Locally O-Minimal Structures With Tame Topological Properties

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.

Tameness of definably complete locally o-minimal structures and definable bounded multiplication

“…A counter example is found in [12, Example 12]. In Section 2, we demonstrate the property (a) and recall its consequences given in [8].…”

“…Suppose that f (I) is discrete. Since f (I) is a bounded discrete closed set by [8,Lemma 2.3], f (I) has the maximal element h = f (x) where x ∈ I by the definable completeness of M. Take y ∈ I with x < y. Since f is strictly increasing on…”

“…The remaining task is to show that X d is discrete and closed. Suppose not, then X d contains an interval J by [8,Lemma 2.3]. We may assume that J is bounded without loss of generality.…”

“…Locally o-minimal structures are studied in a series of papers [18,12,5,6,7,8]. We first recall the definitions of local o-minimality and definably completeness.…”

“…In [8], the first author demonstrated that definably complete locally o-minimal structures satisfying the following property (a) have tame dimension theory and decomposition theorem into good-shaped definable subsets called quasi-special submanifolds.…”