Types, transversals and definable compactness in o-minimal structures

TL;DR

This paper characterizes definable compactness in o-minimal structures using types, generalizes known results about definable sets, and establishes a connection between dividing and definable types with an elementary proof.

Contribution

It introduces a new notion of definable compactness in o-minimal structures and proves a parameter version of the dividing-definable types connection without relying on forking or VC theory.

Findings

Characterization of definable compactness via types in o-minimal structures

Generalization of results on closed and bounded definable sets

Finiteness of partitions for families with the $(p,q)$-property

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 $p\geq q$ with $q$ large enough, then the family admits a partition into finitely many subfamilies, each of which extends to an $A$-definable type.