A characterization of the topology of compact convergence on C( X)

Abstract

Let C( X) be the algebra of all K-valued continuous functions K = R or C on a normal (and T 1) topological space X, and let τ X be the topology of compact convergence on C( X). The following properties are well known: 1. (i) ( C( X), τ X ) is a locally m-convex K-algebra; 2. (ii) the set C( X): 0 ∉ ( K) is open in ( C( X), τ X ) for each compact subset K of X; 0806 118 3. (iii) the only closed ideals of ( C( X), τ X ) are the ideals of the form C( X): ( C) = 0 0806 118 (with C ⊆ X). In this paper we prove that properties (i), (ii) and (iii) characterize the topology τ X .