Finite intervals in the partial orders of zero-dimensional, Tychonoff and regular topologies

For a set X, let Σz(X), Σt(X) and Σ3(X) denote, respectively, the sets of Hausdorff zero-dimensional, Tychonoff and Hausdorff regular topologies on X, partially ordered by inclusion. We investigate the nature of intervals in these partially ordered sets, with emphasis on finite intervals. We show that all such intervals are lattices, and that basic intervals (i.e., intervals in which the topology changes at at most one point) and finite intervals are sublattices of Σ(X). We show that all finite intervals in Σz(X) and in Σt(X) are of the form 2n for some n, but there is a finite interval in Σ3 which is not of that form.