Sailing over three problems of Koszmider
Félix Cabello Sánchez, Jesús M.F. Castillo, Witold Marciszewski, Grzegorz Plebanek, Alberto Salguero-Alarcón
aa r X i v : . [ m a t h . F A ] A p r SAILING OVER THREE PROBLEMS OF KOSZMIDER
F´ELIX CABELLO S ´ANCHEZ, JES ´US M.F. CASTILLO, WITOLD MARCISZEWSKI,GRZEGORZ PLEBANEK, AND ALBERTO SALGUERO-ALARC ´ON
Abstract.
We discuss three problems of Koszmider on the structure of the spaces of continuousfunctions on the Stone compact K A generated by an almost disjoint family A of infinite subsetsof ω — we present a solution to two problems and develop a previous results of Marciszewskiand Pol answering the third one. We will show, in particular, that assuming Martin’s axiom thespace C ( K A ) is uniquely determined up to isomorphism by the cardinality of A whenever |A| < c ,while there are 2 c nonisomorphic spaces C ( K A ) with |A| = c . We also investigate Koszmider’sproblems in the context of the class of separable Rosenthal compacta and indicate the meaningof our results in the language of twisted sums of c and some C ( K ) spaces. Introduction
Koszmider poses in [26] five problems about the structure of the spaces of continuous functionson the Stone compact K A generated by an almost disjoint family A of infinite subsets of ω .Problem 2, that Koszmider himself solves, is the existence of an almost disjoint family A suchthat under either the Continuum Hypothesis CH or Martin’s Axiom MA , C ( K A ) ≃ c ⊕ C ( K A )is the only possible decomposition of C ( K A ) in two infinite dimensional subspaces. Very recentlyKoszmider and Laustsen have obtained in [27] the same result without any additional set-theoreticassumptions. Problem 1 asks whether a similar separable space exists. Argyros and Raikoftsalis[1] call a Banach space X quasi-prime if there exists an infinite dimensional subspace Y suchthat X ≃ Y ⊕ X is the only possible nontrivial decomposition of X . If, moreover Y is notisomorphic to X then X is called strictly quasi-prime . Argyros and Raikoftsalis [1] show theexistence, for each ℓ p , p ≥
1, (resp. c ) of a separable strictly quasi-prime space X p ≃ ℓ p ⊕ X p (resp. X ≃ c ⊕ X ).We are concerned in this paper with the other three: • Problem 3
Assuming MA , is it true that if |A| = |B| < c then C ( K A ) ≃ C ( K B ) ? • Problem 4
Assuming MA and |A| < c is C ( K A ) ≃ C ( K A ) ⊕ C ( K A )? Mathematics Subject Classification. • Problem 5
Are there two almost disjoint families A , B of the same cardinality such that C ( K A ) is not isomorphic to C ( K B ) ?Koszmider’s questions are mentioned by Hruˇs´ak [22, 9.2] in his survey on applications of almostdisjoint families. Let us point out that Problem 5 was actually solved by Marciszewski and Pol[33] who gave an example of a pair of almost disjoint families A , B of cardinality c such that C ( K A ) C ( K B ) — this is a direct consequence of [33, Theorem 3.4]. It was, moreover, brieflyoutlined in [33, 7.4] how one can prove, using some ideas from [30], that there are 2 c isomorphismtypes of Banach spaces of the form C ( K A ).We will solve affirmatively Problems 3 and 4 and present a detailed, self-contained solution toProblem 5. Summing up, we will obtain Theorem 1.1. (a) Under MA ( κ ), if A and B are almost disjoint families such that |A| = |B| = κ , then C ( K A ) ≃ C ( K B ) .(b) Under MA ( κ ), if A is an almost disjoint family with |A| = κ , then C ( K A ) ≃ C ( K A ) ⊕ C ( K A ) .(c) There are c nonisomorphic spaces C ( K A ) for almost disjoint families A of size c . Here MA ( κ ) denotes Martin’s axiom for ccc partial orders and κ many dense sets (recall thatif we assume MA ( κ ) then, automatically, κ < c ).In the context of part (b) of the above theorem it is worth to recall that the first exampleof an infinite compact space K such that C ( K ) is not isomorphic to C ( K ) ⊕ C ( K ) was givenby Semadeni in [38]. His space K was the interval of ordinals [0 , ω ] equipped with the ordertopology. In [29] an example of an almost disjoint family A was constructed, such that the space C ( K A ) endowed with the weak topology is not homeomorphic to C ( K A ) ⊕ C ( K A ) with the weaktopology, hence the spaces C ( K A ) and C ( K A ) ⊕ C ( K A ) are not isomorphic.Our proof of Theorem 1.1(c) follows a relatively simple counting argument suggested in [33].This approach is applicable to study a more general question, on the number of pairwise non-isomorphic twisted sums of c and Banach spaces of the form C ( K ). In Section 5 we apply thetechnique to K being either the one point compactification of the discrete space c or the classicalnonmetrizable compact space called the double arrow space or the split interval .Recall that Rosenthal compacta form an interesting and important class of topological spacesthat originated in functional analysis, see the survey articles [18] and [31]. Sections 6 and 7 aredevoted to study Rosenthal compacta in the light of our results from the first part of the paper.Our main motivation here is related to the fact that the proof of 1.1(c) does not provide ‘concrete’examples of pairs of nonisomorphic Banach spaces in question. Therefore, we want to indicatemore constructive ways of obtaining such examples. Note that, if we construct an almost disjointfamily A in an ‘effective’ way (i.e., such A , treated as subspace of the Cantor set 2 ω , is Borel oranalytic), then the resulting space K A is a Rosenthal compactum, see [34, Lemma 4.4]. We showin Section 7, that using a completely different argument one can name an uncountable sequenceof almost disjoint families {A ξ : ξ < ω } such that every K A ξ is a Rosenthal compactum and theBanach spaces C ( K A ξ ) are pairwise nonisomorphic. We also prove a similar result on the class AILING OVER THREE PROBLEMS OF KOSZMIDER 3 of twisted sums of c and C ( S ), where S is the double arrows space, a well-known nonmetrizableseparable Rosenthal compactum itself. The latter result partially extend [3, Corollary 4.11]stating that c admits a nontrivial twisted sum with C ( K ) for every nonmetrizable separableRosenthal compactum K . Our considerations presented in Section 7 build on delicate descriptiveproperties of separable Rosenthal compacta; to make the presentation reasonably self-contained,we provide in Section 6 all the relevant facts to be used.The authors are very grateful to the referee for a very careful reading and several commentsthat enabled us to improve the presentation.2. Preliminaries on twisted sums of Banach spaces
We will write A ≃ B to mean that the Banach spaces A and B are isomorphic. An exactsequence z of Banach spaces is a diagram0 −−−→ Y −−−→ Z ρ −−−→ X −−−→ Z is usually called a twistedsum of Y and X . By the open mapping theorem, Y must be isomorphic to a subspace of Z and X to the quotient Z/Y . Two exact sequences z and s are said to be equivalent, denoted z ≡ s ,if there is an operator T : Z → S making commutative the diagram0 −−−→ Y z −−−→ Z ρ z −−−→ X −−−→ (cid:13)(cid:13)(cid:13) y T (cid:13)(cid:13)(cid:13) −−−→ Y −−−→ s S −−−→ ρ s X −−−→ z is said to be trivial , or to split , if the injection admits a left inverse; i.e., thereis a linear continuous projection P : Z → Y along . Equivalently, if z ≡ → Y → Y ⊕ X → X →
0. Given an exact sequence z and an operator γ : X ′ → X the pull-back exact sequence z γ is the lower sequence in the diagram0 −−−→ Y −−−→ Z ρ −−−→ X −−−→ (cid:13)(cid:13)(cid:13) x x γ −−−→ Y −−−→ PB −−−→ X ′ −−−→ { ( z, x ′ ) ∈ Z ⊕ ∞ X ′ : ρz = γx ′ } endowed with the subspace norm. Dually, given anoperator α : Y → Y ′ the push-out exact sequence α z is the lower sequence in the diagram0 −−−→ Y −−−→ Z ρ −−−→ X −−−→ α y y (cid:13)(cid:13)(cid:13) −−−→ Y ′ −−−→ PO −−−→ X −−−→ Y ′ ⊕ Z ) / ∆, where ∆ = { ( αy, − y ) : y ∈ Y } endowed with the quotient norm. Weneed to keep in mind the following two facts: CABELLO, CASTILLO, MARCISZEWSKI, PLEBANEK, AND SALGUERO-ALARC ´ON
Proposition 2.1.
Let z and s be two exact sequences −→ Y −→ Z ρ z −→ X −→ −→ Y −→ S ρ s −→ X −→ (1) If s ρ z ≡ then there is an operator τ : Y → Y such that τ z ≡ s .(2) If s ρ z ≡ ≡ z ρ s then Y ⊕ S ≃ Y ⊕ Z .Proof. The first fact can be seen in [5] and is an easy consequence of the homology sequence; thesecond is the so-called diagonal principle [13]. (cid:3)
The space of exact sequences between two given spaces Y and X modulo equivalence will bedenoted Ext( X, Y ) (in reverse order). We will write Ext(
X, Y ) = 0 to mean that all elements z ∈ Ext(
X, Y ) are z ≡ c and C ( K ), where K is a compactspace. These objects have a long tradition; for instance, if A is an almost disjoint family ofcardinality c , a kind of ℓ p version of C ( K A ) was introduced in [24], and has been sometimes calleda Johnson-Lindenstrauss space [12, Theorem 4.10.a] or [10, 3.1]; even if the standard practice isto call Johnson-Lindenstrauss space to the true ℓ p -versions. The classical Sobczyk theorem, c is complemented in any separable Banach superspace, implies that Ext( C ( K ) , c ) = 0 whenever K is metrizable. The question of whether Ext( C ( K ) , c ) = 0 for every non-metrizable K wasposed in [8, 9] and has been intensively studied [11, 16, 32, 3, 15, 17]. While for some classes ofcompacta K one can demonstrate that Ext( C ( K ) , c ) = 0 more or less effectively, the problemcannot be decided within the usual set theory. Indeed, one has Theorem 2.2. (a) Assuming CH , Ext( C ( K ) , c ) = 0 for every non-metrizable compact space K (see [3, Theorem5.8] ).(b) Assuming MA ( κ ) , Ext( C ( K A ) , c ) = 0 whenever |A| = κ ( [32, Corollary 5.3] ).(c) Assuming MA ( κ ) , Ext( C ( K ) , c ) = 0 whenever K is a separable scattered space of weight κ and finite height (this extension of (b) is due to Correa and Tausk [17, Corollary 4.2] ). Preliminaries on compacta
The most natural method of constructing twisted sums of c and C ( K ) is to consider a compactspace L of the form L = K ∪ ω , containing K as a subspace and additionally containing a countableinfinite set of isolated points denoted simply by ω (we tacitly assume that K ∩ ω = ∅ ). Such aspace L will be called a countable discrete extension of K .Note that if L is a countable discrete extension of K then the subspace Z = { g ∈ C ( L ) : g | K ≡ } of C ( L ) is a natural copy of c , whereas C ( L ) /Z may be identified with C ( K ). Thiswas fully discussed in [32] and [3]. The following useful observation is a particular case of [32,Theorem 2.8(a)] or [3, Theorem 4.13]. Theorem 3.1.
Suppose that L = K ∪ ω is a countable discrete extension of a compact space K such that ω is dense in L (in other words, L is a compactification of ω whose remainder ishomeomorphic to K ). If K is not ccc , then C ( L ) is a nontrivial twisted sum of c and C ( K ) . AILING OVER THREE PROBLEMS OF KOSZMIDER 5
Our solution to Problem 5 is motivated by examining the variety of twisted sums of c and C ( K ), where K is either the Stone space constructed from an almost disjoint family of subsetsof ω , or K = S , where S denotes the classical double arrow space (see below).3.1. The Stone compact of an almost disjoint family.
Recall that a topological spaceis scattered if every of its subsets contains an isolated point (see [39] for the basic propertiesand further references). We consider here some scattered compact spaces K ; K ′ is the firstderivative (the set of nonisolated points in K ). Higher Cantor-Bendixon derivatives K ( α ) aredefined inductively; the height of K is the first ordinal number α for which K ( α ) = ∅ .Recall that a family A of infinite subsets of ω is almost disjoint if A ∩ B is finite for any distinct A, B ∈ A . To every almost disjoint family A one can associate a certain scattered compact space K A of height 3. The space K A may be defined as the Stone space of the algebra of subsets of ω generated by A and all finite sets; alternatively, K A = ω ∪ { A : A ∈ A} ∪ {∞} , where points in ω are isolated, basic open neighborhoods of a given point A ∈ A are of the form { A } ∪ ( A \ F ) with F ⊆ ω finite, and K A is the one point compactification of the locally compactspace ω ∪ A . This means that K A is formed by adding a single point ∞ / ∈ ω ∪ A and declaringthat the local base at ∞ consists of sets of the form {∞} ∪ V , where V is an open subset of ω ∪ A with compact complement.We write A ( κ ) for the (Aleksandrov) one-point compactification of a discrete space of cardi-nality κ . The space K A might be called the Aleksandrov-Urysohn compactum associated to analmost disjoint family A ; this terminology was used in [33], [32] and [3]; see also [20, section2] (most often, spaces of the form K A are called Mr´owka spaces). Hruˇs´ak [22] and Hern´andez-Hern´andez and Hruˇs´ak [23] survey various applications of that construction to topology andfunctional analysis.Given a cardinal number κ , c ( κ ) stands for the Banach space of all functions f : κ → R having the property that the set { α < κ : | f ( α ) | ≥ ε } is finite for every positive ε (with the usualsupremum norm). Note that c ( κ ) may be seen as a hyperplane in C ( A ( κ )) and that c ( κ ) isisomorphic to C ( A ( κ )). Moreover, if |A| = κ then K A is a countable discrete extension of thespace A ( κ ). Hence, the following is a direct consequence of Theorem 3.1. Theorem 3.2.
Given any almost disjoint family A of cardinality κ > ω , the space C ( K A ) is anontrivial twisted sum of c and c ( κ ) . The double arrow space.
Actually, we mention here a single compactum, the doublearrow space, sometimes called the split interval, and later (in Sections 5 and 7) we shall considerthe class of its countable discrete extensions. We denote this classical space by S ; recall that S = (cid:0) (0 , × { } (cid:1) ∪ (cid:0) [0 , × { } (cid:1) , is equipped with the order topology given by the lexicographical order h s, i i ≺ h t, j i if either s < t, or s = t and i < j. CABELLO, CASTILLO, MARCISZEWSKI, PLEBANEK, AND SALGUERO-ALARC ´ON
The space S is a nonmetrizable separable compactum having a countable local base at everypoint. It will be convenient to see S as the Stone space of some algebra of subsets of a countableset. Let Q be a countable dense subset of (0 , x ∈ (0 ,
1) put P x = { q ∈ Q : q ≤ x } .Let A be the algebra of subsets of Q generated by the chain { P x : x ∈ (0 , } . Then the Stonespace ult( A ), of all ultrafilters on the algebra A , is homeomorphic to S .To see this, note first that every F ∈ ult( A ) is uniquely determined by the set I ( F ) = { x ∈ (0 ,
1) : P x ∈ F } , which is a subinterval of (0 ,
1) of the form [ y,
1) or ( y,
1) for some y ∈ [0 , F = F + y if I ( F ) = [ y,
1) and F = F − y if I ( F ) = ( y, h : S → ult( A ) by h ( y, i ) = ( F − y if y ∈ [0 , , i = 1; F + y if y ∈ (0 , , i = 0 . Then h is an homeomorphism of S and ult( A ).4. Solution to Problems 3 and 4
We start by recalling the following well-known observation:
Remark . If K is an infinite scattered compact space then K contains a nontrivial convergentsequence, and so C ( K ) ≃ c ⊕ C ( K ). Proof of Theorem 1.1 (a) and (b).
Pick two almost disjoint families A , B of subsets of ω of car-dinality κ < c . Under MA ( κ ), we haveExt( C ( K A ) , c ) = 0 = Ext( C ( K B ) , c ) , thanks to Theorem 2.2 (b). Hence, if we call a (resp. b ) the two exact sequences in the diagram0 −−−→ c −−−→ C ( K A ) ρ A −−−→ c ( κ ) −−−→ (cid:13)(cid:13)(cid:13) −−−→ c −−−→ C ( K B ) −−−→ ρ B c ( κ ) −−−→ a ρ B ≡ ≡ b ρ A and thus by Proposition 2.1(2) and Remark 4.1 C ( K A ) ≃ c ⊕ C ( K A ) ≃ c ⊕ C ( K B ) ≃ C ( K B ) . Part (b) is consequence of the well-known algebraic identityExt( C ( K A ) ⊕ C ( K A ) , c ) = Ext( C ( K A ) , c ) × Ext( C ( K A ) , c ) , and the following Claim. If X is a twisted sum of c and c ( κ ) so that Ext(
X, c ) = 0 then X ≃ C ( K A ). Proof of the Claim.
Assume the existence of an exact sequence0 −−−→ c −−−→ X ρ −−−→ c ( κ ) −−−→ AILING OVER THREE PROBLEMS OF KOSZMIDER 7
The same argument as before yields X ⊕ c ≃ C ( K A ) ⊕ c ≃ C ( K A ). All that is left to see isthat the space X has a complemented copy of c . Indeed, it follows from [14] that every twistedsum space X as above has Pe lczy´nski’s property (V). Therefore, there is a copy X of c in X such that the restriction ρ | X is an isomorphism. Since ρ ( X ) must be necessarily complementedin c ( κ ), c will also be complemented in X . Hence X ≃ c ⊕ X and the proof concludes. (cid:3) We do not know whether every twisted sum space0 −−−→ c −−−→ X ρ −−−→ c ( κ ) −−−→ C ( K )-space. However, Proposition 4.2.
Let A be an almost disjoint family of subsets of ω such that |A| = κ < c .Under MA ( κ ), every twisted sum space X −−−→ c −−−→ X −−−→ c ( κ ) −−−→ is a quotient of C ( K A ) .Proof. Since Ext( C ( K A ) , c ) = 0 it follows from Proposition 2.1 (1) that there is a commutativediagram 0 −−−→ c −−−→ C ( K A ) −−−→ c ( κ ) −−−→ y y (cid:13)(cid:13)(cid:13) −−−→ c −−−→ X −−−→ c ( κ ) −−−→ −−−→ c −−−→ c ⊕ C ( K A ) −−−→ X −−−→ C ( K A ) ≃ c ⊕ C ( K A ) proves the assertion. (cid:3) The following result contains a generalization of Theorem 1.1(a).
Theorem 4.3.
Assume MA ( κ ) and let K i , i = 0 , be separable scattered compacta of finiteheight and weight κ .(1) If C ( K ′ ) ≃ C ( K ′ ) then C ( K ) ≃ C ( K ) .(2) If C ( K ) ≃ C ( K ) then X ≃ X whenever X i , i = 0 , , is a twisted sum → c → X i → C ( K i ) → .(3) Moreover, all iterated twisted sums of X K i , i = 0 , and c are also isomorphic.Proof. By Remark 4.1, C ( K i ) ≃ c ⊕ C ( K i ). Moreover, one has exact sequences0 −−−→ c −−−→ C ( K i ) R −−−→ C ( K ′ i ) −−−→ R is the natural restriction map, which yields ker R ≃ c . Consider the two exact sequences0 −−−→ c −−−→ C ( K ) αR −−−→ C ( K ′ ) −−−→ (cid:13)(cid:13)(cid:13) −−−→ c −−−→ C ( K ) −−−→ R C ( K ′ ) −−−→ CABELLO, CASTILLO, MARCISZEWSKI, PLEBANEK, AND SALGUERO-ALARC ´ON where α : C ( K ′ ) → C ( K ′ ) is an isomorphism. Let z denote the lower sequence and z theupper sequence. Since K i is a separable scattered compact of finite height, Theorem 2.2(c)implies Ext( C ( K i ) , c ) = 0, and thus z R ≡ ≡ z α R . Therefore, C ( K ) ≃ c ⊕ C ( K ) ≃ c ⊕ C ( K ) ≃ C ( K ) , and this proves (1). To prove (2), consider the two exact sequences 0 → c → X i Q i → C ( K i ) → α : C ( K ) → C ( K ) be an isomorphism. Recall that Ext( X i , c ) = 0 by a 3-spaceargument (see [5]) and thus a similar reasoning as above can be used with the two sequences0 −−−→ c −−−→ X αQ −−−→ C ( K ) −−−→ (cid:13)(cid:13)(cid:13) −−−→ c −−−→ X −−−→ Q C ( K ) −−−→ X i obviously contain complemented copies of c since C ( K i ) does so and the pull-backsequence 0 −−−→ c −−−→ X i Q i −−−→ C ( K i ) −−−→ (cid:13)(cid:13)(cid:13) x x −−−→ c −−−→ c ⊕ c −−−→ c −−−→ X i ≃ c ⊕ X i . (cid:3) Example . The separability assumption in Proposition 4.3 is essential, and for that reasonwe cannot extend the result to higher derived sets in an obvious way. Indeed, take κ < c and let K be the subset of those elements x of the Cantor cube 2 κ for which the support { ξ < κ : x ( ξ ) = 0 } has at most 2 elements. Take any almost disjoint family A of size κ and K A as the second compactum. Then K ′ = A ( κ ) = K ′A ; however, C ( K ) C ( K A ) since C ( K ) isweakly compactly generated (in other words, K is Eberlein compact) while the latter space isnot ( K A is not Eberlein compact since it is separable, but not metrizable).Since each scattered compact space K of height 2 is a finite sum of one-point compactificationsof discrete spaces, the corresponding function space C ( K ) is isomorphic to c ( | K | ). Hence, fromTheorem 4.3 we obtain the following corollaries: Corollary 4.5.
Assuming MA ( κ ) , if K and K are separable scattered compact spaces of height and weight κ , then C ( K ) and C ( K ) are isomorphic. Corollary 4.6.
Assuming MA ( κ ) , if K is a separable scattered compact space of height 3 andweight κ , then c ( C ( K )) (the c –direct sum of C ( K ) ) is isomorphic to C ( K ) . In particular, C ( K ) is isomorphic to its square.Proof. Note that c ( C ( K )) is isomorphic to C ( A ( ω ) × K ) and such compact space has finiteheight, so we infer from [17, Corollary 4.2] that Ext( c ( C ( K )) , c ) = 0. On the other hand, we AILING OVER THREE PROBLEMS OF KOSZMIDER 9 have the following diagram0 −−−→ c −−−→ c ( C ( K )) −−−→ c ( c ( | K | )) −−−→ (cid:13)(cid:13)(cid:13) −−−→ c −−−→ C ( K ) −−−→ c ( | K | )) −−−→ c ⊕ C ( K ) ∼ c ⊕ c ( C ( K )). Since C ( K ) contains c complemented, and so does c ( C ( K )), we are done. In particular, C ( K ) isisomorphic to its square, because c ( C ( K )) has this property. (cid:3) Counting non-isomorphic C ( K ) -spaces For any compact space K we identify, as usual, the dual space C ( K ) ∗ with the space M ( K ), ofall signed regular Borel measures on K having finite variation. We start by the following generalresult on the isomorphism types of C ( K ) spaces. Theorem 5.1.
Let K be a family of compact spaces such that(i) K is separable and | M ( K ) | = c for every K ∈ K ;(ii) For every pair of distinct K, L ∈ K one has C ( K ) ≃ C ( L ) and K, L are not homeomorphic.Then K is of cardinality at most c .Proof. Since every K ∈ K is separable, there is a continuous surjection π K : βω → K . Con-sequently, C ( K ) can be isometrically embedded into C ( βω ) via the mapping π ∗ K ( g ) = g ◦ π K , g ∈ C ( K ).Take K, L ∈ K , K = L . Note that if the condition π K ( p ) = π K ( p ) ⇐⇒ π L ( p ) = π L ( p ) , was satisfied for any p , p ∈ βω then the formula h ( π K ( p )) = π L ( p ) would properly define ahomeomorphism h : K → L . Hence, for instance, there are p , p ∈ βω such that π K ( p ) = π K ( p )while π L ( p ) = π L ( p ). This immediately implies that π ∗ K [ C ( K )] = π ∗ L [ C ( L )] — consider afunction g ∈ C ( L ) that distinguishes π L ( p ) and π L ( p ).Fix now K ∈ K ; for every K ∈ K there is an isomorphism T K : C ( K ) → C ( K ) and therefore S K : C ( K ) → C ( βω ), where S K = π ∗ K ◦ T K is a bounded linear operator. It follows that S K = S K ′ whenever K = K ′ . To conclude the proof, it is therefore sufficient to show that thespace L ( C ( K ) , C ( βω )) of all bounded operators, is of size at most c .Note that any operator R : C ( K ) → C ( βω ) is uniquely determined by the sequence h R ∗ ( δ n ) : n ∈ ω i , of measures from M ( K ), where R ∗ : M ( βω ) → M ( K ) is the adjoint operator. By our assump-tion, | M ( K ) | = c so | M ( K ) | ω = c , and we are done. (cid:3) Let us remark that, in the setting of the above theorem, if we know only that | M ( K ) | = c forsome K ∈ K , then automatically | M ( K ) | = c for every K ∈ K (since C ( K ) ≃ C ( K )). However,separability of the domain is not preserved by isomorphisms between the spaces of continuous functions. For instance, ℓ ∞ ≃ L ∞ [0 ,
1] by Pe lczy´nski theorem, so C ( βω ) ≃ C ( K ), where K isthe Stone space of the measure algebra (recall that such K is not separable). Corollary 5.2.
There are c pairwise nonisomorphic twisted sums of c and c ( c ) .Proof. In what follows, A , B (with possible indices) denote almost disjoint families of subsets of ω of cardinality c . For every almost disjoint family A , K A is a separable compact space and,since K A is scattered, | M ( K A ) | = c .Note first that any homeomorphism h : K A → K B is determined by a permutation of ω .Therefore, we can define a sequence hA ξ : ξ < c i such that the spaces K A ξ are pairwise nothomeomorphic. By a direct application of Theorem 5.1, we conclude that there is I ⊆ c ofcardinality 2 c such that C ( K A ξ ) is not isomorphic to C ( K A η ) whenever ξ, η ∈ I and ξ = η . Now,the assertion follows from Theorem 3.2. (cid:3) The argument that proves Theorem 5.1 and Corollary 5.2 can be generalized to some highercardinals, see Section 8.Recall that, given a scattered compact space K , we have the equality w ( K ) = | K | (see [39]).Since, for infinite compacta K , weights of K and C ( K ) are equal, we obtain the following simpleobservation. Proposition 5.3.
Let A and B be infinite almost disjoint families of subsets of ω . If C ( K A ) and C ( K B ) are isomorphic then |A| = |B| . Using the construction from the proof of Theorem 8.7 from [32], we shall now define a largefamily of compactifications of ω with remainders homeomorphic to S . We follow here the notationintroduced in subsection 3.2; recall, in particular, that for ane x ∈ (0 ,
1) we write P x for { q ∈ Q : q ≤ x } .For every x ∈ (0 , q nx ) n ∈ ω in Q such that lim n q nx = x, q nx < x ,and put S x = { q nx : n ∈ ω } . Take any function θ : (0 , → R θx = ( P x if θ ( x ) = 0 ,P x \ S x if θ ( x ) = 1 . Then R θx ⊆ ∗ R θy whenever x < y since, in such a case, P x ∩ S y is finite.Let B θ be a subalgebra of P ( Q ) generated by { R θx : x ∈ (0 , } ∪ fin, where fin denotes thefamily of all finite subsets of Q ; we let the space L θ be ult( B θ ), the Stone space of the underlyingBoolean algebra.Note that every space L θ may be seen as a compactification of the discrete set Q with theremainder homeomorphic to S . Indeed, we may think that Q ⊆ L θ by identifying q ∈ Q with F ∈ ult( B θ ) containing { q } . If F ∈ ult( B θ ) contains no finite subset of Q then it is uniquelydetermined by the set { x ∈ (0 , R x ∈ F } . Now the point is that the family { R x : x ∈ (0 , } forms a chain with respect to almost inclusion. Hence the Boolean algebra B / fin is isomorphicto A which means that L θ \ Q is homeomorphic to ult( A ) ≃ S (see also [32, Theorem 8.7]). Itfollows that every space C ( L θ ) is a twisted sum of c and C ( S ).. AILING OVER THREE PROBLEMS OF KOSZMIDER 11
Corollary 5.4.
There are c pairwise nonisomorphic twisted sums of c and C ( S ) .Proof. We can follow the idea of the proof of Corollary 5.2. We have a family { L θ : θ ∈ (0 , } of separable compact spaces. For any θ , | M ( L θ ) | = c because L θ may be identified with S ∪ ω and M ( S ) has cardinality c . The latter follows from the fact that a compact separable linearlyordered space carries at most c regular measures, see e.g. Mercourakis [36].Again, for a fixed space L θ , the set { η ∈ (0 , : L θ ≃ L η } is of size at most c (since anyhomeomorphism of such spaces fixes Q and, as before, we can single out 2 c many pairwisenonisomorphic spaces of the form C ( L θ ). (cid:3) Separable Rosenthal compacta
The purpose of this section is to collect several subtle results concerning applications of de-scriptive set theory to separable Rosenthal compacta that will be needed in the next section.A compact space K is Rosenthal compact if embeds into B ( ω ω ), the space of Baire-one func-tions on the Polish space ω ω (homeomorphic to the irrationals), equipped with the topologyof pointwise convergence; recall that a Baire-one function is a pointwise limit of a sequence ofcontinuous functions. We refer to [18] and [31] for basic properties of Rosenthal compacta andfurther references.Recall that in a Polish space T , the Borel σ -algebra Bor ( T ) can be written as Bor ( T ) = [ ≤ α<ω Σ α ( T ) = [ ≤ α<ω Π α ( T ) , where Σ ( T ) and Π are the families of open and closed sets, respectively. Additive classes Σ α are defined inductively as all countable unions of elements from S β<α Π β and so on, see Kechris[25, 11.B] for details. Also recall that a subset of a Polish space is analytic if it is a continuousimage of ω ω . In a Polish space, every Borel set is analytic.If K is any separable compact space, D ⊆ K is any countable dense subset and X is atopological space, then we denote by C D ( K, X ) the space of all continuous functions from K into X , equipped with the topology of pointwise convergence on D . We write C D ( K ) for C D ( K, R ).In other words, C D ( K ) is a subset of R D which is the image of the restriction map C ( K ) ∋ g → g | D ∈ R D .Recall that the dual Banach space C ( K ) ∗ may be identified with the space M ( K ), of regularBorel measures on K of finite variation. We extend the notation introduced above as follows.Consider any countable subset M ⊆ M ( K ) separating the functions in C ( K ); then C M ( K )stands for the space C ( K ) endowed with the weak topology generated by M . Here, we simplyidentify f ∈ C M ( K ) with ( µ ( f )) µ ∈ M ∈ R M .We briefly mention some properties of Rosenthal compacta. Godefroy showed in [21] that if K is Rosenthal compact, then M ( K ), the dual unit ball, is Rosenthal compact in its weak ∗ topology. On the other hand, Bourgain, Fremlin and Talagrand [4] proved that every Rosenthalcompactum K is Fr´echet-Urysohn , i.e., for any A ⊆ K and x ∈ A , there is a sequence ( x n ) n ∈ ω of points from A which converges to x . Those properties, in particular, imply that if D ⊆ K is a countable dense set then every µ ∈ M ( K ) is a Baire-one function on C D ( K ); see the proof ofTheorem 3.1 in [30].The following result can be used as a characterization of separable Rosenthal compacta; part( a ) of Theorem 6.1 is due to Godefroy [21], while part ( b ) is Corollary 2.4 in Dobrowolski andMarciszewski [19]. Theorem 6.1. (a) A separable compact space K is Rosenthal compact if and only if C D ( K ) isanalytic for every countable dense set D ⊆ K .(b) If K is a compact space and D ⊆ K is a countable dense set such that C D ( K ) is analyticthen either K is Rosenthal compact or K contains a copy of βω . Godefroy’s characterization of separable Rosenthal compacta was followed by introducing in[30] a certain index measuring the complexity of such spaces. This kind of Rosenthal’s index willbe denoted here by ri ( · ); note that in [30] the working notation η ( · ) was used. Definition 6.2.
We define the index ri on the class of separable Rosenthal compacta as follows.Set ri ( K ) = ω if C D ( K ) is Borel in R D for no countable dense set D ⊆ K . Otherwise, set ri ( K ) = α , where α is the least ordinal number < ω such that C D ( K ) ∈ Σ α ( R D ) ∪ Π α ( R D ) , for some countable dense D ⊆ K . The reader should be warned that the difference between Definition 6.2 and that from [30] isconnected with the fact that older tradition was to count Borel classes starting from 0 ratherthan 1 (for instance, F σδ is Π ). Recall that ri ( K ) ≥ K is infinite, see [30, Theorem2.1]. The double arrow space S is a classical nonmetrizable Rosenthal compactum with ri ( S ) = 2.The main feature of the index ri is that it is almost preserved by isomorphisms of Banachspaces, as we show next. The following result is a particular case of [30, Corollary 3.2]; Weoutline the main idea of its proof because the argument is a much shorter in our setting. Theorem 6.3. If K and K ′ are separable Rosenthal compacta and C ( K ) ≃ C ( K ′ ) then ri ( K ) ≤ ri ( K ′ ) (and, by symmetry, ri ( K ′ ) ≤ ri ( K )) . In particular, ri ( K ) = ri ( K ′ ) whenever ri ( K ) ≥ ω .Proof. Let T : C ( K ) → C ( K ′ ) be an isomorphism such that c · k g k ≤ k T g k ≤ k g k for every g ∈ C ( K ), where c >
0. Fix countable dense sets D ⊆ K and D ′ ⊆ K ′ realizing the values of ri ( K ) and ri ( K ′ ), respectively, and write ∆ D = { δ d : d ∈ D } , ∆ D ′ = { δ d : d ∈ D ′ } . Claim 1.
There are countable sets
M, M ′ , where ∆ D ⊆ M ⊆ M ( K ) , ∆ D ′ ⊆ M ′ ⊆ M ( K ′ ) such that C M ( K ) ∋ g → T g ∈ C M ′ ( K ′ ) , is a homeomorphism. AILING OVER THREE PROBLEMS OF KOSZMIDER 13
Proof of the claim:
Note that T ∗ sends M ( K ′ ) into M ( K ). Likewise, if we consider S : C ( K ′ ) → C ( K ) given by S = c · T − then S ∗ sends M ( K ) into M ( K ′ ). To define M and M ′ put M (0) = ∆ D and M ′ (0) = ∆ D ′ and define inductively M n +1 = T ∗ [ M ′ ( n )] , M ′ ( n + 1) = S ∗ [ M ( n )] . Then the sets M = S n M ( n ) and M ′ = S n M ′ ( n ) are as required. For instance, suppose that asequence ( g k ) k converges in C M ( K ) and consider any ν ∈ M ′ . Then ν ∈ M ′ ( n ) for some n andtherefore µ = T ∗ ν ∈ M n +1 ⊆ M . Hence ν ( T g k ) = T ∗ ν ( g k ) → T ∗ ν ( g ) = ν ( T g ). (cid:3) Now we can examine the mapping ϕ : C D ( K ) → C D ′ ( K ′ ) closing the following diagram C M ( K ) T / / id (cid:15) (cid:15) C M ′ ( K ′ ) id (cid:15) (cid:15) C D ( K ) ϕ / / C D ′ ( K ′ ) Claim 2. ϕ and ϕ − are mapping of the first Baire class.Proof of the claim: Indeed, while the mapping id : C M ( K ) → C D ( K ) is continuous (as ∆ D ⊆ M ),its inverse is of the first Baire class since every µ ∈ M ( K ) is a weak ∗ limit of a sequence fromthe absolute convex hull of ∆ D . This shows that ϕ is of the first Baire class; the argument for ϕ − is symmetric. (cid:3) The final step is to use Kuratowski’s theorem [28, par. 35 VII] which assures that there are F σδ sets A, B containing C D ( K ) and C D ′ ( K ′ ) respectively, and an extension e ϕ of ϕ to a Baire-one isomorphism A → B . This, together with ri ( K ) , ri ( K ′ ) ≥ ri ( K ) ≤ ri ( K ′ ) and ri ( K ′ ) ≤ ri ( K ). (cid:3) Twisted sums and Rosenthal compacta
Denote by 2 <ω the full dyadic tree and, as usual, 2 ω is the Cantor set. For any x ∈ ω , set B ( x ) = { x | n : n ∈ ω } ⊂ <ω . It is clear that for any Z ⊆ ω ,(2) A Z = { B ( x ) : x ∈ Z } , is an almost disjoint family of infinite subsets of 2 <ω .The mapping x B ( x ) is a homeomorphic embedding of 2 ω into 2 <ω (we identify P (2 <ω )with 2 <ω ). Therefore, for any Z ⊆ ω , A Z is an almost disjoint family (in 2 <ω ) which ishomeomorphic to Z . Moreover, for Borel Z , we have the following (see [30, 4.2]). Theorem 7.1. If Z ∈ Σ α (2 ω ) , where α ≥ , and A Z is an almost disjoint family given by (2),then K A Z is Rosenthal compact and α ≤ ri ( K A Z ) ≤ α + 1 . Using Theorem 7.1 and Theorem 6.3 we arrive at the following.
Corollary 7.2.
There is a family { K ξ : ξ < ω } of separable Rosenthal compacta such that C ( K ξ ) C ( K η ) whenever ξ = η and every C ( K ξ ) is a (nontrivial) twisted sum of c and c ( c ) . Proposition 7.3.
Let −→ c −→ C ( L ) −→ C ( K ) −→ be a twisted sum, where K is an infinite separable Rosenthal compact space, and L is a compactspace. If this twisted sum is trivial, then L is a separable Rosenthal compact space, and ri ( L ) ≤ ri ( K ) .Proof. Triviality of our twisted sum gives us the following string of isomorphisms C ( L ) ≃ c ⊕ C ( K ) ≃ C ( ω + 1) ⊕ C ( K ) ≃ C (( ω + 1) ⊕ K ) . Godefroy [21] proved that the class of separable Rosenthal compacta is preserved by isomorphismsof function spaces; since ( ω + 1) ⊕ K belongs to this class, so does L . By Theorem 6.3 it is enoughto verify that ri (( ω + 1) ⊕ K ) ≤ ri ( K ). Let D be a countable dense subset of K realizing thevalue of ri ( K ). By [30, Theorem 2.1] C D ( K ) is not a G δσ -subset of R D , and it is well knownthat C ω ( ω + 1) is an F σδ -subset of R ω . Hence, for the countable dense subset E = ω ∪ D of( ω + 1) ⊕ K , the space C E (( ω + 1) ⊕ K ) can be identified with the product C ω ( ω + 1) × C D ( K )which is a Borel subset of R ω × R D of the class ri ( K ). (cid:3) We turn to examining twisted sums of c and C ( S ); below we follow the notation introduced in3.2 and Section 5; recall that the space L θ was defined just below formula (1). Here the parameter θ is any function θ : (0 , →
2. It will be now convenient to write θ as χ Z , the characteristicfunction of a set Z ⊆ (0 , L ( Z ) for L θ with θ = χ Z . Lemma 7.4.
For any Z ⊆ (0 , , the space C Q ( L ( Z )) contains a G δ -subset X homeomorphic to [0 , \ ( Z ∪ Q ) .Proof. We will look for X inside the closed subset C Q ( L ( Z ) ,
2) = C Q ( L ( Z )) ∩ Q of C Q ( L ( Z )),which may be seen as traces of clopen subsets of L ( Z ) on Q , i.e., the algebra B θ . Define P = { ( x, ∩ Q : x ∈ [0 , \ Q } ⊆ Q P = { ( x, ∩ Q : x ∈ Q } ⊆ Q P = { [ x, ∩ Q : x ∈ Q } ⊆ Q where we identify P ( Q ) with 2 Q . Observe that the union P = P ∪ P ∪ P is closed in 2 Q . Indeed,we have s ∈ Q \ P if and only if there exists p < q in Q with s ( p ) = 1 and s ( q ) = 0. Since P ∪ P is countable, P is G δ -subset of 2 Q . A routine verification shows that the mapping x ( x, ∩ Q is a homeomorphism of [0 , \ Q onto P . Analyzing the description of the algebra B χ Z defining L ( Z ) one can verify that [( x, ∩ Q ] ∈ P ∩ C Q ( L ( Z )) if and only if x ∈ [0 , \ ( Z ∪ Q ). Hence,the set X = P ∩ C Q ( L ( Z )) has the required properties. (cid:3) Corollary 7.5.
Let Z ⊆ (0 , be such that Z / ∈ Σ ((0 , ∪ Π ((0 , . Then C ( L ( Z )) is anontrivial twisted sum of c and C ( S ) .Proof. Suppose that C ( L ( Z )) represents a trivial twisted sum of c and C ( S ). Since ri ( S ) = 2,from Proposition 7.3 we consclude that L ( Z ) is Rosenthal compact with ri ( L ( Z )) ≤
3. Theorem2.2 from [30] says that, for a separable Rosenthal compact space K , and any two countable dense AILING OVER THREE PROBLEMS OF KOSZMIDER 15 subsets
D, E of K , the Borel classes of C D ( K ) and C E ( K ) can differ by at most 1. Therefore C Q ( L ( Z )) ∈ Σ (( R Q ) ∪ Π (( R Q ). Hence, any G δ –subset of C Q ( L ( Z )) is also Borel of the sameclass, a contradiction with Lemma 7.4 and our assumption on Z . (cid:3) To prove the next Lemma 7.8 we need to employ a more effective description of the sets S x used in Section 5. We now consider Q = Q ∩ (0 , x ∈ (0 , .i x i x . . . be the binaryexpansion of x using infinitely many 1 ′ s, i.e., i xk ∈ { , } , k ∈ ω and x = P ∞ k =0 ( i xk / k +1 ). Wedefine S x = ( n X k =0 i xk / k +1 : n ∈ ω ) \ { } . Using the fact that, for x ∈ (0 , \ Q , the expansion 0 .i x i x . . . also has infinitely many 0 ′ s, one caneasily verify that the mapping x S x is continuous on (0 , \ Q (actually, it is a homeomorphicembedding of (0 , \ Q into 2 Q ).We also need the following two auxiliary results. The first one is well known, and follows easilyfrom the fact that Boolean operations on P ( ω ) are continuous. Proposition 7.6.
Each subalgebra of P ( ω ) with an analytic set of generators is analytic. The second result is probably also well known, it can be derived from the results of Godefroy[21]. For the sake of completeness we include an (alternative) proof of it.
Proposition 7.7.
Let K be a separable zero-dimensional compact space and D a countable densesubspace of K . If C D ( K,
2) = C D ( K ) ∩ D is analytic, so is C D ( K ) .Proof. Since R is homeomorphic to (0 ,
1) and C D ( K, (0 , C D ( K, [0 , ∩ (0 , D is a G δ -subset of C D ( K, [0 , C D ( K, [0 , C D ( K, ω can be identified with the space C D ( K, ω ). Hence, C D ( K, ω ) is also analytic, andit is enough to show that C D ( K, [0 , C D ( K, ω ). By [37, Lemma 1],whose key ingredient is the Mardeˇsic factorization theorem [35], there exists a continuous map φ : 2 ω → [0 ,
1] such that the composition map φ ◦ : C ( K, ω ) −→ C ( K, [0 , g → φ ◦ g is surjective. (cid:3) Lemma 7.8. If θ : (0 , → is Borel then L θ is Rosenthal compact.Proof. Let θ = χ Z , obviously Z is a Borel subset of (0 , Z = Z \ Q and Z = (0 , \ ( Z ∪ Q ).First, we will show that the set { R θx : x ∈ (0 , } ⊂ P ( Q ) of generators of the algebra B θ is analytic. We can neglect the countable set { R θx : x ∈ Q } , so it is enough to verify thatboth sets G i = { R θx : x ∈ Z i } , i = 0 ,
1, are analytic. We have G = { P x \ S x : x ∈ Z } and G = { P x : x ∈ Z } . Since the map A Q \ A is a homeomorphism of P ( Q ) and Q \ P x = ( x, ∩ Q , the argument from the proof of Lemma 7.4 shows that G is homeomorphicto Z , hence analytic. Observe that, by the same argument and the continuity of the mapping x S x , the map ϕ : Z → G defined by ϕ ( x ) = P x \ S x is continuous and onto, therefore G isalso analytic. Lemma 7.6 implies that the algebra B θ , which can be identified with C Q ( L θ , C Q ( L θ ) is also analytic. Now, Theorem 6.1(b) gives us thedesired conclusion. (cid:3) Corollary 7.9.
If the set Z ⊆ (0 , is not coanalytic then(i) L ( Z ) is not Rosenthal compact;(ii) C ( L ( Z )) is a nontrivial twisted sum of c and C ( S ) ;(iii) C ( L ( Z )) is not isomorphic to C ( L θ ) whenever the function θ is Borel.Proof. Part ( i ) follows directly from Lemma 7.4; ( ii ) is a consequence of ( i ) and Proposition 7.3.The last statement follows from Lemma 7.8. (cid:3) It is likely that twisted sums of c and C ( S ) that are of the form C ( K ) with K Rosenthal com-pact can be examined more closely using the Rosenthal index. However, the following problemis open to us.
Problem 7.10.
Let θ : (0 , → be Borel. Is ri ( L θ ) < ω ? Can we find an effective estimate of ri ( L θ ) using the class of θ ? Counting non-isomorphic C ( K ) -spaces one more time We consider here families A consisting of countable infinite subsets of an uncountable set I ,as in Dow and Vaughan [20]. Such A is an almost disjoint family if, again, A ∩ B is finite for anydistinct A, B ∈ A . We can form a version of Aleksandrov-Urysohn space K A , where K A = I ∪ { A : A ∈ A} ∪ {∞} . As before, points in I are isolated, basic open neighborhoods of a given point A ∈ A are of theform { A } ∪ ( A \ F ) with F ⊆ I finite, and the point ∞ compactifies the locally compact space I ∪ { A : A ∈ A} . Then K A is a scattered compact space of height 3 and density | I | .The arguments used in Theorem 5.1 and Corollary 5.2 can be generalized to obtain informationabout the cardinality of Ext( c ( κ ω ) , c ( κ )) for κ satisfying κ < κ ω — we outline it here. Theorem 8.1.
Assume that κ < κ ω and let A be an almost disjoint family of countable subsetsof κ with |A| = κ ω . Then C ( K A ) is a non-trivial twisted sum of c ( κ ) and c ( κ ω ) .Proof. Write K ′A for the derivative of K A , which is equal to A ∪ {∞} . It is enough to check thatthere is no bounded extension operator E : C ( K ′A ) → C ( K A ). Assume the existence of such E ;for any A ∈ A , the singleton { A } is open in K ′A so, writing g A for its characteristic function, wehave g A ∈ C ( K ′A ). Hence, there must be ξ A ∈ κ so that | Eg A ( ξ A ) | > . Since |A| = κ ω > κ , weinfer the existence of a point ξ ∈ κ such that ξ = ξ A n for a sequence of distinct A n ∈ A . Then,for every natural number m we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =1 Eg A j ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 g A j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 1 , which contradicts continuity of E . (cid:3) AILING OVER THREE PROBLEMS OF KOSZMIDER 17
The next results can be proved by similar arguments to those used in 5.1 and 5.2.
Theorem 8.2.
Fix an infinite cardinal κ , and let K be a family of compact spaces such that(i) every K has density κ and | M ( K ) | ≤ κ ;(ii) For every pair of distinct K, L ∈ K one has C ( K ) ≃ C ( L ) and K, L are not homeomorphic.Then K is of cardinality at most κ . Corollary 8.3. If κ ω > κ then there are κ ω pairwise non-isomorphic twisted sums of c ( κ ) and c ( κ ω ) . The above corollary can be applied to any (infinite) κ < c under Martin’s axiom since then2 κ = c . References [1] S. Argyros, T. Raikoftsalis,
Banach spaces with a unique nontrivial decomposition , Proc. Amer. Math. Soc.136 (2008), 3611–3620.[2] A. Avil´es, F. Cabello S´anchez, J.M.F. Castillo, M. Gonz´alez, Y. Moreno
Separably injective Banach spaces ,Lecture Notes in Mathemathics 2132, Springer (2016).[3] A. Avil´es, W. Marciszewski, G. Plebanek,
Twisting c around nonseparable Banach spaces , arxiv1902.07783v1.[4] J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets of Baire-measurable functions , Amer. J.Math. 100 (1978), 845–886.[5] F. Cabello S´anchez, J.M.F. Castillo,
The long homology sequence for quasi Banach spaces, with applications ,Positivity, 8 (2004), 379–394.[6] F. Cabello S´anchez, J.M.F. Castillo,
Uniform boundedness and twisted sums of Banach spaces
Houston J.Math. 30 (2004), 523–536.[7] F. Cabello S´anchez, J.M.F. Castillo,
Homological methods in Banach space Theory , Cambridge Studies inAdvanced Math. Cambridge Univ. Press. Scheduled 2020.[8] F. Cabello S´anchez, J.M.F. Castillo, D. Yost,
Sobczyk’s theorems from A to B , Extracta Math. 15 (2000),391–420.[9] F. Cabello S´anchez, J.M.F. Castillo, N.J. Kalton, D.T. Yost,
Twisted sums with C ( K ) spaces, Trans. Amer.Math. Soc. 355 (2003), 4523–4541.[10] J.M.F. Castillo, P.L. Papini,
Epheastus account on Trojanski’s polyhedral war , Extracta Math. Vol. 29, (2014)35 – 51.[11] J.M.F. Castillo,
Nonseparable C ( K ) -spaces can be twisted when K is a finite height compact , Topology Appl.198 (2016), 107–116.[12] J.M.F. Castillo, M. Gonz´alez, Three-space problems in Banach space theory , Lecture Notes in Math. 1667,Springer 1997.[13] J.M.F. Castillo, Y. Moreno,
On the Lindenstrauss-Rosenthal theorem , Israel J. Math. 140 (2004), 253–270.[14] J.M.F. Castillo, M. Simoes,
Property ( V ) still fails the 3-space property , Extracta Math. (2012), 5–11.[15] C. Correa, Nontrivial twisted sums for finite height spaces under Martin’s Axiom , Fund. Math. 248 (2020)195–204.[16] C. Correa, D.V. Tausk,
Nontrivial twisted sums of c and C ( K ), J. Funct. Anal. 270 (2016), 842–853.[17] C. Correa, D.V. Tausk, Local extension property for finite height spaces , Fund. Math. 245 (2019), 149–165.[18] G. Debs,
Descriptive Aspects of Rosenthal Compacta , in: Recent progress in general topology III, 205–227,Atlantis Press, Paris (2014). [19] T. Dobrowolski, W. Marciszewski,
Classification of function spaces with the pointwise topology determinedby a countable dense set , Fund. Math. 148 (1995), 35–62.[20] A. Dow, J.E. Vaughan,
Mr´owka maximal almost disjoint families for uncountable cardinals , Topology Appl.157 (2010), 1379–1394.[21] G. Godefroy,
Compacts de Rosenthal , Pacific J. Math. 91 (1980), 293–306.[22] M. Hruˇs´ak,
Almost disjoint families and topology , in: Recent progress in general topology III, 601–638,Atlantis Press, Paris (2014).[23] F. Hern´andez-Hern´andez, M. Hruˇs´ak,
Topology of Mr´owka-Isbell spaces , Pseudocompact topological spaces,253–289, Dev. Math. 55, Springer, Cham, 2018.[24] W.B. Johnson, J. Lindenstrauss.
Some remarks on weakly compactly generated Banach spaces.
Israel J. Math.17 (1974), 219–230.[25] A.S. Kechris,
Classical descriptive set theory
Graduate Texts in Mathematics 156, Springer-Verlag, New York,(1995).[26] P. Koszmider,
On decomposition of Banach spaces of continuous functions on Mr´owka’s spaces , Proc. Amer.Math. Soc. 133 (2005), 2137–2146.[27] P. Koszmider, N.J. Laustsen,
A Banach space induced by an almost disjoint family, admitting only fewoperators and decompositions , arXiv:2003.03832.[28] K. Kuratowski,
Topology , Vol. 1, Academic Press and PWN (1966).[29] W. Marciszewski,
A function space C ( K ) not weakly homeomorphic to C ( K ) × C ( K ). Studia Math. 88 (1988),no. 2, 129–137.[30] W. Marciszewski. On a classification of pointwise compact set of the first Baire class functions , Fund. Math.133 (1989), 195–2009.[31] W. Marciszewski,
Rosenthal compacta , in: Encyclopedia of General Topology, Elsevier, 2003, 142–144.[32] W. Marciszewski, G. Plebanek,
Extension operators and twisted sums of c and C ( K ) spaces , J. Func. Anal.274 (2018), 1491–1529.[33] W. Marciszewski, R. Pol, On Banach spaces whose norm-open sets are F σ sets in the weak topology , J. Math.Anal. Appl. 350 (2009), 708–722.[34] W. Marciszewski, R. Pol, On Borel almost disjoint families , Monatsh. Math., (2012), 545–562.[35] S. Mardeˇsic,
On covering dimension and inverse limits of compact spaces , Illinois J. Math. 4(2) (1960),278-–291.[36] S. Mercourakis,
Some remarks on countably determined measures and uniform distribution of sequences ,Monatsh. Math. 121 (1996), 79–111.[37] O. Okunev,
The Lindel¨of number of C p ( X ) × C p ( X ) for strongly zero-dimensional X , Cent. Eur. J. Math. 9(2011), 978—983.[38] Z. Semadeni, Banach spaces non-isomorphic to their Cartesian squares. II , Bull. Acad. Polon. Sci. S´er. Sci.Math. Astronom. Phys. 8 No. 2 (1960) 81–86.[39] L. Soukup,
Scattered spaces , in: Encyclopedia of General Topology, Elsevier, 2003, 350–353.
AILING OVER THREE PROBLEMS OF KOSZMIDER 19
Instituto de Matem´aticas Imuex, Universidad de Extremadura, Avenida de Elvas, 06071-Badajoz, Spain
E-mail address : [email protected], [email protected], [email protected] Institute of Mathematics, University of Warsaw, Banacha 202–097 Warszawa, Poland
E-mail address : [email protected] Mathematical Institute, University of Wroc law, pl.Grunwaldzki, 2/4, 50-384 Wroclaw,Poland,
E-mail address ::