## 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 < ω_{1} 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 ℝ^{2} [3]. As an application of ρ to Banach space geometry, it is proved that for every α < ω_{1}, 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 K_{1}, K_{1} are countable compact metric spaces and S_{i} is the unit sphere in C(K_{i}) with the sup-norm, i = 1,2, then ρ(S_{1}) = ρ(S_{2}) if and only if K_{1} and K_{2} 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 ℝ^{2}.

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