The Distance to a Subset of a Metric Space: Bounds, the Lipschitz Estimate, Continuity, and Vanishing Exactly on the Closure

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(X,d)$ be a metric space and let $A\subseteq X$ be nonempty, with the distance $\operatorname{dist}(x,A)$ from a point $x\in X$ to $A$.

(i) Bounds. For every $x\in X$, $0\le\operatorname{dist}(x,A)\le d(x,a)$ for every $a\in A$; and for every real number $r>\operatorname{dist}(x,A)$ there is $a\in A$ with $d(x,a)<r$.

(ii) Lipschitz estimate. For all $x,y\in X$,

$$\bigl|\operatorname{dist}(x,A)-\operatorname{dist}(y,A)\bigr|\le d(x,y).$$

(iii) Continuity. For every $K\subseteq X$, the function $K\to\mathbb{R}$, $x\mapsto\operatorname{dist}(x,A)$, is continuous on $K$.

(iv) Vanishing. For $x\in X$, $\operatorname{dist}(x,A)=0$ if and only if $x$ is a closure point of $A$.

(v) Positivity off a closed set. If $A$ is closed in $(X,\mathcal{T}_{d})$ and $x\in X\setminus A$, then $\operatorname{dist}(x,A)>0$.