The k-spaces property of the free Abelian topological groups over non-metrizable Lasnev spaces

摘要

Given a Tychonoff space X, let A(X) be the free Abelian topological group over X in the sense of Markov. For every n is an element of N, let An (X) denote the subspace of A(X) that consists of words of reduced length at most n with respect to the free basis X. In this paper, we show that for an arbitrary non-metrizable Lasnev space X, the subspace A(4)(X) is a k-space if and only if A(X) is a k-space, which gives a complementary for one result of K. Yamada's. Under the assumption of b = omega(1), we also show that for an arbitrary non-metrizable Lagnev space X, the subspace A(3) (X) is a k-space if and only if A(X) is a k-space. However, under the assumption of b > omega(1), we provide a non-metrizable Laanev space X such that A(3) (X) is a k-space but A(X) is not a k-space. (C) 2017 Elsevier B.V. All rights reserved.