Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Thin-very tall compact scattered spaces which are hereditarily separable


Authors: Christina Brech and Piotr Koszmider
Journal: Trans. Amer. Math. Soc. 363 (2011), 501-519
MSC (2010): Primary 54G12; Secondary 03E35, 46B26
DOI: https://doi.org/10.1090/S0002-9947-2010-05149-9
Published electronically: August 30, 2010
MathSciNet review: 2719691
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We strengthen the property $ \Delta$ of a function $ f:[\omega_2]^2\rightarrow [\omega_2]^{\leq \omega}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $ K$ as above where $ K^n$ is hereditarily separable for each $ n\in\mathbb{N}$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $ C(K)$ is an Asplund space of density $ \aleph_2$ which has no Fréchet smooth renorming, nor an uncountable biorthogonal system.


References [Enhancements On Off] (What's this?)

  • 1. J. E. Baumgartner, Applications of the proper forcing axiom, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 913-959. MR 776640 (86g:03084)
  • 2. J. E. Baumgartner and S. Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (1987), no. 2, 109-129. MR 874021 (88d:03100)
  • 3. J. M. Borwein and J. D. Vanderwerff, Banach spaces that admit support sets, Proc. Amer. Math. Soc. 124 (1996), no. 3, 751-755. MR 1301010 (96f:46016)
  • 4. R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow, 1993.
  • 5. A. Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (1988), no. 1, 17-72. MR 1031969 (91a:54003)
  • 6. P. Hájek, V. Montesinos Santalucía, J. Vanderwerff, and V. Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 26, Springer, New York, 2008.
  • 7. R. Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (1990), no. 3, 261-268. MR 1041141 (91h:46045)
  • 8. R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1-61. MR 776620 (86j:54007)
  • 9. T. Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded. MR 1940513 (2004g:03071)
  • 10. M. Jiménez Sevilla and J.-P. Moreno, Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal. 144 (1997), no. 2, 486-504. MR 1432595 (98a:46024)
  • 11. I. Juhász and L. Soukup, How to force a countably tight, initially $ \omega\sb 1$-compact and noncompact space?, Topology Appl. 69 (1996), no. 3, 227-250. MR 1382294 (97c:54004)
  • 12. I. Juhász and W. Weiss, On thin-tall scattered spaces, Colloq. Math. 40 (1978/79), no. 1, 63-68. MR 529798 (82k:54005)
  • 13. W. Just, Two consistency results concerning thin-tall Boolean algebras, Algebra Universalis 20 (1985), no. 2, 135-142. MR 806609 (87c:03101)
  • 14. P. Koszmider, Semimorasses and nonreflection at singular cardinals, Ann. Pure Appl. Logic 72 (1995), no. 1, 1-23. MR 1320107 (96i:03043)
  • 15. -, On strong chains of uncountable functions, Israel J. Math. 118 (2000), 289-315. MR 1776085 (2001g:03091)
  • 16. -, Universal matrices and strongly unbounded functions, Math. Res. Lett. 9 (2002), no. 4, 549-566. MR 1928875 (2003g:03078)
  • 17. J. C. Martínez, A consistency result on thin-very tall Boolean algebras, Israel J. Math. 123 (2001), 273-284. MR 1835300 (2003c:03090)
  • 18. S. Mazur, Über schwache Konvergenz in den Raumen $ L^p$, Studia Math. 4 (1933), 129-133.
  • 19. I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), no. 4, 735-750. MR 0390721 (52:11544)
  • 20. S. Negrepontis, Banach spaces and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1045-1142. MR 776642 (86i:46018)
  • 21. A. J. Ostaszewski, A countably compact, first-countable, hereditarily separable regular space which is not completely regular, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 4, 431-435. MR 0377815 (51:13984)
  • 22. M. Rabus, An $ \omega\sb 2$-minimal Boolean algebra, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3235-3244. MR 1357881 (96j:03070)
  • 23. M. Rajagopalan, A chain compact space which is not strongly scattered, Israel J. Math. 23 (1976), no. 2, 117-125. MR 0402701 (53:6517)
  • 24. J. Roitman, Introduction to modern set theory, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1990, A Wiley-Interscience Publication. MR 1028781 (91h:03003)
  • 25. B. È. Shapirovskiĭ, Cardinal invariants in compacta, Seminar on General Topology, Moskov. Gos. Univ., Moscow, 1981, pp. 162-187. (Russian) MR 656957 (83f:54024)
  • 26. S. Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982. MR 675955 (84h:03002)
  • 27. S. Todorcevic, A note on the proper forcing axiom, Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 209-218. MR 763902 (86f:03089)
  • 28. -, Directed sets and cofinal types, Trans. Amer. Math. Soc. 290 (1985), no. 2, 711-723. MR 792822 (87a:03084)
  • 29. -, Biorthogonal systems and quotient spaces via Baire category methods, Math. Ann. 335 (2006), no. 3, 687-715. MR 2221128 (2007d:46016)
  • 30. D. Velleman, Simplified morasses, J. Symbolic Logic 49 (1984), no. 1, 257-271. MR 736620 (85i:03162)
  • 31. W. S. Zwicker, $ P\sb k\lambda $ combinatorics. I. Stationary coding sets rationalize the club filter, Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 243-259. MR 763904 (86e:03046)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 54G12, 03E35, 46B26

Retrieve articles in all journals with MSC (2010): 54G12, 03E35, 46B26


Additional Information

Christina Brech
Affiliation: Institute of Mathematics, Statistics and Scientific Computing, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, SP, Brazil
Address at time of publication: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970, São Paulo, SP, Brazil
Email: christina.brech@gmail.com

Piotr Koszmider
Affiliation: Instytut Matematyki, Politechnika Łódzka, ul. Wólczańska 215; 90-924 Łódź, Poland
Email: pkoszmider.politechnika@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2010-05149-9
Received by editor(s): August 26, 2008
Received by editor(s) in revised form: June 24, 2009
Published electronically: August 30, 2010
Additional Notes: The research was part of Thematic Project FAPESP (2006/02378-7). The first author was supported by scholarships from CAPES (3804/05-4) and CNPq (140426/2004-3 and 202532/2006-2). She would like to thank Stevo Todorcevic and the second author, her Ph.D. advisors at the University of São Paulo and at the University of Paris 7, under whose supervision the results of this paper were obtained.
The second author was partially supported by Polish Ministry of Science and Higher Education research grant N N201 386234.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society