Lebesgue's number lemma

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 $\delta > 0$ such that every subset of $X$ having diameter less than $\delta$; is contained in some member of the cover.

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

Proof

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 $\delta$. The key observation is that $\delta > 0$. 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 ball of radius $\delta$ 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, $Y\subseteq A_i$.

Alternative Proof

Let $\mathcal U$ be an open cover of $X$. For each $\delta > 0$ the set $V_\delta = \{ x \in X \vert \exists U_x \in \mathcal{U}: d(x,x') \le \delta \Rightarrow x'\in U_x\}$ is open, because for each $x \in V_\delta$ there is a positive distance $\epsilon$ between $X\backslash U_x$ and the compact $\delta$-ball around $x$, hence the open $\epsilon$-ball around $x$ is also contained in $V_\delta$.

The collection $\{ V_\delta \vert \delta>0\}$ is also an open cover of $X$. Since $X$ is compact, it is already contained in the union of finitely many $V_\delta$ and since $V_\delta \supseteq V_{\delta'}$ for $\delta < \delta'$, these finitely many are all contained in one, hence $X= V_\delta$ for some $\delta>0$, which is a Lebesgue number.

References

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