Definition:Compact Space/Real Analysis

Definition

Let $\R$ be the real number line considered as a topological space under the Euclidean topology.

Let $H \subseteq \R$.

Definition 1

$H$ is compact in $\R$ if and only if $H$ is closed and bounded.

Definition 2

$H$ is compact in $\R$ if and only if:

when $H$ is the union of a set of neighborhoods which are open in $H$,

then $H$ is also the union of a finite number of these neighborhoods.

Also see

• Equivalence of Definitions of Compact Subset of Real Numbers

• Heine-Borel Theorem, where it is proved that this definition is equivalent to the topological definition when $\R$ is considered with the Euclidean topology.

• Results about compact spaces can be found here.