Countably convex $G_{δ}$ sets

Vladimir Fonf ; Menachem Kojman

Fundamenta Mathematicae, Tome 167 (2001), p. 131-140 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ℝ³ all cliques are scattered, whereas in ℝ⁴ a countably convex $G_{δ}$ set may contain a dense-in-itself clique.

Détail

Publié le : 2001-01-01

Zbl 0980.46007

EUDML-ID : urn:eudml:doc:282192