# Perfect set

### 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.