Heine-Borel Theorem/Real Line
Contents
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 $\R$.
$\blacksquare$
Note
A part of this page has to be extracted as a theorem: Put this into a page of its own: it needs to be proved. |
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
- Bolzano-Weierstrass Theorem
- Heine-Borel Theorem for Metric Spaces
- Compact Subspace of Linearly Ordered Space
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): $\text{III}$: Compactness