Heine-Borel Theorem

From ProofWiki
Jump to: navigation, search

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 $\R$ iff $C$ 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 iff it is compact.


Metric Space

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


Dedekind-Complete Linearly Ordered Space

Let $(X, \preceq, \tau)$ be a Dedekind-complete linearly ordered space.

Let $Y$ be a nonempty subset of $X$.


Then $Y$ is compact iff $Y$ is closed and bounded in $X$.



Source of Name

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


Also see