The specialisation topology, also called the Alexandroff topology, is a natural structure of a topological space induced on the underlying set of a preordered set. This is similar to the Scott topology, which is however coarser.
Spaces with this topology, called Alexandroff spaces and named after Paul Alexandroff (Pavel Aleksandrov), should not be confused with Alexandrov spaces (which arise in differential geometry and are named after Alexander Alexandrov).
Let be a preordered set.
Declare subset of to be an open subset if it is upwards-closed. That is, if and , then .
This defines a topology on , called the specialization topology or Alexandroff topology.
One may also use the convention that the open sets are the downwards-closed subsets; this is the specialisation topology on the opposite .
Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology.
This appears as (Caramello, p. 55).
The functor factors through and exhibits an equivalence of categories
This appears as (Caramello, theorem 4.2).
This appears as (Caramello, remark 4.3).
The original article is
Details on Alexandroff spaces are in
F. Arenas, Alexandroff spaces, Acta Math. Univ. Comenianae Vol. LXVIII, 1 (1999), pp. 17–25 (pdf)
Timothy Speer, A Short Study of Alexandroff Spaces (pdf)
A useful discussion of the abstract relation between posets and Alexandroff locales is in section 4.1 of
See also around page 45 in
A discussion of abelian sheaf cohomology on Alexandroff spaces is in