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

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

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.

**1.**A. V. Arhangel'ski,*Algebraic objects generated by topological structure*, J. Soviet Math.**45**(1989) 956-978.**2.**A. V. Arhangel'ski, O. G. Okunev and V. G. Pestov,*Free topological groups over metrizable spaces*, Topology Appl.**33**(1989) 63-76. MR**90h:22002****3.**E. Hewitt and K. Ross,*Abstract harmonic analysis I*, Academic Press, New York (1963). MR**28:158****4.**A. A. Markov,*On free topological groups*, Izv. Akad. Nauk SSSR Ser. Mat.**9**(1945) 3-64 (in Russian); Amer. Math. Soc. Transl. Ser. 1**8**(1950) 195-272. MR**12:318b****5.**V. Pestov and K. Yamada,*Free topological groups on metrizable spaces and inductive limits*, Topology Appl.**98**(1999) 291-301. MR**2000j:54044****6.**K. Yamada,*Characterizations of a metrizable space such that every is a -space*, Topology Appl.**49**(1994) 75-94. MR**94g:54019****7.**K. Yamada,*Metrizable subspaces of free topological groups on metrizable spaces*, Topology Proc.**23**(2000) 379-409. CMP**2001:06**

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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:
https://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