Definition:Perfect Set

Definition
Let $T = \left({X, \tau}\right)$ be a topological space.

A perfect set is a subset $S \subseteq X$ such that:
 * $S = S\,'$

where $S\,'$ is the derived set of $S$.

That is, where:
 * every point of $S$ is a limit point of $S$ and
 * every limit point of $S$ is a point of $S$.

Alternative definitions:
 * A perfect set is a closed set which has no isolated points.
 * A perfect set is a set $S$ which is dense-in-itself and which contains all its limit points.

Also see

 * Equivalence of Definitions of Perfect Set