The equivariant universality and couniversality of the Cantor cube

Michael G. Megrelishvili ¹   ; Tzvi Scarr ²

¹ Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel
² Department of Applied Mathematics Jerusalem College of Technology 21 Havaad Haleumi St. Jerusalem, Israel 91160

Fundamenta Mathematicae, Tome 167 (2001) no. 3, pp. 269-275

Résumé

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 $.

Détail

DOI : 10.4064/fm167-3-4

Keywords: langle alpha rangle g space where non archimedean having local base identity consisting subgroups second countable topological group zero dimensional compact metrizable space langle aleph aleph tau rangle natural evaluation action full group autohomeomorphisms cantor cube there exists topological group embedding varphi hookrightarrow aleph there exists embedding psi hookrightarrow aleph equivariant respect varphi psi equivariant retract aleph respect varphi psi