Cantor Bendixson Theorem: Every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.
This definition differs a bit from that in wikipedia.
I have proved that 'If $X$ is a separable metric space and $E$ is a uncountable subset and $P$ is the set of all condensation points of $E$, then $P$ is perfect and $P^c \cap E$ is at most countable'.
Then, you can see that 'every uncountable set in a separable metric space is the union of a nonempty perfect set and a set which is at most countable, and sets are disjoint' (Since $E= P\cup (P^c \cap E)$)
Here, i didn't use the condition 'closed' at all! Where did i go wrong?