Lebesgue's number lemma

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In topology, Lebesgue's number lemma states

If the metric space (X, d) is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. The number δ is called the Lebesgue Number of this cover for X.


[edit] Proof

Let \mathcal A be an open cover of X. If X \in \mathcal A, then any δ will work, so assume X \not\in \mathcal A. Choose \{A_1, \dots, A_n\} \subseteq \mathcal A that covers X. For each i, set Ci = X − Ai and define f:X\rightarrow\mathbb R by letting f(x) = \frac{1}{n}\sum_{i=1}^n d(x, C_i). Given x in X, choose i so that x \in A_i. Then choose ε so that the ε-neighborhood of x lies in Ai. Then d(x, C_i) \ge \epsilon, so f(x) \ge \frac{\epsilon}{n}.

Since f is continuous, it has a minimum value δ. We will show that δ is the Lebesgue number. Let B be a subset of X of diameter less than δ. Choose x_0 \in B, thus B lies in the δ-neighborhood of x0. Now \delta \le f(x_0) \le d(x_0, C_m) where d(x0,Cm) is the largest of the number d(x0,Ci). So the δ-neighborhood of x0, and thus B, is contained in the element Am = X − Cm of the covering \mathcal A

[edit] Applications

The Lebesgue number lemma is useful in the study of compact metric spaces and functional metric spaces, since it can often be used to obtain approximations of distances when the space is compact.