Some remarks on the product of two $C_\alpha$-compact subsets

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

Czechoslovak Mathematical Journal, Tome 50 (2000) no. 2, pp. 249-264

Abstract (VO)

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 $f\: X \rightarrow \mathbb R^{\alpha }$, $f[B]$ is a compact subset of $\mathbb R^{\alpha }$. If $B$ is a $C$-compact subset of a space $X$, then $\rho (B,X)$ denotes the degree of $C_{\alpha }$-compactness of $B$ in $X$. A space $X$ is called $\alpha $-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 $X \times X$ 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.

Détail

Classification : 54B10, 54C50, 54D30, 54D35
Keywords: bounded subset; $C_\alpha$-compact; $\alpha$-pseudocompact; degree of $C_\alpha$-pseudocompactness; $\alpha_r$-space