Some functions on hyperspaces of continua
type: Item
instance of: scholarly article
title: Some functions on hyperspaces of continua
zbMATH Open document ID: 0589.54016
DOI: 10.1016/0166-8641(86)90021-0
author: Włodzimierz J. Charatonik
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: For a metric continuum X let \(2^ X\) denote the space of nonempty closed subsets of X with the Hausdorff metric and C(X) the subspace of \(2^ X\) consisting of all nonempty subcontinua of X. In 1977 Goodykoontz examined two functions f,g: \(2^ X\to C(X)\) defined by \[ f(A)=\cap \{K\in C(X):\quad A\subseteq K\}\quad and\quad g(A)=\cap \{K\in C(X):\quad A\subseteq int K\}. \] In this paper the author defines two similar functions F,G: \(2^ X\to C(C(X))\) by \[ F(A)=\{K\in C(X):\quad A\subseteq K\}\quad and\quad G(A)=\{K\in C(X):\quad A\subseteq int K\}. \] Preliminary results include various continuity properties of these functions and their connections with Goodykoontz's functions are examined. These results are then used to characterize smoothness, local connectedness, connectedness im kleinen and the property of Kelley of a continuum.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 3947159
zbMATH Keywords: upper (lower) semicontinuous function
zbMATH Keywords: smoothness
zbMATH Keywords: local connectedness
zbMATH Keywords: connectedness im kleinen
zbMATH Keywords: property of Kelley
reviewed by: Lewis Lum
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90021-0
OpenAlex ID: W2122850358
cites work: Connectedness im Kleinen and local connectedness in \(2^x\) and C(X)
cites work: Some functions on hyperspaces of hereditarily unicoherent continua
cites work: A characterization of solenoids
cites work: On a Certain Type of Homogeneous Plane Continuum
cites work: Hyperspaces of a Continuum
cites work: On smooth continua
cites work: Monotone Decompositions of Hausdorff Continua
cites work: Q4127048
description: scientific article; zbMATH DE number 3947159