# 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 $\R$ if and only if $C$ 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 if and only if it is compact.

### Metric Space

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

### Normed Vector Space

Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space.

A subset $K \subseteq X$ is compact if and only if $K$ is closed and bounded.

### Dedekind-Complete Linearly Ordered Space

Let $T = \struct {X, \preceq, \tau}$ be a Dedekind-complete linearly ordered space.

Let $Y$ be a non-empty subset of $X$.

Then $Y$ is compact if and only if $Y$ is closed and bounded in $T$.

## Source of Name

This entry was named for Heinrich Eduard Heine and Émile Borel.

## Also see

- Compact Subspace of Linearly Ordered Space
- Heine-Borel Theorem/Real Line/Closed and Bounded Interval
- Heine-Borel Theorem/Real Line/Closed and Bounded Set

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Heine-Borel theorem**