I don’t know if there is a factual answer to this, but it seems like there should be a fact based answer. Maybe we don’t know, but we should be able to make some guesses based on what we do know, and not just make a wild ass guess.
Ok, so if the topology (topography?) of the universe was spherical (I think it’s actually flat but work with me here) and we were to somehow develop a telescope powerful enough to see all the way around the curve of the sphere back to our location, would we recognize what we were looking at? I’m not even talking about resolution fine enough to resolve finer than galactic scale objects. Would we be able to pinpoint enough “landmarks” to recognize the Milkyway Galaxy and the local group, cluster, super cluster, filament etc?
It seems to me we would not be able to, the time/distance scale would just be too large if nothing else, but I don’t know if that would actually be true or not.
I seem to recall that due to the expansion of the universe at the current rate, the light around the circumference of the universe would never reach us. I don’t remember where I got that from
So would we be able to recognize what it looked like at some indeterminate extremely early age, perhaps at Big Bang, if we could see it at some early point? The viewing angle is not likely the issue.
From what I understand, it might have been possible back when the universe was smaller.* My personal WAG, based on what I’ve studied, is a donut shaped universe rather than a sphere. I guess that light would travel in a “straight” line around the curvature, and we would be able to see it back around. But that all depends on just how big the universe actually is, and how small it would have been back when such observations would have first been able to be made, something like a few hundred thousand years after the Big Bang at the earliest.*
ETA. Not just because the universe was smaller back in the day, but because its rate of expansion was slower. It could be that there was a period of time when the universe was expanding slower than the speed of light.
ETA 2. Or if we can develop telescopes that can see neutrinos rather than photons. Then we could go back a lot further to when the universe was small.
This is a really hard question to answer, as it’s really only about hypothetical universes rather than the one we live in. My guess is that any universe that at least superficially resembles ours, the distance around the universe would be so great that the appearance of anything would have vastly changed in the time since it the light left, such that you’d really only be able to tell for things that were relatively close to the same distance away in multiple directions. But even in that case, there would likely be gravitational lensing that would cause things to look a bit different in one direction than the other. In the unlikely case that the line of sight was clear in both directions, you’d then have the issue of looking at the exact opposite sides of things.
Thus, while I think it might be possible, it would be very difficult in a universe that resembled ours where it might even theoretically be possible.
You don’t try to look all the way around to your own location. You try to look in two opposite locations, to try to see the same thing from both sides. Not only does this avoid the “things looked different back then” problem, because the age is the same in both directions, but it also effectively nearly doubles the range at which you could see a wrap-around.
The search has been conducted, using the most distant light it’s possible to see. The results were negative: If the Universe does wrap around, it’s on a larger scale than that.
Oh, and a minor point, but “spherical” and “flat” are geometries, not topologies. You could have a flat or even hyperbolic universe that still wraps around, and you could have a spherical one that wrapped around on a shorter scale than the scale implied by the curvature.