\(\mathcal{T}\)-closed sets
type: Item
instance of: scholarly article
title: \(\mathcal{T}\)-closed sets
zbMATH Open document ID: 1369.54007
author: Hussam Abobaker
author: Włodzimierz J. Charatonik
published in: Topology and its Applications
publication date: 2017-05-02T00:00:00Z
review text: A continuum is a compact, connected metric space. Let \({\mathcal T}\) denote Jones' set function. A subset \(A\) of a continuum \(X\) is called a \({\mathcal T}\)-closed set if \({\mathcal T}(A) = A\). \textit{D.~Bellamy} et al. in [Topology Appl. 195, 209--225 (2015; Zbl. 1333.54022)] proved that if \(A\) is a subset of a continuum \(X\) and \(A\) is a \({\mathcal T}\)-closed set, then every component of \(X \setminus A\) is open and continuumwise connected. Then they asked (Question 4.6) if the converse is true if we additionally assume that the set function \({\mathcal T}\) is idempotent on \(X\) (i.e. \({\mathcal T}({\mathcal T}(A)) = {\mathcal T}(A)\) for any subset \(A\) of \(X\)). In this paper the authors provide a negative answer to that question. They construct a continuum \(X\) on which the set function \({\mathcal T}\) is idempotent, and a subcontinuum \(A\) of \(X\) such that every component of \(X \setminus A\) is open and continuumwise connected but \({\mathcal T}(A) \neq A\). Moreover they give a characterization of \({\mathcal T}\)-closed sets.
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 6710357
zbMATH Keywords: continuum
zbMATH Keywords: continuumwise connected
zbMATH Keywords: exhaustive \(\sigma\)-continuum
zbMATH Keywords: Jones' set function \(\mathcal{T}\)
zbMATH Keywords: \(\mathcal{T}\)-closed sets
MaRDI profile type: Publication
full work available at URL: j.topol.2017.02.066
OpenAlex ID: W2589822345
cites work: On \(\mathcal{T}\)-closed sets
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cites work: Q4230677
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cites work: Monotone decompositions into trees of Hausdorff continua irreducible about a finite subset
DOI: 10.1016/J.TOPOL.2017.02.066
description: scientific article; zbMATH DE number 6710357