Invariance of domain in locally o-minimal structures

TL;DR

The paper proves an invariance-of-domain result for definably complete locally o-minimal expansions of ordered groups: if $V\subseteq M^n$ is definable open and $f:V\to M^n$ is definable, continuous, and injective, then $f$ is an open map, i.e., $f(U)$ is open for any open $U\subseteq V$. The authors leverage the tame behavior of the dimension function $\dim(X)$ in this setting, together with definable connectivity/reachability and a curve-selection argument, to show that the boundary-intersection $H:=f(U)\cap\operatorname{bd}(f(U))$ must be empty. They avoid Brouwer degree and cell decomposition, instead relying on dimension-theoretic arguments in the vdD style (FKK) and a key proposition about removing low-dimensional closed subsets from boxes. The result extends naturally to maps between locally definable manifolds, providing an open-mapping principle in this broader geometric context with potential applications to definable topology and structure theory.

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.

Theorems & Definitions (11)

• Theorem 1.1: Invariance of domain
• Lemma 2.1
• Proposition 2.2
• proof
• Remark 2.3
• proof
• Corollary 2.4
• proof
• proof : Proof of Theorem \ref{['thm:open_map']}
• Corollary 2.5