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.
To determine the closure of A, follow these steps:
• Identify all the closed sets that encompass A.
• Take the intersection of these closed sets.
• Identify all the closed sets that encompass A.
• Take the intersection of these closed sets.
The outcome will be the closure of A.
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.