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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## The equivariant universality and couniversality of the Cantor cube

### Tom 167 / 2001

Fundamenta Mathematicae 167 (2001), 269-275 MSC: 54H15, 22A99. DOI: 10.4064/fm167-3-4

#### Streszczenie

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 $\psi$.

#### Autorzy

• Michael G. MegrelishviliDepartment of Mathematics
Bar-Ilan University
52900 Ramat-Gan, Israel
e-mail
• Tzvi ScarrDepartment of Applied Mathematics
Jerusalem College of Technology