Heine-Borel Theorem

Theorem

Real Line

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.

Euclidean Space

For any natural number $n \ge 1$, a subspace $C$ of the Euclidean space $\R^n$ is closed and bounded iff it is compact.

Metric Space

A metric space is compact iff it is both complete and totally bounded.

Dedekind-Complete Linearly Ordered Space

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.