Let \(X=\{1,2,3\}\) have open sets \(\varnothing,\{1\},\{2\},\{1,2\},X\) and \(q:X\to\{a,b\}\) with \(q(1)=q(2)=a\), \(q(3)=b\). What are the open sets of the quotient topology?
The preimage of \(\{a\}\) is \(\{1,2\}\), which is open; that of \(\{b\}\) is \(\{3\}\), which is not. So \(\{a\}\) is open and \(\{b\}\) is not: the topology is neither trivial nor discrete.
The topology \(\mathcal{T}_q\) on \(Y\) determined by a surjective function \(q:X\to Y\) from a space \((X,\mathcal{T})\), whose open sets are the \(V\subseteq Y\) with \(q^{-1}(V)\) open in \(X\).