I have been reading about the nature of space (Brian Greene's and Max Tegmark's recent books) and have a question on the size of space. Both references acknowledge that general relativity equations allow for a finite and infinite spatial extent. Also, since actual density equals critical density within 1%, space is basically flat. Thus, the two primary options for flat finite space "shapes" are 3-D torus or video-game-screen. I do not understand why finite space is not considered in the inflation framework, why always an infinite space?
Thus, the two primary options for flat finite space "shapes" are 3-D torus or video-game-screen.
Not true. It's possible that the universe has a nontrivial topology, but there is no evidence for it, and it's not the most common assumption. The most common assumption is that if the universe is closed, it has the topology of a 3-sphere.
I do not understand why finite space is not considered in the inflation framework, why always an infinite space?
We observe that the universe is nearly flat. General relativity predicts that if the universe is not exactly flat at one time, then as it expands it should get less and less flat. This produces a fine-tuning problem: it's surprising that the early universe could have been so incredibly fine-tuned for flatness that the near-flatness would survive until today. This is called the flatness problem.
One of the main selling points of inflation is that it's supposed to solve the flatness problem. Inflation is compatible with a universe that isn't perfectly flat, but it does predict that the universe should be nearly flat. A flat universe is infinite.