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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 Traditionally, a finite verb (from la, fīnītus, past particip ...
. 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 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 X 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 h ...
on X is a subset \tau of P(X) (the power set of X ) such that # \varnothing \in \tau and X\in \tau . # if U, V \in \tau then U \cup V \in \tau . # if U, V \in \tau then U \cap V \in \tau . In other words, a subset \tau of P(X) is a topology if \tau contains both \varnothing and X 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 ** ''U ...
s. Elements of \tau 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 ...
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 ...
s (and only finitely many
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s). A topology on a finite set can also be thought of as a
sublattice A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...
of (P(X), \subset) which includes both the bottom element \varnothing and the top element X .


Examples


0 or 1 points

There is a unique topology on the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
∅. 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, th ...
. Here the open sets are ∅ and . This topology is both discrete 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 func ...
. There is also a unique continuous function from ''X'' to the singleton space , namely the
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properti ...
to ''a''. In the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
the empty space serves as an initial object in the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continu ...
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) # # # (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 to ...
) 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 named ...
. 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 neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
.


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 with
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special ca ...
s on ''X''. Recall that a preorder on ''X'' is a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
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, th ...
. This preorder is called the ''
specialization preorder In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the ...
'' 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, 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 space In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite restr ...
s. 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 since any open cover 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-class ...
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 T1, 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 neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
if and only if ''x'' ≤ ''y'' and ''y'' ≤ ''x'', where ≤ is the specialization preorder on ''X''. It follows that a space ''X'' is T0 if and only if the specialization preorder ≤ on ''X'' is a partial order. There are numerous partial orders on a finite set. Each defines a unique T0 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 on ''X''. The equivalence classes will be the classes of topologically indistinguishable points. Since the partition topology is pseudometrizable, a finite space is R0 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. 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 ...
T0 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 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 terminals, ...
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 components 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 since the set :\mathopx = \ is a path-connected open
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of ''x'' that is contained in every other neighborhood. In other words, this single set forms a local base 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 In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
if and only if there is a greatest element 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 intersectio ...
if and only if there is a least element 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 collect ...
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 named ...
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 :d(x,y) = \begin0 & x\equiv y \\ 1 & x\not\equiv y\end 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 neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
. A finite topological space is metrizable if and only if it is discrete. Likewise, a topological space is uniformizable if and only if it is R0. 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 unif ...
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 Eucl ...
''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 cl ...
(i.e. it induces an isomorphism of
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s). 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 binar ...
. 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 ...
''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
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special ca ...
s on the set, and T0 topologies are in one-to-one correspondence with partial orders. Therefore, the number of topologies on a finite set is equal to the number of preorders and the number of T0 topologies is equal to the number of partial orders. The table below lists the number of distinct (T0) 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 T0 topologies on a set with ''n'' points, denoted ''T''0(''n''), is related to ''T''(''n'') by the formula :T(n) = \sum_^S(n,k)\,T_0(k) 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 a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
.


See also

* Finite geometry * 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