Definition:Compact Space/Topology/Subspace/Definition 2
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a subset of $S$.
$H$ is compact in $T$ if and only if every open cover $\CC \subseteq \tau$ for $H$ has a finite subcover.
Also see
- Results about compact spaces can be found here.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.2$: Definition of compactness: Definition $5.2.4$
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.5$: Topological spaces
- 2009: W.A. Sutherland: Introduction to Metric and Topological Spaces (2nd ed.): $13$: Compact spaces $\S$ Definition of compactness: Definition $13.6$
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 1.5$: Normed and Banach spaces. Compact sets