Filament sets and decompositions of homogeneous continua
type: Item
instance of: scholarly article
title: Filament sets and decompositions of homogeneous continua
zbMATH Open document ID: 1131.54019
author: J. R. Prajs
author: Keith Whittington
published in: Topology and its Applications
publication date: 2007-05-30T00:00:00Z
review text: This paper is a continuation of another paper [Topology Appl. 154, 1581--1591 (2007; Zbl 1119.54025)] by the same authors in which they defined filament and ample continua. In the present paper, the authors study decompositions of homogeneous continua using these concepts which generalize Jones's aposyndetic decomposition. A continuum is a compact connected metric space. A continuum \(X\) is homogeneous if for any two points \(x_1\) and \(x_2\) of \(X\), there exists a homeomorphism \(h:X\to X\) such that \(h(x_1)=x_2\). A map \(f:X\to Y\) is atomic provided that for every subcontinuum \(K\) of \(X\) either \(f(K)\) is degenerate or \(K=f^{-1}(f(K))\). A decomposition \(\mathcal{D}\) of a continuum \(X\) is atomic if the quotient map \(\pi:X\to X/\mathcal{D}\) is atomic. A decomposition \(\mathcal{D}\) of a continuum \(X\) is respected by homeomorphisms if \(h(D)\in\mathcal{D}\) for every \(D\in\mathcal{D}\). If \(X\) is a continuum, a subcontinuum \(K\) of \(X\) is called filament (in \(X\)) provided that there exists a neighborhood \(N\) of \(K\) in \(X\) such that the component, \(Cnt_N(K)\), of \(N\) containing \(K\) has empty interior. A subset \(F\) of the continuum \(X\) is a filament set if any continuum \(K\) contained in \(F\) is filament in \(X\). The subcontinuum \(K\) of \(X\) is called ample provided that for every open set \(U\) of \(X\) containing \(K\), there exists a subcontinuum \(L\) of \(X\) such that \(K\subset Int(L)\subset L\subset U\). A continuum \(X\) is filament additive if for each two filament subcontinua \(K\) and \(L\) of \(X\) with nonempty intersection, the union \(K\cup L\) is filament. The paper contains many interesting results. We mention some of them. Proposition 3.2. Let \(f:X\to Y\) be an atomic map. (1) If \(K\) is a subcontinuum of \(X\) such that \(f(K)\) is nondegenerate, then \(K\) is indecomposable if and only if \(f(K)\) is indecomposable. (2) \(X\) is hereditarily indecomposable if and only if \(Y\) is hereditarily in…
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 5159479
zbMATH Keywords: homogeneous
zbMATH Keywords: continuum
zbMATH Keywords: filament
reviewed by: Sergio Macías
MaRDI profile type: Publication
full work available at URL: j.topol.2007.01.012
OpenAlex ID: W2020184269
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DOI: 10.1016/J.TOPOL.2007.01.012
description: scientific article; zbMATH DE number 5159479