A Functional Analytic Description of Normal Spaces

Abstract

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).