On extensions of the domain invariance theorem
type: Item
instance of: scholarly article
title: On extensions of the domain invariance theorem
zbMATH Open document ID: 1029.54048
DOI: 10.1016/S0166-8641(02)00223-7
author: Władysław Kulpa
published in: Topology and its Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(X\) be a compact subset of \(\mathbb{R}^n\) and suppose \(x_0\) is in the interior of \(X\). The main result of this well-written paper states that if \(f: X\to\mathbb{R}^n\) is a map such that, for every point \(x\) on \(\partial X\), the boundary of \(X\), the diameter of \(f^{-1}(f(x))\) is less than half the distance from \(x_0\) to \(\partial X\), then \(f(x_0)\) is in the interior of \(f(X)\). Consequences of this result include the Brouwer invariance of the domain theorem as well as theorems of Borsuk and Sitnikov, concerning maps to \(\mathbb{R}^n\) from \(\mathbb{R}^n\) and an open subset \(U\) of \(\mathbb{R}^n\), respectively, in which conditions on the diameter of \(f^{-1}(f(x))\) imply that the map has open image.
reviewed by: Robert F. Brown
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 1956725
zbMATH Keywords: Domain invariance theorem
zbMATH Keywords: Brouwer-Borsuk-Sitnikov's type theorems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1956725