# 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

Let $n \in \N_{> 0}$.

Let $C$ be a subspace of the Euclidean space $\R^n$.

Then $C$ 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.