The Hyperspaces $C_{n}(X)$ of finite ray-graphs

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.