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 $\varphi :X\to Y$ and its induced map ${\varphi }_{\ast }:{\mathcal{M}}^1(X)\to {\mathcal{M}}^1(Y)$ 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 $\varphi$ is an extension of minimal flows, then ${\varphi }_{\ast }$ is semi-open. (2) If $\varphi$ is a quasi-separable fiber-onto extension of minimal semiflows, then $\varphi$ and ${\varphi }_{\ast }$ are semi-open. (3) If $Y$ is metrizable, then $\varphi$ is semi-open if and only if ${\varphi }_{\ast }$ is semi-open. In addition, if $X,Y$ are left-topological groups, $X$ is Lindelöf quasi-regular, $Y$ is Baire and if $\varphi$ is a locally closed continuous onto equivariant mapping, then $\varphi$ is semi-open (This is a generalization of Pontryagin's open-mapping theorem).

28 pages