On the infinite powers of large zero-dimensional metrizable spaces

TL;DR

This paper proves that large powers of certain zero-dimensional metrizable spaces are strongly homogeneous, answering a question by Terada and improving previous results, with additional insights into partitioning such spaces into clopen sets.

Contribution

It demonstrates strong homogeneity of infinite powers of non-separable zero-dimensional metrizable spaces and advances understanding of their partition properties.

Findings

$X^$ is strongly homogeneous for non-separable zero-dimensional metrizable spaces.

Every non-compact weight-homogeneous metrizable space with a clopen c-base can be partitioned into c many clopen sets.

Improves previous results by van Engelen and partially answers Terada's question.

Abstract

We show that $X^\lambda$ is strongly homogeneous whenever $X$ is a non-separable zero-dimensional metrizable space and $\lambda$ is an infinite cardinal. This partially answers a question of Terada, and improves a previous result of the author. Along the way, we show that every non-compact weight-homogeneous metrizable space with a $\pi$-base consisting of clopen sets can be partitioned into $\kappa$ many clopen sets, where $\kappa$ is the weight of $X$. This improves a result of van Engelen.