On Cα-compact subsets

For an infinite cardinal α, we say that a subset B of a space X is Cα-compact in X if for every continuous function f : X → Rα, [B] is a compact subset of Rα. This concept slightly generalizes the notion of α-pseudocompactness introduced by J.F. Kennison: a space X is α-pseudocompact ifX is Cα-compact in itself. If α = ω, then we say C-compact instead of Cω-compact and ω-pseudocompactness agrees with pseudocompactness. We generalize Tamanoʹs theorem on the pseudocompactness of a product of two spaces as follows: let A ⊆ X and B ⊆ Y be such that A is z-embedded in X. Then the following three conditions are equivalent: (1) A × B is Cα-compact in X × Y; (2) A and B are Cα-compact in X and Y, respectively, and the projection map π : X × Y → X is a zα-map with respect to A × B and A; and (3) A and B are Cω-compact in X and Y, respectively, and the projection map π : X × Y → X is a strongly zα-map with respect to A × B and A (the zα-maps are the strongly zα-maps are natural generalizations of the z-maps and the strongly z-maps, respectively). The degree of Cα-compactness of a C-compact subset B of a space X is defined by: ϱ(B,X) = ∞ if B is compact, and if B is not compact, then ϱ(B,X) = supα: B is Cα-compact in X. We estimate the degree of pseudocompactness of locally compact pseudocompact spaces, topological products and Σ-products. We also establish the relation between the pseudocompact degree and some other cardinal functions. In the context of uniform spaces, we show that if A is a bounded subset of a uniform space (X,U), then A is Cα-compact in X̂, where (X̂,Û), is the completion of (X,U) iff f(A) is a compact subset of Rα from every uniformly continuous function from X into Rα; we characterize the Cα-compact subsets of topological groups; and we also prove that if Gi: i I is a set of topological groups 15 and Ai is a Cα-compact subset of Gα for all i I, then ΠiI Ai is a Cα-compact subset of ΠiI Gi.