# Thread: Cantor-Bendixson Theorem, uniqueness

1. ## Cantor-Bendixson Theorem, uniqueness

Prove that the decomposition

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

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

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. $\mathcal{N}$ denotes Baire space. The decomposition comes from the Cantor-Bendixson Theorem:

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

Thanks.

2. Originally Posted by pascal4542
Prove that the decomposition

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

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

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. $\mathcal{N}$ denotes Baire space. The decomposition comes from the Cantor-Bendixson Theorem:

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

Thanks.
I might be able to help but you're definitions are different enough from mine that I feel like I need to ask.

I define the Cantor-Bendixson theorem to be "a closed subset of a Polish space may be written as the disjoint union of a perfect set and an at most countable set", yeah?

3. Originally Posted by Drexel28
I might be able to help but you're definitions are different enough from mine that I feel like I need to ask.

I define the Cantor-Bendixson theorem to be "a closed subset of a Polish space may be written as the disjoint union of a perfect set and an at most countable set", yeah?
It is fine to use this version because we have covered what Polish spaces are, so yes, this version is fine to use.

4. Originally Posted by pascal4542
It is fine to use this version because we have covered what Polish spaces are, so yes, this version is fine to use.
So is your question then to prove the representation is unique?

5. Yes that is what I need help with. I don't see how to do that.

6. Originally Posted by pascal4542
Yes that is what I need help with. I don't see how to do that.
The same question was posted here: S.O.S. Mathematics CyberBoard :: View topic - Cantor-Bendixson Theorem, uniqueness
I wish the poster mentioned such thing - so that they do not waste the time of helpers. I've tried to answer the question there, so far the OP did not answer whether my answer is satisfactory for him.

The terminology follows the books Notes on set theory by Yiannis Moschovakis. (From what I have read so far, I can say it's a good book.) This question is Problem x10.2 from that book.