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

Theorems & Definitions (84)

• Theorem 1.2.2
• Theorem 1.2.3
• Theorem 1.2.5: Thm. \ref{['4.5']}
• Lemma 1.3.2: cf. B79 or V70WoA88
• Lemma 1.3.3: Thm. \ref{['3.2']}
• Theorem 1.3.4: Thm. \ref{['3.4']}
• Theorem 1.3.5: Thm. \ref{['5.2.2']}
• Theorem 1.4.1: Pontryagin's open-mapping theorem; cf. HR63
• Theorem 1.4.3: Thm. \ref{['6.1']}
• Theorem 1.4.4: Cor. \ref{['6.2']}