Localic remote points revisited

Abstract

We consider remote points in general extensions of frames, with an emphasis on perfect extensions. For a strict extension ?XL ? L determined by a set X of filters in L, we show that if there is an ultrafilter in X then the extension has a remote point. In particular, if a completely regular frame L has a maximal completely regular filter which is an ultrafilter, then ?L ? L has a remote point, where ?L is the Stone-C?ch compactification of L. We prove that in certain extensions associated with radical ideals and l-ideals of reduced f-rings, remote points induced by algebraic data are exactly non-essential prime ideals or non-essential irreducible l-ideals. Concerning coproducts, we show that if M1 ? L1 and M2 ? L2 are extensions of T1-frames, then each of these extensions has a remote point if the extensionM1?M2?L1?L2 has a remote point.

A note on weakly pseudocompact locales

“…It is taken from [8,Lemma 4.2] where it is proved for T 1 -locales, and hence is valid for completely regular locales.…”