Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929329734156288 |
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| author | Dai, Xiongping Xie, Yuxun |
| author_facet | Dai, Xiongping Xie, Yuxun |
| contents | Let $f\colon X\rightarrow Y$ be a continuous surjection of compact Hausdorff spaces. By $$f_*\colon\mathfrak{M}(X)\rightarrow\mathfrak{M}(Y),\ μ\mapsto μ\circ f^{-1} \quad{\rm and}\quad 2^f\colon2^X\rightarrow2^Y,\ A\mapsto f[A]$$ 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 $f_*$ is semi-open, then $f$ is semi-open.
(2) If $f$ is semi-open densely open, then $f_*$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12727 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps Dai, Xiongping Xie, Yuxun Dynamical Systems General Topology 37B05, 54B20 Let $f\colon X\rightarrow Y$ be a continuous surjection of compact Hausdorff spaces. By $$f_*\colon\mathfrak{M}(X)\rightarrow\mathfrak{M}(Y),\ μ\mapsto μ\circ f^{-1} \quad{\rm and}\quad 2^f\colon2^X\rightarrow2^Y,\ A\mapsto f[A]$$ 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 $f_*$ is semi-open, then $f$ is semi-open. (2) If $f$ is semi-open densely open, then $f_*$ 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. |
| title | Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps |
| topic | Dynamical Systems General Topology 37B05, 54B20 |
| url | https://arxiv.org/abs/2404.12727 |