  We study the evolution of the hydrogen ionization field during the epoch of
reionization (EoR) using semi-numerical simulations. By calculating the
Minkowski functionals (MFs) of the 21cm brightness temperature field, which
provides a quantitative description of topology of the neutral and ionized
regions, we find that the reionization process can be divided into five stages,
each with different topological structures, corresponding to isolated ionized
regions ("bubbles"); connected ionized region ("ionized fibers") which
percolate through the cosmic volume; a sponge-like configuration with
intertwined neutral and ionized regions after the overlap of bubbles; connected
neutral regions ("neutral fibers") when the ionized region begin to surround
the remaining neutral region; and isolated neutral regions ("islands"), before
the whole space is ionized except for the rare dense clumps in galaxies and
minihalos and the reionization process is completed. We use the MFs and the
size statistics on the ionized or neutral regions to distinguish these
different stages, and find the neutral fractions at which the transition of the
different stages occur. At the later stages of reionization the neutral regions
are more isolated than the ionized regions, this is the motivation for the
island model description of reionization. We compare the late stage topological
evolution in the island model and the bubble model, showing that in the island
model the neutral fibers are more easily broken into multiple pieces due to the
ionizing background.
