With \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\), \(Y=\{a,b,c\}\) and \(q(1)=q(2)=a\), \(q(3)=b\), \(q(4)=c\), let \(Z=\{0,1\}\) have open sets \(\varnothing,\{1\},Z\). How many functions \(f:Y\to Z\) are continuous for the quotient topology?
\(f\) is continuous if and only if \(f^{-1}(\{1\})\) is open in \(Y\), that is, one of \(\varnothing,\{a\},\{a,b\},Y\): 4 functions. This matches \(f\circ q\) being continuous, as the universal property says.
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 \(X\).