  We investigate the cosmological consequences of having quantum fields living
in a space with compactified dimensions. We will show that the equation of
state is not modified by topological effects and so the dynamics of the
universe remains as it is in the infinite volume limit. On the contrary the
thermal history of the universe depends on terms that are associated with
having non-trivial topology. In the conclusions we discuss some issues about
the relationship between the $c=1$ non-critical string-inspired cosmology and
the result obtained with matter given by a hot massless field in $S^{1}\times
\mbox{\bf
  R}$.
