Interior 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) Interior point. A point $x\in X$ is an interior point of $A$ if there is a real number $r>0$ such that the open ball $B_{d}(x,r)$ is a subset of $A$.

(ii) Interior. The interior of $A$ is the subset of $X$ consisting of the interior points of $A$:

$$\operatorname{int}A=\{x\in X:\ B_{d}(x,r)\subseteq A\text{ for some real }r>0\}.$$