Partition properties of ω1 compatible with CH

Uri Abraham; Stevo Todorčević

Fundamenta Mathematicae (1997)

  • Volume: 152, Issue: 2, page 165-181
  • ISSN: 0016-2736

Abstract

top
A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.

How to cite

top

Abraham, Uri, and Todorčević, Stevo. "Partition properties of ω1 compatible with CH." Fundamenta Mathematicae 152.2 (1997): 165-181. <http://eudml.org/doc/212204>.

@article{Abraham1997,
abstract = {A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.},
author = {Abraham, Uri, Todorčević, Stevo},
journal = {Fundamenta Mathematicae},
keywords = {Continuum hypothesis; Suslin hypothesis; Hausdorff gaps; ideal; Suslin trees; coherent sequence; iterated forcing; consistency},
language = {eng},
number = {2},
pages = {165-181},
title = {Partition properties of ω1 compatible with CH},
url = {http://eudml.org/doc/212204},
volume = {152},
year = {1997},
}

TY - JOUR
AU - Abraham, Uri
AU - Todorčević, Stevo
TI - Partition properties of ω1 compatible with CH
JO - Fundamenta Mathematicae
PY - 1997
VL - 152
IS - 2
SP - 165
EP - 181
AB - A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.
LA - eng
KW - Continuum hypothesis; Suslin hypothesis; Hausdorff gaps; ideal; Suslin trees; coherent sequence; iterated forcing; consistency
UR - http://eudml.org/doc/212204
ER -

References

top
  1. [0] U. Abraham, K. J. Devlin and S. Shelah, The consistency with CH of some consequences of Martin's axiom plus non-CH, Israel J. Math. 31 (1978), 19-33. Zbl0382.03040
  2. [1] W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer, Berlin, 1974. 
  3. [2] H. G. Dales and W. H. Woodin, An Introduction to Independence for Analysts, London Math. Soc. Lecture Note Ser. 115, Cambridge University Press, 1987. 
  4. [3] K. J. Devlin and H. Johnsbraten, The Souslin Problem, Lecture Notes in Math. 405, Springer, 1974. Zbl0289.02043
  5. [4] A. Dow, PFA and ω*, Topology Appl. 28 (1988), 127-140. 
  6. [5] F. Galvin, On Gruenhage's generalization of first countable spaces II, Notices Amer. Math. Soc. 24 (1977), A-257. 
  7. [6] F. Galvin, letters of November 12, 1980 and May 18, 1981. 
  8. [7] F. Hausdorff, Die Graduierung nach dem Endverlauf, Abh. König. Sächs. Gesell. Wiss. Math.-Phys. Kl. 31 (1909), 296-334. Zbl40.0446.02
  9. [8] F. Hausdorff, Summen von ℵ 1 Mengen, Fund. Math. 26 (1936), 241-255. 
  10. [9] K. Kunen, (κ,λ*) gaps under MA, note of August 1976. 
  11. [10] M. Magidor and J. Malitz, Compact extensions of L(Q), Ann. Math. Logic 11 (1977), 217-261. 
  12. [11] J. van Mill and G. M. Reed, Open Problems in Topology, North-Holland, Amsterdam, 1990. Zbl0718.54001
  13. [12] A. Ostaszewski, On countably compact perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), 505-516. Zbl0348.54014
  14. [13] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, 1982. 
  15. [14] S. Todorčević, Forcing positive partition relations, Trans. Amer. Math. Soc. 280 (1983), 703-720. Zbl0532.03023
  16. [15] S. Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), 261-294. Zbl0658.03028
  17. [16] S. Todorčević, Partition Problems in Topology, Contemp. Math. 84, Amer. Math. Soc., Providence, 1989. Zbl0659.54001
  18. [17] S. Todorčević, Some applications of S and L combinatorics, Ann. New York Acad. Sci. 705 (1993), 130-167. 
  19. [18] N. M. Warren, Properties of Stone-Čech compactifications of discrete spaces, Proc. Amer. Math. Soc. 33 (1972), 599-606. Zbl0241.54016

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.