A Compact Subset of an Open Set Admits a Uniform Ball Radius

Let $(X,d)$ be a metric space, equipped with the collection of all subsets open in $(X,d)$, which is a topology by The Open Subsets of a Metric Space Form a Topology. Let $\mathbb{R}$ be the real numbers. Write $\bar B_{d}(x,r)$ for the closed ball in $(X,d)$ with centre $x$ and radius $r$.

Let $\Omega\subseteq X$ be open in $(X,d)$ and let $K\subseteq\Omega$ be compact in $X$.

Then there is a real number $r$ with $0<r$ such that

$$\bar B_{d}(x,r)\subseteq\Omega\qquad\text{for every }x\in K .$$