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

TL;DR

This paper studies the hyperspaces of finite ray-graphs, focusing on their topological structure and connected components under Hausdorff and Vietoris topologies.

Contribution

It introduces the class of finite ray-graphs and analyzes the structure and connectedness of their hyperspaces under different topologies.

Findings

Hyperspaces of finite ray-graphs have a finite number of connected components.

The structure of these hyperspaces varies with the topology used (Hausdorff vs. Vietoris).

Results extend understanding of hyperspaces for noncompact graph-like spaces.

Abstract

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.