On semi-openness of fiber-onto extensions of minimal semiflows and quasi-separable maps

Abstract

The purpose of this paper is to find conditions for a continuous onto map $\phi\colon X\rightarrow Y$ and its induced map $\phi_*\colon\mathcal{M}^1(X)\rightarrow\mathcal{M}^1(Y)$ to be semi-open, where $X$, $Y$ are compact Hausdorff spaces and $\mathcal{M}^1(X)$, $\mathcal{M}^1(Y)$ are their Borel probability spaces. For that, we mainly prove the following results by using the structure theory of extensions of semiflows and inverse limit techniques: (1) If $\phi$ is an extension of minimal flows, then $\phi_*$ is semi-open. (2) If $\phi$ is a quasi-separable fiber-onto extension of minimal semiflows, then $\phi$ and $\phi_*$ are semi-open. (3) If $Y$ is metrizable, then $\phi$ is semi-open if and only if $\phi_*$ is semi-open. In addition, if $X,Y$ are left-topological groups, $X$ is Lindel\"{o}f quasi-regular, $Y$ is Baire and if $\phi$ is a locally closed continuous onto equivariant mapping, then $\phi$ is semi-open (This is a generalization of Pontryagin's open-mapping theorem).