 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $q:X\to X/\mathcal R$ send each point to its <equivalence class>. The <quotient topology> is
$$
\boxed{U\subseteq X/\mathcal R\text{ is open}\iff q^{-1}(U)\text{ is open in }X.}
$$
It is the finest <topology> on the quotient set making the canonical surjection $q$ continuous. The preimage test, rather than a choice of representatives, defines its <open sets>.

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