| Title:
|
On FU($p$)-spaces and $p$-sequential spaces (English) |
| Author:
|
Garcia-Ferreira, Salvador |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
32 |
| Issue:
|
1 |
| Year:
|
1991 |
| Pages:
|
161-171 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Following Kombarov we say that $X$ is $p$-sequential, for $p\in\alpha^\ast$, if for every non-closed subset $A$ of $X$ there is $f\in{}^\alpha X$ such that $f(\alpha)\subseteq A$ and $\bar f(p)\in X\backslash A$. This suggests the following definition due to Comfort and Savchenko, independently: $X$ is a {\rm FU($p$)}-space if for every $A\subseteq X$ and every $x\in A^{-}$ there is a function $f\in {}^\alpha A$ such that $\bar f(p)=x$. It is not hard to see that $p \leq {\,_{\operatorname{RK}}} q$ ($\leq {\,_{\operatorname{RK}}}$ denotes the Rudin--Keisler order) $\Leftrightarrow $ every $p$-sequential space is $q$-sequential $\Leftrightarrow $ every {\rm FU($p$)}-space is a {\rm FU($q$)}-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential (for $p\in U(\alpha )$) spaces which are not {\rm FU($p$)}-spaces. We slightly improve a result of Boldjiev and Malykhin by proving that every $p$-sequential (Tychonoff) space is a {\rm FU($q$)}-space $\Leftrightarrow \forall \nu <\omega _1 (p^\nu \leq {\,_{\operatorname{RK}}} q)$, for $p,q \in \omega ^\ast $; and $S_n$ is a {\rm FU($p$)}-space for $p\in \omega ^\ast $ and $1<n<\omega \Leftrightarrow $ every sequential space $X$ with $\sigma (X)\leq n$ is a {\rm FU($p$)}-space $\Leftrightarrow \exists \{p_{n-2}, \dots , p_1\}\subseteq \omega ^\ast (p_{n-2}<{\,_{\operatorname{RK}}} \dots <{\,_{\operatorname{RK}}} p_1 <_{\,l} p)$; hence, it is independent with ZFC that $S_3$ is a {\rm FU($p$)}-space for all $p\in \omega ^\ast $. It is also shown that $|\beta (\alpha )\setminus U(\alpha )|\leq 2^\alpha \Leftrightarrow $ every space $X$ with $t(X)<\alpha $ is $p$-sequential for some $p\in U(\alpha ) \Leftrightarrow $ every space $X$ with $t(X)<\alpha $ is a {\rm FU($p$)}-space for some $p\in U(\alpha )$; if $t(X)\leq \alpha $ and $|X|\leq 2^\alpha $, then $ \exists p\in U(\alpha ) $ ($X$ is a {\rm FU($p$)}-space). (English) |
| Keyword:
|
ultrafilter |
| Keyword:
|
Rudin--Frol\'\i k order |
| Keyword:
|
Rudin--Keisler order |
| Keyword:
|
$p$-compact |
| Keyword:
|
quasi $M$-compact |
| Keyword:
|
strongly $M$-sequential |
| Keyword:
|
weakly $M$-sequential |
| Keyword:
|
$p$-sequential |
| Keyword:
|
FU($p$)-space |
| Keyword:
|
sequential |
| Keyword:
|
$P$-point |
| MSC:
|
03E05 |
| MSC:
|
04A20 |
| MSC:
|
54A25 |
| MSC:
|
54D55 |
| MSC:
|
54D99 |
| idZBL:
|
Zbl 0789.54032 |
| idMR:
|
MR1118299 |
| . |
| Date available:
|
2008-10-09T13:11:38Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/116952 |
| . |
| 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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| 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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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
