S. Garcia-Ferreira, A.H. Tomita
Countable compactness and p-limits

Comment.Math.Univ.Carolinae 42,3 (2001) 521-527.

Abstract:For $\emptyset \not =M \subseteq \omega ^*$,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-\v Cech extensionof $f$. In this context, a space $X$ is countably compact iff$X$ is quasi $\omega ^*$-compact. If $X$ is quasi $M$-compactand $M$ 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 notcountably 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.

Keywords: $p$-limit, $p$-compact, almost $p$-compact, quasi $M$-compact, countably compact
AMS Subject Classification: Primary 54A20, 54A35; Secondary 54B99