Interior of a set

Table of Contents

• Definition
  • Example1
  • Example2

Definition

\begin{array}{l}Let\text{ }(X,\tau )\text{ }be\text{ }a\text{ }topological\text{ }space\text{ }and\text{ }A\subseteq X.\text{ }Then\text{ }the\text{ }interior\\\text{ }of\text{ }A\text{ }is\text{ }the\text{ }union\text{ }of\text{ }all\text{ }open\text{ }subsets\text{ }of\text{ }A.\text{ }It\text{ }is\text{ }denoted\text{ }by\\\text{ }Int\left( A \right)\text{ }or~\text{ }{{A}^{o}}.Int\left( A \right)is\text{ }the\text{ }largest\text{ }open\text{ }subset\text{ }of\text{ }A.\\i-e~\text{        }{{A}^{o}}~\subseteq A\end{array}

OR

The set of all interior points of A is called the interior of set A. it is denoted by Int(A).

Example1

Let X={a,b,c,d,e} and the topology τ={φ,{b},{a,b},{a,b,d},{a,c,d,e},X}

Let A= {a,b,c} ⊆X   and B={a,e} ⊆X

Find int(A) and int(B)

Solution

Open subsets: φ,{b},{a,b},{a,b,d},{a,c,d,e},X

Open subsets of set A are

φ,{b}

int(A)=union of all open subsets of A={a,b,c}

int(A)= φ∪{b}

int(A)={b}

Open subsets of set B 

φ

int(B)=union of all open subsets of B={a,e}

int(B)= φ

Example2

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.

Find interior points of set A.

Solution

Int(A)= set of all interior points of A

The Interior points of set ‘A’ are 1 and 4 (see the solution for finding interior points).

So,

Int(A)={1,4}