Is every open, bounded and simply connected subset of $\mathbb{R}^n$ essentially a ball?

algebraic-topology general-topology examples-counterexamples

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and simply connected. I wonder if the answer to the following question is known:

Is there a homeomorphism $\Omega\to \operatorname{B}_1(0)$, where $\operatorname{B}_R(0)$ is the open ball (in the topology induced by the metric) with radius $R$ around $0\in \mathbb{R}^n$?

The Poincare conjecture comes to mind, but it only concerns manifolds without border, as far as I understand.

Thanks for any hints to literature, theorems or counterexamples etc... :)


$B_1(0)\setminus\{0\}$ is a counterexample when $n>2$.


In $\mathbb R^2$, yes, every open, bounded and simply connected set is homeomorphic to $B_1(0)$.

Look at Are simply connected open sets in $\mathbb{R}^2$ homeomorphic to an open ball? and Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?

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