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
DOI:
https://doi.org/10.1090/S0002-9947-00-02599-X
Published electronically:
June 20, 2000
MathSciNet review:
1707497
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.”
- A. V. Arhangel’skii, W. Just, E. A. Reznichenko, and P. J. Szeptycki; Sharp bases and weakly uniform bases versus point-countable bases, to appear in Topology and its Applications.
- S. W. Davis, G. M. Reed, and M. L. Wage, Further results on weakly uniform bases, Houston J. Math. 2 (1976), no. 1, 57–63. MR 394564
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- Saharon Shelah, On successors of singular cardinals, Logic Colloquium ’78 (Mons, 1978) Stud. Logic Foundations Math., vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 357–380. MR 567680
- Saharon Shelah, Classification of nonelementary classes. II. Abstract elementary classes, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 419–497. MR 1033034, DOI https://doi.org/10.1007/BFb0082243
- Saharon Shelah, Advances in cardinal arithmetic, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 411, Kluwer Acad. Publ., Dordrecht, 1993, pp. 355–383. MR 1261217
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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
MR Author ID:
160185
ORCID:
0000-0003-0462-3152
Email:
shelah@math.huji.ac.il
P. J. Szeptycki
Affiliation:
Department of Mathematics, Ohio University, Athens, Ohio 45701
Email:
szeptyck@math.ohiou.edu
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
