On ordinal-metric intersection topologies
type: Item
instance of: scholarly article
title: On ordinal-metric intersection topologies
zbMATH Open document ID: 0593.54001
DOI: 10.1016/0166-8641(86)90030-1
author: Kenneth Kunen
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: Following G. M. Reed the author considers the class C of all spaces X of size \(\omega_ 1\) for which there exist a separable metric topology \(T_ 1\) on the set X and an order topology \(T_ 2\) induced by some well-order of X of the type \(\omega_ 1\) such that \(\{\) \(U\cap V:\) U is open in \(T_ 1\) and V is open in \(T_ 2\}\) is a base of the topology of X. The author investigates normality and perfectness \((=\) all closed sets are \(G_{\delta}'s)\) in the class C and presents theorems which clear up connections between these properties. He proves that there are no spaces in C which are both perfect and normal. Under MA\(+\neg CH\) all spaces in C are perfect since separable metric spaces of size \(\omega_ 1\) are Q- sets. Hence, under \(MA+\neg CH\) all spaces in C are nonnormal. Under CH the situation is almost reversed: the author proves that for every \(X\subset C\) there exists a club set \(D\subset X\) such that D is normal. Since C is closed under club subsets, this gives a strengthening of a theorem of Reed. It is not known if a model of ZFC exists in which all members of C are neither perfect nor normal. But in this paper the author proves that if \(\omega_ 1\) Cohen reals are added to a model of \(MA+\neg CH\), then in the extension the class C will contain both perfect and normal spaces.
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 3953772
zbMATH Keywords: intersection topology
zbMATH Keywords: stationary set
zbMATH Keywords: normality
zbMATH Keywords: perfectness
zbMATH Keywords: \(MA+\neg CH\)
zbMATH Keywords: CH
zbMATH Keywords: club set
zbMATH Keywords: ZFC
reviewed by: Aleksander Błaszczyk
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90030-1
OpenAlex ID: W1966850583
cites work: On Mike's misnamed intersection topologies
cites work: Q5560150
cites work: Q4151359
cites work: The Intersection Topology W.R.T. The Real Line and the Countable Ordinals
description: scientific article; zbMATH DE number 3953772