Thin-very tall compact scattered spaces which are hereditarily separable

Brech, Christina; Koszmider, Piotr

doi:10.1090/S0002-9947-2010-05149-9

Thin-very tall compact scattered spaces which are hereditarily separable
HTML articles powered by AMS MathViewer

by Christina Brech and Piotr Koszmider

Trans. Amer. Math. Soc. 363 (2011), 501-519

DOI: https://doi.org/10.1090/S0002-9947-2010-05149-9

Published electronically: August 30, 2010

PDF | Request permission

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

James E. Baumgartner, Applications of the proper forcing axiom, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 913–959. MR 776640
James E. Baumgartner and Saharon Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (1987), no. 2, 109–129. MR 874021, DOI 10.1016/0168-0072(87)90077-7
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, DOI 10.1090/S0002-9939-96-03122-X
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.
Alan Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (1988), no. 1, 17–72. MR 1031969
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.
Richard Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (1990), no. 3, 261–268. MR 1041141, DOI 10.1112/blms/22.3.261
R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1–61. MR 776620
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
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, DOI 10.1006/jfan.1996.3014
I. Juhász and L. Soukup, How to force a countably tight, initially $\omega _1$-compact and noncompact space?, Topology Appl. 69 (1996), no. 3, 227–250. MR 1382294, DOI 10.1016/0166-8641(95)00102-6
I. Juhász and W. Weiss, On thin-tall scattered spaces, Colloq. Math. 40 (1978/79), no. 1, 63–68. MR 529798, DOI 10.4064/cm-40-1-63-68
Winfried Just, Two consistency results concerning thin-tall Boolean algebras, Algebra Universalis 20 (1985), no. 2, 135–142. MR 806609, DOI 10.1007/BF01278592
Piotr Koszmider, Semimorasses and nonreflection at singular cardinals, Ann. Pure Appl. Logic 72 (1995), no. 1, 1–23. MR 1320107, DOI 10.1016/0168-0072(93)E0068-Y
Piotr Koszmider, On strong chains of uncountable functions, Israel J. Math. 118 (2000), 289–315. MR 1776085, DOI 10.1007/BF02803525
Piotr Koszmider, Universal matrices and strongly unbounded functions, Math. Res. Lett. 9 (2002), no. 4, 549–566. MR 1928875, DOI 10.4310/MRL.2002.v9.n4.a15
Juan Carlos Martínez, A consistency result on thin-very tall Boolean algebras, Israel J. Math. 123 (2001), 273–284. MR 1835300, DOI 10.1007/BF02784131
S. Mazur, Über schwache Konvergenz in den Raumen $L^p$, Studia Math. 4 (1933), 129–133.
I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), no. 4, 735–750. MR 390721
S. Negrepontis, Banach spaces and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1045–1142. MR 776642
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 (English, with Russian summary). MR 377815
Mariusz Rabus, An $\omega _2$-minimal Boolean algebra, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3235–3244. MR 1357881, DOI 10.1090/S0002-9947-96-01663-7
M. Rajagopalan, A chain compact space which is not strongly scattered, Israel J. Math. 23 (1976), no. 2, 117–125. MR 402701, DOI 10.1007/BF02756790
Judith 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
B. È. Shapirovskiĭ, Cardinal invariants in compacta, Seminar on General Topology, Moskov. Gos. Univ., Moscow, 1981, pp. 162–187 (Russian). MR 656957
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
Stevo Todorčević, 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, DOI 10.1090/conm/031/763902
Stevo Todorčević, Directed sets and cofinal types, Trans. Amer. Math. Soc. 290 (1985), no. 2, 711–723. MR 792822, DOI 10.1090/S0002-9947-1985-0792822-9
Stevo Todorcevic, Biorthogonal systems and quotient spaces via Baire category methods, Math. Ann. 335 (2006), no. 3, 687–715. MR 2221128, DOI 10.1007/s00208-006-0762-7
Dan Velleman, Simplified morasses, J. Symbolic Logic 49 (1984), no. 1, 257–271. MR 736620, DOI 10.2307/2274108
William S. Zwicker, $P_k\lambda$ 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, DOI 10.1090/conm/031/763904