Christopher Gilmour

Cozero bases of frames

A frame L which is generated by a regular -frame (= cozero basis of L) is completely regular. Its cozero part CozL is then the largest cozero basis of L, and we characterize here those L for which it is the only such. Further, we give a similar characterization for the nitary analogue of this situation where regular -frames are replaced by Boolean algebras. In addition, we consider the compactications of a frame L provided by its cozero bases and show that all compactications are of this kind i L is pseudocompact. Finally, as an aside, we characterize the completely regular frames with unique compactication and their zero-dimensional counterparts. c 2001 Elsevier Science B.V. All rights reserved. MSC: 54D35; 54D20; 54D60; 06E99

Oz in pointfree topology

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.

REAL COMPACTIFICATIONS THROUGH ZERO-SET SPACES

Abstract The Alexandroff (= zero-set) spaces were introduced in [l] as the “completely normal spaces”, and have been studied in a number of more recent papers. In this paper we unify the theory of Wallman realcompactifications via the Alexandroff bases and introduce the realcompactfine Alexandroff spaces as particularly relevant to their investigation. These latter spaces are defined analogously to the A-c uniform spaces which are based on a construction of A.W. Hager [25].