2015/09/17 by Piotr Koszmider, Koszmider, Piotr
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Logic (math.LO) #math.FA #math.GN #math.LO
paper · pdf · doi:10.48550/arxiv.1509.05282
arxiv created 2015/09/17 · openalex publication_date 2015/09/17 · arxiv updated 2015/09/18 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L⊆ K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.