On non-separable components of hyperspaces with the Hausdorff metric

Abstract

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 $F(X)$ bethe space of nonvoid closed subsets of $X$ with the Hausdorff distance associated to $d$.We prove that every component of $F(X)$ which contains an unbounded closed subset ishomeomorphic to the Hilbert space $ell^2(dea)$.

Keywords

• metric space
• Hausdorff distance