Convexity numbers of closed sets in
Authors:
Stefan Geschke and Menachem Kojman
Journal:
Proc. Amer. Math. Soc. 130 (2002), 28712881
MSC (1991):
Primary 05A20, 52A05, 03E17, 03E35; Secondary 03E75
Published electronically:
March 25, 2002
MathSciNet review:
1908910
Fulltext PDF Free Access
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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 nonincreasing 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 26, D1495 Berlin
Email:
geschke@math.fuberlin.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/S0002993902064377
PII:
S 00029939(02)064377
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
