FINITE TOPOLOGIES AND DIGRAPHS

In this paper we study the relation between finite topologies and digraphs. We associate a digraph to a topology by means of the “specialization” relation between points in the topology. Reciprocally, we associate a topology to each digraph, taking the sets of vertices adjacent (in the digraph) to v, for all vertex v, as a subbasis of closed sets for the topology. We use these associations to examine the relation between a simple digraph and its homologous topology. We also extend this relation to the functions preserving the structure between these classes of objects. AMS Classification : 54A05, 05C20.


Introduction
The one to one correspondence between finite preorder relations and finite topologies with the same underlying set of points, and also between finite posets and finite T 0 topologies is well known.Then, the one to one correspondence between finite digraphs and topologies is easily deducible.In fact, Evans, Harary and Lynn [3] prove that "There is a 1-1 correspondence between the labeled topologies with n points and the labeled transitive digraphs with n points".They associate a transitive digraph D(T ) to a topology T with the same set of points as follows: "For two distinct points u and v of T , u will be adjacent to v in D(T ) provided u is in every neighborhood of v".Reciprocally, "to each labeled transitive digraph D with n points, there corresponds a unique labeled topology T (D), in which the basic open sets are the sets of points adjacent to v, for all points v".
In this paper, we study the relations between finite topologies and digraphs in a different way.We associate a digraph to a topology by means of the "specialization" relation between points in a topological space: x is a specialization of y if and only if x is in the closure of {y}.This relation was introduced in a pioneering work by Alexandroff [1] and it has been used by Grothendieck and Dieudonné [5] to characterize the generic points of the irreducible components of a topological space as the maximal points of this binary relation.Reciprocally, we associate a topology T to each digraph D (not necessarily transitive) taking the sets of vertices adjacent to v in D, for all vertices v, as a subbasis of closed sets for the topology T .
In section 2 we use these two associations to make a more profound study of the relations between simple digraphs and their homologous topologies.We also extend the relation between these structures to the functions preserving the structure between these classes of objects.
In section 3 we restrict the study to the particular bijective relation between finite acyclic transitive digraphs and T 0 topologies.In this context we consider dual concepts to those used in the preceding section to obtain a minimum basis of open sets for the topology T and to prove that the set of the closures of the points in T is the minimum basis of open sets in the dual situation.This work corresponds to the unpublished first chapter of the author's Ph.D. [6].

Finite digraphs
By a digraph we mean a couple (X, G) where X is a finite nonempty set and G ⊂ X × X − {(x, x) : x ∈ X}, (so our digraph has no loops).The elements in X and G are called points and arcs respectively.For an arc (x, y) we will say that x is adjacent to y.In the following we will denote by xy an arc (x, y).
A sequence x 1 x 2 . . .x q x q+1 , q ≥ 2, of distinct points, except The non existence of loops in a digraph requires the following correction in the standard concepts of transitivity and antisymmetry.
We denote the set of digraphs with a set of points X by G X , the subset of transitive digraphs by G T X and the antisymmetric transitive digraphs by G T A X .
Proposition 2. Let (X, G) be a transitive digraph.Then (X, G) is acyclic if and only if it is antisymmetric.
Proof: An acyclic digraph is antisymmetric because, if there are arcs xy and yx, then it must be the cycle xyx.Reciprocally, if x 1 . . .x q x q+1 is a cycle then, by transitivity, x 1 x q and x q x 1 are arcs, in contradiction with the antisymmetric property.

Finite topological spaces
Let X be a nonempty set whose elements we will call points.Then, a topology T on X is a set of subsets of X, including ∅ and X, that is closed under union and finite intersection and the couple (X, T ) is a topological space on X.The elements in T are called open sets and their complements closed sets.The largest topology on X, T = P(X), is called the discrete topology.
If (X, T ) is a topological space and A ⊂ X, the closure of A is the minimum closed set that contains A, and we denote it by ĀT , or simply by Ā if there is no possible confusion.We also use some other standard topological concepts such as basis and subbasis of open or closed sets, neighborhood, connection,. . .as can be see in [8].
In the following definition we describe some "separation properties" by means of conditions easily related to each other.In this context, we use E d to denote the derived set of E ⊂ X. Definition 3. Let (X, T ) be a topological space.Then we will say that (X, T ) is T 0 if ∀x ∈ X, {x} d is a union of closed sets or, equivalently, if ∀x, y ∈ X, with x 6 = y, then {x} 6 = {y} (Kolmogoroff, 1935) It is well known that, in general, T 1 =⇒ T D =⇒ T 0 , and that the reciprocal is not true.
A topological space (X, T ) is finite if the set X is finite.In this case, the following result can be deduced from the above definitions.Lemma 4. Let (X, T ) be a finite topological space.Then 1. T D ⇐⇒ T 0 .

T 1 ⇐⇒ T is the discrete topology.
Other separation properties, of a general topological space, more restrictive than T 1 (such as T 2 , T 2a , T 3 , T 3a , T 4 , . . . ) are equivalent, in a finite topological space, to T 1 .In this way, T 0 is the only relevant separation property in a finite nondiscrete topological space.

Relations between topological spaces and digraphs
The "specialization" relation between points in a topological space was introduced by Alexandroff [1].Definition 5. Let (X, T ) be a topological space.For each pair of points x, y ∈ X we will say that x is a specialization of y if and only if x ∈ {y}.
The specialization relation is a preorder on X: it is obviously reflexive, and is transitive because But, in general, it is not antisymmetric, because in a topology, distinct points can have the same closure, in which case the points are related in both ways.This relation permits us to associate a digraph over X to each topological space on the set of points X.We denote the set of topological spaces over a same finite set of points X by T X , and the set of T 0 topological spaces over X by T 0 X .
Proposition 6.Let f : T X −→ G X be the function given by f (X, T ) = (X, G) where G is the set of arcs Proof: a) If T and T 0 are distinct topologies over X, then there exists at least a point x with distinct respective closures, that is to say, {x} T 6 = {x} T 0 .Then, there exists a point y with y ∈ {x} T and y / ∈ {x} T 0 (or vice versa) and so yx ∈ G and yx / ∈ G 0 (or vice versa), then f (X, b) It is a consequence of the transitivity of the specialization relation.c) f is not suprajective because the nontransitive digraphs do not procced from any topology.
It is also possible to associate a topological space over X to each digraph with set of points X by means of the following procedure.Definition 7. Let g : G X −→ T X be the function given by g(X, G) = (X, T ) where T is the topology over X generated by the subbasis of closed sets G ↓= {x ↓: x ∈ X}, where x ↓= {y : yx ∈ G} ∪ {x} We also use the notation g(G) = T.
Proposition 8.With the notations as above we have is not the identity and nor does it preserve the inclusion.
Proof: a) For any topological space (X, T ) ∈ T X we consider the digraph (X, G) = f (X, T ) and we will prove that g(X, G) = (X, T ).We have G = {xy : x ∈ {y}, x 6 = y} and g(X, G) is the topological space (X, T 0 ) whose subbasis of closed sets is G ↓= {x ↓: x ∈ X} where x ∈ X} and, as this is the subbasis of closed sets of the topology T , we have Next we give counterexamples proving c), d) and e).
Example 9.For each of the digraphs (X, G) on Figure 1 we construct: a) the family G ↓ of unipoint adjacencies x ↓ (that we will take as a subbasis of closed sets for the topology T = g(G)), b) the topology T given by its closed sets, c) the family T = {{x}, x ∈ X} of unipoint closures of T and d) the digraphs f (X, T ), that we denote by G 0 , given in Figure 2. In this way we have f • g(G) = G 0 according to the scheme Note that G 1 and G 2 are non comparable by the inclusion relation but they have the same image by g and by f • g, proving c) of Proposition 8. G 1 and G 3 verify , proving e) of Proposition 8. G 1 and G 4 verify G 4 ⊂ G 1 and T 4 ⊃ T 1 and, consequently, g(G 4 ) ⊃ g(G 1 ) and f • g(G 4 ) ⊃ f • g(G 1 ), proving d) of Proposition 8. G 3 and G 4 are digraphs with "regular behaviour" under f and g as we shall see now.The non-transitivity is the cause of these anomalies.In the example, only G 3 is transitive and for this one we have f • g(G 3 ) = G 3 .
Proposition 10.Let (X, G) be the digraph that is a cycle.Then a) The topology of g(X, G) associated to the cycle (X, G) is the discrete topology.
b) The digraph f • g(X, G) associated to the discrete topology g(X, G) is totally disconnected.
If we restrict the function g to the set of transitive digraphs, the relation between f and g is finally shown to be clear.
Theorem 11.Let (X, G) be a transitive digraph and let g(X, G) = (X, T ) be its associated topological space. Then x ↓= {x} for each x ∈ X Proof: Because G ↓= {x ↓: x ∈ X} is a subbasis of closed sets for the topology T , the closed sets of T are obtained as intersections of unions of sets x ↓ and also as unions of intersections of sets x ↓ (De Morgan's laws).So the minimal closed sets (by inclusion) are obtained as an intersection of sets x ↓.Then, for each point y ∈ X, we consider the intersection of the closed sets that include y: As C y is the minimum closed set including y, we also have C y = {y}.So, it is sufficient to prove that C y = y ↓.Corollary 12.With the notations as above, the function f : To prove that f • g(X, G) = G, it is sufficient to see that G ↓= T and this was proved in the previous theorem.
Remark 13.Let (X, G) be a transitive digraph and (X, T ) its associated topological space.Then a) y ↓= \ y∈x↓ x ↓= {y} that we denote simply by y. b) Both the digraph and the topology are known by the family G ↓= T = {x : x ∈ X}.In the digraph, x is the set {y : yx ∈ G} ∪ {x}.In the topology, x is the minimum closed set that contains x.
In general, x ↓ ⊂ y ↓ implies xy ∈ G.The reciprocal is not true and characterizes the transitive digraphs.
In the digraph (X, G) with G = {xy, yx, zx, zy} we have x = y = X y z = {z}.Thus, in a transitive digraph, it can happen that distinct points, x 6 = y, have the same closure x = y and so, in general, Card(G ↓) ≤ Card(X).
Corollary 15.Let (X, G) be a transitive digraph and x, y two distinct points in the digraph.Then The property of a topological space that has distinct closures of all its points characterizes the separation property T 0 (Definition 3).The bijections f and g for transitive digraphs make the properties T 0 and antisymmetric equivalent.Thus we have the following result: Theorem 16.Let (X, G) be a transitive digraph and (X, T ) its associated topological space.Then, the following conditions are equivalent: Corollary 17. a) There is a bijective correspondence between transitive digraphs and finite topological spaces.
b) There is a bijective correspondence between acyclic transitive digraphs and finite T 0 topological spaces.
The relation between acyclic transitive digraphs and finite T 0 topological spaces can be extended to the applications preserving the structure between these classes of objects.
Theorem 21.Let (X, T ) and (X 0 , T 0 ) be two finite T 0 topological spaces and (X, G) and (X 0 , G 0 ) their respective associated acyclic transitive digraphs.Let ϕ : X −→ X 0 be a function between their underlying sets of points.Then a) ϕ is a continuous function between the topological spaces if and only if ϕ is a digraph morphism between the respective digraphs.b) ϕ is a homeomorphism between the topological spaces if and only if ϕ is an isomorphism between the respective digraphs.
Corollary 22.With notations as above and denoting by ≈ both the equivalence relations of homeomorphism and isomorphism, we have that the quotient function is bijective.
In [3] it was pointed out that the enumeration of transitive digraphs is a particularly intractable problem.These authors give the number of finite topologies as a function of the number of finite T 0 topologies by means of the number of partitions of an n-set into m parts, also known as Stirling numbers of the second kind.These numbers are currently unknown.For a history of the enumeration of finite order relations and topologies, and for references about it, see the work of Erné and Stege [4].These authors describe an algorithm to compute the connected and non connected, T 0 and non T 0 , topologies on n points for n ≤ 14.These numbers can also be found in the on-line encyclopedia of integer sequences of Sloane [7].More recently, Benoumhani [2] has obtained formulas for the number of labeled topologies on n points having k open sets for k ≤ 12 and for the number of unlabeled T 0 topologies with k open sets for n + 4 ≤ k ≤ n + 6.

Dual digraph
A finite T 0 topological space (X, T ) and its homologous acyclic transitive digraph (X, G) are both described by the family T = G ↓= {x : x ∈ X} as a subbasis of closed sets (remark 13).
The topological duality between the concepts of open and closed sets allows us to describe the topology and the digraph over X by the family {Cx : x ∈ X} as a subbasis of open sets (CA denotes the complementary set of A ⊂ X).However, these open sets do not form the best subbasis for a simple description of the topological space (X, T ), and neither is its homologous digraph concept of interest for a good description of the digraph (X, G).
On the other hand, it is natural to consider a dual concept in the digraph for the x (changing the direction of the arrows) and to interpret its meaning in the topology.Lemma 26.With the hypothesis and the notations as above, we have Cy, for all x ∈ X (3.1) Proof: One inclusion is clear by the nature of the intersections.To prove the other we suppose that there exists z ∈ X such that z ∈ Cy whenever x * ⊂ Cy but that z / ∈ x * .Then, we would have x ∈ Cz and, by lemma 25, x * ⊂ Cz.Therefore, we would have z ∈ Cz, which is absurd.
Note that the formula is preserved when the family of opens Cy in the intersection is empty, since, if there is no y such that x * ⊂ Cy, then the digraph is connected and x is the unique minimal point; in consequence, x * = X is the intersection of the empty family.
Remark 27. 1) The formula (3.1) is the translation in terms of digraphs of the topological statement "An open is the intersection of all the opens that contain it".Note that this lemma proves that the sets x * are opens in the homologous topology to the digraph (X, G).
2) Then, Lemma 25 says that x * is the minimum open that contains x, because the formula (3.1) admits the versions: Cy in which the second members are read as "intersection of the opens that contains x".
Proposition 28.Let (X, G) ∈ G T A X be an acyclic transitive digraph and (X, T ) ∈ T 0 X its homologous T 0 topological space.Then the family T * = {x * : x ∈ X} is a subbasis of open sets for (X, T ).
Proof: {Cx, x ∈ X} is a subbasis of open sets and each Cx is a union of sets y * (Lemma 24), so T * is also a subbasis.
To consider T * as a subbasis of open sets for (X, T ), as opposed to the family {Cx, x ∈ X}, facilitates a digraph interpretation dual to that of T as a subbasis of closed sets.
We shall now see other topological properties of T * that justify its choice.
Lemma 29.With the notations as above, we have that T * is a basis for (X, T ).
Proof: It is sufficient to prove that any intersection of elements of T * can be written as a union of elements of T * .More precisely, let J be any set of indexes, then if The right hand inclusion is clear, since x j k ∈ x * j k .For the other, let Then there exists k such that x ∈ x * j k (or x j k ∈ x) and, by Proposition 14, we have x j k ⊂ x.On the other hand, x j k ∈ \ J x * j , then x j ∈ x j k , for all j ∈ J.In consequence, x j ∈ x, for all j ∈ J, and therefore x ∈ Although each x * is the minimum open set that contains x, this does not necessarily mean that the intersection of elements of T * is an element of T * .The formula x * j k , even though it has been used to prove Lemma 29, has little practical interest since the union of the right hand side, in general, is not disjoint.Even, some of the open sets that make it up can be redundant.In the next example these observations are made clear.
Theorem 30.With the notations as above, we have that T * is the minimum basis for (X, T ).
Proof: We know that T * is a basis and that, for all x ∈ X, we have that x * is the minimum open set including x.If B is another arbitrary basis, then the opens of B that contains x are a fundamental system of neighborhoods of x and, so, keeping in mind that x * is a neighborhood of x, there exists an open V ∈ B such that x ∈ V ⊂ x * .As x * is minimum we have that V = x * and so x * ∈ B, for all x ∈ X.
Corollary 31.Let B be a basis of (X, T ).Then Card(B) ≥ Card(X) and This is a consequence of the previous theorem.
Remark 32.In general, T * is not a minimum subbasis.For the digraph over X = {x, y, z} with a set of arcs G = {xz, yz}, we have that T * = {x * = {x, z}, y * = {y, z}, z * = {z}} is the minimum basis, but it is not the minimum subbasis, since {x * , y * } is also a subbasis.Nor is T the minimum subbasis of closed sets.

\
y∈x↓x ↓ ⊂ y ↓ because y ↓ is one of the sets in the intersection.If z ∈ y ↓ and y ∈ x ↓ (for each x such that y ∈ x ↓), then, by transitivity, z ∈ x ↓ for each x such that y ∈ x ↓, then y ↓ ⊂ \ y∈x↓ x ↓.

Definition 23 .
Let (X, G) ∈ G T A X be an acyclic transitive digraph.For each x ∈ X we will denote x ↑= {y, xy ∈ G} ∪ {x} and x * = {y, x ∈ y} It can be deduced from the definition that x ∈ y ⇐⇒ y ∈ x * and x * = x ↑.Lemma 24.With the hypothesis and the notations as above, we have Cx = [ x/ ∈y * y * , for all x ∈ X Proof: We have z ∈ Cx ⇐⇒ z / ∈ x ⇐⇒ x / ∈ z * so z * is among those that unite and, as z ∈ z * , we have z ∈ [ x/ ∈y * y * .Reciprocally, if z ∈ [ x/ ∈y * y * , then there exists y ∈ X such that z ∈ y * , with x / ∈ y * and so x / ∈ z * .Lemma 25.With the hypothesis and the notations as above, we have x ∈ Cy ⇐⇒ x * ⊂ Cy, for all x ∈ X Proof: If x ∈ Cy and z ∈ x * , we have xy / ∈ G and xz ∈ G. Therefore, by transitivity, we have zy / ∈ G and z ∈ Cy.The reciprocal is clear.