Boundaries of connected components

Question

Let $ K \subset \mathbb{R}^{n}$ a compact set and $L_i$ the connected components of $\mathbb{R}^{n} \backslash K$.

I can't see why $\partial L_i \subset K$.

Answer 1

I’ll assume that $\partial A$ denotes a boundary of a set $A$. The claim can be proved assuming only closedness of the set $K$ in $\Bbb R^n$. Let $x$ be any point of any $\partial L_i$. If $x\in K$, we are done. Otherwise there exists (a unique) $L_j$ containing $x$. Since $L_j$ is an open subset of the space $\Bbb R^n\setminus K$ there exists a open subset $U$ of $\Bbb R^n$ such that $U\cap (\Bbb R^n\setminus K)=U\setminus K=L_j$. Thus $U$ intersects only $L_j$ among $L_k$’s, so $L_i=L_j$. Let $V\subset U$ be any neighborhood of the point $x$. Since $x\in \partial L_i$, $\varnothing\ne V\cap (\Bbb R^n\setminus L_i)\subset V\cap K$. That is $x\in\overline{K}=K$.