On dissolute spaces
type: Item
instance of: scholarly article
title: On dissolute spaces
zbMATH Open document ID: 0734.54001
DOI: 10.1016/0166-8641(91)90058-T
published in: Topology and its Applications
publication date: 1991-01-01T00:00:00Z
review text: The dissolution of a locale X (the term is due to the reviewer) is the locale \(X_ d\) whose frame of opens is the congruence lattice of the frame of opens of X; equivalently, it is characterized by the existence of a continuous map \(X_ d\to X\) inducing a bijection between arbitrary sublocales of X and closed sublocales of \(X_ d\) (that is, one dissolves a locale by declaring everything to be closed). It is thus something like the discrete modification of a topological space, although the construction \(X\to X_ d\) is not idempotent; in particular, locales of the form \(X_ d\) are highly disconnected. Not much is known in general about which locales can occur as \(X_ d\) for some X; this paper makes a start on the characterization of dissolute spaces, i.e. those which occur as spatial sublocales of some \(X_ d\). Sample result: a compact Hausdorff space whose third Hausdorff derivative is empty is homeomorphic with a dissolution (but if you substitute ``fourth'' for ``third'', he result fails).
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54G12
Mathematics Subject Classification ID: 54B99
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 4214903
zbMATH Keywords: sober space
zbMATH Keywords: dissolution locale
zbMATH Keywords: dissolute spaces
author: J. R. Isbell
reviewed by: Peter Johnstone
MaRDI profile type: Publication
cites work: Q3898534
cites work: Atomless Parts of Spaces.
cites work: Fibrewise separation axioms for locales
cites work: Spatial sublocales and essential primes
full work available at URL: 0166-8641(91)90058-t
OpenAlex ID: W2043392906
Recommended article: Q3110668
Recommended article: On \(d\)-spaces
Recommended article: Q3980402
Recommended article: Q3719470
Recommended article: Q4798547
Recommended article: DIFFEOLOGICAL SPACES
Recommended article: Q3723358
Recommended article: On discus spaces
Recommended article: Q5061822
description: scientific article; zbMATH DE number 4214903