Pregeometry over locally o-minimal structure and dimension

TL;DR

This paper introduces a discrete closure operation in locally o-minimal structures, establishing a pregeometry framework that aligns definable set dimensions with rank and topological dimension.

Contribution

It defines a new discrete closure operation in locally o-minimal structures, linking pregeometry, definable set rank, and topological dimension.

Findings

The discrete closure operation forms a pregeometry in locally o-minimal structures.

Definable set dimension equals the rank over parameters.

Dimension rank coincides with topological dimension in the structure.

Abstract

We define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact. A definable set $X$ is of dimension equal to the rank of $X$ over the set of parameters of a formula defining the set $X$. The structure $\mathcal M$ is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure $\mathcal M$ also coincides with its dimension.