Locally o-minimal structures with tame topological properties

TL;DR

This paper explores locally o-minimal structures with tame topological properties, establishing foundational dimension properties and a decomposition theorem into quasi-special submanifolds.

Contribution

It introduces new results on the dimension theory and a decomposition theorem for locally o-minimal structures with tame topological features.

Findings

Dimension equality for definable maps with equi-dimensional fibers

Basic properties of the dimension of definable sets

Decomposition into quasi-special submanifolds

Abstract

We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.