The hyperspace of non-cut subcontinua of graphs and dendrites

TL;DR

This paper investigates the topological properties of the hyperspace of non-cut subcontinua in graphs and dendrites, revealing conditions for properties like compactness and connectedness, and characterizing certain cases as homeomorphic to the Baire space.

Contribution

It provides new characterizations of the hyperspace of non-cut subcontinua for finite graphs and dendrites, including conditions for various topological properties and a homeomorphism result.

Findings

$NC^{*}(X)$ is compact, connected, locally connected, or totally disconnected under specific conditions.

If $X$ is a dendrite with dense endpoints, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrationals.

The paper establishes criteria for topological properties of $NC^{*}(X)$ based on the structure of $X$.

Abstract

Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of $X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)=\{ A \in C(X):X\setminus A\text{ is connected}\}$ when $X$ is a finite graph or a dendrite; in particular, we give conditions under which $NC^{*}(X)$ is compact, connected, locally connected or totally disconnected. Also, we prove that if $X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrational numbers.