Quotient space

Let be a topological space and an equivalence relation on . Define the quotient map with . Equip this space with the quotient topology:

By this definition, is continuous and surjective.

Now suppose and are topological spaces with a relation on . Let and s.t. . Then we can find s.t. .

1. If is continuous, so is
2. If is open, so is

Proof

1. Let be open in . Then is open in . Now is open, so by definition is open.
2. Let be open in . Then is open so is open i.e. is open. But, as is surjective, we have , hence is open.

Corollary

If is surjective, continuous and open and , then is a homeomorphism.