This is well beyond me, so I'll just post something a guy who knows what he's talking about says: [quote]It's not an assumption of math but one of philosophical input into our science. We choose to believe that physics doesn't change with location in the universe because to assume otherwise is unnecessary complication. We haven't seen any evidence that the laws of physics vary, and we philosophically choose to keep the scientific theory that takes the fewest number of unnecessary ideas. So working from the idea that physics itself doesn't change, let us assume that the universe could have a "boundary" in any meaningful sense of the term. You've suggested a few. One boundary idea means that space goes on and on and on, but it's empty. So we would have to ask ourselves... why is it empty? What physical process created matter and energy here but not there? Again this runs into our "unnecessary idea" problem. The assumption brings more problems than it solves (it doesn't really solve anything in fact, just says that there's a finite amount of matter/energy in the universe). Another boundary idea may be that there's a "hard" edge to the universe. Not just hard like diamond, but like... space doesn't exist past some point. But that too really is complicated. What if we shine a light on that edge? what happens to it? What if we throw rocks at it? Again, the laws of physics would have to change over location to determine why you couldn't cross that wall. So we don't think this boundary exists either. I can't think of any other boundary cases, but hopefully I've at least demonstrated why physics being universal implies a universe without boundaries. So next we ask ourselves, okay, no boundaries, what "shape" can the universe have. This has a lot of answers actually. But I'll boil the discussion down to the highlights. The error bars on our measurement haven't yet excluded a positive curvature. The universe could be very slightly positively curved, but the probability of this case is rather quite small. In this case the universe would curve over very long distances (like 200 some observable universes) until it came back to where it started. No edge, see? But let's take the data for what it seems to be pointing to. Flat curvature. There are several "shapes" of flat curvature that don't have boundaries. Some are things like the 3-Torus. A 2-D example would be a pacman or asteroids screen. Pass through one edge, appear on the other side, but the motion is all straight lines and normal geometry. Another example is a tesselation, suppose the universe had some shape that tiles its edges together, like a pentagonal dodecahedron. This universe is flat in its interior but again, the edges "wrap" back around, but in a more complicated pattern than the 3-Torus. So finally we get to the simplest geometry that fits the flat data, and that's the flat Euclidean plane without boundaries. And so what we mean here, is an infinite amount of matter spread over an infinite volume of space. Go to the edge of our observable universe and you'll find another observable universe that looks very similar to our own. and so on ad infinitum. Galaxy clusters and filaments filled with stars and planets. Forever and ever in every direction. [/quote]