On semi-openness of fiber-onto extensions of minimal semiflows and quasi-separable maps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915825378656256 |
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| author | Dai, Xiongping Feng, Li Lv, Congying Xie, Yuxuan |
| author_facet | Dai, Xiongping Feng, Li Lv, Congying Xie, Yuxuan |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03380 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On semi-openness of fiber-onto extensions of minimal semiflows and quasi-separable maps Dai, Xiongping Feng, Li Lv, Congying Xie, Yuxuan Dynamical Systems General Topology 54C10 (Primary), 37B05 (Secondary) 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). |
| title | On semi-openness of fiber-onto extensions of minimal semiflows and quasi-separable maps |
| topic | Dynamical Systems General Topology 54C10 (Primary), 37B05 (Secondary) |
| url | https://arxiv.org/abs/2601.03380 |