Let $X$ be a topological space and $Y$ be a set. Let $p: X \rightarrow Y$ be a surjection. The quotient topology on $Y$ is defined as follows: a subset $V \subset Y$ is open if and only if $p^{-1}(V)$ is open in $X$.

(1) Show that this does indeed define a topology on $Y$, and show that $p$ is continuous when we endow $Y$ with this topology.

(2) Let $Z$ be another topological space and $f: Y \rightarrow Z$ be a map. Show that $f$ is continuous if and only if $f \circ p: X \rightarrow Z$ is continuous.