On weak bases
type: Item
instance of: scholarly article
title: On weak bases
zbMATH Open document ID: 1081.54026
published in: Topology and its Applications
publication date: 2005-06-01T00:00:00Z
review text: \textit{F. Siwiec} [Pac. J. Math. 52, 233-245 (1974; Zbl 0285.54022)] introduced \(g\)-metrizable spaces as ones with a \(\sigma\)-locally finite weak base. This notation is motivated by the classical Nagata-Smirnov metrization theorem. \textit{D. Burke, R. Engelking} and \textit{D. Lutzer} [Proc. Am. Math. Soc. 51, 483-488 (1975; Zbl 0307.54030)] proved that a topological space \(X\) is metrizable if and only if \(X\) has a \(\sigma\)-hereditarily closure-preserving base. Then \textit{Y. Tanaka} [ibid. 112, 283-290 (1991; Zbl 0770.54031)] asked whether a space with a \(\sigma\)-hereditarily closure-preserving weak base is \(g\)-metrizable. In this paper the author gives a positive answer to this question. On the other hand, following \textit{L. Foged's} result [Pac. J. Math. 98, 327-332 (1982; Zbl 0478.54025)], \textit{Y. Tanaka} [op. cit.] and \textit{S. Lin} [Chin. Ann. Math., Ser. A 13, 3, 403-409 (1992; Zbl 0770.54030)] (in Chinese) independently showed that a \(g\)-first countable space is \(g\)-metrizable if it has a \(\sigma\)-hereditarily closure-preserving \(k\)-network. Hence \textit{S. Lin} in [Generalized metric spaces and maps, (Beijing: China) Science Press, (1995; Zbl 0940.54002)] (in Chinese) asked whether every weak base of a topological space is a \(k\)-network. Although a point-countable weak base is a \(k\)-network and for a space each of whose point is a \(G_{\delta}\)-set every weak base is a \(k\)-network, here the author gives a negative example. A typical example of a non \(g\)-first countable space is the sequential fan \(S_{\omega}\), which plays an important role in the investigation of \(g\)-metrizable spaces. In the second part of the paper the author applies this property to the study of mapping theorems on weak bases. He shows that \(\sigma\)-locally countable (\(\sigma\)-compact-finite) weak bases are preserved by closed irreducible sequence-covering mappings. The author also discusses a product theorem of spaces having a point-coun…
reviewed by: Akira Koyama
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 2173256
zbMATH Keywords: \(g\)-first countable spaces
zbMATH Keywords: hereditarily closure-preserving covers
zbMATH Keywords: \(cs\)-networks
author: Chuan Liu
MaRDI profile type: Publication
full work available at URL: j.topol.2004.11.008
OpenAlex ID: W2048463273
cites work: MAPPINGS AND SPACES
cites work: Hereditarily Closure-Preserving Collections and Metrization
cites work: On g-metrizability
cites work: Q4186976
cites work: Q3218783
cites work: Spaces determined by point-countable covers
cites work: Q5617506
cites work: Q4038222
cites work: A note on the Arens' space and sequential fan
cites work: Q4256665
cites work: On spaces with point-countable cs-networks
cites work: Point-countable \(k\)-networks, closed maps, and related results
cites work: Q4004604
cites work: Q4822577
cites work: \(g\)-metrizability and \(S_ \omega\)
cites work: Q5526599
cites work: On Paracompactness in Function Spaces with the Compact-Open Topology
cites work: Sequence-covering and countably bi-quotient mappings
cites work: On defining a space by a weak base
cites work: On quasi-$k$-spaces
cites work: σ-Hereditarily Closure-Preserving k-Networks and g-Metrizability
cites work: Q4305587
cites work: Point-countable \(k\)-networks, \(cs^*\)-networks and \(\alpha_4\)-spaces
DOI: 10.1016/J.TOPOL.2004.11.008
Recommended article: Q4822577
Recommended article: Remarks on weak-bases
Recommended article: Q3351966
Recommended article: Weak one-basedness
Recommended article: Q3169806
Recommended article: Weak bases in modular lattices
Recommended article: Q5743692
Recommended article: Q3841341
Recommended article: Weak bases and metrizability
Recommended article: Q3015329
description: scientific article; zbMATH DE number 2173256