Definition:Compact Space/Real Analysis/Definition 1

Definition

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

Let $H \subseteq \R$.

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

Also see

• Equivalence of Definitions of Compact Subset of Real Numbers

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.9$: Intervals