The specialisation topology
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 a 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.
A preorder is a poset if and only if its specialisation topology is .
A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.
Alexandroff topological spaces
Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology.
The specialization topology embeds the category of preordered sets fully-faithfully in the category Top of topological spaces.
If we restrict to a finite underlying set, then the categories and of finite prosets and finite topological spaces are equivalent in this way.
Write for the non-full subcategory of Locale whose
objects are Alexandroff locales, that is locales of the form for with ;
morphisms are those morphisms of locales , for which the dual inverse image morphism of frames has a left adjoint .
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 (arXiv:0708.2136)
A useful discussion of the abstract relation between posets and Alexandroff locales is in section 4.1 of
- Olivia Caramello, A topos-theoretic approach to Stone-type dualities (arXiv:1103.3493)
See also around page 45 in
A discussion of abelian sheaf cohomology on Alexandroff spaces is in
- Morten Brun, Winfried Bruns, Tim Römer, Cohomology of partially ordered sets and local cohomology of section rings Advances in Mathematics 208 (2007) 210–235 (pdf)