With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\), which of these subsets of \(S\times T\) is open?
\(\{(1,a),(1,b)\}\) equals \(\{1\}\times\{a,b\}\), a product of open sets. The other three are not unions of products of open sets: none contains an open product around its point with first coordinate 0.
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.