Directed sets and topological spaces definable in o-minimal structures

TL;DR

This paper investigates the properties of directed sets and topological spaces within o-minimal structures, revealing conditions under which definable sets admit certain curves and characterizing definable compactness and countability properties.

Contribution

It establishes the existence of cofinal definable curves in expansions of ordered fields and extends results to definable families with the finite intersection property using tame pairs.

Findings

Directed sets in ordered fields admit cofinal definable curves.

Definable topological spaces exhibit properties similar to first countability.

A generalized notion of definable compactness is characterized.

Abstract

We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all definable topological spaces display properties akin to first countability, and give several characterizations of a notion of definable compactness due to Peterzil and Steinhorn generalized to this setting.