Trying to prove that is a manifold.
I think I can see why it's Hausdorff, take two points on the sphere of distance from each other, then 'draw' two open balls around both these points of radius less than , so it should be Hausdorff.
I'm not sure about proving 2nd countability, draw open balls of radius 1 around all the points with rational coordinates?
And I've no idea about proving it's locally Euclidean.
And finally, just to clarify I'm not doing this wrong, then open sets of are the same open sets used in ? (i.e. the open sets as given by the distance metric?)