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 $M\left(\pi \right):M\left(X\right)\to M\left(Y\right)$ and $H\left(\pi \right):H\left(X\right)\to H\left(Y\right)$ between the spaces of regular Borel probability measures, and the spaces of closed subsets, respectively. It is well known that $H\left(\pi \right)$ is irreducible if and only if π is irreducible. We show that $M\left(\pi \right)$ is irreducible if and only if π is irreducible. Furthermore, we show that whenever π is almost one-to-one then $M\left(\pi \right)$ and $H\left(\pi \right)$ 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 $H\left(\pi \right)$ is almost one-to-one and a similar statement about $M\left(\pi \right)$. Finally, we give alternative proofs for some results in [8] regarding semi-open maps.