Around definable types in $p$-adically closed fields
| Autor: | Guerrero, Pablo Andujar, Johnson, Will |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Annals of Pure and Applied Logic 175(10): 103484 (December 2024) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.apal.2024.103484 |
| Popis: | 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. Comment: 39 pages; fixed two minor typos |
| Databáze: | arXiv |
| Externí odkaz: |
|