Prove: For any topologies T_{1}and T_{2}for X, if B_{1}is a base for T_{1}, and if B_{1}⊂ T_{2}, then T_{1}⊂ T_{2}.

Using:

Thm 2.1: A is open in (X,T) iff For every p ∈ A, there is G ∈T such that p ∈ G and G ⊂ A

Cor 8.2: B is a base for T iff: B ⊂ T and for every G ∈ T, if p ∈G, then there is A ∈ B such that p is an element of A ⊂ G