A note on the structural stability of almost one-to-one maps

Abstract

A continuous surjection $\pi:X\to Y$ between compact Hausdorff spaces induces continuous surjections $\mathcal{M}(\pi)\colon \mathcal{M}(X)\to\mathcal{M}(Y)$ and $\mathcal{H}(\pi): \mathcal{H}(X)\to\mathcal{H}(Y)$ between the spaces of regular Borel probability measures, and the spaces of closed subsets, respetively. It is well known that $\mathcal{H}(\pi)$ is irreducible if and only if $\pi$ is irreducible. We show that $\mathcal{M}(\pi)$ is irreducible if and only if $\pi$ is irreducible. Furthermore, we show that whenever $\pi$ is almost one-to-one then $\mathcal{M}(\pi)$ and $\mathcal{H}(\pi)$ are almost one-to-one. In particular, we observe that continuous surjections between compact metric spaces are almost one-to-one if and only if $\mathcal{H}(\pi)$ is almost one-to-one and a similar statement about $\mathcal{M}(\pi)$. Finally, we give alternative proofs for some results in 'Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps' by Xiongping Dai and Yuxuan Xie regarding semi-open maps.