About closed subsets definable in Hensel minimal structures
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918439685193728 |
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| author | Nowak, Krzysztof Jan |
| author_facet | Nowak, Krzysztof Jan |
| contents | The main purpose is to establish two theorems about closed 0-definable subsets $A$ of an affine space $K^{n}$ over a Hensel minimal field $K$. The first, being a non-Archimedean counterpart of one from o-minimal geometry, states that every such subset $A$ is the zero locus of a continuous 0-definable function on $K^{n}$. The second is a definable, non-Archimedean version of the Tietze-Urysohn extension theorem. The proofs use ubiquity of clopen sets in non-Archimedean geometry and a description of definable sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | About closed subsets definable in Hensel minimal structures Nowak, Krzysztof Jan Logic Algebraic Geometry 03C65, 54C20, 12J25, 03C98 The main purpose is to establish two theorems about closed 0-definable subsets $A$ of an affine space $K^{n}$ over a Hensel minimal field $K$. The first, being a non-Archimedean counterpart of one from o-minimal geometry, states that every such subset $A$ is the zero locus of a continuous 0-definable function on $K^{n}$. The second is a definable, non-Archimedean version of the Tietze-Urysohn extension theorem. The proofs use ubiquity of clopen sets in non-Archimedean geometry and a description of definable sets. |
| title | About closed subsets definable in Hensel minimal structures |
| topic | Logic Algebraic Geometry 03C65, 54C20, 12J25, 03C98 |
| url | https://arxiv.org/abs/2403.08039 |