The semicontinuous quasi-uniformity of a frame

Abstract

The aim of this note is to show how various facts in classical topology connected with semicontinuous functions and the semicontinuous quasi-uniformity have their counterparts in pointfree topology. In particular, we introduce the localic semicontinuous quasi-uniformity, which generalizes the semicontinuous quasiuniformity of a topological space (known to be one of the most important examples of transitive compatible quasi-uniformities). We show that it can be characterized in terms of the so called spectrum covers, via a construction introduced by the authors in a previous paper. Several consequences are derived.