Past exam of the topology course of the University of Oxford / 2014 / 1 / c / i / Solution

The quotient set $X/\mathcal{R}$ consists of the equivalence classes. Let $p(x)=[x]$. Equip that set with the quotient topologyPreimages preserve the empty set, whole set, arbitrary unions and finite intersections, so this is a topology. The canonical projection is continuous and surjective, and the displayed equivalence specifies the quotient space completely.