Adapted from P. Coles, 1999, The Routledge Critical Dictionary of the New Cosmology, Routledge Inc., New York. Reprinted with the author's permission. To order this book click here: http://www.routledge-ny.com/books.cfm?isbn=0415923549
Any cosmological model in which the curvature of spacetime is positive. In such a universe the normal rules of Euclidean geometry do not necessarily hold. For example, the sum of the interior angles of a triangle is greater than 180°, and parallel lines can actually intersect. Among the family of Friedmann models, the particular cases describing closed universes are those in which the density parameter > 1 and the deceleration parameter q > 0.5. These models are finite in physical size. They also recollapse in the future: the deceleration generated by gravity eventually causes the expansion of the Universe to cease, and go into reverse. Eventually these models produce a second singularity, sometimes called the Big Crunch, in which the density of all matter again becomes infinite.
It was often thought that there could be an infinite series of bigbangs followed by big crunches, so that a closed model could, in some sense, be eternal. It is now thought, however, that an infinite oscillating universe of this type is not possible because each cycle becomes progressively more disordered than the previous one as a consequence of the second law of thermodynamics. Eventually the oscillations would peter out, rather like a bouncing ball which gradually comes to rest as its energy dissipates.
Most theoretical treatments of quantum cosmology suggest that the Universe should be closed, but this is difficult to reconcile with present determinations of the density parameter, which suggest the strong possibility that we live in an open universe.