Heine-Borel Theorem/Real Line

Theorem

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.

Proof

Necessary Condition

Let $C$ be closed and bounded in $\R$.

Then, by Closed Bounded Subset of Real Numbers is Compact, $C$ is compact.

$\Box$

Sufficient Condition

Let $C$ be compact in $\R$.

Then, by Compact Subspace of Real Numbers is Closed and Bounded, $C$ is closed and bounded in $\R$.

$\blacksquare$

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This does not apply in the general metric space.

A trivial example is $\left({0 \,.\,.\, 1}\right)$ as a subspace of itself.

It is closed and bounded but not compact.

Also see

Source of Name

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

The theorem is sometimes called the Borel-Lebesgue Theorem, for Émile Borel and Henri Léon Lebesgue.

Sources

1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness