Let be a topological space.
Let = is a neighborhood of
Then is a local base at if for every neighborhood of , , where .
Also, is a base if 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.