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

Abstract

We study subposets of the lattice L 1 (X) of all T 1 -topologies on a set X, namely Σt(X), Σ3(X) and Σ lc (X), being respectively the collections of all Tychonoff, all T 3 and all locally compact Hausdorff topologies on X, with a view to deciding which elements of these partially ordered sets have and which do not have covers, that is to say immediate successors, in the respective posets. In the final section we discuss the subposet Σ G of all Hausdorff group topologies on a group G.