From 'baby' Rudin.
I've seen that a set is closed iff it contains all of its limit points. In Rudin, $(d)$ says if every limit point of E is a point of E, then $E$ is closed. He also says $(h)$: $E$ is perfect if $E$ is closed and if every point of $E$ is a limit point of $E$.
But Closed $\implies$ contains all of its limit points. So, is every closed set a perfect set?
Best Answer
No finite set is perfect but every finite set is closed; a finite set has no limit points and thus all of its limit points (all zero of them) belong to it, so it's closed.