Universal cover of the pinched sphere? Consider the sphere $S^2$ and identify its north and south poles to get a "pinched" sphere. What is the universal cover of this space?

The universal cover of this space is an infinite chain of spheres (in both directions) - one sphere for each integer - the north pole of each identified with the south pole of the next. 
To verify that this is indeed the universal cover, first check that the space described is simply connected. Now consider the obvious covering map; the projection map on each sphere. Verify that this is indeed a covering map. That does it for us; the infinite chain of spheres is the universal cover of the pinched sphere.
