quotient map

Definition

For two topological spaces X and Y, a quotient map $p:X\to Y$ is a surjective (important) map such that

$U\subset Y\text{open}\iff p^{-1}(U)\subset X\text{open}$

Notes

• A quotient map is always a continuous map, since the quotient condition is stronger.
• If a map is continous and open (or closed) then it is a quotient map (but the converse it not always true)