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)
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