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?