Interior points of a set

• Definition of interior points:
  • Example1:
  • solution:

Definition of interior points:

Let (X,τ) be a topological space and A⊆X Then, a point x∈A (x is an element of set A) is said to be an interior point of A if there exists an open set ‘U’ containing ‘x’ such that

                                X∈U⊆A

Example1:

Consider the set X={1,2,3,4} and the topology τ={φ,{1},{1,2,3},{4},{1,4},X}

let A= {1,3,4} ⊆ X. what are the interior points of set A?

solution:

To find the interior points of set A, we check all the elements of set A one by one.

For 1:(there exists 4 open sets ‘U’ containing ‘1’)

1∈{1}⊆{1,3,4}                     1∈{1,4}⊆{1,3,4}    

1∈{1,2,3}⊈{1,3,4}              1∈X ⊈{1,3,4}           

1 is an interior point of set A.

For 3:

3∈{1,2,3}⊈{1,3,4}               3∈X ⊈{1,3,4}           

3 is not an interior point of set A.

For 4:

4∈{4}⊆{1,3,4}                     4∈{1,4}⊆{1,3,4}    

4∈X ⊈{1,3,4}    

4 is an interior point of set A.