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 $V_{\delta }=\{x\in X\vert \exists U_{x}\in {\mathcal {U}}:d(x,x')\leq \delta \Rightarrow x'\in U_{x}\}$ 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 $V_{\delta }$.

For $\delta <\delta '$ we have $V_{\delta }\supseteq V_{\delta '}$ and moreover $X$ is covered by the union of all $V_{\delta }$ ($\delta >0$). By the compactness of $X$ there are finitely many $V_{\delta }$, that already cover $X$ and by the monotony of $\delta \mapsto V_{\delta }$ these finitely many sets 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