Product topology on $X\times Y$

Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces. The product topology on \(X\times Y\) is the topology generated by the basis \[ \mathcal B\coloneqq\ts{U\times V \st U\in\tau_X,\ V\in\tau_Y}: \] a subset \(W\subseteq X\times Y\) is open if and only if it is a union of elements of \(\mathcal B\).