Counting spaces of functions on separable compact lines

COUNTING SP A CES OF FUNCTIONS ON SEP ARABLE COMP A CT LINES MA CIEJ K ORP ALSKI, PIOTR K OSZMIDER, AND WITOLD MAR CISZEWSKI Abstract. W e in v estigate the follo wing general problem, closely related to the prob- lem of isomorphic classification of Banac h spaces C ( K ) of con tinuous real-v alued func- tions on a compact space K , equipp ed with the suprem um norm: Let K b e a class of compact spaces. Ho w man y isomorphism types of Banac h spaces C ( K ) are there, for K ∈ K ? W e pro ve that for any uncountable regular cardinal num b er κ , there exist exactly 2 κ isomorphism types of spaces C ( K ) for compact spaces of weigh t κ . W e sho w that, for the class L ω 1 of separable compact linearly ordered spaces of weigh t ω 1 , the answ er to the ab o ve question dep ends on additional set-theoretic axioms. In particular, assuming the contin uum hypothesis, there are 2 ω 1 isomorphism types of C ( L ), for L ∈ L ω 1 , and assuming a certain axiom prop osed by Baumgartner, there is only one type. 1. Introduction One of the most fundamen tal and natural c hallenges of mathematics is the problem of classifying ob jects within a given class. An important part of suc h a c hallenge is an attempt to determine how man y isomorphism t yp es of ob jects w e hav e in that class. In this pap er, w e consider the following case of this general problem: Let K b e a class of compact spaces. Ho w man y isomorphism t yp es of Banach spaces C ( K ) of real-v alued con tinuous functions on K , equipp ed with the suprem um norm, do w e hav e for K ∈ K ? Recall that for an infinite compact space K , the w eigh t of K is equal to the w eigh t and the densit y of C ( K ). Hence, if the Banach spaces C ( K ) and C ( L ) are isomorphic, then the compact spaces K and L m ust ha v e equal w eights. It is also known that compact spaces K and L with isomorphic function spaces ha ve equal cardinalities [7]. Therefore, 2020 Mathematics Subje ct Classific ation. Primary 03E35, 03E65, 03E75, 46B03, 46E15, 54F05. The second-named and the third-named authors were partially supp orted b y the NCN (National Science Centre, Poland) researc h gran t no. 2020/37/B/ST1/02613. 1 2 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI it is natural that in the problem of counting isomorphism t yp es of spaces C ( K ), w e restrict ourselves to the classes of compacta K of fixed w eight or fixed cardinalit y . The classical results of Miljutin, Bessaga and Pe lczy ´ nski giv e us a complete isomorphic classification of spaces C ( K ) for metrizable compacta K , i.e., for compacta of coun table w eigh t. In particular, Bessaga and Pe lczy ´ nski (cf. [35]) pro ved that there are exactly ω 1 isomorphism types of C ( K ), for countable (hence metrizable) K , and Miljutin (cf. [35]) sho w ed that, for uncountable metrizable K , there is only one isomorphic type of C ( K ) space. Summarizing, w e ha v e ω 1 isomorphism types of spaces C ( K ) for compacta K of w eigh t ω (regardless of the size of the contin uum 2 ω ). Recall that we ha v e 2 ω top ological t yp es of compacta of weigh t ω [24]. F or the classes of nonmetrizable compacta, the chances of obtaining such complete classification results seem to b e v ery slim, for a simple reason - we usually ha ve to o man y isomorphism t yp es of spaces C ( K ). It is known that even if w e restrict ourselv es to some concrete classes of compact spaces of weigh t 2 ω , we still ha v e 2 2 ω isomorphism t yp es of C ( K ). In particular, it is true for the class of separable scattered compacta of heigh t 3 (cf. [6]), or the class of separable, compact linearly ordered spaces (cf. [23]). Note that, for a compact K of weigh t κ , the cardinality of isomorphism t yp es of C ( K ) is b ounded by 2 κ (cf. remarks in the last section). In this pap er, we concen trate on the case of compact spaces of w eigh t ω 1 . W e prov e that there are 2 ω 1 isomorphism types of C ( K ), for K of suc h weigh t. Actually , we prov e the following more general result: Theorem 1.1. Supp ose that κ is an unc ountable r e gular c ar dinal. Ther e is a family of c ar dinality 2 κ of c omp act sp ac es K of weight κ , such that the c orr esp onding Banach sp ac es C ( K ) ar e p airwise nonisomorphic. Most of our inv estigations deal with the class of separable, compact linearly ordered spaces (in short: separable compact lines). Banac h spaces C ( K ) asso ciated with such compacta K play an imp ortant role in Banac h space theory . They ha v e pro vided many in teresting examples (see [15], [16], [18], [29]), and ha v e b een an ob ject of in vestigations in numerous pap ers (see [2], [8], [17], [21], [23], [25], [34]). Let L κ b e the class of separable, compact linearly ordered spaces of w eigh t κ (for our top ological terminology see [11]). W e sho w that the cardinalit y of isomorphism t yp es of C ( L ), for L ∈ L ω 1 , dep ends on additional set-theoretic axioms. In particular, assuming the contin uum h yp othesis (i.e., 2 ω = ω 1 ), there are 2 2 ω isomorphism types of C ( L ), for L ∈ L ω 1 . More precisely , w e prov e the following: COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 3 Theorem 1.2. If κ ≤ 2 ω < 2 κ , then ther e exist 2 κ p airwise nonisomorphic sp ac es C ( L ) for L ∈ L κ . On the other hand, assuming a certain axiom ( BA ) prop osed b y Baumgartner, there is only one type of C ( L ) for L ∈ L ω 1 : Theorem 1.3 ( BA ) . If K and L ar e sep ar able c omp act lines of weight ω 1 , then the Banach sp ac es C ( K ) and C ( L ) ar e isomorphic. F rom this theorem w e will deriv e the follo wing: Corollary 1.4 ( BA ) . If K is a sep ar able c omp act line of weight ω 1 , then the Banach sp ac e C ( K ) is isomorphic with the dir e ct sum C ( K ) ⊕ C ( K ) . The ab o v e corollary should b e compared with a recen t result of Kuc harski [22] who pro v ed that there exists a separable compact line K of weigh t 2 ω suc h that the Banach space C ( K ) is not isomorphic to C ( K ) ⊕ C ( K ). In particular, this sho ws that Corollary 1.4 requires some additional set-theoretic hypothesis. Let us recall that another example of a class K of compact spaces of w eight ω 1 , suc h that the cardinalit y of isomorphism t yp es of C ( K ), for K ∈ K , dep ends on additional set-theoretic axioms, was giv en in [6] ( K is the class of separable scattered compacta of heigh t 3). The pap er is organized as follo ws: In Section 2 we explain our terminology , recall some definitions and auxiliary facts. Section 3 con tains the pro of of Theorem 1.1 and its prerequisites. In Section 4 we presen t some general top ological prop erties of separable compact lines, in particular coun ting their homeomorphic types and recalling some handy c haracterizations. The main result of Section 5, dedicated to spaces of contin uous functions on separable compact lines, is Lemma 5.4, whic h states that C ( K ) is isomorphic to C ( K ) ⊕ C ( M ) for an y uncountable separable compact line K and an y metrizable compact space M . Sections 6 and 7 are dedicated to pro ving Theorems 1.2 and 1.3, resp ectively . In Section 8, we turn our atten tion to spaces of contin uous functions on finite pro ducts of separable compact lines. Using the results of Mic halak [25] and assuming the Baum- gartner’s Axiom, we provide a complete isomorphic classification of such C ( K ) spaces of weigh t ω 1 , see Theorem 8.5. The last section, Section 9, is dedicated to some scattered though ts and remarks about the sub ject of the pap er. 4 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI 2. Preliminaries 2.1. Set-theoretic terminology. W e denote by cf ( α ) the cofinality of the ordinal α . Let κ b e a regular cardinal, i.e., cf ( κ ) = κ . Recall that a set A ⊆ κ is a club set if it is unbounded in κ and closed in the order top ology . A set S ⊆ κ is stationary if it has a nonempty in tersection with all club subsets of κ . By ω 1 w e denote the first uncoun table cardinal and usually write 2 ω for the con tin uum, the cardinality of the set of real n um b ers. In general, our set-theoretic terminology follows Jec h’s bo ok [14]. 2.2. Banac h spaces of con tin uous functions and their duals . Each compact and lo cally compact space we consider is Hausdorff. F or a lo cally compact space X w e denote by C 0 ( X ) the space of real-v alued con tinuous functions on X v anishing at infinit y , considered with the suprem um norm. F or a compact space K , we write C ( K ) instead of C 0 ( K ), as it is the Banac h space of all real-v alued con tin uous functions on K . It is a classical fact that the density and w eight of C ( K ) is equal to the weigh t of K . F or a set A ⊆ K , w e denote the characteristic function of A by 1 A . The dual space C ( K ) ∗ , due to the Riesz represen tation theorem, can be seen as M ( K ), the Banach space of signed Radon measures on K with absolute v ariation as the norm. There is a similar representation of C 0 ( X ) ∗ for an y lo cally compact space X , where ev ery elemen t of C 0 ( X ) ∗ can b e represen ted as the in tegration with resp ect to a signed Radon measure on the one-p oin t compactification αX of X that v anishes at the added p oin t (see Theorem 18.4.1 of [35]). In the space M ( K ), equipp ed with the w eak* top ology induced by C ( K ), there is a linearly dense top ological copy of K , which is the set of Dirac delta measures ∆ K = { δ x : x ∈ K } (w e will sometimes identify K and ∆ K ). This, in particular, means that if the space K is separable, then M ( K ) is also separable in the weak* top ology . W e write supp ( µ ) for the supp ort of the measure µ ∈ M ( K ). F or tw o Banac h spaces X , Y w e write X ≃ Y if X and Y are isomorphic. By X | Y w e mean that Y contains a complemen ted isomorphic copy of X , i.e., there is a Banac h space X ′ suc h that Y ≃ X ⊕ X ′ . F or a countable family of Banach spaces { X n : n ∈ ω } , their c 0 -sum is the Banach space consisting of all sequences ( x n ) n ∈ ω ∈ Q n ∈ ω X n suc h that for ev ery ε > 0, the set { n ∈ ω : ∥ x n ∥ > ε } is finite, equipp ed with the supremum norm ∥ ( x n ) n ∈ ω ∥ = s up n ∈ ω ∥ x n ∥ . The follo wing fact is standard, see e.g. [35, Prop osition 21.5.8]. Note that all h yp er- planes of a Banach space are m utually isomorphic, see [12, Exercise 2.9]. COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 5 F act 2.1. L et K b e a c omp act sp ac e with a nontrivial c onver gent se quenc e. Then the Banach sp ac e C ( K ) is isomorphic to its hyp erplanes. In general, our terminology concerning Banach spaces follo ws bo oks [12] and [35]. 2.3. Scattered spaces. Recall that a topological space F is scattered if ev ery nonempt y set A ⊆ F contains a relative isolated p oin t. F or a scattered space X , b y the Can tor-Bendixson heigh t (or simply the heigh t) ht ( X ) of X we mean the least ordinal α suc h that the Can tor-Bendixson deriv ative X ( α ) of the space X is empt y . The heigh t of a compact scattered space is alwa ys a nonlimit ordinal. Let us make a general remark ab out the b ounded linear functionals on C 0 ( X ) for an y lo cally compact scattered space X . By the Rudin theorem (Theorem 19.7.6 of [35]), if α X is scattered, then all Radon measures on the one-p oin t compactification αX are purely atomic. F rom the Riesz representation theorem men tioned ab o v e it follows that an y b ounded linear functional φ on C 0 ( X ) is defined b y φ ( f ) = X n ∈ ω a n f ( x n ) , where P n ∈ ω | a n | < ∞ and { x n : n ∈ ω } ⊆ X are distinct, and f ∈ C 0 ( X ). 2.4. Separable compact linearly ordered spaces. Consider an arbitrary closed sub- set K of the unit in terv al I = [0 , 1] and an y subset A ⊆ K . Denote K A =  K × { 0 }  ∪  A × { 1 }  and consider this set with the order top ology giv en by the lexicographic order on [0 , 1] × { 0 , 1 } . If A = (0 , 1) and K = I , then the space K A is the w ell-kno wn double arro w space. Some authors use this name for the space I I ; others call the space I I the split in terv al. F or an y sets A, K as ab ov e, the space K A is a separable compact line of top ological w eigh t | A | (if the set A is infinite), and thus it is nonmetrizable whenev er the set A is uncoun table. If A is dense in K , then the space K A is zero-dimensional. It turns out that spaces of the form K A exhaust the class of separable compact lines: Theorem 2.2 (Ostaszewski, [31]) . The sp ac e L is a sep ar able c omp act line arly or der e d sp ac e if and only if L is or der isomorphic (henc e home omorphic) to the sp ac e K A for some close d set K ⊆ [0 , 1] and a subset A ⊆ K . Due to Ostaszewski’s result, in this pap er we ma y assume that any separable compact line is of the form K A . 6 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI 3. Counting isomorphism types of C ( K ) of a given weight The main result of this Section is Theorem 1.1, in whic h we count isomorphism types of Banac h spaces of contin uous functions on compact spaces of regular weigh t κ . Before w e pro ceed further, let us introduce some notation. F or an uncoun table cardinal κ , w e denote by E κ ω = { α < κ : cf ( α ) = ω } the set of ordinals b elo w κ of countable cofinalit y . F urthermore, if A, B are an y sets of ordinals, then A < B means that α < β for all α ∈ A , β ∈ B . Similarly , if α is an ordinal, then α < A ( A < α ) means that { α } < A ( A < { α } ). The following space, known, at least for κ = ω 1 , as the ladder system space, seems to b elong to the top ological folklore since the early 1970s. Perhaps its b est-known early app earances are in [9] and [33]. Definition 3.1. Consider an uncountable regular cardinal κ . Let L = { L α : α ∈ E κ ω } b e a system of ladders, i.e., for α ∈ E κ ω w e demand • L α ⊆ { β + 1 : β < α } , • L α ∩ β is finite for all β < α , • sup( L α ) = α . F or a subset S of E κ ω , we define the ladder system space X S ( L ) = X S on κ : • The p oin ts of X S are elements of κ . • The p oin ts of κ \ S are isolated in X S . • The basic neighborho o ds of α ∈ S are declared as all sets of the form ( L α \ F ) ∪ { α } , where F ⊆ L α is finite. The following should b e clear: Prop osition 3.2. Supp ose that κ is an unc ountable r e gular c ar dinal. Any ladder system sp ac e on κ is a lo c al ly c omp act, sc atter e d sp ac e of weight κ and height 2 . The one-point compactification of X S will be denoted K L S . It is quite easy to pro ve that the space C 0 ( X S ) is isomorphic to C ( K L S ). Lemma 3.3. Supp ose that κ is an unc ountable r e gular c ar dinal and X S is a ladder system sp ac e on κ for some S ⊆ E κ ω . Then the Banach sp ac es C 0 ( X S ) and C ( K L S ) ar e isomorphic. COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 7 Pr o of. By the Stone-W eierstrass theorem, C 0 ( X S ) is a h yp erplane of C ( K L S ). The space K L S is scattered and infinite, so it admits a non trivial con vergen t sequence. The rest follo ws from F act 2.1. □ Lemma 3.4. Supp ose that κ is an unc ountable r e gular c ar dinal and X R , X S ar e two ladder system sp ac es on κ for some sets R, S ⊆ E κ ω . If T : C 0 ( X R ) → C 0 ( X S ) is a b ounde d line ar op er ator, then for every γ ∈ κ we have |{ α ∈ X S : T ∗ ( δ α )( { γ } )  = 0 }| < κ. Pr o of. Assume tow ards a con tradiction that there is γ ∈ X R suc h that cardinality the set A = { α ∈ X S : T ∗ ( δ α )( { γ } )  = 0 } equals κ . By passing to a subset of A of cardinality κ , we may assume that there is ε > 0 such that | T ∗ ( δ α )( { γ } ) | > ε for all α ∈ A . If γ ∈ X R is isolated in X R , then 1 { γ } is contin uous and then | T (1 { γ } )( α ) | = | T ∗ ( δ α )( { γ } ) | > ε for all α ∈ A , which is imp ossible since A is noncompact as a subset of κ of cardinality κ , but con tin uous functions in C 0 ( X S ) must v anish at infinit y . Otherwise, γ is not isolated in X R . Let L γ b e the ladder at γ . By passing to a subset of A of cardinalit y κ , using the regularit y of κ and the regularity of the measure T ∗ ( δ α ), w e ma y assume that there is a cofinite subset L of L γ suc h that | T ∗ ( δ α ) | ( L ) < ε/ 2 for ev ery α ∈ A . Now 1 L ∪{ γ } is contin uous on X R and | T (1 L ∪{ γ } )( α ) | = | T ∗ ( δ α )( L ∪ γ ) | > ε − ε/ 2 = ε/ 2 for all α ∈ A , whic h is again imp ossible, as contin uous functions in C 0 ( X S ) m ust v anish at infinity . □ The following lemma will b e the main to ol for proving Theorem 1.1. Lemma 3.5. Supp ose that κ is an unc ountable r e gular c ar dinal and that R, S ⊆ E κ ω ar e two sets such that S \ R is stationary in κ . Then ther e is no b ounde d line ar op er ator fr om C 0 ( X R ) into C 0 ( X S ) with a dense r ange. Conse quently, ther e is no b ounde d line ar op er ator C ( K L R ) into C ( K L S ) with a dense r ange. 8 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI Pr o of. Supp ose that T : C 0 ( X R ) → C 0 ( X S ) is a b ounded linear op erator. W e will show that it do es not hav e a dense range. Consider the adjoint op erator T ∗ : C 0 ( X S ) ∗ → C 0 ( X R ) ∗ . Let F , G : κ → κ b e nondecreasing functions suc h that supp ( T ∗ ( δ α )) ⊆ F ( α ) , [ { supp ( T ∗ ( δ β )) : G ( α ) < β < κ } ∩ α = ∅ for ev ery α < κ . The existence of F follows from the fact that the supp orts of Radon measures on αX R that v anish on the one p oin t added in the compactification are coun t- able and suc h sets m ust b e b ounded in κ since it is an uncountable regular cardinal. The pro of of the existence of G uses the fact that T is b ounded: if the v alue G ( α ) could not b e found, there would exist A ⊆ κ of cardinality κ and a fixed γ < α suc h that T ∗ ( δ β )( { γ } )  = 0 for all β ∈ A , which con tradicts Lemma 3.4. Using the standard closure argumen t one can easily verify that, for an y function H : κ → κ , the set { α < κ : H [ α ] ⊆ α } is a club set in κ . Since the in tersection of t w o club sets in κ is a club set in κ we obtain a club set C ⊆ κ whic h is closed under F and G , that is, for every α ∈ C we ha v e F [ α ] , G [ α ] ⊆ α . Fix α ∈ C ∩ ( S \ R ). By recursion, construct a strictly increasing sequence β ′ n ∈ L α suc h that F ( β ′ n ) , G ( β ′ n ) < β ′ n +1 . This can b e done since C is closed under F and G , and L α is cofinal in α . The definitions of F and G imply that supp ( T ∗ ( δ β ′ n +1 )) ⊆ [ β ′ n , β ′ n +2 ) for every n ∈ ω . No w define β n = β ′ 2 n and note that the supp orts of T ∗ ( δ β n ) satisfy supp ( T ∗ ( δ β n )) < β ′ 2 n +1 < supp ( T ∗ ( δ β n +1 )) < α . W e can see that ( β n ) n ∈ ω con v erges to α in X S as it is a strictly increasing sequence included in L α . Hence ( δ β n ) n ∈ ω w eakly ∗ con v erges to δ α in C 0 ( X S ) ∗ . The weak ∗ con tin uity of T ∗ yields the w eak ∗ con v ergence of ( T ∗ ( δ β n )) n ∈ ω to T ∗ ( δ α ) in C 0 ( X R ) ∗ . W e will show that in fact ( T ∗ ( δ β n )) n ∈ ω w eakly ∗ con v erges to 0 in C 0 ( X R ), which will imply T ∗ ( δ α ) = 0 showing that T ∗ is not injective and hence T cannot hav e a dense range. Indeed, when f ∈ C 0 ( X R ), then for ev ery ε > 0 there is β < α suc h that all v alues of f | [ β , α ) are in the in terv al ( − ε, ε ). Otherwise, there would be a strictly increasing and cofinal sequence ( α n ) n ∈ ω of p oin ts in α satisfying | f ( α n ) | ≥ ε . As the p oin t α is isolated COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 9 in X R and the basic neigh b orho od of other points in X R ha v e b ounded in tersections with α , the sequence ( α n ) n ∈ ω is divergen t to infinity , which is a con tradiction. It follows from the ab o ve and the fact that T ∗ ( δ β n ) form a b ounded sequence of mea- sures that the sequence T ∗ ( δ β n )( f ) conv erges to 0 for each f ∈ C 0 ( X R ). Thus, w e ha v e that the sequence ( T ∗ ( δ β n )) n ∈ ω w eakly ∗ con v erges to 0 in C 0 ( X S ) ∗ and T ∗ ( δ α ) = 0 as needed to conclude that T do es not ha v e dense range. □ W e can now pro ceed to the pro of of Theorem 1.1. Pr o of of The or em 1.1. Fix κ as in the statemen t. By a theorem of Solov a y (Theorem 8.10 of [14] also cf. Lemma 8.8 of [14]) there is a pairwise disjoin t family { S ξ : ξ < κ } of stationary subsets of E κ ω . F or every A ⊆ κ consider S ( A ) = [ ξ ∈ A S ξ . Note that S ( A ) △ S ( B ) is stationary for an y tw o distinct A, B ⊆ κ . So, b y Lemmas 3.5 and 3.3 the family { C ( K L S ( A ) ) : A ⊆ κ } is the desired family of Banach spaces. □ 4. Topological proper ties of sep arable comp a ct lines In this section we use our conv ention that we identify separable compact lines with the spaces of the form K A , where K ⊆ [0 , 1] and A ⊆ K , see subsection 2.4. First, w e will show that for an y cardinal num b er κ satisfying ω ≤ κ ≤ 2 ω , there are 2 κ pairwise nonhomeomorphic separable compact lines K A of weigh t κ (recall that, for infinite A , the weigh t of K A is equal to | A | ). W e start with the follo wing observ ation of v an Douw en [10, Section 10]. Prop osition 4.1. L et K A , L B b e sep ar able c omp act lines that ar e home omorphic. Then ther e exist c ountable sets C ⊆ K and D ⊆ L such that K \ ( A ∪ C ) and L \ ( B ∪ D ) ar e home omorphic. The ab o ve prop osition together with a standard fact that each subset of I can b e homeomorphic to at most contin uum man y subsets of I , and some routine cardinal calculations, easily implies that for a fixed separable compact line K A w e ha v e at most 2 ω man y separable compact lines L B homeomorphic to K A (cf. [10, Section 10], [23, Section 2]). This immediately allows us to obtain the follo wing v ariation of v an Douw en’s result that there are 2 2 ω separable compact lines. Corollary 4.2. If a c ar dinal numb er κ ≤ 2 ω satisfies 2 κ > 2 ω , then ther e ar e 2 κ p airwise nonhome omorphic sep ar able c omp act lines K A of weight κ . 10 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI W e will show that the ab ov e result also holds true when 2 κ = 2 ω . Theorem 4.3. F or any infinite c ar dinal numb er κ ≤ 2 ω , ther e ar e 2 κ p airwise nonhome- omorphic sep ar able c omp act lines K A of weight κ . Pr o of. By Corollary 4.2 it is enough to consider the case when 2 κ = 2 ω , and construct a family K of size 2 ω consisting of pairwise nonhomeomorphic spaces K A of weigh t κ . By a classical result of Mazurkiewicz and Sierpi ´ nski [24] there is a family L of size 2 ω consisting of pairwise nonhomeomorphic compact subsets of the interv al [0 , 1 / 3]. If κ = ω , then w e are done. Otherwise, we tak e a subset A of (2 / 3 , 1) suc h that | A ∩ P | = κ , for an y nonempty op en subin terv al P of (2 / 3 , 1), and w e put K = { L ∪ [2 / 3 , 1] A : L ∈ L} . The family K has the required prop erties, since each p oin t of [2 / 3 , 1] A has only nonmetriz- able neigh b orho ods, hence L ∪ [2 / 3 , 1] A and L ′ ∪ [2 / 3 , 1] A ∈ K are homeomorphic if and only if the spaces L and L ′ are homeomorphic. □ Prop osition 4.4. Each sep ar able c omp act line is her e ditarily sep ar able, her e ditarily Lin- del¨ of, and F r ´ echet. Pr o of. Every separable compact line has to b e F r ´ ec het, since it is a first countable space. Eac h separable compact line is of the form K A , so it is a con tinuous image of the double arro w space I (0 , 1) and all these prop erties are preserv ed by con tinuous images. The double arrow space is hereditarily separable, hereditarily Lindel¨ of, as it is a union of tw o disjoin t copies of the Sorgenfrey line, whic h is suc h. □ Observ e that the ab o ve prop osition implies, in particular, that the class of separable compact lines is closed under taking closed subsets. F or an ordinal α , X ( α ) is the α th Can tor-Bendixson deriv ative of the space X . The follo wing result follo ws quite easily from Prop osition 4.4. Corollary 4.5. L et L b e a sep ar able c omp act line and α = min { β : L ( β +1) = L ( β ) } . Then α < ω 1 and | L \ L ( α ) | ≤ ω . If a separable compact line is p erfect (without isolated p oints), we can mo dify it b y a homeomorphism to a more con v enien t form. Prop osition 4.6. L et L b e a sep ar able c omp act line. If L has no isolate d p oints, then ther e exists B ⊆ (0 , 1) such that L is or der isomorphic (henc e home omorphic) to I B . Pr o of. Without loss of generalit y , we can assume that L = K A , where A ⊆ K ⊆ I , and K is a closed set suc h that inf K = 0 and sup K = 1. COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 11 En umerate the family of all connectedness comp onents of I \ K as { ( a n , b n ) : n ∈ N } , where N = ω or N ∈ ω . Observ e that, for eac h n ∈ N , the set { a n , b n } is disjoin t from A , since K A has no isolated p oin ts. No w consider an equiv alence relation on K defined by a n ∼ b n for all n ∈ N and let q : K → K/ ∼ b e the quotient map. Notice that the relation ∼ is a closed relation (see [11]), as every increasing or decreasing sequence of a n ’s will hav e the same limit as a sequence of b n ’s with the same indicies. One can easily c heck that the resulting quotient space K / ∼ is a linearly ordered connected metrizable compact space, hence it is order isomorphic to the unit interv al (cf. [11, 6.3.2]). Denote by h : K / ∼ → I the order isomorphism satisfying h ( q (0)) = 0. One can v erify that K A can b e identified with I B where B = h [ q [ A ∪ { a n : n ∈ N } ]]. □ Prop osition 4.7. L et L b e a sep ar able zer o-dimensional c omp act line. L et C ⊆ I b e the ternary Cantor set. If L has no isolate d p oints, then ther e exists B ⊂ C such that L is or der isomorphic (henc e home omorphic) to C B . Pr o of. By Prop osition 4.6, we can assume that L = I A for some A ⊆ I . As L is zero- dimensional, A is dense in I . W e can find a copy of the dyadic num bers Q = { a s : s ∈ 2 <ω } dense in A , so that a s < a s ′ (in A ) if and only if s < s ′ (in 2 <ω with the lexicographical order). Let { ( q 0 s , q 1 s ) ⊆ I : s ∈ 2 <ω } b e the usual, order preserving en umeration of the interv als remov ed during the construction of the Cantor set C ⊆ I . Define a function h : Q × { 0 , 1 } → C b y h ( a s , b ) = q b s for s ∈ 2 <ω , b ∈ { 0 , 1 } . Since h is order-preserving (with resp ect to the lexicographic order on Q × { 0 , 1 } ), Q is dense in I , and { q 0 s , q 1 s : s ∈ 2 <ω } is dense in C , we can extend h to an order isomorphism e h : I A → C B , where B × { 1 } = e h  ( A \ Q  × { 1 } ]. □ 5. General proper ties of sp a ces C ( K A ) F or a closed subset F of a compact space K , w e denote by C 0 ( K ∥ F ) the subspace { f ∈ C ( K ) : f | F ≡ 0 } of C ( K ). Prop osition 5.1. L et F b e a close d subset of a F r´ echet c omp act sp ac e K , such that K \ F is infinite. Then C 0 ( K ∥ F ) ≃ C ( K/F ) . Pr o of. The space C 0 ( K ∥ F ) is isomorphic to a h yp erplane of C ( K /F ). F rom our hypoth- esis, it follo ws that K /F contains a nontrivial conv ergen t sequence, th us the statement follo ws from F act 2.1. □ 12 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI Corollary 5.2. L et F b e a close d subset of a sep ar able c omp act line K . Then ther e is a c omp act sp ac e L such that C 0 ( K ∥ F ) ≃ C ( L ) . Pr o of. If K \ F is infinite, then we can apply the ab o ve prop osition. Otherwise, the space C 0 ( K ∥ F ) is finite-dimensional, and we can take as L a discrete finite space of appropriate size. □ The next result follows from a theorem of Heath and Lutzer [13, Theorem 2.4] which pro vides the existence of an extension op erator T : C ( L ) → C ( K ) for compact lines L ⊆ K (see [23, Lemma 4.3] for a self-con tained pro of ). Lemma 5.3 (Marciszewski, [23, Lemma 4.3]) . L et K A b e a sep ar able c omp act line. F or e ach nonempty close d subset F of K A , the sp ac e C ( K A ) is isomorphic to C ( F ) ⊕ C 0 ( K A ∥ F ) . The follo wing result is a more general version of [23, Lemma 4.6]. Note that it is true in ZF C, without an y additional set-theoretic h yp othesis. The proof of this lemma follo ws closely the argumen t in [23], we presen t it here for the conv enience of readers. Lemma 5.4. F or every unc ountable sep ar able c omp act line K A and every nonempty metrizable c omp act sp ac e M the sp ac es C ( K A ) and C ( K A ) ⊕ C ( M ) ar e isomorphic. Pr o of. First, let us sho w that C (2 ω ) | C ( K A ). As K is an uncountable closed subset of the unit in terv al, w e can find a copy of the Can tor set C ⊆ K . Put B = C ∩ A and see that C B is a closed subspace of K A . By Lemma 5.3, w e hav e C ( C B ) | C ( K A ). Now, it is sufficien t to sho w that C (2 ω ) | C ( C B ). W rite p 1 : C B → C for the pro jection. Fix a homeomorphism h : C → C 1 × C 2 , where b oth C 1 , C 2 are copies of 2 ω and b y π 1 , π 2 denote the pro jections from C 1 × C 2 on to C 1 , C 2 , resp ectiv ely . Our goal is to construct a regular a veraging op erator for the map φ = π 1 ◦ h ◦ p 1 : C B → C 1 , i.e., a p ositive linear op erator T : C ( C B ) → C ( C 1 ) satisfying T (1 C B ) = 1 C 1 and T ( g ◦ φ ) = g for ev ery g ∈ C ( C 1 ). The existence of suc h an op erator implies that C ( C 1 ) | C ( L ), see [32, Prop osition 8.2]. The set C 2 is a copy of 2 ω , so there is a standard pro duct measure on C 2 , denote it b y µ . Define the op erator T : C ( C B ) → C ( C 1 ) by the formula T ( f )( x ) = Z C 2 f ( h − 1 ( x, y ) , 0) d µ ( y ) COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 13 for f ∈ C ( C B ) and x ∈ C 1 . It is a well-kno wn fact that, for every function f ∈ C ( C B ), the function f 0 ( t ) = f ( t, 0) has at most countably man y p oin ts of discontin uity (see e.g. [23, Pro of of Thm. 3.7]) and is therefore measurable. It follows that T ( f )( x ) is w ell defined for ev ery x ∈ C 1 , it is also clear that T is p ositiv e and T (1 C B ) = 1 C 1 . Additionally , for every g ∈ C ( C 1 ), x ∈ C 1 and y ∈ C 2 w e hav e g ◦ φ ( h − 1 ( x, y ) , 0) = g ( x ), hence T ( g ◦ φ ) = g . It remains to chec k that T ( f ) is contin uous on C 1 for every f ∈ C ( C B ). Consider an y conv ergent sequence x n → x in C 1 . Then for every y ∈ C 2 w e hav e h − 1 ( x n , y ) → h − 1 ( x, y ), as h is a homeomorphism. Consider again the function f 0 ( t ) = f ( t, 0) and let P b e the set of points of discon tinuit y of f 0 ( P is coun table). Denote f n ( y ) = f 0 ( h − 1 ( x n , y )) and e f ( y ) = f 0 ( h − 1 ( x, y )). If R = π 2 ◦ h ( P ), then for every p oin t y ∈ C 2 \ R w e hav e f n ( y ) → e f ( y ), thus the sequence ( f n ) n ∈ ω is µ -a.e. con vergen t to e f on C 2 (note that µ ( R ) = 0, as R is countable and the measure µ is nonatomic). As all functions | f n | are b ounded b y the norm of f , from the Leb esgue dominated conv ergence theorem it follows that T ( f )( x n ) → T ( f )( x ). No w consider an y nonempt y metrizable compact space M . It is a w ell-known fact that C ( M ) ⊕ C (2 ω ) ≃ C (2 ω ) (e.g. by Miljutin’s theorem). Since C (2 ω ) is a factor of C ( K A ), it follows that C ( M ) ⊕ C ( K A ) ≃ C ( K A ), as C ( M ) ⊕ C ( K A ) ≃ C ( M ) ⊕ C (2 ω ) ⊕ E ≃ C (2 ω ) ⊕ E ≃ C ( K A ) , for E suc h that C ( K A ) ≃ C (2 ω ) ⊕ E . □ Corollary 5.5. L et L b e an unc ountable sep ar able c omp act line and α = min { β : L ( β +1) = L ( β ) } . Then the sp ac es C ( L ) and C ( L ( α ) ) ar e isomorphic. Pr o of. By Corollary 4.5 w e hav e | L \ L ( α ) | ≤ ω . Let K b e a compact space given by Corollary 5.2 such that C 0 ( L ∥ L ( α ) ) ≃ C ( K ). Then either K = L/L ( α ) or K is finite, in b oth cases the space K is metrizable since L \ L ( α ) is coun table. Therefore, we can use Lemmas 5.3 and 5.4 to conclude that C ( L ) ≃ C ( L ( α ) ) ⊕ C 0 ( L ∥ L ( α ) ) ≃ C ( L ( α ) ) ⊕ C ( K ) ≃ C ( L ( α ) ) . □ The next theorem follows immediately from the corollary abov e and Proposition 4.6. Theorem 5.6. L et L b e an unc ountable sep ar able c omp act line. Then ther e exists A ⊆ (0 , 1) such that C ( L ) ≃ C ( I A ) . 14 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI 6. Counting sp aces C ( K A ) of weight κ when 2 κ > 2 ω It is a w ell-kno wn fact that, consistently (e.g. under CH ), 2 ω 1 can b e greater than 2 ω . The purp ose of this Section is the pro of of Theorem 1.2, whic h we will derive from a result obtained in [6]. First, we count the n umber of p oten tial isomorphisms b et w een spaces C ( K A ). Let us recall the follo wing, v ery general theorem due to Cab ello-S´ anchez, Castillo, Marciszewski, Plebanek and Salguero-Alarc´ on. Theorem 6.1. [6, Theorem 5.1] L et K b e a family of c omp act sp ac es such that • K is sep ar able and | M ( K ) | = 2 ω for every K ∈ K ; • F or every p air of distinct K , L ∈ K one has C ( K ) ≃ C ( L ) and K , L ar e not home omorphic. Then K is of c ar dinality at most 2 ω . Pr o of of The or em 1.2. It is a kno wn fact that for any separable compact line K A w e ha v e | M ( K A ) | = 2 ω , see [18]. Due to Theorem 4.3, there exists a family K of size 2 κ consisting of pairwise nonhome- omorphic separable compact lines K A of weigh t κ . F or a fixed K A ∈ K , by Theorem 6.1, w e hav e at most 2 ω spaces L B ∈ K suc h that C ( K A ) ≃ C ( L B ). No w, the statemen t of Theorem 1.2 easily follows from the h yp othesis that 2 κ > 2 ω . □ 7. Counting sp aces C ( K A ) under Baumgar tner ’s Axiom In this Section we will pro v e Theorem 1.3, i.e., we will show that consistently w e can ha v e a situation completely opp osite to the one describ ed by Theorem 1.2. The pro of is presen ted at the end of this section, after all the necessary lemmas. Note that by Theorem 4.3 there exist (in ZFC ) 2 ω 1 pairwise nonhomeomorphic sepa- rable compact lines of w eigh t ω 1 . Classical Cantor’s isomorphism theorem states that all countable dense subsets of R are order isomorphic. The same cannot b e said ab out uncoun table subsets, but, after adding a very natural condition, a similar statement can b e prov ed to b e consistent with ZF C . F or formulating this statement w e need the following definition. Definition 7.1. Let κ b e an infinite cardinal. W e say that a subset A of a topological space X is κ -dense in X , if for an y nonempt y op en subset U of X we hav e | U ∩ A | = κ . W e will mainly use this definition in the case when X is a subspace of the real line R , so w e will consider only cardinals κ ≤ 2 ω . Obviously , a subset A of R is κ -dense in R if, COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 15 for every nonempty op en interv al I ⊆ R , we ha v e | I ∩ A | = κ . W e refer to the following statemen t ab out κ -dense sets in R as Baumgartner’s Axiom. BA ( κ ): Every t w o sets of reals that are κ -dense in R are order isomorphic. This axiom was in tro duced as a natural consequence of a classical theorem of Baum- gartner [3], who prov ed that consistently all ω 1 -dense sets of reals are order isomorphic, i.e., using the abov e language he show ed the following (later on, we will denote BA ( ω 1 ) simply as BA ). Theorem 7.2 (Baumgartner) . BA ( ω 1 ) is consisten t with ZF C . On the other hand, it should b e clear that BA ( c ) is false, so consisten tly BA ( ω 1 ) is false (e.g. under CH ). T o dorˇ cevi´ c in [37] even show ed that BA ( b ) is also false. Theorem 7.2 can b e prov en by forcing (as in the original pro of ), or using PF A as in [4, Theorem 6.9, p. 945]. The consistency of BA ( ω 2 ) is an imp ortan t op en problem in set theory , see [28]. Using Dedekind cuts, one can easily obtain the following lemma. Lemma 7.3. Every or der isomorphism f : A → B b etwe en ω 1 -dense sets A, B ⊆ R extends to an or der isomorphism e f : R → R . W e can easily transfer such results from the real line to the unit interv al (0 , 1), since these t wo spaces are order isomorphic. Therefore, we can use Lemma 7.3 to pro v e some structural results ab out separable compact lines. Corollary 7.4. If ω 1 -dense sets A, B ⊆ (0 , 1) ar e or der isomorphic, then the sp ac es I A , I B ar e home omorphic. Pr o of. Supp ose that there exists an order isomorphism f : A → B . By Lemma 7.3 and the remark following it, we extend it to an order isomorphism e f : (0 , 1) → (0 , 1), which in turn, can b e extended in an obvious w a y to an order isomorphism ˆ f : [0 , 1] → [0 , 1]. Consider the map φ : I A → I B defined b y the formula φ ( x, i ) = ( ˆ f ( x ) , i ) for ( x, i ) ∈ I A . Clearly , φ is an order isomorphism b et w een I A and I B , equipp ed with the lexicographic orders, hence it is a homeomorphism. □ As a consequence, it should b e clear that under BA w e obtain the following. Corollary 7.5 ( BA ) . If sets A, B ⊆ (0 , 1) ar e ω 1 -dense, then the sp ac es I A and I B ar e home omorphic. 16 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI W e need one more lemma which do es not require an y additional set-theoretic axioms. Lemma 7.6. F or any sep ar able c omp act line K of weight ω 1 , the sp ac e C ( K ) is isomor- phic to a sp ac e C ( I B ) for some ω 1 -dense set B ⊆ (0 , 1) . Pr o of. Using Theorem 5.6 we can assume that K = I A , where A is a subset of (0 , 1) of size ω 1 . Let U = [ { ( a, b ) ⊆ (0 , 1) : a, b ∈ Q and | ( a, b ) ∩ A | ≤ ω } . Clearly , | U ∩ A | ≤ ω and A \ U is ω 1 -dense in a compact set M = I \ U . W e will consider a separable compact line L = M A \ U . First, note that L is a closed subset of K , hence, b y Lemma 5.3 we ha v e C ( K ) ≃ C ( L ) ⊕ C 0 ( K ∥ L ) . (1) Next, let us verify that the space C 0 ( K ∥ L ) = C 0 ( I A ∥ M A \ U ) is separable. The space C ( I A ∩ U ) is separable, since A ∩ U is countable; therefore its subspace C 0 ( I A ∩ U ∥ M × { 0 } ) is also separable. Let φ : I A → I A ∩ U b e a natural contin uous surjection defined b y the form ula: φ (( x, i )) =    ( x, 0) for ( x, i ) ∈ ( A \ U ) × { 1 } , ( x, i ) for ( x, i ) ∈ I A \ [( A \ U ) × { 1 } ] . One can easily chec k that the isometric embedding T φ : C ( I A ∩ U ) → C ( I A ) generated by φ , i.e., an op erator sending each f ∈ C ( I A ∩ U ) to the comp osition f ◦ φ , maps the space C 0 ( I A ∩ U ∥ M × { 0 } ) onto C 0 ( I A ∥ M A \ U ), hence the latter space is also separable. By Corollary 5.2 the space C 0 ( K ∥ L ) = C 0 ( I A ∥ M A \ U ) is isomorphic to the space C ( N ) for some compact space N . Since C 0 ( K ∥ L ) is separable, N m ust b e metrizable, so we can use Lemma 5.4 and prop ert y (1) to conclude that C ( K ) and C ( L ) are isomorphic. Let α = min { β : L ( β +1) = L ( β ) } . By Corollary 4.5 w e ha v e | L \ L ( α ) | ≤ ω . The space L ( α ) has no isolated p oin ts, therefore, b y Prop osition 4.6 it is order isomorphic (hence homeomorphic) to a space I B for some B ⊆ (0 , 1). The set A \ U is ω 1 -dense in M , hence eac h nonempty op en interv al in L = M A \ U (an interv al with resp ect to lexicographic order in L ) contains ω 1 man y gaps, i.e., empty in terv als of the form  ( a, 0) , ( a, 1)  , where a ∈ A \ U . Since | L \ L ( α ) | ≤ ω , the spaces L ( α ) and I B ha v e the same prop ert y , that is, every nonempty order op en interv al contains ω 1 gaps. This means that the set B is ω 1 -dense in I . By Corollary 5.5 the spaces C ( L ) and C ( L ( α ) ) are isomorphic, hence w e obtain the desired conclusion C ( K ) ≃ C ( L ) ≃ C ( L ( α ) ) ≃ C ( I B ) . □ COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 17 W e can now complete the pro of of the main result of this section. Pr o of of The or em 1.3. The theorem follo ws immediately from Corollary 7.5 and Lemma 7.6. □ Corollary 1.4 easily follows from Theorem 1.3: Pr o of of Cor ol lary 1.4. Assume BA , and let K b e an arbitrary separable compact line of w eigh t ω 1 . The direct sum C ( K ) ⊕ C ( K ) can b e identified with the space C ( K ⊔ K ), where K ⊔ K is a disjoint union of tw o copies of the space K . Since K ⊔ K is homeomorphic to a separable compact line of w eight ω 1 , Theorem 1.3 implies that C ( K ) and C ( K ) ⊕ C ( K ) are isomorphic. □ 8. Function sp aces on finite pr oducts of sep arable comp a ct lines In this section, using Theorem 1.3, the results of Michalak from [25], and assuming Baumgartner’s Axiom, w e consistently obtain a complete isomorphic classification of spaces C ( K ), where K is a nonmetrizable finite pro duct of separable compact lines of w eigh t ≤ ω 1 . F or the entiret y of this Section, fix some set A ⊆ (0 , 1) of cardinality ω 1 . Define a family of compact spaces K A = { ( I A ) n : n ≥ 1 } ∪ { ( I A ) n × [0 , 1] : n ≥ 1 } ∪ { ( I A ) n × ( ω ω α + 1) : n ≥ 1 , ω 1 > α > 0 } . The goal of this Section is to pro v e that the class K A (regardless of whic h set A ⊆ (0 , 1) of cardinality ω 1 is chosen) exhausts all isomorphism types of C ( K ), where K is an y nonmetrizable finite pro duct of separable compact lines of w eigh t ≤ ω 1 . First, let us note that the spaces C ( K ) for K ∈ K A are pairwise nonisomorphic due to the following result of Mic halak. Theorem 8.1 ([25, Corollary 5.3]) . L et n, k ∈ N . L et K 1 , . . . , K n , L 1 , . . . , L k b e non- metrizable sep ar able c omp act lines. (a) If M 1 , M 2 ar e infinite c omp act metric sp ac es such that the sp ac es C ( Q n i =1 K i × M 1 ) and C ( Q k i =1 L i × M 2 ) ar e isomorphic, then n = k and the sp ac es C ( M 1 ) and C ( M 2 ) ar e isomorphic. (b) If M is a c omp act metric sp ac e such that the sp ac es C ( Q n i =1 K i × M ) and C ( Q k i =1 L i ) ar e isomorphic, then n = k and the sp ac e C ( M ) is finite-dimensional or isomorphic to c 0 . 18 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI Let us recall some notation and simple facts regarding the spaces of vector-v alued functions. F or a compact space K and a Banac h space ( X , ∥ · ∥ X ), b y C ( K , X ) w e denote the space of contin uous functions f : K → X with the suprem um norm ∥ f ∥ = sup k ∈ K ∥ f ( k ) ∥ X . F act 8.2 ([35, Prop osition 7.7.5]) . If K, L ar e c omp act sp ac es, then C ( K × L ) ≃ C ( K , C ( L )) . The next fact is ob vious. F act 8.3. F or any c omp act sp ac e K and isomorphic Banach sp ac es X , Y , we have C ( K , X ) ≃ C ( K , Y ) . T o fully classify the spaces of contin uous functions on compact spaces of the family K A , we also need the following result. Lemma 8.4 ( BA ) . If a set A is ω 1 -dense in (0 , 1) , then the Banach sp ac es C ( I A ) and C ( I A , c 0 ) ar e isomorphic. Pr o of. First, recall a simple observ ation that, for any compact space K , the space C ( K , c 0 ) is isometric to the c 0 -sum of ω copies of C ( K ). Indeed, c 0 is isometric to C 0 ( ω ), hence C ( K , c 0 ) is isometric to C ( K , C 0 ( ω )), which in turn is isometric to C 0 ( K × ω ), see [35, Prop osition 7.7.6]. Clearly , this latter space is isometric to the c 0 -sum of ω copies of C ( K ). Let a 0 = 1 and ( a n ) n ≥ 1 b e a decreasing sequence in A which con v erges to 0. Put K n = [( a n +1 , 1) , ( a n , 0)] ⊆ I A for n ∈ ω (an interv al with resp ect to lexicographic order in I A ). F or ev ery n ∈ ω , the set A ∩ K n is ω 1 -dense in ( a n +1 , a n ), so by Corollary 7.5, C ( K n ) is isometric to C ( I A ). As I A is F r ´ ec het, due to F act 2.1, w e hav e C ( I A ) ≃ C 0 ( I A ∥{ (0 , 0) } ). What remains is a standard observ ation that C 0 ( I A ∥{ (0 , 0) } ) is isometric to the c 0 -sum of spaces C ( K n ) , n ∈ ω . □ No w we ha ve all the tools to prov e the main result of this Section. Theorem 8.5 ( BA ) . L et K b e a nonmetrizable finite pr o duct of sep ar able c omp act lines of weight ≤ ω 1 . Then the sp ac e C ( K ) is isomorphic to exactly one sp ac e C ( L ) for L ∈ K A . COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 19 Pr o of. By Theorem 8.1, for t w o distinct spaces K 1 , K 2 ∈ K A , we hav e C ( K 1 ) ≃ C ( K 2 ). Th us, it is enough to pro v e that, for any m ≥ 1, and any separable compact lines K 1 , . . . , K m of weigh t ≤ ω 1 , if at least one of them is nonmetrizable, then there is some L ∈ K A suc h that C ( Q m i =1 K i ) ≃ C ( L ). Without loss of generalit y , assume that for some 1 ≤ n ≤ m , we hav e w ( K i ) = ω 1 ⇐ ⇒ i ≤ n . Then M = Q m i = n +1 K i is a metrizable compact space. By F acts 8.2, 8.3 w e ha v e C m Y i =1 K i ! ≃ C n Y i =1 K i , C m Y i = n +1 K i !! ≃ C n Y i =1 K i , C ( M ) ! . Using Theorem 1.3 we can also see that C n Y i =1 K i ! ≃ C n − 1 Y i =1 K i , C ( K n ) ! ≃ C n − 1 Y i =1 K i , C ( I A ) ! ≃ C I A × n − 1 Y i =1 K i ! , so by con tin uing this pro cess further w e obtain C ( Q n i =1 K i ) ≃ C (( I A ) n ). It follo ws that C m Y i =1 K i ! ≃ C  ( I A ) n × M  . By the results of Bessaga, Pe lczy ´ nski [5] and Miljutin [26] we kno w that C ( M ) is iso- morphic to C ( M ′ ), where M ′ ∈ {{ 0 , 1 , . . . , n − 1 } : n ∈ ω } ∪ { [0 , 1] } ∪ { ω ω α + 1 : 0 ≤ α < ω 1 } . By F acts 8.2 and 8.3 we ha ve C  ( I A ) n × M  ≃ C  ( I A ) n , C ( M )  ≃ C  ( I A ) n , C ( M ′ )  ≃ C  ( I A ) n × M ′  . Additionally , if C ( M ′ ) is finite-dimensional (i.e., M ′ is finite), then we ha v e C  ( I A ) n × M ′  ≃ C  ( I A ) n − 1 × I A × M ′  ≃ C  ( I A ) n − 1 , C ( I A × M ′ )  ≃ C  ( I A ) n − 1 , C ( I A )  ≃ C (( I A ) n ) due to Theorem 1.3 and the fact that the space I A × M ′ is homeomorphic to a separable compact line. If C ( M ′ ) is isomorphic to c 0 (i.e., M ′ = ω + 1), then C (( I A ) n × M ′ ) ≃ C (( I A ) n − 1 × I A × M ′ ) ≃ C (( I A ) n − 1 , C ( I A × M ′ )) ≃ C (( I A ) n − 1 , C ( I A , C ( M ′ ))) ≃ C (( I A ) n − 1 , C ( I A , c 0 )) ≃ C (( I A ) n − 1 , C ( I A )) ≃ C (( I A ) n ) 20 M. KORP ALSKI, P . KOSZMIDER, AND W. MARCISZEWSKI due to Lemma 8.4. □ 9. Final remarks 9.1. Compact spaces of w eigh t κ . Theorem 1.1 should b e compared with the corol- lary from the Miljutin, Bessaga and P e lczy ´ nski’s classification of separable Banac h spaces of the form C ( K ) which sa ys that there are only ω 1 m utually nonisomorphic such spaces regardless of the relation b et w een ω 1 and 2 ω (although the spaces ℓ p for 1 < p < ∞ form a family of cardinality 2 ω consisting of mutually nonisomorphic separable Banach spaces). Our metho d of pro ving Prop osition 1.1 w orks only for regular cardinals. In particular, we do not kno w the answ er to the following: Question 1. Supp ose that κ is a singular c ar dinal. Is it pr ovable without any additional hyp othesis that ther e ar e 2 κ mutual ly nonisomorphic Banach sp ac es of the form C ( K ) (of any form) of density κ ? Note that the answ er to the ab o ve questions is p ositive if we assume the Generalized Con tin uum Hyp othesis ( GCH ). Indeed, Kisljako v’s classification of spaces of the form C ([0 , α ]) for α < κ + ([19]) shows that without an y additional h yp othesis w e hav e κ + m utually nonisomorphic Banac h spaces of the form C ( K ) for an y uncoun table cardinal κ . As GCH means that 2 κ = κ + for every infinite cardinal κ , w e obtain the p ositive answ er to the ab o ve questions under GCH . Also note that w e cannot ha ve more than 2 κ m utually nonisometric (and so noniso- morphic) Banach spaces of the form C ( K ). This follo ws from the fact that all compact spaces of weigh t not exceeding κ embed homeomorphically into [0 , 1] κ and this space has at most 2 κ closed subsets as they all are complemen ts of some unions of elements of the basis. This also implies that there are at most 2 κ m utually nonisometric general Banach spaces of density κ . Indeed, each suc h space isometrically embeds into some space of the form C ( K ) of densit y κ and there are only 2 κ closed subspaces of such a C ( K ) as they are again complemen ts of unions of some op en balls around the p oin ts in the dense set of cardinality κ . It should also be noted that for ev ery uncoun table cardinal κ (including singular cardi- nals), there are 2 κ m utually nonhomeomorphic compact spaces of weigh t κ . This follows from the Stone duality and the existence of man y examples of families of cardinalities 2 κ of mutually nonisomorphic Bo olean algebras of cardinalit y κ presen ted in [27]. In particular, it is shown in Theorem 1.3 of [27] that for ev ery uncountable κ there are 2 κ m utually nonisomorphic interv al Bo olean algebras. This together with Theorem 15.7 COUNTING SP ACES OF FUNCTIONS ON SEP ARABLE COMP ACT LINES 21 from [20] implies that there are 2 κ m utually nonhomeomorphic linearly ordered compact spaces for eac h uncountable cardinal κ . Let us also men tion that the p ossibilit y of using families of pairwise disjoin t stationary subsets of ω 1 to build 2 ω 1 differen t structures of “sizes” ω 1 as in the pro of of Theorem 1.1 has b een well kno wn. F or example, this argumen t shows that there are man y non- homeomorphic 2-dimensional nonmetrizable manifolds (Examples 6.2 of [30]) or is used in [36] in the con text of con tin ua. 9.2. Separable compact lines. The whole pro of of Theorem 1.3 relies on Baumgart- ner’s Theorem (Theorem 7.2). In fact, we do not need its full form, but only a w eaker v ersion stating that for tw o ω 1 -dense subsets A, B of (0 , 1) spaces C ( I A ) and C ( I B ) are isomorphic. This statement ma y p erhaps b e consistently pro ven under weak er as- sumptions, for example, that every tw o ω 1 -dense subsets A, B of (0 , 1) there is an order em b edding of A to B . Many in teresting facts ab out this and similar axioms can be found in [1, Section 6]. The follo wing commen t is purely theoretical (as it is not kno wn whether ev en BA ( ω 2 ) is consisten t), but can still b e of some v alue (in particular, it indicates a sligh tly differen t w a y of proving Theorem 1.3). Under the ass umption of BA ( κ ), for κ of uncoun table cofinality , it is p ossible to prov e the analogue of Theorem 1.3 for separable compact lines of weigh t κ . The pro of, in v olving P e lczy ´ nski decomp osition, would use a counterpart of Lemma 8.4, for κ instead of ω 1 , and the follo wing tw o facts. Lemma 9.1 ( BA ( κ )) . F or every sep ar able c omp act line L of weight κ and every κ -dense set B ⊆ (0 , 1) the sp ac e C ( L ) c an b e emb e dde d in C ( I B ) as a c omplemente d subsp ac e. Pr o of. By rep eating the pro of of Lemma 7.6, the space C ( L ) is isomorphic to C ( L ′ ) ⊕ C ( N ) for a 0-dimensional perfect separable compact line L ′ of w eigh t κ and some metriz- able compact space N . But b y Lemma 5.4, we hav e C ( L ′ ) ≃ C ( L ′ ) ⊕ C ( N ) ≃ C ( L ) . By Proposition 4.7, the space L ′ is homeomorphic to C A for some set A ⊆ C of cardinality κ . Note that for κ > ω 1 the analogue of Corollary 7.4 also holds. The set ( B \ C ) ∪ A is still κ -dense in (0 , 1), so the spaces I B and I ( B \ C ) ∪ A are homeomorphic. It is standard to chec k that C A is a closed subset of I ( B \ C ) ∪ A , so b y Lemma 5.3 we ha v e C ( C A ) | C ( I ( B \ C ) ∪ A ) , 22 M. K ORP ALSKI, P . K OSZMIDER, AND W. MARCISZEWSKI whic h ends the pro of, as C ( L ) ≃ C ( L ′ ) ≃ C ( C A ) and C ( I ( B \ C ) ∪ A ) ≃ C ( I B ). □ Lemma 9.2 ( BA ( κ )) . If κ has unc ountable c ofinality then, for every sep ar able c omp act line L of weight κ and every κ -dense set B ⊆ (0 , 1) , the sp ac e C ( I B ) c an b e emb e dde d in C ( L ) as a c omplemente d subsp ac e. Pr o of. By Lemma 5.3 it is enough to sho w that L contains a topological cop y of I B . T o ac hiev e this we need again to use some ideas from the pro of of Lemma 7.6. As usual, w e can assume that L = I A , where A is a subset of (0 , 1) of size κ . Let U = [ { ( a, b ) ⊆ (0 , 1) : a, b ∈ Q and | ( a, b ) ∩ A | < κ } . Clearly , | U ∩ A | < κ (b y cf ( κ ) > ω ) and A \ U is κ -dense in a compact set M = I \ U . W e consider a separable compact line K = M A \ U ⊆ L . Let α = min { β : K ( β +1) = K ( β ) } . 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Instytut Ma tema tyczny, Uniwersytet Wroc la wski, pl. Grunw aldzki 2/4, 50-384 Wroc- la w, Poland Email addr ess : Maciej.Korpalski@math.uni.wroc.pl Institute of Ma thema tics of the Polish Academy of Sciences, ul. ´ Sniadeckich 8, 00- 656 W arsza w a, Poland Email addr ess : piotr.math@proton.me Institute of Ma thema tics, University of W arsa w, Banacha 2, 02-097 W arsza w a, Poland Email addr ess : W.Marciszewski@mimuw.edu.pl