Prove that the decomposition
from the Cantor-Bendixson Theorem of a closed pointset into a perfect set and a countable set determines uniquely and .
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. denotes Baire space. The decomposition comes from the Cantor-Bendixson Theorem:
Every closed subset of can be decomposed uniquely into two disjoint subsets where , the kernel of , is perfect and , the scattered part of , is countable.