Originally Posted by

**pascal4542** Prove that the decomposition

$\displaystyle F = P \cup S, P \cap S = \emptyset $

from the Cantor-Bendixson Theorem of a closed pointset $\displaystyle F$ into a perfect set $\displaystyle P$ and a countable set $\displaystyle S$ determines uniquely $\displaystyle P$ and $\displaystyle S$.

I am not sure how to prove this. I would appreciate a few hints or suggestions. In our book, pointsets are subsets of Baire space. Also, a pointset is perfect if it is the body of a splitting tree. $\displaystyle \mathcal{N}$ denotes Baire space. The decomposition comes from the Cantor-Bendixson Theorem:

Every closed subset $\displaystyle F$ of $\displaystyle \mathcal{N}$ can be decomposed uniquely into two disjoint subsets $\displaystyle F = P \cup S, P \cap S = \emptyset $ where $\displaystyle P$, the kernel of $\displaystyle F$, is perfect and $\displaystyle S$, the scattered part of $\displaystyle F$, is countable.

Thanks.