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I had a question on the horizon problem (explanation for apparent thermal equilibrium between distal points on the CMB, anisotropies notwithstanding). Just as background, I just have an amateur interest in cosmology with no formal training (biophysist by training and career), so please forgive and inform if I'm totally off-base somewhere.

Prior to my actual question on the horizon, I have a preliminary question on the nature of the initial singularity post-big bang. As I understand it, although the current universe appears spatially flat inflation could have minimized any curvature present, and driven the universe to apparent flatness. Thus the early universe could have been significantly curved.

Going back in time, what would be the nature of space as we approach the initial singularity in the three different curvature regimes? Intuitively, for a closed universe I can envision everything collapsing towards a single point (perhaps with non-trivial topology) as t-->0; a closed bounded universe space then just expands in all directions from all points.

For flat or open universes, however, what would be the nature of the initial singularity? Both of those universes, as I understand it, are infinite in scale. No matter how many fold you shrink infinity, it'll still be infinity, so as t-->0 in those universes does the initial singularity which it approaches still have infinite scope/volume in our 3d space?

I'll hop back on with my actual horizon question once I get some feedback on the above, as the questions are intertwined. ;-D