Characterizations of a metrizable space \(X\) such that every \(A_ n(X)\) is a \(k\)-space
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)].
scientific article; zbMATH DE number 221873