The Product Metric Induces the Product Topology

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Let $\mathcal{T}_X$ be the collection of all subsets open in $(X,d_X)$ and let $\mathcal{T}_Y$ be the collection of all subsets open in $(Y,d_Y)$; both are topologies by Metric Open Sets Form a Topology. Let $d_{X\times Y}$ be the product metric on $X\times Y$, which is a metric by claim 1 of The Product Metric is a Metric.

Then a subset $W\subseteq X\times Y$ is open in the metric space $(X\times Y,d_{X\times Y})$ if and only if $W$ belongs to the product topology determined by $\mathcal{T}_X$ and $\mathcal{T}_Y$. Consequently the collection of subsets of $X\times Y$ open in $(X\times Y,d_{X\times Y})$ is exactly that product topology.