Compact Subsets of a Metric Space are Closed and Borel

Let $(X,d)$ be a metric space, let $\mathcal{T}_{d}$ be the collection of subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology, and let $\mathcal{B}(X)$ be the Borel $\sigma$-algebra of $(X,d)$. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_{d})$.

Then the following hold.

1. (Closedness)¶ $K$ is a closed subset of the topological space $(X,\mathcal{T}_{d})$.

2. (Borel measurability)¶ $K\in\mathcal{B}(X)$ and $X\setminus K\in\mathcal{B}(X)$.