Are positively separated sets separated?

Question

I was reading this answer where the author does a clever trick to prove that the Lebesgue outer measure is a metric outer measure. I know how to prove that the Lebesgue outer measure is a metric outer measure the "long way," but how does one show that $\text{dist}(A, B) > 0$ implies $\overline{A} \cap B = \emptyset$? Sorry, I only know a tiny bit of topology and whenever topology shows up in my measure theory course I get incredibly lost.

Answer 1

Assume that's not the case, i.e. there is some $x\in \overline{A}\cap B$. Since $x\in\overline{A}$ then there is a sequence $(a_n)\subseteq A$ such that $a_n\to x$. Then

$$d(a_n,x)\to d(x,x)=0$$

And so $d(a,b)$ over $a\in A$ and $b\in B$ is arbitrarily small. Meaning $\text{dist}(A,B)=0$.