  We use the framework of noncommutative geometry to define a discrete model
for fluctuating geometry. Instead of considering ordinary geometry and its
metric fluctuations, we consider generalized geometries where topology and
dimension can also fluctuate. The model describes the geometry of spaces with a
countable number $n$ of points. The spectral principle of Connes and
Chamseddine is used to define dynamics.We show that this simple model has two
phases. The expectation value $<n>$, the average number of points in the
universe, is finite in one phase and diverges in the other. Moreover, the
dimension $\delta$ is a dynamical observable in our model, and plays the role
of an order parameter. The computation of $<\delta>$ is discussed and an upper
bound is found, $<\delta> < 2$. We also address another discrete model defined
on a fixed $d=1$ dimension, where topology fluctuates. We comment on a possible
spontaneous localization of topology.
