Arc-reduced forms for Peano continua
type: Item
instance of: scholarly article
title: Arc-reduced forms for Peano continua
zbMATH Open document ID: 1255.55003
published in: Topology and its Applications
publication date: 2012-10-11T00:00:00Z
review text: In [Topology Appl. 159, No. 15, 3253--3262 (2012; Zbl 1255.55002)], the authors introduced the deforestation of a Peano contionuum as a certain strong deformation retraction. The deforestation yields the unique minimal deformation retract if the space is one-dimensional and non-contractible. In this paper the authors simplify the homotopy representative of a Peano continuum even more by introducing the arc reduced forms. The one dimensional set \(I(X)\) of a space \(X\) is the set of points with one-dimensional neighborhoods. The set of bad points \(B(X)\) of a space \(X\) is the set of points which possess no simply connected neighborhood. The space \(X\) is arc-reduced if \(I(X)-B(X)\) is either a disjoint union of open arcs or a finite boquet of circles. The main theorem: Every Peano continuum is homotopy equivalent to an arc-reduced continuum. As an application the authors discuss the case of one-dimensional Peano continua.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 6092998
zbMATH Keywords: Peano continua
zbMATH Keywords: homotopy equivalence
zbMATH Keywords: reduced forms
zbMATH Keywords: one-dimensional
Wikidata QID: Q41641037
author: G. Conner
author: M. Meilstrup
reviewed by: Ziga Virk
MaRDI profile type: Publication
full work available at URL: j.topol.2012.08.015
OpenAlex ID: W2058977626
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DOI: 10.1016/J.TOPOL.2012.08.015
description: scientific article; zbMATH DE number 6092998