The Lebesgue number of an open covering $\omega$ of a metric space $X$ is any number $\epsilon>0$ such that if a subset $A$ of $X$ has diameter $<\epsilon$, then $A$ is contained in at least one element of $\omega$. 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 $\lambda$ of a metric space $X$ is any number $\epsilon>0$ such that if a set $A\subset X$ of diameter $\leq\epsilon$ intersects all the elements of some subsystem $\lambda'$ of $\lambda$, 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://www.encyclopediaofmath.org/index.php?title=Lebesgue_number&oldid=33437