Zero-dimensional metrizable CDH space $X$ such that $X^2$ is not CDH

TL;DR

This paper constructs a specific zero-dimensional metrizable space that is countable dense homogeneous, yet its square has only continuum many countable dense subsets, addressing an open question in topology.

Contribution

It provides the first example of a zero-dimensional metrizable CDH space with a non-CDH square, using $eta$-sets and consistent co-analytic construction.

Findings

Constructed a zero-dimensional metrizable CDH space with a non-CDH square.

Showed the space can be constructed as a consistent co-analytic set.

Answered an open question by Medini regarding the properties of such spaces.

Abstract

In this paper a construction of a metrizable zero-dimensional CDH space $X$ such that $X^2$ has exactly $\mathfrak{c}$ countable dense subsets is provided. Furthermore, it is shown that the space can be constructed consistently co-analytic. Thus answering an open question asked by Medini. To do so we use the notion of $\lambda$-sets.