nLab specialization 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.

Idea

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).

Definition

Definition

Let $P$ be a preordered set.

Declare a subset $A$ of $P$ to be an open subset if it is upwards-closed. That is, if $x \leq y$ and $x \in A$, then $y \in A$. (Equivalently, closed subsets are the downwards-closed subsets.)

This defines a topology on $P$, called the specialization topology or Alexandroff topology.

Remark

One may also use the convention that the open sets are the downwards-closed subsets; this is the specialisation topology on the opposite $P^\op$.

Examples

• Sierpinski space is the poset of truth values with the specialisation topology.

Properties

General

Proposition

Every finite topological space is an Alexandroff space.

Proposition

A preorder $P$ is a poset if and only if its specialisation topology is $T_0$.

Proposition

A preorder $P$ is a equivalence relation if and only if its specialisation topology is $R_0$.

Proposition

The specialisation topology of equality is $T_1$.

Proposition

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

Definition

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

Write

$$AlexTop \hookrightarrow Top$$

for the full subcategory of Top on the Alexandroff spaces.

Proposition

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

Corollary

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

$$Proset \hookrightarrow Top \,.$$

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

Alexandroff locales

Definition

Write $AlexLocale$ for the non-full subcategory of Locale whose

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

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

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

This appears as (Caramello, p. 55).

Remark

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

Proposition

The functor $Alex\colon Poset \to Locale$ factors through $AlexLocale$ and exhibits an equivalence of categories

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

The inverse functor

$$AlexLocale \to Poset$$

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

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

Proposition

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

This appears as (Caramello, remark 4.3).

Related concepts

• order topology

• Scott topology

References

• Wikipedia (English), Alexandrov topology

The original article is

• Paul Alexandroff, Diskrete Räume (web) (1973)

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

• Peter Johnstone, Stone Spaces

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)