Covering a Set with Balls in a Continuously Varying Way

Question

Say $U \subset \mathbb{R}^n$ is open and $X \subset U$ is compact. I want to find a number $\epsilon > 0$ such that for all $x \in X$, $B_\epsilon(x) \subset U$. My plan is to define a function $\rho : X \to [0.\infty)$, where $\rho(x) = \sup\{\epsilon \in (0,1] : B_\epsilon(x) \subset U\}$. If this function is continuous, we can find its minimum because $X$ is compact, and then we're set. But is this function actually continuous?

Answer 1

The continuity of the distance function $d(x,U^c)$ as a function of $x\in \mathbb R^n$ is a routine exercise. Restricting $x\in X$ only has the effect of making this function positive (because $X$ is (closed) compact and contained in (open) $U$).

The only additional ingredient to consider is the limitation of $\rho(x)$ to take values that are at most $1$. So the Reader is asked to show that:

$$ \rho(x) := \min \{1,d(x,U^c)\} $$

is accordingly a continuous function of $x\in X$.