Definition:Compact Space/Real Analysis/Definition 2
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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:
- 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
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.1$: Motivation: Provisional definition $5.1.3$