On C-alpha-compact subsets

Abstract

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 Y

(2) A and B are C-alpha-compact in X and Y, respectively, and the projection map pi:X x Y --> X is a z(alpha)-map with respect to A x B and A

and (3) A and B are C-alpha-compact in X and Y, respectively, and the projection map pi:X x Y --> X is a strongly z(alpha)-map with respect to A x B and A (the z(alpha)-maps and the strongly z(alpha)-maps are natural generalizations of the z-maps and the strongly z-maps, respectively). The degree of C alpha-compactness of a C-compact subset B of a space X is defined by: rho(B, X) = infinity if B is compact, and if B is not compact, then rho(B, X) = sup{alpha:B is C-alpha-compact in X}. We estimate the degree of pseudocompactness of locally compact pseudocompact spaces, topological products and Sigma-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-alpha-compact in (X) over cap, where ((X) over cap, (U) over cap) is the completion of (X, U) iff f(A) is a compact subset of R-alpha from every uniformly continuous function from X into R-alpha

we characterize the C-alpha-compact subsets of topological groups

and we also prove that if {G(i):i is an element of I} is a set of topological groups and A(i) is a C-alpha-compact subset of G(alpha) for all i is an element of I, then Pi(i is an element of) I A(i) is a C-alpha-compact subset of Pi(i is an element of I) G(i). (C) 1997 Elsevier Science B.V.