Definition:Compact/Topology
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Definition
A topological space $X$ is compact if every open cover of $X$ has a finite subcover.
See also the other equivalent definitions of compactness.
A subset $Y \subseteq X$ is said to be compact (in $X$) if the topological subspace $Y$ is.
For subsets of Euclidean space, compactness is equivalent to being closed and bounded by the Heine-Borel Theorem.
Also see
- Results about compact spaces can be found here.
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$
