Around definable types in $p$-adically closed fields
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916326585401344 |
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| author | Guerrero, Pablo Andujar Johnson, Will |
| author_facet | Guerrero, Pablo Andujar Johnson, Will |
| contents | We prove some technical results on definable types in $p$-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable $n$-type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if $X$ is an interpretable set, then the space of global definable types on $X$ is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye. Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if $G$ is a definable group with definable f-generics ($dfg$), and $G$ acts on a definable set $X$, then the quotient space $X/G$ is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao. Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_05815 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Around definable types in $p$-adically closed fields Guerrero, Pablo Andujar Johnson, Will Logic 03C60 (Primary), 12L12 (Secondary) We prove some technical results on definable types in $p$-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable $n$-type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if $X$ is an interpretable set, then the space of global definable types on $X$ is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye. Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if $G$ is a definable group with definable f-generics ($dfg$), and $G$ acts on a definable set $X$, then the quotient space $X/G$ is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao. Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families. |
| title | Around definable types in $p$-adically closed fields |
| topic | Logic 03C60 (Primary), 12L12 (Secondary) |
| url | https://arxiv.org/abs/2208.05815 |