Characterizing realcompact locales via remainders

Abstract

We prove that a completely regular locale L is realcompact if and only if the “remainder” {\beta L\smallsetminus L} is the join of the zero-sublocales of {\beta L} that miss L. This extends a result of Mrówka which characterizes realcompact spaces in terms of their remainders in Stone–Čech compactifications. We prove that {\beta L\smallsetminus L} is Lindelöf if and only if L is of countable type, where the latter is defined for locales exactly as for spaces, subject to replacing subspaces with sublocales.

Remainders in pointfree topology

“…(e) [31] has a few more special formulas for the supplements in particular classes of frames like the T 1 -spatial or subfit ones, where they are used to show that the system of all joins of closed sublocales of L is the Booleanization of SpLq. Recently, in [16], T. Dube uses these formulas to compute remainders βL L, βL υL and βL λL for any completely regular L (see Section 7 below for details about υL and λL).…”

“…Some of the results in this paper were presented for the first time by J. Picado at the conference held at the University of Cape Town in March 2016 to celebrate Bernhard Banaschewski's 90th birthday. We were pleased to see in a very recent paper ( [16]) that T. Dube uses our approach to characterize realcompact locales.…”