Open + simply connected + antipodal + proper set in the sphere

Question

It seems to me that if $U\subseteq\mathbb S^n$ is open, simply connected and proper, there must exist some point $x\in U$ such that $-x\notin U$. Otherwise, $U=-U$, and $U$ would satisfy the conditions of the title. Could such a subset exist?

Answer 1

$\Bbb S^n$ minus its two poles is a proper, open, antipodal subset of $\Bbb S^n$. As it is homeomorphic to $\Bbb R^n$ minus a point, it is also simply connected as soon as $n\ge 3$.

I suppose, you do not want that all $\Bbb S^1\to U$ can be contracted to a point but rather all $\Bbb S^{n-1}\to U$?