A note on the structural stability of almost one-to-one maps
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915517384622080 |
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| author | Cortez, María Isabel Hauser, Till |
| author_facet | Cortez, María Isabel Hauser, Till |
| contents | A continuous surjection $π:X\to Y$ between compact Hausdorff spaces induces continuous surjections $\mathcal{M}(π)\colon \mathcal{M}(X)\to\mathcal{M}(Y)$ and $\mathcal{H}(π): \mathcal{H}(X)\to\mathcal{H}(Y)$ between the spaces of regular Borel probability measures, and the spaces of closed subsets, respetively. It is well known that $\mathcal{H}(π)$ is irreducible if and only if $π$ is irreducible. We show that $\mathcal{M}(π)$ is irreducible if and only if $π$ is irreducible. Furthermore, we show that whenever $π$ is almost one-to-one then $\mathcal{M}(π)$ and $\mathcal{H}(π)$ are almost one-to-one. In particular, we observe that continuous surjections between compact metric spaces are almost one-to-one if and only if $\mathcal{H}(π)$ is almost one-to-one and a similar statement about $\mathcal{M}(π)$. Finally, we give alternative proofs for some results in 'Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps' by Xiongping Dai and Yuxuan Xie regarding semi-open maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23015 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the structural stability of almost one-to-one maps Cortez, María Isabel Hauser, Till Dynamical Systems General Topology 54C10 (Primary) 54B20, 37B05, 54B10 (Secondary) A continuous surjection $π:X\to Y$ between compact Hausdorff spaces induces continuous surjections $\mathcal{M}(π)\colon \mathcal{M}(X)\to\mathcal{M}(Y)$ and $\mathcal{H}(π): \mathcal{H}(X)\to\mathcal{H}(Y)$ between the spaces of regular Borel probability measures, and the spaces of closed subsets, respetively. It is well known that $\mathcal{H}(π)$ is irreducible if and only if $π$ is irreducible. We show that $\mathcal{M}(π)$ is irreducible if and only if $π$ is irreducible. Furthermore, we show that whenever $π$ is almost one-to-one then $\mathcal{M}(π)$ and $\mathcal{H}(π)$ are almost one-to-one. In particular, we observe that continuous surjections between compact metric spaces are almost one-to-one if and only if $\mathcal{H}(π)$ is almost one-to-one and a similar statement about $\mathcal{M}(π)$. Finally, we give alternative proofs for some results in 'Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps' by Xiongping Dai and Yuxuan Xie regarding semi-open maps. |
| title | A note on the structural stability of almost one-to-one maps |
| topic | Dynamical Systems General Topology 54C10 (Primary) 54B20, 37B05, 54B10 (Secondary) |
| url | https://arxiv.org/abs/2509.23015 |