More on Uniform Paracompactness in Pointfree Topology

Abstract

Abstract We revisit uniformly paracompact uniform frames and show that, in analogy with their spatial counterparts, they have a characterisation in terms of a “completeness property”. Namely, they are precisely those in which every weakly Cauchy filter clusters. We also give another characterisation in terms of the Čech-Stone compactification of the underlying frame. By tweaking the definition of uniformly paracompact frames, we define uniformly para-Lindelöf frames (analogously to same-named uniform spaces) and characterise them in terms of the Lindelöf coreflection of the underlying frame. This latter characterisation has no spatial counterpart.