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 exist 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}.

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