Exercise 7, § 5

Let $X$ and $Y$ be two non-empty topological spaces and $f : X \to Y$ a surjective map. Show that $f$ is closed if and only if the mapping $y \to (f^{-1}y)$ of $Y$ into $\mathfrak{P}_0(X)$ is continuous when $\mathfrak{P}_0(X)$ carries the topology $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) (§ 2, Exercise 7).

Deduce that the topology induced on a quotient set $X/R$ of a non-empty topological space $X$ by the topology $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) coincides with the quotient of the topology of $X$ by $R$ if and only if $R$ is closed (resp. open) (cf. § 3, Exercise 16).