Products and h-homogeneity

TL;DR

This paper investigates the properties of h-homogeneity in zero-dimensional spaces, demonstrating its productivity, existence of non-trivial products, and conditions under which infinite powers are h-homogeneous, thus advancing the understanding of zero-dimensional topology.

Contribution

It generalizes previous results on h-homogeneity, proves its productivity in zero-dimensional spaces, and relates key open questions to new partial answers.

Findings

h-homogeneity is productive in zero-dimensional spaces

Existence of non-empty zero-dimensional Y such that X×Y is h-homogeneous

Infinite powers of spaces with dense isolated points are h-homogeneous

Abstract

Building on work of Terada, we prove that h-homogeneity is productive in the class of zero-dimensional spaces. Then, by generalizing a result of Motorov, we show that for every non-empty zero-dimensional space $X$ there exists a non-empty zero-dimensional space $Y$ such that $X\times Y$ is h-homogeneous. Also, we simultaneously generalize results of Motorov and Terada by showing that if $X$ is a space such that the isolated points are dense then $X^\kappa$ is h-homogeneous for every infinite cardinal $\kappa$. Finally, we show that a question of Terada (whether $X^\omega$ is h-homogeneous for every zero-dimensional first-countable $X$) is equivalent to a question of Motorov (whether such an infinite power is always divisible by 2) and give some partial answers.