text: As stated in the introduction, investigations within the study of the global structure of the universe include:
  whether the universe is infinite or finite in extent,
  whether the geometry of the global universe is flat, positively curved, or negatively curved, and,
  whether the topology is simply connected (for example, like a sphere) or else multiply connected (for example, like a torus).
  Infinite or finite
  One of the unanswered questions about the universe is whether it is infinite or finite in extent. For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled up with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. Mathematically, the question of whether the universe is infinite or finite is referred to as boundedness. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance , there are points that are of a distance at least  apart. A finite universe is a bounded metric space, where there is some distance  such that all points are within distance  of each other. The smallest such  is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale".