Closed Interval $[a,b]$ is Compact in $\mathbb{R}$

Let $a,b$ be real numbers with $a\le b$ in the order of the ordered field $\mathbb{R}$, and let $[a,b]$ be the closed interval determined by $a$ and $b$.

Let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, whose metric coincides with the Euclidean distance on $\mathbb{R}^{1}$ identified with $\mathbb{R}$ by The Euclidean Distance on the Real Line is the Absolute Value Metric, and let $\mathcal{T}_{d_{\mathbb{R}}}$ be the collection of subsets of $\mathbb{R}$ that are open in $(\mathbb{R},d_{\mathbb{R}})$, which is a topology on $\mathbb{R}$ by Metric Open Sets Form a Topology.

Then $[a,b]$ is compact in $(\mathbb{R},\mathcal{T}_{d_{\mathbb{R}}})$.