Pregeometry over locally o‐minimal structures and dimension

Abstract

Abstract We define a discrete closure operator for definably complete locally o‐minimal structures . The pair of the underlying set of 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 ‐dimension. A definable set X is of dimension equal to the ‐dimension of X . The structure is simultaneously a first‐order topological structure. The dimension rank of a set definable in the first‐order topological structure also coincides with its dimension.