Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-16T09:21:15.709Z Has data issue: false hasContentIssue false
  • English
  • Français

Compactness in Hom(G, H)

Published online by Cambridge University Press:  20 November 2018

H. H. Corson  and
I. Glicksberg
H. H. Corson
Affiliation:
University of Washington, Seattle, Washington
I. Glicksberg
Affiliation:
University of Washington, Seattle, Washington
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a locally compact abelian group with Bohr compactification Ga. Then [3, Theorem 1.2] any subset F of G compact in Ga is necessarily compact in G; alternatively, any closed non-compact subset F of G has its closure F– in Ga ≠ F; hence F –\F ≠ ø. One of our aims in the present note is to give a result (Corollary 6) which asserts that F –\F has no points which are Gδs, so that F–\F is a perfect set. Another aim is to give an extension of a cited result of [3] in which commutativity and local compactness are essentially irrelevant, and to unify the proofs.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 164 - 170
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Bourbaki, N., Eléments de mathématique. I:Les structures fondamentales de l'analyse. Fasc. VIII. Livre III : Topologie générale. Chapitre 9: Utilisation des nombres réels en topologie générale, 2e éd., Actualités Sci. Indust., No. 1045 (Hermann, Paris, 1958).Google Scholar
2. Day, M. M., Normed linear spaces (Springer, Berlin, 1958).Google Scholar
3. Glicksberg, I., Uniform boundedness for groups, Can. J. Math. 14 (1962), 269–276.Google Scholar
4. Grothendieck, A., Critères de compacité dans les espaces fonctionelles généraux, Amer. J. Math. 74 (1952), 168–186.Google Scholar