Type: Article

Some remarks on the product of two C_alpha-compact subsets

S. García-Ferreira; Manuel Sanchis; S. Watson;

Abstract:

For a 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. If B is a C-compact subset of a space X, then rho(B,X) denotes the degree of Gor-compactness of B in X. A space X is called cu-pseudocompact if X is C-alpha-compact into itself. For each cardinal alpha, we give an example of an alpha-pseudocompact space X such that X x X is not pseudocompact: this answers a question posed by T. Retta in "Some cardinal generalizations of pseudocompactness" Czechoslovak Math. J. 43 (1993), 385-390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.