Lebesgue number lemma

Statement

Let $(X, d)$ be a compact metric space and let $(U_i)_{i \in I}$ be an open cover of $X$. Then there is a real $\delta > 0$ such that each subset $A \subseteq X$ with diameter $\operatorname{diam}(A) < \delta$ is contained in $U_i$ for some $i \in I$.

Remarks

Such a number $\delta$ is a Lebesgue number of the cover. Equivalently, for each $x \in X$, the ball $B(x, \delta / 2)$ lies in one of the sets $U_i$. Compactness is needed: the cover of $(0, 1)$ by the sets $(1/n, 1)$ has no Lebesgue number.

Explanations

L1 human

In a compact metric space, each open cover has a uniform size: each small enough set fits completely into one set of the cover. You can think of $\delta$ as a safe size for a window that you move around the space.