  The distribution of ionised hydrogen during the epoch of reionization (EoR)
has a complex morphology. We propose to measure the three-dimensional topology
of ionised regions using the Betti numbers. These quantify the topology using
the number of components, tunnels and cavities in any given field. Based on the
results for a set of reionization simulations we find that the Betti numbers of
the ionisation field show a characteristic evolution during reionization, with
peaks in the different Betti numbers characterising different stages of the
process. The shapes of their evolutionary curves can be fitted with simple
analytical functions. We also observe that the evolution of the Betti numbers
shows a clear connection with the percolation of the ionized and neutral
regions and differs between different reionization scenarios. Through these
properties, the Betti numbers provide a more useful description of the topology
than the widely studied Euler characteristic or genus. The morphology of the
ionisation field will be imprinted on the redshifted 21-cm signal from the EoR.
We construct mock image cubes using the properties of the low-frequency element
of the future Square Kilometre Array and show that we can extract the Betti
numbers from such datasets if an observation time of 1000 h is used. Even for a
much shorter observation time of 100 h, some topological information can be
extracted for the middle and later stages of reionization. We also find that
the topological information extracted from the mock 21-cm observations can put
constraints on reionization models.
