Heine-Borel Theorem
Theorem
Real Line
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.
Euclidean Space
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.
Metric Space
A metric space is compact if and only if it is both complete and totally bounded.
Normed Vector Space
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.
Dedekind-Complete Linearly Ordered Space
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 $T$.
Source of Name
This entry was named for Heinrich Eduard Heine and Émile Borel.
Also see
- Compact Subspace of Linearly Ordered Space
- Heine-Borel Theorem/Real Line/Closed and Bounded Interval
- Heine-Borel Theorem/Real Line/Closed and Bounded Set
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Heine-Borel theorem