The hyperspace of noncut subcontinua of graphs and dendrites
Volume 173 / 2023
Colloquium Mathematicum 173 (2023), 57-75
MSC: Primary 54F50; Secondary 54B20, 54E50, 54F15, 54F65.
DOI: 10.4064/cm8947-9-2022
Published online: 28 March 2023
Abstract
Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of $X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)=\{ A \in C(X):X\setminus A$ 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.
