Definition:Perfect Set/Definition 1

Definition
A perfect set of a topological space $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that:
 * $H = H'$

where $H'$ is the derived set of $H$.

That is, where:
 * every point of $H$ is a limit point of $H$

and
 * every limit point of $H$ is a point of $H$.

Also see

 * Equivalence of Definitions of Perfect Set