Notes on definably complete locally o-minimal expansions of ordered groups

TL;DR

This paper investigates properties of definably complete locally o-minimal expansions of ordered groups, establishing uniform continuity, an Arzela-Ascoli-type theorem, and a decomposition of definable sets into special submanifolds.

Contribution

It introduces a framework for understanding definable functions and sets in these structures, including new decomposition techniques and continuity results.

Findings

Definable continuous functions on closed, bounded sets are uniformly continuous.

Established an Arzela-Ascoli-type theorem for these structures.

Decomposed any definable set into finitely many special submanifolds with tubular neighborhoods.

Abstract

We study definably complete locally o-minimal expansions of ordered groups in this paper. A definable continuous function defined on a closed, bounded and definable set behave like a continuous function on a compact set. We demonstrate uniform continuity of a definable continuous function on a closed, bounded and definable set and Arzela-Ascoli-type theorem. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.