# Exterior Point of a Set

Let $\left( {X,\tau } \right)$ be a topological space and $A$ be a subset of $X$, then a point $x \in X$, is said to be an exterior point of $A$ if there exists an open set $U$, such that

$x \in U \in {A^c}$

In other words, let $A$ be a subset of a topological space $X$. A point $x \in {A^c}$ is said to be an exterior point of $A$ if there exists an open set $U$ containing $x$ such that

$U \in {A^c}$

Exterior of a Set

The set of all exterior points of $A$ is said to be the exterior of $A$ and is denoted by ${\text{Ext}}\left( A \right)$.

Remark:

It may be noted that an exterior point of $A$ is an interior point of ${A^c}$.

Theorems

• If $A$ is a subset of a topological space $X$, then (1) ${\text{Ext}}\left( A \right) = {\text{Int}}\left( {{A^c}} \right)$ (2) ${\text{Ext}}\left( {{A^c}} \right) = {\text{Int}}\left( A \right)$.
• If $A$ is a subset of a topological space $X$, then ${\text{Ext}}\left( A \right) \cap {\text{Int}}\left( A \right) = \phi$.
• In a topological space $X$, (1) ${\text{Ext}}\left( \phi \right){\text{ = Int}}\left( X \right)$ (2) ${\text{Ext}}\left( X \right){\text{ = Int}}\left( \phi \right)$.
• If $A$ is a subset of a topological space $X$, then (1) ${\text{Ext}}\left( A \right) \subseteq {A^c}$ (2) ${\text{Ext}}\left( {{A^c}} \right) \subseteq A$.
• If $A$ is a subset of a topological space $X$, then ${\text{Ext}}\left( A \right)$ is the largest open subset of ${A^c}$.