Analysis of a topological basis problem
type: Item
instance of: scholarly article
title: Analysis of a topological basis problem
author: Yanyan Li
published in: Acta Mathematica Hungarica
publication date: 2022-11-09T00:00:00Z
review text: Recall that a subclass \(\mathcal{B}\) of a class \(\mathcal{K}\) of topological spaces is a basis provided each member of \(\mathcal{K}\) contains a member of \(\mathcal{B}\) as a subspace. This paper works toward a small basis for the class of uncountable, regular, first countable spaces, assuming PFA. Given a topological space \((X,\tau)\) and a collection \(\mathcal{C}\) of subsets of \(X\) two new topologies, called inner and outer, induced by \(\mathcal{C}\) are defined on \(X\): for example the inner, \(\tau^{I,\mathcal{C}}\) has base \(\{\{x\}\cup O^{I,\mathcal{C}}\ /\ x\in O\in\tau\}\), where \(O^{I,\mathcal{C}}=\cup\{C\in\mathcal{C}\ /\ C\subset O\}\). A couple of sample results assuming PFA: if \(X\) is regular and the inner topology is hereditarily Lindelöf then \(X\) either has a countable network or contains an uncountable Sorgenfrey subset; if \(X\) has cardinality \(\aleph_1\), is first countable and Hausdorff and the inner topology induced by a countable collection is hereditarily Lindelöf then \(X\) contains an uncountable subset that is metrisable or Sorgenfrey. There are also results involving perfect preimages and D-spaces.
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 03E75
zbMATH DE Number: 7614055
zbMATH Keywords: basis problem
zbMATH Keywords: countable network
zbMATH Keywords: Sorgenfrey space
zbMATH Keywords: hereditarily Lindelöf
zbMATH Keywords: D-space
reviewed by: David B. Gauld
MaRDI profile type: Publication
full work available at URL: s10474-022-01253-y
OpenAlex ID: W4287378166
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DOI: 10.1007/S10474-022-01253-Y
description: scientific article; zbMATH DE number 7614055