## Abstract

We investigate countably convex G_{δ} 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 G_{δ} 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 G_{δ} subsets of finite-dimensional spaces. The results distinguish dimension d ≤ 3 from dimension d ≥ 4: in a countably convex G_{δ} subset of ℝ_{3} all cliques are scattered, whereas in ℝ_{4} a countably convex G_{δ} set may contain a dense-in-itself clique.

Original language | English |
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Pages (from-to) | 131-140 |

Number of pages | 10 |

Journal | Fundamenta Mathematicae |

Volume | 168 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2001 |

## ASJC Scopus subject areas

- Algebra and Number Theory

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