On FU( p )-spaces and p -sequential spaces

Salvador García-Ferreira

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 1, page 161-171
  • ISSN: 0010-2628

Abstract

top
Following Kombarov we say that X is p -sequential, for p α * , if for every non-closed subset A of X there is f α X such that f ( α ) A and f ¯ ( p ) X A . This suggests the following definition due to Comfort and Savchenko, independently: X is a FU( p )-space if for every A X and every x A - there is a function f α A such that f ¯ ( p ) = x . It is not hard to see that p RK q ( RK denotes the Rudin–Keisler order) every p -sequential space is q -sequential every FU( p )-space is a FU( q )-space. We generalize the spaces S n to construct examples of p -sequential (for p U ( α ) ) spaces which are not FU( p )-spaces. We slightly improve a result of Boldjiev and Malykhin by proving that every p -sequential (Tychonoff) space is a FU( q )-space ν < ω 1 ( p ν RK q ) , for p , q ω * ; and S n is a FU( p )-space for p ω * and 1 < n < ω every sequential space X with σ ( X ) n is a FU( p )-space { p n - 2 , , p 1 } ω * ( p n - 2 < RK < RK p 1 < l p ) ; hence, it is independent with ZFC that S 3 is a FU( p )-space for all p ω * . It is also shown that | β ( α ) U ( α ) | 2 α every space X with t ( X ) < α is p -sequential for some p U ( α ) every space X with t ( X ) < α is a FU( p )-space for some p U ( α ) ; if t ( X ) α and | X | 2 α , then p U ( α ) ( X is a FU( p )-space).

How to cite

top

García-Ferreira, Salvador. "On FU($p$)-spaces and $p$-sequential spaces." Commentationes Mathematicae Universitatis Carolinae 32.1 (1991): 161-171. <http://eudml.org/doc/247273>.

@article{García1991,
abstract = {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 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 $p \le \{\,_\{\operatorname\{RK\}\}\} q$ ($\le \{\,_\{\operatorname\{RK\}\}\}$ denotes the Rudin–Keisler order) $\Leftrightarrow $ every $p$-sequential space is $q$-sequential $\Leftrightarrow $ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential (for $p\in U(\alpha )$) spaces which are not FU($p$)-spaces. We slightly improve a result of Boldjiev and Malykhin by proving that every $p$-sequential (Tychonoff) space is a FU($q$)-space $\Leftrightarrow \forall \nu <\omega _1 (p^\nu \le \{\,_\{\operatorname\{RK\}\}\} q)$, for $p,q \in \omega ^\ast $; and $S_n$ is a FU($p$)-space for $p\in \omega ^\ast $ and $1<n<\omega \Leftrightarrow $ every sequential space $X$ with $\sigma (X)\le n$ is a FU($p$)-space $\Leftrightarrow \exists \lbrace p_\{n-2\}, \dots , p_1\rbrace \subseteq \omega ^\ast (p_\{n-2\}<\{\,_\{\operatorname\{RK\}\}\} \dots <\{\,_\{\operatorname\{RK\}\}\} p_1 <_\{\,l\} p)$; hence, it is independent with ZFC that $S_3$ is a FU($p$)-space for all $p\in \omega ^\ast $. It is also shown that $|\beta (\alpha )\setminus U(\alpha )|\le 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 FU($p$)-space for some $p\in U(\alpha )$; if $t(X)\le \alpha $ and $|X|\le 2^\alpha $, then $ \exists p\in U(\alpha ) $ ($X$ is a FU($p$)-space).},
author = {García-Ferreira, Salvador},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {ultrafilter; Rudin–Frolík order; Rudin–Keisler order; $p$-compact; quasi $M$-compact; strongly $M$-sequential; weakly $M$-sequential; $p$-sequential; FU($p$)-space; sequential; $P$-point; -sequential space; Rudin-Frolík order; -space; Rudin- Keisler order; ultrafilter},
language = {eng},
number = {1},
pages = {161-171},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On FU($p$)-spaces and $p$-sequential spaces},
url = {http://eudml.org/doc/247273},
volume = {32},
year = {1991},
}

TY - JOUR
AU - García-Ferreira, Salvador
TI - On FU($p$)-spaces and $p$-sequential spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 1
SP - 161
EP - 171
AB - 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 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 $p \le {\,_{\operatorname{RK}}} q$ ($\le {\,_{\operatorname{RK}}}$ denotes the Rudin–Keisler order) $\Leftrightarrow $ every $p$-sequential space is $q$-sequential $\Leftrightarrow $ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential (for $p\in U(\alpha )$) spaces which are not FU($p$)-spaces. We slightly improve a result of Boldjiev and Malykhin by proving that every $p$-sequential (Tychonoff) space is a FU($q$)-space $\Leftrightarrow \forall \nu <\omega _1 (p^\nu \le {\,_{\operatorname{RK}}} q)$, for $p,q \in \omega ^\ast $; and $S_n$ is a FU($p$)-space for $p\in \omega ^\ast $ and $1<n<\omega \Leftrightarrow $ every sequential space $X$ with $\sigma (X)\le n$ is a FU($p$)-space $\Leftrightarrow \exists \lbrace p_{n-2}, \dots , p_1\rbrace \subseteq \omega ^\ast (p_{n-2}<{\,_{\operatorname{RK}}} \dots <{\,_{\operatorname{RK}}} p_1 <_{\,l} p)$; hence, it is independent with ZFC that $S_3$ is a FU($p$)-space for all $p\in \omega ^\ast $. It is also shown that $|\beta (\alpha )\setminus U(\alpha )|\le 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 FU($p$)-space for some $p\in U(\alpha )$; if $t(X)\le \alpha $ and $|X|\le 2^\alpha $, then $ \exists p\in U(\alpha ) $ ($X$ is a FU($p$)-space).
LA - eng
KW - ultrafilter; Rudin–Frolík order; Rudin–Keisler order; $p$-compact; quasi $M$-compact; strongly $M$-sequential; weakly $M$-sequential; $p$-sequential; FU($p$)-space; sequential; $P$-point; -sequential space; Rudin-Frolík order; -space; Rudin- Keisler order; ultrafilter
UR - http://eudml.org/doc/247273
ER -

References

top
  1. Arhangel'skii A.V., Martin's axiom and the construction of homogeneous bicompacta of countable tightness, Soviet Math. Dokl. 17 (1976), 256-260. (1976) 
  2. Arhangel'skii A.V., Franklin S.P., Ordinal invariants for topological spaces, Michigan Math. J. 15 (1968), 313-320. (1968) MR0240767
  3. Balogh Z., On compact Hausdorff spaces of countable tightness, Proc. Amer. Math. Soc. 105 (1989), 755-764. (1989) Zbl0687.54006MR0930252
  4. Bernstein A.R., A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193. (1970) Zbl0198.55401MR0251697
  5. Boldjiev B., Malykhin V., The sequentiality is equivalent to the -Fréchet-Urysohn property, Comment. Math. Univ. Carolinae 31 (1990), 23-25. (1990) Zbl0696.54020MR1056166
  6. Booth D.D., Ultrafilters on a countable set, Ann. Math. Logic 2 (1970), 1-24. (1970) Zbl0231.02067MR0277371
  7. Comfort W.W., Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977), 417-455. (1977) MR0454893
  8. Comfort W.W., Negrepontis S., On families of large oscillation, Fund. Math. 75 (1972), 275-290. (1972) Zbl0235.54005MR0305343
  9. Comfort W.W., Negrepontis S., The Theory of Ultrafilters, Grundlehren der Mathematischen Wissenschaften Vol. 211, Springer-Verlag, 1974. Zbl0298.02004MR0396267
  10. Fedorčuk V.V., Fully closed mappings and the compatibility of some theorems of general topology with the axioms of set-theory, Math. USSR Sbornik 28 (1976), 1-26. (1976) 
  11. Garcia-Ferreira S., Various Orderings on the Space of Ultrafilters, Doctoral Dissertation, Wesleyan University, 1990. 
  12. Garcia-Ferreira S., Three Orderings on β ( ø m e g a ) ø m e g a , preprint. Zbl0791.54032MR1227550
  13. Kombarov A.P., On a theorem of A. H. Stone, Soviet Math. Dokl. 27 (1983), 544-547. (1983) Zbl0531.54007
  14. Kombarov A.P., Compactness and sequentiality with respect to a set of ultrafilters, Moscow Univ. Math. Bull. 40 (1985), 15-18. (1985) Zbl0602.54025MR0814266
  15. Mills Ch., An easier proof of the Shelah P -point independence theorem, Rapport 78, Wiskundig Seminarium, Free University of Amsterdam. 
  16. Savchenko I.A., Convergence with respect to ultrafilters and the collective normality of products, Moscow Univ. Math. Bull. 43 (1988), 45-47. (1988) Zbl0687.54004MR0938072
  17. Wimmers E.L., The Shelah P -point independence theorem, Israel J. Math. 43 (1982), 28-48. (1982) Zbl0511.03022MR0728877

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.