Countable Fréchet \(\alpha_ 1\)-spaces may be first countable
type: Item
instance of: scholarly article
title: Countable Fréchet \(\alpha_ 1\)-spaces may be first countable
zbMATH Open document ID: 0798.03052
DOI: 10.1007/BF01270393
author: Alan Dow
author: Juris Steprāns
published in: Archive for Mathematical Logic
publication date: 1994-01-23T00:00:00Z
review text: An \(\alpha_ 1\)-space is one in which the following condition holds: If \(\{S_ n\): \(n\in\omega\}\) is a countable collection of sequences converging to a point \(x\) then there is one sequence \(S\) that converges to \(x\) and is such that \(S_ n\subseteq^*S\) for all \(n\). It should be clear that first-countable spaces are \(\alpha_ 1\); the converse is not true as witnessed by a consistent examples of a Fréchet-Urysohn \(\alpha_ 1\)-space that is not first-countable [the first author, Proc. Am. Math. Soc. 108, No. 1, 241-247 (1990; Zbl 0675.54029)]. The authors construct a model of set theory in which all countable Fréchet-Urysohn \(\alpha_ 1\)-spaces are first-countable, thus eliminating the most obvious candidates for counterexamples, namely spaces of the form \(\omega\cup \{x\}\). The result may also be interpreted in terms of the space \(\omega^*\). For consider the trace \({\mathcal F}\) of the neighbourhood filter of \(x\) on \(\omega\) and its associated closed subset \(F=\bigcap \{A^*\): \(A\in {\mathcal F}\}\) of \(\omega^*\). It is not too hard to check that \(\omega\cup \{x\}\) is first-countable iff \(F\) is a \(G_ \delta\)-set; it is Fréchet-Urysohn iff \(F\) is regularly closed and that it is \(\alpha_ 1\) iff for every countable family \({\mathcal O}\) of clopen subsets of \(\omega^*\) such that \(\bigcup {\mathcal O} \subseteq F\) there is one clopen set \(C\) such that \(\bigcup {\mathcal O} \subseteq C\subseteq F\). We see that in the authors' model the \(G_ \delta\)-subsets of \(\omega^*\) have an -- as it were -- internal characterization: A set is a \(G_ \delta\)-set iff it is regularly closed and the family if its clopen subsets if countably directed. With every filter \({\mathcal F}\) on \(\omega\) the authors associate a partial order \(\mathbb{M}({\mathcal F})\) which shoots an infinite subset of \(\omega\) below the filter \({\mathcal F}\). If the \({\mathcal F}\) is Fréchet-Urysohn and \(\alpha_ 1\) then \(\mathbb{M}({\mathcal F})\) …
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 484791
zbMATH Keywords: Fréchet-Urysohn space
zbMATH Keywords: \(\alpha_ 1\)-space
zbMATH Keywords: first-countable space
zbMATH Keywords: proper forcing
reviewed by: K. P. Hart
MaRDI profile type: Publication
cites work: Classes of topological groups
cites work: Q4732462
cites work: Two Classes of Frechet-Urysohn Spaces
cites work: A model in which countable Fréchet α1-spaces are first countable
cites work: Metrizability and the Frechet-Urysohn Property in Topological Groups
cites work: Proper forcing
description: scientific article; zbMATH DE number 484791