| Title:
|
On $p$-sequential $p$-compact spaces (English) |
| Author:
|
Garcia-Ferreira, Salvador |
| Author:
|
Tamariz-Mascarua, Angel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
2 |
| Year:
|
1993 |
| Pages:
|
347-356 |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is shown that a space $X$ is $L({}^{\mu }p)$-Weakly Fréchet-Urysohn for $p\in \omega ^{\ast }$ iff it is $L({}^{\nu }p)$-Weakly Fréchet-Urysohn for arbitrary $\mu ,\nu <\omega _1$, where ${}^{\mu }p$ is the $\mu $-th left power of $p$ and $L(q)=\{{}^{\mu }q:\mu <\omega _1\}$ for $q\in \omega ^{\ast }$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L({}^{\nu }p)$-Weakly Fréchet-Urysohn space with $\nu <\omega _1$, are equivalent; consequently if $X$ is $p$-compact and $\nu <\omega _1$, then $X$ is $p$-sequential iff $X$ is ${}^{\nu }p$-sequential (Boldjiev and Malyhin gave, for each $P$-point $p\in \omega ^{\ast }$, an example of a compact space $X_p$ which is $^2p$-Fréchet-Urysohn and it is not $p$-Fréchet-Urysohn. The question whether such an example exists in ZFC remains unsolved). (English) |
| Keyword:
|
$p$-compact |
| Keyword:
|
$p$-sequential |
| Keyword:
|
$\operatorname{FU}(p)$-space |
| Keyword:
|
Rudin-Keisler order |
| Keyword:
|
tensor product of ultrafilters |
| Keyword:
|
left power of ultrafilters |
| Keyword:
|
$\operatorname{SMU}(M)$-space |
| Keyword:
|
$\operatorname{WFU}(M)$-space |
| MSC:
|
03E05 |
| MSC:
|
04A20 |
| MSC:
|
54A25 |
| MSC:
|
54D55 |
| idZBL:
|
Zbl 0791.54034 |
| idMR:
|
MR1241743 |
| . |
| Date available:
|
2009-01-08T18:03:55Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118587 |
| . |
| Reference:
|
[Be] Bernstein A.R.: A new kind of compactness for topological spaces.Fund. Math. 66 (1970), 185-193. Zbl 0198.55401, MR 0251697 |
| Reference:
|
[Bl] Blass A.R.: Kleene degrees of ultrafilters.in: Recursion Theory Weak (OberWolfach, 1984), 29-48, Lecture Notes in Math. 1141, Springer, Berlin-New York, 1985. Zbl 0573.03020, MR 0820773 |
| Reference:
|
[Bo] Booth D.D.: Ultrafilters on a countable set.Ann. Math. Logic 2 (1970), 1-24. Zbl 0231.02067, MR 0277371 |
| Reference:
|
[BM] Boldjiev B., Malyhin V.: The sequentiality is equivalent to the $\Cal F$-Fréchet-Urysohn property.Comment. Math. Univ. Carolinae 31 (1990), 23-25. MR 1056166 |
| Reference:
|
[CN] Comfort W.W., Negrepontis S.: The Theory of Ultrafilters.Grundlehren der Mathematichen Wissenschaften, vol. 211, Springer-Verlag, 1974. Zbl 0298.02004, MR 0396267 |
| Reference:
|
[F] Frolík Z.: Sums of ultrafilters.Bull. Amer. Math. Soc. 73 (1967), 87-91. MR 0203676 |
| Reference:
|
[G-F$_1$] Garcia-Ferreira S.: On ${FU}(p)$-spaces and $p$-sequential spaces.Comment. Math. Univ. Carolinae 32 (1991), 161-171. Zbl 0789.54032, MR 1118299 |
| Reference:
|
[G-F$_2$] Garcia-Ferreira S.: Three orderings on $\beta (ømega)\setminus ømega $.Top. Appl., to appear. Zbl 0791.54032, MR 1227550 |
| Reference:
|
[K] Katětov M.: Products of filters.Comment. Math. Univ. Carolinae 9 (1968), 173-189. MR 0250257 |
| Reference:
|
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| Reference:
|
[Ko] Kombarov A.P.: On a theorem of A.H. Stone.Soviet Math. Dokl. 27 (1983), 544-547. Zbl 0531.54007 |
| Reference:
|
[M] Malyhin V.I.: On countable space having no bicompactification of countable tightness.Soviet Math. Dokl. 13 (1972), 1407-1411. MR 0320981 |
| Reference:
|
[V] Vopěnka P.: The construction of models of set-theory by the method of ultraproducts.Z. Math. Logik Grundlagen Math. 8 (1962), 293-306. MR 0146085 |
| . |
