Manifolds such as these are actually defined by the maps used from a linear space. Two manifolds (i.e. two sets of maps) are considered the same (isomorphic) if the maps of one set can be "morphed" into the other and vice versa.

The flashlight demonstrates how the manifold's map projects into the linear space. See stereographic projection.

That's kind of part of a larger point actually: There is no 3d vector space in reality. It's a made up construct used to make sense of the world around us.