Products and countable dense homogeneity

TL;DR

This paper constructs a specific zero-dimensional separable metrizable space that is countable dense homogeneous but whose square is not, under Martin's Axiom, revealing nuanced properties of such spaces.

Contribution

It introduces a novel example of a space with unique homogeneity properties, extending previous work by Baldwin and Beaudoin under set-theoretic assumptions.

Findings

Constructed a space where X is countable dense homogeneous but X^2 is not

Showed such a space cannot be Borel under Martin's Axiom

Demonstrated the space can be made homogeneous and completely Baire

Abstract

Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hru\v{s}\'ak and Zamora Avil\'es that such a space $X$ cannot be Borel. Furthermore, $X$ can be made homogeneous and completely Baire as well.