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Fréchet-Urysohn spaces in free topological groups

Author: Kohzo Yamada
Journal: Proc. Amer. Math. Soc. 130 (2002), 2461-2469
MSC (1991): Primary 54H11, 54A35, 54A25
Published electronically: February 4, 2002
MathSciNet review: 1897473
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Abstract: Let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group on a Tychonoff space $X$. For every natural number $n$ we denote by $F_n(X)$ ($A_n(X)$) the subset of $F(X)$($A(X)$) consisting of all words of reduced length $\leq n$. It is well known that if a space $X$ is not discrete, then neither $F(X)$ nor $A(X)$ is Fréchet-Urysohn, and hence first countable. On the other hand, it is seen that both $F_2(X)$ and $A_2(X)$ are Fréchet-Urysohn for a paracompact Fréchet-Urysohn space $X$. In this paper, we prove first that for a metrizable space $X$, $F_3(X)$ ($A_3(X)$) is Fréchet-Urysohn if and only if the set of all non-isolated points of $X$ is compact and $F_5(X)$ is Fréchet-Urysohn if and only if $X$ is compact or discrete. As applications, we characterize the metrizable space $X$ such that $A_n(X)$ is Fréchet-Urysohn for each $n\geq3$ and $F_n(X)$ is Fréchet-Urysohn for each $n\geq3$ except for $n=4$. In addition, however, there is a first countable, and hence Fréchet-Urysohn subspace $Y$ of $F(X)$ ($A(X)$) which is not contained in any $F_n(X)$ ($A_n(X)$). We shall show that if such a space $Y$is first countable, then it has a special form in $F(X)$ ($A(X)$). On the other hand, we give an example showing that if the space $Y$ is Fréchet-Urysohn, then it need not have the form.

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Additional Information

Kohzo Yamada
Affiliation: Department of Mathematics, Faculty of Education, Shizuoka University, Shizuoka, 422 Japan

Keywords: Free topological group, free Abelian topological group, Fr\'echet-Urysohn space, first countable space, semidirect product
Received by editor(s): June 20, 2000
Received by editor(s) in revised form: March 7, 2001
Published electronically: February 4, 2002
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society