Fréchet--Urysohn for finite sets
type: Item
instance of: scholarly article
title: Fréchet--Urysohn for finite sets
zbMATH Open document ID: 1085.54016
author: Gary Gruenhage
author: Paul J. Szeptycki
published in: Topology and its Applications
publication date: 2005-06-10T00:00:00Z
review text: This paper provides a variety of results and examples related to topological spaces which are Fréchet-Urysohn. Reznichenko and Sipacheva defined a space \(X\) to be \textit{Fréchet-Urysohn for finite sets} abbreviated \(FU_{\text{fin}}\) if for each \(x\) in \(X\) and collection \(P\) of finite subsets of \(X\) with every neighborhood of \(x\) containing a member of \(P\), a subfamily of \(P\) converges to \(x\). The authors consider spaces called \textit{boundedly-}\(FU_{\text{fin}}\) which limits the choices of \(P\) . It is shown that although each \(FU_{\text{fin}} \) space is boundedly-\(FU_{\text{fin}}\) , there are spaces which are boundedly-\(FU_{\text{fin}}\) but not \(FU_{\text{fin}}\). The study includes various examples of \(FU_{\text{fin}}\) spaces related to \(2^{\omega}\) and \(\omega^{\omega}\) and relations to \(\alpha_i\)-spaces introduced by Arhangel'skii. A natural association between \(FU_{\text{fin}}\) spaces and Fréchet-Urysohn groups is explored. Further, \(FU_{\text{fin}}\) spaces are characterized in terms of game theory and consequences for cartesian products are investigated.
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 2174918
zbMATH Keywords: Fréchet-Urysohn
zbMATH Keywords: topological group
reviewed by: William A. Feldman
MaRDI profile type: Publication
full work available at URL: j.topol.2003.09.014
OpenAlex ID: W1963588177
cites work: Q3921418
cites work: Two Classes of Frechet-Urysohn Spaces
cites work: Countable Fréchet \(\alpha_ 1\)-spaces may be first countable
cites work: Infinite games and generalizations of first-countable spaces
cites work: Baireness of \(C_ k(X)\) for locally compact \(X\)
cites work: Some properties of C(X). I
cites work: Fréchet-Urysohn for finite sets, II
cites work: On the consistency of Borel's conjecture
cites work: The Cantor Tree, the γ-Property, and Baire Function Spaces
cites work: Q3715868
cites work: Subsets of \({}^ \omega\omega\) and the Fréchet-Urysohn and \(\alpha_ i\)-properties
cites work: Q4369847
cites work: Fréchet-Urysohn type properties in topological spaces, groups, and locally convex vector spaces
cites work: Q4004868
cites work: Spaces Fréchet-Urysohn with respect to families of subsets
cites work: Products of compact Fréchet spaces
DOI: 10.1016/J.TOPOL.2003.09.014
description: scientific article; zbMATH DE number 2174918