Rigid zerodimensional dyadic spaces
type: Item
instance of: scholarly article
title: Rigid zerodimensional dyadic spaces
zbMATH Open document ID: 0770.54023
DOI: 10.1016/0166-8641(92)90030-4
author: Eric K. van Douwen
published in: Topology and its Applications
publication date: 1993-04-01T00:00:00Z
review text: A space \(X\) is called rigid if the identity is the only autohomeomorphism of \(X\). It was shown by Kuratowski that there exists a dense subset \(X\) of the real line such that \(\beta X\) is both zerodimensional and rigid. The main result of the paper under review says that for every uncountable cardinal \(\kappa\) there are \(2^ \kappa\) pairwise totally different rigid zerodimensional dyadic spaces of weight \(\kappa\) as well as \(2^ \kappa\) pairwise nonhomeomorphic rigid connected dyadic spaces of weight \(\kappa\); here spaces are called totally different provided that they do not contain homeomorphic open sets. The result itself is very interesting in the light of the fact that dyadic spaces are continuous images of Cantor cubes which are homogeneous. In the zero-dimensional case the result was announced in [the author with \textit{J. D. Monk} and \textit{M. Rubin}, Algebra Univers. 11, 220-243 (1980; Zbl 0451.06014)]. It was shown by \textit{Sabine Koppelberg} [in ``Handbook of Boolean algebras'', pp. 741-775 (1989; Zbl 0671.06001)] that if \(\aleph_ \omega\) is less than the continuum, then there exists a rigid space of weight \(\aleph_ \omega\) which is a retract of the Cantor cube of weight \(\aleph_ \omega\).
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 06E05
Mathematics Subject Classification ID: 54G15
Mathematics Subject Classification ID: 54B15
zbMATH DE Number: 149081
zbMATH Keywords: rigid zerodimensional dyadic spaces
zbMATH Keywords: Cantor cube
reviewed by: Aleksander Błaszczyk
MaRDI profile type: Publication
full work available at URL: 0166-8641(92)90030-4
OpenAlex ID: W2044562635
cites work: Some questions about Boolean algebras
cites work: Stone-Cech Compactifications of Products
description: scientific article; zbMATH DE number 149081