With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\), how many points does the closure of \(\{(0,a)\}\) in \(S\times T\) have?
The closure of a product is the product of the closures: \(\overline{\{0\}}=\{0\}\) in \(S\) and \(\overline{\{a\}}=T\) in \(T\). This gives \(\{0\}\times T\), with 3 points.
The topology on the product \(X\times Y\) of two spaces \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) that has as basis \(\mathcal{B}\) the sets \(U\times V\) with \(U\) and \(V\) open.