Closure and Interior in a Metric Space: Containments, the Closure is Closed, the Interior is Open

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 $A\subseteq X$, with closure $\operatorname{cl}A$ and interior $\operatorname{int}A$.

(i) Containments.

$$\operatorname{int}A\subseteq A\subseteq\operatorname{cl}A.$$

(ii) The closure is closed. $\operatorname{cl}A$ is closed in $(X,\mathcal{T}_{d})$.

(iii) The interior is open. $\operatorname{int}A$ is open in $(X,d)$.

(iv) Closed sets. $A$ is closed in $(X,\mathcal{T}_{d})$ if and only if $\operatorname{cl}A=A$.

(v) Open sets. $A$ is open in $(X,d)$ if and only if $\operatorname{int}A=A$.