Heine-Borel Theorem/Euclidean Space

Theorem

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.


Proof

Necessary Condition

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

$\Box$


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.

$\blacksquare$


Source of Name

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


Sources

  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $2$. Heine-Borel Theorem
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): compact
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