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A Functional Analytic Description of Normal Spaces

Published online by Cambridge University Press:  20 November 2018

E. Binz  and
W. Feldman
E. Binz
Affiliation:
Universität Mannheim, Mannheim, West Germany
W. Feldman
Affiliation:
Queen's University, Kingston, Ontario
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Throughout the paper, X will denote a completely regular (Hausdorff) topological space and C(X) the R-algebra of all real-valued continuous functions on X. When this algebra carries the continuous convergence structure [1], we write CC(X). We note that CC(X)is a complete [5] convergence R-algebra [1].

Our description of normality reads as follows. A completely regular topological space X is normal if and only if CC(X)/J (endowed with the obvious quotient structure; see § 1) is complete for every closed ideal J ⊂ CC(X).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 45 - 49
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Binz, E. and Keller, H. H., Funktionenräume in der Kategorie der Limesräume, Ann. Acad. Sci. Fenn. Ser. AI 383 (1966), 1–21.Google Scholar
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3. Binz, E., Zu den Beziehungen zwischen c-einbettbaren Limesräumen und ihren limitierten Funktionenalgebren, Math. Ann. 181 (1969), 45–52.Google Scholar
4. Binz, E., On closed ideals in convergence function algebras, Math. Ann. 182 (1969), 145–153.Google Scholar
5. Binz, E., Notes on a characterization of function algebras, Math. Ann. 186 (1970), 314–326.Google Scholar
6. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, N.J. 1960).Google Scholar
7. Schroder, M., Continuous convergence in a Gelfand theory for topological algebras, Ph.D. thesis, Queen's University, Kingston, Ontario, 1971.Google Scholar