Continuous images of arcs: extensions of Cornette's theorem
type: Item
instance of: scholarly article
title: Continuous images of arcs: extensions of Cornette's theorem
zbMATH Open document ID: 1330.54038
author: N. E. Zubov
published in: Topology and its Applications
publication date: 2015-11-09T00:00:00Z
review text: A continuum is a compact connected Hausdorff space. A cyclic element in a locally connected continuum \(X\) is either (a) a connected subset \(M\) of \(X\) which is maximal with respect to the property that no point disconnects \(M\) or (b) a cutpoint of \(X\). A family \(G\) of sets in a continuum \(X\) is null if for each open cover \(W\) of \(X\) there are at most finitely many \(g \in G\) such that \(g\) is not contained in \(U\) for any \(U \in W\). It is well known that each cyclic element is a retract of \(X\); the family of cyclic elements of \(X\) is null and two distinct cyclic elements can intersect in at most one point. An arc is a continuum with a linear order such that its topology is given by the order topology. A continuum \(X\) is an IOC if it is the continuous image of an arc. Clearly, IOCs are locally connected continua and in the metric case, by the classical Hahn-Mazurkiewicz theorem, metric IOCs are the locally connected, metric continua. In 1960, \textit{S. Mardešić} [Proc. Am. Math. Soc. 11, 929--937 (1961; Zbl 0100.19003)] constructed a locally connected continuum which is not connected by arcs. \textit{J. L. Cornette} [Trans. Am. Math. Soc. 199, 253-267 (1974; Zbl 0291.54039)] proved that a locally connected continuum is an IOC if and only if each of its cyclic elements is an IOC. In the paper under review, the authors extend the result of Cornette by proving the following theorems. Theorem 2. Let \(X\) be a locally connected continuum and \(G\) be a null family which covers \(X\). Suppose that {\parindent=6mm \begin{itemize} \item[(i)] each \(g\) in \(G\) is an IOC (possibly degenerate), \item [(ii)] if \(h\), \(k\) are disjoint elements in \(G\), then there exists an element \(g\) of \(G\) and a closed subset \(A\) of \(g\) that separates \(h\) and \(k\) in \(X\), and \item [(iii)] for every \(g\), \(h\) in \(G\), \( g\smallsetminus h\) is contained in a component of \(X\smallsetminus h\). \end{itemize}} Then \(X\) is an IOC. Theorem 3. L…
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 6506271
zbMATH Keywords: images of arcs
zbMATH Keywords: locally connected
zbMATH Keywords: cyclic element
zbMATH Keywords: null family
MaRDI profile type: Publication
full work available at URL: j.topol.2015.09.030
OpenAlex ID: W2189687623
cites work: A characterization of continuous images of compact ordered spaces and the Hahn-Mazurkiewicz problem
cites work: "Image of a Hausdorff Arc" is Cyclically Extensible and Reducible
cites work: Q5596563
cites work: Characterizing local connectedness in inverse limits
cites work: Cyclic Element Theory in Connected and Locally Connected Hausdorff Spaces
cites work: Continuous images of linearly ordered continua and compacta
cites work: Q3276310
cites work: Cyclic Elements in Topology, A History
cites work: Q3903703
cites work: Images of arcs - a nonseparable version of the Hahn-Mazurkiewicz theorem
cites work: The Hahn-Mazurkiewicz theorem for hereditary locally connected continua
cites work: A locally connected rim-countable continuum which is the continuous image of no arc
cites work: Mapping arcs and dendritic spaces onto netlike continua
cites work: A characterization of spaces that are the continuous image of an arc
cites work: Q3918740
cites work: A Generalization of the Hahn-Mazurkiewicz Theorem
cites work: Q5837858
DOI: 10.1016/J.TOPOL.2015.09.030
Recommended article: On continuous images of ultra-arcs
Recommended article: Continuous Images of Arcs
Recommended article: Images of arcs - a nonseparable version of the Hahn-Mazurkiewicz theorem
Recommended article: Continuous images of arcs and inverse limit methods
Recommended article: A note on inverse limits of continuous images of arcs
Recommended article: Q5751581
Recommended article: Arc-like and circle-like almost continuous images of Peano continua
Recommended article: Continuous mappings between spaces of arcs
Recommended article: A characterization of spaces that are the continuous image of an arc
Recommended article: Q3797867
description: scientific article; zbMATH DE number 6506271