Rigid zerodimensional dyadic spaces
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 \(\aleph_ \omega\).
scientific article; zbMATH DE number 149081