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

Published electronically:
March 25, 2002

MathSciNet review:
1908910

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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 .

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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:
http://dx.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