subcover

[suhb-kuhv-er]

noun

Mathematics.

1. a set of subsets of a cover of a given set that also is a cover of the set.

Word History and Origins

Origin of subcover¹

sub- + cover

Example Sentences

The real definition of compactness is that a space is compact if every open cover of the space has a finite subcover.

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So the number line is not compact because we have found an open cover that does not have a finite subcover.

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That’s the point of the finite subcover in the definition of compactness.

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Proving noncompactness only requires producing one counterexample, while proving compactness requires showing that every single open cover of a space, no matter how oddly constructed, has a finite subcover.

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If you’ve taken a topology class before, you might have seen the definition of the topological property called compactness: a set is compact if every open cover of the set has a finite subcover. The topologist’s sine curve is not compact, but the closed topologist’s sine curve is.

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