# Boundary Point of a Set

Let be a subset of a topological space , a point is said to be boundary point or frontier point of if each open set containing at intersects both and .

The set of all boundary points of a set is called the boundary of or the frontier of . It is denoted by . Since, by definition, each boundary point of is also a boundary point of and vice versa, so the boundary of is the same as that of , i.e. .

**Theorems**

• If is a subset of a topological space , then .

• If is a subset of a topological space , then .

• If is a subset of a topological space , the is open .

• A subset of a topological space is closed if and only if it contains its boundary.

• A subset of a topological space has an empty boundary if and only if it is both open and closed.

• The boundary of a closed set is nowhere dense in a topological space.

• Let be a topological space. Then any closed subset of is the disjoint union of its interior and its boundary, in the sense that it contains these sets, they are disjoint, and it is their union.