TITLE: Is every Closed set a Perfect set?
QUESTION [9 upvotes]: 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?

REPLY [7 votes]: Closed means all limit points are in E.  But that doesn't mean all points in E are limit points.  Any closed set with a point that is not a limit point will not be perfect.
The easiest counter example is a set with a single point.  That set is closed but its one point isn't a limit point.
Less trivial and less contrived is $D = \{a + 1/n| a \in \mathbb Z,n \in \mathbb N\} $.  Every integer is a limit point.  No other point is a limit point.  All integers are in D (because $a + 1/1$ is an integer) so D is closed.  But for all $n > 1$ then $a + 1/n $ is in D but is not a limit point.  So D is not perfect.