Pregeometry over locally o-minimal structure and dimension

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.