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The inductive limit of uncountably many compact Abelian groups

Published online by Cambridge University Press:  24 October 2008

D. L. Salinger
D. L. Salinger
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Cambridge
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Extract

In (2), Varopoulos examined the structure of topological groups that are the inductive limits of countably many locally compact Abelian groups. The purpose of this note is to show that the theory does not extend to the case of uncountably many groups. We give two examples, the first to show that the strict inductive limit of uncountably many compact Abelian groups need not be complete, the second to show it need not be separated by its continuous characters. The treatment of this latter half follows closely that given for topological linear spaces by Douady in (l).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 4 , October 1968 , pp. 985 - 987
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Douady, A.Topologie induite par une topologie localement convexe limite inductive. G.R. Acad. Sci., Paris 259 (1964), 2946–2947.Google Scholar
(2)Varopoutlos, N. Th.Studies in harmonic analysis. Proc. Cambridge Philos. Soc. 60 (1964), 465–516.CrossRefGoogle Scholar