Characterizations of a metrizable space \(X\) such that every \(A_ n(X)\) is a \(k\)-space
type: Item
instance of: scholarly article
title: Characterizations of a metrizable space \(X\) such that every \(A_ n(X)\) is a \(k\)-space
zbMATH Open document ID: 0817.54020
DOI: 10.1016/0166-8641(93)90130-6
author: Kohzo Yamada
published in: Topology and its Applications
publication date: 1993-06-29T00:00:00Z
review text: The author considers the free abelian topological group \(A(X)\) over a metrizable space \(X\) and proves the following interesting results: Theorem 1. For a metrizable space \(X\), the following conditions are equivalent: (a) \(A_ n(x)\) is a \(k\)-space for each \(n\in \mathbb{N}^ +\); (b) \(A_ 4(X)\) is a \(k\)-space; (c) either \(X\) is locally compact and the set \(X'\) of all non-isolated points of \(X\) is separable, or \(X'\) is compact. Theorem 2. The following conditions are equivalent for a metrizable space \(X\): (a) \(A_ 3 (X)\) is a \(k\)-space; (b) \(X\) is locally compact or \(X'\) is compact. Reviewer's note: Theorem 2.3 of the paper coincides with Theorem 1 proved by the reviewer in [Sov. Math., Dok. 27, No. 2, 341-345 (1983); translation from Dokl. Akad. Nauk SSSR 269, 299-303 (1983; Zbl 0521.22002)].
Mathematics Subject Classification ID: 54D50
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 221873
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
full work available at URL: 0166-8641(93)90130-6
OpenAlex ID: W1971665674
cites work: Q5577640
cites work: Algebraic objects generated by topological structure
cites work: Free topological groups over metrizable spaces
cites work: Q3994935
cites work: The free topological group over the rationals
cites work: Q5723435
cites work: Free Topological Groups and Dimension
cites work: Free Topological Groups and the Projective Dimension of a Locally Compact Abelian Group
cites work: Uniform free topological groups and Samuel compactifications
cites work: Q3697250
cites work: Q4043263
cites work: Q3671393
cites work: On topologies of free groups
cites work: Q5186348
cites work: Q4885599
description: scientific article; zbMATH DE number 221873