Oz in pointfree topology

Abstract

Oz frames are the natural pointfree counterpart of Oz spaces, that is, those topological spaces in which every open set is z-embedded. We give here the point-free analogues of known characterisations of Oz spaces and show that the Lindelöf coreflection of an Oz frame is Oz. The latter result has no spatial version, but has implications for a number of well-known frame and spatial extensions, characterizations and properties of these frames. Extremally disconnected frames are investigated in relation to Oz frames and weak Oz frames, the latter being a very natural generalisation of Oz.