A characterization of Erd\H{o}s space factors

TL;DR

This paper characterizes Erdős space factors among almost zero-dimensional spaces using Sierpiński stratifications of C-sets, and explores their stability properties with new examples and counterexamples.

Contribution

It provides a new characterization of Erdős space factors via Sierpiński stratifications and investigates their stability through novel examples, answering open questions.

Findings

Characterization of Erdős space factors using Sierpiński stratification of C-sets.

Existence of almost zero-dimensional $F_{\sigma\delta}$-spaces that are not Erdős space factors.

Erdős space $rak E$ is shown to be unstable with new examples.

Abstract

We prove that an almost zero-dimensional space $X$ is an Erd\H{o}s space factor if and only if $X$ has a Sierpi\'{n}ski stratification of C-sets. We apply this characterization to spaces which are countable unions of C-set Erd\H{o}s space factors. We show that the Erd\H{o}s space $\mathfrak E$ is unstable by giving strongly $\sigma$-complete and nowhere $\sigma$-complete examples of almost zero-dimensional $F_{\sigma\delta}$-spaces which are not Erd\H{o}s space factors. This answers a question by Dijkstra and van Mill.