# Polish spaces are continuous images of the Baire space

I'm having some troubles understanding the proof of Theorem 7.9 (pag. 39) in Kechris' "Classical Descriptive Set Theory":

There are two points of the proof proposed that I don't quite understand. First of all, when we define the Lusin scheme, why do we need to specify that every $$F_s$$ is going to be a $$F_\sigma$$ if right afterwards we set $$F_s = \bigcup_i \overline{F_{s^\smallfrown i}}$$, making it a $$F_\sigma$$ set by definition. Moreover I truly don't get how does he manage, at the end of the proof, to write $$C_{i+1} \setminus C_i = \bigcup_j E_j^{(i)}$$ with $$E_j^{(i)}$$ being pairwise disjoint $$F_\sigma$$ sets of diameter $$< \epsilon$$. Where do $$(E_j^{(i)})_j$$ come from? How do we know that $$C_{i+1} \setminus C_i$$ can be covered by pairwise disjoint $$F_\sigma$$ sets of diameter $$< \epsilon$$?

(ii) is somewhat superfluous. (iii) is already the intended construction: each $$F_s$$ is partitioned by its successors $$F_{s\smallfrown i}$$ and these sets are all $$F_\sigma$$ too. (ii) is just to reinforce and anticipate (iii), I think. (iii) is a statement of intent, not the definition of $$F_s$$.
The final point is more subtle: $$C_{i+1}\setminus C_i$$ is a relatively open set of the Polish space $$C_{i+1}$$. Every open subset of a Polish space can be written as a pairwise disjoint countable union of small diameter $$F_\sigma$$ sets. This follows as we can take a cover by small (diameter) open sets and as the set is hereditarily Lindelöf (being second countable) we can find a countable subcover of it, which we enumerate. After that, taking the standard trick of subtracting all previous sets, we get a countable disjoint family of sets that are all $$F_\sigma$$ (the difference of open sets in a metric space, using open sets are $$F_\sigma$$) and still small.
Ok, I just read the same proof on these notes, and after a while I realized that this proof actually answers my doubts. The fact is that given a $$F_\sigma$$ set $$F$$ and having $$F = \bigcup_i (C_{i}\setminus C_{i-1})$$, we can always find a countable open cover (since Polish spaces are second-countable and therefore Lindelöf) $$(U_n^{(i)})$$ of $$D_i = C_{i}\setminus C_{i-1}$$ of diameter $$< \epsilon$$. Then we can rearrange it in: $$D_i = \bigcup_n [D_i \cap (U_n^{(i)} \setminus \bigcup_{k=1}^{n-1} U_k^{(i)})]$$ If we call $$F_{i,n} = D_i \cap (U_n^{(i)} \setminus \bigcup_{k=1}^{n-1} U_k^{(i)})$$ then it is easy to see that these new pairwise disjoint sets $$(F_{i,n})_{i,n}$$ are $$F_\sigma$$ sets (since we are in a metrizable space) of diameter $$< \epsilon$$, and also we have that $$\forall i,n \quad \overline{F_{i,n}} \subseteq \bigcup_{j=1}^i D_j = C_i \subseteq F$$. So this proves the claim.