##  [Lebesgue Number Lemma](/lebesgue-number-lemma-0) 

 Definition

A lemma stating that for every open cover of a compact metric space there exists a positive real number (a Lebesgue number) such that every subset of the space with diameter less than that number is contained in some member of the cover.

 

 

 

 

 

 





## Principle

Principle

Compactness in metric spaces implies a uniform scale below which the cover is locally trivial: compactness guarantees a positive lower bound on the sizes of subsets needed to ensure containment in a single cover element.

 

 

 

 

 





## Demonstration

Demonstration

For the unit interval [0,1] with an open cover by finitely many intervals, choose the minimum radius among Lebesgue numbers coming from a finite subcover; concretely, if [0,1] is covered by open intervals each of length > ε_i, a positive δ bounded by the minimum of these lengths works so that any subset of diameter < δ lies in one interval.

 

 

 

 

## Misapplication

Misapplication

Assuming a Lebesgue number exists for arbitrary open covers of noncompact metric spaces (for example the open cover of (0,1) by intervals {(1/n,1): n∈N} has no positive Lebesgue number) or for general topological spaces without a metric; doing so yields false uniformity claims.

 

 

 

 

 





## Consequence

Consequence

Enables uniform local control over covers: underlies constructions such as partitions of unity subordinate to a cover, guarantees existence of sufficiently fine meshes for triangulations, and is a standard tool in proofs that require passing from local to global data on compact metric spaces.

 

 

 

 

## Reversal

Reversal

If no positive Lebesgue number exists for a cover, then the space cannot be compact (in the metric sense) relative to that cover; conversely, exhibiting a positive Lebesgue number for every open cover characterizes compactness in metric spaces.

 

 

 

 

 





## Boundary

Boundary

Applies to compact metric spaces and their open covers; it does not hold in general for noncompact metric spaces or for covers in non-metrizable topologies without further hypotheses (local compactness or paracompactness are insufficient alone to guarantee a uniform positive number).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Sometimes conflated with notions of mesh or Lebesgue covering dimension; the Lebesgue number is a quantitative local scale for a specific cover, while covering dimension and mesh are global or combinatorial invariants of covers.

 

 

 

 

 





## Synthesis

Synthesis

The Lebesgue Number Lemma states that compactness in metric spaces implies the existence of a uniform positive scale (the Lebesgue number) below which every small-diameter subset is contained in a single element of any given open cover, providing a bridge between local metric size and global covering structure.