Definition 11

In a topological space $X$, a point $x$ is said to be a frontier point of a set $A$ if $x$ lies in the closure of $A$ and in the closure of $\overline{CA}$. The set of frontier points of $A$ is called the frontier of $A$.

The frontier of $A$ is therefore the set $\overline{A} \cap \overline{\overline{CA}}$, which is closed. A frontier point $x$ of $A$ is characterized by the property that every neighbourhood of $x$ contains at least one point of $A$ and at least one point of $\overline{CA}$; $x$ may or may not belong to $A$. The frontier of $A$ is the same as the frontier of $\overline{CA}$. The interior of $A$, the exterior of $A$ and the frontier of $A$ are mutually disjoint and their union is the whole space $X$.