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 ρ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ρ 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 a < ω1 so that ρ(χ) < α for all χ ∈ S. Classification of countably convex closed subsets of Polish linear spaces follows then easily. A similar classification (by a different rank function) was previously known for closed subset of ℝ2 [3]. As an application of ρ to Banach space geometry, it is proved that for every α < ω1, the unit sphere of C(ωα) with the sup-norm has rank α. Furthermore, a countable compact metric space K is determined by the rank of the unit sphere of C(K) with the natural sup-norm: THEOREM. If K1, K1 are countable compact metric spaces and Si is the unit sphere in C(Ki) with the sup-norm, i = 1,2, then ρ(S1) = ρ(S2) if and only if K1 and K2 are homeomorphic. Uncountably convex closed sets are also studied in dimension n > 2 and are seen to be drastically more complicated than uncountably convex closed subsets of ℝ2.
Original language | English |
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Pages (from-to) | 143-163 |
Number of pages | 21 |
Journal | Fundamenta Mathematicae |
Volume | 164 |
Issue number | 2 |
DOIs | |
State | Published - 1 Jan 2000 |
Keywords
- Cantor-Bendixson degree
- Continuum hypothesis
- Convexity
- Convexity number
- Polish vector space
ASJC Scopus subject areas
- Algebra and Number Theory