Katětov revisited: a frame-theoretic excursion

Abstract

The Katětov extension of a frame is, as in the case of spaces, compact only for compact completely regular frames. This result is purely topological because, subject to appropriate foundations, compact regular frames are topologies. In this paper we find a necessary and sufficient condition for the Katětov extension of a frame (and hence, as a corollary, of a topological space) to be compact-like. We also characterize frames the Katětov extensions of which are one-point extensions in the sense of Banaschewski and Gilmour. Next, the Fomin extension of a frame is defined, and several characterizations of frames whose Fomin extensions coincide with their Stone-Čech compactifications are given. These characterizations include frames which are not topologies, and therefore strictly transcend topology.