The hyperspace of non-cut subcontinua of graphs and dendrites

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.

Figures (1)

• Figure 1: Model of $NC^{*}(X)$ when $X$ is a tree.

Theorems & Definitions (34)

• Lemma 2.1
• Theorem 2.2
• Lemma 3.1
• Theorem 3.2
• Theorem 3.5
• Corollary 3.6
• Corollary 3.7
• Corollary 3.8
• Theorem 3.9
• Claim 1