Product topology in the hyperspace of subcontinua
type: Item
instance of: scholarly article
title: Product topology in the hyperspace of subcontinua
zbMATH Open document ID: 0957.54005
DOI: 10.1016/S0166-8641(99)00065-6
author: Alejandro Illanes
author: Verónica Martínez-de-la-Vega
published in: Topology and its Applications
publication date: 2001-03-22T00:00:00Z
review text: Let \(X\) be a metrizable continuum with its hyperspace of nondegenerate subcontinua \(C(X)\). If \(A\) and \(B\) are (possibly) empty finite subsets of \(X\) then \(U(A,B)=\{K\in C(X) : A\subseteq K\) and \(K\cap B=\emptyset\}\). The collection of all sets of the form \(U(A,B)\) serves as the basis of a topology on \(C(X)\), called the product topology \(\tau_P\) by the authors. Let \(\tau_H\) denote the usual Vietoris topology on \(C(X)\). The authors prove that the topological properties of a dendroid \(X\) are close to those of its hyperspace \(C(X)\) endowed with the product topology. They prove for a dendroid \(X\) that \(((C(X),\tau_p)\) has countable \(\pi\)-weight if and only if \(X\) has countably many endpoints only. In addition, the product topology on \(C(X)\) is separable if and only if every pairwise disjoint family of arcs in \(X\) is countable, and \(\tau_H\subseteq \tau_P\) if and only if \(X\) is a dendrite.
reviewed by: Jan van Mill
Mathematics Subject Classification ID: 54B20
zbMATH DE Number: 1501778
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1501778