The equivariant universality and couniversality of the Cantor cube
Volume 167 / 2001
Fundamenta Mathematicae 167 (2001), 269-275
MSC: 54H15, 22A99.
DOI: 10.4064/fm167-3-4
Abstract
Let $\langle G,X,\alpha \rangle $ be a $G$-space, where $G$ is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and $X$ is a zero-dimensional compact metrizable space. Let $\langle H(\{ 0,1\} ^{\aleph _0}),\{ 0,1\} ^{\aleph _0},\tau \rangle $ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then
(1) there exists a topological group embedding $\varphi :G \hookrightarrow H(\{ 0,1\} ^{\aleph _0})$;
(2) there exists an embedding $\psi :X \hookrightarrow \{ 0,1\} ^{\aleph _0}$, equivariant with respect to $\varphi $, such that $\psi (X)$ is an equivariant retract of $\{ 0,1\} ^{\aleph _0}$ with respect to $\varphi $ and $\psi $.
