TD spaces

In any topological space, the closure of any one-element set {x} is also its downward closure ↓x with respect to the specialization preordering. A T_D space is a topological space in which, for every point x, ↓x – {x} is closed, too. This seemingly weird concept was introduced by Aull and Thron in a 1962 paper, but it has funny and interesting applications, notably in the comparison of the notions of subspaces and of sublocales, and in Thron’s so-called lattice equivalence problem. I will also mention the Skula topology again… Read the full post.