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.
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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) |
Borel-Lebesgue covering theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel-Lebesgue_covering_theorem&oldid=22169