The Product Topology is a Topology

Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces, let $X\times Y$ be their Cartesian product, and let $\mathcal{T}_{X\times Y}$ be the product topology on $X\times Y$. Then $(X\times Y,\mathcal{T}_{X\times Y})$ is a topological space.