  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.
