Is the Universe infinite?

Here’s another light-hearted topic you can discuss with family and
friends instead of football8. If it turns out that the overall density
of the Universe, due to all the visible and invisible matter and
energy, is still not enough to make it closed, then conventional
wisdom would suggest that it has to be infinite (to avoid having
an edge you could fall off). Of course it may be that what appears
to us to be an infinite flat universe may still be closed and just too
huge for any curvature to ever be detected. In such a universe, the
value of omega would be very close to one.
Most people, cosmologists included, would much rather not
have to deal with an infinite universe. Over the past few years
a new field of study called cosmic topology has emerged which
is the study of the Universe’s shape. One result of work in this field that I was not aware of until recently was that even an open
universe, whether flat or hyperbolic, can still have a finite size. In
fact, and here is the fun part, even if the Universe is flat, it might
turn out to be shaped like the higher dimensional equivalent of
the surface of a doughnut (the ones with the hole in the middle).
I know you must be thinking that the surface of a doughnut is
hardly flat. But that is because it is only an approximation to the
shape I am talking about. First of all, the surface of a doughnut
is only two-dimensional. Secondly, even the 2D equivalent of the
space I amreferring to could not possibly exist imbedded in our 3D
space. The correct name for such a shape is a Euclidean torus, and
has the property, just like a doughnut’s surface, of having more
than one line joining any two points on it.
Of course if the Universe really is doughnut shaped then the
missing mass is most likely sugar or cinnamon.