The intersection of two topological balls

Question

Let $B_1, B_2 \subset \mathbb{R}^n$ be two topological balls in the usual metric topology with non empty interior.

How to prove or disprove the connected components non empty interior of $B_1 \cap B_2$ are again topological balls?

Thanks in advance.

Answer 1

I assume topological ball means any subspace which is topologically equivalent to a ball.

Let's consider if it's possible to disprove. For a counterexample, we would need to intersect two topological balls, and get a connected component which is not a ball. For simplicity, shoot for having only one connected component, and the simplest non-ball open set would I guess be a solid torus. Is it possible to intersect two topological balls and get a topological solid torus?

(Hint: I'm thinking red blood cells.)