S₂ and the Fréchet property of free topological groups

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• 作者：Zhangyong Cai^a ; ^{zycaigxu2002@126.com" class="auth_mail" title="E-mail the corresponding author} ; Shou Lin^b ; ^{shoulin60@163.com" class="auth_mail" title="E-mail the corresponding author} ; Chuan Liu^c ; ^{liuc1@ohio.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：22A05 ; 54A20 ; 54D50 ; 54D55
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：204
• 期：Complete
• 页码：103-111
• 全文大小：348 K

文摘

Let F(X)$F(X)$ denote the free topological group over a Tychonoff space X  , F_n(X)$F_n(X)$ denote the subspace of F(X)$F(X)$ that consists of all words of reduced length ≤n with respect to the free basis X for every non-negative integer n   and E_n(X)=F_n(X)∖F_n−1(X)$E_n(X)=F_n(X)\setminus F_{n-1}(X)$ for n≥1$n\ge 1$. In this paper, we study topological properties of free topological groups in terms of Arens' space S₂$S_2$. The following results are obtained.

(1) If the free topological group F(X)$F(X)$ over a Tychonoff space X   contains a non-trivial convergent sequence, then F(X)$F(X)$ contains a closed copy of S₂$S_2$, equivalently, F(X)$F(X)$ contains a closed copy of S_ω$S_{\omega }$, which extends [6, Theorem 1.6].

(2) Let X   be a topological space and A={n₁,...,n_i,...}$A=\{n_1,...,n_i,...\}$ be an infinite subset of N$\mathbb{N}$. If C=⋃_i∈NE_ni(X)$C={\bigcup }_{i\in \mathbb{N}}{E}_{n_i}(X)$ is κ  -Fréchet–Urysohn and contains no copy of S₂$S_2$, then X is discrete, which improves [15, Proposition 3.5].

(3) If X is a μ  -space and F₅(X)$F_5(X)$ is Fréchet–Urysohn, then X is compact or discrete, which improves [15, Theorem 2.4].

At last, a question posed by K. Yamada is partially answered in a shorter alternative way by means of a Tanaka's theorem concerning Arens' space S₂$S_2$.