nLab specialization topology

The specialisation topology

This entry is about topologized preorders also known as Alexandroff spaces, named after Paul Alexandroff. For length spaces behaving like having sectional curvature bounded below see instead at Alexandrov space, named after Alexander Alexandrov.



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

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 PP be a preordered set.

Declare a subset AA of PP to be an open subset if it is upwards-closed. That is, if xyx \leq y and xAx \in A, then yAy \in A. (Equivalently, closed subsets are the downwards-closed subsets.)

This defines a topology on PP, 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 P opP^\op.





Every finite topological space is an Alexandroff space.


A preorder PP is a poset if and only if its specialisation topology is T 0T_0.


A preorder PP is a equivalence relation if and only if its specialisation topology is R 0R_0.


The specialisation topology of equality is T 1T_1.


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


An Alexandroff space is a topological space for which arbitrary (as opposed to just finite) intersections of open subsets are still open.


AlexTopTop AlexTop \hookrightarrow Top

for the full subcategory of Top on the Alexandroff spaces.


Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology.


The specialization topology embeds the category Proset\Proset of preordered sets fully-faithfully in the category Top of topological spaces.

ProsetTop. Proset \hookrightarrow Top \,.

If we restrict to a finite underlying set, then the categories FinPros\Fin\Pros and FinTop\Fin\Top of finite prosets and finite topological spaces are equivalent in this way.

Alexandroff locales


Write AlexLocaleAlexLocale for the non-full subcategory of Locale whose

  • objects are Alexandroff locales, that is locales of the form AlexPAlex P for PPosetP\in Poset with Open(Alex(P))=UpSets(P)Open(Alex(P)) = UpSets(P).

    Equivalently, Alexandroff locales can be defined as locales that admit a base consisting of supercompact opens, i.e., opens aa for which any open cover of aa contains aa itself.

  • morphisms are those morphisms of locales f:AlexPAlexQf\colon Alex P \to Alex Q, for which the dual inverse image morphism of frames f *:UpSet(Q)UpSet(P)f^*\colon UpSet(Q) \to UpSet(P) has a left adjoint f !:UpSet(P)UpSet(Q)f_!\colon UpSet(P) \to UpSet(Q).

This appears as (Caramello, p. 55).


By the definition of the 2-category Locale (see there), this means that AlexPosetAlexPoset consists of those morphisms which have right adjoints in Locale.


The functor Alex:PosetLocaleAlex\colon Poset \to Locale factors through AlexLocaleAlexLocale and exhibits an equivalence of categories

Alex:PosetAlexLocale. Alex \colon Poset \stackrel{\simeq}{\to} AlexLocale \,.

The inverse functor

AlexLocalePoset AlexLocale \to Poset

sends a locale LL to the poset of supercompact elements of LL (defined above) and a morphism f:LLf\colon L\to L' to the restriction of f !f_! to supercompact elements (which are preserved by f !f_!).

This appears as (Caramello, theorem 4.2). Dually, reversing the direction of arrows of Alexandroff locales,


The category of Alexandroff frames is equivalent to that of completely distributive algebraic lattices.

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

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)

Last revised on June 2, 2024 at 06:03:57. See the history of this page for a list of all contributions to it.