Scott spaces and sober spaces
type: Item
instance of: scholarly article
title: Scott spaces and sober spaces
zbMATH Open document ID: 0943.54021
DOI: 10.1016/S0166-8641(98)00089-3
author: Elias David
published in: Topology and its Applications
publication date: 2000-09-04T00:00:00Z
review text: It is well known that every \(T_0\) topology on a set induces a partial ordering (the specialization ordering) on the same set. Furthermore, if the topology is sober, then the partial ordering has directed joins, and the topology is contained in the Scott topology for the induced ordering (though the Scott topology need not be sober in general). In this paper the author investigates what happens to the notion of sobriety if you jettison the \(T_0\) axiom, and what happens to the Scott topology if you jettison the assumption that the underlying pre-order (is a partial order and) has directed joins. As is to be expected, there is some fragmentation of conditions which are equivalent under the assumptions thus jettisoned, but there turns out to be a fair amount that can still be said.
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 06A99
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 06B35
zbMATH DE Number: 1395056
reviewed by: Peter Johnstone
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00089-3
OpenAlex ID: W2005981376
description: scientific article; zbMATH DE number 1395056