Borel-Lebesgue covering theorem

Let be a bounded closed set in and let be an open covering of it, i.e. a system of open sets the union of which contains ; then there exists a finite subsystem of sets , , in (a subcovering) which is also a covering of , i.e.

The Borel–Lebesgue theorem has a converse: If and if a finite subcovering may be extracted from any open covering of , then is closed and bounded. The possibility of extracting a finite subcovering out of any open covering of a set is often taken to be the definition of the set to be compact. According to such a terminology, the Borel–Lebesgue theorem and the converse theorem assume the following form: For a set to be compact it is necessary and sufficient for to be bounded and closed. The theorem was proved in 1898 by E. Borel [1] for the case when is a segment and is a system of intervals; the theorem was given its ultimate form by H. Lebesgue [2] in 1900–1910. Alternative names for the theorem are Borel lemma, Heine–Borel lemma, Heine–Borel theorem.

References

[1]	E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1928)
[2]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)