Kechris, p. 32, Exercise (6.6) (A)
Let be a Polish space. Decompose uniquely into , where is perfect, is countable open. This decomposition is possible by Cantor-Bendixson Theorem.
Exercise (6.6): is the largest perfect subset of .
Suppose is perfect (so closed and perfect in its relative topology). From the proof of Theorem (6.4), and in the notation defined there, we have , hence . Now suppose , and further that ; notice that , being a set, is also a Polish space.
Claim: is a perfect space.
Let . Let be an open nbhd of . That is, there exists open in such that . Then is also open in , since is open in . Since is perfect, there exists . Therefore, is perfect and nonempty, hence uncountable by Theorem (6.2).
But this contradicts that is countable. Hence, , and thus .