Open hemisphere
= Open hemisphere
{title2=$\{x\in S^d:u\cdot x>0\}$}
For a nonzero vector $u$, the open hemisphere of $S^d$ with direction $u$ is $\{x\in S^d:u\cdot x>0\}$. Its boundary is excluded, so opposite <open hemispheres> are disjoint.
= Open hemisphere
{title2=$\{x\in S^d:u\cdot x>0\}$}
For a nonzero vector $u$, the open hemisphere of $S^d$ with direction $u$ is $\{x\in S^d:u\cdot x>0\}$. Its boundary is excluded, so opposite <open hemispheres> are disjoint.