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How Big Was the Universe When It Was First Born?
StartsWithABang writes: Looking out at the distant stars, galaxies and radiation in the Universe today, we've been able to determine not only what it's made out of, but how long it's been since the Big Bang: 13.8 billion years. Put all that information together, and you can also figure out how large the observable part of that Universe is today. From our point of view, it appears to extend for 46.1 billion light years in all directions. So what if you extrapolate backwards, to the very end of inflation and the start of the hot, dense state we identify with the Big Bang, and ask how large that 46.1 billion light year "size" was back then? How big would it be? Depending on the particulars of when inflation came to an end, the answer is somewhere between the size of a soccer ball and the size of a city block, no smaller and no larger.
You've managed to argue that a definitely finite 2-d surface is infinite. The infinite surface would be mapped onto a flat plane.
The relevant 3-d surfaces are hyperspheres, flat surfaces, and hyperboloids (ie saddles). The first is definitely finite, the other two ae infinite pending any further topological assumptions. However, a flat surface can still lead to a finite universe for a given topology, the simplest and most common example being a torus; this looks like Pacmanland, given that if you walk in any direction you get wrap round.
It's worth noticing that as a local theory, general relativity (and indeed all metric theories of gravity) says nothing about global topology.