Closure of a Subset of a Metric Space

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 $A\subseteq X$.

(i) Closure point. A point $x\in X$ is a closure point, or adherent point, of $A$ if every open ball centred at $x$ meets $A$: $B_{d}(x,r)\cap A\neq\emptyset$ for every real number $r>0$.

(ii) Closure. The closure of $A$ is the subset of $X$ consisting of the closure points of $A$:

$$\operatorname{cl}A=\{x\in X:\ B_{d}(x,r)\cap A\neq\emptyset\text{ for every real }r>0\}.$$