Small semi-Eberlein compacta and inverse limits
aa r X i v : . [ m a t h . GN ] F e b SMALL SEMI-EBERLEIN COMPACTA AND INVERSE LIMITS
CLAUDIA CORREA, TOMMASO RUSSO, AND JACOPO SOMAGLIA
Abstract.
We study properties of semi-Eberlein compacta related to inverse limits. Weconcentrate our investigation on an interesting subclass of small semi-Eberlein compactawhose elements are obtained as inverse limits of a certain kind of inverse systems. Introduction
The notion of semi-Eberlein compact space was introduced by Kubiś and Leiderman in[KL], as a natural generalization of the classical notion of Eberlein compact. We say that acompact space K is semi-Eberlein if there exists a homeomorphic embedding h : K → R Γ such that h − [ c (Γ)] is dense in K , where c (Γ) := { x ∈ R Γ : ( ∀ ε > |{ γ ∈ Γ : | x ( γ ) | > ε }| < ω } ⊂ R Γ . Clearly the class of semi-Eberlein compacta contains every Eberlein compact space andit is contained in the class of Valdivia compacta. It is easy to see that the generalizedCantor cube κ is an example of a semi-Eberlein compact space that is not Eberlein and in[KL, Corollary 5.3] it was shown that the Valdivia space [0 , ω ] is not semi-Eberlein. It isworth mentioning that even though Eberlein and Valdivia compact spaces have been widelystudied (see for example [AL, BRW, DG, Ka, KM] and more recently [CCS, CT, S]), theclass of semi-Eberlein compacta has not been thoroughly investigated yet. Indeed, after itsintroduction in [KL], this class was studied only in the papers [CCS] and [HRST, Section4.3].The goal of this work is to investigate properties of semi-Eberlein compacta relatedto inverse limits. In [KM] those properties were investigated in the context of Valdiviacompact spaces. It was shown in [KM, Proposition 2.6] that every Valdivia compact spacecan be obtained as the inverse limit of a certain kind of inverse system. In Theorem 2.4, we Date : February 23, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Semi-Eberlein compact space, Inverse system, retractional skeleton, semi-openretraction.Research of C. Correa was supported by Fundação de Amparo à Pesquisa do Estado de Sao Paulo(FAPESP) grants 2018/09797-2 and 2019/08515-6.Research of T. Russo was supported by the GAČR project 20-22230L; RVO: 67985840 and by GruppoNazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionaledi Alta Matematica (INdAM), Italy.Research of J. Somaglia was supported by Università degli Studi di Milano, and by Gruppo Nazionaleper l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di AltaMatematica (INdAM), Italy. present the semi-Eberlein version of this result. The key ingredient in the proof of Theorem2.4 is the characterization of semi-Eberlein compacta in terms of retractional skeletonspresented in [CCS, Theorem B]. Moreover in [KM] a characterization of small Valdiviacompact spaces via inverse limits was established as a consequence of [KM, Proposition 2.6]and [KM, Corollary 4.3]. More precisely, a small compact space is Valdivia if and only if itis the inverse limit of a continuous inverse system of compact metric spaces whose bondingmaps are retractions. In Theorem 2.6, we establish a version of this characterization in thecontext of small semi-Eberlein compacta. Recall that a topological space K is said to be small if its weight w ( K ) is ω .Finally, inspired by a stability result for semi-Eberlein compact spaces presented in [KL],in Section 3 we introduce a subclass of the class of small semi-Eberlein compacta. Moreprecisely, it was shown in [KL, Theorem 4.2] that every inverse limit of a continuous inversesystem of compact metric spaces whose bonding maps are semi-open retractions is semi-Eberlein. We define RS as the class comprising all such inverse limits. Having in mindthe aforementioned characterization of small Valdivia compacta, it was quite natural toconjecture that every small semi-Eberlein compactum belongs to RS . Rather surprisingly,it turns out that this is not the case. For instance, in Section 3 we show that RS doesnot even contain every small Eberlein compact space. More precisely, in Corollary 3.8,we show that if K is a nonmetrizable scattered Eberlein compact space, then K does notbelong to RS . In Subsection 3.1, we present some stability results for the class RS .2. Relations between semi-Eberlein compacta and inverse limits
In this section we establish relations between semi-Eberlein compacta and inverse limits.In order to do so, we use the characterization of semi-Eberlein spaces in terms of retractionalskeletons presented in [CCS] and explore the deep connection between retractional skeletonsand inverse limits. To understand this connection, we need to recall some notions andproperties of those objects. Here all topological spaces are assumed to be Hausdorff andthe following monographs contain basic definitions and results that are used without specificreference: [E], [J], and [KKL]. Recall that an up-directed partially ordered set Σ is saidto be σ -complete if every countable and up-directed subset of Σ admits supremum in Σ ;equivalently, every increasing sequence in Σ admits supremum in Σ . Definition 2.1. A retractional skeleton on a compact space K is a family of continuousinternal retractions s = ( R s ) s ∈ Σ on K indexed by an up-directed and σ -complete partiallyordered set Σ , such that:(i) R s [ K ] is a metrizable compact space, for every s ∈ Σ ,(ii) if s, t ∈ Σ and s ≤ t then R s = R t ◦ R s = R s ◦ R t ,(iii) given an increasing sequence ( s n ) n ∈ ω in Σ , if s = sup n ∈ ω s n ∈ Σ , then R s ( x ) =lim n →∞ R s n ( x ) , for every x ∈ K ,(iv) for every x ∈ K , x = lim s ∈ Σ R s ( x ) .We say that S s ∈ Σ R s [ K ] is the set induced by the retractional skeleton s and we denote itby D ( s ) . MALL SEMI-EBERLEIN COMPACTA AND INVERSE LIMITS 3
Note that condition (iv) implies that D ( s ) is dense in K . Definition 2.2. An inverse system of compact spaces indexed by an up-directed partiallyordered set Σ is a pair S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) , where K s is a compact space, for every s ∈ Σ , p st : K t → K s is a continuous map, for every s, t ∈ Σ with s ≤ t and the followingconditions are satisfied(i) p ss is the identity of K s , for every s ∈ Σ ;(ii) If s ≤ s ≤ s , then p s s = p s s ◦ p s s .The maps p st are called bonding maps . We say that a pair (cid:0) K, ( p s ) s ∈ Σ (cid:1) is a cone over S if K is a compact space, p s : K → K s is a continuous map, for every s ∈ Σ and it holdsthat p s = p st ◦ p t , for every s ≤ t . The maps p s are called projections . An inverse limit of S is a cone (cid:0) K, ( p s ) s ∈ Σ (cid:1) over S such that given any cone (cid:0) L, ( q s ) s ∈ Σ (cid:1) over S , there exists aunique continuous function f : L → K such that q s = p s ◦ f , for every s ∈ Σ .Recall that every inverse system of compact spaces admits an inverse limit, which isunique up to homeomorphisms (more precisely, cone homeomorphisms). Throughout thiswork, we will always assume that the bonding maps are onto. This implies that theprojections from the inverse limit are also onto [E, Corollary 3.2.15]. It is not hard to seethat a cone (cid:0) K, ( p s ) s ∈ Σ (cid:1) over an inverse system of compact spaces S is the inverse limit of S if and only if the family { p s : s ∈ Σ } separates the points of K . Indeed, this follows fromthe explicit representation of the inverse limit of S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) as K S = { ( x s ) s ∈ Σ ∈ Π s ∈ Σ K s : p st ( x t ) = x s , ∀ s ≤ t } , equipped with the usual projections. Note that if S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) is an inversesystem and T is an up-directed subset of Σ , then S | T := (cid:0) ( K s ) s ∈ T , ( p st ) s ≤ t (cid:1) is again aninverse system. Moreover if T is a cofinal subset of Σ and (cid:0) K, ( p s ) s ∈ Σ (cid:1) is the inverse limitof S , then (cid:0) K, ( p t ) t ∈ T (cid:1) is the inverse limit of S | T . Here a central role is played by continuousand σ -complete inverse systems. We say that an inverse system S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) is continuous if for every up-directed subset T of Σ that admits supremum s = sup T in Σ ,it holds that (cid:0) K s , ( p ts ) t ∈ T (cid:1) is the inverse limit of S | T . We say that S is σ -complete if Σ is σ -complete and for every countable and up-directed subset T of Σ with s = sup T in Σ , itholds that (cid:0) K s , ( p ts ) t ∈ T (cid:1) is the inverse limit of S | T .The theory developed in [CCS] allows us to show in Lemma 2.3 that every compactspace that admits a retractional skeleton is the inverse limit of a continuous inverse systemsuch that the bonding maps and the projections are retractions with nice properties. Thekey ingredient of Lemma 2.3 is the notion of canonical retractions associated to suitablemodels that was introduced in [CCS, Definition 12]. Lemma 2.3.
Let K be a compact space with weight κ and s = ( R s ) s ∈ Σ be a retractionalskeleton on K .(1) There exists a family of sets ( M α ) α ∈ κ satisfying conditions (Ra)-(Rd) of [CCS,Proposition 17] and for each α ∈ κ , there exists the canonical retraction r α : K → K associated to M α , K and D ( s ) . Moreover, conditions (R1)-(R10) of [CCS, Propo-sition 17] are satisfied. C. CORREA, T. RUSSO, AND J. SOMAGLIA (2) For every α ∈ κ , set K α = r α [ K ] and for every α, β ∈ κ with α ≤ β , define r αβ : K β → K α as r αβ = r α | K β . Then S = (cid:0) ( K α ) α ∈ κ , ( r αβ ) α ≤ β (cid:1) is a continuousinverse system of compact spaces whose bonding maps are retractions, (cid:0) K, ( r α ) α ∈ κ (cid:1) is the inverse limit of S and w ( K α ) ≤ max( ω, | α | ) , for every α ∈ κ .Proof. Item (1) follows from Skolem’s Theorem ([CCS, Theorem 4]) and [CCS, Proposi-tion 17]. Now let us prove (2). Clearly, (R1) implies that S is an inverse system whosebonding maps are retractions and (R5) ensures that w ( K α ) ≤ max( ω, | α | ) . Note that(R2) ensures that the family { r α : α ∈ κ } separates the points of K , which implies that (cid:0) K, ( r α ) α ∈ κ (cid:1) is the inverse limit of S , since it is a cone for S and the projections are onto.Finally, using (R3) and a similar argument, we conclude that S is continuous. (cid:4) A fundamental concept involved in the proofs of Theorems 2.4 and 2.6 is the shrink-ingness of a retractional skeleton that was introduced in [CCS, Definition 29]. Recallthat given a retractional skeleton s = ( R s ) s ∈ Σ on a compact space K , a bounded sub-set A of C ( K ) and a subset D of K , we say that s is A -shrinking with respect to D if for every x ∈ D and every increasing sequence ( s n ) n ∈ ω in Σ , if s = sup n ∈ ω s n , then lim n →∞ sup f ∈A | f ( R s n ( x )) − f ( R s ( x )) | = 0 . As usual, C ( K ) denotes the Banach spaceof real-valued continuous functions defined on the compact space K , endowed with thesupremum norm. Theorem 2.4. If K is a semi-Eberlein compact space with weight κ , then there exists acontinuous inverse system of compact spaces S = (cid:0) ( K α ) α ∈ κ , ( r αβ ) α ≤ β (cid:1) such that K is theinverse limit of S , each r αβ is a retraction and each K α is semi-Eberlein with w ( K α ) ≤ max( ω, | α | ) .Proof. Let s = ( R s ) s ∈ Σ be the retractional skeleton on K , D ⊂ D ( s ) be the dense subset of K and A be the bounded and separating subset of C ( K ) given by [CCS, Theorem B] andlet ( M α ) α ∈ κ , ( r α ) α ∈ κ and S = (cid:0) ( K α ) α ∈ κ , ( r αβ ) α ≤ β (cid:1) be given by Lemma 2.3. To concludethe result, it remains to prove that each K α is semi-Eberlein. Fixed α ∈ κ , it follows from(R5) that s α := ( R s | K α ) s ∈ (Σ ∩ M α ) σ is a retractional skeleton on K α with D ( s α ) = D ( s ) ∩ K α .Moreover if we set D α = D ∩ K α , then it is clear that D α ⊂ D ( s α ) . Note that (R8) andcondition (c) of [CCS, Theorem B] ensure that r α [ D ] ⊂ D α , which implies that D α is densein K α , since r α [ D ] is dense in K α . It follows from condition (c) of [CCS, Theorem B] that forevery x ∈ D α and every up-directed subset Σ ′ of (Σ ∩ M α ) σ it holds that lim s ∈ Σ ′ R s ( x ) ∈ D α .Finally, set A α = { f | K α : f ∈ A} . It is clear that A α is a bounded and separatingsubset of C ( K α ) and it is easy to see that s α is A α -shrinking with respect to D α and that g ◦ R s | K α ∈ A α , for every g ∈ A α and every s ∈ (Σ ∩ M α ) σ . Therefore, [CCS, Theorem B]ensures that K α is semi-Eberlein. (cid:4) Note that if a compact space is small, then the characterization of when it is semi-Eberlein presented in [CCS, Corollary 32] is much simpler then the general characteri-zation given by [CCS, Theorem B]. This allows us to obtain in Theorem 2.6 an inter-esting characterization of small semi-Eberlein spaces in terms of inverse limits. In orderto do so, we introduce Definition 2.5 that is the translation of shrinkingness to the con-text of inverse limits. First let us recall the notion of right inverse of an inverse system.
MALL SEMI-EBERLEIN COMPACTA AND INVERSE LIMITS 5
Given an inverse system S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) , we say that a family of continuous maps I = { i st : K s → K t , s ≤ t } is a right inverse of S if i st is a right inverse of p st , for every s ≤ t and i sr = i tr ◦ i st , for every s ≤ t ≤ r . In this case [KM, Lemma 3.1] ensures that if (cid:0) K, ( p s ) s ∈ Σ (cid:1) is the inverse limit of S , then there exists a unique family of continuous maps { i s : K s → K, s ∈ Σ } such that i s is a right inverse of p s , for every s ∈ Σ and i s = i t ◦ i st ,for every s ≤ t . We say that { i s : s ∈ Σ } is the right inverse of (cid:0) K, ( p s ) s ∈ Σ (cid:1) with respectto I . Definition 2.5.
Let S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) be a σ -complete inverse system, I be a rightinverse of S , (cid:0) K, ( p s ) s ∈ Σ (cid:1) be the inverse limit of S , A be a bounded subset of C ( K ) and D be a subset of K . We say that (cid:0) K, ( p s ) s ∈ Σ (cid:1) is ( A , I ) -shrinking with respect to D if forevery increasing sequence ( s n ) n ∈ ω in Σ with s = sup n ∈ ω s n and every x ∈ D it holds that lim n →∞ sup f ∈A | f ◦ i s n ◦ p s n ( x ) − f ◦ i s ◦ p s ( x ) | = 0 , where { i s : s ∈ Σ } is the right inverseof (cid:0) K, ( p s ) s ∈ Σ (cid:1) with respect to I . Theorem 2.6.
Let K be a small compact space. Then the following conditions are equiv-alent:(1) K is semi-Eberlein.(2) There exist a bounded and separating subset A of C ( K ) , a dense subset D of K , acontinuous inverse system of compact metric spaces S = (cid:0) ( K α ) α ∈ ω , ( r αβ ) α ≤ β (cid:1) witha right-inverse I = { i αβ : α ≤ β } and a family of retractions { r α : K → K α , α ∈ ω } such that (cid:0) K, ( r α ) α ∈ ω (cid:1) is the inverse limit of S , it is ( A , I ) -shrinking with respect to D and D ⊂ S α ∈ ω i α [ K α ] , where { i α : α ∈ ω } is the right inverse of (cid:0) K, ( r α ) α ∈ ω (cid:1) with respect to I .Proof. Assume that K is semi-Eberlein. Let s = ( R s ) s ∈ Σ be the retractional skeleton on K , D ⊂ D ( s ) be the dense subset of K and A be the bounded and separating subset of C ( K ) given by [CCS, Corollary 32] and let ( M α ) α ∈ ω , ( r α ) α ∈ ω and S = (cid:0) ( K α ) α ∈ ω , ( r αβ ) α ≤ β (cid:1) begiven by Lemma 2.3. It is clear that each K α is metrizable and that I = { i αβ : α ≤ β } isa right inverse of S , where i αβ is the inclusion of K α into K β , for every α ≤ β . Moreover,for each α ∈ ω , if we define i α as the inclusion of K α into K , then { i α : α ∈ ω } is the right inverse of (cid:0) K, ( r α ) α ∈ ω (cid:1) with respect to I . Note that it follows from (R1),(R2), (R3) and (R5) that ( r α ) α ∈ ω is a retractional skeleton on K . Therefore, using [CCS,Theorem 15(ii)(c)] and [C, Lemma 3.2], we conclude that S α ∈ ω r α [ K ] = D ( s ) and thus D ⊂ S α ∈ ω r α [ K ] . Finally, the fact that (cid:0) K, ( r α ) α ∈ ω (cid:1) is ( A , I ) -shrinking with respect to D follows from the A -shrinkingness of s with respect to D and [CCS, Theorem 15(ii)(c)],having in mind that ( M α ) α ∈ ω is an increasing and continuous family of countable sets.Now assume that (2) holds and for each α ∈ ω , define q α : K → K as q α = i α ◦ r α .We claim that s = ( q α ) α ∈ ω is a retractional skeleton on K . Indeed, conditions (i) and(ii) of Definition 2.1 are clearly satisfied. Note that [KM, Lemma 3.3 (2)] ensures thatcondition (iv) holds. Finally, condition (iii) follows from the continuity of S and [KM,Lemma 3.3 (2)]. Since D ⊂ S α ∈ ω i α [ K α ] , we have that D ⊂ D ( s ) and it is easy to see that ( q α ) α ∈ ω is A -shrinking with respect to D . Therefore [CCS, Corollary 32] ensures that K is semi-Eberlein. (cid:4) C. CORREA, T. RUSSO, AND J. SOMAGLIA
Remark . Using the characterization of small Eberlein compacta in terms of retractionalskeletons also contained in [CCS, Corollary 32], one can obtain a similar characterizationof small Eberlein compact spaces in terms of inverse limits through an argument analogousto the one presented in the proof of Theorem 2.6.3.
A special class of semi-Eberlein compacta
This section is dedicated to the introduction and study of the class RS . Following [KL],we say that a function f : X → Y between topological spaces is semi-open if f [ U ] hasnonempty interior, for every nonempty open subset U of X . Definition 3.1.
We say that a compact space K belongs to RS if K is the inverse limit ofa continuous inverse system of compact metric spaces S = (cid:0) ( K α ) α ∈ ω , ( p αβ ) α ≤ β (cid:1) such that p αβ is a semi-open retraction, for every α ≤ β .In Corollary 3.5 we show that in the definition of RS it is enough to require that p αα +1 is a semi-open retraction, for every α ∈ ω . Lemma 3.2.
Let X , Y and Z be topological spaces, q : X → Y be a continuous and ontomap and h : X → Z be a semi-open map. If there exists ¯ h : Y → Z such that ¯ h ◦ q = h ,then ¯ h is semi-open.Proof. Let U be a nonempty open subset of Y . Since q is continuous and onto, we havethat q − [ U ] is a nonempty and open subset of X . Thus h (cid:2) q − [ U ] (cid:3) = ¯ h [ U ] has nonemptyinterior, since h is semi-open. (cid:4) Lemma 3.3.
Let S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) be an inverse system of compact spaces and (cid:0) K, ( p s ) s ∈ Σ (cid:1) be the inverse limit of S . Then the following conditions are equivalent:(a) p s is semi-open, for every s ∈ Σ .(b) p st is semi-open, for every s ≤ t .Proof. To see that (a) implies (b), fix s ≤ t and apply Lemma 3.2 to q = p t , h = p s and ¯ h = p st . Now assume (b) and fix s ∈ Σ . Since { s ∈ Σ : s ≤ s } is a cofinal subset of Σ ,[E, Proposition 2.5.5] ensures that S s ≥ s { p − s [ W ] : W is open in K s } is an open basis of K and therefore to conclude that p s is semi-open, it is enough to show that for every s ≥ s and every nonempty open subset W of K s the set p s (cid:2) p − s [ W ] (cid:3) has nonempty interior. Thisfollows from the fact that p s s is semi-open and p s s [ W ] ⊂ p s (cid:2) p − s [ W ] (cid:3) . (cid:4) Proposition 3.4.
Let κ be a cardinal and S = (cid:0) ( K α ) α ∈ κ , ( p αβ ) α ≤ β (cid:1) be a continuous inversesystem of compact spaces. If p αα +1 is semi-open, for every α ∈ κ , then p αβ is semi-open,for every α ≤ β .Proof. Let us prove by induction on α ∈ κ that p γ γ is semi-open, for every γ < γ ≤ α .Suppose that the result holds for α and let γ < γ ≤ α + 1 . If γ ≤ α , then the resultfollows from the induction hypothesis. Otherwise, we have that γ = α + 1 and thus p γ γ is semi-open, since p γ α +1 = p γ α ◦ p αα +1 and p γ α and p αα +1 are semi-open. Now fix a limitordinal α ∈ κ and assume that the result holds for every ordinal strictly smaller than α . MALL SEMI-EBERLEIN COMPACTA AND INVERSE LIMITS 7
Note that the induction hypothesis ensures that p γ γ is semi-open, for every γ < γ < α .Moreover, it follows from the continuity of S that (cid:0) K α , ( p γα ) γ ∈ α (cid:1) is the inverse limit of S | α and therefore Lemma 3.3 ensures that p γα is semi-open, for every γ ∈ α . (cid:4) Corollary 3.5. If S = (cid:0) ( K α ) α ∈ ω , ( p αβ ) α ≤ β (cid:1) is a continuous inverse system of compactmetric spaces such that p αα +1 is a semi-open retraction, for every α ∈ ω , then the inverselimit of S belongs to RS .Proof. The result follows from Proposition 3.4, having in mind that [KM, Lemma 3.2]ensures that p αβ is a retraction, for every α ≤ β . (cid:4) Given the tight connection between inverse limits and retractional skeletons alreadyexplored in Section 2, we can now provide a useful equivalent description of the class RS via retractional skeletons. Lemma 3.6. If K is a compact space, then the following conditions are equivalent:(1) K belongs to RS (2) K admits a retractional skeleton ( R α ) α ∈ ω such that R α : K → R α [ K ] is semi-open,for every α ∈ ω .Proof. Assume that K belongs to RS and let S = (cid:0) ( K α ) α ∈ ω , ( p αβ ) α ≤ β (cid:1) be a continuousinverse system of compact metric spaces such that each p αβ is a semi-open retractionand let { p α : K → K α , α ∈ ω } be such that (cid:0) K, ( p α ) α ∈ ω (cid:1) is the inverse limit of S .Let { i α : K α → K, α ∈ ω } be the family of maps whose existence is ensured by [KM,Lemma 3.1 and Lemma 3.2] and for every α ∈ ω set R α := i α ◦ p α : K → K . Itfollows from the continuity of S and [KM, Lemma 3.3] that ( R α ) α ∈ ω is a retractionalskeleton on K . Moreover, since Lemma 3.3 ensures that each p α is semi-open, it easy tosee that R α : K → R α [ K ] is semi-open. Now assume (2) and for each α ≤ β , define theretraction p αβ : R β [ K ] → R α [ K ] as p αβ := R α | R β [ K ] . It follows from conditions (i) and(iii) of Definition 2.1 and [KM, Lemma 3.4] that (cid:0) K, ( R α ) α ∈ ω (cid:1) is the inverse limit of thecontinuous inverse system of compact metric spaces (cid:0) ( R α [ K ]) α ∈ ω , ( p αβ ) α ≤ β (cid:1) . Finally, thefact that each p αβ is semi-open follows from Lemma 3.3. (cid:4) Quite clearly, compact metric spaces and the cubes ω and [ − , ω are examples ofcompacta that belong to RS . More generally, retracts of such cubes also belong to RS ,[KL, Corollary 4.3]. On the other hand, we shall now show that RS is indeed a propersubclass of the class of small semi-Eberlein compacta. Given a set Γ and x ∈ Γ , we denoteby suppt( x ) the support of x , i.e., suppt( x ) = { γ ∈ Γ : x ( γ ) = 0 } . Moreover, given atopological space K , we denote the set of its isolated points by I ( K ) Theorem 3.7.
Let K ⊂ Γ be a compact space. Assume that there exists an uncountablesubset A of I ( K ) such that suppt( x ) is finite, for every x ∈ A . Then K does not belong to RS .Proof. Suppose by contradiction that K ∈ RS . Let ( R α ) α ∈ ω be the retractional skeletongiven by Lemma 3.6 and for each α ∈ ω , set K α = R α [ K ] . In what follows, we identify C. CORREA, T. RUSSO, AND J. SOMAGLIA subsets of Γ with their characteristic functions. It follows from the ∆ -system Lemma thatthere exist a finite subset ∆ of Γ and an uncountable subset B of A such that b ∩ b = ∆ , forevery b , b ∈ B with b = b . We claim that there exists α ∈ ω such that ∆ ∈ K α \ I ( K α ) .Indeed, it is easy to see that any injective sequence of elements of B converges to ∆ in Γ . Fix an injective sequence ( b n ) n ∈ ω of elements of B . Since S α ∈ ω K α is dense in K ,we have that I ( K ) ⊂ S α ∈ ω K α and therefore, for every n ∈ ω , there exists α n ∈ ω suchthat b n ∈ K α n . If α = sup n ∈ ω α n , then b n ∈ K α , for every n ∈ ω , which implies that ∆ ∈ K α and of course ∆ / ∈ I ( K α ) . It follows from the fact that R α : K → K α is semi-openthat R α ( z ) ∈ I ( K α ) , for every z ∈ I ( K ) . Thus we have that B = S x ∈ I ( K α ) R − α [ { x } ] ∩ B ,which implies that there exists x ∈ I ( K α ) such that R − α [ { x } ] ∩ B is uncountable, since I ( K α ) is countable. Let ( b n ) n ∈ ω be an injective sequence of elements of R − α [ { x } ] ∩ B . Asargued above, we have that ( b n ) n ∈ ω converges to ∆ in K and thus (cid:0) R α ( b n ) (cid:1) n ∈ ω convergesto R α (∆) . But this is a contradiction, because R α (∆) = ∆ and R α ( b n ) = x , for every n ∈ ω . (cid:4) We recall that for a set Γ , we define Σ(Γ) = { x ∈ R Γ : suppt( x ) is countable } and thatthe small σ -product of real lines is defined by σ (Γ) = { x ∈ R Γ : suppt( x ) is finite } . Plainly,every compact space K ⊂ R Γ such that σ (Γ) ∩ K is dense in K is a fortiori semi-Eberlein. Corollary 3.8. If K ⊂ Γ is a nonmetrizable scattered compact space such that σ (Γ) ∩ K isdense in K , then K does not belong to RS . In particular, nonmetrizable scattered Corsoncompacta do not belong to RS .Proof. According to Theorem 3.7 and using that I ( K ) ⊂ σ (Γ) ∩ K , in order to concludethat K does not belong to RS , it is enough to show that I ( K ) is uncountable. Note thatif I ( K ) is countable, then K ⊂ Σ(Γ) , since K is scattered, I ( K ) ⊂ Σ(Γ) and
Σ(Γ) iscountably closed in R Γ . But this implies that K is a separable Corson compact space andtherefore metrizable. Now assume that K is a nonmetrizable scattered Corson compactspace. Then the result follows from [A, Corollary 1], since it ensures that K is stronglyEberlein, i.e., we may assume that K ⊂ Γ ∩ σ (Γ) , for some set Γ . (cid:4) Quite on the opposite extreme of the connectedness spectrum, we now show that the unitball of the Hilbert space ℓ ( ω ) , endowed with the weak topology, does not belong to RS .Recall that ℓ ( ω ) denotes the Hilbert space { x : ω → R : P γ ∈ ω | x ( γ ) | < ∞} , endowedwith the norm k·k given by k x k := ( P γ ∈ ω | x ( γ ) | ) / . It follows from the reflexivityof ℓ ( ω ) that its closed unit ball B ℓ ( ω ) , endowed with the weak topology, is compactand thus it is an Eberlein compact space. Finally, recall that the weak topology and theproduct topology coincide on B ℓ ( ω ) ⊂ [ − , ω . Theorem 3.9. If K = B ℓ ( ω ) ⊂ [ − , ω , then K does not belong to RS .Proof. Towards a contradiction, assume that K ∈ RS . Then by Lemma 3.6 there existsa retractional skeleton s = ( R α ) α ∈ ω on K such R α : K → R α [ K ] is semi-open, for every α ∈ ω . Moreover it is easy to see that s ′ = ( Q α ) α ∈ ω is also a retractional skeleton on K , MALL SEMI-EBERLEIN COMPACTA AND INVERSE LIMITS 9 where Q α : K → K is given by Q α ( x )( γ ) = ( x ( γ ) γ < α γ ≥ α ( x ∈ K ) . Since K is Eberlein, [C, Theorem 3.11] ensures that D ( s ) = D ( s ′ ) = K . Therefore itfollows from [CCS, Theorem 21] applied to s and s ′ that there exist α, β < ω suchthat α is infinite and Q α = R β , which implies that Q α : K → Q α [ K ] is semi-open.However, we shall show that this is not the case. Fix γ ∈ ω with α < γ and considerthe nonempty open subset W = { x ∈ K : x ( γ ) = 0 } of K . Let U be a nonempty basicopen subset of Q α [ K ] ; then there exist u ∈ Q α [ K ] , α , . . . , α n ∈ α and ε > such that U = { x ∈ Q α [ K ] : | x ( α i ) − u ( α i ) | < ε, i = 1 , . . . , n } . Note that there exists x ∈ U suchthat k x k = 1 . On the other hand, for every y ∈ W , we have that ≥ k y k ≥ k Q α ( y ) k + | y ( γ ) | > k Q α ( y ) k , which implies that k Q α ( y ) k < . Therefore, there is no y ∈ W with Q α ( y ) = x and thus U Q α [ W ] . Since U was arbitrary, we conclude that Q α [ W ] has empty interior in Q α [ K ] ,as desired. (cid:4) Remark . For the reader inclined to Banach space theory, we observe that the aboveresult actually holds true more in general. Indeed, a similar argument shows the following:if X is a WLD Banach space and X ∗ is strictly convex, then the dual unit ball of X ,endowed with the weak ∗ topology, does not belong to RS . In particular, this applies tothe unit ball of ℓ p ( ω ) , endowed with the weak topology, for < p < ∞ . Finally, the sameproof as for ℓ ( ω ) also gives that the unit ball of ℓ ( ω ) , endowed with the weak ∗ topologyinduced by c ( ω ) , does not belong to RS either.3.1. Stability results.
The class RS enjoys several stability properties. Needless to say,such properties yield several further examples of compacta in this class. Proposition 3.11. If K belongs to RS and L is a clopen subset of K , then L belongs to RS .Proof. Let s = ( R α ) α ∈ ω be the retractional skeleton given by Lemma 3.6. Since L is open,we have that D ( s ) ∩ L is dense in L , therefore it follows from [C, Lemma 3.5] that theset T = { α ∈ ω : R α [ L ] ⊂ L } is a cofinal and σ -closed subset of ω and ( R α | L ) α ∈ T is aretractional skeleton on L . Clearly the fact that L is open ensures that R α | L : L → R α [ L ] is semi-open, for every α ∈ T . Therefore, the result follows from Lemma 3.6, since T isorder-isomorphic to ω . (cid:4) Remark . Note that the class RS is not stable for closed subspaces in general. Forinstance, every small compact space embeds homeomorphically into the cube [0 , ω and [0 , ω belongs to RS . Proposition 3.13. If { K α : α ∈ ω } is a family of elements of RS , then K = Q α ∈ ω K α belongs to RS . Proof.
If any K α is empty, then the result is trivial. For every α ∈ ω , let ( R αβ ) β ∈ ω bethe retractional skeleton on K α given by Lemma 3.6 and for each α ∈ ω , fix an element x α ∈ R α [ K α ] . Fixed β ∈ ω , define P β : K → K as P β ( y )( α ) = R αβ ( y ( α )) if α < β and P β ( y )( α ) = x α otherwise, for every y = ( y ( α )) α ∈ ω ∈ K . It is straightforward tocheck that ( P β ) β ∈ ω is a retractional skeleton on K . Now fix β ∈ ω and let us show that P β : K → P β [ K ] is semi-open. Let F be a finite subset of ω and consider the basicopen set U = Q α ∈ F U α × Q α/ ∈ F K α , where U α is a nonempty open subset of K α , forevery α ∈ F . Note that P β [ U ] = Q α ∈ F ∩ β R αβ [ U α ] × Q α ∈ β \ F R αβ [ K α ] × Q α ≥ β { x α } and that P β [ K ] = Q α ∈ β R αβ [ K α ] × Q α ≥ β { x α } . Therefore, it is easy to see that P β [ U ] has nonemptyinterior in P β [ K ] , since R αβ : K α → R αβ [ K α ] is semi-open, for every α ∈ F ∩ β . The resultfollows from Lemma 3.6. (cid:4) Given a family of topological spaces { X i : i ∈ I } , we denote by F i ∈ I X i its topologicalsum. Proposition 3.14. If { K n : n ∈ ω } is a family of elements of RS , then the one-pointcompactification K = F n ∈ ω K n ∪ {∞} of F n ∈ ω K n belongs to RS . In particular, everyfinite topological sum of elements of RS belongs to RS .Proof. For every n ∈ ω , let ( R nβ ) β ∈ ω be the retractional skeleton on K n given by Lemma3.6. For each β ∈ ω , define P β : K → K by P β ( x ) = R nβ ( x ) , if x ∈ K n and P β ( ∞ ) = ∞ .We claim that ( P β ) β ∈ ω is a retractional skeleton on K . In fact it is straightforward to checkthat it satisfies conditions (ii), (iii) and (iv) of Definition 2.1 and condition (i) follows fromthe fact that the one-point compactification of a locally compact and σ -compact metricspace is metrizable [Ke, Theorem 5.3]. Now fix β ∈ ω and let us show that P β : K → P β [ K ] is semi-open. If U is a nonempty open subset of K , then there exists n ∈ ω such that U ∩ K n is a nonempty open subset of K n . Therefore, the semi-openness of R nβ : K n → R nβ [ K n ] easily implies that P β [ U ∩ K n ] has nonempty interior in P β [ K ] . Using Lemma 3.6, weconclude the proof. Now fix a finite family { K i : i = 1 , . . . , k } of elements of RS and notethat F ki =1 K i is a clopen subset of the one-point compactification of the topological sum ofa countable family of elements of RS and therefore, Proposition 3.11 ensures that F ki =1 K i belongs to RS . (cid:4) Next, we study the stability of RS for continuous images. In Theorem 3.19 we showthat if a Valdivia compact space L is a continuous and semi-open image of an elementof RS , then L belongs to RS . As a consequence of this result, we conclude in Corollary3.20 that the class RS is stable for semi-open retractions. Proposition 3.15 is the keyto establish Theorem 3.19 and its proof relies mostly on the theory of suitable modelspresented in [CCS, Section 3.1]. Recall that a subset T of a σ -complete and up-directedpartially ordered set Σ is said to be σ -closed in Σ if the supremum of M belongs to T , forevery countable and up-directed subset M of T . Clearly if T is an up-directed and σ -closedsubset of Σ and S is a σ -complete inverse system indexed in Σ , then S | T is also σ -complete. Proposition 3.15.
Let S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) and S = (cid:0) ( L s ) s ∈ Σ , ( q st ) s ≤ t (cid:1) be σ -completeinverse systems of compact metric spaces. Let (cid:0) K, ( p s ) s ∈ Σ (cid:1) be the inverse limit of S and MALL SEMI-EBERLEIN COMPACTA AND INVERSE LIMITS 11 (cid:0) L, ( q s ) s ∈ Σ (cid:1) be the inverse limit of S . If f : K → L is a continuous and semi-open map,then there exists a cofinal and σ -closed subset T of Σ such that for every t ∈ T , there existsa continuous and semi-open map f t : K t → L t such that q t ◦ f = f t ◦ p t .Proof. Using [KM, Proposition 2.2], we may assume without loss of generality that forevery s ∈ Σ , there exists a continuous map f s : K s → L s satisfying q s ◦ f = f s ◦ p s . Define: T = { t ∈ Σ : ∀ U ⊂ K t open, U = ∅ , ∃ V ⊂ L t open, V = ∅ , s.t. q − t [ V ] ⊂ f (cid:2) p − t [ U ] (cid:3) } . It is easy to see that f t is semi-open, for every t ∈ T . To see that T is σ -closed in Σ ,let ( t n ) n ∈ ω be an increasing sequence of elements of T and t = sup n ∈ ω t n ∈ Σ . Since S is σ -complete, we have that (cid:0) K t , ( p t n t ) n ∈ ω (cid:1) is the inverse limit of S | { t n : n ∈ ω } and therefore[E, Proposition 2.5.5] ensures that S n ∈ ω { p − t n t [ W ] : W ⊂ K t n is open } is an open basis of K t . Thus, fixed a nonempty open subset U of K t , there exist n ∈ ω and a nonempty opensubset W of K t n such that p − t n t [ W ] ⊂ U . Since t n ∈ T , there exists a nonempty opensubset V ′ of L t n such that q − t n [ V ′ ] ⊂ f (cid:2) p − t n [ W ] (cid:3) . It is easy to see that if V := q − t n t [ V ′ ] ,then q − t [ V ] ⊂ f (cid:2) p − t [ U ] (cid:3) . Now let us show that T is cofinal in Σ . Fix s ∈ Σ and set S ′ = S ∪ { s, f, K, L, Σ , ϕ K , ϕ L , ψ K , ψ L } , where S is the union of the countable sets fromthe statements of [CCS, Lemma 7] and [CCS, Lemma 8] and ϕ K , ϕ L , ψ K and ψ L arethe maps defined on Σ given by ϕ K ( t ) = p t , ϕ L ( t ) = q t , ψ K ( t ) = B K t and ψ L ( t ) = B L t ,where B K t and B L t are fixed countable open basis of K t and L t , respectively. Let Φ be the union of the finite lists of formulas from the statements of [CCS, Lemma 7] and[CCS, Lemma 8] enriched by the formulas (and their subformulas) marked as (*) in theproof below. According to Skolem’s Theorem [CCS, Theorem 4] there exists a countableset M such that M ≺ (Φ; S ′ ) . Note that [CCS, Lemma 8 (1)] implies that there exists δ = sup(Σ ∩ M ) ∈ Σ and clearly s ≤ δ . It follows from the σ -completeness of S and [E,Proposition 2.5.5] that to conclude that δ ∈ T it is enough to show that for every t ∈ Σ ∩ M and every W ∈ B K t nonempty, there exists a nonempty open subset V of L δ such that q − δ [ V ] ⊂ f (cid:2) p − t [ W ] (cid:3) . Fix t ∈ M ∩ Σ and W ∈ B K t with W = ∅ . Note that it followsfrom [CCS, Lemma 7 (2)] and [CCS, Lemma 7 (4)] that W ∈ M . Since f is semi-open, if U := p − tδ [ W ] , then there exists a nonempty open subset W ′ of L such that W ′ ⊂ f (cid:2) p − δ [ U ] (cid:3) and thus W ′ ⊂ f (cid:2) p − t [ W ] (cid:3) . We claim that we may assume that W ′ ∈ M . Indeed, considerthe following formula ∃ W ′ ⊂ L open (cid:0) W ′ = ∅ and W ′ ⊂ f [ p − t [ W ]] (cid:1) . ( ∗ ) Since all free variables of this formula belong to M , by absoluteness we conclude that thereexists W ′ ∈ M satisfying ( ∗ ) . It follows from the fact that (cid:0) L, ( q s ) s ∈ Σ (cid:1) is the inverse limitof S and [E, Proposition 2.5.5] that there exist r ∈ Σ and A ∈ B L r such that A = ∅ and q − r [ A ] ⊂ W ′ . Note that the absoluteness of the following formula (and its subformulas)for M ensures that we may assume that r ∈ M ∃ r ∈ Σ (cid:0) ∃ A ∈ B L r s.t. A = ∅ and q − r [ A ] ⊂ W ′ (cid:1) . ( ∗ ) This concludes the proof, since q − r [ A ] = q − δ (cid:2) q − rδ [ A ] (cid:3) . (cid:4) Remark . It is worth mentioning that Proposition 3.15 is a version of [KM, Proposi-tion 2.2] with semi-open maps in place of open ones. The definition of T in our proof ismore technical than the one in [KM, Proposition 2.2], since we have to circumvent one flawpresent there. Indeed, the set T defined there might fail to be σ -closed; we offer an instanceof this phenomenon in Example 3.17 below. The statement of [KM, Proposition 2.2] is inany case correct and it can be proved with an argument similar to ours above. Example 3.17.
For every n ≥ , define K n = {− , − / , , . . . , − /n, , /n, . . . , / , } ,endowed with the discrete topology and for every n ≤ m , define p nm : K m → K n as p nm ( x ) = x , if x ∈ K n and p nm ( x ) = 0 , otherwise. Set K ω = {− /n : n ≥ } ∪ { } ∪ { /n : n ≥ } , endowed with the subspace topology of R and for each n ∈ ω , let p nω : K ω → K n be given by p nω ( x ) = x , if x ∈ K n and p nω ( x ) = 0 , otherwise. Finally, set K ω +1 = K ω ,let p ωω +1 be the identity of K ω and p nω +1 = p nω , for every n ∈ ω . It is easy to seethat S = (cid:0) ( K α ) α ∈ [1 ,ω +1] , ( p αβ ) α ≤ β (cid:1) is a σ -complete inverse system whose inverse limit is (cid:0) K ω +1 , ( p αω +1 ) α ∈ [1 ,ω +1] (cid:1) . Now for every n ≥ , define L n = { , /n, . . . , / , } , endowedwith the discrete topology and for every n ≤ m , define q nm : L m → L n as q nm ( x ) = x ,if x ∈ L n and q nm ( x ) = 0 , otherwise. Set L ω = { } ∪ { /n : n ≥ } , endowed with thesubspace topology of R and for each n ∈ ω , define q nω : L ω → L n as q nω ( x ) = x , if x ∈ L n and q nω ( x ) = 0 , otherwise. Set L ω +1 = K ω +1 and let q ωω +1 : L ω +1 → L ω be given by q ωω +1 ( x ) = x , if x ∈ L ω and q ωω +1 ( x ) = 0 , otherwise. Finally, define q nω +1 = q nω ◦ q ωω +1 ,for every n ∈ ω . It is easy to see that S = (cid:0) ( L α ) α ∈ [1 ,ω +1] , ( q αβ ) α ≤ β (cid:1) is a σ -complete inversesystem whose inverse limit is (cid:0) L ω +1 , ( q αω +1 ) α ∈ [1 ,ω +1] (cid:1) . Now let f : K ω +1 → L ω +1 be theidentity of K ω +1 and, as in [KM, Proposition 2.2], set: T = { α ∈ [1 , ω + 1] : ∃ f α : K α → L α such that f α is open and q αω +1 ◦ f = f α ◦ p αω +1 } . Note that ω ⊂ T , since for every n ∈ ω , the open map f n : K n → L n given by f n ( x ) = x ,if x ∈ L n and f n ( x ) = 0 , otherwise, satisfies q nω +1 ◦ f = f n ◦ p nω +1 . However, ω = sup n ∈ ω n does not belong to T , since q ωω +1 is not open and this is the only map that could witnessthat ω ∈ T . Therefore T is not σ -closed in [1 , ω + 1] . Corollary 3.18.
Let S = (cid:0) ( K s ) s ∈ Σ , ( p st ) s ≤ t (cid:1) and S = (cid:0) ( L s ) s ∈ Σ , ( q st ) s ≤ t (cid:1) be σ -completeinverse systems of compact metric spaces. Let (cid:0) K, ( p s ) s ∈ Σ (cid:1) be the inverse limit of S , (cid:0) L, ( q s ) s ∈ Σ (cid:1) be the inverse limit of S and f : K → L be a continuous, onto and semi-openmap. If p st is semi-open, for every s ≤ t , then there exists a cofinal and σ -closed subset T of Σ such that q st is semi-open, for every s, t ∈ T with s ≤ t .Proof. Let T be the cofinal and σ -complete subset of Σ given by Proposition 3.15. Notethat for every t ∈ T , it holds that q t ◦ f = f t ◦ p t is semi-open, since Lemma 3.3 ensuresthat p t is semi-open and composition of semi-open maps is semi-open. Thus it followsfrom Lemma 3.2 applied to q = f , h = q t ◦ f and ¯ h = q t that q t is semi-open, for every t ∈ T . Finally, the result follows from Lemma 3.3, since (cid:0) L, ( q t ) t ∈ T (cid:1) is the inverse limit of S | T . (cid:4) Theorem 3.19.
Let K be an element of RS and L be a Valdivia compact space. If thereexists a continuous, onto and semi-open map f : K → L , then L belongs to RS . MALL SEMI-EBERLEIN COMPACTA AND INVERSE LIMITS 13
Proof.
Let S = (cid:0) ( K α ) α ∈ ω , ( p αβ ) α ≤ β (cid:1) be a continuous inverse system of compact metricspaces such that p αβ is a semi-open retraction, for every α ≤ β and let { p α : K → K α , α ∈ ω } be such that (cid:0) K, ( p α ) α ∈ ω (cid:1) is the inverse limit of S . Note that w ( L ) ≤ w ( K ) ≤ ω and therefore, the only interesting case is when the weight of L is ω . It follows from [KM,Proposition 2.6] that there exist a continuous inverse system of compact metric spaces S = (cid:0) ( L α ) α ∈ ω , ( q αβ ) α ≤ β (cid:1) such that each q αβ is a retraction and a family { q α : L → L α , α ∈ ω } such that (cid:0) L, ( q α ) α ∈ ω (cid:1) is the inverse limit of S . Let T be the cofinal and σ -complete subsetof ω given by Corollary 3.18. Since L is the inverse limit of S | T , T is order-isomorphic to ω and S | T is continuous, we conclude that L belongs to RS . (cid:4) Corollary 3.20.
Let K be an element of RS , L be a compact space and f : K → L be acontinuous map.(1) If f is a semi-open retraction, then L belongs to RS ;(2) If L is zero-dimensional and f is an open surjection, then L belongs to RS .Proof. Note that [KM, Theorem 4.4] implies that L is Valdivia in (1) and (2). Thereforethe result follows from Theorem 3.19. (cid:4) References [A] K. Alster,
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Centro de Matemática, Computação e Cognição, Universidade Federal doABC, Avenida dos Estados, 5001, Santo André, Brasil
Email address : [email protected], [email protected] (T. Russo) Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague1, Czech Republic; and Department of Mathematics, Faculty of Electrical Engineering,Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic
Email address : [email protected], [email protected] (J. Somaglia) Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano,Via Cesare Saldini 50, 20133 Milano, Italy
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