# 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.