Let \(X\) and \(Y\) be topological spaces. A map \(q\colon X\to Y\) is a quotient map if \(q\) is surjective and, for every \(U\subseteq Y\), \(U\) is open in \(Y\) if and only if \(\inverseof{q}(U)\) is open in \(X\).
The implication from \(U\) open to \(\inverseof{q}(U)\) open is continuity of \(q\); a quotient map is a continuous surjection that also satisfies the converse implication.
A surjective map \(q\colon X\to Y\) is a quotient map if and only if \(q\) is continuous and \(q(V)\) is open in \(Y\) for every open subset \(V\subseteq X\) that is saturated with respect to \(q\). The same holds with “open” replaced by “closed” throughout.
See [@Mun00].