Lebesgue's number lemma

Lebesgue's number lemma is a result in metric topology asserting that every open cover of a compact metric space admits a uniform positive scale subordinate to the cover. More precisely, if ((X,d)) is a compact metric space and (\mathcal U) is an open cover of (X), then there exists a real number (\delta>0) such that every subset of (X) having diameter less than (\delta) is contained in at least one member of (\mathcal U). Any such (\delta) is called a Lebesgue number of the cover.

The lemma converts the local condition expressed by openness into a uniform metric condition. Compactness is responsible for this passage from point-dependent neighborhoods to a single positive number valid throughout the space. The result is used in arguments concerning uniform continuity, simplicial approximation, and the subdivision of singular chains.

Statement

For a nonempty subset (A\subseteq X), its diameter is

[ \operatorname{diam}(A)=\sup{d(x,y):x,y\in A}. ]

A positive number (\delta) is a Lebesgue number for (\mathcal U) when

[ \operatorname{diam}(A)<\delta \quad\Longrightarrow\quad A\subseteq U ]

for some (U\in\mathcal U). The quantification includes subsets that are neither open nor closed.

An equivalent formulation uses metric balls. There exists (r>0) such that, for every (x\in X), the ball (B(x,r)) lies in some member of the cover. The two formulations can require different numerical constants. A ball condition with radius (r) implies the diameter condition for every (\delta\leq r), while a diameter condition with constant (\delta) implies a ball condition after replacing (\delta) by a sufficiently smaller radius.

The lemma remains unchanged when the original cover is replaced by an open refinement. It also applies directly to finite covers, although finiteness is not part of the hypothesis because compactness supplies a finite subcover.

Proof

Assume that no Lebesgue number exists. For every positive integer (n), there is then a subset (A_n\subseteq X) satisfying

[ \operatorname{diam}(A_n)<\frac1n ]

that is not contained in any member of (\mathcal U). Select a point (x_n\in A_n). Compactness gives a convergent subsequence ((x_{n_k})) with limit (x\in X).

Because (\mathcal U) covers (X), some (U\in\mathcal U) contains (x). Openness provides an (\varepsilon>0) for which (B(x,\varepsilon)\subseteq U). For sufficiently large (k),

[ d(x_{n_k},x)<\frac{\varepsilon}{2} \qquad\text{and}\qquad \operatorname{diam}(A_{n_k})<\frac{\varepsilon}{2}. ]

Every (y\in A_{n_k}) consequently satisfies

[ d(y,x)\leq d(y,x_{n_k})+d(x_{n_k},x)<\varepsilon. ]

Thus (A_{n_k}\subseteq B(x,\varepsilon)\subseteq U), contradicting the defining property of (A_{n_k}). A positive Lebesgue number therefore exists.

A second proof expresses the same compactness argument through distance functions. After passing to a finite subcover (U_1,\ldots,U_m), define

[ f(x)=\max_{1\leq i\leq m} d\bigl(x,X\setminus U_i\bigr). ]

When none of the (U_i) equals (X), each term is a continuous real-valued function. At every point (x), at least one term is positive because (x) belongs to some open set (U_i). Hence (f(x)>0) on (X). The extreme value theorem gives a positive minimum (a), and every (\delta<a) is a Lebesgue number. If one member of the cover equals (X), every positive number is already a Lebesgue number.

Historical development

The lemma developed from early twentieth-century work connecting compactness with dimension theory. Henri Lebesgue used uniform scales subordinate to coverings in his study of topological dimension, and the resulting covering principle acquired his name as it entered the general language of metric topology.

In 1911, You Watanabe formulated the diameter version for arbitrary open covers of compact metric spaces. Her treatment separated the finite-subcover argument from the dimension-theoretic setting and identified the resulting positive scale as a property of the cover rather than of a particular construction. This formulation is equivalent to the modern statement.

The terminology stabilized after metric spaces became a standard framework for topology. Maurice Fréchet had introduced the abstract metric-space formalism in which diameter and convergence could be treated independently of Euclidean coordinates. Felix Hausdorff subsequently organized related compactness and neighborhood principles within general topology.

Cover-based methods were later incorporated systematically into dimension theory by Pavel Alexandrov and Pavel Urysohn. In that context, the lemma relates metric mesh to the purely topological operation of refining a cover. The eponym continues to refer to Lebesgue because the covering-scale argument originated in his work on dimension.

Interpretation by refinements

For a family (\mathcal V) of subsets of (X), the mesh is

[ \operatorname{mesh}(\mathcal V) =\sup_{V\in\mathcal V}\operatorname{diam}(V). ]

If (\delta) is a Lebesgue number of (\mathcal U), every family (\mathcal V) with mesh less than (\delta) refines (\mathcal U). The lemma may therefore be stated as the existence of a scale below which all metric decompositions are automatically subordinate to the given cover.

This interpretation distinguishes the lemma from the finite-subcover property. A finite subcover reduces the number of relevant open sets, whereas a Lebesgue number controls the size of subsets that can cross their boundaries. The proof based on distance functions shows how the former property yields the latter in a compact metric space.

The cover also determines the function

[ \lambda_{\mathcal U}(x) =\sup{r>0:B(x,r)\subseteq U \text{ for some }U\in\mathcal U}. ]

Openness implies (\lambda_{\mathcal U}(x)>0) for every (x). Without compactness, the infimum of these pointwise radii can equal zero. On a compact space, the lemma states that a positive uniform lower scale exists, although the supremal admissible radius need not itself satisfy every strict or closed-ball version of the definition.

Role of compactness

Compactness cannot be omitted from the standard theorem. Consider (X=(0,1)) with its usual metric and the open cover

[ \mathcal U=\left{\left(\frac1n,1\right):n\geq 2\right}. ]

For every (\delta>0), a sufficiently short interval of the form ((0,a)\subseteq X) has diameter less than (\delta). It is not contained in any member of (\mathcal U), because it contains points below (1/n) for every fixed (n). The cover therefore has no Lebesgue number.

The converse does not characterize compactness among all metric spaces. An infinite space equipped with the discrete metric is not compact, but every open cover has a Lebesgue number smaller than (1). Metric spaces in which every open cover has a Lebesgue number form a broader class sometimes studied under the Lebesgue covering property or as Atsuji spaces.

Compactness nevertheless gives a general sufficient condition without additional restrictions on isolated points or on the behavior of sequences at infinity. It also makes the result independent of the particular compatible metric on a compact metrizable space, since any two compatible metrics on such a space determine the same uniform structure.

Applications

In the compact-domain theorem for uniform continuity, a continuous map

[ f:X\longrightarrow Y ]

from a compact metric space to a metric space determines, for each prescribed (\varepsilon>0), an open cover of (X) on whose members the oscillation of (f) is less than (\varepsilon). A Lebesgue number for that cover supplies a single positive (\delta) satisfying

[ d_X(x,y)<\delta \quad\Longrightarrow\quad d_Y(f(x),f(y))<\varepsilon. ]

Thus the lemma isolates the covering argument underlying the Heine–Cantor theorem.

In algebraic topology, repeated barycentric subdivision decreases the mesh of the simplices in a finite complex. Once that mesh is below a Lebesgue number for a cover, every subdivided simplex lies in one member of the cover. This fact connects local data defined on covering sets with global constructions involving simplicial maps or chains.

The same scale control appears in the proof of the simplicial approximation theorem. The inverse images of open stars form a cover of the domain, while a Lebesgue number determines a subdivision level at which each simplex is mapped into a suitable star. The topological cover and the combinatorial subdivision are thereby compared through a metric bound.