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| In [[mathematics]], a '''finite topological space''' is a [[topological space]] for which the underlying [[set (mathematics)|point set]] is [[finite set|finite]]. That is, it is a topological space for which there are only finitely many points.
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| While topology has mainly been developed for infinite spaces, finite topological spaces are often used to provide examples of interesting phenomena or [[counterexample]]s to plausible sounding conjectures. [[William Thurston]] has called the study of finite topologies in this sense "an oddball topic that can
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| lend good insight to a variety of questions."<ref>{{cite book | last = Thurston | first = William P. | authorlink = William Thurston |date=April 1994 | title = On Proof and Progress in Mathematics | journal = [[Bulletin of the American Mathematical Society]] | volume = 30 | issue = 2 | pages = 161–177 | arxiv = math/9404236 | doi = 10.1090/S0273-0979-1994-00502-6 }}</ref>
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| ===As a bounded sublattice===
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| A [[topology (structure)|topology]] on a set ''X'' is defined as a subset of ''P''(''X''), the [[power set]] of ''X'', which includes both ∅ and ''X'' and is closed under finite [[intersection (set theory)|intersection]]s and arbitrary [[union (set theory)|union]]s.
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| Since the power set of a finite set is finite there can be only finitely many [[open set]]s (and only finitely many [[closed set]]s). Therefore one only need check that the union of a finite number of open sets is open. This leads to a simpler description of topologies on a finite set.
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| Let ''X'' be a finite set. A topology on ''X'' is a subset τ of ''P''(''X'') such that
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| #∅ ∈ τ and ''X'' ∈ τ
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| #if ''U'' and ''V'' are in τ then ''U'' ∪ ''V'' ∈ τ
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| #if ''U'' and ''V'' are in τ then ''U'' ∩ ''V'' ∈ τ
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| A topology on a finite set is therefore nothing more than a [[sublattice]] of (''P''(''X''), ⊂) which includes both the bottom element (∅) and the top element (''X'').
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| Every finite [[bounded lattice]] is [[complete lattice|complete]] since the [[meet and join|meet or join]] of any family of elements can always be reduced to a meet or join of two elements. It follows that in a finite topological space the union or intersection of an arbitrary family of open sets (resp. closed sets) is open (resp. closed).
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| ===Specialization preorder===
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| Topologies on a finite set ''X'' are in [[one-to-one correspondence]] with [[preorder]]s on ''X''. Recall that a preorder on ''X'' is a [[binary relation]] on ''X'' which is [[reflexive relation|reflexive]] and [[transitive relation|transitive]].
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| Given a (not necessarily finite) topological space ''X'' we can define a preorder on ''X'' by
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| :''x'' ≤ ''y'' if and only if ''x'' ∈ cl{''y''}
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| where cl{''y''} denotes the [[closure (topology)|closure]] of the [[singleton set]] {''y''}. This preorder is called the ''[[specialization preorder]]'' on ''X''. Every open set ''U'' of ''X'' will be an [[upper set]] 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 ≤.
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| 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 ≤.
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| 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]]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.
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| ==Examples==
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| ===0 or 1 points===
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| There is a unique topology on the [[empty set]] ∅. The only open set is the empty one. Indeed, this is the only subset of ∅.
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| Likewise, there is a unique topology on a [[singleton set]] {''a''}. Here the open sets are ∅ and {''a''}. This topology is both [[discrete space|discrete]] and [[trivial topology|trivial]], 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.
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| For any topological space ''X'' there is a unique [[continuous function (topology)|continuous function]] from ∅ to ''X'', namely the [[empty function]]. There is also a unique continuous function from ''X'' to the singleton space {''a''}, namely the [[constant function]] to ''a''. In the language of [[category theory]] the empty space serves as an [[initial object]] in the [[category of topological spaces]] while the singleton space serves as a [[terminal object]].
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| ===2 points===
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| Let ''X'' = {''a'',''b''} be a set with 2 elements. There are four distinct topologies on ''X'':
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| #{∅, {''a'',''b''<nowiki>}}</nowiki> (the [[trivial topology]])
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| #{∅, {''a''}, {''a'',''b''<nowiki>}}</nowiki>
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| #{∅, {''b''}, {''a'',''b''<nowiki>}}</nowiki>
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| #{∅, {''a''}, {''b''}, {''a'',''b''<nowiki>}}</nowiki> (the [[discrete topology]])
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| 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]]. So, in fact, there are only three inequivalent topologies on a two point set: the trivial one, the discrete one, and the Sierpiński topology.
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| The specialization preorder on the Sierpiński space {''a'',''b''} with {''b''} open is given by: ''a'' ≤ ''a'', ''b'' ≤ ''b'', and ''a'' ≤ ''b''.
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| ===3 points===
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| Let ''X'' = {''a'',''b'',''c''} be a set with 3 elements. There are 29 distinct topologies on ''X'' but only 9 inequivalent topologies:
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| #{∅, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''c''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''a'',''b''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''c''}, {''a'',''b''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''c''}, {''b'',''c''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''c''}, {''a'',''c''}, {''b'',''c''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''a''}, {''b''}, {''a'',''b''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''b''}, {''c''}, {''a'',''b''}, {''b'',''c''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| #{∅, {''a''}, {''b''}, {''c''}, {''a'',''b''}, {''a'',''c''}, {''b'',''c''}, {''a'',''b'',''c''<nowiki>}}</nowiki>
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| The last 5 of these are all [[T0 space|T<sub>0</sub>]]. The first one is trivial, while in 2, 3, and 4 the points ''a'' and ''b'' are [[topologically indistinguishable]].
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| ==Properties==
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| ===Compactness and countability===
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| Every finite topological space is [[compact space|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.
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| Every finite topological space is also [[second-countable]] (there are only finitely many open sets) and [[separable space|separable]] (since the space itself is [[countable set|countable]]).
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| ===Separation axioms===
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| If a finite topological space is [[T1 space|T<sub>1</sub>]] (in particular, if it is [[Hausdorff space|Hausdorff]]) then it must, in fact, be discrete. This is because the [[complement (set theory)|complement]] of a point is a finite union of closed points and therefore closed. It follows that each point must be open.
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| Therefore, any finite topological space which is not discrete cannot be T<sub>1</sub>, Hausdorff, or anything stronger.
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| However, it is possible for a non-discrete finite space to be [[T0 space|T<sub>0</sub>]]. In general, two points ''x'' and ''y'' are [[topologically indistinguishable]] if and only if ''x'' ≤ ''y'' and ''y'' ≤ ''x'', where ≤ is the specialization preorder on ''X''. It follows that a space ''X'' is T<sub>0</sub> 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 T<sub>0</sub> topology.
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| Similarly, a space is [[R0 space|R<sub>0</sub>]] 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 space|pseudometrizable]], a finite space is R<sub>0</sub> if and only if it is [[completely regular]].
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| Non-discrete finite spaces can also be [[normal space|normal]]. The [[excluded point topology]] on any finite set is a [[completely normal space|completely normal]] T<sub>0</sub> space which is non-discrete.
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| ===Connectivity===
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| 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 (graph theory)|connectivity]] of the associated graph Γ.
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| In any topological space, if ''x'' ≤ ''y'' then there is a [[path (topology)|path]] 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]]s of a finite topological space are precisely the (weakly) [[connected component (graph theory)|connected component]]s of the associated graph Γ. That is, there is a topological path from ''x'' to ''y'' if and only if there is an [[path (graph theory)|undirected path]] between the corresponding vertices of Γ.
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| Every finite space is [[locally path-connected]] since the set
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| :<math>\mathop{\uarr}x = \{y \in X : x \leq y\}</math>
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| is a path-connected open [[neighborhood (topology)|neighborhood]] of ''x'' that is contained in every other neighborhood. In other words, this single set forms a [[local base]] at ''x''.
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| Therefore, a finite space is [[connected space|connected]] if and only if it is path-connected. The connected components are precisely the path components. Each such component is both [[clopen set|closed and open]] in ''X''.
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| Finite spaces may have stronger connectivity properties. A finite space ''X'' is
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| *[[hyperconnected space|hyperconnected]] 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''.
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| *[[Ultraconnected space|ultraconnected]] 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''.
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| For example, the [[particular point topology]] on a finite space is hyperconnected while the [[excluded point topology]] is ultraconnected. The [[Sierpiński space]] is both.
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| ===Additional structure===
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| A finite topological space is [[pseudometrizable space|pseudometrizable]] if and only if it is [[R0 space|R<sub>0</sub>]]. In this case, one possible [[pseudometric space|pseudometric]] is given by
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| :<math>d(x,y) = \begin{cases}0 & x\equiv y \\ 1 & x\not\equiv y\end{cases}</math>
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| where ''x'' ≡ ''y'' means ''x'' and ''y'' are [[topologically indistinguishable]]. A finite topological space is [[metrizable space|metrizable]] if and only if it is discrete.
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| Likewise, a topological space is [[uniformizable space|uniformizable]] if and only if it is R<sub>0</sub>. The [[uniform structure]] will be the pseudometric uniformity induced by the above pseudometric.
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| ===Algebraic topology===
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| Perhaps surprisingly, there are finite topological spaces with nontrivial [[fundamental group]]s. 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]] ''S''<sup>1</sup> to ''X'' which is a [[weak homotopy equivalence]] (i.e. it induces an [[isomorphism]] of [[homotopy group]]s). It follows that the fundamental group of the pseudocircle is [[infinite cyclic group|infinite cyclic]].
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| More generally it has been shown that for any finite [[abstract simplicial complex]] ''K'', there is a finite topological space ''X''<sub>''K''</sub> and a weak homotopy equivalence ''f'' : |''K''| → ''X''<sub>''K''</sub> where |''K''| is the [[geometric realization]] of ''K''. It follows that the homotopy groups of |''K''| and ''X''<sub>''K''</sub> are isomorphic. In fact, the underlying set of ''X''<sub>''K''</sub> can be take to be ''K'' itself, with the topology associated to the inclusion partial order.
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| ==Number of topologies on a finite set==
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| As discussed above, topologies on a finite set are in one-to-one correspondence with [[preorder]]s on the set, and [[T0 space|T<sub>0</sub> topologies]] are in one-to-one correspondence with [[partial order]]s. Therefore the number of topologies on a finite set is equal to the number of preorders and the number of T<sub>0</sub> topologies is equal to the number of partial orders.
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| The table below lists the number of distinct (T<sub>0</sub>) topologies on a set with ''n'' elements. It also lists the number of inequivalent (i.e. [[homeomorphic|nonhomeomorphic]]) topologies.
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| {|class=wikitable style="margin: auto; text-align:right;"
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| |+style="margin-bottom:1ex;"| Number of topologies on a set with ''n'' points
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| !''n''!!Distinct<br>topologies!!Distinct<br>T<sub>0</sub> topologies!!Inequivalent<br>topologies!!Inequivalent<br>T<sub>0</sub> topologies
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| |-
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| !0
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| |1 || 1 || 1 || 1
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| |-
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| !1
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| |1 || 1 || 1 || 1
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| |-
| |
| !2
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| |4 || 3 || 3 || 2
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| |-
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| !3
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| |29 || 19 || 9 || 5
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| |-
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| !4
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| |355 || 219 || 33 || 16
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| |-
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| !5
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| |6942 || 4231 || 139 || 63
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| |-
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| !6
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| |209527 || 130023 || 718 || 318
| |
| |-
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| !7
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| |9535241 || 6129859 || 4535 || 2045
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| |-
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| !8
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| |642779354 || 431723379 || 35979 || 16999
| |
| |-
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| !9
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| |63260289423 || 44511042511 || 363083 || 183231
| |
| |-
| |
| !10
| |
| |8977053873043 || 6611065248783 || 4717687 || 2567284
| |
| |- style="text-align:center;"
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| ![[OEIS]]
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| | [[OEIS:A000798|A000798]] || [[OEIS:A001035|A001035]] || [[OEIS:A001930|A001930]] || [[OEIS:A000112|A000112]]
| |
| |}
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| 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.
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| The number of distinct T<sub>0</sub> topologies on a set with ''n'' points, denoted ''T''<sub>0</sub>(''n''), is related to ''T''(''n'') by the formula
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| :<math>T(n) = \sum_{k=0}^{n}S(n,k)\,T_0(k)</math>
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| where ''S''(''n'',''k'') denotes the [[Stirling number of the second kind]].
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| ==See also==
| |
| | |
| *[[Finite geometry]]
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| *[[Finite metric space]]
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| *[[Topological combinatorics]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| *[http://www.maths.ed.ac.uk/~aar/papers/stong2.pdf Finite topological spaces], RE Stong - Trans. Amer. Math. Soc, 1966
| |
| *Singular homology groups and homotopy groups of finite topological spaces, Michael C. McCord, Duke Math. J. Volume 33, Number 3 (1966), 465-474.
| |
| *{{cite book
| |
| | last = Barmak | first = Jonathan | author-link =
| |
| | isbn = 978-3-642-22002-9
| |
| | publisher = Springer
| |
| | title = Algebraic Topology of Finite Topological Spaces and Applications
| |
| | year = 2011}}
| |
| *{{cite book
| |
| | last1 = Merrifield | first1 = Richard | author-link1 =
| |
| | last2 = Simmons | first2 = Howard E. | author-link2 = Howard Ensign Simmons, Jr.
| |
| | title = Topological Methods in Chemistry
| |
| | year = 1989
| |
| | publisher = Wiley
| |
| | url = http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471838179.html
| |
| | isbn = 978-0-471-83817-3
| |
| }}
| |
| | |
| ==External links==
| |
| *[http://math.uchicago.edu/~may/finite.html Notes and reading materials on finite topological spaces], J.P. MAY
| |
| | |
| [[Category:Topological spaces]]
| |
| [[Category:Combinatorics]]
| |
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