A semifilter approach to selection principles II: τ * -covers

Lubomyr Zdomsky

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 3, page 539-547
  • ISSN: 0010-2628

Abstract

top
Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property fin ( 𝒪 , T * ) provided ( 𝔲 < 𝔤 ) , and every space with the property fin ( 𝒪 , T * ) is Hurewicz provided ( Depth + ( [ ω ] 0 ) 𝔟 ) . Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties P and Q [do not] coincide, where P and Q run over fin ( 𝒪 , Γ ) , fin ( 𝒪 , T ) , fin ( 𝒪 , T * ) , fin ( 𝒪 , Ω ) , and fin ( 𝒪 , 𝒪 ) .

How to cite

top

Zdomsky, Lubomyr. "A semifilter approach to selection principles II: $\tau ^\ast $-covers." Commentationes Mathematicae Universitatis Carolinae 47.3 (2006): 539-547. <http://eudml.org/doc/249868>.

@article{Zdomsky2006,
abstract = {Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property $\bigcup _\{\operatorname\{fin\}\}(\mathcal \{O\}, \operatorname\{T\}^\ast )$ provided $(\mathfrak \{u\}<\mathfrak \{g\})$, and every space with the property $\bigcup _\{\operatorname\{fin\}\}(\mathcal \{O\}, \operatorname\{T\}^\ast )$ is Hurewicz provided $(\operatorname\{Depth\}^+([\omega ]^\{\aleph _0\})\le \mathfrak \{b\})$. Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties $\text\{P\}$ and $\text\{Q\}$ [do not] coincide, where $\text\{P\}$ and $\text\{Q\}$ run over $\bigcup _\{\operatorname\{fin\}\}(\mathcal \{O\},\Gamma )$, $\bigcup _\{\operatorname\{fin\}\}(\mathcal \{O\}, \operatorname\{T\})$, $\bigcup _\{\operatorname\{fin\}\}(\mathcal \{O\}, \operatorname\{T\}^\ast )$, $\bigcup _\{\operatorname\{fin\}\}(\mathcal \{O\}, \Omega )$, and $\bigcup _\{\operatorname\{fin\}\}(\mathcal \{O\}, \mathcal \{O\})$.},
author = {Zdomsky, Lubomyr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {selection principle; semifilter; small cardinals; selection principle; semifilter; small cardinals},
language = {eng},
number = {3},
pages = {539-547},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A semifilter approach to selection principles II: $\tau ^\ast $-covers},
url = {http://eudml.org/doc/249868},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Zdomsky, Lubomyr
TI - A semifilter approach to selection principles II: $\tau ^\ast $-covers
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 3
SP - 539
EP - 547
AB - Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property $\bigcup _{\operatorname{fin}}(\mathcal {O}, \operatorname{T}^\ast )$ provided $(\mathfrak {u}<\mathfrak {g})$, and every space with the property $\bigcup _{\operatorname{fin}}(\mathcal {O}, \operatorname{T}^\ast )$ is Hurewicz provided $(\operatorname{Depth}^+([\omega ]^{\aleph _0})\le \mathfrak {b})$. Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties $\text{P}$ and $\text{Q}$ [do not] coincide, where $\text{P}$ and $\text{Q}$ run over $\bigcup _{\operatorname{fin}}(\mathcal {O},\Gamma )$, $\bigcup _{\operatorname{fin}}(\mathcal {O}, \operatorname{T})$, $\bigcup _{\operatorname{fin}}(\mathcal {O}, \operatorname{T}^\ast )$, $\bigcup _{\operatorname{fin}}(\mathcal {O}, \Omega )$, and $\bigcup _{\operatorname{fin}}(\mathcal {O}, \mathcal {O})$.
LA - eng
KW - selection principle; semifilter; small cardinals; selection principle; semifilter; small cardinals
UR - http://eudml.org/doc/249868
ER -

References

top
  1. Banakh T., Zdomsky L., Coherence of semifilters; http://www.franko.lviv.ua/faculty/mechmat/Departments/Topology/ booksite.html, . 
  2. Banakh T., Zdomsky L., Selection principles and infinite games on multicovered spaces and their applications, in preparation. 
  3. Bartoszyński T., Shelah S., Tsaban B., Additivity properties of topological diagonalizations, J. Symbolic Logic 68 (2003), 1254-1260; (Full version: http://arxiv.org/abs/math.LO/0112262). (2003) Zbl1071.03031MR2017353
  4. Blass A., Combinatorial cardinal characteristics of the continuum, in Handbook of Set Theory (M. Foreman et al., Eds.), to appear. Zbl1198.03058MR2768685
  5. Chaber J., Pol R., A remark on Fremlin-Miller theorem concerning the Menger property and Michael concentrated sets, preprint. 
  6. Dordal P., A model in which the base-matrix tree cannot have cofinal branches, J. Symbolic Logic 52 (1987), 651-664. (1987) Zbl0637.03049MR0902981
  7. Dow A., Set theory in topology, in Recent Progress in General Topology (M. Hušek et al., Eds.), Elsevier Sci. Publ., Amsterdam, 1992, pp.168-197. Zbl0796.54001MR1229125
  8. Miller A., Fremlin D., On some properties of Hurewicz, Menger, and Rothberger, Fund. Math. 129 (1988), 17-33. (1988) Zbl0665.54026MR0954892
  9. Gerlits J., Nagy Zs., Some properties of C ( X ) , I, Topology Appl. 14 2 (1982), 151-1613. (1982) Zbl0503.54020MR0667661
  10. Hurewicz W., Über die Verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1925), 401-421. (1925) Zbl51.0454.02
  11. Just W., Miller A., Scheepers M., Szeptycki S., The combinatorics of open covers II, Topology Appl. 73 (1996), 241-266. (1996) Zbl0870.03021MR1419798
  12. Laflamme C., Equivalence of families of functions on natural numbers, Trans. Amer. Math. Soc. 330 (1992), 307-319. (1992) Zbl0759.03025MR1028761
  13. Marczewski E. (Szpilrajn), The characteristic function of a sequence of sets and some of its applications, Fund. Math. 31 (1938), 207-233. (1938) 
  14. Menger K., Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte, Abt. 2a, Mathematic, Astronomie, Physic, Meteorologie und Mechanic (Wiener Akademie) 133 (1924), 421-444. Zbl50.0129.01
  15. Scheepers M., Combinatorics of open covers I: Ramsey Theory, Topology Appl. 69 (1996), 31-62. (1996) Zbl0848.54018MR1378387
  16. Shelah S., Tsaban B., Critical cardinalities and additivity properties of combinatorial notions of smallness, J. Appl. Anal. 9 (2003), 149-162; http://arxiv.org/abs/math.LO/0304019. (2003) Zbl1052.03026MR2021285
  17. Solomon R., Families of sets and functions, Czechoslovak Math. J. 27 (1977), 556-559. (1977) Zbl0383.04002MR0457218
  18. Talagrand M., Filtres: Mesurabilité, rapidité, propriété de Baire forte, Studia Math. 74 (1982), 283-291. (1982) Zbl0503.04003MR0683750
  19. Tsaban B., Selection principles and the minimal tower problem, Note Mat. 22 2 (2003), 53-81; http://arxiv.org/abs/math.LO/0105045. (2003) Zbl1176.03022MR2112731
  20. Tsaban B. (eds.), [unknown], SPM Bulletin 3 (2003; http://arxiv.org/abs/math.GN/0303057). (2003; http://arxiv.org/abs/math.GN/0303057) Zbl1071.03031
  21. Tsaban B., Zdomsky L., Scales, fields, and a problem of Hurewicz, submitted to J. Amer. Math. Soc.; http://arxiv.org/abs/math.GN/0507043. MR2421163
  22. Vaughan J., Small uncountable cardinals and topology, in Open Problems in Topology (J. van Mill, G.M. Reed, Eds.), Elsevier Sci. Publ., Amsterdam, 1990, pp.195-218. MR1078647
  23. Zdomsky L., A semifilter approach to selection principles, Comment. Math. Univ. Carolin. 46 (2005), 525-539; http://arxiv.org/abs/math.GN/0412498. (2005) Zbl1121.03060MR2174530

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.