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Keywords:
$p$-limit; $p$-compact; almost $p$-compact; quasi $M$-compact; countably compact
$p$-limit; $p$-compact; almost $p$-compact; quasi $M$-compact; countably compact
Summary:
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.
References:
[Be] Bernstein A.R.: A new kind of compactness for topological spaces. Fund. Math. 66 (1970), 185-193. MR 0251697 | Zbl 0198.55401
[Bl] Blass A.: Near coherence of filters I: Cofinal equivalence of models of arithmetic. Notre Dame J. Formal Logic 27 (1986), 579-591. MR 0867002 | Zbl 0622.03040
[BL] Blass A., Laflamme C.: Consistency results about filters and the number of inequivalent growth types. J. Symbolic Logic 54 (1989), 50-56. MR 0987321 | Zbl 0673.03038
[BS1] Blass A., Shelah S.: Near coherence of filters III: A simplified consistency proof. to appear. MR 1036674 | Zbl 0702.03030
[BS2] Blass A., Shelah S.: There may be simple $P_{\aleph_1}$ and $P_{\aleph_2}$ points and the Rudin-Keisler ordering may be downward directed. Ann. Pure Appl. Logic 33 (1987), 213-243. MR 0879489
[Co] Comfort W.W.: Ultrafilters: some old and some new results. Bull. Amer. Math. Soc. 83 (1977), 417-455. MR 0454893
[CN] Comfort W., Negrepontis S.: The Theory of Ultrafilters. Springer-Verlag, Berlin, 1974. MR 0396267 | Zbl 0298.02004
[G] Garcia-Ferreira S.: Quasi $M$-compact spaces. Czechoslovak Math. J. 46 (1996), 161-177. MR 1371698 | Zbl 0914.54019
[GJ] Gillman L., Jerison M.: Rings of continuous functions. Lectures Notes in Mathematics No. 27, Springer-Verlag, 1976. MR 0407579 | Zbl 0327.46040
[GS] Ginsburg J., Saks V.: Some applications of ultrafilters in topology. Pacific J. Math. 57 (1975), 403-418. MR 0380736 | Zbl 0288.54020
[Ku] Kunen K.: Weak $P$-points in $ømega^*$. Colloquia Math. Soc. János Bolyai 23 (1978), North-Holland, Amsterdam, pp.741-749. MR 0588822
[vM] van Mill J.: An introduction to $\beta(ømega)$. in K. Kunen and J.E. Vaughan, eds., Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp.503-567. MR 0776630 | Zbl 0555.54004
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