  We show that if an open set in $\mathbb{R}^d$ can be fibered by unit
$n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be
pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of
$n$, we construct unit $n$-sphere fibrations in $\mathbb{R}^{2n+1}$.
