With \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\), \(Y=\{a,b,c\}\) and \(q(1)=q(2)=a\), \(q(3)=b\), \(q(4)=c\), which of these subsets of \(Y\) is open in the quotient topology?
The preimage of \(\{a,b\}\) is \(\{1,2,3\}\), open in \(X\). The preimages of \(\{b\}\), \(\{c\}\) and \(\{a,c\}\) are \(\{3\}\), \(\{4\}\) and \(\{1,2,4\}\), none of which is open.
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 \(X\).