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

García-Ferreira, Salvador, Sanchis, Manuel, and Watson, Stephen W.. "Some remarks on the product of two $C_\alpha $-compact subsets." Czechoslovak Mathematical Journal 50.2 (2000): 249-264. <http://eudml.org/doc/30559>.

@article{García2000,
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 $f\: X \rightarrow \mathbb \{R\}^\{\alpha \}$, $f[B]$ 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.},
author = {García-Ferreira, Salvador, Sanchis, Manuel, Watson, Stephen W.},
journal = {Czechoslovak Mathematical Journal},
keywords = {bounded subset; $C_\alpha $-compact; $\alpha $-pseudocompact; degree of $C_\alpha $-pseudocompactness; $\alpha _r$-space; bounded subset; -compact; -pseudocompact; degree of -pseudocompactness; -space},
language = {eng},
number = {2},
pages = {249-264},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some remarks on the product of two $C_\alpha $-compact subsets},
url = {http://eudml.org/doc/30559},
volume = {50},
year = {2000},
}

TY - JOUR
AU - García-Ferreira, Salvador
AU - Sanchis, Manuel
AU - Watson, Stephen W.
TI - Some remarks on the product of two $C_\alpha $-compact subsets
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 2
SP - 249
EP - 264
AB - 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.
LA - eng
KW - bounded subset; $C_\alpha $-compact; $\alpha $-pseudocompact; degree of $C_\alpha $-pseudocompactness; $\alpha _r$-space; bounded subset; -compact; -pseudocompact; degree of -pseudocompactness; -space
UR - http://eudml.org/doc/30559
ER -