Countable compactness and $p$-limits

García-Ferreira, Salvador, and Tomita, Artur Hideyuki. "Countable compactness and $p$-limits." Commentationes Mathematicae Universitatis Carolinae 42.3 (2001): 521-527. <http://eudml.org/doc/248786>.

@article{García2001,
abstract = {For $\emptyset \ne M \subseteq \omega ^*$, we say that $X$ is quasi $M$-compact, if for every $f: \omega \rightarrow X$ there is $p \in M$ such that $\overline\{f\}(p) \in X$, where $\overline\{f\}$ is the Stone-Čech extension of $f$. In this context, a space $X$ is countably compact iff $X$ is quasi $\omega ^*$-compact. If $X$ is quasi $M$-compact and $M$ is either finite or countable discrete in $\omega ^*$, then all powers of $X$ are countably compact. Assuming $CH$, we give an example of a countable subset $M \subseteq \omega ^*$ and a quasi $M$-compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi $M$-compact space is $p$-compact (= quasi $\lbrace p\rbrace $-compact) for some $p \in \omega ^*$, whenever $M \in [\omega ^*]^\{< \{\mathfrak \{c\}\}\}$. We prove that if $\emptyset \notin \lbrace T_\xi :\, \xi < 2^\{\{\mathfrak \{c\}\}\} \rbrace \subseteq [\omega ^*]^\{< 2^\{\{\mathfrak \{c\}\}\}\}$, then there is a countably compact space $X$ that is not quasi $T_\xi $-compact for every $\xi < 2^\{\{\mathfrak \{c\}\}\}$; hence, if $2^\{\{\mathfrak \{c\}\}\}$ is regular, then there is a countably compact space $X$ such that $X$ is not quasi $M$-compact for any $M \in [\omega ^*]^\{< 2^\{\{\mathfrak \{c\}\}\}\}$. We list some open problems.},
author = {García-Ferreira, Salvador, Tomita, Artur Hideyuki},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$p$-limit; $p$-compact; almost $p$-compact; quasi $M$-compact; countably compact; -limit; -compact; almost -compact; quasi -compact; countably compact},
language = {eng},
number = {3},
pages = {521-527},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Countable compactness and $p$-limits},
url = {http://eudml.org/doc/248786},
volume = {42},
year = {2001},
}

TY - JOUR
AU - García-Ferreira, Salvador
AU - Tomita, Artur Hideyuki
TI - Countable compactness and $p$-limits
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 3
SP - 521
EP - 527
AB - For $\emptyset \ne M \subseteq \omega ^*$, we say that $X$ is quasi $M$-compact, if for every $f: \omega \rightarrow X$ there is $p \in M$ such that $\overline{f}(p) \in X$, where $\overline{f}$ is the Stone-Čech extension of $f$. In this context, a space $X$ is countably compact iff $X$ is quasi $\omega ^*$-compact. If $X$ is quasi $M$-compact and $M$ is either finite or countable discrete in $\omega ^*$, then all powers of $X$ are countably compact. Assuming $CH$, we give an example of a countable subset $M \subseteq \omega ^*$ and a quasi $M$-compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi $M$-compact space is $p$-compact (= quasi $\lbrace p\rbrace $-compact) for some $p \in \omega ^*$, whenever $M \in [\omega ^*]^{< {\mathfrak {c}}}$. We prove that if $\emptyset \notin \lbrace T_\xi :\, \xi < 2^{{\mathfrak {c}}} \rbrace \subseteq [\omega ^*]^{< 2^{{\mathfrak {c}}}}$, then there is a countably compact space $X$ that is not quasi $T_\xi $-compact for every $\xi < 2^{{\mathfrak {c}}}$; hence, if $2^{{\mathfrak {c}}}$ is regular, then there is a countably compact space $X$ such that $X$ is not quasi $M$-compact for any $M \in [\omega ^*]^{< 2^{{\mathfrak {c}}}}$. We list some open problems.
LA - eng
KW - $p$-limit; $p$-compact; almost $p$-compact; quasi $M$-compact; countably compact; -limit; -compact; almost -compact; quasi -compact; countably compact
UR - http://eudml.org/doc/248786
ER -