Decomposing topological spaces into two rigid homeomorphic subspaces
type: Item
instance of: scholarly article
title: Decomposing topological spaces into two rigid homeomorphic subspaces
zbMATH Open document ID: 0683.54026
DOI: 10.1007/BF02765038
author: S. Shelah
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: A topological space A is said to be rigid if the only homemorphism of A onto itself is the identity. A topological space X can be rigidly halved if X is the disjoint union of two subsets A and B so that A is homeomorphic with B and each of A and B is rigid. In the 1982 North- Holland Calendar, J. van Mill asked if the set of real numbers could be rigidly halved. \textit{J. van Mill} and \textit{E. Wattel} have shown [Colloq. Math. 50, 95-102 (1985; Zbl 0588.54031)] that the circle can be rigidly halved, but their technique fails for the set of real numbers. Independent affirmative answers to this question about the real numbers were given by the author [Results in mathematical logic, II, Abstr. Am. Math. Soc. 4, 190 (1983)] and by \textit{F. van Engelen} [Topology Appl. 17, 275-285 (1984; Zbl 0538.54011)]. The paper gives the author's proof that the set of real numbers can be rigidly halved. Furthermore, the proof holds for many separable normed vector spaces over the rationals.
Mathematics Subject Classification ID: 54C99
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 54B05
zbMATH DE Number: 4119211
zbMATH Keywords: rigid sets
zbMATH Keywords: partition
reviewed by: David G. Wright
MaRDI profile type: Publication
cites work: A decomposition of \({\mathbb{R}}\) into two homeomorphic rigid parts
cites work: Partitioning spaces into homeomorphic rigid parts
full work available at URL: https://doi.org/10.1007/bf02765038
OpenAlex ID: W2080288002
description: scientific article; zbMATH DE number 4119211