Let A be a subset of a topological space X, a point is said to be boundary point or frontier point of A if each open set containing at intersects both and.
            The set of all boundary points of a set A is called the boundary of A or the frontier of A. It is denoted by. Since, by definition, each boundary point of A is also a boundary point of and vice versa, so the boundary of A is same as that of, i.e. .

Theorems:

• If A is a subset of a topological space X, then.
• If A is a subset of a topological space X, then.
• If A is a subset of a topological space X, the A is open.
• A subset of a topological space X is closed if and only if it contains its boundary.
• A subset of a topological space has 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 X be a topological space. Then any closed subset of X 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.