# 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 the 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 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 $$X$$ 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.