Here is The Cantor-Bendixon theorem In Kechris Books page 32 “Every Polish space can be uniquely written as a disjoint union of a $P$ perfect set and Countable open $C$”
While Kechris show being uniqueness he says “if $Y$ is a perfect Polish space, $Y$ equals the condensation points of it”
While showing that part here is his attempt
Let $Y$ be a perfect Polish space And $x\in Y$ and $U$ is any open neighborhood of $x$, then $U\cap Y$ is perfect and so uncountable, so x is a condensation point of Y because $U\cap Y$ is uncountable.
How can we say that $U\cap Y$ is perfect as subspace? My wondering is to prove this: “If $Y$ is a perfect Polish space then for any open neighborhood U of any point $x$ of $Y$, $U\cap Y$ is perfect”
Here is my attempt:
“Let’s assume that $U\cap Y$ is not perfect and thus it has an isolated point $x_{0}$. So there is an open neighborhood $V$ in $U\cap Y$ of $x_{0}$ such that $V\cap(U\cap Y)=\{x_{0}\}$.
On the other hand, there is a $G$ open in $Y$ such that $V=G\cap(U\cap Y)$ so from here $G\cap U$ is open in $Y$. And $(G\cap U)\cap Y=\{x_{0}\}$. So $x_{0}$ is an isolated point of $Y$, then it contradicts $Y$ is perfect.
It might be wrong, any help or hint will be greatly appreciated. Thank you in advance.