Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-25T11:35:04.665Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on B- and Br- incomplete topological Abelian groups

Published online by Cambridge University Press:  24 October 2008

L. J. sulley
L. J. sulley
Affiliation:
University College, Swansea
Get access Rights & Permissions [Opens in a new window]

Extract

From results of Baker (2) it appeared to be unlikely that, in Hausdorff topological Abelian groups, completeness would be implied by Br- or B-completeness (for definitions, see below). We show here (Corollary 1) that the group of all complex roots of unity, though not complete, is B-complete. Another example, for the suggestion of which we are indebted to Dr J. W. Baker, is used to show (Corollary 2) that B-completeness is not a consequence of Br-completeness. It is proved, however, that B- and Br-complete Hausdorff topological Abelian groups are embedded in their completions in special ways in relation to the closed subgroups of the completions. Also, the completions of B- and Br-complete Hausdorff topological Abelian groups are shown to be, respectively, B- and Br-complete.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 66 , Issue 2 , September 1969 , pp. 275 - 279
Copyright
Copyright © Cambridge Philosophical Society 1969

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Baker, J. W.Topological groups and the closed-graph theorem. J. London Math. Soc. 42 (1967), 217–225.CrossRefGoogle Scholar
(2)Baker, J. W.Inversion theorems and compactness, J. London Math. Soc. 42 (1967), 409–415.CrossRefGoogle Scholar
(3)Bourbaki, N.Éléments de mathématique Topologie génerale, Livre III, 3rd ed. chapter 3 (Hermann; Paris, 1960).Google Scholar
(4)Pontryagin, L. S.Topological groups (Princeton University Press; Princeton, 1939).Google Scholar
(5)Pták, V.Completeness and the open mapping theorem, Bull. Soc. Math. France, 86 (1958), 41–74.CrossRefGoogle Scholar