  We study the topology of the network of ionized and neutral regions that
characterized the intergalactic medium during the Epoch of Reionization. Our
analysis uses the formalism of persistent homology, which offers a highly
intuitive and comprehensive description of the ionization topology in terms of
the births and deaths of topological features. Features are identified as
$k$-dimensional holes in the ionization bubble network, whose abundance is
given by the $k$th Betti number: $\beta_0$ for ionized bubbles, $\beta_1$ for
tunnels, and $\beta_2$ for neutral islands. Using semi-numerical models of
reionization, we investigate the dependence on the properties of sources and
sinks of ionizing radiation. Of all topological features, we find that the
tunnels dominate during reionization and that their number is easiest to
observe and most sensitive to the astrophysical parameters of interest, such as
the gas fraction and halo mass necessary for star formation. Seen as a phase
transition, the importance of the tunnels can be explained by the entanglement
of two percolating clusters and the fact that higher-dimensional features arise
when lower-dimensional features link together. We also study the relation
between the morphological components of the bubble network (bubbles, tunnels,
islands) and those of the cosmic web (clusters, filaments, voids), describing a
correspondence between the $k$-dimensional features of both. Finally, we apply
the formalism to mock observations of the 21-cm signal. Assuming 1000
observation hours with HERA Phase II, we show that astrophysical models can be
differentiated and confirm that persistent homology provides additional
information beyond the power spectrum.
