Invariance of domain in locally o-minimal structures

TL;DR

This paper proves that in locally o-minimal structures, definable continuous injective maps from open sets into Euclidean spaces are necessarily open maps, extending classical invariance of domain results.

Contribution

It establishes the invariance of domain property within locally o-minimal structures, a significant generalization of classical topology results.

Findings

Injective continuous maps are open in locally o-minimal structures

Extends classical invariance of domain to a broader logical setting

Supports the topological robustness of locally o-minimal structures

Abstract

Definable continuous injective maps defined on definable open sets into the Euclidean spaces of the same dimension are open maps in definably complete locally o-minimal expansions of ordered groups.