In
mathematics, a finite topological space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
for which the underlying
point set is
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are often used to provide examples of interesting phenomena or
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
s to plausible sounding conjectures.
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thursto ...
has called the study of finite topologies in this sense "an oddball topic that can
lend good insight to a variety of questions".
Topologies on a finite set
Let
be a finite set. A
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on
is a subset
of
(the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of
) such that
#
and
.
# if
then
.
# if
then
.
In other words, a subset
of
is a topology if
contains both
and
and is closed under finite
intersections and arbitrary
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
s. Elements of
are called
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s. Since the power set of a finite set is finite there can be only finitely many
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s (and only finitely many
closed sets).
A topology on a finite set can also be thought of as a
sublattice of
which includes both the bottom element
and the top element
.
Examples
0 or 1 points
There is a unique topology on the
empty set ∅. The only open set is the empty one. Indeed, this is the only subset of ∅.
Likewise, there is a unique topology on a
singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
. Here the open sets are ∅ and . This topology is both
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
and
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
, although in some ways it is better to think of it as a discrete space since it shares more properties with the family of finite discrete spaces.
For any topological space ''X'' there is a unique
continuous function from ∅ to ''X'', namely the
empty function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
. There is also a unique continuous function from ''X'' to the singleton space , namely the
constant function to ''a''. In the language of
category theory the empty space serves as an
initial object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
in the
category of topological spaces while the singleton space serves as a
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
.
2 points
Let ''X'' = be a set with 2 elements. There are four distinct topologies on ''X'':
# (the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
)
#
#
# (the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
)
The second and third topologies above are easily seen to be
homeomorphic. The function from ''X'' to itself which swaps ''a'' and ''b'' is a homeomorphism. A topological space homeomorphic to one of these is called a
Sierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.
It is the smallest example of a topological space which is neither trivial nor discrete. It is name ...
. So, in fact, there are only three inequivalent topologies on a two-point set: the trivial one, the discrete one, and the Sierpiński topology.
The specialization preorder on the Sierpiński space with open is given by: ''a'' ≤ ''a'', ''b'' ≤ ''b'', and ''a'' ≤ ''b''.
3 points
Let ''X'' = be a set with 3 elements. There are 29 distinct topologies on ''X'' but only 9 inequivalent topologies:
#
#
#
#
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
The last 5 of these are all
T0. The first one is trivial, while in 2, 3, and 4 the points ''a'' and ''b'' are
topologically indistinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhood (topology), neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborh ...
.
4 points
Let ''X'' = be a set with 4 elements. There are 355 distinct topologies on ''X'' but only 33 inequivalent topologies:
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
# (
T0)
The last 16 of these are all
T0.
Properties
Specialization preorder
Topologies on a finite set ''X'' are in
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
with
preorders on ''X''. Recall that a preorder on ''X'' is a
binary relation on ''X'' which is
reflexive and
transitive.
Given a (not necessarily finite) topological space ''X'' we can define a preorder on ''X'' by
:''x'' ≤ ''y'' if and only if ''x'' ∈ cl
where cl denotes the
closure of the
singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
. This preorder is called the ''
specialization preorder'' on ''X''. Every open set ''U'' of ''X'' will be an
upper set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
with respect to ≤ (i.e. if ''x'' ∈ ''U'' and ''x'' ≤ ''y'' then ''y'' ∈ ''U''). Now if ''X'' is finite, the converse is also true: every upper set is open in ''X''. So for finite spaces, the topology on ''X'' is uniquely determined by ≤.
Going in the other direction, suppose (''X'', ≤) is a preordered set. Define a topology τ on ''X'' by taking the open sets to be the upper sets with respect to ≤. Then the relation ≤ will be the specialization preorder of (''X'', τ). The topology defined in this way is called the
Alexandrov topology determined by ≤.
The equivalence between preorders and finite topologies can be interpreted as a version of
Birkhoff's representation theorem
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).''
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice ...
, an equivalence between finite distributive lattices (the lattice of open sets of the topology) and partial orders (the partial order of equivalence classes of the preorder). This correspondence also works for a larger class of spaces called
finitely generated spaces. Finitely generated spaces can be characterized as the spaces in which an arbitrary intersection of open sets is open. Finite topological spaces are a special class of finitely generated spaces.
Compactness and countability
Every finite topological space is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
since any
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
must already be finite. Indeed, compact spaces are often thought of as a generalization of finite spaces since they share many of the same properties.
Every finite topological space is also
second-countable (there are only finitely many open sets) and
separable (since the space itself is
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
).
Separation axioms
If a finite topological space is
T1 (in particular, if it is
Hausdorff) then it must, in fact, be discrete. This is because the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of a point is a finite union of closed points and therefore closed. It follows that each point must be open.
Therefore, any finite topological space which is not discrete cannot be T
1, Hausdorff, or anything stronger.
However, it is possible for a non-discrete finite space to be
T0. In general, two points ''x'' and ''y'' are
topologically indistinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhood (topology), neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborh ...
if and only if ''x'' ≤ ''y'' and ''y'' ≤ ''x'', where ≤ is the specialization preorder on ''X''. It follows that a space ''X'' is T
0 if and only if the specialization preorder ≤ on ''X'' is a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
. There are numerous partial orders on a finite set. Each defines a unique T
0 topology.
Similarly, a space is
R0 if and only if the specialization preorder is an equivalence relation. Given any equivalence relation on a finite set ''X'' the associated topology is the
partition topology
In mathematics, the partition topology is a topological space, topology that can be induced on any set X by Partition of a set, partitioning X into disjoint subsets P; these subsets form the basis (topology), basis for the topology. There are two i ...
on ''X''. The equivalence classes will be the classes of topologically indistinguishable points. Since the partition topology is
pseudometrizable, a finite space is R
0 if and only if it is
completely regular
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
.
Non-discrete finite spaces can also be
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
. The
excluded point topology In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection
:T = \ \cup \
of subsets of ''X'' is then the excluded ...
on any finite set is a
completely normal
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
T
0 space which is non-discrete.
Connectivity
Connectivity in a finite space ''X'' is best understood by considering the specialization preorder ≤ on ''X''. We can associate to any preordered set ''X'' a
directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pa ...
Γ by taking the points of ''X'' as vertices and drawing an edge ''x'' → ''y'' whenever ''x'' ≤ ''y''. The connectivity of a finite space ''X'' can be understood by considering the
connectivity
Connectivity may refer to:
Computing and technology
* Connectivity (media), the ability of the social media to accumulate economic capital from the users connections and activities
* Internet connectivity, the means by which individual terminal ...
of the associated graph Γ.
In any topological space, if ''x'' ≤ ''y'' then there is a
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
from ''x'' to ''y''. One can simply take ''f''(0) = ''x'' and ''f''(''t'') = ''y'' for ''t'' > 0. It is easily to verify that ''f'' is continuous. It follows that the
path component
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
s of a finite topological space are precisely the (weakly)
connected components of the associated graph Γ. That is, there is a topological path from ''x'' to ''y'' if and only if there is an
undirected path between the corresponding vertices of Γ.
Every finite space is
locally path-connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness ...
since the set
:
is a path-connected open
neighborhood of ''x'' that is contained in every other neighborhood. In other words, this single set forms a
local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
at ''x''.
Therefore, a finite space is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
if and only if it is path-connected. The connected components are precisely the path components. Each such component is both
closed and open in ''X''.
Finite spaces may have stronger connectivity properties. A finite space ''X'' is
*
hyperconnected if and only if there is a
greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
with respect to the specialization preorder. This is an element whose closure is the whole space ''X''.
*
ultraconnected
In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.PlanetMath Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersecti ...
if and only if there is a
least element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
with respect to the specialization preorder. This is an element whose only neighborhood is the whole space ''X''.
For example, the
particular point topology In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collec ...
on a finite space is hyperconnected while the
excluded point topology In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection
:T = \ \cup \
of subsets of ''X'' is then the excluded ...
is ultraconnected. The
Sierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.
It is the smallest example of a topological space which is neither trivial nor discrete. It is name ...
is both.
Additional structure
A finite topological space is
pseudometrizable if and only if it is
R0. In this case, one possible
pseudometric is given by
:
where ''x'' ≡ ''y'' means ''x'' and ''y'' are
topologically indistinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhood (topology), neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborh ...
. A finite topological space is
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
if and only if it is discrete.
Likewise, a topological space is
uniformizable
In mathematics, a topological space ''X'' is uniformizable if there exists a uniform structure on ''X'' that induces the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is homeomorphic to a uniform space (equipped with ...
if and only if it is R
0. The
uniform structure
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
will be the pseudometric uniformity induced by the above pseudometric.
Algebraic topology
Perhaps surprisingly, there are finite topological spaces with nontrivial
fundamental groups. A simple example is the
pseudocircle, which is space ''X'' with four points, two of which are open and two of which are closed. There is a continuous map from the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
''S''
1 to ''X'' which is a
weak homotopy equivalence In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.
A model category is a category with cla ...
(i.e. it induces an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of
homotopy groups). It follows that the fundamental group of the pseudocircle is
infinite cyclic
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
.
More generally it has been shown that for any finite
abstract simplicial complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
''K'', there is a finite topological space ''X''
''K'' and a weak homotopy equivalence ''f'' : , ''K'', → ''X''
''K'' where , ''K'', is the
geometric realization of ''K''. It follows that the homotopy groups of , ''K'', and ''X''
''K'' are isomorphic. In fact, the underlying set of ''X''
''K'' can be taken to be ''K'' itself, with the topology associated to the inclusion partial order.
Number of topologies on a finite set
As discussed above, topologies on a finite set are in one-to-one correspondence with
preorders on the set, and
T0 topologies are in one-to-one correspondence with
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
s. Therefore, the number of topologies on a finite set is equal to the number of preorders and the number of T
0 topologies is equal to the number of partial orders.
The table below lists the number of distinct (T
0) topologies on a set with ''n'' elements. It also lists the number of inequivalent (i.e.
nonhomeomorphic) topologies.
Let ''T''(''n'') denote the number of distinct topologies on a set with ''n'' points. There is no known simple formula to compute ''T''(''n'') for arbitrary ''n''. The
Online Encyclopedia of Integer Sequences presently lists ''T''(''n'') for ''n'' ≤ 18.
The number of distinct T
0 topologies on a set with ''n'' points, denoted ''T''
0(''n''), is related to ''T''(''n'') by the formula
:
where ''S''(''n'',''k'') denotes the
Stirling number of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to Partition of a set, partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) ...
.
See also
*
Finite geometry
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
*
Finite metric space
*
Topological combinatorics The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics.
History
The discipline of combinatorial topology used combinatorial concepts in top ...
References
*
*
*
*
External links
*{{cite web , url=http://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf , title=Notes and reading materials on finite topological spaces , first=J.P. , last=May , date=2003 , work=Notes for REU
Topological spaces
Combinatorics