Countably convex Gδ sets

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We investigate countably convex Gδ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition is also necessary. We show that for countably convex Gδ subsets of infinite-dimensional Banach spaces there are no necessary limitations on cliques and semi-cliques. Various necessary conditions on cliques and semi-cliques are obtained for countably convex Gδ subsets of finite-dimensional spaces. The results distinguish dimension d ≤ 3 from dimension d ≥ 4: in a countably convex Gδ subset of ℝ3 all cliques are scattered, whereas in ℝ4 a countably convex Gδ set may contain a dense-in-itself clique.

Original languageEnglish
Pages (from-to)131-140
Number of pages10
JournalFundamenta Mathematicae
Issue number2
StatePublished - 1 Jan 2001

ASJC Scopus subject areas

  • Algebra and Number Theory


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