Pregeometry over locally o-minimal structures and dimension

Abstract

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