A characterization of X for which spaces C p (X) are distinguished and its applications
aa r X i v : . [ m a t h . GN ] N ov A CHARACTERIZATION OF X FOR WHICH SPACES C p ( X ) ARE DISTINGUISHED AND ITS APPLICATIONS
JERZY KA¸ KOL AND ARKADY LEIDERMAN
Abstract.
We prove that the locally convex space C p ( X ) of continuous real-valued functions on a Tychonoff space X equipped with the topology of point-wise convergence is distinguished if and only if X is a ∆-space in the sense of[21]. As an application of this characterization theorem we obtain the followingresults:1) If X is a ˇCech-complete (in particular, compact) space such that C p ( X )is distinguished, then X is scattered. 2) For every separable compact space ofthe Isbell–Mr´owka type X , the space C p ( X ) is distinguished. 3) If X is thecompact space of ordinals [0 , ω ], then C p ( X ) is not distinguished.We observe that the existence of an uncountable separable metrizable space X such that C p ( X ) is distinguished, is independent of ZFC. We explore alsothe question to which extent the class of ∆-spaces is invariant under basictopological operations. Introduction
Following J. Dieudonn´e and L. Schwartz [8] a locally convex space (lcs) E is called distinguished if every bounded subset of the bidual of E in the weak ∗ -topologyis contained in the closure of the weak ∗ -topology of some bounded subset of E .Equivalently, a lcs E is distinguished if and only if the strong dual of E (i.e. thetopological dual of E endowed with the strong topology) is barrelled , (see [19,8.7.1]). A. Grothendieck [17] proved that a metrizable lcs E is distinguished if andonly if its strong dual is bornological . We refer the reader to survey articles [5] and[6] which present several more modern results about distinguished metrizable andFr´echet lcs.Throughout the article, all topological spaces are assumed to be Tychonoff andinfinite. By C p ( X ) and C k ( X ) we mean the spaces of all real-valued continuousfunctions on a Tychonoff space X endowed with the topology of pointwise conver-gence and the compact-open topology, respectively. By a bounded set in a topolog-ical vector space (in particular, C p ( X )) we understand any set which is absorbedby every 0-neighbourhood.For spaces C p ( X ) we proved in [13] the following theorem (the equivalence (1) ⇔ (4) has been obtained in [11]). Theorem 1.1.
For a Tychonoff space X , the following conditions are equivalent : Date : December 1, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Distinguished locally convex space, scattered compact space, ∆-set,Isbell–Mr´owka space.The research for the first named author is supported by the GAˇCR project 20-22230L andRVO: 67985840. He thanks also the Center For Advanced Studies in Mathematics of Ben-GurionUniversity of the Negev for financial support during his visit in 2019. (1) C p ( X ) is distinguished. (2) C p ( X ) is a large subspace of R X , i.e. for every bounded set A in R X thereexists a bounded set B in C p ( X ) such that A ⊂ cl R X ( B ) . (3) For every f ∈ R X there is a bounded set B ⊂ C p ( X ) such that f ∈ cl R X ( B ) . (4) The strong dual of the space C p ( X ) carries the finest locally convex topology. Several examples of C p ( X ) with(out) distinguished property have been providedin papers [11], [12] and [13]. The aim of this research is to continue our initial workon distinguished spaces C p ( X ).The following concept plays a key role in our paper. We show its applicabilityfor the studying of distinguished spaces C p ( X ). Definition 1.2. ([21])
A topological space X is said to be a ∆ -space if for everydecreasing sequence { D n : n ∈ ω } of subsets of X with empty intersection, there isa decreasing sequence { V n : n ∈ ω } consisting of open subsets of X , also with emptyintersection, and such that D n ⊂ V n for every n ∈ ω . We should mention that R. W. Knight [21] called all topological spaces X satis-fying the above Definition 1.2 by ∆-sets. The original definition of a ∆-set of thereal line R is due to G. M. Reed and E. K. van Douwen (see [27]). In this paper,for general topological spaces satisfying Definition 1.2 we reserve the term ∆ -space .The class of all ∆-spaces is denoted by ∆.In Section 2 we give an intrinsic description of all spaces X ∈ ∆. One of themain results of our paper, Theorem 2.1 says that X is a ∆-space if and only if C p ( X ) is a distinguished space. This characterization theorem has been appliedsystematically for obtaining a range of results from our paper.Our main result in Section 3 states that a ˇCech-complete (in particular, compact) X ∈ ∆ must be scattered. A very natural question arises what are those scatteredcompact spaces X ∈ ∆. In view of Theorem 2.1, it is known that a Corson compact X belongs to the class ∆ if and only if X is a scattered Eberlein compact space[13]. With the help of Theorems 2.1 and 3.8 we show that the class ∆ containsalso all separable compact spaces of the Isbell–Mr´owka type. Nevertheless, as wedemonstrate in Section 3, there are compact scattered spaces X / ∈ ∆ (for example,the compact space [0 , ω ]).Section 4 deals with the questions about metrizable spaces X ∈ ∆. We noticethat every σ -scattered metrizable space X belongs to the class ∆. For separablemetrizable spaces X , our analysis reveals a tight connection between distinguished C p ( X ) and well-known set-theoretic problems about special subsets of the real line R . We observe that the existence of an uncountable separable metrizable space X ∈ ∆ is independent of ZFC and it is equivalent to the existence of a separablecountably paracompact nonnormal Moore space. We refer readers to [24] for thehistory of the normal Moore problem.In Section 5 we study whether the class ∆ is invariant under the basic topologicaloperations: subspaces, (quotient) continuous images, finite/countable unions andfinite products. We pose several new open problems.2. Description Theorem
In this section we provide an intrinsic description of X ∈ ∆. For the reader’sconvenience we recall some relevant terminology.(a) A disjoint cover { X γ : γ ∈ Γ } of X is called a partition of X . CHARACTERIZATION OF X ∈ ∆ 3 (b) A collection of sets { U γ : γ ∈ Γ } is called an expansion of a collection ofsets { X γ : γ ∈ Γ } in X if X γ ⊆ U γ ⊆ X for every index γ ∈ Γ.(c) A collection of sets { U γ : γ ∈ Γ } is called point-finite if no point belongs toinfinitely many U γ -s. Theorem 2.1.
For a Tychonoff space X , the following conditions are equivalent :(1) C p ( X ) is distinguished. (2) Any countable partition of X admits a point-finite open expansion in X . (3) Any countable disjoint collection of subsets of X admits a point-finite openexpansion in X . (4) X is a ∆ -space.Proof. Observe that every collection of pairwise disjoint subsets of X , { X γ : γ ∈ Γ } can be extended to a partition by adding a single set X ∗ = X \ S { X γ : γ ∈ Γ } .If the obtained partition admits a point-finite open expansion in X , then removingone open set we get a point-finite open expansion of the original disjoint collection.This shows evidently the equivalence (2) ⇔ (3).Assume now that (3) holds. Let { D n : n ∈ ω } be a decreasing sequence subsets of X with empty intersection. Define X n = D n \ D n +1 for each n ∈ ω . By assumption,a disjoint collection { X n : n ∈ ω } admits a point-finite open expansion { U n : n ∈ ω } in X . Then { V n = S { U i : i ≥ n } : n ∈ ω } is an open decreasing expansion in X with empty intersection. This proves the implication (3) ⇒ (4).Next we show (4) ⇒ (2). Let { X n : n ∈ ω } be any countable partition of X .Define D = X and D n = X \ S { X i : i < n } . Then X n ⊂ D n for every n , thesequence { D n : n ∈ ω } is decreasing and its intersection is empty. Assuming (4),we find an open decreasing expansion { U n : n ∈ ω } of { D n : n ∈ ω } in X such that T { U n : n ∈ ω } = ∅ . For every x ∈ X there is n such that x / ∈ U m for each m > n ,it means that { U n : n ∈ ω } is a point-finite expansion of { X n : n ∈ ω } in X . Thisfinishes the proof (3) ⇒ (4) ⇒ (2) ⇔ (3).Now we prove the implication (1) ⇒ (2). Let { X n : n ∈ ω } be any countablepartition of X . Fix any function f ∈ R X which satisfies the following conditions:for each n ∈ ω and every x ∈ X n the value of f ( x ) is greater than n . By assumption,there is a bounded subset B of C p ( X ) such that f ∈ cl R X ( B ). Hence, for every n ∈ ω and every point x ∈ X n , there exists f x ∈ B such that f x ( x ) > n . But f x is a continuous function, therefore there is an open neighbourhood U x ⊂ X of x such that f x ( y ) > n for every y ∈ U x . We define an open set U n ⊂ X as follows: U n = S { U x : x ∈ X n } . Evidently, X n ⊆ U n for each n ∈ ω . If we assume that theopen expansion { U n : n ∈ ω } is not point-finite, then there exists a point y ∈ X such that there are infinitely many numbers n with y ∈ U x n for some x n ∈ X n .This means that sup { g ( y ) : g ∈ B } = ∞ , which contradicts the boundedness of B .It remains to prove (2) ⇒ (1). By Theorem 1.1, we need to show that for everymapping f ∈ R X there is a bounded set B ⊂ C p ( X ) such that f ∈ cl R X ( B ). Ifthere exists a constant r > {| f ( x ) | : x ∈ X } < r , then we take B = { h ∈ C ( X ) : sup {| h ( x ) | : x ∈ X } < r } . It is easy to see that B is as required.Let f ∈ R X be unbounded. Denote by Y = ∅ and Y n = { x ∈ X : n − ≤| f ( x ) | < n } for each non-zero n ∈ ω . Define ϕ : X → ω by the rule: if Y n = ∅ then ϕ ( x ) = n for every x ∈ Y n . So, | f | < ϕ . Put X n = ϕ − ( n ) for each n ∈ ω . Notethat some sets X n might happen to be empty, but the collection { X n : n ∈ ω } isa partition of X with countably many nonempty X n -s. By our assumption, thereexists a point-finite open expansion { U n : n ∈ ω } of the partition { X n : n ∈ ω } . JERZY KA¸ KOL AND ARKADY LEIDERMAN
Define F : X → ω by F ( x ) = max { n : x ∈ U n } . Obviously, f < F . Finally, wedefine B = { h ∈ C p ( X ) : | h | ≤ F } . Then f ∈ cl R X ( B ), because for every finitesubset K ⊂ X there is a function h ∈ B such that f ↾ K = h ↾ K . Indeed, givena finite subset K ⊂ X , let { V x : x ∈ K } be the family of pairwise disjoint opensets such that x ∈ V x ⊂ U ϕ ( x ) for every x ∈ K . For each x ∈ K , fix a continuousfunction h x : X → [ − ϕ ( x ) , ϕ ( x )] such that h x ( x ) = f ( x ) and h x is equal to theconstant value 0 on the closed set X \ V x . One can verify that h = Σ x ∈ K h x ∈ B isas required. (cid:3) Below we present a straightforward application of Theorem 2.1.
Corollary 2.2. ([13])
Let Z be any subspace of X . If X belongs to the class ∆ ,then Z also belongs to the class ∆ .Proof. If { Z γ : γ ∈ Γ } is any collection of pairwise disjoint subsets of Z andthere exists a point-finite open expansion { U γ : γ ∈ Γ } in X , then obviously { U γ ∩ Z : γ ∈ Γ } is a point-finite expansion consisting of the sets relatively open in Z . It remains to apply Theorem 2.1. (cid:3) The last result can be reversed, assuming that X \ Z is finite. Proposition 2.3.
Let Z be a subspace of X such that Y = X \ Z is finite. If Z belongs to the class ∆ , then X belongs to ∆ as well.Proof. Let { X n : n ∈ ω } be any countable collection of pairwise disjoint subsets of X . Denote by F the set of those n ∈ ω such that X n T Y = ∅ . There might beonly finitely many X n -s which intersect the finite set Y , hence F ⊂ ω is finite. If n ∈ F , then we simply declare that U n is equal to X . Consider the subcollection { X n : n ∈ ω \ F } . It is a countable collection of pairwise disjoint subsets of Z . Since Z ∈ ∆, by Theorem 2.1, there is a point-finite open expansion { U n : n ∈ ω \ F } in Z . Observe that Z is open in X , therefore all those U n -s remain open in X .Bringing all U n -s of both sorts together we obtain a point-finite open expansion { U n : n ∈ ω } in X . Finally, X ∈ ∆, by Theorem 2.1. (cid:3) Remark 2.4.
The following applicable concept has been re-introduced in [13].A family {N x : x ∈ X } of subsets of a Tychonoff space X is called a scant cover for X if each N x is an open neighbourhood of x and for each u ∈ X the set X u = { x ∈ X : u ∈ N x } is finite. Our Theorem 2.1 generalizes one of the results obtained in [13] stating that if X admits a scant cover {N x : x ∈ X } then C p ( X ) is distinguished. Indeed, let { X γ : γ ∈ Γ } be any collection of pairwise disjoint subsets of X . Define U γ = S {N x : x ∈ X γ } . It is easily seen that { U γ : γ ∈ Γ } is a point-finite open expansionin X , by definition of a scant cover. Applying Theorem 2.1, we conclude that C p ( X ) is distinguished.3. Applications to compact spaces X ∈ ∆First we recall a few definitions and facts (probably well-known) which will beused in the sequel. A space X is said to be scattered if every nonempty subset A of X has an isolated point in A . Denote by A (1) the set of all non-isolated (in A ) The referee kindly informed the authors that this notion also is known in the literature underthe name the point-finite neighbourhood assignment . CHARACTERIZATION OF X ∈ ∆ 5 points of A ⊂ X . For ordinal numbers α , the α -th derivative of a topological space X is defined by transfinite induction as follows. X (0) = X ; X ( α +1) = ( X ( α ) ) (1) ; X ( γ ) = T α<γ X ( α ) for limit ordinals γ .For a scattered space X , the smallest ordinal α such that X ( α ) = ∅ is called the scattered height of X and is denoted by ht ( X ). For instance, X is discrete if andonly if ht ( X ) = 1.The following classical theorem is due to A. Pe lczy´nski and Z. Semadeni. Theorem 3.1. ([28, Theorem 8.5.4])
A compact space X is scattered if and onlyif there is no continuous mapping of X onto the segment [0 , . A continuous surjection π : X → Y is called irreducible (see [28, Definition 7.1.11])if for every closed subset F of X the condition π ( F ) = Y implies F = X . Proposition 3.2. ([28, Proposition 7.1.13])
Let X be a compact space and let π : X → Y be a continuous surjection. Then there exists a closed subset F of X such that π ( F ) = Y and the restriction π ↾ F : F → Y is irreducible. Proposition 3.3. ([28, Proposition 25.2.1])
Let X be a compact space and let π : X → Y be a continuous surjection. Then π is irreducible if and only if whenever E ⊂ X and π ( E ) is dense in Y , then E is dense in X . Recall that a Tychonoff space X is ˇCech-complete if X is a G δ -set in some(equivalently, any) compactification of X , (see [9, 3.9.1]). It is well known that everylocally compact space and every completely metrizable space is ˇCech-complete.Next statement resolves an open question posed in [13]. Theorem 3.4.
Every ˇCech-complete (in particular, compact) ∆ -space is scattered.Proof. Step 1: X is compact . On the contrary, assume that X is not scattered.First, by Theorem 3.1, there is a continuous mapping π from X onto the segment[0 , F of X such that π ( F ) = [0 ,
1] and the restriction π ↾ F : F → [0 ,
1] is irreducible. Since X ∈ ∆ thecompact space F also belongs to ∆, by Corollary 2.2. For simplicity, without lossof generality we may assume that F is X itself and π : X → [0 ,
1] is irreducible.Let { X n : n ∈ ω } be a partition of [0 ,
1] into dense sets. Put Y n = S k ≥ n X k , and Z n = π − ( Y n ) for all n ∈ ω . Then all sets Z n are dense in X by Proposition 3.3and the intersection T n ∈ ω Z n is empty. Every compact space X is a Baire space,i.e. the Baire category theorem holds in X , hence if { U n : n ∈ ω } is any openexpansion of { Z n : n ∈ ω } , then the intersection T n ∈ ω U n is dense in X . In view ofour Theorem 2.1 this conclusion contradicts the assumption X ∈ ∆, and the prooffollows. Step 2: X is any ˇCech-complete space . By the first step we deduce that everycompact subset of X is scattered. But any ˇCech-complete space X is scatteredif and only if every compact subset of X is scattered. A detailed proof of thisprobably folklore statement can be found in [29]. (cid:3) Proposition 3.5. If X is a first-countable compact space, then X ∈ ∆ if and onlyif X is countable.Proof. If X ∈ ∆, then X is scattered, by Theorem 3.4. By the classical theorem ofS. Mazurkiewicz and W. Sierpi´nski [28, Theorem 8.6.10], a first-countable compactspace is scattered if and only if it is countable. This proves (i) ⇒ (ii). The converse JERZY KA¸ KOL AND ARKADY LEIDERMAN is known [13] and follows from the fact that any countable space X = { x n : n ∈ ω } admits a scant cover. Indeed, define X n = { x i : i ≥ n } . Then the family { X n : n ∈ ω } is a scant cover of X . Now it suffices to mention Remark 2.4. (cid:3) Remark 3.6.
Theorem 3.4 extends also a well-known result of B. Knaster and K.Urbanik stating that every countable ˇCech-complete space is scattered [20]. It iseasy to see that a countable Baire space contains a dense subset of isolated points,but in general does not have to be scattered. We don’t know whether every Baire∆-space must have isolated points.Recall that an Eberlein compact is a compact space homeomorphic to a subsetof a Banach space with the weak topology. A compact space is said to be a Corsoncompact space if it can be embedded in a Σ-product of the real lines. Every Eberleincompact is Corson, but not vice versa. However, every scattered Corson compactspace is a scattered Eberlein compact space [1].
Theorem 3.7. ([13])
A Corson compact space X belongs to the class ∆ if and onlyif X is a scattered Eberlein compact space. Bearing in mind Theorem 3.4, to show Theorem 3.7 it suffices to use the factthat every scattered Eberlein compact space admits a scant cover (the latter followsfrom the proof of [4, Lemma 1.1]) and then apply Remark 2.4.Being motivated by the previous results one can ask if there exist scatteredcompact spaces X ∈ ∆ which are not Eberlein compact. The next question is alsocrucial: Does there exist a compact scattered space X / ∈ ∆? Below we answer bothquestions positively.We need the following somewhat technical Theorem 3.8.
Let Z = C ∪ C be a Tychonoff space such that (1) C ∩ C = ∅ . (2) C is an open F σ subset of Z . (3) both C and C belong to the class ∆ .Then Z also belongs to the class ∆ .Proof. By assumption, C = S { F n : n ∈ ω } , where each F n is closed in Z . Let { X n : n ∈ ω } be any countable collection of pairwise disjoint subsets of Z . Ourtarget is to define open sets U n ⊇ X n , n ∈ ω in such a way that the collection { U n : n ∈ ω } is point-finite. We decompose the sets X n = X n ∪ X n , where X n = X n ∩ C and X n = X n ∩ C . By Theorem 2.1, the collection { X n : n ∈ ω } expands to a point-finite open collection { U n : n ∈ ω } in C . The set C is open in Z , therefore U n are open in Z as well.Now we consider the disjoint collection { X n : n ∈ ω } in C . By assumption, C ∈ ∆, therefore applying Theorem 2.1 once more, we find a point-finite expansion { V n : n ∈ ω } in C consisting of sets which are open in C . Every set V n is a traceof some set W n , which is open in Z , i.e. V n = W n ∩ C , and every W n is open in Z .We refine the sets W n by the formula U n = W n \ S { F i : i ≤ n } . Since all sets F i areclosed in Z , the sets U n remain open in Z . Since all sets F i are disjoint with C , thecollection { U n : n ∈ ω } remains to be an expansion of { X n : n ∈ ω } . Furthermore,the collection { U n : n ∈ ω } is point-finite, because { V n : n ∈ ω } is point-finite, andevery point z ∈ C belongs to some F n , hence z / ∈ U m for every m ≥ n . Finally, wedefine U n = U n ∪ U n . The collection { U n : n ∈ ω } is a point-finite open expansionof { X n : n ∈ ω } , and the proof is complete. (cid:3) CHARACTERIZATION OF X ∈ ∆ 7 This yields the following
Corollary 3.9.
Let Z be any separable scattered Tychonoff space such that itsscattered height ht ( Z ) is equal to 2. Then Z ∈ ∆ .Proof. The structure of Z is the following. Z = C ∪ C , where C is a countabledense in Z set consisting of isolated in Z points and C consists of all accumulationpoints. Moreover, the space C with the topology induced from Z is discrete. Allconditions of Theorem 3.8 are satisfied, and the result follows. (cid:3) Our first example will be the one-point compactification of an Isbell–Mr´owkaspace Ψ( A ). We recall the construction and basic properties of Ψ( A ). Let A be analmost disjoint family of subsets of the set of natural numbers N and let Ψ( A ) bethe set N ∪ A equipped with the topology defined as follows. For each n ∈ N , thesingleton { n } is open, and for each A ∈ A , a base of neighbourhoods of A is thecollection of all sets of the form { A } ∪ B , where B ⊂ A and | A \ B | < ω . The spaceΨ( A ) is then a first-countable separable locally compact Tychonoff space. If A isa maximal almost disjoint (MAD) family, then the corresponding Isbell–Mr´owkaspace Ψ( A ) would be in addition pseudocompact. (Readers are advised to consult[18, Chapter 8] which surveys various topological properties of these spaces). Theorem 3.10.
There exists a separable scattered compact space X with the fol-lowing properties: (a) The scattered height of X is equal to 3. (b) X ∈ ∆ . (c) X is not an Eberlein compact space.Proof. Let A be any uncountable almost disjoint (in particular, MAD) family ofsubsets of N and let Z be the corresponding first-countable separable locally com-pact Isbell–Mr´owka space Ψ( A ). It is easy to see that Z = Ψ( A ) satisfies theassumptions of Corollary 3.9. Hence, Z ∈ ∆. Now, denote by X the one-pointcompactification of the separable locally compact space Z . Then the scatteredheight of X is equal to 3. Note that X ∈ ∆ by Proposition 2.3. Moreover, X isnot an Eberlein compact space, since every separable Eberlein compact space ismetrizable, while Ψ( A ) is metrizable if and only if A is countable. (cid:3) Now we show that there exist scattered compact spaces which are not in theclass ∆. We will use the classical Pressing Down Lemma. Let [0 , ω ) be the set ofall countable ordinals equipped with the order topology. For simplicity, we identify[0 , ω ) with ω . A subset S of ω is called a stationary subset if S has nonemptyintersection with every closed and unbounded set in ω . A mapping ϕ : S → ω iscalled regressive if ϕ ( α ) < α for each α ∈ S . The proof of the following fundamentalstatement can be found for instance in [22]. Theorem 3.11.
Pressing Down Lemma.
Let ϕ : S → ω be a regressivemapping, where S is a stationary subset of ω . Then for some γ < ω , ϕ − { γ } isa stationary subset of ω . It is known that there are plenty of stationary subsets of ω . In particular, everystationary set can be partitioned into countably many pairwise disjoint stationarysets [22]. Note that ω is a scattered locally compact and first-countable space.Next statement resolves an open question posed in [13]. JERZY KA¸ KOL AND ARKADY LEIDERMAN
Theorem 3.12.
The compact scattered space [0 , ω ] is not in the class ∆ .Proof. It suffices to show that ω does not belong to the class ∆. Assume, onthe contrary, that ω ∈ ∆. Denote by L the set of all countable limit ordinals.Evidently, L is a closed unbounded set in ω . Take any representation of L asthe union of countably many pairwise disjoint stationary sets { S n : n ∈ ω } . ByTheorem 2.1, there exists a point-finite open expansion { U n : n ∈ ω } in ω .Fr every α ∈ U n there is an ordinal β ( α ) < α such that [ β ( α ) , α ] ⊂ U n . In fact,for every n ∈ ω we can define a regressive mapping ϕ n : S n → ω by the formula: ϕ n ( α ) = β ( α ). Since S n is a stationary set for every n , we can apply to ϕ n thePressing Down Lemma. Hence, for each n there are a countable ordinal γ n and anuncountable subset T n ⊂ S n with the following property: [ γ n , α ] ⊂ U n for every α ∈ T n . Denote γ = sup { γ n : n ∈ ω } ∈ ω . Because all T n are unbounded, forall natural n we have an ordinal α n ∈ T n such that γ < α n and [ γ n , α n ] ⊂ U n .This implies that γ ∈ U n for every n ∈ ω . However, a collection { U n : n ∈ ω } ispoint-finite. The obtained contradiction finishes the proof. (cid:3) The function space C k ( X ) is called Asplund if every separable vector subspaceof C k ( X ) isomorphic to a Banach space, has the separable dual. Proposition 3.13.
If a Tychonoff space X belongs to the class ∆ , then the space C k ( X ) is Asplund. The converse conclusion fails in general.Proof. Let K ( X ) be the family of all compact subset of X . By the assumptionand Corollary 2.2, each K ∈ K ( X ) belongs to the class ∆. Clearly, C k ( X ) isisomorphic to a (closed) subspace of the product Π = Q K ∈K ( X ) C k ( K ) of Banachspaces C k ( K ). Assume that E is a separable vector subspace of C k ( X ) isomorphicto a Banach space. Observe that E is isomorphic to a subspace of the finite product Q j ∈ F C k ( K j ) for K j ∈ K ( X ) and j ∈ F . Indeed, let B be the unit (bounded) ball ofthe normed space E . Then there exists a finite set F such that T j ∈ F π − j ( U j ) ∩ Π ⊂ B , where U j are balls in spaces C k ( K j ), j ∈ F , and π j are natural projections from E onto C k ( K j ). Let π F be the (continuous) projection from Π onto Q ∈ F C k ( K j ).Then π F ↾ E is an injective continuous and open map from E onto ( π F ↾ E )( E ) ⊂ Q ∈ F C k ( K j ). The injectivity of π F ↾ E follows from the fact that B is a boundedneighbourhood of zero in E . It is easy to see that the image ( π F ↾ E )( B ) is anopen neighbourhood of zero in Q j ∈ F C k ( K j ). On the other hand, Q j ∈ F C k ( K j ) isisomorphic to the space C k ( L j ∈ F K j ) and the compact space L j ∈ F K j is scattered.By the classical [10, Theorem 12.29] E must have the separable dual E ∗ . Hence, C k ( X ) is Asplund. The converse fails, as Theorem 3.12 shows for X = [0 , ω ]. (cid:3) Since every infinite compact scattered space X contains a nontrivial convergingsequence, for such X the Banach space C ( X ) is not a Grothendieck space, (see [7]). Corollary 3.14. If X is an infinite compact and X ∈ ∆ , then the Banach space C ( X ) is not a Grothendieck space. The converse fails, as X = [0 , ω ] applies. For non-scattered spaces X Theorem 3.4 implies immediately the following
Corollary 3.15. If X is a non-scattered space, the Stone- ˇCech compactification βX is not in the class ∆ . Proposition 3.16.
Let X = βZ \ Z , where Z is any infinite discrete space. Then X is not in the class ∆ . CHARACTERIZATION OF X ∈ ∆ 9 Proof. βZ \ Z does not have isolated points for any infinite discrete space Z . (cid:3) It is known that X = [0 , ω ] is the Stone- ˇCech compactification of [0 , ω ). Weshowed that X / ∈ ∆. Also, βZ / ∈ ∆ for any infinite discrete space Z . Everyscattered Eberlein compact space belongs to the class ∆ by Theorem 3.7, howeverno Eberlein compact X can be the Stone- ˇCech compactification βZ for any propersubset Z of X by the Preiss–Simon theorem (see [2, Theorem IV.5.8]). All thesefacts provides a motivation for the following result. Example 3.17.
There exists an Isbell–Mr´owka space Z which is almost compact in the sense that the one-point compactification of Z coincides with βZ (see [18,Theorem 8.6.1]). Define X = βZ . Then X ∈ ∆, by Theorem 3.10.4. Metrizable spaces X ∈ ∆In this section we try to describe constructively the structure of nontrivial metriz-able spaces X ∈ ∆. Note first that every scattered metrizable X is in the class∆ since every such space X homeomorphically embeds into a scattered Eberleincompact [3], and then Theorem 3.7 and Corollary 2.2 apply. We extend this resultas follows.A topological space X is said to be σ -scattered if X can be represented as acountable union of scattered subspaces and X is called strongly σ -discrete if it is aunion of countably many of its closed discrete subspaces. Strongly σ -discretenessof X implies that X is σ -scattered, for any topological space. For metrizable X ,by the classical result of A. H. Stone [30], these two properties are equivalent. Proposition 4.1.
Any σ -scattered metrizable space belongs to the class ∆ .Proof. In view of aforementioned equivalence, every subset of X is F σ . If everysubset of X is F σ , then X ∈ ∆. This fact apparently is well-known (see alsoa comment after Claim 4.2). For the sake of completeness we include a directargument. We show that X satisfies the condition (2) of Theorem 2.1. Let { X n : n ∈ ω } be any countable disjoint partition of X . Denote X n = S { F n,m : m ∈ ω } ,where each F n,m is closed in X . Define open sets U n as follows: U = X and U n = X \ S { F k,m : k < n, m < n } for n ≥
1. Then { U n : n ∈ ω } is a point-finiteopen expansion of { X n : n ∈ ω } in X . (cid:3) A metrizable space A is called an absolutely analytic if A is homeomorphic to aSouslin subspace of a complete metric space X (of an arbitrary weight), i.e. A isexpressible as A = S σ ∈ N N T n ∈ N A σ | n , where each A σ | n is a closed subset of X . It isknown that every absolutely analytic metrizable space X (in particular, every Borelsubspace of a complete metric space) either contains a homeomorphic copy of theCantor set or it is strongly σ -discrete. Therefore, for absolutely analytic metrizablespace X the converse is true: X ∈ ∆ implies that X is strongly σ -discrete [13].However, the last structural result can not be proved in general for all (separable)metrizable spaces without extra set-theoretic assumptions. Let us recall severaldefinitions of special subsets of the real line R (see [23], [27]).(a) A Q -set X is a subset of R such that each subset of X is F σ , or, equivalently,each subset of X is G δ in X .(b) A λ -set X is a subset of R such that each countable A ⊂ X is G δ in X . (c) A ∆-set X is a subset of R such that for every decreasing sequence { D n : n ∈ ω } subsets of X with empty intersection there is a decreasing expansion { V n : n ∈ ω } consisting of open subsets of X with empty intersection. Claim 4.2.
The existence of an uncountable separable metrizable ∆ -space is equiv-alent to the existence of an uncountable ∆ -set.Proof. Note that every separable metrizable space homeomorphically embeds intoa Polish space R ω and the latter space is a one-to-one continuous image of the setof irrationals P . Therefore, if M is an uncountable separable metrizable space, thenthere exist an uncountable set X ⊂ R and a one-to-one continuous mapping from X onto M . It is easy to see that X is a ∆-set provided M is a ∆-space. (cid:3) Note that in the original definition of a ∆-set, G. M. Reed used G δ -sets instead ofopen sets and E. van Douwen observed that these two versions are equivalent [27].From the original definition it is obvious that each Q -set must be a ∆-set. The factthat every ∆-set is a λ -set is known as well. K. Kuratowski showed that in ZFCthere exist uncountable λ -sets. The existence of an uncountable Q -set is one of thefundamental set-theoretical problems considered by many authors. F. Hausdorffshowed that the cardinality of an uncountable Q -set X has to be strictly smallerthan the continuum c = 2 ℵ , so in models of ZFC plus the Continuum Hypothesis(CH) there are no uncountable Q -sets. Let us outline several known most relevantfacts.(1) Martin’s Axiom plus the negation of the Continuum Hypothesis (MA + ¬ CH)implies that every subset X ⊂ R of cardinality less than c is a Q -set (see [15]).(2) It is consistent that there is a Q -set X such that its square X is not a Q -set[14].(3) The existence of an uncountable Q -set is equivalent to the existence of anuncountable strong Q -set, i.e. a Q -set all finite powers of which are Q -sets [25].(4) No ∆-set X can have cardinality c [26]. Hence, under MA, every subset of R that is a ∆-set is also a Q -set. Recently we proved the following claim: If X hasa countable network and | X | = c , then C p ( X ) is not distinguished [13]. In view ofour Theorem 2.1 this fact means that no ∆-space X with a countable network canhave cardinality c . (5) It is consistent that there exists a ∆-set X that is not a Q -set [21]. Of course,there are plenty of nonmetrizable ∆-spaces with non- G δ subsets, in ZFC.(6) An uncountable ∆-set exists if and only if there exists a separable countablyparacompact nonnormal Moore space (see [16] and [26]).Summarizing, the following conclusion is an immediate consequence of our The-orem 2.1 and the known facts about ∆-sets listed above. Corollary 4.3. (1)
The existence of an uncountable separable metrizable space such that C p ( X ) is distinguished, is independent of ZFC. (2) There exists an uncountable separable metrizable space X such that C p ( X ) is distinguished, if and only if there exists a separable countably paracompactnonnormal Moore space. The referee kindly informed the authors that the last result can be derived easily from theactual argument of [26].
CHARACTERIZATION OF X ∈ ∆ 11 Basic operations in ∆ and open problems In this section we consider the question whether the class ∆ is invariant underthe following basic topological operations: subspaces, continuous images, quotientcontinuous images, finite/countable unions, finite products.
1. Subspaces . Trivial because of Corollary 2.2.
2. (Quotient) continuous images . Evidently, every topological space is a contin-uous image of a discrete one. The following assertion is a consequence of a knownfact about MAD families (see [18, Chapter 8]).
Proposition 5.1.
There exists a first-countable separable pseudocompact locallycompact Isbell–Mr´owka space Ψ( A ) which admits a continuous surjection onto thesegment [0 , . Thus, the class ∆ is not invariant under continuous images even for separable lo-cally compact spaces. However, one can show that every uncountable quotient con-tinuous image of any Isbell–Mr´owka space Ψ( A ) satisfies the conditions of Corollary3.9, therefore it is a ∆-space. Note also that a class of scattered Eberlein compactspaces preserves continuous images. We were unable to resolve the following majoropen problem. Problem 5.2.
Let X be any compact ∆ -space and Y be a continuous image of X .Is Y a ∆ -space? Even a more general question is open.
Problem 5.3.
Let X be any ∆ -space and Y be a quotient continuous image of X .Is Y a ∆ -space? Towards a solution of these problems we obtained several partial positive results.
Proposition 5.4.
Let X be any ∆ -space and ϕ : X → Y be a quotient continuoussurjection with only finitely many nontrivial fibers. Then Y is also a ∆ -space.Proof. By assumption, there exists a closed subset K ⊂ X such that ϕ ( K ) is finiteand ϕ ↾ X \ K : X \ K → Y \ ϕ ( K ) is a one-to-one mapping. Both sets X \ K and Y \ ϕ ( K ) are open in X and Y , respectively. Since ϕ is a quotient continuousmapping, it is easy to see that ϕ ↾ X \ K is a homeomorphism. X \ K is a ∆-space,hence Y \ ϕ ( K ) is also a ∆-space. Finally, Y is a ∆-space, by Proposition 2.3. (cid:3) Proposition 5.5.
Let X be any ∆ -space and ϕ : X → Y be a closed continuoussurjection with finite fibers. Then Y is also a ∆ -space.Proof. Let { Y n : n ∈ ω } be a partition of Y . By assumption, the partition { ϕ − ( Y n ) : n ∈ ω } admits a point-finite open expansion { U n : n ∈ ω } in X .Clearly, ϕ ( X \ U n ) are closed sets in Y . Define V n = Y \ ϕ ( X \ U n ) for each n ∈ ω .We have that { V n : n ∈ ω } is an open expansion of { Y n : n ∈ ω } in Y . It remainsto verify that the family { V n : n ∈ ω } is point-finite. Indeed, let y ∈ Y be anypoint. Each point in the fiber ϕ − ( y ) belongs to a finite number of sets U n . Sincethe fiber ϕ − ( y ) is finite, y is contained only in a finite number of sets V n whichfinishes the proof. (cid:3)
3. Finite/countable unions . Proposition 5.6.
Assume that X is a finite union of closed subsets X i , where each X i belongs to the class ∆ . Then X also belongs to ∆ . In particular, a finite unionof compact ∆ -spaces is also a ∆ -space.Proof. Denote by Z the discrete finite union of ∆-spaces X i . Obviously, Z is a ∆-space which admits a natural closed continuous mapping onto X . Since all fibersof this mapping are finite, the result follows from Proposition 5.5. (cid:3) We recall a definition of the Michael line. The Michael line X is the refinement ofthe real line R obtained by isolating all irrational points. So, X can be representedas a countable disjoint union of singletons (rationals) and an open discrete set.Nevertheless, the Michael line X is not in ∆ [13]. This example and Proposition5.6 justify the following Problem 5.7.
Let X be a countable union of compact subspaces X i such that each X i belongs to the class ∆ . Does X belong to the class ∆ ?4. Finite products . We already mentioned earlier that the existence of a Q -set X ⊂ R such that its square X is not a Q -set, is consistent with ZFC. Problem 5.8.
Is the existence of a ∆ -set X ⊂ R such that its square X is not a ∆ -set, consistent with ZFC? It is known that the finite product of scattered Eberlein compact spaces is ascattered Eberlein compact.
Problem 5.9.
Let X be the product of two compact spaces X and X such thateach X i belongs to the class ∆ . Does X belong to the class ∆ ? Our last problem is inspired by Theorem 3.10.
Problem 5.10.
Let X be any scattered compact space with a finite scattered height.Does X belong to the class ∆ ? Acknowledgements.
The authors thank Michael Hruˇs´ak for a useful informa-tion about Isbell–Mr´owka spaces.
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Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Pozna´n,Poland and Institute of Mathematics Czech Academy of Sciences, Prague, CzechRepublic
Email address : [email protected] Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva,Israel
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