Let (X,T) be a topological space and suppose that A and B are subsets of X then (Int(A)∪Int(B))⊆Int(A∪B).

Proof;

As we know 

A⊆(A∪B) and B⊆(A∪B)

⇒ Int(A)⊆Int(A∪B) and In(B)⊆Int(A∪B)

∵ If A⊆B then Int(A)⊆Int(B)

⇒ (Int(A)∪Int(B))⊆Int(A∪B) 

∵ If A⊆C and B⊆C then (A∪B)⊆C

Hence proved that (Int(A)∪Int(B))⊆Int(A∪B), where A and B are subsets of a topological space (X, T).