Submission date: 6 November 2021

Abstract:

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.

Mathematics Subject Classification: 03C64 (Primary), 54A05, 54D30 (Secondary)

Keywords and phrases:

Full text arXiv 2111.03802: pdf, ps.