= Solution
For the <canonical projection of a quotient set> $p$, define
$$
\boxed{\mathcal T_q=\{U\subseteq X/\mathcal R:p^{-1}(U)\in\mathcal T\}.}
$$
The <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.
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