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

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 Journal of Mathematics, 2 (1976) 57-63. MR**52:15365****3.**A. Hajnal, I. Juhász, and S. Shelah [HJSh249];*Splitting strongly almost disjoint families,*TAMS 295 (1986) 369-387. MR**87i:03098****4.**R. W. Heath and W. F. Lindgren;*Weakly uniform bases,*Houston J. Math. 2 (1976) 85-90. MR**52:15364****5.**S. Shelah [Sh108];*On Successors of Singular Cardinals,*in: Logic Colloquium 78 (M. Boffa, D. van Dalen, K. McAloon, eds.), 357-380, North-Holland 1979. MR**82d:03079****6.**S. Shelah [Sh88a];*Appendix: On stationary sets (to ``Classification of nonelementary classes II. Abstract elementary classes''),*in: Classification Theory, Proceedings of the USA-Israel Conference on Classification Theory, Chicago, December 1985, J. T. Baldwin, ed., LNM 1292, 483-495, Springer 1987. MR**91h:03046****7.**S. Shelah [Sh420];*Advances in Cardinal Arithmetic,*Finite and Infinite Combinatorics in Sets and Logic (N. W. Sauer et al. eds.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 411, Kluwer 1993, 355-383. MR**95h:03112**

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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:
https://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