2021/11/06 by Pablo Andújar Guerrero, Guerrero, Pablo Andújar
Mathematics · #03C64 (Primary) #54A05 #54D30 (Secondary) #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2111.03802
openalex publication_date 2021/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in o-minimal expansions of ordered groups. Along the way we prove a parameter version for o-minimal theories of the connection between dividing and definable types known in the more general dp-minimal context [SS14], through an elementary proof that avoids the use of existing forking and VC literature. In particular we show that, if an A-definable family of sets has the (p,q)-property, for some p≥ q with q large enough, then the family admits a partition into finitely many subfamilies, each of which extends to an A-definable type.