Prove that if A and B are subsets of a topological space (X, T) then cl(A∪B) = cl(A) ∪ cl(B).

Proof;

To prove cl(A∪B) = cl(A) ∪ cl(B) where A and B are Subsets of topological (X, T), we have need to show that cl(A) ∪ cl(B) ⊆ cl(A ∪ B) and cl(A ∪ B) ⊆ cl(A) ∪ cl(B).

As we know that

A ⊆ (A ∪ B) and B ⊆ (A ∪ B)

⇒    cl(A) ⊆ cl(A ∪ B) And  cl(B) ⊆ cl(A ∪ B)

∵ if A ⊆ B ⇒ cl(A) ⊆ cl(B)

⇒ cl(A) ∪ cl(B) ⊆ cl(A ∪ B) ----> (1)

∵ if A ⊆ C and B ⊆ C ⇒ (A ∪ B) ⊆ C

⇒ (A ∪ B) ⊆ cl(A) ∪ cl(B)

∵ A ⊆ cl(A) and B ⊆ cl(B)

⇒ cl(A ∪ B) ⊆ cl(A) ∪ cl(B) --- > (2)

∵ cl(A) ∪ cl(B) is the closed super set of (A ∪ B) and cl(A ∪ B) is the smallest closed super set of (A ∪ B) 

From (1) and (2) we have

cl(A ∪ B) = cl(A) ∪ cl(B)

Hence proved that if A and B are Subsets of Topological Space (X, T) then cl(A ∪ B) = cl(A) ∪ cl(B).