Prove that 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. (Rudin: Principles of Mathematical Analysis, 2nd ed.)
Every separable metric space has a countable base.
The Attempt at a Solution
Let F be closed. Using the above fact, I've shown that the isolated points of F are at most countable, likewise their closure. I'm trying to construct a perfect set by removing non-limit points of F' points from F', the set of limit points of F, but it's not quite falling into place yet. Is this a good direction to go?