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

Let $f\text{\hspace{0.17em}⁣}:X\to Y$ be a continuous surjection of compact Hausdorff spaces. By $$f_{\ast }\text{\hspace{0.17em}⁣}:\mathfrak{M}(X)\to \mathfrak{M}(Y),\mu \mapsto \mu \circ f^{-1}\mathrm{a}\mathrm{n}\mathrm{d}{2}^f\text{\hspace{0.17em}⁣}:2^X\to 2^Y,A\mapsto f[A]$$ 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 $f_{\ast }$ is semi-open, then $f$ is semi-open. (2) If $f$ is semi-open densely open, then $f_{\ast }$ 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.