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.

Thanks.