[I originally wrote this post for the UiS Cosmology Facebook page .]
Observations, in light of the standard cosmological model, seem to strongly suggest that the Universe as we know it today started in a point-like event some good 13 billion years ago. Since then it has grown and grown, and now we observe living in a flat (at least quite flat…) Universe. Thus it seems as if the Universe currently has infinite spatial extension! But is a point-like big bang event within finite past really reconcilable with the present state of infinite spatial extension? W. Rindler discusses this in a paper with the catching title “Finite foliations of open FRW universes and the point-like big bang”  published in year 2000. The solution is not simple (so we won’t work out any details!), but yet the mathematics is clear.
The clue lies in the space of simultaneity and choice of reference frames. By using the “Milne model” Rindler shows that the simultaneity space of an observer in one set of coordinates has infinite spatial extension, whereas the spatial section of simultaneity in another set of coordinates (co-moving coordinates) has finite spatial sections of simultaneity. Thus it is actually possible, concludes Rindler, that we now live in a universe whose space of simultaneity is infinite — yet it all came from the same point in finite past.
Note that this “point” where the universe originated is in itself not a trivial concept, since spacetime itself had –for all what we know — no extension beyond this point. Think of the surface of a balloon that starts off very small. The surface of the balloon is analogous to our world (our world is 3-dimensional though…). After the big bang (“big blow” in this case!), the balloon is much bigger. Note that the big blow event did not happen anywhere on the current surface of the ballon. It cannot have, since the ballon has expanded away from the “point” where the big blow took place. But if the big bang was truly a point-event — then one could also say that it happened everywhere (all the points on the current surface of the balloon came from the same super-small balloon in the past). So! One needs to be precise about what one means when discussing the singularities of space-time theories.
The bottom line is that yes: a big-bang event in finite past is in principle reconcilable with an open universe model!