# Isolated Point of a Set

Let be a subset of a topological space , then a point is said to be an isolated point of if there exists an open set containing which does not contain any point of different from . In other words, a point is said to be an isolated point of if there exists an open set containing such that . It is obvious from the definition of an isolated point of a set that an isolated point of can never be the limit point of . The set of all isolated points of is usually denoted by .

**Theorem**

Any closed subset of a topological space is the disjoint union of its set of isolated points and its set of limit points in the sense that it contains these sets, they are disjoint, and it is their union.

**Perfect Set**

A subset of a topological space is said to be a perfect set if it is equal to its derived set. Thus, a subset of a topological space is said to be a perfect set if .

**Theorem**

A subset of a topological space is perfect if and only if it is closed and has no isolated points.

**Proof**

Let be a perfect subset of a topological space , then . Since is the set of all limit points of and a limit point is not an isolated point, so has no isolated points.

Conversely, let be closed and have no isolated points. Then is equal to its derived set, i.e., , so is perfect.