Convexity numbers of closed sets in

Authors:
Stefan Geschke and Menachem Kojman

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2871-2881

MSC (1991):
Primary 05A20, 52A05, 03E17, 03E35; Secondary 03E75

DOI:
https://doi.org/10.1090/S0002-9939-02-06437-7

Published electronically:
March 25, 2002

MathSciNet review:
1908910

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For let be the -ideal in generated by all sets which do not contain equidistant points in the usual metric on . For each a set is constructed in so that the -ideal which is generated by the convex subsets of restricted to the convexity radical is isomorphic to . Thus is equal to the least number of convex subsets required to cover -- the *convexity number* of .

For every non-increasing function we construct a model of set theory in which for each . When is strictly decreasing up to , uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of . It is conjectured that , but never more than , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of . This conjecture is true for and .

**1.**T. Bartoszynski, H. Judah,*Set Theory: On the structure of the real line.*A K Peters, Ltd., Wellesley, MA, 1995. MR**96k:03002****2.**A. Blass,*Simple cardinal characteristics of the continuum.*Set theory of the reals (Ramat Gan, 1991), 63-90, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993. MR**94i:03098****3.**S. Geschke, M. Kojman, W. Kubis, R. Schipperus,*Convex decompositions in the plane and continuous pair colorings of the irrationals*, submitted.**4.**M. Goldstern and S. Shelah,*Many simple cardinal invariants of the continuum*, Arch. Math. Logic 32 (1993), no. 3, 203-221. MR**94c:03064****5.**T. Jech,*Multiple Forcing*, Cambridge Tracts in Mathematics, 88. Cambridge University Press, Cambridge, 1986. MR**89h:03001****6.**L. Newelski, A. Roslanowski,*The ideal determined by the unsymmetric game*, Proc. Amer. Math. Soc. 117 (1993), no. 3, 823-831. MR**93d:03053****7.**S. Shelah, J. Steprans,*The covering numbers of Mycielski ideals are all equal*, J. Symbolic Logic**66**(2001), 707-718.**8.**J. Steprans,*Decomposing Euclidean space with a small number of smooth sets.*, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1461-1480. MR**99f:04002****9.**J. Zapletal,*Isolating cardinal invariants*, preprint.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
05A20,
52A05,
03E17,
03E35,
03E75

Retrieve articles in all journals with MSC (1991): 05A20, 52A05, 03E17, 03E35, 03E75

Additional Information

**Stefan Geschke**

Affiliation:
Freie Universität Berlin, Arnimallee 2-6, D-1495 Berlin

Email:
geschke@math.fu-berlin.de

**Menachem Kojman**

Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

Email:
kojman@math.bgu.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-02-06437-7

Keywords:
Convex cover,
convexity number,
$n$-space,
forcing extension,
covering number

Received by editor(s):
April 19, 2001

Received by editor(s) in revised form:
May 31, 2001

Published electronically:
March 25, 2002

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2002
American Mathematical Society