Definition:Lebesgue Number

Definition

Let $M = \struct {A, d}$ be a metric space.

Let $\UU$ be an open cover of $M$.

A fixed strictly positive real number $\epsilon \in \R_{>0}$ is called a Lebesgue number for $\UU$ if and only if:

$\forall x \in A: \exists \map U x \in \UU: \map {B_\epsilon} x \subseteq \map U x$

where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$ in $M$.

Also see

• Number Smaller than Lebesgue Number is also Lebesgue Number
• Open Cover may not have Lebesgue Number

Source of Name

This entry was named for Henri Léon Lebesgue.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Definition $7.2.11$