Definable compactness in o-minimal structures

TL;DR

This paper characterizes the concept of definable compactness within o-minimal structures, clarifying its various definitions and establishing their equivalence, thus advancing the understanding of topological properties in this setting.

Contribution

It provides a comprehensive characterization of definable compactness, resolving open questions and unifying multiple existing definitions in o-minimal structures.

Findings

Equivalence of definitions of definable compactness

Characterization using definable curves and types

Resolution of questions by Peterzil, Steinhorn, and Johnson

Abstract

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various definitions of definable compactness in the literature, including those in terms of definable curves, definable types, and definable downward directed families of closed sets.