自由拓扑群中可数性公理的两种弱形式
Two Weak Forms of Countability Axioms in Free Topological Groups
摘要：
给定一个Tychonoff空间$X$，设$F(X)$和$A(X)$分别是$X$上的自由拓扑群和自由Abel拓扑群.对于每个$n\in\mathbb{N}$，令$F_{n}(X)$（分别$A_n(X)$）表示$F(X)$的子空间（分别$A(X)$），它由相对于自由基$X$而言最多$n$长度减小的字组成。本文讨论了$F(X)$或$A(X)$中可数性公理的两种弱形式，即$csf$ -可数性和$snf$ -可数性.本文给出了空间$X$中$F(X)$和$A(X)$的$csf$ -可数性和$snf$ -可数性的一些刻画.此外，对于$n=2, 3, 4$，我们还研究了$F_n(X)$或$A_n(X)$的$csf$ -可数性和$snf$ -可数性.推广了Arhangel'ski \v在\cite{A1980}和Yamada在\cite{Y1998}中的一些结果.对Li等人在\cite{LLL}中提出的一个开放性问题给出了肯定的回答。
英文摘要：
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 $n\in\mathbb{N}$, 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 \cite{A1980} and Yamada in \cite{Y1998} are generalized. An affirmative answer to an open question posed by Li et al. in \cite{LLL} is provided.
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