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 $${A^c}$$.