# Interior Point of a Set

Let be the topological space and , then a point is said to be an interior point of set , if there exists an open set such that

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

**Interior of a Set**

Let be a topological space and be a subset of , then the interior of is denoted by or is defined to be the union of all open sets contained in .

In other words let be a topological space and be a subset of . The interior of is the union of all open subsets of , and a point in the interior of is called an interior point of .

**Remarks:**

• The interior of is the union of all open sets contained in . The union of open sets is again an open set. Hence the interior of is the largest open set contained in .

• and

• The interior of sets is always open.

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**Example:**

Let with topology . If , then find . Since there is no open set containing and a subset of , so is not an interior point of . Similarly, is not an interior point of . Since is an open set containing and is a subset of , so is an interior point of . Hence .

**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 of a topological space is the union of all open subsets of .

• The subset of topological space is open if and only if .

• If is a subset of a topological space , then .

• Let be a topological space and and are subsets of , then **(1)** **(2)** **(3)**