Let $M$ be the projective cover (e.g, [Gleason1958](https://doi.org/10.1215/ijm/1255454110)) of the Cantor set $\{-1,1\}^{\mathbb{N}}$. Let $\textrm{homeo}(M)$ denote the group of all homeomorphisms of $M$.
Some of the $\gamma\in\textrm{homeo}(M)$ are extensions of the homeomorphisms of the Cantor set. Some others are the extensions of the homeomorphisms of Polish spaces $K$ for which $M$ is the projective cover of $K$.



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> **Q:** Is there a complete explicit description of the homeomorphisms of $M$?
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I'd be grateful if you point me to a reference about the properties of $\textrm{homeo}(M)$.


Thanks in advance.



Homeomorphisms of the projective cover of the Cantor set
I claim that every autohomeomorphism of $M$ is conjugate to an extension of an autohomeomorphism of the Cantor space.


Let $B$ be the Stone space of the Cantor space. Let $\overline{B}$ denote its Boolean completion. We shall show that every automorphism of $\overline{B}$ is conjugate to an automorphism of $B$. Recall that $\overline{B}$ is up-to-isomorphism the only complete atomless Boolean algebra with a countable basis (by a basis, I mean a subset $A\subseteq\overline{B}$ where $\overline{B}=\{\bigvee R:R\subseteq A\}$; for example, $B$ is a countable basis for $\overline{B}$). Suppose that $h:\overline{B}\rightarrow \overline{B}$ is an automorphism. Then let $D\subseteq \overline{B}$ be the smallest Boolean subalgebra closed under the operations $h,h^{-1}$ and which contains the countable basis $A$. Then $h$ restricts to an automorphism $g:D\rightarrow D$, and $h$ is the unique automorphism of $\overline{B}$ that extends $g$. Recall that every countable atomless Boolean algebra is isomorphic (the theory of countable atomless Boolean algebras is $\omega$-categorical), so $B$ and $D$ are isomorphic. Let $\phi:B\rightarrow D$ be an isomorphism. Then $\phi$ extends to an automorphism $\Phi:\overline{B}\rightarrow\overline{B}$. In this case, $\Phi^{-1}h\Phi:\overline{B}\rightarrow\overline{B}$ is an automorphism that uniquely extends an automorphism of $B$.


One can extend this argument to conclude that if $(h\_n)\_{n=0}^\infty$ is a collection of automorphisms of $\overline{B}$, then there is some
automorphism $\Phi:\overline{B}\rightarrow\overline{B}$ where for each $j$, the mapping $\Phi^{-1}h\_j\Phi$ uniquely extends an automorphism of $B$.


