How do we resolve a flat spacetime and the cosmological principle? As I've said elsewhere, I've not had the opportunity to take a class in general relativity.  Nonetheless, I understand that two major aspects of the standard cosmological model are the cosmological principle and the observation of a flat space.  To get where I'm coming from, I'll try to give a brief description of my understanding of these concepts:


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*cosmological principle - This principle states that there is no privileged position within the universe.  In other words, wherever any observer is located, s/he will observe approximately the same thing.  Obviously the specific celestial bodies observed will change, but the expansion of the universe will be judged the same and the universe will essentially appear isotropic.

*flat space - This observation, tested and largely verified by the WMAP satellite, shows that the large scale universe is not curved.
My natural inclination is that these two things cannot be simultaneously true.  The reason it seems this way to me is that if any observer can see roughly the same amount of the universe in any direction, and the universe is of finite size, the observable portions must overlap somewhere.  If the observable portions overlap, it must be possible to continue traveling in one direction and eventually end up where you started.  To me, this seems to be what curvature is.
How do we reconcile these two concepts?
$\dagger$  I have read some of the articles on the subject such as those on Minkowski space and multidimensional toruses.  I believe I can reconcile the two concepts and imagine a higher dimensional flat torus, but the concept is still a difficult one for me and I would love some clarification.
 A: I'm not sure I understand what you don't understand. I am adding a answer since this would be too long for a comment... 
Given the cosmological principle and the flat space time observation, the idea is that the flat spacetime is infinite, or at least very much larger than our horizon.  So, yes, an observer that is, say 7 billion light years away from us would see parts of the universe that are beyond our horizon, and we would see parts of the universe beyond his horizon.  There is no contradiction in this.  This other observer at 7 billion years would, for example, also see a CMB with properties similar to ours but all the ripples would be different.
It is true that this could also be a flat spacetime that has the topological property of a torus but that is not at all required.  This would give a flat spacetime that is finite.  Do you think the universe cannot be infinite?
By the way, the cosmological principle is more than a principle, there is observational support for it.  For example the CMB looks the same in all directions and the large scale structure of galaxy cluster seems uniform on the largest scales.  
EDIT:  Thinking about it more, you probably are thinking of the big bang as being one point (or very small region) in space at $t=0$ that then "explodes" into our universe.  I had that same problem / misconception when I started learning about this.  The problem is, we cannot say anything about t=0 of the big bang, because that would effectively be like a singularity filling all of space.  Instead think of $t=\epsilon$, just very slightly after the big bang.  At that time, you would have an infinite space filled with extremely high energy density at a very high temperature and the space would be expanding extremely rapidly.  As time goes on from there the expansion rate slows, the density and temperature go down and you get the infinite universe we live in.  Does that help?
