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Open subgroups of free abelian topological groups

Published online by Cambridge University Press:  24 October 2008

Sidney A. Morris  and
Vladimir G. Pestov
Sidney A. Morris
Affiliation:
University of Wollongong, Wollongong, N.S.W. 2522, Australia
Vladimir G. Pestov
Affiliation:
Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand
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Extract

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 3 , November 1993 , pp. 439 - 442
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Brown, R.. Some non-projective subgroups of free topological groups. Proc. Amer. Math. Soc. 52 (1975), 433–441.CrossRefGoogle Scholar
[2]Brown, R. and Hardy, J. P. L.. Subgroups of free topological groups and free topological products of topological groups. J. London Math. Soc. (2) 10 (1975), 431–440.CrossRefGoogle Scholar
[3]Graev, M. I.. Free topological groups. Amer. Math. Soc. Transl. (1) 8 (1962), 305–364.Google Scholar
[4]Gul'ko, S. P.. On uniform homeomorphisms of spaces of continuous functions. Proc. Baku Intern. Topol. Conf. (1987), Trudy Mat. Inst. Steklova 193 (1990) (in Russian).Google Scholar
[5]Hunt, D. C. and Morris, S. A.. Free subgroups of free topological groups. Proc. Second Internat. Conf. Theory of Groups (Canberra 1973), Lect. Notes Math. 372, (Springer-Verlag, 1974), 377–387.CrossRefGoogle Scholar
[6]Katz, E., Morris, S. A. and Nickolas, P.. Free abelian topological groups on spheres. Quart. J. Math. Oxford (2) 35 (1984), 173–181.CrossRefGoogle Scholar
[7]Katz, E., Morris, S. A. and Nickolas, P.. Free subgroups of free abelian topological groups. Math. Proc. Camb. Philos. Soc. 100 (1986), 347–353.CrossRefGoogle Scholar
[8]Leiderman, A. G., Morris, S. A. and Pestov, V. G.. The free abelian topological group and the free locally convex space on the unit interval (to appear).Google Scholar
[9]Markov, A. A.. On free topological groups. Doklady Akad. Nauk. SSSR 31 (1941), 299–301 (in Russian).Google Scholar
[10]Markov, A. A.. On free topological groups. Amer. Math. Soc. Transl. 30 (1950), 11–88.Google Scholar
[11]Morris, S. A.. Free abelian topological groups. Categorical Topology, Proc. Conference Toledo, Ohio, 1983 (Heldermann-Verlag, 1984), 375–391.Google Scholar
[12]Nickolas, P.. Subgroups of the free topological group on [0, 1]. J. London Math. Soc. (2) 12 (1976), 199–205.CrossRefGoogle Scholar
[13]Pestov, V. G.. The coincidence of the dimensions dim of 1-equivalent topological spaces. Soviet Math. Dokl. 26 (1982), 380–382.Google Scholar