A characterization of Erdős space factors

Abstract

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