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 | References | Similar Articles | Additional Information

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of for singular is proved. CECA is used to show that certain ``almost point-'' families can be refined to point- 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 -space with a weakly uniform base has a point-countable base.''

**1.**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.**2.**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**0394564****3.**A. Hajnal, I. Juhász, and S. Shelah,*Splitting strongly almost disjoint families*, Trans. Amer. Math. Soc.**295**(1986), no. 1, 369–387. MR**831204**, 10.1090/S0002-9947-1986-0831204-9**4.**R. W. Heath and W. F. Lindgren,*Weakly uniform bases*, Houston J. Math.**2**(1976), no. 1, 85–90. MR**0394563****5.**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****6.**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**, 10.1007/BFb0082243**7.**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

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

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