Definable separability and second-countability in o-minimal structures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915340909281280 |
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| author | Guerrero, Pablo Andújar |
| author_facet | Guerrero, Pablo Andújar |
| contents | We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and definable second-countability -- which make sense in a wider model-theoretic context. We prove that, within o-minimality, these notions have the desired properties, including their equivalence among definable metric spaces, and conjecture a definable version of Urysohn's Metrization Theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Definable separability and second-countability in o-minimal structures Guerrero, Pablo Andújar Logic General Topology 03C64 (Primary), 54A05, 54D65, 54D70 (Secondary) We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and definable second-countability -- which make sense in a wider model-theoretic context. We prove that, within o-minimality, these notions have the desired properties, including their equivalence among definable metric spaces, and conjecture a definable version of Urysohn's Metrization Theorem. |
| title | Definable separability and second-countability in o-minimal structures |
| topic | Logic General Topology 03C64 (Primary), 54A05, 54D65, 54D70 (Secondary) |
| url | https://arxiv.org/abs/2405.07114 |