Subsets of \({}^ \omega\omega\) and the Fréchet-Urysohn and \(\alpha_ i\)-properties
type: Item
instance of: scholarly article
title: Subsets of \({}^ \omega\omega\) and the Fréchet-Urysohn and \(\alpha_ i\)-properties
zbMATH Open document ID: 0774.54019
DOI: 10.1016/0166-8641(92)90021-Q
author: Peter Nyikos
published in: Topology and its Applications
publication date: 1993-05-16T00:00:00Z
review text: A space is Fréchet-Urysohn if \(x\) in the closure of \(A\) implies \(A\) has a subsequence converging to \(x\). Arkhangelskij defined subclasses by adding the restriction that the spaces have one of the so-called \(\alpha_ i\)-properties \((0\leq i\leq 4)\) in order to get better behaviour (in products for example). A countable Fréchet-Urysohn \(\alpha_ 2\)-space which is not first countable \((\alpha_ 0)\) is constructed. Under a variety of additional set-theoretic hypotheses, the classes \(\alpha_ 2\), \(\alpha_ 1\) and \(\alpha_ 0\) are distinguished. It is now known that it is consistent that \(\alpha_ 2\) implies \(\alpha_ 1\) and \(\alpha_ 1\) implies \(\alpha_ 0\), but not both simultaneously as the above mentioned example demonstrates. An example of a countable \(\alpha_ 2\)-topological group that is not first countable is given. It is shown to be Fréchet-Urysohn if \({\mathfrak p}={\mathfrak b}\). It is unknown if it is consistent that each countable Fréchet-Urysohn topological group is first countable (hence metrizable).
reviewed by: Alan Dow
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03E65
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 54D65
zbMATH DE Number: 166414
zbMATH Keywords: Fréchet-Urysohn
zbMATH Keywords: \(\alpha_ i\)-properties
zbMATH Keywords: combinatorics on \(\omega\)
MaRDI profile type: Publication
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full work available at URL: 0166-8641(92)90021-q
OpenAlex ID: W1997355585
description: scientific article; zbMATH DE number 166414