Perfect Sets

Definition

Given $P \subseteq R$ then $P$ is perfect if $P$ is closed and $P$ has no Isolated Points.

For example, if $a < b$ :

$P = [a, b]$ is a perfect set.
In contrast if $P = [a, b] \cup {c}$ ( $c > b$ ), then this set is not perfect since it has an isolated point.