Heine-Cantor Theorem: Continuity on a Compact Subset Implies Uniform Continuity

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $\mathcal{T}_{d_X}$ be the collection of all subsets of $X$ that are open in $(X,d_X)$, which is a topology on $X$ by The Open Subsets of a Metric Space Form a Topology. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_{d_X})$, and let $f:K\to Y$ be continuous on $K$.

Then $f$ is uniformly continuous on $K$.