In this paper, we revisit pattern mining and study the distribution underlying a binary dataset thanks to the closure structure which is based on passkeys, i.e., minimum generators in equivalence classes robust to noise. We introduce ∆-closedness, a generalization of the closure operator, where ∆ measures how a closed set differs from its upper neighbors in the partial order induced by closure. A ∆-class of equivalence includes minimum and maximum elements and allows us to characterize the distribution underlying the data. Moreover, the set of ∆classes of equivalence can be partitioned into the so-called ∆closure structure. In particular, a ∆-class of equivalence with a high ∆ is supported by more observations and thus is more stable. In the experiments, we study the ∆-closure structure of several real-world datasets and show that this structure is very stable for large ∆ and does not substantially depend on the data sampling used for the analysis.