Benjamin Horowitz's answer covered a lot of the key points, but it's worth adding that the question of the topology of the universe has been investigated by astrophysical observations. If the universe is multiply connected, and if the length scale is shorter than the horizon scale, then we should be able to see evidence of it.
To take a simple example, imagine that the universe is geometrically flat but has the geometry of a 3-torus. Specifically, take a cubical volume, and identify opposite faces, so that if you "leave" the cube through one face you reenter through the opposite face. If the length of an edge of the cube is sufficiently small, then you could see multiple copies of any given object. Of course, if the length is much larger than the horizon, then there's no way to tell the difference between this model and one in which space is infinite.
The best way to test these models is the "circles in the sky" technique, in which you look for correlated circles in different directions in maps of the microwave background radiation. The result is negative: we don't live in a multiply-connected universe with a sufficiently short length scale to be observable.