Countable compactness and $p$-limits
García-Ferreira, Salvador ; Tomita, Artur Hideyuki
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 521-527 / Harvested from Czech Digital Mathematics Library
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For $\emptyset \neq M \subseteq \omega^*$, 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 $\{p\}$-compact) for some $p \in \omega^*$, whenever $M \in [\omega^*]^{< {\frak c}}$. We prove that if $\emptyset \notin \{ T_\xi :\, \xi < 2^{{\frak c}} \} \subseteq [\omega^*]^{< 2^{{\frak c}}}$, then there is a countably compact space $X$ that is not quasi $T_\xi$-compact for every $\xi < 2^{{\frak c}}$; hence, if $2^{{\frak c}}$ is regular, then there is a countably compact space $X$ such that $X$ is not quasi $M$-compact for any $M \in [\omega^*]^{< 2^{{\frak c}}}$. We list some open problems.

Publié le : 2001-01-01
Classification:  54A20,  54A35,  54B99,  54D20,  54D30 
@article{119266,
     author = {Salvador Garc\'\i a-Ferreira and Artur Hideyuki Tomita},
     title = {Countable compactness and $p$-limits},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {521-527},
     zbl = {1053.54003},
     mrnumber = {1860240},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119266}
}

García-Ferreira, Salvador; Tomita, Artur Hideyuki. Countable compactness and $p$-limits. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 521-527. http://gdmltest.u-ga.fr/item/119266/

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