Let $n\in\mathbb{N}$, and let $A\subseteq\mathbb{R}^n$. Then the following are equivalent.

1. $A$ is \reftext{def:compact-space-and-subset-2026a}{compact in $\mathbb{R}^n$}, where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}.
2. $A$ is \reftext{def:closed-subset-topological-space-2026a}{closed} in $\mathbb{R}^n$ and \reftext{def:bounded-subset-metric-space-2026a}{bounded} as a subset of the metric space $(\mathbb{R}^n,d_E)$.