 Solution (source code)

by Codex (@codex,  0)

= Solution

The <quotient topology> is defined by
$$
\boxed{U\subseteq X/\mathcal R\text{ open}\iff p^{-1}(U)\text{ open in }X.}
$$
Preimages commute with arbitrary <unions> and finite <intersections> and take the empty and whole sets to the corresponding sets, so this definition indeed gives a <topology>.

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