How closely does the analogy of a 2-sphere hold in cosmology? Two dimensional creatures living on the surface of an inflating 2-sphere are often used to explain general relativity, curved space, and big bang cosmology.  For years, though, I have wanted to ask: The 2-sphere is closed.  What about the actual universe?  Is it closed in the same sense?  When cosmologists talk about a "closed" or "open" universe, it is not clear to me that they are answering the question that I want to ask.
So I think I have figured out how to ask my question:  Let's say we take a space-like hypersurface defined by the points where the proper time since the big bang singularity is, say, 13 billion years.  What are some of the global properties of this hypersurface?  Does it close back in on itself, like the 2-sphere, or does it extend to inifinity?  Does it have a volume, or is it infinite in volume?  Does it have a finite number of, say, galaxies in it, or are they infinite in number?
 A: This is (almost) exactly what cosmologist mean when the talk about the Universe being open or closed.
Observational evidence points to the universe being almost perfectly flat.* This means that the spatial slices of the universe are not 3-spheres. Or, at least if they are, the radius of curvature of this 3-sphere would have to be much larger than the observable universe.
Being flat, however, does not necessarily imply that the spatial slices are infinite. For example, they could also be 3-tori, in which case the volume and total number of stars would be finite. Thus far we have found no real evidence that the universe is period in some way. This suggests that any periodicity length scale cannot be much smaller than the size of the observable universe.
*Recently is has been suggested (see 2003.04935) that allowing a small positive spatial curvature may help resolve the tension between different measurements of the Hubble rate. This would suggest that the spatial slices are in fact (very big) 3-spheres. However, this is not widely accepted.
