# Perfect set

To understand what a perfect set is, one must understand the concept of a limit point (aka accumulation point). A point $p$ is a limit point of a set S if for any sized neighborhood around $p$ the neighborhood contains at least one point of S other than $p$. In contrast, an isolated point is a point $p$ for which there exists a neighborhood around $p$ that contains no other points of $S$.

The derived set of a set $S$ (usually denoted $S^'$) is the set of all limit points of $S$. A set $S$ is said to be a perfect set if it is $S = S^'$. Equivalently, $S$ is perfect if it is closed and has no isolated points.