quotient topology

Definition

For any topological space X and a set A with a surjective map $f:X\to A$. 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 $f:X\to A$, a subset $U\subset A$ is open in the quotient topology if $f^{-1}(U)\subset X$ is open.

Or in other words, to get an open set in $A$, take one $U\subset X$ and $f(U)\subseteq A$ 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