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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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′skiĭ, O. G. Okunev, and V. G. Pestov,*Free topological groups over metrizable spaces*, Topology Appl.**33**(1989), no. 1, 63–76. MR**1020983**, 10.1016/0166-8641(89)90088-6**3.**Edwin Hewitt and Kenneth A. Ross,*Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations*, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR**0156915****4.**A. A. Markov,*Three papers on topological groups: I. On the existence of periodic connected topological groups. II. On free topological groups. III. On unconditionally closed sets*, Amer. Math. Soc. Translation**1950**(1950), no. 30, 120. MR**0037854****5.**Vladimir Pestov and Kohzo Yamada,*Free topological groups on metrizable spaces and inductive limits*, Topology Appl.**98**(1999), no. 1-3, 291–301. II Iberoamerican Conference on Topology and its Applications (Morelia, 1997). MR**1720007**, 10.1016/S0166-8641(98)00103-5**6.**Kohzo Yamada,*Characterizations of a metrizable space 𝑋 such that every 𝐴_{𝑛}(𝑋) is a 𝑘-space*, Topology Appl.**49**(1993), no. 1, 75–94. MR**1202877**, 10.1016/0166-8641(93)90130-6**7.**K. Yamada,*Metrizable subspaces of free topological groups on metrizable spaces*, Topology Proc.**23**(2000) 379-409. CMP**2001:06**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
54H11,
54A35,
54A25

Retrieve articles in all journals with MSC (1991): 54H11, 54A35, 54A25

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