The Boundary of a Subset of a Metric Space Is Closed and Borel

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(S,d)$ be a metric space with Borel $\sigma$-algebra $\mathcal{B}(S)$, and let $A\subseteq S$ with boundary $\partial A$.

(i) The boundary is closed. $\partial A$ is closed in $(S,\mathcal{T}_{d})$, and hence $\partial A\in\mathcal{B}(S)$.