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 and be respectively the free topological group and the free Abelian topological group on a Tychonoff space . For every natural number we denote by () the subset of () consisting of all words of reduced length . It is well known that if a space is not discrete, then neither nor is Fréchet-Urysohn, and hence first countable. On the other hand, it is seen that both and are Fréchet-Urysohn for a paracompact Fréchet-Urysohn space . In this paper, we prove first that for a metrizable space , () is Fréchet-Urysohn if and only if the set of all non-isolated points of is compact and is Fréchet-Urysohn if and only if is compact or discrete. As applications, we characterize the metrizable space such that is Fréchet-Urysohn for each and is Fréchet-Urysohn for each except for . In addition, however, there is a first countable, and hence Fréchet-Urysohn subspace of () which is not contained in any (). We shall show that if such a space is first countable, then it has a special form in (). On the other hand, we give an example showing that if the space 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

Email:
eckyama@ipc.shizuoka.ac.jp

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06343-8

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