Sur les composantes non separables des hyperespaces avec la distance de Hausdorff

Let $(X,d)$ be a connected non compact metric space. Suppose the metric $d$ convex and such that every closed bounded subset of $X$ is compact. Let $mathcal F(X)$ be the space of nonvoid closed subsets of $X$ with the Hausdorff distance associated to $d$. We prove that every component of $mathcal F(X)$ which contains an unbounded closed subset is homeomorphic to the Hilbert space $ell^2(2^{aleph_0})$.