yes, GR models the Universe. there are 3 types of universe open, closed and flat. open and flat are infinite according to theory. whether they actually are or just really big in another matter.
Curvature
The curvature of the Universe places constraints on the topology. If the spatial geometry is spherical, i.e. possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite. Many textbooks erroneously state that a flat universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe. For example, Euclidean space is flat, simply connected and infinite, but the torus is flat, multiply connected, finite and compact.
In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.
The latest research shows that even the most powerful future experiments (like SKA, Planck..) will not be able to distinguish between flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10−4. If the true value of the cosmological curvature parameter is larger than 10−3 we will be able to distinguish between these three models even now.
Universe with zero curvature
In a universe with zero curvature, the local geometry is flat. The most obvious global structure is that of Euclidean space, which is infinite in extent. Flat universes that are finite in extent include the torus and Klein bottle. Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable. The most familiar is the aforementioned 3-Torus universe.
In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching zero. With dark energy, the expansion rate of the Universe initially slows down, due to the effect of gravity, but eventually increases. The ultimate fate of the universe is the same as that of an open universe.
A flat universe can have zero total energy.
Universe with positive curvature
A positively curved universe is described by spherical geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold (such as the Poincaré dodecahedral space), all of which are quotients of the 3-sphere.
Poincaré dodecahedral space, a positively curved space, colloquially described as “soccerball-shaped”, as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by Jean-Pierre Luminet and colleagues in 2003 and an optimal orientation on the sky for the model was estimated in 2008.
Universe with negative curvature
A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood. For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called horn topologies, so called because of the shape of the pseudosphere, a canonical model of hyperbolic geometry.An example is the Picard horn, a negatively curved space, colloquially described as “funnel-shaped”.
Curvature: Open or closed
When cosmologists speak of the Universe as being “open” or “closed”, they most commonly are referring to whether the curvature is negative or positive. These meanings of open and closed are different from the mathematical meaning of open and closed used for sets in metric spaces and for the mathematical meaning of open and closed manifolds, which gives rise to ambiguity and confusion. In mathematics, there are definitions for a closed manifold (i.e. compact without boundary) and open manifold (i.e. one that is not compact and without boundary,). A “closed universe” is necessarily a closed manifold. An “open universe” can be either a closed or open manifold. For example, the Friedmann–Lemaître–Robertson–Walker (FLRW) model the Universe is considered to be without boundaries, in which case “compact universe” could describe a universe that is a closed manifold.
