From Math Images
To understand what a perfect set is, one must understand the concept of a limit point (aka accumulation point). A point is a limit point of a set S if for any sized neighborhood around the neighborhood contains at least one point of S other than . In contrast, an isolated point is a point for which there exists a neighborhood around that contains no other points of .
The derived set of a set (usually denoted ) is the set of all limit points of . A set is said to be a perfect set if it is . Equivalently, is perfect if it is closed and has no isolated points.