The Heine-Borel Theorem states that a subset of Euclidean space is compact if and only if it is both closed and bounded. It gives the standard working definition of compactness in finite-dimensional spaces and is named for Eduard Heine and Emile Borel, who established special cases of the result before it was stated in full generality.
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StatementA subset of Euclidean space is compact if and only if it is both closed and bounded. 1 Classification
Statement FormCharacterization Theorem 1 Heine-Borel Theorem
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Source Heine-Borel Theorem (Wikipedia)
Source MacTutor History of Mathematics: Emile Borel (MacTutor)
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Source Heine-Borel Theorem (Wikipedia)
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1. Heine-Borel Theorem (Wikipedia)
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Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine-Borel theorem.
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In real analysis in mathematics, the Heine-Borel theorem, named after Eduard Heine and Émile Borel, states:
Principal Result Of: Eduard Heine, Lead sentence of the article on the result
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.
Proved By: Peter Gustav Lejeune Dirichlet, Lead
Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.
View the Source MacTutor History of Mathematics: Emile Borel (MacTutor)
MacTutor History of MathematicsPrincipal Result Of: Emile Borel, Biography, 1892 memoirQuote, Principal Result Of: Emile Borel, Biography, 1892 memoir
acquired the name of the Heine -Borel theorem
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