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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Strongly almost disjoint sets and weakly uniform bases


Authors: Z. T. Balogh, S. W. Davis, W. Just, S. Shelah and P. J. Szeptycki
Journal: Trans. Amer. Math. Soc. 352 (2000), 4971-4987
MSC (2000): Primary 03E05, 03E35, 03E75, 54D70
Published electronically: June 20, 2000
MathSciNet review: 1707497
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Abstract:

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of $\Box_\lambda$ for singular $\lambda$ is proved. CECA is used to show that certain ``almost point-$<\tau$'' families can be refined to point-$< \tau$ families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable $T_1$-space with a weakly uniform base has a point-countable base.''


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Additional Information

Z. T. Balogh
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: ztbalogh@miavx1.acs.muohio.edu

S. W. Davis
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: swdavis@miavx1.muohio.edu

W. Just
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
Email: just@math.ohiou.edu

S. Shelah
Affiliation: Institute of Mathematics, The Hebrew University, Givat Ram, 91904 Jerusalem, Israel
Email: shelah@math.huji.ac.il

P. J. Szeptycki
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
Email: szeptyck@math.ohiou.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02599-X
PII: S 0002-9947(00)02599-X
Keywords: GCH, $\Box$, strongly almost disjoint families, weakly uniform base, point countable base
Received by editor(s): March 17, 1998
Published electronically: June 20, 2000
Additional Notes: The first author’s research was partially supported by NSF grant DMS-9623391. The third author’s research was done during visits at Rutgers University and The Hebrew University, Jerusalem, which were supported by NSF grant DMS-9704477 and the Landau Center. The fourth author was partially supported by the Israel Basic Research Fund. This is publication number 674 in Shelah’s publication list.
Article copyright: © Copyright 2000 American Mathematical Society