Product Topology

Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces. The product topology on the Cartesian product $X\times Y$ is the collection of all subsets $W\subseteq X\times Y$ with the following property:

for every point $(x,y)\in W$, there exist sets $U\in\mathcal{T}_X$ and $V\in\mathcal{T}_Y$ such that