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

TL;DR

This paper explores the properties of definably complete locally o-minimal structures, establishing tame dimension theory, decomposition into quasi-special submanifolds, and extending classical theorems like Łojasiewicz's inequality and Michael's selection theorem within this framework.

Contribution

It proves that projections of discrete definable sets are discrete, develops a tame dimension theory, and extends key classical theorems to definably complete locally o-minimal structures with bounded multiplication.

Findings

Projection of discrete definable sets remains discrete.

Dimension theory and decomposition into quasi-special submanifolds are established.

Classical theorems like Łojasiewicz's inequality and Michael's selection theorem hold in this setting.

Abstract

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, in the latter part of this paper, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fields, {\L}ojasiewicz's inequality, Tietze extension theorem and affiness of psudo-definable spaces hold true for such structures under the extra assumption that the domains of definition and the psudo-definable spaces are definably compact. Here, a pseudo-definable space is a topological spaces having finite definable atlases. We also demonstrate Michael's selection theorem for definable set-valued functions with definably compact domains of definition.