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Lebesgue number

From Encyclopedia of Mathematics
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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.

How to Cite This Entry:
Lebesgue number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_number&oldid=11603
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article