There are no hereditary productive \(\gamma \)-spaces
type: Item
instance of: scholarly article
title: There are no hereditary productive \(\gamma \)-spaces
zbMATH Open document ID: 1160.54013
author: Francis Jordan
published in: Topology and its Applications
publication date: 2008-09-23T00:00:00Z
review text: Hereditarily Lindelöf topological spaces in which every open set is a countable union of clopen sets are the subject of investigation. The main result says that every uncountable productive \(\gamma\)-space contains a co-countable subspace which is not Hurewicz, and thus not a \(\gamma\)-space. Several results interesting by their own are proved on the way to the main result. Every co-countable subspace of \(X\) is Hurewicz if and only if \(X\) is a Hurewicz \(\lambda\)-space. Unions of countably many productive \(\gamma\)-spaces are productive \(\gamma\)-spaces. A question of Miller is answered by showing that unions of increasing sequences of \(\gamma\)-spaces are \(\gamma\)-spaces. The Hurewicz \(\lambda\)-spaces \(X\) are characterized in terms of some continuous functions \(f:X\to (\omega\cup\{\infty\})^{\omega}\). Results concerning preservation with respect to unions are based on a theorem which gives that the suprema of (increasing) countable sequences of (productive) Fréchet filters are (productive) Fréchet filters.
reviewed by: Petr Holický
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54F65
zbMATH DE Number: 5346280
zbMATH Keywords: \(\gamma\)-space
zbMATH Keywords: \(\lambda\)-space
zbMATH Keywords: Hurewicz space
MaRDI profile type: Publication
full work available at URL: j.topol.2008.05.016
OpenAlex ID: W2094776454
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DOI: 10.1016/J.TOPOL.2008.05.016
description: scientific article; zbMATH DE number 5346280