Interpolation of $\kappa$-compactness and PCF

Juhász, István, and Szentmiklóssy, Zoltán. "Interpolation of $\kappa $-compactness and PCF." Commentationes Mathematicae Universitatis Carolinae 50.2 (2009): 315-320. <http://eudml.org/doc/32501>.

@article{Juhász2009,
abstract = {We call a topological space $\kappa $-compact if every subset of size $\kappa $ has a complete accumulation point in it. Let $\Phi (\mu ,\kappa ,\lambda )$ denote the following statement: $\mu < \kappa < \lambda = \operatorname\{cf\} (\lambda )$ and there is $\lbrace S_\xi : \xi < \lambda \rbrace \subset [\kappa ]^\mu $ such that $|\lbrace \xi : |S_\xi \cap A| = \mu \rbrace | < \lambda $ whenever $A \in [\kappa ]^\{<\kappa \}$. We show that if $\Phi (\mu ,\kappa ,\lambda )$ holds and the space $X$ is both $\mu $-compact and $\lambda $-compact then $X$ is $\kappa $-compact as well. Moreover, from PCF theory we deduce $\Phi (\operatorname\{cf\} (\kappa ), \kappa , \kappa ^+)$ for every singular cardinal $\kappa $. As a corollary we get that a linearly Lindelöf and $\aleph _\omega $-compact space is uncountably compact, that is $\kappa $-compact for all uncountable cardinals $\kappa $.},
author = {Juhász, István, Szentmiklóssy, Zoltán},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {complete accumulation point; $\kappa $-compact space; linearly Lindelöf space; PCF theory; complete accumulation point; -compact space; linearly Lindelöf space; PCF theory},
language = {eng},
number = {2},
pages = {315-320},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Interpolation of $\kappa $-compactness and PCF},
url = {http://eudml.org/doc/32501},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Juhász, István
AU - Szentmiklóssy, Zoltán
TI - Interpolation of $\kappa $-compactness and PCF
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 2
SP - 315
EP - 320
AB - We call a topological space $\kappa $-compact if every subset of size $\kappa $ has a complete accumulation point in it. Let $\Phi (\mu ,\kappa ,\lambda )$ denote the following statement: $\mu < \kappa < \lambda = \operatorname{cf} (\lambda )$ and there is $\lbrace S_\xi : \xi < \lambda \rbrace \subset [\kappa ]^\mu $ such that $|\lbrace \xi : |S_\xi \cap A| = \mu \rbrace | < \lambda $ whenever $A \in [\kappa ]^{<\kappa }$. We show that if $\Phi (\mu ,\kappa ,\lambda )$ holds and the space $X$ is both $\mu $-compact and $\lambda $-compact then $X$ is $\kappa $-compact as well. Moreover, from PCF theory we deduce $\Phi (\operatorname{cf} (\kappa ), \kappa , \kappa ^+)$ for every singular cardinal $\kappa $. As a corollary we get that a linearly Lindelöf and $\aleph _\omega $-compact space is uncountably compact, that is $\kappa $-compact for all uncountable cardinals $\kappa $.
LA - eng
KW - complete accumulation point; $\kappa $-compact space; linearly Lindelöf space; PCF theory; complete accumulation point; -compact space; linearly Lindelöf space; PCF theory
UR - http://eudml.org/doc/32501
ER -