Past exam of the topology course of the University of Oxford / 2015 / 2 / b

For the canonical projection of a quotient set $p$, defineThe inverse images of the empty set and whole target are $\varnothing$ and $X$. Inverse images commute with arbitrary unions and finite intersections, so the topology axioms follow from those of $\mathcal{T}$. This is the quotient topology, and it makes $p$ continuous by definition.

The crucial criterion is an equivalence: $U$ is open precisely when $p^{-1}(U)$ is open. This is stronger than continuity alone and is what makes the next part work.

If $g$ is continuous, then so is $g\circ p$, as a composition of continuous functions. Conversely, suppose $g\circ p$ is continuous. For an open subset $W\subseteq Z$,is open in $X$. The quotient topology criterion therefore says $g^{-1}(W)$ is open in $X/\mathcal{R}$. This proves continuity of $g$.This is the universal property of the quotient topology.