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 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

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

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