Affiness of topological space definable in a definably complete uniformly locally o-minimal structure of the second kind

TL;DR

This paper characterizes when a one-dimensional definable topological space in a specific o-minimal structure is affine, providing a precise condition based on regularity, Hausdorff property, and bounded multiplication.

Contribution

It establishes a necessary and sufficient condition for such spaces to be affine within the framework of definably complete uniformly locally o-minimal structures of the second kind.

Findings

Characterization of affine spaces in the given structure

Necessary and sufficient condition involving regularity and Hausdorff property

Role of definable bounded multiplication in affineness

Abstract

We give a necessary and sufficient condition for a one-dimensional regular and Hausdorff topological space definable in a definably complete uniformly locally o-minimal structure of the second kind having definable bounded multiplication being affine.