Countably convex $G_{\delta }$ sets

Vladimir Fonf ¹   ; Menachem Kojman ¹

¹ Department of Mathematics Ben Gurion University of the Negev Beer Sheva, Israel

Fundamenta Mathematicae, Tome 168 (2001) no. 2, pp. 131-140

Résumé

We investigate countably convex $G_{\delta }$ 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 $P\subseteq S$ so that for every $x\in P$ and open neighborhood $u$ of $x$ there exists a finite set $X\subseteq P\cap u$ such that $\mathop {\rm conv}(X)\not \subseteq S$. For closed sets this condition is also necessary. We show that for countably convex $G_{\delta }$ 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 $G_{\delta }$ subsets of finite-dimensional spaces. The results distinguish dimension $d\le 3$ from dimension $d\ge 4$: in a countably convex $G_{\delta }$ subset of ${\mathbb R}^{3}$ all cliques are scattered, whereas in ${\mathbb R}^4$ a countably convex $G_{\delta }$ set may contain a dense-in-itself clique.

Détail

DOI : 10.4064/fm168-2-4

Keywords: investigate countably convex delta subsets banach spaces subset linear space countably convex represented countable union convex sets known sufficient condition countable convexity arbitrary subset separable normed space does contain semi clique semi clique set subset subseteq every neighborhood there exists finite set subseteq cap mathop conv subseteq closed sets condition necessary countably convex delta subsets infinite dimensional banach spaces there necessary limitations cliques semi cliques various necessary conditions cliques semi cliques obtained countably convex delta subsets finite dimensional spaces results distinguish dimension dimension countably convex delta subset mathbb cliques scattered whereas mathbb countably convex delta set may contain dense in itself clique