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Heine–Borel theorem
Subset of Euclidean space is compact if and only if it is closed and bounded
Heine–Borel theorem
In real analysis in mathematics, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S {\displaystyle S} of Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , the following two statements are equivalent: S {\displaystyle S} is compact, that is, every open cover of S {\displaystyle S} has a finite subcover S {\displaystyle S} is closed and bounded. The theorem is sometimes also called the Borel–Lebesgue lemma.
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