Sometimes basic elements are not easy to understand. It is well known result that every perfect can be written as pairwise disjoint perfect sets. Assume $P\subset\Bbb R$ be a perfect set. and let $\Bbb P$ be the set of all perfect subsets of $\Bbb R.$ Then, Let $$\mathcal P:=\{P^K_{\xi}\colon \xi<\mathfrak c\ \ \&\ \ K\in\Bbb P\}$$ be a partition of $P$ into perfect sets. I have no problem about that. My question about the indexes:
Does that mean for every $\xi<\mathfrak c$ and for every $K\in\Bbb P$ I have pair $\langle \xi, K\rangle$ index for some perfect set such that $P^K_{\xi}$ in $\mathcal P$? Or not. Also I know the partition can be written as $\{P_{\xi}\colon \xi<\mathfrak c\}$ but I am asking in case as in $\mathcal P$. As I said my question seems simiply but I just want to make sure I am in right path.
Any help will be appreciated greatly.
Yes, if you gave a set of size $\mathfrak{c}$ (like your partition of $P$) you can index it by another set of size $\mathfrak{c}$ (the product of $\mathfrak{c}$ and $\Bbb P$) and we can devote each member of that partition as some (unique) $P^K_\xi$ as you did. But why one would is unclear, it's more common to just enumerate them as indexed by ordinals (for transfinite recursion e.g.). It could be for a bookkeeeping reason perhaps.