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

Adjacency in subposets of the lattice of T1-topologies on a set

“…However, the results we present here concerning Σ 3 and Σ t are quite similar but need somewhat different proofs. The following result is Lemma 22 of [14].…”

“…In a recent paper [14], McIntyre and Watson have studied a variety of subsets of L 1 (X) with a view to describing the structure of intervals in these partially ordered sets. Among those subsets studied are Σ t (X), Σ z (X) and Σ r (X), being respectively, the collections of all zero-dimensional, Tychonoff and regular topologies on X; these partially ordered sets (posets) are not sublattices of L 1 (X).…”

“…We say that τ 2 is a P-successor of τ 1 , τ 1 is a P-predecessor of τ 2 , τ 1 is a P-lower topology and τ 2 is a P-upper topology. In the terminology of [14], τ 2 is a cover of τ 1 in Σ P (X). A space is P-minimal if there is no weaker topology with P .…”

“…In [14], it was shown that the structure of basic intervals in Σ 3 (X) is essentially different from those of Σ t (X) in that not every finite interval is isomorphic to the power set of a finite ordinal. However, the results we present here concerning Σ 3 and Σ t are quite similar but need somewhat different proofs.…”

The structure of the poset of regular topologies on a set

“…In [11], it was shown that the structure of basic intervals in Σ 3 is essentially different from those of the poset Σ t of Tychonoff spaces in that not every finite interval is isomorphic to the power set of a finite ordinal. The following result is Lemma 22 of [11].…”