If the universe is actually globally homogeneous and isotropic, that means that each point is equivalent to every other point within the universe. This type of symmetry can easily be attained by a sphere, in which any point can be transformed into any other point by a simple rotation of the sphere. If we accept that the universe is flat, it cannot be spherical. A flat plane can also satisfy these conditions (homogeneous and isotropic), but only if it is infinite--in this case any point can be transformed into any other point by translation.

As a chemist, though, I have to point out that this is analogous to our theoretical models of crystals. There are 219 (or 230) space groups--the types of symmetry a crystal structure can have. In each there is an assumption that the entire crystal can be translated in one direction or another to line up with itself again. These models are extremely accurate in their predictions of the properties of all types of crystals. However, the models assume an infinite crystal lattice, which we know is not an accurate depiction of any crystal ever observed/characterized (most of those studied are less than 1mm on a side, >1000000 atoms across, though there are single crystals as large as a meter on a side, and possible even larger). It also turns out that while the models work very well at predicting what goes on inside a crystal, they are very poor models of the boundaries of crystals, which we invoke other models for.

My point is, from the viewpoint of an atom near the center of a crystal, the whole universe is an infinite perfect crystal--the model works perfectly as far as the atom can "see" and beyond. But eventually there is a boundary that is completely inexplicable given a perfect crystal model.

The observable universe appears to be flat, homogeneous and isotropic. But in my opinion, there could very well be inhomogeneity, anisotropy or curvature beyond our observable bubble.