Critical sets in the plane
type: Item
instance of: scholarly article
title: Critical sets in the plane
zbMATH Open document ID: 0785.58016
author: Alec Kercheval
author: Charles Pugh
published in: Michigan Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: For a smooth \((C^ \infty)\) function \(f:\mathbb{R}^ m\to\mathbb{R}\) consider its critical set cp\((f)=\{x\in\mathbb{R}^ m:(Df)_ x=0\}\). A subset \(C\subset\mathbb{R}^ m\) is critical if \(C=\text{cp}(f)\) for some smooth \(f:\mathbb{R}^ m\to\mathbb{R}\), and it is properly critical if such an \(f\) exists which is proper. This very interesting paper deals with the following nice problem: which topological properties have a critical set, respectively a properly critical set? I mention here the main results: A compact nonempty subset of \(\mathbb{R}^ 2\) is critical if and only if it is properly critical if and only if the components of its complement are multiply connected (Theorem 1). The noncompact case is also studied: A closed set \(K\subset\mathbb{R}^ 2\) is critical if and only if \(\infty\) is arcwise accessible in each simply connected subset of \(K^ c\), the complement of \(K\) (Theorem 5). The authors mention some results concerning the above problem in general case \(m\geq 3\).
reviewed by: Dorin Andrica
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 59885
zbMATH Keywords: critical set
zbMATH Keywords: properly critical set
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004393
OpenAlex ID: W1988509213
DOI: 10.1307/MMJ/1029004393
description: scientific article; zbMATH DE number 59885