Previous |  Up |  Next


$p$-compact; $p$-sequential; $\operatorname{FU}(p)$-space; Rudin-Keisler order; tensor product of ultrafilters; left power of ultrafilters; $\operatorname{SMU}(M)$-space; $\operatorname{WFU}(M)$-space
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).
[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.R.: Kleene degrees of ultrafilters. in: Recursion Theory Weak (OberWolfach, 1984), 29-48, Lecture Notes in Math. 1141, Springer, Berlin-New York, 1985. MR 0820773 | Zbl 0573.03020
[Bo] Booth D.D.: Ultrafilters on a countable set. Ann. Math. Logic 2 (1970), 1-24. MR 0277371 | Zbl 0231.02067
[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
[CN] Comfort W.W., Negrepontis S.: The Theory of Ultrafilters. Grundlehren der Mathematichen Wissenschaften, vol. 211, Springer-Verlag, 1974. MR 0396267 | Zbl 0298.02004
[F] Frolík Z.: Sums of ultrafilters. Bull. Amer. Math. Soc. 73 (1967), 87-91. MR 0203676
[G-F$_1$] Garcia-Ferreira S.: On ${FU}(p)$-spaces and $p$-sequential spaces. Comment. Math. Univ. Carolinae 32 (1991), 161-171. MR 1118299 | Zbl 0789.54032
[G-F$_2$] Garcia-Ferreira S.: Three orderings on $\beta (ømega)\setminus ømega $. Top. Appl., to appear. MR 1227550 | Zbl 0791.54032
[K] Katětov M.: Products of filters. Comment. Math. Univ. Carolinae 9 (1968), 173-189. MR 0250257
[Koč] Kočinac L.D.: A generalization of chain net spaces. Publ. Inst. Math. (Beograd) 44 (58) (1988), 109-114. MR 0995414
[Ko] Kombarov A.P.: On a theorem of A.H. Stone. Soviet Math. Dokl. 27 (1983), 544-547. Zbl 0531.54007
[M] Malyhin V.I.: On countable space having no bicompactification of countable tightness. Soviet Math. Dokl. 13 (1972), 1407-1411. MR 0320981
[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
Partner of
EuDML logo