Abstract
A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function % is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then % is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists α 2 and are seen to be drastically more complicated than uncountably convex closed subsets of R2.
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