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 open sets does the product topology on \(S\times T\) have?
The open sets are all unions of the 7 basic products; computing them gives 10 distinct sets. The 7 products are only the basis, not the whole topology.
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.