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VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

Published online by Cambridge University Press:  01 December 2008

CAROLYN E. MCPHAIL
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia (email: sandison@uow.edu.au)
SIDNEY A. MORRIS*
Affiliation:
School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat, VIC 3353, Australia (email: s.morris@ballarat.edu.au)
*
For correspondence; e-mail: s.morris@ballarat.edu.au
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Abstract

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The variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Type
Research Article
Copyright
Copyright © 2009 Australian Mathematical Society

References

[1]Brooks, M. S., Morris, S. A. and Saxon, S. A., ‘Generating varieties of topological groups’, Proc. Edinburgh Math. Soc. (2) 18 (1973), 191197.CrossRefGoogle Scholar
[2]Engelking, R., General Topology (PWN—Polish Scientific Publishers, Warszawa, Poland, 1977).Google Scholar
[3]Graev, M. I., ‘Free topological groups’, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 279324 (Russian); English translation, Amer. Math. Soc. Transl. no. 35 (1951); reprint, Amer. Math. Soc. Transl. 8(1) (1962), 305–364.Google Scholar
[4]Hall, Jr., M., The Theory of Groups (The Macmillan Company, New York, 1959).Google Scholar
[5]Kuratowski, K., Topology, Vol. I (PWN – Polish Scientific Publishers, Warszawa, Poland, 1966).Google Scholar
[6]Mack, J., Morris, S. A. and Ordman, E. T., ‘Free topological groups and the projective dimension of a locally compact abelian group’, Proc. Amer. Math. Soc. 40 (1973), 303308.CrossRefGoogle Scholar
[7]McPhail, C. E. and Morris, S. A., ‘Identifying and Distinguishing various varieties of abelian topological groups’, Dissertationes Math. to appear.Google Scholar
[8]Morris, S. A., ‘Varieties of topological groups’, Bull. Austral. Math. Soc. 1 (1969), 145160.CrossRefGoogle Scholar
[9]Morris, S. A., ‘Varieties of topological groups and left adjoint functors’, J. Austral. Math. Soc. 16 (1973), 220227.CrossRefGoogle Scholar
[10]Morris, S. A., Pontryagin Duality and the Structure of Locally Compact Abelian Groups (Cambridge University Press, Cambridge, 1977).CrossRefGoogle Scholar
[11]Morris, S. A., ‘Varieties of topological groups — a survey’, Colloq. Math. 46 (1982), 147165.CrossRefGoogle Scholar
[12]Morris, S. A., ‘Free abelian topological groups’, in: Proc. Internat. Conf. Categorical Topology (Toledo, Ohio, 1983) (Heldermann, Berlin, 1984), pp. 375391.Google Scholar
[13]Steenrod, N. E., ‘A convenient category of topological spaces’, Mich. Math. J. 14 (1967), 133152.CrossRefGoogle Scholar
[14]Uspenskiĭ, V., Private communication, 27 March 1999.Google Scholar