Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps

TL;DR

This paper explores how properties of continuous surjective maps between compact Hausdorff spaces are characterized by the induced maps on measure and hyperspace structures, establishing equivalences and implications for openness and semi-openness.

Contribution

It provides new characterizations of open and semi-open maps via their induced maps on measure and hyperspaces, linking topological properties across different spaces.

Findings

f_* semi-open implies f semi-open

f semi-open densely open implies f_* semi-open densely open

f is open iff 2^f is open

Abstract

Let $f\colon X\rightarrow Y$ be a continuous surjection of compact Hausdorff spaces. By $$f_*\colon\mathfrak{M}(X)\rightarrow\mathfrak{M}(Y),\ \mu\mapsto \mu\circ f^{-1} \quad{\rm and}\quad 2^f\colon2^X\rightarrow2^Y,\ A\mapsto f[A]$$ we denote the induced continuous surjections on the probability measure spaces and hyperspaces, respectively. In this paper we mainly show the following facts: (1) If $f_*$ is semi-open, then $f$ is semi-open. (2) If $f$ is semi-open densely open, then $f_*$ is semi-open densely open. (3) $f$ is open iff $2^f$ is open. (4) $f$ is semi-open iff $2^f$ is semi-open. (5) $f$ is irreducible iff $2^f$ is irreducible.