Let \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\). How many distinct sets of the form \(U\times V\) are there, with \(U\) open in \(S\) and \(V\) open in \(T\)?
There are \(2\times 3=6\) nonempty products, since \(S\) has 2 nonempty open sets and \(T\) has 3; every product involving \(\varnothing\) is the same empty set, counted once: \(6+1=7\).
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.