With \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\), \(Y=\{a,b,c\}\) and \(q(1)=q(2)=a\), \(q(3)=b\), \(q(4)=c\), how many open sets does the quotient topology on \(Y\) have?
A subset \(V\subseteq Y\) is open if \(q^{-1}(V)\) is open in \(X\). This holds for \(\varnothing\), \(\{a\}\) (preimage \(\{1,2\}\)), \(\{a,b\}\) (preimage \(\{1,2,3\}\)) and \(Y\): 4 in total.
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 \(X\).