Exercise 16, § 3

Let $X$ be a non-empty topological space and $R$ an equivalence relation on $X$. Show that if $X/R$ is regarded as a subset of $\mathfrak{P}_0(X)$, then the quotient topology on $X/R$ is coarser than the topologies induced on $X/R$ by the topologies $\mathfrak{T}_\Omega$ and $\mathfrak{T}_\Phi$ on $\mathfrak{P}_0(X)$ (§ 2, Exercise 7).