{'fraction_non_alphanumeric': 0.06751463403899476, 'fraction_numerical': 0.024236022475536582, 'mean_word_length': 3.1621786652011967, 'pattern_counts': {'":': 0, '<': 20, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 127, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Poset splitting and minimality of finite models

We develop a novel technique, which we call poset splitting, that allows us to solve two open problems regarding minimality of finite models of spaces: the nonexistence of a finite model of the real projective plane with fewer than 13 points and the nonexistence of a finite model of the torus with fewer than 16 points. Indeed, we prove much stronger results from which we also obtain that there does not exist a finite model of the Klein bottle with fewer than 16 points and that the integral homology groups of finite spaces with fewer than 13 points are torsion-free, settling a conjecture of Hardie, Vermeulen and Witbooi. Furthermore, we also apply our technique to give a complete characterization of the minimal finite models of the real projective plane, the torus, and the Klein bottle. In addition, we show that the poset splitting technique has an intrinsic interest giving original topological results that can be obtained from its application, such as a generalization of Hurewicz's theorem for non-simply-connected spaces and a generalization of a result of R. Brown on the fundamental group of a space.


Introduction
A well-known result of McCord [15] asserts that for every finite simplicial complex K there exists a finite T 0 -space X (K) and a weak homotopy equivalence |K| → X (K). Hence, every compact CW-complex X admits finite models, that is, finite topological spaces which are weak equivalent to X. Moreover, the poset of cells of a finite and regular CW-complex X is a finite model of X. Finite models permit the study of homotopical invariants of spaces by means of combinatorial tools and serve as a method to tackle open problems in algebraic topology from a different point of view [3].
A natural problem regarding finite models is to find minimal finite models for a given compact CW-complex X, that is finite models of X of the minimum possible cardinality. The first question in this line was stated by J.P. May in [14], where he conjectures that the n-fold non-Hausdorff suspension of the 0-sphere, denoted S n S 0 , is a minimal finite model of the n-sphere. This question was answered positively by Barmak and Minian in [4], where they also prove that S n S 0 is the only minimal finite model of the n-sphere. In the same article they give a characterization of the minimal finite models of finite graphs.
Despite the simplicity of its formulation, it turns out that finding minimal finite models of spaces is a very hard problem due to the lack of tools which relate homotopy invariants of a finite T 0 -space with the combinatorial information present in its Hasse diagram. Also, as the number of (unlabeled) posets on n points seems to grow exponentially on n 2 [8] and since there does not exist an effective algorithm to describe posets of n points, even a computer-aided brute force method becomes impractical for posets with more than 12 points. The intrinsic difficulty of the minimality problem is reflected in the fact that, prior to this article, the only known minimal finite models were those of the spheres, the graphs and the homotopically trivial spaces.
Several open questions about minimal finite models appear in the literature. For example, a finite model of the real projective plane of 13 points is exhibited in [3,Example 7.1.1] as the poset of cells of a certain regular CW-complex structure and is also constructed in [13] from a finite version of the mapping cone. Its minimality was raised as an open question in [13] and in [3] and verified by M. Adamaszek with a computer-assisted proof which required the analysis of approximately 10 8 cases [1].
In this article we give the first purely mathematical proof of this result showing that if X is a finite and connected space with fewer than 13 points then π 1 (X) is a free group. Moreover, we prove that there exist (up to homeomorphism) exactly two minimal finite models of the real projective plane: the aforementioned one and its opposite.
In [13], Hardie, Vermeulen and Witbooi state a stronger conjecture: that the finite model of the real projective plane mentioned above is the smallest finite space with 2torsion in any integral homology group. We will also prove this conjecture showing that if X is a finite space such that H n (X) has torsion for some n ∈ N then X has at least 13 points.
Also, in section 6 we prove that there does not exist a finite model of the torus with fewer than 16 points settling another open problem [3, p.44]. Indeed, we prove a stronger result from which we also derive the non-existence of a finite model of the Klein bottle with fewer than 16 points. Finite models of the Klein bottle had not been studied before. Furthermore, we prove that there exist (up to homeomorphism) exactly two minimal finite models of the torus: SS 0 × SS 0 and the space T 2 1,1 of page 23 of this work. The first one was known, but not the second one, which was obtained from our methods. Moreover, in the same proof we also obtain that there exist (up to homeomorphism) exactly two minimal finite models of the Klein bottle: the spaces K 1,0 and K 0,1 of figures 4 and 6. Remarkably, these four minimal finite models are posets of height 2 obtained by suitably combining two posets of height one in four possible ways, resembling the way the torus and the Klein bottle are obtained as a quotient of a cylinder.
All these original results are achieved by developing a novel technique which we call poset splitting and which turns out to be interesting in itself. This technique is developed in section 3 where we show some new topological results that can be obtained from it. In particular, we obtain a generalization of a result of R. Brown [7] regarding the fundamental group and a generalization of Hurewicz's theorem for non-simply-connected spaces.
Finally, in the last section of this article we exhibit finite models for real projective spaces, tori and Moore spaces of type (Z n , k) raising open questions about their minimality.

Preliminaries
In the first part of this section we will recall the basic notions of the theory of finite topological spaces, fix notation, and give some basic results which were not found in the classical literature. For a comprehensive exposition on finite spaces the reader may consult [3]. In the second part of this section we will review the categorical approach to homotopy theory of posets with some new insights which are related to works of C. Cibils and J. MacQuarrie [10] and of J. Barmak and G. Minian [5].
Classical definitions and results on homotopy theory of posets. The theory of finite topological spaces is based in the well-known functorial correspondence between Alexandroff T 0 -spaces and posets given by Alexandroff which preserves the underlying set and is induced by the inclusion relation between minimal open neighbourhoods [2]. In this way, any Alexandroff T 0 -space is naturally endowed with a partial order. Indeed, if X is an Alexandroff T 0 -space and x ∈ X then the minimal open set which contains x is U x = {a ∈ X / a ≤ x} and the minimal closed set which contains x is F x = {x} = {a ∈ X / a ≥ x}. Under this correspondence, continuous map between Alexandroff T 0 -spaces correspond to order-preserving morphisms between posets.
If X is an Alexandroff T 0 -space it is standard to denote U x = {a ∈ X / a < x}, F x = {a ∈ X / a > x}, C x = U x ∪ F x and C x = C x − {x}. In the case that several topological spaces are considered at the same time, we will denote U x by U X x to indicate the space in which the minimal open set is considered. We will use similar notations for F x , C x , U x , F x and C x .
Stong gives in [17] a simple algorithm to decide whether two finite T 0 -spaces are homotopy equivalent or not. Following May's terminology [14], if X is a finite T 0 -space we will say that a point x ∈ X is an up beat point (resp. down beat point) of X if the subposet F x has a minimum (resp. if the subposet U x has a maximum). Stong proves in the aforementioned article that if x is a beat point of X then X − {x} is a strong deformation retract of X and that two finite T 0 -spaces are homotopy equivalent if and only if one obtains homeomorphic spaces after successively removing their beat points. Using the results of Stong it is easy to prove that if X is a finite T 0 -space and x ∈ X then C x is contractible.
A different approach to the study of finite topological spaces was given by McCord in [15] where he proves that the face poset X (K) of a simplicial complex K (which is the poset of simplices of K ordered by inclusion) is weak homotopy equivalent to |K|. More generally, if K is a regular CW-complex then the face poset X (K) of K is weak homotopy equivalent to K.
McCord also proves in [15] that if X is an Alexandroff T 0 -space then there exists a weak homotopy equivalence |K(X)| → X. Here, K(X) is the order complex of X, that is, the simplicial complex of the non-empty chains of X and |K(X)| denotes its geometric realization.
Note that K(X (K)) is the barycentric subdivision of K. This leads to the definition of the barycentric subdivision of a poset X as X ′ = X (K(X)). It follows that X and X ′ are weak homotopy equivalent.
McCord also shows that for every Alexandroff space X there exists a quotient map q : X → Y such that Y is an Alexandroff T 0 -space and q is a homotopy equivalence [15,Theorem 4].
From the results of McCord it follows that the singular homology groups of an Alexandroff T 0 -space X are naturally isomorphic to the simplicial homology groups of K(X). Hence, the homology groups of an Alexandroff T 0 -space X can be described in terms of the chains of X.
This idea is exploited in [9] where we obtain a spectral sequence which converges to the integral homology groups of a locally finite T 0 -space in which the differentials have an explicit and simple description. Under certain conditions, the first page of this spectral sequence reduces to a chain complex, which motivates the following definition.
Definition 2.1. Let X be a locally finite T 0 -space and let A ⊆ X be a subspace. We say that (X, A) is a relative quasicellular pair if A is open in X and there exists an order preserving map ρ : X − A −→ N 0 , which will be called quasicellular morphism for (X, A), such that (1) The set {x ∈ X − A : ρ(x) = n} is an antichain for every n ∈ N 0 .
(2) For every x ∈ X − A, the reduced integral homology of U X x is concentrated in degree ρ(x) − 1. We say that a locally finite T 0 -space is quasicellular if (X, ∅) is a relative quasicellular pair.
Note that if (X, A) is a relative quasicellular pair, ρ is a quasicellular morphism for (X, A) and x, y ∈ X − A, then x < y implies that ρ(x) < ρ(y).
Note also that if K is a regular CW-complex then X (K) is quasicellular. In particular, if X is an Alexandroff T 0 -space then X ′ is quasicellular.
Hereafter, homology will mean integral homology and hence the group of coefficients will be omitted from the notation.
In [9] we obtained the following result, which will be applied in section 3.
Proposition 2.2. Let (X, A) be a relative quasicellular pair and let ρ be a quasicellular morphism for (X, A). For each n ∈ N 0 , let J n = {x ∈ X − A : ρ(x) = n}. Let C(X, A) = (C n (X, A), d n ) n∈Z be the chain complex defined by for each n ∈ N 0 and C n (X, A) = 0 for n < 0.
• For each n ∈ Z, d n : where for every x ∈ D n , l x ∈ N, and for every i = {1, . . . , l x }, a x i ∈ Z and s x i is an (n − 1)-chain of U x . Then, H n (X, A) = H n (C(X, A)) for all n ∈ N 0 .
McCord also defines the non-Hausdorff suspension of a topological space X as the space S(X) whose underlying set is X ∐ {+, −} and whose open sets are those of X together with X ∪ {+}, X ∪ {−} and X ∪ {+, −}, and proves that for every space X there exists a weak homotopy equivalence between the suspension of X and S(X) [15].
Recall that if X and Y are finite T 0 -spaces, the non-Hausdorff join X ⊛ Y is defined as the poset whose underlying set is the disjoint union X ∐ Y and whose partial order is defined keeping the partial orders within X and Y and setting x ≤ y for all x ∈ X and y ∈ Y . Note that S(X) = X ⊛ S 0 .
If X is a poset, X op will denote the poset X with the inverse order and will be called the opposite space of X. Note that |K(X op )| = |K(X)| and hence X and X op are weak homotopy equivalent.
Finally, recall that a finite model of a topological space Z is a finite space which is weak homotopy equivalent to Z. For example, the n-fold non-Hausdorff suspension of the 0-sphere, S n S 0 , is a finite model of the n-sphere [15]. A minimal finite model of a topological space Z is a finite model of Z of minimum cardinality. By [15,Theorem 4], minimal finite models are T 0 -spaces.
We will give now some basic results that were not found in the literature and will be needed later.
Hence, we may assume that a and b are incomparable. Let It is easy to check that the map r is order-preserving and hence continuous. Also, ri = Id and ir ≥ Id. Thus, by [3, Corollary 1.2.6], r is a strong deformation retraction.
We ought to mention that the map r of the previous proof already appears in the proof of proposition 11.2.3 of [3]. In a similar way, the following proposition is related to the notion of qc-reductions [3, p.140].
Proposition 2.4. Let X be a finite T 0 -space and let a and b be maximal elements of X. If U a ∪ U b is homotopically trivial then the quotient map q : X → X/{a, b} is a weak homotopy equivalence. The following simple lemma constitutes a key step for the results of this article.
Lemma 2.5. Let X be an Alexandroff T 0 -space and let A X. Then Proof.
(1) Follows easily from the definition of barycentric subdivision of a poset.
(2) Let σ ∈ X ′ − A ′ and let η = σ ∩ (X − A). Then η ∈ (X − A) ′ and Categorical approach to homotopy theory of posets. When considered as a poset, an Alexandroff T 0 -space X can be regarded as a small category in the standard way, that is, the objects of the category are the elements of X and for x, y ∈ X there is a (unique) morphism from x to y if and only if x ≤ y. It is well known that the nerve of this category is the simplicial set associated to the simplicial complex K(X) and hence the classifying space of X is homeomorphic to |K(X)|. In particular, the categorical definition of homotopy and homology groups of X agrees with the usual definition for topological spaces. Given a small category C we can consider its localization with respect to all its morphisms C [Mor(C ) −1 ]. We will write LC for this category, which is clearly a groupoid, and ι C : C → LC for the natural localization functor. Note that a functor F : C → D between small categories induces a unique groupoid morphism LF : LC → LD in a functorial way [12].
D. Quillen proved in [16] that the fundamental group of a small category C at an object c 0 is canonically isomorphic to Aut LC (c 0 ), the group of automorphisms of c 0 in the groupoid LC . So we can canonically identify the fundamental group of an Alexandroff T 0 -space X at x 0 with the group of automorphisms of x 0 in LX. Now let G be a groupoid and let N be a subgroupoid of G. Following [6], we say that N is a normal subgroupoid of G if Obj(N ) = Obj(G) and for all x, y ∈ Obj(G) and g ∈ Hom G (x, y) we have that g −1 Aut N (y)g = Aut N (x). In this case we can identify objects x, y ∈ Obj(G) if Hom N (x, y) = ∅ and morphisms α, β ∈ Mor(G) if there exist morphisms n 1 , n 2 ∈ Mor(N ) such that α = n 1 βn 2 . It is easy to see that the classes of objects of G form a category whose morphisms are the classes of morphisms of G with composition induced by the composition in G. This category is clearly a groupoid, called the quotient groupoid, and will be denoted by G/N . The canonical projection q : G → G/N that maps each object and morphism of G to its equivalence class is a functor and hence a morphism of groupoids. It is easy to prove that, with the notations above, if F : G → H is a morphism of groupoids such that F sends morphisms of N to identity maps then there exists a unique groupoid morphism F : G/N → H such that F q = F .
Recall that an indiscrete category is a category C such that for any two objects x, y ∈ Obj(C ), the set Hom C (x, y) consists of exactly one morphism. Note that any indiscrete category is a connected groupoid. We say that a category T is a tree if LT is an indiscrete category and that a category is a forest if it is a disjoint union of trees. If C is a connected small category, a maximal tree in C is a subcategory T ⊆ C such that T is a tree and Obj(T ) = Obj(C ). Now, let C be a connected small category and let T be a maximal tree in C . Clearly LT is a normal subgroupoid of LC and, since LT is connected, the quotient LC /LT is actually a group, which is easily seen to be canonically isomorphic to Aut LC (c 0 ) for every c 0 ∈ Obj(C ).
Let C and D be connected small categories, let T C and T D be maximal trees in C and D respectively and let F : C → D be a functor such that F (T C ) ⊆ T D . Given c 0 ∈ Obj(C ), there is a commutative diagram where q C and q D are the canonical projections. Now, given a connected small category C , a maximal tree T in C and a group homomorphism α : π 1 (C ) → G, we have a functor F T,α : C → G induced by T and α given by It is not difficult to prove that the morphism induced by F T,α in the fundamental groups is α.
Note that if A is a connected subcategory of C we can choose a maximal tree T A of A and extend it to a maximal tree T of C . Let i : A → C be the inclusion functor and let α : π 1 (C ) → G be a morphism of groups. From the diagram above, it follows that the functor F T A ,αi * : A → G is the restriction of F T,α : C → G. In particular, if i * = 0, or more generally if Im i * ⊆ ker α, then F T,α | A is trivial. If the subcategory A is not connected, the same reasoning applies to each connected component of A.
Now, let X be a connected Alexandroff T 0 -space, let * be the only object of G and let F = F T,α . An explicit computation of the homotopy fiber F/ * shows that Obj(F/ * ) = X × G and, for x, y ∈ X there exists a unique morphism in F/ * from (x, g) to (y, h) if an only if x ≤ y and hF(φ) = g, where φ is the only morphism in X from x to y. Therefore F/ * is a poset and if X is a locally finite T 0 -space then F/ * is the poset associated to a locally finite T 0 -space. The projection p : F/ * → X is easily seen to be a covering of spaces so F/ * is a cover of X.
The change of fiber g * : F/ * → F/ * induced by g ∈ G is just the isomorphism (x, h) → (x, gh) and hence, by Quillen's theorem B, π 1 (F/ * ) ∼ = ker α and π 0 (F/ * ) = G/Im α. In particular, F/ * is connected if and only if α is an epimorphism, in which case F/ * is the (connected) cover of X corresponding to ker α.

Poset splitting
In this section we will introduce the poset splitting technique and give many interesting results that can be obtained from its application. Moreover, we will develop the necessary results to achieve in the next two sections the main goal of this article: a thorough study of the minimal finite models of the real projective plane, the torus and the Klein bottle which, in its first step, gives answers to two open questions.
The splitting technique consists in dividing a poset X into two (not necessarily connected) subspaces, say C and its complement, and from homotopical (or homological) information of C and X − C obtain homotopical or homological information of X. This seems odd at first sight since one would expect a different situation, such as covering the space X with two open subspaces. However, applying lemma 2.5 we will be able to extend arguments for open covers {U, V } to any cover {C, D} of X.
Before getting into the details of this extension, note that if X is a poset and U and V are open subspaces of X such that X = U ∪ V then X is the non-Hausdorff homotopy pushout [11, Definition 2.1] of the diagram U ←֓ U ∩ V ֒→ V and thus, by Thomason's theorem [18], |K(X)| is the homotopy pushout of the diagram |K(U )| ←֓ |K(U ∩ V )| ֒→ |K(V )| (cf. [11]). Note also that (|K(X)|; |K(U )|, |K(V )|) is a CW-triad. Hence, several tools may be applied to obtain homotopical information of X from homotopical information of U and V . And considering opposite spaces one can apply the previous argument also if U and V are closed subspaces of X.
However, the requirement that U and V be open (or closed) subspaces of X is too restrictive to tackle some combinatorial problems such as the minimality of finite models. To get rid of this hypothesis we may apply subdivision of posets as follows.
Let C and D be subspaces of X such that X = C ∪ D. Then C ′ and D ′ are open subspaces of X ′ and X ′ = (X ′ − C ′ ) ∪ (X ′ − D ′ ). By the argument above, we might be able to obtain homotopical information of X from X ′ − C ′ and X ′ − D ′ , which by lemma 2.5, are weak homotopy equivalent to (X − C) ′ and (X − D) ′ respectively. Hence, if C and D are disjoint subspaces of X such that X = C ∪ D one may expect to obtain homotopical information of X from C ′ and D ′ , and hence from C and D.
As a first simple example consider the following result.
Proposition 3.1. Let X be an Alexandroff T 0 -space. Suppose that there exist subspaces C, D ⊆ X such that X = C ∪ D and such that the morphisms H 1 (C) → H 1 (X) and H 1 (D) → H 1 (X) induced by the inclusion maps are trivial. Then H 1 (X) is a free abelian group.
Proof. We may assume that C, D X. Observe that the inclusion ( , considering the opposite spaces we obtain the following portion of the Mayer-Vietoris exact sequence A more sophisticated example of application of poset splitting is given by the following theorem about the homology groups of a cover of a poset which is obtained combining the splitting idea with theorem 5.1 of [9] and with the arguments of coverings of posets given in the second part of section 2. Theorem 3.2. Let X be a connected Alexandroff T 0 -space. Let C X be a subspace and let i : X − C → X be the inclusion map. Let G be a group and let α : π 1 (X) → G be an epimorphism such that i * (π 1 (X − C, x 0 )) ⊆ ker α for all x 0 ∈ X − C. Let p : X → X be a covering map (with X connected) such that p * (π 1 ( X)) = ker α. Then, Proof. By lemma 2.5, taking the barycentric subdivisions of the posets X and C we may suppose that C is an open subspace of X and that X is a quasicellular poset and hence that (X, C) is a relative quasicellular pair. By the arguments in the previous section, there exists a functor F : X → G such that F| X−C is trivial and such that the homotopy fiber F/ * is a locally finite T 0 -space which is the covering space of X corresponding to the subgroup ker α of π 1 (X). We may assume that X = F/ * .
Let ρ be a quasicellular morphism for (X, C). Note that ( X, p −1 (C)) is a relative quasicellular pair with quasicellular morphism induced by ρ.
Since F| X−C is trivial we obtain that where d n and d n are the differentials defined in 2.2 for the relative quasicellular pairs (X, C) and ( X, p −1 (C)) respectively. The result follows.
From this theorem we obtain the following corollary which will be applied later.
X be a subspace such that the inclusion of each connected component of X − C in X induces the trivial morphism between the fundamental groups. Let p : X → X be the universal cover of X. Then This corollary motivates the following definition.
Definition 3.4. Let X be a connected Alexandroff T 0 -space and let C and D be nonempty subspaces of X such that X = C ∪ D. We say that the 3-tuple (X; C, D) is a splitting triad if the inclusion of each connected component of D in X induces the trivial morphism between the fundamental groups.
Observe that a splitting triad (X; C, D) gives rise to a triad (X; C, X − C) that splits X into two disjoint subspaces satisfying the hypothesis of the previous corollary. In this case, (X; C, X − C) is also a splitting triad.
We will be specially interested in splitting triads (X; C, X − C) such that the subspace C satisfies additional hypotheses. Under adequate conditions we will be able to obtain concrete homotopical information of X as the following Hurewicz-type theorem shows.
Theorem 3.5. Let n ∈ N with n ≥ 2 and let (X; C, D) be a splitting triad such that • π r (C, c 0 ) ∼ = H r (C) = 0 for all 2 ≤ r ≤ n and for all c 0 ∈ C • π 1 (C, c 0 ) is an abelian group for all Proof. Let p : X → X be the universal cover of X.
Let i : C → X be the inclusion map. Since π 1 (C, c 0 ) is an abelian group for all c 0 ∈ C and i * : Hence, for every connected component C ′ of C and every connected component D of p −1 (C ′ ) we obtain that p| D : D → C ′ is a universal cover and thus D is n-connected since π r (C, c 0 ) = 0 for all 2 ≤ r ≤ n and for all c 0 ∈ C.
Therefore, from the long exact sequence in homology for the topological pair ( X, p −1 (C)) we obtain that H r ( X) ∼ = H r ( X, p −1 (C)) for all 2 ≤ r ≤ n.
On the other hand, since the inclusion of each connected component of X − C in X induces the trivial morphism between the fundamental groups, from 3.2 we obtain that Finally, H r (X, C) ∼ = H r (X) for all 2 ≤ r ≤ n since H r (C) = 0 for all 2 ≤ r ≤ n and i * : Note that the hypotheses of the previous theorem are fulfilled if all the connected components of C are 2-connected.
The following is a simple example of application of 3.5.
Example 3.6. Consider the poset Z defined by the following Hasse diagram We wish to compute π 2 (Z). Clearly, the inclusion of each connected component of Z − U a in Z induces the trivial morphism between the fundamental groups, and since U a is contractible, the previous theorem applies and yields Now, it is easy to verify that H 2 (Z) ∼ = Z (cf. example 5.4 of [9]). On the other hand, the fundamental group of Z is also easy to compute. Considering the open sets U a ∪ U c and U b and applying the van Kampen theorem we obtain that Here, the first isomorphism holds since the map π 1 (U b ∩(U a ∪U c )) → π 1 (U a ∪U c ) induced by the inclusion is trivial as it can be factorized through π 1 (U a ). And the second isomorphism holds since U c ∪ {a} can be obtained from U a ∪ U c by removing beat points.
The following theorem is another example of application of splitting triads. As a corollary we will obtain a generalization of a result of R. Brown.
Theorem 3.7. Let X be a connected Alexandroff T 0 -space. Suppose that there exist nonempty subspaces C, D ⊆ X such that X = C ∪ D and such that the inclusions of each connected component of C and D in X induce the trivial morphism between the fundamental groups. Then π 1 (X) is a free group.
Proof. We may assume that C, D X. Since X − C ⊆ D then the inclusion of each connected component of X −C in X induces the trivial morphism between the fundamental groups. Thus, by lemma 2.5, the inclusion of each component of C ′ and X ′ − C ′ in X ′ induces the trivial morphism between the fundamental groups. It follows that, without loss of generality, we can assume that X is a locally finite T 0 -space and that C is an open subspace of X.
Choosing a maximal tree on every connected component of C and X − C we obtain a forest T which can be extended to a maximal tree T of X. Consider the functor F = F T ,Id : X → π 1 (X) associated to T and the identity morphism Id : π 1 (X) → π 1 (X). Since the inclusion of each connected component of C and X − C induces the trivial morphism between fundamental groups, by the argument in page 7 we obtain that F is trivial on C and X − C.
We define an equivalence relation on X as follows. Two objects x, y ∈ Obj(X) are equivalent if and only if they both lie in the same connected component of C or X − C and two arrows α, β ∈ Mor(X) are equivalent if and only if their domains and codomains are respectively equivalent and F(α) = F(β). For x ∈ Obj(X) and α ∈ Mor(X) we will denote the equivalence class of x by [x] and the equivalence class of α by [α].
Let C be the category whose objects are the equivalence classes of objects of X and whose morphisms are the equivalence classes of arrows of X, that is, Observe that since F is trivial on every connected component of C and X − C, then the domain and codomain of every non-identity morphism in C are the class of an element of C and the class of an element of X − C respectively. In particular, C does not have non-trivial compositions. It follows that every non-degenerate simplex of the (simplicial) nerve of C has dimension at most 1 and hence the classifying space of C is homotopically equivalent to a wedge sum of circumferences. Consequently the fundamental group of C is a free group.
Let φ : X → C be the function that maps every object and every morphism of X to its class. It is easy to check that φ is a well defined functor from X to C.
It is clear that F factors through C as F = F φ. Since F induces the identity on π 1 (X) we obtain that φ * : π 1 (X) → π 1 (C) is a monomorphism. And since π 1 (C) is a free group, it follows that π 1 (X) is also a free group as desired.
It is not hard to prove that the categories LC and LX/LT of the previous proof are isomorphic groupoids, and that π 1 (X) ∼ = π 1 (C).
It is important to observe that results 3.2, 3.3, 3.5 and 3.7 also hold for arbitrary topological spaces replacing the corresponding splitting triads by excisive triads (X; X 1 , X 2 ) such that the inclusion X 2 ֒→ X induces the trivial map between the fundamental groups for any choice of the base point of X 2 . Indeed, such an excisive triad may be replaced by a weak equivalent CW-triad (Z; A, B) such that Z is a regular CW-complex, to obtain then a splitting triad (X (Z); X (A), X (B)).
In particular, 3.7 yields the following generalization of corollary 3.7 of [7].
Corollary 3.8. Let X be a path-connected topological space, let x 0 ∈ X and let X 1 and X 2 be subspaces of X such that their interiors cover X and such that for j ∈ {1, 2} the inclusion of each path-connected component of X j in X induces the trivial morphism between the fundamental groups. Then, π 1 (X, x 0 ) is a free group.
And from 3.5 we obtain the following generalization of Hurewicz's theorem.
Theorem 3.9. Let X be a topological space and let n ∈ N with n ≥ 2. Suppose that there exist non-empty subspaces C, D ⊆ X such that their interiors cover X and such that they satisfy the following: • The subspace C is n-connected.
• The inclusion of each connected component of D into X induces the trivial map between the fundamental groups. If l ∈ N is such that 2 ≤ l ≤ n and H j (X) = 0 for all 2 ≤ j ≤ l − 1 then π j (X) = 0 for all 2 ≤ j ≤ l − 1 and π l (X) Splitting properties. We will now define certain properties on splitting triads of finite and connected T 0 -spaces which will be called splitting properties. This properties are essential for the results of the next two sections since they will provide the link between Topology and Combinatorics that is needed to solve the problems on minimality of finite models we address in this article.
Definition 3.10. Let X be a finite and connected T 0 -space and let (X; C, D) be a splitting triad. We define the following properties on (X; C, D): (S1) The inclusion of each connected component of C in X induces the trivial morphism between the fundamental groups, that is, the triad (X; D, C) is also a splitting triad. (S2) At most one of the connected components of C is weak homotopy equivalent to S 1 and the rest of them are homotopically trivial. We will also say that a finite and connected T 0 -space X satisfies (S1) (resp. (S2)) if there exists a splitting triad (X; C, D) that satisfies (S1) (resp. (S2)).
Note that, by 3.7, if X satisfies (S1) then π 1 (X) is a free group. The following remark sums up easy facts about conditions (S1) and (S2). Recall that if X is a finite poset, the height of X, denoted h(X), is one less than the maximum cardinality of a chain of X. Note that h(X) = dim(K(X)).
Remark 3.11. Let X be a finite and connected T 0 -space.
(2) If h(X) = 1, then X satisfies (S1) and (S2). Indeed, in this case if C is the set of maximal points of X then the splitting triad (X; C, X − C) satisfies (S1) and (S2). (3) If X is the union of two contractible subspaces then X satisfies (S1) and (S2). In particular, if X has at most 2 maximal points (or at most 2 minimal points) then X satisfies (S1) and (S2).
Example 3.12. If X is a finite model of the real projective plane then X does not satisfy (S1) since its fundamental group is not free. In particular, the space P 2 2 of figure 2 of page 15 does not satisfy (S1). However, it satisfies (S2) since the splitting triad (P 2 2 ; P 2 2 − U a 1 , U a 1 ) satisfies (S2). Proposition 3.13. Let X be a finite and connected T 0 -space and let a be a beat point of X. If X − {a} satisfies (S1) (resp. (S2)) then X satisfies (S1) (resp. (S2)).
Proof. Follows from the fact that if a is a beat point of X and C ⊆ X − {a} then either a is a beat point of C ∪ {a} or a is a beat point of X − C.
Proposition 3.14. Let X be a finite T 0 -space such that X is an Eilenberg-MacLane space of type (G, 1). If X satisfies (S2) then H 1 (X) is free abelian and H n (X) = 0 for all n ≥ 2.
Proof. Note that the group G has finite cohomological dimension over Z since |K(X)| is a finite aspherical CW-complex with fundamental group isomorphic to G. Hence, G is torsion-free.
Let (X; C, D) be a splitting triad that satisfies (S2) and let i : C → X be the inclusion map. If the inclusion of each connected component of C in X induces the trivial map between the fundamental groups then, from 3.7 we obtain that π 1 (X) is a free group. Thus, X is homotopy equivalent to a wedge of circumferences and hence H 1 (X) is free abelian and H n (X) = 0 for all n ≥ 2.
Let p : X → X be the universal cover of X.
Since i * : π 1 (C, c 0 ) → π 1 (X, c 0 ) is a monomorphism for all c 0 ∈ C, for every connected component C ′ of C and every connected component D of p −1 (C ′ ) we obtain that p| D : D → C ′ is a universal cover and thus D homotopically trivial since π r (C, c 0 ) = 0 for all r ≥ 2 and for all c 0 ∈ C.
On the other hand, since the inclusion of each connected component of X − C in X induces the trivial morphism between the fundamental groups, from 3.3 we obtain that Hence, H r (X, C) = 0 for all r = 1 and H 1 (X, C) is free abelian. Thus, H 1 (X) is free abelian and H n (X) = 0 for all n ≥ 2.
From the previous proposition we obtain the following corollary which will be used in section 6 to characterize the minimal finite models of the torus and the Klein bottle. Condition (S2) permits to give yet another Hurewicz-type theorem which will be applied in example 3.17.
Proposition 3.16. Let X be a finite and connected T 0 -space such that π 2 (X) = 0 and such that X satisfies (S2). If n ≥ 3 is such that H l (X) = 0 for all 3 ≤ l ≤ n − 1 then π l (X) = 0 for all 3 ≤ l ≤ n − 1 and π n (X) = H n (X) ⊗ Z[π 1 (X)]. In particular, Proof. Let (X; C, X − C) be a splitting triad that satisfies (S2) and let p : X → X be the universal cover of X. For every connected component C ′ of C and every connected component D of p −1 (C ′ ) we obtain that p| D : D → C ′ is a covering map and thus D is either homotopically trivial or weak homotopy equivalent to S 1 . In any case, H r (D) = 0 for all r ≥ 2. Therefore, from the long exact sequence in homology for the topological pair ( X, p −1 (C)) we obtain that H r ( X) ∼ = H r ( X, p −1 (C)) for all r ≥ 3.
On the other hand, since the inclusion of each connected component of X − C in X induces the trivial morphism between the fundamental groups, from 3.2 we obtain that H r (X) for all r ≥ 3. The result follows.
The following example shows a a finite and connected T 0 -space which satisfies (S1) and does not satisfy (S2). Recall that S 0 denotes the 0-sphere.
Note that X is a finite model of S 1 × S 3 . It follows that π 1 (X) ∼ = Z, π 2 (X) = 0 and π 3 (X) ∼ = Z. Thus, from 3.16 we obtain that X does not satisfy (S2). Now, let w be a minimal element of SS 0 and let C = {w} × S 3 S 0 . It is easy to check that C and X − C are both homotopy equivalent to S 3 S 0 . Thus X satisfies (S1).

Setting up
In this short section we will introduce the specific notation that will be used in the next two sections. We will also define two relations which will be applied to study the problems of minimality of finite models.
If X is a poset, mxl(X) and mnl(X) will denote the subsets of maximal and minimal points of X respectively. Also, if a ∈ mxl(X), we define V X a = U X a ∪ mnl(X) and W X a = U X a ∪ mnl(X) ∪ mxl(X).
Definition 4.2. Let X be a finite and connected T 0 -space. We define We shall frequently write m, n and l in place of m X , n X and l X when there is no risk of confusion.
Note also that, if X does not have beat points then α b ≥ 2 and β b ≥ 2 for all b ∈ B X by 4.1.
Finally, if m ∈ N, D m will denote the discrete space of cardinality m.

Minimal finite models of the real projective plane
In this section we will show that the poset of figure 1(b) below is a minimal finite model of the real projective plane, answering an open question of Barmak [3,p. 94]. Indeed, we will prove a stronger result which states that if X is a poset with fewer than 13 points then X satisfies (S1) (see definition 3.10 above) and thus, by 3.7, π 1 (X) is a free group. In addition, we will prove that if X is a poset with fewer than 13 points then H n (X) is torsion-free for all n ∈ N settling a conjecture of Hardie, Vermeulen and Witbooi [13,Remark 4.2].
Furthermore, we will show that the poset of figure 1(b) below and its opposite are the only minimal finite models of the real projective plane.
Recall that the finite model of the real projective plane of figure 1  and was also obtained in [13] from a finite version of the mapping cone. This poset will be denoted by P 2 1 and the real projective plane will be denoted by P 2 . Clearly, the space (P 2 1 ) op is another finite model of P 2 and will be called P 2 2 (see figure  2(b)). It can also be obtained as the face poset of the regular CW-complex structure of P 2 given in figure 2(a). To prove the minimality of these finite models we need the following lemmas.
Lemma 5.1. Let X be a finite T 0 -space which does not satisfy condition (S1). Let a ∈ mxl(X). Then there exist two incomparable elements a is a (possibly empty) chain and hence the connected components of X − V X a are contractible. Since the connected components of V X a are also contractible we see that X satisfies condition (S1). Now, if X − W X a = {b 1 , b 2 } and z ∈ U b 1 ∩ U b 2 then (X; V X a , F z ∪ mxl(X)) is a splitting triad which satisfies (S1).
Lemma 5.2. Let X be a finite and connected T 0 -space. If X does not satisfy condition (S1), then for all (x, y) ∈ M X there exists b ∈ B X such that b R X (x, y).
Proof. Suppose that there exists (x, y) ∈ M X such that R −1 X (x, y) = ∅ and let G y = F y ∪ mxl(X). Then V x ∪ G y = X. Since the connected components of the subspaces V x and G y are contractible we obtain that the splitting triad (X; V x , G y ) satisfies (S1). Hence, X satisfies (S1). Lemma 5.3. Let X be a finite and connected T 0 -space without beat points. Suppose that X does not satisfy condition (S1). Then, and the equality holds if and only if α b = 2 for all b ∈ B X and #S −1 X (a) = 2 for all a ∈ mxl(X).

The result follows.
Lemma 5.4. Let X be a finite and connected T 0 -space without beat points. Suppose that X does not satisfy condition (S1). Then, and the equality holds only if α b = β b = 2 for all b ∈ B X .
Lemma 5.5. Let X be a finite and connected T 0 -space without beat points. Suppose that X does not satisfy condition (S1) and that there exist c, d ∈ B X such that c < d. Then and the equality holds only if α b = β b = 2 for all b ∈ B X .
Proof. As in the previous proofs, #mxl(X) ≥ 3 and #mnl(X) ≥ 3. Since c < d, Therefore, applying 5.2 we obtain that #R X ≥ m X n X + (m X − α c )(n X − β d ). On the other hand, The result follows.
The following theorem is one of the main results of this article.
Theorem 5.6. Let X be a finite and connected T 0 -space which does not satisfy condition (S1). Then #X ≥ 13.
Proof. By 3.13 we may assume that X does not have beat points. Since #X = m X + n X + l X , from 5.4 we obtain that #X ≥ m X + n X + m X n X (m X −2)(n X −2) . Let a = m X − 2 and b = n X − 2. Then where the last inequality holds by the inequality of arithmetic and geometric means, with equality holding if and only if a = b = 2. Thus, if m X = 4 or n X = 4 we obtain that #X ≥ 13 as desired. Suppose now that m X = n X = l X = 4. By 5.5, B X must be an antichain and from 5.4 we obtain that α b = β b = 2 for all b ∈ B X . From 5.3 it follows that #S −1 X (a) = 2 for all a ∈ mxl(X). Thus, #(U x ∩ B X ) = 2 for all x ∈ mxl(X). Let x 1 ∈ mxl(X) and suppose that U is contractible and hence X satisfies (S1), which contradicts the hypothesis. Thus That is to say that X−mnl(X) is homeomorphic to SD 2 ∐SD 2 . In a similar way we obtain that X −mxl(X) is homeomorphic to SD 2 ∐SD 2 . And since X is connected, without loss of generality we may assume that U b 1 = U b 3 = {y 1 , y 3 } and U b 2 = U b 4 = {y 2 , y 4 }, where y 1 , y 2 , y 3 and y 4 are the minimal points of X. Now note that the connected components of {x 1 , x 2 , x 3 , x 4 , b 1 , b 4 } and {b 2 , b 3 , y 1 , y 2 , y 3 , y 4 } are contractible and hence X satisfies (S1), which contradicts the hypothesis.
As a corollary of the previous theorem we obtain a purely mathematical proof for a result of Adamaszek which states that the first homology group of any poset with at most 12 elements is torsion-free [1]. We ought to mention that Adamaszek gave a computeraided proof which needed to analyse approximately 90 million cases and took more than 200 hours of processor time.
Theorem 5.7. Let X be a finite T 0 -space such that #X ≤ 12. Then π 1 (X) is a free group. In particular, H 1 (X) is torsion-free.
Proof. Without loss of generality, we may assume that X is connected. By 5.6, X satisfies (S1). Thus, by 3.7, π 1 (X) is a free group. Hence, H 1 (X) is torsion-free.
As a corollary of the results above we obtain an affirmative answer to the question about the minimality of a finite model of the real projective plane raised in [3, Example 7.1.1].
Corollary 5.8. Let X be a finite model of the real projective plane. Then #X ≥ 13.
We will now prove a conjecture of Hardie, Vermeulen and Witbooi which states that posets with fewer than 13 points do not have 2-torsion in integral homology (cf. [13,Remark 4.2]). Indeed, we will prove the following stronger result.
Theorem 5.9. Let X be a finite T 0 -space such that H k (X) has torsion for some k ∈ N. Then #X ≥ 13.
Proof. We may assume that X does not have beat points and that the integral homology groups of every proper subspace of X are torsion-free. Choose k ∈ N such that H k (X) has torsion. By 5.7 we may assume that k ≥ 2.
Let p ∈ X. From [9, Proposition 3.2] we obtain that there is a long exact sequence and H k−1 ( C p ) are free abelian by our assumptions we obtain that H k (X) is free abelian. Thus, H k ( C p ) = 0. In particular, h(C p ) ≥ k + 1 and hence every point p ∈ X belongs to a chain of height k + 1. Let a ∈ mxl(X). Suppose that X − U a is an antichain. Then, by [9, Proposition 3.2] which is torsion-free by one of the assumptions above. Hence, X − U a is not an antichain. Thus, X − U a − mxl(X) = ∅. Let d ∈ mxl(X − U a − mxl(X)). By 4.1, there exists b, c ∈ mxl(X) such that b = c, b ≥ d and c ≥ d. Note that b = a and c = a. Applying the same reasoning as above we obtain that X − U b is not an antichain and thus there exists a point e ∈ mxl(X − U b − mxl(X)). Again, by 4.1 we obtain that #(F e ∩ mxl(X)) ≥ 2. Let X 1 = mxl(X) ∪ mxl(X − mxl(X)). If F e ∩ mxl(X) {a, b, c} then #mxl(X) ≥ 4 and thus #X 1 ≥ 6. Otherwise, F e = {a, c} and since X − U c is not an antichain, there exists a point f ∈ mxl(X − U c − mxl(X)) which must, of course, be different from d and e. Therefore, #X 1 ≥ 6 in any case.
Let X 2 = mnl(X) ∪ mnl(X − mnl(X)). Applying the argument above to X op , we obtain that #X 2 ≥ 6. Since H k (X) has torsion it follows that h(X) ≥ k + 1 ≥ 3. And since every point of X belongs to a chain of height k + 1 we obtain that X 1 ∩ X 2 = ∅. Therefore, #X ≥ 12.
Suppose now that #X = 12. Then X = X 1 ∪ X 2 and #X 1 = #X 2 = 6. In particular, h(X) = 3 and then k = 2. Following the previous argument and with the notation used before, we have that either F e ∩ mxl(X) {a, b, c} or F e = {a, c}.
Suppose that F e ∩ mxl(X) {a, b, c}. It follows that #mxl(X) = 4. Let g be the only element of F e ∩ mxl(X) − {a, b, c}. Suppose that d ≤ g. Since X − U g is not an antichain, there exists a point h ∈ mxl(X − U g − mxl(X)) which is clearly different from d and e. Note that h ∈ X 1 . Therefore, #X 1 ≥ 7 which entails a contradiction. Therefore d ≤ g.
In a similar way we obtain that e ≤ c.
Thus, F e ⊆ {a, g} and hence, by 4.1, F e = {a, g}. Therefore, X 1 is the following poset which will be called V . But, in this case, if y is a maximal point of X 2 then ∅ = F y ⊆ X 1 and then the connected components of F y are contractible (since every subspace of V satisfies this). Thus, C y = U y ⊛ F y is homotopy equivalent to the non-Hausdorff join of two discrete spaces and hence H 2 ( C y ) = 0, which entails a contradiction.
Therefore, F e = {a, c} and from 4.1 we obtain that F f = {a, b}. Then X 1 is the following poset • e • f which will be called H. Hence, the poset X 1 is homeomorphic to H and thus, the poset X 2 is homeomorphic to H op which is homeomorphic to H. By the previous argument, if y is a maximal point of X 2 then F y can not be homotopy equivalent to a discrete space. And since F y ⊆ X 1 the only possibility is that F y = X 1 for all y ∈ mxl(X 2 ). It follows that X = X 2 ⊛ X 1 , that is, X is the following poset It is easy to prove that U is contractible. Then, by [9, Proposition 3.2], which is free abelian by our initial assumptions. Thus, this case is not possible either. Therefore, #X ≥ 13. Now we will prove that there are just two finite models of P 2 with 13 points: the spaces P 2 1 and P 2 2 of figures 1(b) and 2(b) respectively. Theorem 5.10. Let X be a finite T 0 -space such that X is a finite model of the real projective plane and #X = 13. Then X is homeomorphic to either P 2 1 or P 2 2 . Proof. By 5.8, X does not have beat points. Also, by considering the opposite space X op , we may assume that #mxl(X) ≤ #mnl(X). Note also that #mxl(X) ≥ 3 by 3.11 and that mxl(X) ∩ mnl(X) = ∅ since X is connected and #X ≥ 2.
We will consider several cases.
In a similar fashion, we obtain that there exist incomparable points b 3 , b 4 ∈ U a 1 ∩ U a 3 − U a 2 and incomparable points b 5 , b 6 ∈ U a 2 ∩ U a 3 − U a 1 which are not minimal in X. Note that {b 1 , b 2 , b 3 , b 4 , b 5 , b 6 } is an antichain with six elements.
Since #X = 13, we obtain that #mnl(X) = 4. Let mnl( Therefore, applying 4.1 we obtain that #(U b 1 ∩ mnl(X)) = #(U b 2 ∩ mnl(X)) = 2 and U b 1 ∪ U b 2 = mnl(X) and thus F c 1 contains exactly one of the points b 1 , b 2 . A similar result holds for the points b 3 and b 4 and for the points b 5 and b 6 .
Without loss of generality, we may assume that b 2 , b 4 and b 6 belong to F c 4 . We will prove now that #( is homotopy equivalent to {b 1 , b 3 , c 1 , c 2 } which is included in the contractible subspace {b 1 , b 3 , c 1 , c 2 , a 1 }. Thus, the inclusion of X − F c 4 in X induces the trivial map between the fundamental groups and hence X satisfies (S1). If c 3 b 5 then c 1 ≤ b 5 and c 2 ≤ b 5 and thus the connected components of X − F c 4 are {c 3 } and {b 1 , b 3 , b 5 , c 1 , c 2 }. Note that the latter is included in the contractible subspace {b 1 , b 3 , b 5 , c 1 , c 2 , a 1 , a 3 }. Hence, X satisfies (S1). Therefore, In a similar way, #(U b 1 ∩ U b 5 ) = 1 and #(U b 3 ∩ U b 5 ) = 1. Hence, without loss of generality we may assume Note that in this case X is homeomorphic to the space P 2 2 of figure 2(b). Case 2: #mxl(X) = 4. Let a 1 , a 2 , a 3 and a 4 be the maximal points of X. Case 2.1: #mnl(X) = 4. Then #B X = 5. If there exist d 1 , d 2 ∈ B X such that d 1 < d 2 then, from 5.5, we obtain that α b = β b = 2 for all b ∈ B X . Hence, F d 1 ∩ mxl(X) = F d 2 ∩ mxl(X). Without loss of generality, we may assume that F d 1 ∩ mxl(X) = {a 1 , a 2 }.
Since X does not have beat points, F d 1 does not have minimum element. Thus, #mnl( F d 1 ) ≥ 2 and it is easy to prove that mnl( F d 1 ) ⊆ B X . Let e 1 , e 2 be distinct elements of mnl( F d 1 ). Note that F e 1 ∩ mxl(X) = F e 2 ∩ mxl(X) = {a 1 , a 2 }. In a similar way, #mxl( U e 1 ) ≥ 2 and mxl( U e 1 ) ⊆ B X . Thus, #(U a 1 ∩ B X ) ≥ 4 which contradicts 5.1.
Therefore, B X must be an antichain. Since #(F b ∩ mxl(X)) ≥ 2 for all b ∈ B X it follows that there exists a ∈ mxl(X) such that #(U a ∩ B X ) ≥ 3. Without loss of generality, we may assume that #(U a 1 ∩ B X ) ≥ 3. By 5.1, there exist incomparable points b 1 , b 2 ∈ X − W X a 1 . Hence, #(U a 1 ∩ B X ) = 3 and B X − U a 1 = {b 1 , b 2 }. Thus, U b 1 ∩ U b 2 = ∅ by 5.1. Hence, from 4.1 it follows that #(U b 1 ∩ mnl(X)) = 2 and #(U b 2 ∩ mnl(X)) = 2. Now note that, by 4.1, is contractible and hence X satisfies (S1). Thus #(F b 1 ∩ F b 2 ) ≥ 2. Without loss of generality we may assume that It follows that U a j ⊇ mnl(X) for j ∈ {3, 4}. We will prove now that the same holds for j ∈ {1, 2}. Note that #(U a ∩ B X ) ≤ 3 for all a ∈ mxl(X) by 5.1. If #(U a k ∩ B X ) = 3 for some k ∈ {3, 4} then applying the reasoning above to a k shows that U a j ⊇ mnl(X) for , or there exist distinct points z 1 , z 2 ∈ mnl(X) such that each of them belongs to exactly two of the subsets U b 3 , U b 4 and U b 5 and such that {b 3 , b 4 , b 5 } ⊆ F z 1 ∪ F z 2 . Note that in both cases the subspace {a 1 , a 2 , b 3 , b 4 , b 5 } is included in a contractible subspace of X. It follows that the splitting triad (X; X −V a 4 , V a 4 ) satisfies (S1) which entails a contradiction. Therefore, U a ⊇ mnl(X) for all a ∈ mxl(X). Now, note that if b ∈ B X is such that α b ≥ 3 and β b ≥ 3 then the connected components of X − C b are contractible and hence X satisfies (S1). Thus, for all b ∈ B X either α b = 2 or β b = 2. Hence, Thus, it follows that the Euler characteristic of X is Hence, X can not be a finite model of the real projective plane. Case 2.2: #mnl(X) ≥ 5. From 5.3 it follows that #B X = 4 and that α b = 2 for all b ∈ B X and #S −1 X (a) = 2 for all a ∈ mxl(X). Since #B X = 4 we obtain that #mnl(X) = 5. From 5.5 it follows that B X must be an antichain. Proceeding as in the proof of 5.6 we obtain that X − mnl(X) is homeomorphic to SD 2 ∐ SD 2 .
Without loss of generality we may assume that Let i ∈ {1, 2} and j ∈ {3, 4}. If #(U b i ∩ U b j ) ≤ 1 then the connected components of {b i , b j }∪mnl(X) and its complement are contractible and hence X satisfies (S1). Therefore Case 3: #mxl(X) ≥ 5. Let a 1 ∈ mxl(X). By 5.1, there exist incomparable points b 1 , b 2 ∈ X − W X a 1 . Moreover, we may assume that F b 1 ∩ F b 2 = ∅ since F y ∩ F z = ∅ for all y, z ∈ X − W X a 1 implies that the connected components of X − V X a 1 are contractible and hence X satisfies (S1).
Let a 2 ∈ mxl(X) such that a 2 ∈ F b 1 ∩ F b 2 . By 5.1, there exist incomparable points b 3 , b 4 ∈ X − W X a 2 which must, of course, be different from b 1 and b 2 . Therefore #X ≥ 14.

Minimal finite models of the torus and the Klein bottle
In this section we will prove that the space SS 0 × SS 0 is a minimal finite model of the torus, answering another open question of Barmak [3,p. 44]. This will be obtained as a corollary of theorem 6.3 which states that if a poset X does not satisfy (S2) then #X ≥ 16. From this theorem we will also infer that there do not exist finite models of the Klein bottle with fewer than 16 points.
With further development of our techniques we will also find all the minimal finite models of the torus and the Klein bottle, which is a much harder task. Indeed, we will prove that there exist exactly two minimal finite models of the torus: the spaces T 2 0,0 and T 2 1,1 of figures 3(b) and 5(b). It is worth mentioning that the space T 2 1,1 was not previously known and appears naturally in the proof of 6.10. In the same proof, we will also show that there exist exactly two minimal finite models of the Klein bottle: the space K 1,0 of figure 4(b) and its opposite, the space K 0,1 of figure 6(b).
We will begin by describing the aforementioned finite models of the torus and the Klein bottle. Consider the regular CW-complex structure of the torus T 2 given in figure 3(a) with 2-cells a i , 1-cells b j , and 0-cells c k , where the 0-cells and the 1-cells are identified in the standard way. The face poset of this regular CW-complex, given in figure 3(b), is a finite model of T 2 . We will refer to this space as T 2 0,0 , name that follows from the proof of theorem 6.10 below. This finite space is homeomorphic to SS 0 × SS 0 .
In a similar way, we can construct the finite model of the Klein bottle of figure 4(b) as the face poset of the regular CW-complex structure of figure 4(a). The Klein bottle will be denoted by K. As was mentioned above, another finite model of the torus will be found in the proof of 6.10: the space T 2 1,1 of figure 5(b). Observe that T 2 1,1 is the face poset of the regular CW-complex structure of T 2 given in figure 5(a) On the other hand, it is clear that K op 1,0 is another finite model of the Klein bottle. It is easy to check that K op 1,0 is homeomorphic to the space K 0,1 of figure 6(b), which is the face poset of the regular CW-complex structure of K given in figure 6(a).
We will give now two lemmas that will be used several times in this section.
Lemma 6.1. Let X be a finite and connected T 0 -space without beat points. If X does not satisfy condition (S2), then for all (x, y) ∈ M X either #R −1 Proof. As in the proof of 5.2 we obtain that R −1 X (x, y) = ∅ for all (x, y) ∈ M X . Suppose that there exist (x, y) ∈ M X and b ∈ B X such that R −1 X (x, y) = {b} and α b = 2. By 2.3 F b ∪ F y is homotopy equivalent to S(F b ∩ F y ). If F b ∩ F y is connected then the connected components of F b ∪ F y ∪ mxl(X) have trivial fundamental group and hence (X; V x , F b ∪F y ∪mxl(X)) is a splitting triad that satisfies (S2), which entails a contradiction. Thus . Then F b ∩ F y has, at most, two maximal points. But since F b ∩ F y is not connected, F b ∩ F y has exactly two maximal points each of which must be the maximum of its connected component. Thus F b ∩ F y is homotopy equivalent to D 2 and hence F b ∪ F y is homotopy equivalent to SD 2 . Then, the splitting triad (X; F b ∪ F y ∪ mxl(X), V x ) satisfies (S2). Lemma 6.2. Let X be a finite and connected T 0 -space without beat points. Let B ′ X = {b ∈ B X / min{α b , β b } ≥ 3} and let B ′′ X = B X − B ′ X . Suppose that X does not satisfy condition (S2). Then, with the notations above, ≥ 2m X n X and the equality holds only if α b = β b = 2 for all b ∈ B ′′ X . In addition, if the equality holds and min{m X , n X } ≥ 4 then α b = β b = 3 for all b ∈ B ′ X . Proof. Note that #mxl(X) ≥ 3 and #mnl(X) ≥ 3 by 3.11.
Let M ′ X = {(x, y) ∈ M X / #R −1 X (x, y) = 1} and let M ′′ X = M X − M ′ X . By 6.1, On the other hand, The result follows.
The following theorem is another major result of this article. Theorem 6.3. Let X be a finite and connected T 0 -space without beat points and which does not satisfy condition (S2). Then #X ≥ 16.
Proof. As before, #mxl(X) ≥ 3 and #mnl(X) ≥ 3 by 3.11. Now, by 6.2, where the penultimate inequality holds by the inequality of arithmetic and geometric means.
, let a = m − 2 and b = n − 2. Then where, as before, the penultimate inequality holds by the inequality of arithmetic and geometric means. Therefore, #X ≥ 16.
As a corollary of the previous theorem we obtain an affirmative answer to the question posed in [3, p.44] about the minimality of SS 0 × SS 0 as a finite model of the torus. Proof. Let X be a finite model of the torus. Without loss of generality we may suppose that X does not have beat points. From 3.15 we obtain that X does not satisfy (S2). Therefore, #X ≥ 16 by 6.3. Corollary 6.5. Let X be a finite T 0 -space. If X is a finite model of the Klein bottle then #X ≥ 16.
Proof. By 3.15, X does not satisfy (S2). The result follows from 6.3. Now we will find all the minimal finite models of the torus and the Klein bottle. Let X be a finite and connected T 0 -space without beat points which does not satisfy (S2). For each a ∈ mxl(X) we define Remark 6.6. Note that from 6.1 it follows that if X is a finite and connected T 0 -space without beat points which does not satisfy (S2) then #S −1 X (a) ≥ 2 for all a ∈ mxl(X). Lemma 6.7. Let X be a finite and connected T 0 -space without beat points which does not satisfy (S2). Let a ∈ mxl(X).
a are contractible and hence the splitting triad (X; X − V X a , V X a ) satisfies (S2). Thus, F b 1 ∩ F b 2 = ∅ and {b 1 , b 2 } is an antichain. Hence, the connected components of Thus, β b 2 ≥ 3 by 6.1. In a similar way we obtain that β b 1 ≥ 3.
Lemma 6.8. Let X be a finite and connected T 0 -space without beat points which does not satisfy (S2) and such that m X ≥ 4. Let a ∈ mxl(X). Then σ a ≥ min Lemma 6.9. Let X be a finite and connected T 0 -space without beat points which does not satisfy condition (S2) and such that m X ≥ 4. Then If, in addition, n X ≤ 5 then #B X ≥ 2m X m X −3 .
Proof. Note that The result now follows from 6.8.
The following theorem describes all the minimal finite models of the torus and the Klein bottle.
Theorem 6.10. Let X be a finite T 0 -space such that #X = 16.
(1) If X is a finite model of the torus then X is homeomorphic to either T 2 0,0 or T 2 1,1 .
(2) If X is a finite model of the Klein bottle then X is homeomorphic to either K 1,0 or K 0,1 .
Proof. Suppose that X is either a finite model of the torus or a finite model of the Klein bottle. Hence, by 3.15, X does not satisfy (S2). Thus, by 3.11, #mxl(X) ≥ 3. Also, mxl(X) ∩ mnl(X) = ∅ since X is connected. Without loss of generality, we may assume that #mxl(X) ≤ #mnl(X). From 6.4 and 6.5 we obtain that X does not have beat points. Case 1: #mxl(X) = 3. By 6.2, #B X ≥ 6n n−2 . Thus, where the penultimate inequality holds by the inequality of arithmetic and geometric means. Case 2: #mxl(X) = 4. Case 2.1: #mnl(X) = 4 or #mnl(X) = 5. From 6.9 we obtain that #B X ≥ 8. Hence, #B X = 8 and #mnl(X) = 4. From 6.2 we obtain that 2#B ′ X + 4#B ′′ X ≥ 32. Hence, #B ′′ X = 8, #B ′ X = 0 and the equality holds. Thus, B ′′ X = B X and from 6.1 and the proof of 6.2 it follows that α b = β b = 2 for all b ∈ B X and #R −1 X (x, y) = 2 for all (x, y) ∈ M X . We will prove now that #(U a ∩ B X ) ≤ 4 for all a ∈ mxl(X). Let a ∈ mxl(X). Let y 1 ∈ mnl(X) and let b 1 and b 2 be the elements of R −1 X (a, y 1 ). Since y 1 / ∈ U b 1 ∪ U b 2 and β b 1 = β b 2 = 2 it follows that there exists y 2 ∈ mnl(X) such that y 2 ∈ U b 1 ∩ U b 2 . Since #R −1 X (a, y 2 ) = 2, it follows that #S −1 X (a) ≥ 4 and hence #(U a ∩ B X ) ≤ 4. Since α b = 2 for all b ∈ B X and #(U a ∩ B X ) ≤ 4 for all a ∈ mxl(X) we obtain that #(U a ∩ B X ) = 4 for all a ∈ mxl(X). In a similar way we obtain that #(F y ∩ B X ) = 4 for all y ∈ mnl(X). And since #R −1 X (x, y) = 2 for all (x, y) ∈ M X it follows that #(U x ∩ F y ∩ B X ) = 2 for all (x, y) ∈ M X . In particular, x > y for all (x, y) ∈ M X .
We claim that B X is an antichain. Indeed, suppose that there exist b 1 , b 2 ∈ B X with b 1 < b 2 . Since α b 1 = α b 2 = 2 we obtain that F b 1 ∩ mxl(X) = F b 2 ∩ mxl(X). Since b 1 is not an up beat point of X, there exists b 3 ∈ F b 1 ∩ B X such that b 3 = b 2 . As before, we obtain that F b 3 ∩ mxl(X) = F b 1 ∩ mxl(X). Let x 0 ∈ F b 1 ∩ mxl(X) and let y 0 ∈ U b 1 ∩ mnl(X). Then, {b 1 , b 2 , b 3 } ⊆ U x 0 ∩ F y 0 ∩ B X which entails a contradiction.
We will prove now that #(U a 1 ∩ U a 2 ∩ B X ) ≤ 3 for all a 1 , a 2 ∈ mxl(X) with a 1 = a 2 . Let a 1 , a 2 ∈ mxl(X) with a 1 = a 2 and suppose that #(U a 1 ∩ U a 2 ∩ B X ) = 4. Thus U a 1 ∩ B X = U a 2 ∩ B X . Let a 3 and a 4 be the remaining maximal points of X. It follows that U a 3 ∩B X = U a 4 ∩B X = B X −U a 1 since α b = 2 for all b ∈ B X . Note that #(F y ∩U a j ∩B X ) = 2 for all y ∈ mnl(X) and for all j ∈ {1, 3}. Let w ∈ mnl(X) and let A = F w ∪ B X . Since H 0 (A) = 0 there is an epimorphism H 1 (X) → H 1 (X, A). From proposition 3.2 of [9] we obtain that H 1 (X, A) ∼ = z∈X−A H 0 ( F z ) ∼ = Z 3 . Thus, rk(H 1 (X)) ≥ 3 which can not be possible since X is a finite model of either the torus or the Klein bottle. Therefore, #(U a 1 ∩ U a 2 ∩ B X ) ≤ 3 for all a 1 , a 2 ∈ mxl(X) with a 1 = a 2 .
Applying this argument to X op we obtain that #(F y 1 ∩ F y 2 ∩ B X ) ≤ 3 for all y 1 , y 2 ∈ mnl(X) with y 1 = y 2 .
We will prove now that indeed #(U a 1 ∩ U a 2 ∩ B X ) ≤ 2 for all a 1 , a 2 ∈ mxl(X) with a 1 = a 2 . Let a 1 , a 2 , a 3 and a 4 be the maximal points of X and suppose that #(U a 1 ∩ U a 2 ∩ B X ) = 3. Let b 1 , b 2 and b 3 be the elements of U a 1 ∩ U a 2 ∩ B X , let b 4 be the only element of U a 1 − U a 2 and let b 5 be the only element of U a 2 − U a 1 . Let b 6 , b 7 and b 8 be the remaining elements of B X . Clearly, Let c 2 and c 3 be the elements of U b 4 ∩ mnl(X). Then, for each j ∈ {2, 3} we have that # Let c 1 and c 4 be the remaining minimal points of X.
Then W is contractible and hence U a 1 ∪U a 2 is contractible by 2.3. Thus, from 2.4 it follows that there exists a finite model of either the torus or the Klein bottle with fewer than 16 points, which contradicts either 6.4 or 6.5. Therefore, #(F W c 1 ∩ F W c 4 ) = 2. In a similar way, it follows that #(F c 1 ∩F c 4 ∩{b 6 , b 7 , b 8 }) = 2. Hence, #(F c 1 ∩F c 4 ∩B X ) = 4 which entails a contradiction.
Since there are six pairs of maximal points and eight points in B X and α b = 2 for all b ∈ B X we obtain that there exist distinct points a 1 , a 2 ∈ mxl(X) such that #(U a 1 ∩ U a 2 ∩ B X ) = 2. Let a 3 and a 4 be the remaining maximal points of X. Observe that Let b 1 and b 2 be the elements of U a 1 ∩ U a 2 ∩ B X , let b 3 and b 4 be the elements of U a 1 − U a 2 , let b 5 and b 6 be the elements of U a 2 − U a 1 and let b 7 and b 8 be the elements of U a 3 ∩ U a 4 ∩ B X . Since α b 3 = α b 4 = 2, without loss of generality we may assume that one of the following two cases hold: In the first case, we obtain that b 5 < a 4 and b 6 < a 4 and thus X −mnl(X) is homeomorphic to the following space which will be called Q 0 .
In the second case, without loss of generality, we may assume that b 5 < a 3 and b 6 < a 4 and thus X − mnl(X) is homeomorphic to the following space which will be called Q 1 .
Now, we claim that in any of the cases above, since , 2, 3, 4} we obtain that z < b 7 and z < b 8 . Thus, if z 1 and z 2 are distinct elements of U b 1 ∩ U b 2 then #(F z 1 ∩ F z 2 ∩ B X ) = 4 which entails a contradiction. Therefore, which is the disjoint union of two contractible spaces. Hence, by 2.3, U a 1 ∪ U a 2 is homotopy equivalent to SD 2 . Note that the inclusion of X − (U a 1 ∪ U a 2 ) = {a 3 , a 4 , b 7 , b 8 } into X induces the trivial map between the fundamental groups since X − (U a 1 ∪ U a 2 ) is included in the contractible subspace {a 3 , a 4 , b 7 , b 8 , z 1 }. Therefore, U b 1 ∩ U b 2 = ∅ as desired. Clearly, the same conclusion holds for the pair b 7 , b 8 in both cases and for the pairs b 3 , b 4 and b 5 , b 6 in the first case. Applying the same reasoning to X op it follows that if y 1 , y 2 ∈ mnl(X) are such that F y 1 ∩ F y 2 ∩ B X = {b ′ , b ′′ } (with b ′ = b ′′ ) then F b ′ ∩ F b ′′ = ∅.
There are now two possibilities: c 3 < b 5 or c 3 < b 6 . If c 3 < b 5 then c 4 < b 6 and X − mxl(X) is homeomorphic to Q op 0 . In this case X is homeomorphic to the space T 2 0,0 of page 22. If c 3 < b 6 then c 4 < b 5 and X − mxl(X) is homeomorphic to Q op 1 . In this case X is homeomorphic to the space K 0,1 of page 24.

Open questions
In this section, we will exhibit finite models for very well-known families of spaces and raise open questions about their minimality. 7.1. Real projective spaces.
Notation. For n ∈ N we will write P n for the n-dimensional real projective space. Proof. Let Λ be the finite T 0 -space whose underlying set is {−1, 0, 1} with the topology associated to the order generated by 0 > 1 and 0 > −1.
Let n ∈ N. Let S n be the subspace of the product Λ n+1 obtained by removing its maximum (0) n+1 i=1 . We define the following relation in S n : It is easy to see that ∼ is an equivalence relation. Let P n = S n / ∼ with the topology induced by the canonical projection q : S n → P n . It is not difficult to prove that P n is a T 0 -space and that U Pn q(a) = q(U Sn a ) for all a ∈ S n . Consider the function ϕ : P n → P n that sends [(x i ) n+1 i=1 ] to q (sgn(x i )) n+1 i=1 . It is clear that ϕ is well defined. We claim that ϕ is a weak homotopy equivalence, and thus P n is a finite model of P n .
Let p be the canonical projection of R n+1 − {0} on P n and let a = q((a i ) n+1 i=1 ) ∈ P n . For 1 ≤ i ≤ n + 1 we define It is easy to prove that ϕ −1 (U a ) = p(J) and that p(J) is contractible and open in P n . By McCord's theorem ( [15, theorem 6]) it follows that ϕ is a weak homotopy equivalence.
Remark 7.2. The space P 1 constructed in the previous proof is homeomorphic to SS 0 , and hence it is the minimal finite model of P 1 ∼ = S 1 . It is easy to check that the space P 2 is homeomorphic to the space P 2 1 of figure 1(b) which is a minimal finite model of P 2 by 5.8. The previous remark leads naturally to the following question. Open question 7.3. Let n ∈ N, n ≥ 3. Is the space P n of the proof of 7.1 a minimal finite model of P n ? 7.2. Tori.
Notation. For n ∈ N we will write T n for the n-dimensional torus  Proof. Let X = SS 0 which is a finite model of S 1 with four points. Then |K(X n )| ≃ |K(X)| n ∼ = n i=1 S 1 ∼ = T n and hence X n is a finite model of T n with 4 n points.
Remark 7.5. Note that SS 0 is a minimal finite model of T 1 ∼ = S 1 with four points. Note also that the space SS 0 × SS 0 is homeomorphic to the space T 2 0,0 of figure 3(b) which, by 6.4, is a minimal finite model of T 2 with 16 points.
As above, the previous remark leads to the following question.
Open question 7.6. Let n ∈ N, n ≥ 3. Is the space (SS 0 ) n a minimal finite model of T n ? 7.3. Moore spaces M (Z n , k).
Proposition 7.7. Let n, k ∈ N with n ≥ 2. There exists a finite model of M (Z n , k) with 4n + 2k + 3 points.
Proof. Let C n be a regular 2n-gon with vertices c 0 , . . . , c 2n−1 and center d. Let c 2n = c 0 . We regard C n as regular CW-complex with 2-cells c t c t+1 d for 0 ≤ t ≤ 2n − 1, 1-cells c t d and c t c t+1 for 0 ≤ t ≤ 2n − 1 and 0-cells d and c t for 0 ≤ t ≤ 2n − 1.
We obtain a CW-model of M (Z n , 1), which will be denoted X n , by identifying (directed) edges c 2t c 2t+1 for 0 ≤ t ≤ n − 1 and c 2t−1 c 2t for 1 ≤ t ≤ n.
Now, X n inherits from C n a regular CW-complex structure which has 2n 2-cells, 2n + 2 1-cells and 3 0-cells. Let Z n be the face poset of X n . It follows that Z n is a finite model of M (Z n , 1) which has 4n + 5 points. Hence S k−1 Z n is a finite model of M (Z n , k) with 4n + 2k + 3 points.
Remark 7.8. The space X 2 constructed in the proof above is homeomorphic to P 2 . Moreover, it has the regular CW-complex structure of figure 1(a) and thus, the space Z 2 is homeomorphic to the space P 2 1 of figure 1(b). By 5.8, Z 2 is a minimal finite model of M (Z 2 , 1). Open question 7.9. Let n, k ∈ N with n ≥ 2. If n ≥ 3 or k ≥ 2, is the space S k−1 Z n a minimal finite model of M (Z n , k)?
We finish this article stating a conjecture which is related to the previous question and to the conjecture of Hardie, Vermeulen and Witbooi that we proved in this work (cf. 5.9). Conjecture 7.10. Let X be a finite T 0 -space and let k ∈ N such that k ≥ 2. If H k (X) has torsion then #X ≥ 2k + 11.
Clearly, if this conjecture is true then the finite models of M (Z 2 , k) constructed above will be minimal finite models and thus we will obtain an affirmative answer to the case n = 2 of the previous question.