quotient topology

Definition

For any topological space X and a set A with a surjective map . There is a unique topology on A for which f is a quotient map, this topology is called the quotient topology

Notes

Usually, this definition is useless, so instead we use to it build topologies. Given a topological space X and a set A with a surjective map , a subset is open in the quotient topology if is open.

Or in other words, to get an open set in , take one and must be open in the quotient topology.

Geometric craziness

This is especially useful to build some geometric shapes from known topological spaces such as:

• torus