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
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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 $P$ 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.
One may also use the convention that the open sets are the downwards-closed subsets; this is the specialisation topology on the opposite $P^\op$.
Every finite topological space is an Alexandroff space.
A preorder $P$ is a equivalence relation if and only if its specialisation topology is $R_0$.
The specialisation topology of equality is $T_1$.
A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.
An Alexandroff space is a topological space for which arbitrary (as opposed to just finite) intersections of open subsets are still open.
Write
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 $\Proset$ of preordered sets fully-faithfully in the category Top of topological spaces.
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.
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).
By the definition of the 2-category Locale (see there), this means that $AlexPoset$ consists of those morphisms which have right adjoints in Locale.
The functor $Alex\colon Poset \to Locale$ factors through $AlexLocale$ and exhibits an equivalence of categories
The inverse functor
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,
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
Last revised on June 2, 2024 at 06:03:57. See the history of this page for a list of all contributions to it.