Tame topology in Hensel minimal structures
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916513457373184 |
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| author | Nowak, Krzysztof Jan |
| author_facet | Nowak, Krzysztof Jan |
| contents | We are concerned with topology of Hensel minimal structures on non-trivially valued fields $K$, whose axiomatic theory was introduced in a recent paper by Cluckers-Halupczok-Rideau. We additionally require that every definable subset in the imaginary sort $RV$, binding together the residue field $Kv$ and value group $vK$, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets; among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Lojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_01836 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Tame topology in Hensel minimal structures Nowak, Krzysztof Jan Algebraic Geometry 03C65, 32B20, 32P05, 03C98, 12J25, 57N35 We are concerned with topology of Hensel minimal structures on non-trivially valued fields $K$, whose axiomatic theory was introduced in a recent paper by Cluckers-Halupczok-Rideau. We additionally require that every definable subset in the imaginary sort $RV$, binding together the residue field $Kv$ and value group $vK$, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets; among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Lojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces. |
| title | Tame topology in Hensel minimal structures |
| topic | Algebraic Geometry 03C65, 32B20, 32P05, 03C98, 12J25, 57N35 |
| url | https://arxiv.org/abs/2103.01836 |