  We use the results of large-scale simulations of reionization to explore
methods for characterizing the topology and sizes of HII regions during
reionization. We use four independent methods for characterizing the sizes of
ionized regions. Three of them give us a full size distribution: the
friends-of-friends (FOF) method, the spherical average method (SPA) and the
power spectrum (PS) of the ionized fraction. These latter three methods are
complementary: While the FOF method captures the size distribution of the small
scale H II regions, which contribute only a small amount to the total
ionization fraction, the spherical average method provides a smoothed measure
for the average size of the H II regions constituting the main contribution to
the ionized fraction, and the power spectrum does the same while retaining more
details on the size distribution. Our fourth method for characterizing the
sizes of the H II regions is the average size which results if we divide the
total volume of the H II regions by their total surface area, (i.e. 3V/A),
computed in terms of the ratio of the corresponding Minkowski functionals of
the ionized fraction field. To characterize the topology of the ionized
regions, we calculate the evolution of the Euler Characteristic. We find that
the evolution of the topology during the first half of reionization is
consistent with inside-out reionization of a Gaussian density field. We use
these techniques to investigate the dependence of size and topology on some
basic source properties, such as the halo mass-to-light ratio, susceptibility
of haloes to negative feedback from reionization, and the minimum halo mass for
sources to form. We find that suppression of ionizing sources within ionized
regions slows the growth of H II regions, and also changes their size
distribution. Additionally, the topology of simulations including suppression
is more complex. (abridged)
