Disconnected compact subsets of $\mathbb{R^{n}}$

Question

Consider $K \subset \mathbb{R^{n}}$ compact and disconnected. Then $K = A \cup B$ for some $A, B \subset \mathbb{R^{n}}$, with $A \cap \bar{B} = \bar{A} \cap B = \emptyset.$ Is it necessarily true that $d(A,B) > 0$? If so how do we prove it?

Answer 1

Yes, it is true. Suppose otherwise. Then, for each $n\in\mathbb N$, there are $a_n\in A$ and $b_n\in B$ such that $d(a_n,b_n)<\frac1n$. Since $K$ is compact, we can assume without loss of generality that $\lim_{n\in\mathbb N}a_n=k\in K$. But then $\lim_{n\to\infty}b_n=k$ too. Then $k\in A$ or $k\in B$. But if $k\in A$, then, since $\lim_{n\to\infty}b_n=k$, $k\in\overline B$. And if $k\in B$, $k\in\overline A$. But this is impossible, since we are assuming that $A\cap\overline B=\overline A\cap B=\emptyset$.