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 of \(S\times T\) are NOT of the form \(U\times V\)?
There are 10 open sets in total and 7 are products (counting \(\varnothing\)), so 10−7=3 are unions of products that are not products. For example, \((\{1\}\times T)\cup(S\times\{a\})\) is not a product.
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.