# Article

 Title: Some remarks on the product of two $C_\alpha$-compact subsets (English) Author: García-Ferreira, S. Author: Sanchis, Manuel Author: Watson, S. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 50 Issue: 2 Year: 2000 Pages: 249-264 Summary lang: English . Category: math . Summary: 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. (English) Keyword: bounded subset Keyword: $C_\alpha$-compact Keyword: $\alpha$-pseudocompact Keyword: degree of $C_\alpha$-pseudocompactness Keyword: $\alpha_r$-space MSC: 54B10 MSC: 54C50 MSC: 54D30 MSC: 54D35 idZBL: Zbl 1050.54016 idMR: MR1761385 . Date available: 2009-09-24T10:32:22Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/127567 .

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