Definable one dimensional topologies in o-minimal structures

TL;DR

This paper characterizes when one-dimensional definable topological spaces in o-minimal structures are homeomorphic to affine definable spaces, showing that regularity and finite decomposition into connected components suffice.

Contribution

It provides equivalent conditions for definable topological spaces to be homeomorphic to affine spaces in o-minimal structures, highlighting the role of regularity and connected component decomposition.

Findings

Regularity and finite connected component decomposition imply affine definability.

Several equivalent conditions characterize when a definable topological space is affine.

The main result simplifies the classification of one-dimensional definable topologies.

Abstract

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.