Definition of exterior boundary of compact set

Question

In potential theory, there is a result which states,

Support of equilibrium measure of compact set belongs to exterior boundary of the compact set.

But what is the definition of exterior boundary of a set?

Answer 1

Let $K$ be a compact subset of $\mathbb{R}^n$, $n\ge 2$. Its complement $U = \mathbb{R}^n\setminus K$ is an open set. For sufficiently large $R>0$, the set $V = \{x:|x|>R\}$ is contained in $U$. Since $V$ is connected, there exists a connected component of $U$ that contains $V$. You may want to notice that this is the unique unbounded connected component of $U$.

By definition, the exterior boundary of $K$ is the boundary of the unbounded connected component of $\mathbb{R}^n\setminus K$.