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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 C C(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).

Information

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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7. Schroder, M., Continuous convergence in a Gelfand theory for topological algebras, Ph.D. thesis, Queen's University, Kingston, Ontario, 1971.Google Scholar