  In this paper we consider the hyperspace $C_{n}(X)$ of non-empty and closed
subsets of a base space $X$ with up to $n$ connected components. We consider a
class of base spaces called finite ray-graphs, which are a noncompact variation
on finite graphs. We prove two results about the structure of these hyperspaces
under different topologies (Hausdorff and Vietoris), in particular, about their
number of connected components.
