Compactness in pointfree topology

Abstract

Our discussion starts with the study of convergence and clustering of filters initiated in pointfree setting by Hong, and then characterize compact and almost compact frames in terms of these filters. We consider the strict extension and show that tQL is a zerodimensional compact frame, where Q denotes the set of filters in L. Furthermore, we study the notion of general filters introduced by Banaschewski and characterize compact frames and almost compact frames using them. For filter selections, we consider F−compact and strongly F−compact frames and show that lax retracts of strongly F−compact frames are also strongly F−compact. We study further the ideals Rs(L) and RK(L) of the ring of realvalued continuous functions on L, RL. We show that Rs(L) and RK(L) are improper ideals of RL if and only if L is compact. We consider also fixed ideals of RL and showthat L is compact if and only if every ideal of RL is fixed if and only if every maximalideal of RL is fixed. Of interest, we consider the class of isocompact locales, which is larger that the class of compact frames. We show that isocompactness is preserved by nearly perfect localic surjections. We study perfect compactifications and show that the Stone-Cˇech compactifications and Freudenthal compactifications of rim-compact frames are perfect. We close the discussion with a small section on Z−closed frames and show that a basically disconnected compact frame is Z−closed.