| Title:
|
Countable compactness and $p$-limits (English) |
| Author:
|
García-Ferreira, S. |
| Author:
|
Tomita, A. H. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
521-527 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
$p$-limit |
| Keyword:
|
$p$-compact |
| Keyword:
|
almost $p$-compact |
| Keyword:
|
quasi $M$-compact |
| Keyword:
|
countably compact |
| MSC:
|
54A20 |
| MSC:
|
54A35 |
| MSC:
|
54B99 |
| MSC:
|
54D20 |
| MSC:
|
54D30 |
| idZBL:
|
Zbl 1053.54003 |
| idMR:
|
MR1860240 |
| . |
| Date available:
|
2009-01-08T19:12:21Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119266 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[BS1] Blass A., Shelah S.: Near coherence of filters III: A simplified consistency proof.to appear. Zbl 0702.03030, MR 1036674 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| . |
