(Note: I asked this question at MSE over a day ago and received no answer, so I'm now reposting it here. Link: https://math.stackexchange.com/questions/853500/homeomorphism-of-compact-hausdorff-spaces)
In the preprint "A REMARK ON CANTOR DERIVATIVE" (http://arxiv.org/pdf/1104.0287v1.pdf), there is the next proof:
We show that two countable locally compact Hausdorff spaces $X$ and $Y$ of same Cantor-Bendixson rank and degree are homeomorphic.
Suppose first that $X$ and $Y$ be compact of rank $\alpha + 1$. Note that they are the disjoint union of finitely many compact spaces of degree 1, so one may assume that their degree is 1. We build a homeomorphism from $X$ to $Y$ by induction on the rank. Let $X_1$, $X_2$,... and $Y_1$, $Y_2$,... be two sequences of clopen sets roughly partitioning $X\smallsetminus X^\alpha$ and $Y\smallsetminus Y^\alpha$ respectively. As $X_1$ has smaller rank or degree than some finite union of $Y_i$, we may assume that X1 has smaller rank or degree than $Y_1$, and that $Y_1$ has smaller rank or degree that $X_2$ etc. We then build a back and forth: by induction hypothesis, there is sequence $f_1$, $g_1^{−1}$, $f_2$, $g^{−1}_2$,... of homeomorphism respectively from $X_1$ to some clopen $\widetilde Y_1 \subseteq Y_1$, from $Y_1 \smallsetminus \widetilde Y_1$ to some clopen set $\widetilde X_2\smallsetminus X_2$, from $X_2 \smallsetminus \widetilde X_2$ to $\widetilde Y_3 \smallsetminus Y_3$ etc. We call $f$ be the union of all $f_i$ and $g_i$, union one more map $f_\omega$ from $X$ to $Y$ and show that $f$ is continuous."
I can't understand how choose the partition $X_1$, $X_2$,... and $Y_1$, $Y_2$,... and how choose the sequence $f_1$, $g_1^{−1}$, $f_2$, $g^{−1}_2$,... can anybody help me please?
