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Varieties of topological groups III

Published online by Cambridge University Press:  17 April 2009

Sidney A. Morris
Sidney A. Morris
Affiliation:
The Flinders University of South Australia, Bedford Park, South Australia, and University of Adelaide, Adelaide, South Australia.
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Abstract

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This paper continues the invèstigation of varieties of topological groups. It is shown that the family of all varieties of topological groups with any given underlying algebraic variety is a class and not a set. In fact the family of all β-varieties with any given underlying algebraic variety is a class and not a set. A variety generated by a family of topological groups of bounded cardinal is not a full variety.

The varieties V(R) and V(T) generated by the additive group of reals and the circle group respectively each with its usual topology are examined. In particular it is shown that a locally compact Hausdorff abelian group is in V(T) if and only if it is compact. Thus V(R) properly contains V(T).

It is proved that any free topological group of a non-indiscrete variety is disconnected. Finally, some comments are made on topologies on free groups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 2 , April 1970 , pp. 165 - 178
Copyright
Copyright © Australian Mathematical Society 1970

References

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