Heine-Borel Theorem

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.

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.

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

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.

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 $T$.

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

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