly Menger;

3. If (X, TC) has the Hurewicz property, then (X, C) is uniformly Hurewicz. Arguments similar to those in 3 and 4 give: 11 Note. There is a uniform space (X, C) which is uniformly Hurewicz, but topological space (X, TC) has no the Hurewicz property. 12 Note. A regular topological space X has the Hurewicz property if and only if its ﬁne uniformity has the uniform Hurewicz property. From 4 and 12 we conclude that the classes of uniformly Menger and uniformly Hurewicz spaces are diﬀerent. The following two results are similar to the corresponding results in Section 1. 13 Theorem. If a uniform space Y is a uniformly continuous image of a uniformly Hurewicz space X, then Y is also uniformly Hurewicz. 14 Theorem. Every subspace of a uniformly Hurewicz uniform space (X, C) is uniformly Hurewicz. The next three results concerning the uniform Hurewicz property are varia- tions on some properties of totally bounded uniform spaces.

15 Theorem. If a uniformly Hurewicz space (X, CX ) is dense in a uniform space (Y, C), then Y is also uniformly Hurewicz. Selection principles in uniform spaces 135

Proof. Let (µ n N) be a sequence in C and let for each n α = n | ∈ n µ X := V X V µ . Then (α n N) is a sequence in C . Choose n ∧ { ∩ | ∈ n} n | ∈ X for each n a member ν C for which ν∗ < µ and let β = ν X. Since X n ∈ n n n n ∧ is uniformly Hurewicz, there are finite sets A X, n N, such that each x in n ⊂ ∈ X belongs to all but finitely many St(A , β )’s. Let us show that (A n N) n n n | ∈ witnesses for (µ n N) that Y is uniformly Hurewicz. n | ∈ Let y Y . Let n be big enough. Then there is x X St(y, ν ). Since X ∈ ∈ ∩ n is uniformly Hurewicz, x belongs to all but finitely many sets St(Ak, βk), i.e. there is k such that for each k > k there exists a A with x St(a , β ) 0 0 k ∈ k ∈ k k ⊂ St(a , ν ). But ν∗ < µ and so for all k > max n, k we have y St(a , µ ). k k k k { 0} ∈ k k This means that (Y, CY ) has indeed the uniform Hurewicz property. QED 16 Corollary. A uniform space X is uniformly Hurewicz if and only if its completion X˜ is uniformly Hurewicz. 17 Theorem. The product (X Y, C C ) of uniformly Hurewicz uniform × X × Y spaces (X, CX ) and (Y, CY ) is uniformly Hurewicz. Proof. Let (γ n N) be a sequence in C C ; one may suppose n | ∈ X × Y that all elements in each γ are of the form U V , with U α C , n n × n n ∈ n ∈ X V β C . For each n let µ C and ν C be such that µ∗ < α and n ∈ n ∈ Y n ∈ X n ∈ Y n n ν∗ < β . There is a sequence (A n N) of finite subsets of X and a sequence n n n | ∈ (B n N) of finite subsets of Y such that each x X and each y Y one n | ∈ ∈ ∈ has x St(A , µ ) and y St(B , ν ) for all but finitely many n. We prove ∈ n n ∈ n n that the sets C = A B , n N, show that X Y uniformly Hurewicz. n n × n ∈ × Let (x, y) X Y . There is n N such that for each n > n we have ∈ × 0 ∈ 0 x St(A , µ ) and y St(B , ν ), i.e for some a A and some b B we ∈ n n ∈ n n n ∈ n n ∈ n have x St(a , µ ), y St(b , ν ). Further, there is k such that for all k > k ∈ n n ∈ n n 0 0 we have a St(A , µ ) and b St(B , ν ). Since µ∗ < α and ν∗ < β it n ∈ k k n ∈ k k n n n n follows that for n > max n , k we have x St(A , α ) and y St(B , β ), { 0 0} ∈ n n ∈ n n i.e. (x, y) St(A B , γ ). QED ∈ n × n n Let us remark that in a similar way one can prove the following result. 18 Theorem. The product of a uniformly Menger space X and a uniformly Hurewicz space Y is uniformly Menger. We end this section showing a result related to the uniform Hurewicz property. Before that we need the following notion [11]. An open cover of a space X is called groupable if it can be expressed as U a countable union of finite, pairwise disjoint subfamilies , n N, such that Un ∈ each x X belongs to for all but finitely many n. ∈ ∪Un An open cover of a space X is a γ-cover [6] if it is infinite and each x X U ∈ belongs to all but finitely many elements of . U 136 L. D. R. Koˇcinac

19 Theorem. For a uniform space (X, C) the following statements are equivalent:

(1) For each sequence (α n N) of uniform covers of X there is a sequence n | ∈ (F n N) of ﬁnite subsets of X such that St(F , α ) n N is a n | ∈ { n n | ∈ } γ-cover of X;

(2) For each sequence (α n N) C there is a sequence (β n N) such n | ∈ ⊂ n | ∈ that for each n β is a ﬁnite subset of α and the set St( β , α ) n n n { ∪ n n | ∈ N is a γ-cover of X; } (3) For each sequence (α n N) C there is a sequence (β n N) such n | ∈ ⊂ n | ∈ that for each n β is a ﬁnite subset of α and St( β , α ) n N is n n { ∪ n n | ∈ } a groupable cover of X;

(4) For each sequence (α n N) C there is a sequence (F n N) of n | ∈ ⊂ n | ∈ ﬁnite subsets of X such that St(F , α ) n N is a groupable cover of { n n | ∈ } X.

Proof. (1) (2): Let (α n N) be a sequence of covers from C. By (2) ⇒ n | ∈ there is a sequence (F n N) of finite subsets of X such that St(F , α ) n | ∈ { n n | n N is a γ-cover X. For each n and each x F take an element A(x, n) α ∈ } ∈ n ∈ n containing x and then put β = A(x, n) n F . It is easy to see that the n { | ∈ n} sequence (β n N) guarantees that (3) is true for X. n | ∈ (2) (3): Because each γ-cover is groupable it follows that this implication ⇒ is true. (3) (4): Let (α n N) be a sequence of uniform covers of X. For each ⇒ n | ∈ n N choose a uniform cover β which is star inscribed in α . Applying (3) to ∈ n n the sequence (β n N) one finds a sequence (µ n N) such that each µ is n | ∈ n | ∈ n a finite subset of β and the family St( µ , β ) n N is a groupable cover of n { ∪ n n | ∈ } X. This means that there is a sequence n < n < < n < in N such that 1 2 · · · k · · · each x X belongs to St( µ , β ) n i < n for all but finitely many ∈ ∪{ ∪ i i | k ≤ k+1} k. For each n N and each M µ pick a set A in α containing St(M, β ) ∈ ∈ n M n n and then a point x A . If we put F = x M µ we obtain a M ∈ M n { M | ∈ n} sequence (F n N) of finite subsets of X such that St(F , α ) n N is a n | ∈ { n n | ∈ } groupable cover for X; the sequence n < n < < n < shows this fact. 1 2 · · · k · · · (4) (1): Let (α n N) be a sequence of covers from C. For each ⇒ n | ∈ n N choose a β C such that β∗ < α . Apply now (4) to the sequence ∈ n ∈ n i≤n i (β n N) and for each n N choose a finite subset F of X such that n | ∈ ∈ V n St(F , β ) n N is a groupable cover of X, i.e. there is a sequence n < { n n | ∈ } 1 n < < n < of positive integers such that each x X belongs to 2 · · · k · · · ∈ St(F , β ) n i < n for all but finitely many k. Further, we define ∪{ i i | k ≤ k+1} Selection principles in uniform spaces 137

Φ = F , for each n < n , n i

By analogy with the deﬁnition of the Rothberger property in topological spaces we introduce the next notion. A uniform space (X, C) has the uniform Rothberger property if for each sequence (α n N) of elements of C there is a sequence (U n N) such n | ∈ n | ∈ that for each n N U α and N U = X. ∈ n ∈ n n∈ n 20 Theorem. For a uniformSspace (X, C) the following are equivalent: (a) X has the uniform Rothberger property;

(b) For each sequence (α n N) C there is a sequence (x n N) of n | ∈ ⊂ n | ∈ elements of X such that X = n∈NSt(xn, αn). (c) For each sequence (α n N)S C there is a sequence (A n N) such n | ∈ ⊂ n | ∈ that for each n A α and X = NSt(A , α ). n ∈ n n∈ n n Proof. (a) (b): Let (α n N) bSe a sequence of covers from C. By (a), ⇒ n | ∈ for each n choose a member U in α such that U n N is a cover of X. n n { n | ∈ } For each n choose an element x U . Then n ∈ n

X = Un St(xn, αn), N ⊂ N n[∈ n[∈ i.e. (b) holds. (b) (c): It is clear. ⇒ (c) (a): Let (α n N) be a sequence of uniform covers of X. Choose a ⇒ n | ∈ sequence (γ n N) C such that for each n γ is star inscribed in α γ∗ < n | ∈ ⊂ n n | n α . Apply (c) to the sequence (γ n N); there is a sequence (G n N) n n | ∈ n | ∈ such that for each n G γ and NSt(G , γ ) = X. Let for each n A be n ∈ n n∈ n n n an element of α such that St(G , γ ) A . Then X = N A . QED n n Sn ⊂ n n∈ n Notice that each uniformly Rothberger uniform spaceSis uniformly Menger but these two classes of spaces are distinct. It follows from the fact (that can be veriﬁed in a similar manner as in 4 and 12) that a regular topological space X has the Rothberger property if and only if its ﬁne uniformity has the uniform Rothberger property. 138 L. D. R. Koˇcinac

4 Uniform γ-sets

A topological space X is a γ-set [6] if for each sequence ( n N) of Un | ∈ ω-covers of X there is a sequence (U n N) such that for each n, U n | ∈ n ∈ Un and the set U n N is a γ-cover of X. { n | ∈ } We deﬁne uniform γ-sets in the following way. A uniform space (X, C) is said to be a uniform γ-set if for each sequence (α n N) of uniform covers of X there is a sequence (x n N) of elements n | ∈ n | ∈ of X such that each x is contained in St(xn, αn) for all but ﬁnitely many n. 21 Theorem. For a uniform space (X, C) the following assertions are equivalent:

(1) X is a uniform γ-set;

(2) For each sequence (α n N) of uniform covers of X there is a sequence n | ∈ (A n N) such that for each n A α and the set St(A , α ) n N n | ∈ n ∈ n { n n | ∈ } is a γ-cover of X.

Proof. To prove (1) implies (2) we need to argue as in the proof of (2) implies (3) of Theorem 9. So, we have to prove only (2) (1). ⇒ Let (α n N) be a sequence of uniform covers of X. For each n choose a n | ∈ β C such that β∗ < α . We apply (2) to the sequence (β n N) and select n ∈ n n n | ∈ for each n an element B β such that the set St(B , β ) n N is a γ-cover n ∈ n { n n | ∈ } of X. For each n N let A be a member of α satisfying St(B , β ) A . ∈ n n n n ⊂ n For each n take a point x A . Then we have the sequence (x n N) n ∈ n n | ∈ witnessing for (α n N) that X is a uniform γ-set. QED n | ∈

5 Closing remarks

To each selection principle for topological spaces it is naturally associated the corresponding game and often selection principles can be characterized game- theoretically (see [16]). In uniform case to each uniform selection principle one can assign also the corresponding game. For example, the game associated to the uniform Menger property is defined in the following way. Two players, ONE and TWO, play a round for each positive integer. In the n-th round ONE chooses a uniform cover α and TWO responds by choosing a finite set F X. TWO wins n n ⊂ a play α , F ; α , F , if X = N St(F , α ); otherwise ONE wins. The 1 1 2 2 · · · n∈ n n other games corresponding to the uniform Hurewicz property and the uniform S Rothberger property are defined similarly. Evidently, if ONE does not have Selection principles in uniform spaces 139 a winning strategy in these games, then the corresponding uniform selection principle holds. Find examples that show that no converse is true in each of these three situations.

References

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