Lebesgue number lemma refinement lemma

Refinement lemma (used for Lebesgue numbers). Let $(X,d)$ be a metric space, let $K\subseteq X$ be compact, and let $\mathcal U$ be an open cover of $K$ . For each $x\in K$ , choose $U_x\in\mathcal U$ with $x\in U_x$ . Since $U_x$ is open, there exists $r_x>0$ such that

B(x,r_x)\subseteq U_x.

Then there exist points $x_1,\dots,x_N\in K$ such that

K\subseteq \bigcup_{i=1}^N B\!\left(x_i,\frac{r_{x_i}}{2}\right).

In particular, if $\delta=\min_{1\le i\le N}r_{x_i}/2$ , then $\delta>0$ and for every $x\in K$ there exists $U\in\mathcal U$ with $B(x,\delta)\subseteq U$ .

This is the standard compactness step that produces a uniform scale from pointwise local containment. See also Lebesgue number lemma.