On open and \(\alpha\)-covering mappings
type: Item
instance of: scholarly article
title: On open and \(\alpha\)-covering mappings
zbMATH Open document ID: 1398.54030
DOI: 10.1134/S0037446618020180
author: K. V. Storozhuk
published in: Siberian Mathematical Journal
publication date: 2018-07-19T00:00:00Z
review text: Let \(0<\alpha<\infty\). A mapping between metric spaces \(f: X\to Y\) is \(\alpha\)-covering if the image of each open ball of radius \(r\) of the space \(X\) centered at a point \(p\) contains the ball of radius \(\alpha r\) of \(Y\) centered at \(f(p)\): \(f(O^X_r(p))\supset O^Y_{\alpha r}(f(p))\). In this paper, the author considers the following question: under what conditions can we make a given open mapping of compacta \(f:X\rightarrow Y\) 1-covering by replacing the metric of \(X\) or \(Y\) with a topologically equivalent metric? In particular the author obtains the following interesting results: (1) An open mapping \(f\) of a compactum \(X\) onto \(Y\) can be made 1-covering by replacing the metric on \(Y\) with an equivalent metric. (2) Among all such metrics on \(Y\), there exists a unique minimal metric. The author also gives some examples demonstrating the obtained results. In particular, an example is provided in which a mapping \(f\) cannot be made 1-covering by changing the metric of \(X\), and some examples are given showing that the condition that \(X\) is compact is substantial.
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54E45
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 6908384
zbMATH Keywords: open mapping
zbMATH Keywords: covering mapping
zbMATH Keywords: compact metric space
zbMATH Keywords: submetry
zbMATH Keywords: multivalued mapping
reviewed by: Shou Lin
MaRDI profile type: Publication
cites work: Q5553887
cites work: Stability of coincidence points and properties of covering mappings
cites work: Continuous dependence of coincidence points on a parameter
cites work: A metric characterization of Riemannian submersions
cites work: Covering \(\mathbb R\)-trees, \(\mathbb R\)-free groups, and dendrites
cites work: Q5327715
cites work: Extension of range of functions
full work available at URL: s0037446618020180
OpenAlex ID: W2800433195
description: scientific article; zbMATH DE number 6908384