Compactness is a topological property that appears in a wide variety of contexts. In particular, it is a "tameness property" that tells you that the objects you are dealing with are in some sense well-behaved.
Let be a topological space and let .
The set is said to be compact if and only if for every open cover of , there exists a finite set such that is also an open cover of . This is often expressed in the sentence, "A set is compact if and only if every open cover admits a finite subcover."
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