Editing Shape of the universe (section)

=== 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 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 [[bounded metric space|boundedness]]. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance {{mvar|d}}, there are points that are of a distance at least {{mvar|d}} apart. A finite universe is a bounded metric space, where there is some distance {{mvar|d}} such that all points are within distance {{mvar|d}} of each other. The smallest such {{mvar|d}} is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale".

==== With or without boundary ====
Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a [[disk (mathematics)|disc]], have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the [[3-sphere]] and [[3-torus]], that have no edges. Mathematically, these spaces are referred to as being [[compact space|compact]] without boundary. The term compact means that it is finite in extent ("bounded") and [[complete metric space|complete]]. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a [[differentiable manifold]]. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a [[closed manifold]]. The 3-sphere and 3-torus are both closed manifolds.

==== Observational methods ====
In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed<ref name="Luminet1995" /> and applied to cosmological observations.<ref name="Nat03" /><ref name="RBSG08" />

In the 2000s and 2010s, it was shown that, since the universe is inhomogeneous as shown in the [[observable universe#Large-scale structure|cosmic web of large-scale structure]], acceleration effects measured on local scales in the patterns of the movements of galaxies should, in principle, reveal the global topology of the universe.<ref name="RBBSJ2007">{{cite Q|Q68598777}}</ref><ref name="RR09">{{cite Q|Q68676519}}</ref><ref name="ORB12">{{cite Q|Q96692451}}</ref>