On the problem of compact totally disconnected reflection of nonmetrizability

TL;DR

This paper constructs a nonmetrizable compact space with all totally disconnected closed subspaces metrizable, explores whether such spaces must have nonmetrizable totally disconnected images, and analyzes related set-theoretic conditions affecting their existence.

Contribution

It provides a ZFC example of a nonmetrizable compact space with only metrizable totally disconnected subspaces and investigates the set-theoretic conditions influencing the existence of nonmetrizable images.

Findings

Constructed a ZFC example of a nonmetrizable compact space with all totally disconnected subspaces metrizable.

Analyzed the link between the space's structure and the Banach space $C(K)$.

Showed that certain set-theoretic axioms prevent the existence of specific counterexamples.

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.