Two weak forms of countability axioms in free topological groups

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• 作者：Fucai Lin^a ; ^{linfucai2008@aliyun.com" class="auth_mail" title="E-mail the corresponding author} ; Chuan Liu^b ; ^{liuc1@ohio.edu" class="auth_mail" title="E-mail the corresponding author} ; Jiling Cao^c ; ^{jiling.cao@aut.ac.nz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 54H11 ; 22A05 ; secondary ; 54E20 ; 54E35 ; 54D50 ; 54D55
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 July 2016
• 年：2016
• 卷：207
• 期：Complete
• 页码：96-108
• 全文大小：422 K

文摘

Given a Tychonoff space X  , let F(X)$F(X)$ and 18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)$A(X)$ be respectively the free topological group and the free Abelian topological group over X   in the sense of Markov. For every n∈N$n\in \mathbb{N}$, let 18ee" title="Click to view the MathML source">F_n(X)$F_n(X)$ (resp. A_n(X)$A_n(X)$) denote the subspace of F(X)$F(X)$ (resp. 18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)$A(X)$) 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)$F(X)$ or 18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)$A(X)$, namely the csf-countability and snf-countability. We provide some characterizations of the csf-countability and snf  -countability of F(X)$F(X)$ and 18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)$A(X)$ for various classes of spaces X. In addition, we also study the csf-countability and snf  -countability of 18ee" title="Click to view the MathML source">F_n(X)$F_n(X)$ or A_n(X)$A_n(X)$, for n=2,3,4$n=2,3,4$. Some results of Arhangel'skiı̌ in [1] and Yamada in [20] are generalized. An affirmative answer to an open question posed by Li et al. in [11] is provided.