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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2001 | 168 | 2 | 131-140

## Countably convex $G_{δ}$ sets

EN

### Abstrakty

EN
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.

131-140

wydano
2001

### Twórcy

autor
• Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel
autor
• Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel