  We first consider interval partitions whose complements are Lebesgue-null and
introduce a complete metric that induces the same topology as the Hausdorff
distance (between complements). This is done using correspondences between
intervals. Further restricting to interval partitions with alpha-diversity, we
then adjust the metric to incorporate diversities. We show that this second
metric space is Lusin. An important feature of this topology is that
path-continuity in this topology implies the continuous evolution of
diversities. This is important in related work on tree-valued stochastic
processes where diversities are branch lengths.
