Lebesgue's number lemma

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In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It 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 having diameter less than δ is contained in some member of the cover.

Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.

Proof[edit]

Let \mathcal U be an open cover of X. Since X is compact we can extract a finite subcover \{A_1, \dots, A_n\} \subseteq \mathcal U.

For each i \in \{1, \dots, n\}, let C_i := X \setminus A_i and define a function f : X \rightarrow \mathbb R by f(x) := \frac{1}{n} \sum_{i=1}^n d(x,C_i).

Since f is continuous on a compact set, it attains a minimum \delta. If Y is a subset of X of diameter less than \delta, then there exist x_0\in X such that Y\subseteq B_\delta(x_0), where B_\delta(x_0) denotes the radius \delta ball centered at x_0 (namely, one can choose as x_0 any point in Y). Since f(x_0)\geq \delta there must exist at least one i such that d(x_0,C_i)\geq \delta. But this means that B_\delta(x_0)\subseteq A_i and so, in particular, Y\subseteq A_i.

References[edit]

Munkres, James R. (1974), Topology: A first course, p. 179, ISBN 978-0-13-925495-6