Two Weak Forms of Countability Axioms in Free Topological Groups

2015/07/15 by Lin, Fucai, Liu, Chuan, Cao, Jiling · 1 citation
#22A05 (Primary) #54D50 #54D55 (Secondary) #54E20 #54E35 #54H11 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1507.04087

Abstract

Given a Tychonoff space X, let F(X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. For every n∈ℕ, let F_n(X) (resp. A_n(X)) denote the subspace of F(X) (resp. A(X)) that consists of words of reduced length at most n with respect to the free basis X. In this paper, we discuss two weak forms of countability axioms in F(X) or A(X), namely the csf-countability and snf-countability. We provide some characterizations of the csf-countability and snf-countability of F(X) and A(X) for various classes of spaces X. In addition, we also study the csf-countability and snf-countability of F_n(X) or A_n(X), for n=2, 3, 4. Some results of Arhangel'ski\vı in \citeA1980 and Yamada in \citeY1998 are generalized. An affirmative answer to an open question posed by Li et al. in \citeLLL is provided.

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