Nonseparable C(K)-spaces can be twisted when K is a finite height compact

Abstract

We show that, given a nonmetrizable compact space K having ω-derived set empty, there always exist nontrivial exact sequences 0 → c 0 → E → C(K) → 0. This partially solves a problem posed in several papers: Is Ext(C(K), c 0 ) = 0 for K a nonmetrizable compact set?

Extension operators and twisted sums of c0 and C(K) spaces

“…Building on this simple idea we construct nontrivial twisted sums of c 0 and C(K) for spaces K from several classes of nonmetrizable compacta: dyadic spaces (Section 7), linearly ordered compact spaces (Section 8), scattered compacta of finite height (Section 9). In this way we extend results from Castillo [7] and Correa and Tausk [10] or present their alternative proofs. It turns out that the problem of constructing a countable discrete extension of a given space K without the corresponding extension operator is related to some problems of infinitary combinatorics; for instance, our Example 7.7 uses multiple gaps of Avilés and Todorcevic [2].…”

“…There are several classes of nonmetrizable compacta for which Problem 1.1 has an affirmative answer, cf. [7], [10]. We prove, however, that under Martin's axiom (MA) and the negation of the continuum hypothesis (CH) there are nonmetrizable compacta K with the property that every twisted sum of c 0 and C(K) is trivial, see Corollary 5.2 and 5.3.…”

“…It turns out that the problem of constructing a countable discrete extension of a given space K without the corresponding extension operator is related to some problems of infinitary combinatorics; for instance, our Example 7.7 uses multiple gaps of Avilés and Todorcevic [2]. Note also that our Theorem 9.7 shows that the main result of Castillo [7] (proved under CH) does require some additional set-theoretic assumptions.…”

“…Using Theorem 9.1 we can also provide an alternative, more topological proof of the following result of Castillo. Theorem 9.3 (Castillo [7]). Under CH, if K is a nonmetrizable scattered compact space of finite height, then there exists a nontrivial twisted sum of c 0 and C(K).…”

Local extension property for finite height spaces

“…Assume that X is locally compact Hausdorff and zerodimensional and let D be a dense subset of X . Given an antichain (v γ ) γ∈Γ in Clop D (X )/ Clop D c (X ), by the topology induced on X ∪Γ by this antichain we mean the topology induced by the family of filters (F γ ) γ∈Γ , where F γ is the filter whose basis is the image of v γ under the inverse of isomorphism (4).…”