The closure of set A within a topological space X is the smallest closed set that encompasses A. It can be described as the intersection of all the closed sets that include A.

• Identify all the closed sets that encompass A.

• Take the intersection of these closed sets.

The outcome will be the closure of A.

How do you find the closure of a set in topological space? |

## Examples:

### 1) Consider T = { Φ, X , {1, 3}, {5, 7} } be topology defined on X = { 1, 3, 5, 7} and A = {1, 3} is a subset of X. Then find Cl(A).

#### Solution;

All closed sets are

Φ, X, {1, 7}, {3, 5}

All closed super sets of A are

X, {1, 3}

Cl(A) = X ∩ {1, 3} = {1, 3}

### 2) Let X be the set of real numbers, and B = {x ∈ X | 0 ≤ x ≤ 1}, the closed interval from 0 to 1. Then find Cl (B).

#### Solution;

The closure of a subset B of set of real numbers is itself B, as closure of a subset is determine by taking intersection of its closed super sets and since B is already a closed subset that contains itself. When we take intersection of closed super sets of a subset B we obtain subset B as a closure.

The concept of closure aids in defining other topological ideas like a set's interior, boundary, and limit points.