Trying to brush up on some geometric and algebraic topology, I got a little confused about the following:
Suppose we have the standard unit sphere $S^2$, but we remove its north and south poles. Is this topological space homeomorphic or homotopic to $S^1 \times \mathbb{R}$? I would think that they are not homotopic since I don't think both spaces are deformation retracts, are they? Now I do know that the stereographic projection is a map from $S^2$ to the plane, but that just involves the removal of either the north or the south pole, correct?