A note on the Arens' space and sequential fan
type: Item
instance of: scholarly article
title: A note on the Arens' space and sequential fan
zbMATH Open document ID: 0885.54019
DOI: 10.1016/S0166-8641(97)00031-X
published in: Topology and its Applications
publication date: 1998-04-22T00:00:00Z
review text: A cover \({\mathcal P}\) of a space is a \(k\)-network if, for any compact subset \(K\) and any open subset \(U\) with \(K\subset U\), there exists a finite \({\mathcal P}^*\subset{\mathcal C}\) such that \(K\subset \bigcup{\mathcal P}^*\subset U\). Lashnev spaces, quotient \(s\)-images of metric spaces, or CW-complexes are \(k\)-spaces with a point-countable \(k\)-network. This paper obtains the following results and others by means of sequential coreflections \(\sigma(X)\). The results below improve the reviewer's results, where spaces have \(G_\delta\)-points. Here, let us assume that spaces are regular and \(T_1\). Theorem: Let \(X\) be a \(k\)-space with a point-countable \(k\)-network. Then, \(X\) is a Fréchet space iff \(X\) contains no closed copy of the Arens' space \(S_2\). Corollary: Let \(X\) be a \(k\)-space with a point-countable \(k\)-network. Then, (a) \(X\) has a point-countable base iff \(X\) contains no closed copy of \(S_2\), and no sequential fan \(S_\omega\). (b) \(X^\omega\) is a \(k\)-space iff \(X\) has a point-countable base.
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 54E99
zbMATH DE Number: 1103050
zbMATH Keywords: \(k\)-space
zbMATH Keywords: \(k\)-network
zbMATH Keywords: Arens' space
zbMATH Keywords: sequential fan
author: Shou Lin
reviewed by: Y. Tanaka
MaRDI profile type: Publication
cites work: Note on Convergence in Topology
cites work: MAPPINGS AND SPACES
cites work: Q5184124
cites work: Q3928648
cites work: Ordinal invariants for topological spaces
cites work: Characterizations of \(\aleph\)-spaces
cites work: Spaces in which sequences suffice II
cites work: Spaces determined by point-countable covers
cites work: Point-countable \(k\)-networks, closed maps, and related results
cites work: \(g\)-metrizability and \(S_ \omega\)
cites work: A quintuple quotient quest
cites work: Metrizability of topological groups having weak topologies with respect to good covers
cites work: Sequential order of product spaces
cites work: Spaces which contain a copy of \(S_{\omega}\) or \(S_ 2\) and their applications
cites work: On defining a space by a weak base
cites work: Products of Sequential Spaces
cites work: Metrizabillity of certain quotient space
cites work: Q4305587
description: scientific article; zbMATH DE number 1103050