Interior Point of a Set

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

x \in U \subseteq A

In other words let A be a subset of a topological space X, a point x \in A is said to be an interior points of A if x is in some open set contained in A.

 

Interior of a Set

Let \left( {X,\tau } \right) be a topological space and A be a subset of X, then the interior of A is denoted by {\text{Int}}\left( A \right) or {A^o} is defined to be the union of all open sets contained in A.

In other words let \left( {X,\tau } \right) be a topological space and A be a subset of X. The interior of A is the union of all open subsets of A, and a point in the interior of A is called an interior point of A.

 

Remarks:
• The interior of A is the union of all open sets contained in A. The union of open sets is again an open set. Hence the interior of A is the largest open set contained in A.
{\phi ^o} = \phi and {X^o} = X
• The interior of sets is always open.
{A^o} \subseteq A

 

Example:

Let X = \left\{ {a,b,c,d,e} \right\} with topology \tau = \left\{ {\phi ,\left\{ b \right\},\left\{ {a,d} \right\},\left\{ {a,b,d} \right\},\left\{ {a,c,d,e} \right\},X} \right\}. If A = \left\{ {a,b,c} \right\}, then find {A^o}. Since there is no open set containing a and a subset of A, so a is not an interior point of A. Similarly, c is not an interior point of A. Since \left\{ b \right\} is an open set containing b and is a subset of A, so b is an interior point of A. Hence {A^o} = \left\{ b \right\}.

 

Theorems
• Each point of a non empty subset of a discrete topological space is its interior point.
• The interior of a subset of a discrete topological space is the set itself.
• The interior of a subset A of a topological space X is the union of all open subsets of A.
• The subset A of topological space X is open if and only if A = {A^o}.
• If A is a subset of a topological space X, then {\left( {{A^o}} \right)^o} = {A^o}.
• Let \left( {X,\tau } \right) be a topological space and A and B are subsets of X, then (1) A \subseteq B \Rightarrow {A^o} \subseteq {B^o} (2) {\left( {A \cap B} \right)^o} = {A^o} \cap {B^o} (3) {\left( {A \cup B} \right)^o} \supseteq {A^o} \cap {B^o}