Heine-Borel Theorem/Euclidean Space

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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.


Necessary Condition

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


Sufficient Condition

Let $C \subseteq \R^n$ be compact.

From Compact Subspace of Metric Space is Bounded, it follows that $C$ is bounded.

From Metric Space is Hausdorff, it follows that $\R^n$ is a Hausdorff space.

Then Compact Subspace of Hausdorff Space is Closed shows that $C$ is closed.


Source of Name

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