On the problem of compact totally disconnected reflection of nonmetrizability

Abstract

We construct a ZFC example of a nonmetrizable compact space $K$ such that every totally disconnected closed subspace $L\subseteq K$ 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 Holszty\'nski'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.