Vanishing of the Distance to a Set Characterizes the Closure

Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of all subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology. Let $A\subseteq X$ be nonempty and let $x\in X$, and write $\operatorname{dist}_d(x,A)$ for the distance from $x$ to $A$ in $(X,d)$. Let $0$ be the additive identity of the ordered field of real numbers.

Then $\operatorname{dist}_d(x,A)=0$ if and only if $x$ belongs to the closure $\operatorname{cl}_X(A)$ of $A$ in the topological space $(X,\mathcal{T}_d)$.