On C-alpha-compact subsets

For an infinite cardinal alpha, we say that a subset B of a space X is C-alpha-compact in X if for every continuous function f:X --> R-alpha, f[B] is a compact subset of R-alpha. This concept slightly generalizes the notion of alpha-pseudocompactness introduced by J.F. Kennison: a space X is alpha-pseudocompact if X is C-alpha-compact in itself. If alpha = omega, then we say C-compact instead of C-omega-compact and omega-pseudocompactness agrees with pseudocompactness. We generalize Tamano's theorem on the pseudocompactness of a product of two spaces as follows: let A subset of or equal to X and B subset of or equal to Y be such that A is z-embedded in X. Then the following three conditions are equivalent: (1) A x B is C-alpha-compact in X x