For an infinite cardinal α, we say that a subset B of a space X is C α - compact in X if for every continuous function f : X → R α , [ B] is a compact subset of R α. This concept slightly generalizes the notion of α-pseudocompactness introduced by J.F. Kennison: a space X is α-pseudocompact if X is C α - compact in itself. If α = ω, then we say C- compact instead of C ω - compact and ω-pseudocompactness agrees with pseudocompactness. We generalize Tamano's 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 C α - compact in X × Y; (2) A and B are C α - compact in X and Y, respectively, and the projection map π : X × Y → X is a z α - 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 z α- map with respect to A × B and A (the z α - maps are the strongly z α - maps are natural generalizations of the z- maps and the strongly z- maps, respectively). The degree of C α - 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 C α- 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 C α - compact in X ̂ , where ( X ̂ , U ̂ ), is the completion of ( X, U) iff f( A) is a compact subset of R α from every uniformly continuous function from X into R α; we characterize the C α - compact subsets of topological groups; and we also prove that if G i : i I is a set of topological groups 15 and A i is a C α - compact subset of G α for all i I, then Π iI A i is a C α - compact subset of Π iI G i .