We study the evolution of the hydrogen ionization field during the epoch of reionization (EoR) using seminumerical simulations. By calculating the Minkowski functionals (MFs) of the 21 cm 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 regions ("ionized fibers") that 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 begins 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 transitions 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 owing to the ionizing background.