On definably proper maps

TL;DR

This paper characterizes definably proper maps in o-minimal structures, showing their equivalence to proper morphisms and exploring their properties and invariance under extensions.

Contribution

It provides new characterizations of definably proper maps, including limit-based criteria, and establishes their fundamental properties and invariance in o-minimal settings.

Findings

Definably proper maps are equivalent to proper morphisms in definable spaces.

Existence of limits of definable types characterizes definably proper maps.

Definably proper maps are invariant under elementary extensions.

Abstract

In this paper we work in o-minimal structures with definable Skolem functions and show that a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is proper morphism in the category of definable spaces. We give several other characterizations of definably proper including one involving the existence of limits of definable types. We also prove the basic properties of definably proper maps and the invariance of definably proper in elementary extensions and o-minimal expansions.