Functions with pseudocompact support

AbstractLet X be a completely regular Hausdorff space, C(X) the ring of real-valued continuous functions on X, CK the ideal of functions with compact support, I the intersection of the free maximal ideals of C(X), and Cψ the ideal of functions with pseudocompact support. For any space, CK ⊆ I ⊆ Cψ. When CK = I, or I = Cψ, or CK = Cψ , it is said that X is μ-compact, η-compact, or ψ-compact, respectively. These ideals and spaces are characterized in terms of the ideal structure of C(X) and the topological structure of βX. Also, μ-, η-, and ψ-compactifications are constructed. Examples and counterexamples are given