Characteristic property of the quotient topology/Statement

Statement

$\xymatrix{ X \ar[d]_{q} \ar[dr]^{f\circ q} & \\ Y \ar[r]_f & Z }$
In this commutative diagram $f$ is continuous $\iff$ $f\circ q$ is continuous

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces and let $q:X\rightarrow Y$ be a quotient map. Then:

• For any topological space, $(Z,\mathcal{ H })$ a map, $f:Y\rightarrow Z$ is continuous if and only if the composite map, $f\circ q$, is continuous