Countably convex $G_{\delta }$ sets

Volume 168 / 2001

Vladimir Fonf, Menachem Kojman Fundamenta Mathematicae 168 (2001), 131-140 MSC: Primary 46B20, 51M05, 52A07, 52A37; Secondary 51M20, 52B10, 52B05, 52A15. DOI: 10.4064/fm168-2-4


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.


  • Vladimir FonfDepartment of Mathematics
    Ben Gurion University of the Negev
    Beer Sheva, Israel
  • Menachem KojmanDepartment of Mathematics
    Ben Gurion University of the Negev
    Beer Sheva, Israel

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