Boundary Point of a Set

Let A be a subset of a topological space X, a point x \in X is said to be boundary point or frontier point of A if each open set containing at x intersects both A and {A^c}.

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 {F_r}\left( A \right). Since, by definition, each boundary point of A is also a boundary point of {A^c} and vice versa, so the boundary of A is same as that of {A^c}, i.e. {F_r}\left( A \right) = {F_r}\left( {{A^c}} \right).

Theorems:

• If A is a subset of a topological space X, then {F_r}\left( A \right) = \overline A \cap \overline {{A^c}} .

• If A is a subset of a topological space X, then {F_r}\left( A \right) = \overline A - {A^o}.

• If A is a subset of a topological space X, the A is open  \Leftrightarrow A \cap {F_r}\left( A \right) = \phi .

• 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.