Convexity Numbers of Closed Sets in R
نویسندگان
چکیده
For n > 2 let In be the σ-ideal in P(nω) generated by all sets which do not contain n equidistant points in the usual metric on nω. For each n > 2 a set Sn is constructed in Rn so that the σ-ideal which is generated by the convex subsets of Sn restricted to the convexity radical K(Sn) is isomorphic to In. Thus cov(In) is equal to the least number of convex subsets required to cover Sn — the convexity number of Sn. For every non-increasing function f : ω \ 2 → {κ ∈ card : cf(κ) > א0} we construct a model of set theory in which cov(In) = f(n) for each n ∈ ω \ 2. When f is strictly decreasing up to n, n− 1 uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of Rn. It is conjectured that n, but never more than n, different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of Rn. This conjecture is true for n = 1 and n = 2.
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