Definition 4.1:

(i) Given a topological space $X$, a set $Y$ and a surjective map $f : X \longrightarrow Y$, the topology ${\cal T}$ defined by (4.1) is called the quotient topology on $Y$ induced by the function $f$. By construction $f$ is continuous with this topology on $Y$.

(ii) Given a map $f : X \longrightarrow Y$ between topological spaces $X$ and $Y$, we say $f$ is a quotient map if the given topology on $Y$ agrees with the quotient topology on $Y$ induced by $f$.

The quotient topology enjoys a universal property which is easy to prove but extremely useful.