On topological spaces whose topology is induced by a binary relation

Abstract

We study the problem of characterizing which topologies on a nonempty set are generated by a binary relation by means of its lower and upper contour sets. In this direction we consider different classical categories of topological spaces whose topology is defined by at least one binary relation. Given a topology defined by some binary relation on a set, we also analyze if the binary relation could belong to some particular category. We furnish some examples of topological spaces whose topology cannot be induced by any binary relation. We analyze some of these items in the context of pointfree topology and extend these questions to the study of bitopological spaces.