Fréchet-Urysohn spaces in free topological groups
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- by Kohzo Yamada PDF
- Proc. Amer. Math. Soc. 130 (2002), 2461-2469 Request permission
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 $n\geq 3$ and $F_n(X)$ is Fréchet-Urysohn for each $n\geq 3$ 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.References
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Additional Information
- Kohzo Yamada
- Affiliation: Department of Mathematics, Faculty of Education, Shizuoka University, Shizuoka, 422 Japan
- Email: eckyama@ipc.shizuoka.ac.jp
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2461-2469
- MSC (1991): Primary 54H11, 54A35, 54A25
- DOI: https://doi.org/10.1090/S0002-9939-02-06343-8
- MathSciNet review: 1897473