Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, what are some necessary conditions on $A$ such that $\mathbb{R}^n - A$ is disconnected?
I don't think the Jordan-Brouwer Separation Theorem can be used as $A$ cannot be homeomorphic to $\mathbb{S}^{n - 1}$.
My speculation
I think $A$ must be unbounded; in fact, it must stretch to infinity in all directions, otherwise any two points can be linked by a path bypassing $A$.
I think $A$ must also be simply connected, or else any two points can be connected by a path piercing a hole in $A$.
A related (unsolved) question: A question about connected sets in $\mathbb{R}^2$
