# Countable compactness and $p$-limits

Salvador García-Ferreira; Artur Hideyuki Tomita

Commentationes Mathematicae Universitatis Carolinae (2001)

- Volume: 42, Issue: 3, page 521-527
- ISSN: 0010-2628

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topGarcía-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 -

## References

top- Bernstein A.R., A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193. (1970) Zbl0198.55401MR0251697
- Blass A., Near coherence of filters I: Cofinal equivalence of models of arithmetic, Notre Dame J. Formal Logic 27 (1986), 579-591. (1986) Zbl0622.03040MR0867002
- Blass A., Laflamme C., Consistency results about filters and the number of inequivalent growth types, J. Symbolic Logic 54 (1989), 50-56. (1989) Zbl0673.03038MR0987321
- Blass A., Shelah S., Near coherence of filters III: A simplified consistency proof, to appear. Zbl0702.03030MR1036674
- 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. (1987) MR0879489
- Comfort W.W., Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977), 417-455. (1977) MR0454893
- Comfort W., Negrepontis S., The Theory of Ultrafilters, Springer-Verlag, Berlin, 1974. Zbl0298.02004MR0396267
- Garcia-Ferreira S., Quasi $M$-compact spaces, Czechoslovak Math. J. 46 (1996), 161-177. (1996) Zbl0914.54019MR1371698
- Gillman L., Jerison M., Rings of continuous functions, Lectures Notes in Mathematics No. 27, Springer-Verlag, 1976. Zbl0327.46040MR0407579
- Ginsburg J., Saks V., Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403-418. (1975) Zbl0288.54020MR0380736
- Kunen K., Weak $P$-points in $\xf8meg{a}^{*}$, Colloquia Math. Soc. János Bolyai 23 (1978), North-Holland, Amsterdam, pp.741-749. (1978) MR0588822
- van Mill J., An introduction to $\beta \left(\xf8mega\right)$, in K. Kunen and J.E. Vaughan, eds., Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp.503-567. Zbl0555.54004MR0776630

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