On structural numbers of topological spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915167458033664 |
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| author | Chatyrko, Vitalij Karassev, Alexandre |
| author_facet | Chatyrko, Vitalij Karassev, Alexandre |
| contents | Zero-dimensional structural numbers $Z_0^{\mathrm{ind}}$ and $Z_0^{\mathrm{dim}}$ w.r.t. dimensions $\mathrm{ind}$ and $\mathrm{dim}$ were introduced by Georgiou, Hattori, Megaritis, and Sereti. Somewhat similarly, we define structural numbers $\mathrm{Sn}^{A}$ for different subclasses $A$ of the class of hereditarily normal $T_1$-spaces. In particular, we show that:
(a) for any metrizable space $X$ with $\dim X = n \geq 0$ we have $1 \leq \mathrm{Sn}^{M_{dim}}X \leq n+1$;
(b) for any countable-dimensional metrizable space $Y$ we have $1 \leq \mathrm{Sn}^{M_{dim}}Y \leq \aleph_0$,
where $ M_{dim}$ is the class of metrizable spaces $Z$ with $\mathrm{dim}\, Z = 0.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16354 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On structural numbers of topological spaces Chatyrko, Vitalij Karassev, Alexandre General Topology Primary 54F45, Secondary 54A10 Zero-dimensional structural numbers $Z_0^{\mathrm{ind}}$ and $Z_0^{\mathrm{dim}}$ w.r.t. dimensions $\mathrm{ind}$ and $\mathrm{dim}$ were introduced by Georgiou, Hattori, Megaritis, and Sereti. Somewhat similarly, we define structural numbers $\mathrm{Sn}^{A}$ for different subclasses $A$ of the class of hereditarily normal $T_1$-spaces. In particular, we show that: (a) for any metrizable space $X$ with $\dim X = n \geq 0$ we have $1 \leq \mathrm{Sn}^{M_{dim}}X \leq n+1$; (b) for any countable-dimensional metrizable space $Y$ we have $1 \leq \mathrm{Sn}^{M_{dim}}Y \leq \aleph_0$, where $ M_{dim}$ is the class of metrizable spaces $Z$ with $\mathrm{dim}\, Z = 0.$ |
| title | On structural numbers of topological spaces |
| topic | General Topology Primary 54F45, Secondary 54A10 |
| url | https://arxiv.org/abs/2502.16354 |