Lebesgue number
The Lebesgue number of an open covering of a metric space
is any number
such that if a subset
of
has diameter
, then
is contained in at least one element of
. For any open covering (cf. Covering (of a set)) of a compactum there is at least one Lebesgue number; one can construct a two-element covering of the straight line for which there is no Lebesgue number.
The Lebesgue number of a system of closed subsets of a metric space
is any number
such that if a set
of diameter
intersects all the elements of some subsystem
of
, then the intersection of the elements of the system is not empty. Any finite system of closed subsets of a compactum has at least one Lebesgue number.
Lebesgue number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_number&oldid=33437