Metrics on sets of interval partitions with diversity

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 …