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.

History

The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed and bounded interval is uniformly continuous. 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. He used this proof in his 1852 lectures, which were published only in 1904. 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. His formulation was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.