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**.