aa r X i v : . [ m a t h . GN ] A p r Ascoli’s theorem for pseudocompact spaces
S. Gabriyelyan
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, P.O. 653, Israel
Abstract
A Tychonoff space X is called ( sequentially ) Ascoli if every compact subset (resp. convergentsequence) of C k ( X ) is equicontinuous, where C k ( X ) denotes the space of all real-valued continuousfunctions on X endowed with the compact-open topology. The classical Ascoli theorem states thateach compact space is Ascoli. We show that a pseudocompact space X is Asoli iff it is sequentiallyAscoli iff it is selectively ω -bounded. Keywords: C k ( X ), Ascoli, sequentially Ascoli, selectively ω -bounded, pseudocompact,compact-covering map
1. Introduction
All topological spaces in the article are assumed to be Tychonoff. We denote by C k ( X ) thespace C ( X ) of all continuous real-valued functions on a space X endowed with the compact-opentopology. One of the basic theorems in Analysis is the Ascoli theorem which states that if X is a k -space, then every compact subset of C k ( X ) is equicontinuous. For the proof of the Ascoli theoremand various its applications see for example the classical books [8], [9] or [23]. The Ascoli theoremmotivates us in [3] to introduce and study the class of Ascoli spaces. A space X is called Ascoli if every compact subset of C k ( X ) is equicontinuous. In [24], Noble proved that every k R -space isAscoli (recall that a space X is called a k R -space if a real-valued function f on X is continuousif and only if its restriction f ↾ K to any compact subset K of X is continuous). However, thereare Ascoli spaces which are not k R , see [3]. Being motivated by the classical notion of c -barrelledlocally convex spaces and the fact that in many highly important cases in Analysis only convergentsequences are considered (as in the Lebesgue Dominated Convergence Theorem), we defined in[13] a space X to be sequentially Ascoli if every convergent sequence in C k ( X ) is equicontinuous.Clearly, every Ascoli space is sequentially Ascoli, but the converse is not true in general (everynon-discrete P -space is sequentially Ascoli but not Ascoli, see [13]). Ascoli and sequentially Ascolispaces in various classes of topological, function and locally convex spaces are thoroughly studiedin [2, 3, 11, 12, 14, 15, 16, 17, 18, 24]. The next diagram shows the relationships between theaforementioned classes of topological spaces k -space + k R -space + Ascoli + sequentially Ascoli . By the Ascoli theorem every compact space is Ascoli. Although the compact spaces form themost important class of topological spaces, there are other classes of compact-type topologicalspaces (as sequentially compact or countably compact spaces etc.) which play a considerable role
Email address: [email protected] (S. Gabriyelyan)
Preprint submitted to Elsevier April 29, 2020 oth in Analysis and General Topology, see for example [9], [19], [23] or the articles [22, 25]. Themost general class of compact-type spaces is the class of pseudocompact spaces. Recall that a space X is called pseudocompact if every continuous function on X is bounded. So the following questionarises naturally: Which pseudocompact spaces X are ( sequentially ) Ascoli ? A partial answer to thisquestion was obtained in [15] where we showed that totally countably compact spaces and nearsequentially compact spaces are sequentially Ascoli, however, there are countably compact spaceswhich are not sequentially Ascoli (for definitions see Section 2).Let X be a pseudocompact space. We denote by βX the Stone- ˇCech compactification of X ,and let β : X → βX be the canonical embedding. Then the adjoint (or restriction) map β ∗ : C ( βX ) → C k ( X ), β ∗ ( f ) = f ◦ β , is a continuous linear isomorphism from the Banach space C ( βX )onto C k ( X ). One of the most important properties of continuous functions is the property of beingcompact-covering. A continuous function f : X → Y between topological space X and Y is called compact-covering if for every compact subset K of Y there is a compact subset C of X such that f ( C ) = K . It is well known that perfect mappings are compact-covering ([9, Theorem 3.7.2]),and compact-covering functions are important for the study of functions spaces as C k ( X ), see [21].Therefore one can ask: For which pseudocompact spaces X the adjoint map β ∗ : C ( βX ) → C k ( X ) is compact-covering ?The following class of pseudocompact spaces plays a crucial role to answer both questions. Definition 1.1.
A space X is called selectively ω -bounded if for any sequence { U n } n ∈ ω of nonemptyopen subsets of X there exists a sequence ( x n ) n ∈ ω ∈ Q n ∈ ω U n containing a subsequence ( x n k ) k ∈ ω with compact closure. (cid:3) Our terminology is explained by the possibility to “select” special (sub)sequences and the classicalnotion of ω -bounded spaces (recall that a space X is ω -bounded if every sequence in X has compactclosure). Clearly, ω -bounded (in particular, compact) spaces and sequentially compact spaces areselectively ω -bounded, and every selectively ω -bounded space is pseudocompact. In Lemma 2.1below we show that the class of selectively ω -bounded spaces coincides with Frol´ık’s class P ∗ introduced in [10]. In the next section we also compare the class of selectively ω -bounded spaceswith other important classes of pseudocompact spaces.The following theorem proved in the next section is the main result of the paper. Theorem 1.2.
For a pseudocompact space X the following assertions are equivalent: (i) X is selectively ω -bounded; (ii) the adjoint map β ∗ : C ( βX ) → C k ( X ) is compact-covering; (iii) X is an Asoli space; (iv) X is a sequentially Asoli space. It is worth mentioning that Kato constructed in [20] a space X in the class P ∗ which is not a k R -space. This example, Lemma 2.1 and Theorem 1.2 show that there is even a pseudocompactAscoli space which is not a k R -space.
2. Proof of Theorem 1.2
In [10], Frol´ık defined the class P ∗ consisting of spaces with the property: each infinite collectionof disjoint open sets has an infinite subcollection each of which meets some fixed compact set. Belowwe show that the class P ∗ coincides with the class of all selectively ω -bounded spaces.2 emma 2.1. A space X belongs to P ∗ if and only if it is selectively ω -bounded. Proof.
It is clear that every selectively ω -bounded space belongs to P ∗ . Conversely, assume that X belongs to P ∗ . Fix a sequence { U n } n ∈ ω of nonempty open subsets of X . We have to showthat there exists a sequence ( x n ) n ∈ ω ∈ Q n ∈ ω U n containing a subsequence ( x n k ) k ∈ ω with compactclosure. Set A := { n ∈ ω : U n is infinite } and consider two cases. Case 1. The family A is finite. Then, without loss of generality, we can assume that U n = { x n } for every n ∈ ω . Put S := { x n } n ∈ ω . If S is finite, then the sequence ( x n ) ∈ Q n ∈ ω U n has compactclosure. If S is infinite, take a subsequence { x n k } k ∈ ω of S consisting of pairwise distinct points.Since X ∈ P ∗ there is a compact set K such that the set J := { j ∈ ω : U n j ∩ K = ∅} is infinite.Then the subsequence ( x n j ) j ∈ J of ( x n ) n ∈ ω has compact closure. Case 2. The family A is infinite. Then, passing to a subsequence if needed, we can assumethat all U n are infinite. By induction on n ∈ ω , we can choose pairwise distinct points z n suchthat z n ∈ U n for every n ∈ ω . Once again by induction, one can choose a subsequence { z n k } k ∈ ω of { z n } n ∈ ω and a sequence { V k } k ∈ ω of open sets in X such that z n k ∈ V k ⊆ U n k and V k ∩ V j = ∅ for alldistinct k, j ∈ ω . Since X ∈ P ∗ and all V k are pairwise disjoint, there is a compact set K such thatthe set J := { k ∈ ω : K ∩ V k = ∅} is infinite. For every j ∈ J , choose a point x n j ∈ K ∩ V j and,for every n
6∈ { n j : j ∈ J } , let x n := z n . It is clear that ( x n ) n ∈ ω ∈ Q n ∈ ω U n and its subsequence( x n j ) j ∈ J has compact closure witnessing the property of being a selectively ω -bounded space. (cid:3) Now we compare the class of selectively ω -bounded spaces with other important classes ofpseudocompact spaces. We recall that a space X is called • sequentially compact if every sequence in X has a convergent subsequence; • totally countably compact if every sequence in X has a subsequence with compact closure; • near sequentially compact if for any sequence { U n } n ∈ ω of open subsets of X there exists asequence ( x n ) n ∈ ω ∈ Q n ∈ ω U n containing a convergent subsequence ( x n k ) k ∈ ω ; • countably compact if every sequence in X has a cluster point.Near sequentially compact spaces were introduced and studied by Dorantes-Aldama and Shakhma-tov [7], who called them selectively sequentially pseudocompact spaces. Later those spaces wereapplied in [4] to the study of the Josefson–Nissenzweig property in the realm of locally convexspaces. Totally countably compact spaces, introduced by Frol´ık [10], were intensively studied byVaughan in [27]. Evidently, totally countably compact spaces and near sequentially compact spacesare selectively ω -bounded.By Example 2.6 of [7], the Mr´owka–Isbell space associated with a maximal almost disjoint family A on the discrete space ω is near sequentially compact. This example and Examples 2.11 and 2.14from [27] show that none of the implications in the following diagram is in general reversible ω -bounded + totally countablycompact + + ❖❖❖❖❖❖❖❖❖❖❖ ❖❖❖❖❖❖❖❖❖❖❖ countablycompact + pseudocompactsequentiallycompact + ; ♦♦♦♦♦♦♦♦♦♦♦ ♦♦♦♦♦♦♦♦♦♦♦ near sequentiallycompact + selectively ω -bounded < ♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣ Moreover, in [5] we constructed an example of a selectively ω -bounded space X which is countablycompact but not totally countably compact. 3et X be a (Tychonoff) space. Then the sets of the form[ K ; ε ] := { f ∈ C ( X ) : | f ( x ) | < ε for all x ∈ K } , where K ⊆ X is compact and ε >
0, form a base of the compact-open topology on C ( X ). Thespace C ( X ) endowed with the pointwise topology is denoted by C p ( X ).Recall that a continuous function f : X → Y between topological spaces X and Y is called sequence-covering if for every convergent sequence S in Y (with the limit point) there is a convergentsequence C ⊆ X such that f ( C ) = K . Proposition 2.2.
Let X be a subspace of a space Y such that the adjoint map i ∗ : C k ( Y ) → C k ( X ) of the identical embedding i : X ֒ → Y is surjective. If i ∗ is compact (sequence) covering and Y is a(resp. sequentially) Ascoli space, then so is X . Proof.
Let K be a compact subset (or a convergent sequence) in C k ( X ). We have to show that K is equicontinuous. Fix a point x ∈ X and ε >
0. Choose a compact subset (or a convergentsequence) C in C k ( Y ) such that i ∗ ( C ) = K . Since Y is (sequentially) Ascoli, there is an openneighborhood U of x in Y such that | g ( y ) − g ( x ) | < ε for all y ∈ U and g ∈ C. (2.1)For every x ∈ U ∩ X and each f ∈ K , take g ∈ C such that f = g ◦ i and then (2.1) implies | f ( x ) − f ( x ) | = (cid:12)(cid:12) g (cid:0) i ( x ) (cid:1) − g (cid:0) i ( x ) (cid:1)(cid:12)(cid:12) < ε. Thus K is equicontinuous. (cid:3) Now we are able to prove our main result.
Proof of Theorem 1.2 . (i) ⇒ (ii) Assume that X is selectively ω -bounded, and let K be a compactsubset of C k ( X ). We have to show that the closed subset C := ( β ∗ ) − ( K ) of the Banach space C ( βX ) is compact. Suppose for a contradiction that C is not compact. Since C ( βX ) is completeand C is closed, it follows that C is not precompact in C ( βX ). Therefore, by [6, Theorem 5], thereexist a sequence { f n } n ∈ ω ⊆ C and ε > k f n − f m k ∞ > ε for all distinct n, m ∈ ω, (2.2)where k f k ∞ denotes the sup-norm of f ∈ C ( βX ).It is clear that K is compact also in the space C p ( X ). Since X is pseudocompact, Theo-rem III.4.22 of [1] implies that K is an Eberlein compact, and hence K is Fr´echet–Urysohn by [1,Theorem III.3.6]. Therefore, passing to a subsequence if needed, we can assume that the sequence { f n } n ∈ ω converges in C k ( X ) to some function g ∈ K . Replacing K by K − g , we can also supposethat g = is the zero function.Since X is dense in βX , (2.2) implies that for every n ∈ ω , the open set U n := { x ∈ X : | f n ( x ) − f n +1 ( x ) | > ε } is not empty. As X is selectively ω -bounded, there exists a sequence ( x n ) n ∈ ω ∈ Q n ∈ ω U n containinga subsequence ( x n k ) k ∈ ω whose closure S := { x n k : k ∈ ω } is a compact subset of X .Now, since f n → in C k ( X ), there is an m ∈ ω such that f n ∈ (cid:2) S ; ε (cid:3) for all n ≥ m . Inparticular, we have (cid:12)(cid:12) f n k ( x n k ) − f n k +1 ( x n k ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) f n k ( x n k ) (cid:12)(cid:12) + (cid:12)(cid:12) f n k +1 ( x n k ) (cid:12)(cid:12) < ε (2.3)4or all sufficiently large k ∈ ω . But since x n k ∈ U n k for all k ∈ ω , (2.3) contradicts to the choice ofthe open sets U n . This contradiction shows that C is compact in C ( βX ), and hence the map β ∗ iscompact-covering.The implication (ii) ⇒ (iii) follows from Proposition 2.2 applied to X and Y = βX , and theimplication (iii) ⇒ (iv) is trivial.(iv) ⇒ (i) Assume that X is a sequentially Asoli space. We have to show that X is selectively ω -bounded. Suppose for a contradiction that X is not a selectively ω -bounded space. Then there existsa sequence { U n } n ∈ ω of nonempty open subsets of X such that for every sequence ( z n ) n ∈ ω ∈ Q n ∈ ω U n there is no subsequence ( z n k ) k ∈ ω whose closure is compact.For every n ∈ ω , choose a point x n ∈ U n and a continuous function f n : X → [0 ,
1] such that f n ( x n ) = 1 and f n ( X \ U n ) ⊆ { } . We claim that f n → in C k ( X ). Indeed, fix a compact subset K of X and ε >
0. Then the choice of the sequence { U n } n ∈ ω implies that the set A := { n ∈ ω : U n ∩ K = ∅} is finite (indeed, otherwise, we could choose a point z n ∈ U n ∩ K for every n ∈ A andan arbitrary point z n ∈ U n for every n ∈ ω \ A , and then the closure of the subsequence { z n : n ∈ A } of { z n } n ∈ ω would be compact that contradicts the choice of the sequence { U n } n ∈ ω ). This meansthat f n ∈ [ K ; ε ] for every n ∈ ω \ A . Thus f n → . Set S := { f n } n ∈ ω ∪ { } , so S is a convergentsequence in C k ( X ).For every n ∈ ω , set V n := { x ∈ X : f n ( x ) > } ; so x n ∈ V n ⊆ U n . Since X is pseudocompact,the family { V n } n ∈ ω is not locally finite (see [9, Theorem 3.10.22]), and therefore there is a point z ∈ X such that for every neighborhood W of z , the set { n ∈ ω : V n ∩ W = ∅} is infinite.Finally, to get a desired contradiction we show that the sequence S is not equicontinuous. Since X is sequentially Ascoli, Theorem 2.7 of [15] states that S is equicontinuous if and only if S isevenly continuous, i.e. the evaluation map ψ : S × X → R , ψ ( f, x ) := f ( x ), is continuous (seealso Lemma 2.1 of [11]). Therefore it is sufficient to show that the map ψ is not continuous at thepoint ( , z ). To this end, fix a k ∈ ω and an open neighborhood W of the point z . Since the set { n ∈ ω : V n ∩ W = ∅} is infinite, there is an m > k such that V m ∩ W contains some point t m . Bythe definition of V m we obtain | ψ ( f m , t m ) − ψ ( , z ) | = f m ( t m ) > . Thus ψ is not continuous at( , z ). (cid:3) It immediately follows from Theorem 1.2 that if X is a selectively ω -bounded space, then everycompact subset of C k ( X ) is metrizable. However, the converse is not true in general as the followingexample shows. Example 2.3.
There is a countably compact non-selectively ω -bounded space X such that all com-pact subsets of C k ( X ) (even of C p ( X ) ) are metrizable. Proof.
In [26], Terasaka constructed a separable countably compact space X whose square X × X is not pseudocompact. By Theorem 3.5 of [10], the product of a selectively ω -bounded space and apseudocompact space is pseudocompact. Therefore the space X is not selectively ω -bounded. Let D be a countable dense subspace of X . Then the restriction map C p ( X ) → C p ( D ) is continuousand injective. Since C p ( D ) is a metric space, it follows that all compact subsets of C p ( X ) and henceof C k ( X ) are metrizable. (cid:3) Acknowledge:
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