Homeomorphism of compact Hausdorff spaces

(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?

• Please give the MSE link anyway, so that people who nevertheless think it belongs there and not here can post an answer there. – Todd Trimble Jul 2 '14 at 23:30
• In general waiting just one day isn't enough before cross posting. Either post on MO from the start, and see if it gets migrated; or wait for a few days and then flag your question for moderator attention and request they manually migrate the question here. In either case, both cross posting is bad and the waiting period after asking a question on one site should be longer than a day. Closer to a week perhaps. – Asaf Karagila Jul 3 '14 at 2:13