Submission date: 4 April 2022

Abstract:

We define a discrete closure operation for definably complete locally o-minimal structures M. The pair of the underlying set of 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 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.

Mathematics Subject Classification: 03C64

Keywords and phrases:

Full text arXiv 2204.01895: pdf, ps.