A characterization of locally compact spaces with homeomorphic one-point compactifications
type: Item
instance of: scholarly article
title: A characterization of locally compact spaces with homeomorphic one-point compactifications
zbMATH Open document ID: 1033.54008
DOI: 10.1016/S0166-8641(01)00112-2
author: Juan J. Font
author: Manuel Sanchis
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Let \(X\) and \(Y\) be locally compact noncompact spaces. It is first shown that the set of extreme points of the closed unit ball of \(C_0(X)*\) consists exactly of \(\{\epsilon_x, - \epsilon_x, \epsilon_x - \epsilon_y : x,y \in X, x \neq y\}\), where \(\epsilon_x\) denotes the evaluation functional at \(x\). Using this result it is shown that there exist basically two types of diameter preserving linear bijections defined between a Borel continuous subspace \(A\) of \(C_0(X)\) and such a subspace \(B\) of \(C_0(Y).\) Type 1 consists of those which can be written as \(\psi(f)(y) = \tau(f \circ \phi)(y),\) where \(\tau \in \{-1, 1\},\) and \(\phi : Y \to X\) is a surjective homeomorphism. Type (2) consists of those which can be represented as \[ \psi(f)(y) = \left\{ \begin{matrix} \tau(f \circ \phi)(y) - \tau f(x_0), & {y \neq y_0},\\ - \tau f(x_0), & {y = y_0}, \end{matrix} \right. \] for some \(x_0 \in X\), \(y_0 \in Y\), where \(\phi : Y \backslash \{y_0\} \to X \backslash \{x_0\} \) is a homeomorphism which can be extended to the one-point compactifications of \(Y\) and \(X\). Finally, a characterization is given of locally compact spaces with homeomorphic one-point compactifications as those which admit diameter-preserving linear bijections like the ones described above.
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 54D45
zbMATH DE Number: 1787518
zbMATH Keywords: locally compact
zbMATH Keywords: extreme point
zbMATH Keywords: one-point compactification
zbMATH Keywords: diameter-preserving bijections
reviewed by: Srinivasa Swaminathan
MaRDI profile type: Publication
cites work: Linear isometries between subspaces of continuous functions
cites work: Q4049588
cites work: Diameter preserving linear maps and isometries
cites work: On extremely regular function spaces
cites work: Automatic continuity of certain isomorphisms between regular Banach function algebras
cites work: Diameter preserving linear bijections of \(C(X)\)
cites work: Q5667036
cites work: Q4266789
full work available at URL: s0166-8641(01)00112-2
OpenAlex ID: W2008022472
description: scientific article; zbMATH DE number 1787518