let $S$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show this is a topological manifold. for starters one needs to define a suitable topology on it. i was thinking let a set $U$ be open in $S$ iff $U \cap S^2$ (intersection with sphere) is open in $S^2$ in the subspace topology? am i going in the right direction? I'd really appreicate some help. thank you
this is really about topological manifold, sorry for not stating it earlier.
here is the definition:
a manifold is a second countable Hausdorff space that is locally Euclidean.