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Locally compact products and coproducts in categories of topological groups

Published online by Cambridge University Press:  17 April 2009

Karl Heinrich Hofmann  and
Sidney A. Morris
Karl Heinrich Hofmann
Affiliation:
Department of Mathematics, Tulane University, New Orleans, Louisiana, USA;
Sidney A. Morris
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Victoria.
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Abstract

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In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 17 , Issue 3 , December 1977 , pp. 401 - 417
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Brown, Ronald, Higgins, Philip J. and Morris, Sidney A., “Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties”, Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32.CrossRefGoogle Scholar
[2]Semadeni, Z., “Projectivity, injectivity and duality”, Rozprawy Mat. 35 (1963), 47pp.Google Scholar
[3]Semadeni, Z., Probléme P490, Colloq. Math. 13 (1964–1965), 127.Google Scholar