Here is a question which seems true to me but I can't rigorously show. Suppose $K$ is a compact subset of $\mathbb{R}^n$ such that $\mathbb{R}^n\setminus K$ is connected, does it follow that for any connected open set $U\subset \mathbb{R}^n$ such that $U\supset K$, $U\setminus K$ is also connected?



A question about connectedness in Euclidean space
Yes. Let $C$ be the closed complement of $U$, then by excision of $C$ we have $H\_1(\mathbb{R}, \mathbb{R} - K) = H\_1(U, U - K)$; since $H\_1(\mathbb{R})=0$, you also have in fact $H\_1(\mathbb{R}, \mathbb{R}-K)= H\_1(U, U-K)= 0$ when $\mathbb{R}-K$ is connected. 


So $H\_0(U-K)$ injects into $H\_0(U)$ and $U-K$ must be connected.


