Let be a topological space.

Let = is a neighborhood of

Then is alocal baseat if for every neighborhood of , , where .

Also, is abaseif for every open set , where .

Prove that

(1) If is a local base for , then is a local base for .

(2) If is a local base at , then both and are local bases for .

(3) if is a local base for , then is a base for the topology on X.