Favorite application of the fact that disjoint compact sets are distant (i.e. $A$ compact, $B$ closed, $A \cap B = \varnothing\implies d(A,B)>0$)? This problem is quite popular (A and B disjoint, A compact, and B closed implies there is positive distance between both sets has currently 70 upvotes, not to mention the endless horde of repeats that one can find of this question on this site), and I myself have seen it for homework at least once or twice. I am wondering if this theorem is widely used. In essence, I am asking the same question as Your favourite application of the Baire Category Theorem but for this "compact sets are distant" theorem instead of BCT.
I'll go first (added to the answers): I've seen it used in a key way on pg. 27 of Shlomo Sternberg's notes introductory notes on Lebesgue measure
http://people.math.harvard.edu/~shlomo/212a/11.pdf. Basically we have that although Lebesgue outer measure $\mu^*$ is not additive in general, it's easy to show that for sets with positive distance between them, $\mu^*$ IS additive, meaning that for disjoint compact sets $\mu^*$ is additive.
 A: One that comes to my mind is the following (closely related to what you proposed above):
Consider an outer measure $\mu$ defined on the class $\mathcal{P}(X)$ of subsets of a metric space $(X, d). \mu$ is a Caratheodory outer measure, more often called metric outer measure if
$$
\mu(A \cup B)=\mu(A)+\mu(B)
$$
for every pair of sets $A, B \subset X$ which have positive distance.
Every time we have a compact and a closed set that are disjoint we can use additivity of the measure thanks to the fact you mention. One can show Hausdorff measures are metric outer measures and this theorem is therefore ubiquitously applied in the field of geometric measure theory.
A: I've seen it used in a key way on pg. 27 of Shlomo Sternberg's notes introductory notes on Lebesgue measure
http://people.math.harvard.edu/~shlomo/212a/11.pdf. Basically we have that although Lebesgue outer measure $\mu^*$ is not additive in general, it's easy to show that for sets with positive distance between them, $\mu^*$ IS additive, meaning that for disjoint compact sets $\mu^*$ is additive.
A: I think the following problem can be immediately answered using the theorem.

Let $(X, d)$ be a metric space. $A$,$B$ are disjoint compact subsets of X.
There exist disjoint open sets $U$, $V$ such that $A \subset U$ and $B \subset V$.

Consider union of open balls of distance $\frac{\delta}{4}$ centered at each point in $A$. This works as the required open set. Similarly, construct for $B$.
In fact, if we are just working in a metric space, only one of $A$ and $B$ need to be compact the other one needs to be just closed.
A: *

*If you are in Euclidean (or Manifold) setting, you can use this to show that there are $C^\infty$ functions that are constant 1 on one set and constant zero on the other.


*If A and B are at a positive distance from each other then the map that takes $(a,b) \in A \times B$ to the line connecting a to b is continuous -- as a map into projective space. I used this in a particular situation together with dimension arguments to show that I can slide one set completely away from the second one by flowing it  parallel to a line that is not in the image of the map above, so, effectively proving that they are not topologically tangled up!
