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Introduction to the Special Issue “Aspects of Contemporary Topology II”

More on Configurations in Priestley Spaces, and Some New Problems

The Stone–Čech Compactification and the Cozero Lattice in Pointfree Topology

Completion of Semi-uniform Spaces

Joins and Meets of Symmetric Idempotents

Idempotency of Extensions via the Bicompletion

Conway’s Question: The Chase for Completeness

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The Stone–Čech Compactification and the Cozero Lattice in Pointfree Topology

Content Provider	Springer Nature Link
Author	Banaschewski, B.
Copyright Year	2007
Abstract	The existence of the pointfree Stone–Čech compactification, here understood as the compact, completely regular coreflection of a frame, has been established long ago in several ways modelled after certain descriptions of the classical Stone–Čech compactification (Johnstone, Stone Spaces, Cambridge Stud. Adv. Math. 3. Cambridge University Press, Cambridge, 1982; Banaschewski and Mulvey, Houston J. Math. 6:301–312, 1980, J. Pure Appl. Algebra 33:107–122, 1984), but – somewhat surprisingly – one of the most familiar representations of the latter, the one popularized by Gillman and Jerison’s influential book as the space of maximal z-filters, has only recently been recognized also to carry over to the pointfree setting, as is implicit in Wei (Appl. Categ. Structures 12:197–202, 2004). The purpose of this note is to present an alternative account of this fact which places the problem into the wider context of the saturation quotients of compact frames and what are introduced here as completely normal lattices. Specifically, we first show that the saturation quotient S( A) of the frame A of ideals of a completely normal lattice A is compact completely regular (Proposition 1) which is then brought to bear on the question at hand by establishing that the cozero lattice CozL of a frame L is completely normal and finally leads to the result that S (CozL) is the Stone–Čech compactification of L (Proposition 2). Given that familiar results concerning saturation quotients exhibit the spectrum of the latter frame as the space of maximal z-filters of X for the cozero set lattice of a topological space X, this is then exactly the pointfree form of the desired kind. With regard to foundations, it should be emphasized that, in notable contrast with most other work involving CozL, the present arguments are constructively valid in the sense of topos theory.
Ending Page	479
Page Count	7
Starting Page	473
File Format	PDF
ISSN	09272852
e-ISSN	15729095
Journal	Applied Categorical Structures
Issue Number	5-6
Volume Number	15
Language	English
Publisher	Springer Netherlands
Publisher Date	2007-07-07
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Rings and algebras of continuous, differentiable or analytic functions Compactification cozero lattice Convex and Discrete Geometry Saturation quotient of a compact frame Real-valued functions Completely normal lattice Theory of Computation Stone–Čech Mathematical Logic and Foundations
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Algebra and Number Theory Computer Science

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The Stone–Čech Compactification and the Cozero Lattice in Pointfree Topology

The Stone–Čech Compactification and the Cozero Lattice in Pointfree Topology

Lattices of quotients of completely distributive lattices

On the ideal of functions with compact support in pointfree function rings

On the Function Ring Functor in Pointfree Topology

A Pseudocompact Completely Regular Frame which is not Spatial

Pointfree Functorial Polar Functions

Notes on Pointfree Disconnectivity with a Ring-theoretic Slant

Remarks on Completely Regular Lindelöf Reflection of Locales

The Stone–Čech Compactification and the Cozero Lattice in Pointfree Topology

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