How many homeomorphisms are there between the $G_ {\delta}$ sets of the Cantor set? I have been stuck in the following problem for a few weeks and I have not been able to prove it.
Fix $D_{1}$ and $D_{2}$ disjoint countable dense subsets of $2^{\omega}$ such that $\mathcal{D}=D_{1}\cup D_{2}$ is an independent family. Why are there at least $\mathfrak{c}$ homeomorphisms $f_{\eta}:G_{\eta}\longrightarrow G_{\eta}$ such that $f[D_{1}]=D_{2}$, where $G_{\eta}\supseteq\mathcal{D}$ is a $G_{\delta}$ subset of $2^{\omega}$?
 A: In the following, I assume that the sets $G_\eta$ are not fixed. At least, given the subindex $\eta$, it seems to be the case.
Since $D_1$ and $D_2$ are dense in $\mathcal{D}$, which in turn is dense in $2^\omega$, it follows that they are perfect, countable metric spaces and thus homeomorphic to $\mathbb{Q}$. On the other hand, any homeomorphism between $D_1$ and $D_2$ extends to a homeomorphism between $G_\delta$ subsets, by Lavrentiev's theorem. Hence, it is enough to construct $\mathfrak{c}$ homeomorphisms from $\mathbb{Q}$ onto itself.
Let $z\in \left\{\frac{1}{3},\frac{2}{3}\right\}^\omega$, and define
$$
H_z(x) :=\begin{cases}
x & x<0 \\
n+2z_n(x-n) & x\in \left[n,n+\frac{1}{2}\right]\cap\mathbb{Q} \\
n+1+2(1-z_n)(x-n-1) & x\in \left[n+\frac{1}{2},1\right]\cap\mathbb{Q},
\end{cases}
$$
where $n$ ranges over the nonnegative integers (you can see this idea put to use in this paper on homeomorphisms of the real line).
Then $\{H_z\}_z$ is a family of $\mathfrak{c}$ homeomorphisms of $\mathbb{Q}$.
