Let \(X=\{1,2,3,4\}\) have the discrete topology and \(q:X\to\{a,b\}\) be given by \(q(1)=q(2)=a\), \(q(3)=q(4)=b\). How many open sets does the quotient topology have?
Since every subset of \(X\) is open, every preimage is open and every subset of \(\{a,b\}\) is open. A 2-point set has \(2^2=4\) subsets: the quotient is 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 \(X\).