A Product of Compact Subsets is Compact in the Product Metric

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, each equipped with the collection of its subsets that are open in the respective metric space, a topology by Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $X$ and let $L\subseteq Y$ be compact in $Y$. Equip $X\times Y$ with the product metric $d_{X\times Y}$, a metric by claim 1 of The Product Metric is a Metric, and with the topology of its open subsets. Let $d_{K\times L}$ denote the restriction of $d_{X\times Y}$ to $K\times L$, a metric on $K\times L$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology.

Then $K\times L$ is compact in $X\times Y$. Equivalently, the topological space consisting of the set $K\times L$ together with the collection of its subsets open in the metric space $(K\times L,d_{K\times L})$ is compact.