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?