On spaces with point-countable cs-networks
type: Item
instance of: scholarly article
title: On spaces with point-countable cs-networks
zbMATH Open document ID: 0869.54036
DOI: 10.1016/S0166-8641(96)00043-0
published in: Topology and its Applications
publication date: 1997-09-01T00:00:00Z
review text: A cover \({\mathcal P}\) of a space \(X\) is called a \(cs\)-network (resp. \(cs^*\)-network) if, for any open neighbourhood \(U\) of \(x\), and for any sequence \(L\) converging to \(x\), there exists \(P\in{\mathcal P}\) such that \(x\in P\subset U\), and \(P\) contains \(L\) eventuelly (resp. frequently). This paper obtains the results (A)\(\sim\)(D) below related to the following questions. \((\text{Q}_1)\) is whether every sequential space with a point-countable \(cs^*\)-network is a compact-covering quotient \(s\)-image of a metric space, and \((\text{Q}_2)\) is whether Fréchet spaces with point-countable \(cs^*\)-networks are preserverd by pseudo-open \(s\)-maps. \((\text{Q}_1)\) Is every quotient \(s\)-image of a metric space a compact-covering quotient \(s\)-image of a metric space? (\textit{E. Michael} and \textit{K. Nagami}'s question in [Proc. Am. Math. Soc. 37, 260-266 (1973; Zbl 0228.54008, Zbl 0245.54023)]). \((\text{Q}_2)\) Are pseudo-open \(s\)-images of metric spaces preserved by pseudo-open \(s\)-maps? (\textit{G. Gruenhage}, \textit{E. Michael} and \textit{Y. Tanaka}'s question in [Pac. J. Math. 113, 303-332 (1984; Zbl 0561.54016)]). \((\text{Q}_3)\) For quotient \(s\)-images \(X\) and \(Y\) of metric spaces, what is a necessary and sufficient condition for \(X\times Y\) to be a \(k\)-space? (\textit{Y. Tanaka}'s question in [Quest. Answers Gen. Topology 1, 36-50 (1983; Zbl 0545.54020)]). (A) \(X\) is a sequential space with a point-countable \(cs\)-network iff \(X\) is a compact-covering, sequence-covering, quotient and \(s\)-image of a metric space. (B) Every closed map from a space \(X\) with a point-countable \(k\)-network is compact-covering if each point of \(X\) is a \(G_\delta\)-set. (C) Every quotient \(s\)-image of a Fréchet space with a point-countable \(cs\)-network has a point-countable \(cs^*\)-network. (D) Let \(X\) and \(Y\) be sequential spaces with point-countable \(cs\)-networks. Then \(X\times Y\) is a \(k\)-space iff one of t…
reviewed by: Y. Tanaka
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 54D50
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D55
zbMATH DE Number: 987911
zbMATH Keywords: \(cs\)-network
author: Shou Lin
author: Chuan Liu
MaRDI profile type: Publication
cites work: MAPPINGS AND SPACES
cites work: On certain point-countable covers
cites work: Q3776238
cites work: k-Spaces and Products of Closed Images of Metric Spaces
cites work: Spaces determined by point-countable covers
cites work: A characterization of ℵ0-spaces
cites work: Point-countable \(k\)-networks, closed maps, and related results
cites work: A note on closed maps and compact sets
cites work: Q5526599
cites work: Bi-quotient maps and Cartesian products of quotient maps
cites work: A quintuple quotient quest
cites work: Compact-Covering Images of Metric Spaces
cites work: On Paracompactness in Function Spaces with the Compact-Open Topology
cites work: Sequence-covering and countably bi-quotient mappings
cites work: Q3862722
cites work: Metrizabillity of certain quotient space
cites work: Q3334606
cites work: Q3831701
cites work: Q4305587
full work available at URL: s0166-8641(96)00043-0
OpenAlex ID: W2154821379
Recommended article: Weak neighborhoods and Michael-Nagami's question
Recommended article: Q4256073
Recommended article: Q3413870
Recommended article: Sequence-covering cs-images of metric spaces
Recommended article: Q3154186
Recommended article: Q4821661
Recommended article: Some sequence-covering locally countable images of locally separable metric spaces
Recommended article: Q4778316
Recommended article: Q4700917
Recommended article: On sequence-covering strong s-mappings
description: scientific article; zbMATH DE number 987911