Topologizing interpretable groups in p-adically closed fields

2022/05/02 by William Johnson, Johnson, Will
Mathematics · #03C60 #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2205.00749

openalex publication_date 2022/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kⁿ. We show every interpretable set has at least one admissible topology, and every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over ℚ_p is definably isomorphic to a definable group.

Related

• Abelian groups definable in p-adically closed fields
• Definably semisimple groups interpretable in p-adically closed fields
• A note on fsg groups in p-adically closed fields
• On groups and fields interpretable in NTP₂ fields
• On groups with definable f-generics definable in p-adically closed fields