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 .

*Proof*.

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 .

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