A non-normal topology generated by a two-point selection
type: Item
instance of: scholarly article
title: A non-normal topology generated by a two-point selection
zbMATH Open document ID: 1144.54006
published in: Topology and its Applications
publication date: 2008-06-19T00:00:00Z
review text: For a set \(X\) and a two-point selection \(f:[X]^{2}\to X\), the topology \(\tau_{f}\) generated by \(f\) is Hausdorff and regular. This suggests the question whether or not \(\tau_{f}\) is a collectionwise Hausdorff topology on \(X\), or even a normal topology as asked by \textit{V. Gutev} and \textit{T. Nogura} [Topology Appl. 153, No. 5--6, 900--911 (2005; Zbl 1089.54005)]. In this paper, the authors construct a two-point selection \(f:[\mathbb{P}]^{2}\to\mathbb{P}\), where \(\mathbb{P}\) is the set of the irrational numbers, such that the topology \(\tau_{f}\) is not normal and not collectionwise Hausdorff either. It is also given that if \(f:[X]^{2}\to X\) is a two-point selection such that the topology \(\tau_{f}\) has countable pseudocharacter, then \(\tau_{f}\) is a Tychonoff topology.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 5290941
zbMATH Keywords: two-point section
zbMATH Keywords: non-normal space
zbMATH Keywords: Vietoris topology
author: Salvador Garcia-Ferreira
author: Artur Hideyuki Tomita
reviewed by: Shou Lin
MaRDI profile type: Publication
full work available at URL: j.topol.2008.01.013
OpenAlex ID: W2042396581
cites work: Q3994935
cites work: Selections and order-like relations
cites work: A topology generated by selections
cites work: Selections generating new topologies
cites work: Topologies on Spaces of Subsets
DOI: 10.1016/J.TOPOL.2008.01.013
description: scientific article; zbMATH DE number 5290941