Let $\R$ be the real number line considered as a Euclidean space.
Let $C \subseteq \R$.
Then $C$ is closed and bounded in $\R$ iff $C$ is compact.
For any natural number $n \ge 1$, a subspace $C$ of the Euclidean space $\R^n$ is closed and bounded iff it is compact.
A metric space is compact iff it is both complete and totally bounded.
Let $(X, \preceq, \tau)$ be a Dedekind-complete linearly ordered space.
Let $Y$ be a nonempty subset of $X$.
Then $Y$ is compact iff $Y$ is closed and bounded in $X$.
Source of Name
This entry was named for Heinrich Eduard Heine and Émile Borel.