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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
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