  We present a new formalism for studying the topology of HII regions during
the Epoch of Reionisation, based on persistent homology theory. With persistent
homology, it is possible to follow the evolution of topological features over
time. We introduce the notion of a persistence field as a statistical summary
of persistence data and we show how these fields can be used to identify
different stages of reionisation. We identify two new stages common to all
bubble ionisation scenarios. Following an initial pre-overlap and subsequent
overlap stage, the topology is first dominated by neutral filaments (filament
stage) and then by enclosed patches of neutral hydrogen undergoing outside-in
ionisation (patch stage). We study how these stages are affected by the degree
of galaxy clustering. We also show how persistence fields can be used to study
other properties of the ionisation topology, such as the bubble size
distribution and the fractal-like topology of the largest ionised region.
