Alexandrov topology
In
general topology the open sets of a
topological space satisfy by definition the conditions:
# The
union of arbitrarily many open sets is open.# The
intersection of finitely many open sets is open.
The obvious asymmetry in these conditions leads one to ask:
"What happens when the intersection of arbitrarily many open sets is open?" The answer is, the
Alexandrov topology.
Alexandrov topologies have numerous characterizations:
Let
X = <
X,
T> be a topological space. Then the following are equivalent
*
Open and closed set characterizations:**
Open set characterization. An arbitrary intersection of open sets in
X is open.
**
Closed set characterization. An arbitrary union of closed sets in
X is closed.
*
Neighbourhood characterizations:**
Smallest neighbourhood characterization. Every point of
X has a smallest neighbourhood.
**
Neighbourhood filter characterization. The neighbourhood filter of every point in
X is closed under arbitrary intersections.
*
Interior and closure algebraic characterizations:**
Interior operator characterization. The
interior operator of
X distributes over arbitrary intersections of subsets.
**
Closure operator characterization. The
closure operator of
X distributes over arbitrary unions of subsets.
*
Preorder characterizations:**
Specialization preorder characterization. T is the finest topology consistent with the
specialization preorder of
X i.e. the finest topology giving the preorder ≤ satisfying
x ≤
y if and only if
x is in the closure of {
y} in
X.
**
Open up-set characterization. There is a
preorder ≤ such that the open sets of
X are precisely those that are upwardly closed i.e if
x is in the set and
x ≤
y then
y is in the set. (This preorder will be precisely the specialization preorder.)
**
Closed down-set characterization. There is a preorder ≤ such that the closed sets of
X are precisely those that are downwardly closed i.e if
x is in the set and
y ≤
x then
y is in the set. (This preorder will be precisely the specialization preorder.)
**
Upward interior characterization. A point
x lies in the interior of a subset
S of
X if and only if there is a point
y in
S such that
y ≤
x where ≤ is the specialization preorder i.e.
y lies in the closure of {
x}.
**
Downward closure characterization. A point
x lies in the closure of a subset
S of
X if and only if there is a point
y in
S such that
x ≤
y where ≤ is the specialization preorder i.e.
x lies in the closure of {
y}.
*
Finite generation and category theoretic characterizations:**
Finite closure characterization. A point
x lies within the closure of a subset
S of
X if and only if there is a finite subset of
F of
S such at
x lies in the closure of
F.
**
Finite subspace characterization. T is the finest topology consistent with the topologies of the finite subspaces of
X.
**
Finite inclusion map characterization. The inclusion maps
fi :
Xi â†'
X of the finite subspaces of
X form a
final sink.
**
Finite generation characterization. X is
finitely generated i.e. it is in the
final hull of the finite spaces. (This means that there is a
final sink fi :
Xi â†'
X where each
Xi is a finite topological space.)
Topological spaces satisfying the above equivalent characterizations are called
finitely generated spaces or
Alexandrov spaces and their topology
T is called the
Alexandrov topology, named after the Russian mathematician
Pavel Alexandrov who first investigated them.
The Alexandrov topology on a preordered set
Given a preordered set
X = <
X, ≤> we can define an Alexandrov topology
T on
X by choosing the open sets to be the
up-sets:
T = {
S ⊆
X : for all
x,
y ∈
X,
x ∈
S and
x≤
y implies
y ∈
S }
We thus obtain a topological space
T(
X) = <
X,
T>. The corresponding closed sets are the
down-sets:
{
S ⊆
X : for all
x,
y ∈
X,
x ∈
S and
y≤
x implies
y ∈
S }
The specialization preorder on a topological space
Given a topological space
X = <
X,
T> the specialization preorder on
X is defined by:
x≤
y if and only if
x is in the closure of {
y}.
We thus obtain a preordered set
W(
X) = <
X, ≤>.
Equivalence between preorders and Alexandrov topologies
For every preordered set
X = <
X, ≤> we always have
W(
T(
X)) =
X, i.e. the preorder of
X is recovered from the topological space
T(
X) as the specialization preorder.Moreover for every
Alexandrov space X, we have
T(
W(
X)) =
X, i.e. the Alexandrov topology of
X is recovered as the topology induced by the specialization preorder.
However for a topological space in general we do
not have
T(
W(
X)) =
X.
Given a
monotone function f :
Xâ†
Y
between two preordered sets (i.e. a function
f : Xâ†Y
between the underlying sets such that
x≤
y in
X implies
f(
x)≤
f(
y) in
Y), let
T(
f) :
T(
X)â†
T
(Y
)
be the same map as f considered as a map between the corresponding Alexandrov spaces. Then
T
(f) : T
(X
)â†T(
Y)
is a continuous map.
Conversely given a continuous map
f :
Xâ†
Y
between two topological spaces, let
W
(f) : W
(X
)â†W(
Y)
be the same map as
f considered as a map between the corresponding preordered sets. Then
W(
f) :
W(
X)â†
W
(Y
)
is a monotone function.
Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding Alexandrov spaces. Conversely a map between two Alexandrov spaces is continuous if and only if it is a monotone function between the corresponding preordered sets.
Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between two topological spaces that is not continuous but which is nevertheless still a monotone function between the corresponding preordered sets. (To see this consider a non-Alexandrov space X
and consider the identity map
i : X
â†T(
W(
X)).)
Let
Set denote the
category of sets and
maps. Let
Top denote the
category of topological spaces and
continuous maps; and let
Pro denote the category of
preordered sets and
monotone functions. Then
T :
Proâ†
'Top and
W :
Topâ†
'Pro are
concrete functors over
Set which are
left and right adjoints respectively.
Let
Alx denote the
full subcategory of
Top consisting of the Alexandrov spaces. Then the restrictions
T :
Proâ†
'Alx and
W :
Alxâ†
'Pro are inverse
concrete isomorphisms over
Set.
Alx is in fact a
bicoreflective subcategory of
Top with bicoreflector
Tâ—¦
W :
Topâ†
'Alx. This means that given a topological space
X, the identity map
i :
T(
W(
X))â†
X
'is continuous and for every continuous map
f :
Yâ†
X
where Y
is an Alexandrov space, the composition
i -1â—¦f : Y
â†T(
W(
X))
is continuous.
Given a preordered set
X, the
interior operator and
closure operator of
T(
X) are given by:
Int(
S) = {
x ∈ X : there exists a
y ∈ S with
y≤
x } or equivalently {
x ∈ X : for all
y ∈ X,
x≤
y implies
y ∈ S }, for all
S ⊆
XCl(
S) = {
x ∈ X : there exists a
y ∈ S with
x≤
y } for all
S ⊆
XConsidering the interior operator and
closure operator to be
modal operators on the
power set Boolean algebra of
X, this construction is a special case of the construction of a
modal algebra from a
modal frame i.e. a set with a single
binary relation. (The latter construction is itself a special case of a more general construction of a
complex algebra from a
relational structure i.e a set with relations defined on it.) The class of modal algebras that we obtain in the case of a preordered set is the class of
interior algebras - the algebraic abstractions of topological spaces.
Computer science
Physics
Alexandrov spaces were first introduced in 1937 by
P. S. Alexandrov under the name
discrete spaces in [Ale37] where he provided the characterizations in terms of sets and neighbourhoods. The name
discrete spaces later came to be used for topological spaces in which every subset is open and the original concept lay forgotten. With the advancement of
categorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of
finite generation was applied to general topology and the name
finitely generated spaces was adopted for them. Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from
denotational semantics and
domain theory in
computer science.
In [McC66], M.C. McCord had observed that there was a duality between
partially ordered sets and spaces which were precisely the
T0 versions of the spaces that Alexandrov had introduced. P. Johnstone referred to such topologies as
Alexandrov topologies in [Joh82]. In [Are99], F. G. Arenas independently proposed this name for the general version of these topologies.
It was also a well known result in the field of
modal logic that a duality exists between finite topological spaces and preorders on finite sets (the finite
modal frames for the modal logic
S4). In [Nat91], C. Naturman extended these results to a duality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as the
interior and closure algebraic characterizations.
A systematic investigation of these spaces from the point of view of general topology which had been neglected since the original paper by Alexandrov, was taken up by F.G. Arenas in [Are99].
Inspired by the use of Alexandrov topologies in computer science, applied mathematicians and physicists in the late 1990's began investigating the Alexandrov topology corresponding to
causal sets which arise from a preorder defined on
spacetime modeling
causality.
* [Ale37] Alexandroff, P.,
Diskrete Räume, Mat. Sb. (N.S.) 2 (1937), 501-518.
* [Are99] Arenas, F.G.,
Alexandroff spaces, Acta Math. Univ. Comenianae Vol. LXVIII, 1 (1999), pp. 17-25
* [Joh82] Johnstone, P.T.,
Stone spaces, Cambridge University Press (1982), 1986 edition
* [McC66] McCord, M. C.,
Singular homology and homotopy groups of finite topological spaces, Duke Math. Jour, 33 (1966), 465-474.
* [Nat91] Naturman, C.A.,
Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Mathematics, (1991)
