Reconstruction of Compacta by Finite Approximations and Inverse

Home , Compact space, Finite topological space, Hausdorff space, Metric space

Reconstruction of compacta by ﬁnite

approximations and1 Inverse Persistence2 Departamento de Matemática Aplicada y Estadística Diego Mondéjar Ruiz and Manuel A. Morón 1 Universidad San Pablo CEU, Madrid, Spain [email protected] Departamento de Álgebra,Geometría y Topología

2 Universidad Complutense de Madrid and Instituto de Matematica Interdisciplinar, Madrid, Spain [email protected]

Abstract

The aim of this paper is to show how the homotopy type of compact metric spaces can be reconstructed by the inverse limit of an inverse sequence of ﬁnite approximations of the corresponding space. This recovering allows us to deﬁne inverse persistence as a new kind of persistence process.

1 Introduction In this paper we deal with approximations of metric compacta. The approximation and reconstruction of topological spaces using simpler ones is an old theme in geometric topology. One would like to construct a very simple space as similar as possible to the original space. Since it is very diﬃcult (or does not make sense) to obtain

arXiv:1802.04331v3 [math.GT] 27 Feb 2018 a homeomorphic copy, the goal will be to ﬁnd an space preserving some (algebraic) topological properties such as compactness, connectedness, separation axioms, homotopy type, homotopy and homology groups, etc.

The first candidates to act as the simple spaces reproducing some properties of the original space are polyhedra. See the survey [41] for the main results. In the very beginnings of this idea, we must recall the studies of Alexandroff around 1920, relating the dimension of compact metric spaces with dimension of polyhedra by means of maps Mathematics Subject Classification. withKey controlled words and (in phrases. terms of distance) images or preimages. In a more modern framework, 2010 54B20, 54C56, 54C60, 54D10, 55P55, 55Q07, 55U05. Finite topological space, Alexandroff, shape theory, homotopy, persistence..

1 the idea of approximation can be carried out constructing a simplicial complex, based on our space, such as the Vietoris-Rips complex or the Čech complex, and compare its realization with it. In this direction, for example, we find the classical Nerve Lemma [13, 21] which claims that for a “good enough" open cover of the space (meaning an open covering with contractible or empty members and intersections), the nerve of the cover has the homotopy type of our original space. The problem is to find those good covers (if they exist). For Riemannian manifolds, there are some results concerning its approximation by means of the Vietoris-Rips complex. Hausmann showed [34] that the realization of the Vietoris-Rips complex of the manifold, for a small enough parameter choice, has the homotopy type of the manifold. In [37], Latschev proved a conjecture made by Hausmann: The homotopy type of the manifold can be recovered using only a (dense enough) finite set of points of it, for the Vietoris-Rips complex. The results of Petersen [55], comparing the Gromov-Hausdorff distance of metric compacta with their homotopy types, are also interesting. Here, polyhedra are just used in the proofs, not in the results.

Another important point, concerning this topic, are finite topological spaces. It could be expected that finite topological spaces are too simple to capture any topological property, but this is far from reality and comes from thinking about finite spaces as discrete ones. Another obstructionT to the use of finite spaces is that a very basic observation reveals that they have very poor separation properties. Any finite topological 1 space satisfying just the axiom of separation is really a discrete space. Since non- Hausdorff spaces seem to be less manageable, finite spaces could represent themselves a more difficult problem to study that the spaces we want to approximate with. There were two papers of Stong [60] and McCord [46] that were a breakthrough in finite topological spaces. Stong studied the homeomorphism and homotopy type of finite spaces.core Among other results, he showed that the homeomorphism types are in bijectiveT correspondence with certain equivalence classes of matrices and that every finite space 0 has a , which is homotopy equivalent to it. McCord defined a functor from finite spaces to polyhedra preserving the homotopy and homology groups (defining a weak homotopy equivalence between them). This is a very important result, since we obtain that the homotopy and homology groups of every compact polyhedron can be obtained as the groups of a finite space. The essential property of finite spaces, making possible this result, is thatT the arbitrary intersection of open sets is open (every space satisfying this property is called Alexandroff space) and hence they have minimal basis. If the 0 finite space is , the minimal basis gives the space a structure of a poset (first noticed in [2]) which is used in the cited result. Both papers were retrieved in a series of very instructive notes by May [45, 44], where these results are adequately valued. Based on the theorems and relations proved in those papers, Barmak and Minian [10, 11, 8] introduced a whole algebraic topology theory over finite spaces.

2 One step further is to make use of the inverse limit construction. If we cannot obtain the desired approximation using only one simple space, we can try to obtain it as some kind of limit of an infinite process of refinement by good spaces. That idea is accomplished by the notion of inverse limit. It is similar in spirit to the use of the Taylor series to approximate a function. For the origins of using inverse limits to approximateT compacta, we should go back, again, to the work of Alexandroff [1], where it is shown 0 that every compact metric space has an associate inverse sequence of finite spaces such that there is a subspace of the inverse limit homeomorphic to the original one. We also have to mention Freudenthal, who showed [30] that every compact metric space is Hausdorffthe inverse reflection limit of an inverse sequence of polyhedra. More recent resultsT were obtained by Kopperman et al [35, 36].calming They showed map that every compact Hausdorff space is the 0 of the inverse limit of an inverse sequence of finite spaces. Also they define the concept of and show that if the maps in this sequence are calming, then an inverse sequence of polyhedra can be associated and its limit is homeomorphic to the original space. Those are good results, although the technical concepts of Hausdorff reflection and calming map, make its real computation hard to achieve. AnotherT important result is the one obtained by Clader [19], who proved that every compact polyhedron has the homotopy type of the inverse limit of an inverse 0 sequence of finite spaces.

Shape theory makes use of this notion of approximation by inverse limits. This theory was founded in 1968 with Borsuk’s paper [14]. It is a theory developed to extend homotopy theory for spaces where it does not work well, because of its pathologies (for example, bad local properties). Although Borsuk’s original approach does not make explicit use of inverse limits, they are in the underlying machinery. The idea of Bor- suk was to enlarge the set of morphisms between metric compacta by embedding the spaces into the Hilbert cube and define some kind of morphisms between the open neighborhoods of those embedded spaces. Later, Mardesic and Segal initiated in [42] the inverse system approach to Shape theory. Here, the approximative sense of Shape theory is clear: Every compact Hausdorff space can be written as the inverse limit of an inverse system (or an inverse sequence if the space is metric) of compact ANR’s, which act as the good spaces. Then, the new morphisms are essentially definedexpansions as maps betweenresolutions the systems. Shape theory, in its invese system approach, is then defined and developed [43] for more general topological spaces and new concepts, as and , have to take the role of the inverse limit for technical reasons, but the point of view is similar. It is evident that the inverse limit approximation point of view for spaces is closely related with Shape theory. There are several shape invariants. Among others, we have the Čech homology, which is the inverse limit of the singular homology groups and the induced maps in homology of the inverse system defining the shape of the space.

3 In the last years, there has been a renewed interest in the approximation and reconstruction of topological spaces, in part because the development of the Computational Topology and more concretely the Topological Data Analysis (read the excellent survey of Carlsson [17] as an introduction for this topic). Here, the idea is to recapture the topological properties of some spacepoint using cloud partial or defective (sometimes called noisy) information about it. Usually we only know a finite set of points and the distances between them (this is known as ) which is a sample of an unknown topological space, and the goal is to reconstruct thewitness topologyDelaunay of the space or, at least, alphabe able shapes to detect some topological properties. Besides the classical Vietoris-Rips and Čech complexes, several other complexes (as the , complexes or the [25]) are defined with this purpose. Some important results in this setting were obtained by Niyogi et al [54, 53], where they give conditions to reconstruct the homotopy type and the homology of the manifold when only a finite set of points (possibly with noise) lying in a submanifold of some euclidean space, is known. They also use probability distributions in their results. There are a large amount of recent papers devoted to this kind of reconstructions. For instance, Attali et al [7], in a more computational approach, give conditions inpersistent which a Vietoris-Rips homology complex of a point cloud in an euclidean space recovers the homotopy type of the sampled space. Among other techniques, we have to highlight the . The idea here is as easy as effective: Instead of considering only one polyhedron based on the point cloud to recover the topology of the hidden space, consider a family of polyhedra constructed from the data and natural maps induced by the inclusion connecting them. Then, we do not choose one concrete resolution to analyze the point cloud, but we consider all possible values of the parameter and their connections at once and use them together to determine the evolution of the topology of the point cloud along the parameter changes.

The first link between Shape theory and Persistent Homology was made in 1999 bypersistent Vanessa Robins Betti number [57]. There, she proposes to use the machinery of shape theory to approximate compact metric spaces from finite data sets. She introduced the concept of , which is the evolution of the Betti numbers in the inverse sequence of polyhedra at different scales (or resolution)ε of approximation. Her approach is the following: Given a sample (finite setε of points, possibly with noise)α of an unknown topological space, construct an inverse system of -neighborhoods of the finite set and inclusion maps. Then, triangulate the -neighborhoods using the -shapes and we obtain an inverse system of polyhedra based on the sample. Then, track the Betti numbers over this system. For some examples arising from dynamical systems, she is able to give bounds for the behavior of the Betti numbers, when the resolution parameter tends to infinity, and hence the sample is more accurate. Her guess is that the more accurate the sample is, the more exactness in the prediction can be made, and is here where shape theory is proposed as a theory to support this and other similar methods.

4 main construction

In 2008, following1 this direction, Morón et al [3] introduced what they called the . This is an inverse sequence of finite topological spaces constructed from more and more tight approximations of a given compact metric space. The finite spaces are not exactly the approximations but some subspaces of the hyperspace of the approximations with the upper semifite topology. This is necessary in order to define continuous maps between these approximations. These maps are defined in terms of proximity betweenAlexandroff-McCord points of consecutive correspondence approximations. Hence, they are not the inclusion (because the finiteT spaces are not necesarilly nested). At this point, they make use of the so called , which is the functor assigning 0 a polyhedron to every finite space, mentioned above. The functoriality is used to define maps between the induced polyhedra and hence we obtain an inverse sequence of polyhedra. The way that this sequence is constructed, using finite approximations, induces them togeneral conjeture principle that the inverse limit of the inverse sequence of polyhedra is somehow related with the topology of the original compact metric space. This conjecture is stated as the , proposing this sequence to detect the shape properties of the space such as the Čech homology. Our work is placed here, understanding and expanding the properties of the main construction. In a forthcoming paper [48] it is shown that this inverse sequence represents this space in terms of shape, namely that it is an HPol-expansion of the original space, and hence it recovers all the shape information about it.

We show here that the inverse limit of every inverse sequence of finite spaces defined by the main construction has the homotopy type of the original space, and it contains an homeomorphic copy as a (strong) deformation retract. We identify explicitly this subspace. After that, we study some properties of the main construction and the result of performing the main construction to some specific classes of spaces as dense subspaces, countable and ultrametric spaces. For the last, we obtain that in this case we can choose a suitable construction such that the inverse limit of the finite spaces is homeomorphic to the ultrametric space. We compare our results with that of Clader and Kopperman et al (previously cited). We can obtain Clader’s result as a corollary of our main theorem. In the other case, their approximations are made for Hausdorff compact spaces, and we do not obtain their generality. In contrast, for the case of metric compacta,T we obtain the same consecuences, and we also deduce that every compact metric space has the 0 homotopy type of the inverse limit of an inverse sequence of finite spaces. The same resultT has been generalized very recently by Bilski [12] in a more generalT setting, namely for the class of locally compact, paracompact, Hausdorff spaces, using inverse 0 0 systems of Alexandroff spaces instead of inverse sequences of finite spaces. The inverse system is defined in terms of the partial order defined in the set of all locally finite open coverings of the space and is related in some sense to the construction 1 This was not the first paper of this research group in this topic. From another point of view, this theme is treated in [33].

5 in [36]. As a consecuence, the results of Clader [19] are generalized using barycentric subdivisions of Alexandroff spaces. Also, we show that the HausdorffT reflection preserves the shape type and hence the results of Kopperman et al implies that every Hausdorff 0 compact space has the same shape as an inverse sequence of finite spaces. The main construction allows us to outline an algorithm to obtain persistence modules as finite inversesequences persistence extracted from an inverse sequence of polyhedra. These persistence modules are obtained in a different way from the usual ones so we call this new point of view . We outline the basics of this process in section 4 and perform it in our paradigmatic example, the Warsaw circle. The constructibility of our process makes it suitable for computational purposes. All the results shown here are part of the first authors’ thesis dissertation [49].

The rest of this section is devoted to present the basic elements needed in the paper.

1.1 Shape Theory Shape theory is a suitable extension of homotopyW theory for topological spaces with bad local properties, where this theory does not give any information about the space. x ; π The paradigmatic example is the Warsaw circle : It is the graph of the function 1 2 ; − sin ;in the interval (0 ] adding its closure (that is, the segment joining (0 1) and (0 1)) and closing the space by; − any simpleπ ; (not intersecting itself or the rest of the space) arc joining the pointsW (0 1) and ( 2 1). See ﬁgure 1. It is readily seen that the fundamental group of is trivial. Moreover,W so are all its homology and homotopy groups. But it is also easy to see that has not the homotopy type of Figure 1: The Warsaw circle. a point (for example, it decomposes the plane in two connected components), so it W has some homotopy type information that the homotopy and homology groups are not able to capture. It is then evident that homotopy theory does not work well for . Shape theory was initiated by Karol Borsuk in 1968 to overcome these limitations, deﬁning a new category, containing the same information about well behaved topological spaces, but giving some information about spacesQ with bad local properties. The idea is that, no matter how bad the spaceW is, its neighborhoods when it is embedded intoS a larger spaceW (for example the Hilbert cube ) are notS too bad. In our example, it is easy1 to seeS thatW the neighborhoods of are annuli, having1 then the homotopy type of . The space1 share some global properties with . There are no non-trivial maps from to , so the method will be to compare them in terms of maps between

6 its neighborhoods. fSpecifically,n Q → Q Borsuk defined a new class of morphism between metric compacta embedded in the Hilbert cube, called fundamental sequences, as sequences of continuous maps : satisfying some homotopy conditions on the neighborhoods of the spaces embedded in the Hilbert cube. He introduced a notion of homotopy among fundamental sequences, setting the shape category of metric compacta as the homotopy classes for this homotopy relation. It is shown that the new category differs only formally from the homotopy category when the space under consideration is an ANR. For the details, see the original source [14], or the books [16, 15]. After Borsuk’s description of the shape category for metric compacta, there was a lot of work in shape theory, such as different descriptions of shape, extensions to more general spaces (for instance, Fox’s extension of shape for metric spaces [29]), classifications of shape types or shape invariants. As general references, we recommend the books [16, 15, 43, 23] and the surveys [39, 40]. The inverse system approach is the most widely used and it will be the one used here. It was initiated by Mardesic and Segal for compact Hausdorff spaces in [42], and it was developed by them and some other authors for more general situations. The best reference for this approach, is the book by the same authors [43] and all the proofs inverse sequence {Xn}n∈N omitted here canp ben foundXn → there.Xn− n ∈ N {Xn; pn;n } An of topological spaces is a countable set of spaces and p ; p ; 1 pn− ;n pn;n pn ;n +1 continuous maps X :←−− X ←−− :::for ←−−− X.n We←−−− denoteXn it←−−−− by ::: or 1 2 2 3 1 +1 +1 +2 1 2 +1 inverse limit X ⊂ X n∈N n x ; x ; : : : ; x ; x ;::: p x x n n n;n n n Q nThe∈ of an inverse sequence is the subset of the product space N 1 2 +1 +1 +1 consisting of the points ( X ) satisfying ( ) = for every

. {Xn; pn} X Given a compact metric space , we consider an inverse sequence of compact ANRs (or polyhedra) with as inverse limit (it always exists). This inverse sequences up to a notion of equivalence are the new objetcs of our category. To define morphisms,2 we consider a map between inverse sequences and, again, a notion of equivalence . Thus we have a new category where essentially we substitute compat metric spaces by their associated inverse sequences. This new category is able to detect some non trivial topological properties that homotopy is not. For example, the homology groups of the warsaw circle described above are trivial, but the equivalent groups in the shape category, named the Čech homology groups are not because they are detected in some way by the inverse sequence representing the warsaw circle. The idea is that we approximate spaces with a poor local behaviourZ with sequences of “nice” spaces. In our example, the inverse sequence is a sequence of circles an the identity map and its first Čech homology group is isomorphic to because it is the inverse limit of the induced inverse sequence of the first homology groups of the inverse sequence representing the 2 We do not include these technical definitions here for the sake of simplicity.

7 warsaw circle. In the shape category, we have equivalent generalizations of homotopy and homology groups. It also have its own invariants such as the movability. This theory can be deﬁned with more complex machinery for every topological space. Hence, we have an extension of the homotopy category, enlarging the set of morphisms. Not every shape morphism is represented by a continuous function, but we have that every continuous function induces a shape morphism. From [38], we have the following useful characterizationLet X forand a functionY be topological to induce spaces an isomorphism and f X in→ theY a shape continuous category. map. Then f is a shape equivalence (that is, the shape morphism induced by f is an isomorphism Theoremin the shape 1. category) if and only if, for every CW-complex: (or equivalently ANR or polyhedron) P, the function

3 f Y;P −→ X;P h 7−→ h · f :[ ] [ ] is a bijection. [ ] [ ]

There is another approach to shape that we shall use here. This is the multivalued theory of shape for metric compacta, initiated by Sanjurjo in [58]. The key and acute idea of multivalued shape theory is to replace the shape morphisms by sequences of multivalued maps with decreasing diameters of their images, which is, in some sense, a very natural way of defining them, but hard to formalize. We do not give the details here, but the idea under this approach will be present in this paper. The equivalence of this definition of shape theory and the usual one is shown in [58]. The importance of this theory lies on the fact that it is internal. That is, we do not make use of external elements (such as the Hilbert cube or polyhedra) to describe the morphisms, as in other shape theories. We just use maps between the metric compacta to define the morphisms. This multivalued theory of shape was reinterpreted later by Alonso-Morón and González Gómez in [6] by the use of hyperspaces with the upper semifinite topology, which seems turns out to be a very adequate topology for our purposes. We define it in the following section.

1.2 Hyperspaces with the upper semifinite topology The idea of hyperspaces isX to define new spaceshyperspace from oldX with a related topology. As a general reference for hyperspaces, we recommend the paper [47] and the book [52]. Given a topological space , weX define{C ⊂ theX C of}: as the set of its non-empty closed subsets, Z;R Z;R 2 = : is closed 3 Z R h Z → R h Notation: For topological spaces , [ ] is the set of homotopy classes of continuous functions from to . For a map : , we represent by [ ] its homotopy class.

8 X X X X ∈ X fat X T There are some distinguished elements of 2 : The{x} subset x is∈ alwaysX a closed{ subspacex} ∈ X 1 of , so it is a point 2 , sometimes named the point. If is , then every point is closed, so we can consider every{{x} singletonx ∈ X} ⊂ ,X with; , as a point 2 . The subset canonical copy X X : 2

X is the of in 2 . Itφ isX the−→ imageH of the inclusion map x 7−→ {x}: : 2

Hausdorﬀ distanceWe can endow hyperspaces withC;D a number∈ X of topologies. If (X,d) is a compact metric space, the most common topology for the hyperspace is the one induced by the

, deﬁned for twodH pointsC;D {ε2 > as C ⊂ Dε;D ⊂ Cε} ;

( ) = inf 0 :

Cε {x ∈ X x; C < ε} where generalized ball ε X X ; = : d( ) H H φ is the φ X of radius . With this metric, 2 = (2 Xd ) is a compact metricX space and the inclusionX map is an isometry. That means, in particular, that the H canonical copy ( ) is homeomorphic to the original space or, in other words, is embedded in 2 . More results about hyperspaces with the Hausdorff metric and its relations with the base space can be seen in [5]. We next define a more general topology for hyperspaces that will be used widely in this paper. It isX a very easy-to-use topology but, on the otherU ⊂ hand,X the hyperspace has very poor topological properties (for example, in general it will be a non-Hausdorff topology). Let be a topologicalB U space.C ∈ ForX C every⊂ U open⊂ setX : , define ( ) = 2 : 2 B {B U U ⊂ X } The family upper semifinite topology X = ( ): open X u T X is a baseC ∈ forX the for the hyperspace 2 , which will be writen 1 2 . The closure operator of this topology is very easy to describe: Given a space and 2 , the closure of the{C set} constistingD ∈ X ofC just⊂ D this: point is = 2 :

The general references for hyperspaces contain only the deﬁnition and some properties for this topology. We add two more references [4, 6] about this topology and some of its properties, that will be used here. We list some of them here.

9 Let X;Y be Tychonoﬀ spaces. We have the following.

X i The set X is the unique closed point in u . Proposition 1. X ii The space u is a compact connected space. ) X2 Y iii X is homeomorphic to Y if and only if u is homeomorphic to u . ) 2 X iv If X is non-degenerate , u is a T but not T space. ) 4 2 2 X 0 T 1 ) 2

1 X In this context, we also have that, ifφ Xis a−→ spaceu the inclusion map x 7−→ {x}; : 2

is a topological embedding. ε > X In the caseX of metric compacta, we have some extra properties. Let (X,d) be a compact X metric space. Consider, for everyUε C ∈0, the subspaceC < of ε2 : consisting of all the closed subsets of = 2 : diam( )

The following result is key in the use of the upper semiﬁnite topology for hyperspaces The family U {U } is a base of open neighborhoods of the canon- in this text and its proof can be foundε ε> in [6]. ical copy φ X inside X . u 0 Proposition 2. = Remark . X ( ) 2 H

Remark 1. This result is also shown for the hyperspace 2 with the Hausdorff metric in [5]. {εn}n∈N {Uεn }n∈N 2X Note that if we consider any decreasing and tending to zero sequence of φ X u positive real numbers , we have that is a nestedf X countable→ Y base of ( ) in 2 . f X Y elevation induced f u → u Consider aX continuousf map between compact metric spaces : . We define C ∈ u C c∈C f c the by as the function 2 : 2 2 defined in the natural way: S 5 For 2 , 2 ( ) = ( ). This is a continuousX;Ymap. Moreover, for every map from a topologicalY space to a hyperspace (of the same space or a differentX Y one), we can f X → u extension F u → u consider an extension to the whole hyperspace. Let be compact metric spaces. If : 2 is a continuous map, itsF C isf x the: function : 2 2 , given by x∈C [ ( ) = ( ) f X X u F It is an extension∗ in the senseY that∗ we can consider that is actually a continuous map f φ X → u f {x} f x f from the canonical copy of in 2 . That is, strictly speaking, would be the extension of the map : ( ) 2 , with ( ) = ( ), which is continuous because is. This X T is Lemma4 3 in [6]. X Y 1 5Actually, just need to be a non-degenerate space to satisfy this property. In weaker topological assumptions for the spaces and , this is not always true.

10 The extension of every continuous map Y f X → u is well deﬁned and continuous. Lemma 1 (Continuity of the extension map). : 2

1.3 Finite spaces and the Alexandroff-McCord correspondence AlexandroffX spaces are topologicalAlexandroff spaces satisfying a topological condition that makes them very special. This notion was introduced by Alexandroff in [2]. Afinite topological space is said to be provided arbitrary intersections of open sets are open. Obviously, the most important case of Alexandroff spaces are the topological ones. Many of the hyperspaces considered in our results are finite. A good reference for Alexandroff and finite topological spaces are the notes of May [45, 44]. We also recommend two papers about Alexandroff and finite spaces [60, 46] that were essential in its development. Finite topological spaces have captured a lot of attention in the last years because of the developments of digital and computational topology. In a series of papers, Barmak and Minian have shown very interesting theorems about the algebraic topology of finite topological spaces (for example, generalizating notions such as collapsibility and simple homotopy type to finite topological spaces). See, for instance, [10, 9, 11] or Barmak’s book [8]. One could have the intuition that a topological space with a finite set of points cannot contain a deep geometric information, but this is shown not to be the case. Concerning Alexandroff spaces, is good to haveT in mind finite topological spaces, for simplicity. WeT can not require too strong separation prop- 1 erties to Alexandroff spaces, because they would turn trivial: An Alexandroff space 0 is discrete. But, on the otherT hand, finite spaces have some geometric interest, since they have, at least, one closed point. Moreover, in terms of algebraic topology, we can 0 consider onlyLet AlexandroffX be an Alexandroffspaces because space. There of the exists following a quotient theoremT ofspace McCordqX [46].X → X homotopically equivalent to X (q is a homotopy equivalence). Moreover, for every 0 Theoremmap between 2. Alexandroff spaces, f X → Y there is a unique map f X →: Y , 0 between T Alexandroff spaces, such that qY f f qX . 0 0 0 : : 0 0 = X x ∈ X Alexandroff spaces and posets The most important property of an Alexandroff space

is that it has a distinguished basis.Bx For every U , we can consider the intersection x∈U \ = x open minimal neighborhood x x of all the open sets containing , which is open and{Bx it isx ∈ calledX} the of X, because, byminimal definition, basis it isX contained in every open set containing . It can be shown, that the set ofX minimalx; neighborhoods, y ∈ X x 6 y : Bx ⊂ Byis a base for the topology of , called the of . This minimal basis defines a reflexive and transitive relation on the space . For , say if . This relation is a partial

11 X T

X Ux {y ∈ X y x} 0 6 order if and only if is . On the other hand, every reflexive and transitive relation on a set determines an Alexandroff topology, with basis the sets = : . For every set, its Alexandroff topologies are in bijective correspondence So, we have the following correspondence. with its reflexive and transitive relations. The topology is T if and only if the relation isProposition a partial order. 3. 0 X 6 poset T We call a set with a partial order a . The last proposition tells us that 0 Alexandroff spaces (sometimes called A-spaces) andf posetsX → Y are the same thing. In what follows we will use both points of view withoutx distinction.6 y Withf x this6 f notation,y continuous maps are easily characterized. A function : of Alexandroff spaces is continuous if and only if is order preserving, that is, implies ( ) ( ).

Alexandroff-McCord correspondence We recall the correspondence proved by Mc-T Cord [46] (wesimplicial call it the complex Alexandroff-McCordvertices correspondenceV becausesimplexes it was AlexandroffK ⊂ V 0 who first worked on it) in which simplicialτ ⊂ complexesσ ∈ K are relatedτ with∈ K Alexandroff spaces.f AK → L is a set of and a finiteK set of L 2 satisfying that any subset of a simplex polyhedronis a simplexX . A simplicial map : is a function between simplicial complexesK and sending vertices to verticesX (and|K | hence simplexes to simplexes). A is af topologicalK → L space obtained as the realization of a simplicial|f complex| |K | → |asL| a subset of an Euclidean space = (see references for details).X EveryK X simplicial map : defines a continuous map betweenX its realizations : turning combinatoricsx 6 ::: 6 intoxs topology. GivenX an A-space space f , defineX → Y ( ) as the abstract simplicial complex 0 Khavingf K asX vertex→ K setY and as simplices the finite totally ordered subsets

ψof theψX poset|K X .| → A continuousX map : z ∈of |K A-spacesX | defines a simplicial map ( ): ( ) (σ ), since it is order preserving. Now, we can definex the6 followingx 6 ::: 6 mapxs = : X( ) as follows.ψ z x Every point ( ) is contained in the interior of 0 1 a unique simplex spanned by a strictly increasing finite sequence The map0 ψ is a weak homotopy equivalence. Moreover, of points of . We define ( ) = , andX the following theorem holds. given a map f X → Y of A-spaces, the induced simplicial map K f makes the followingTheorem diagram3 (McCord commutative. [46]). : ( ) f X / Y O O

ψX ψY

|K X | / |K Y | |K f |

( ) Example . ( ) X {a; b; c; d( })

1 Considerτ the ﬁnite{{a}; space{c}; {a;= c}; {a; b; c}; {witha; c; properd}} : open sets

12 Its minimal basis{B isa {a};Bb {a; b; c};Bc {c};Bd {a; c; d}} :

X = a 6=b; d c 6 b; d= = K X a; b; c; d ha; bi; ha; di; hc; bi; hc; di Hence, is a poset with S , X . The corresponding simplicial complex ( ) has verticesS and1 simplices , whose realization is homeomorphic1 to a sphere . Hence has theK homotopy and singular homologyX K groups of . K σ 6 τ σOn⊂ theτ other direction, given a simplicial complex ,g weK can→ deﬁneL an A-space ( ) whose points are the simplicesX g ofK →and L the relation is given as X K if and only if as simplices. Also, from any simplicial map : itK isX evidentK thatK 0 we obtain a continuous map ( ): of A-spaces. Now, since ( ) in an A-space, we can apply the previous theorem to obtain the simplicial complex ( ( )) = and 0 the weak homotopyφ equivalenceK ψX K |K | |K | |K X K | −→ X K :

( ) = g: X =→ Y = ( ( )) ( )

g Again, for every simplicial map :|K | we have/ that|L| the following diagram commutes up to homotopy. φK φL X K / X L X g

( ) ( ) ( )

So, there is a mutual correspondence of simplicial complexes and A-spaces (or posets) preserving homotopy and singular homology groups.T Note that this means that there are A-spaces with the same homotopy and singular homology groups as every possible 0 simplicial complex. Concretely, there areK ﬁnite spaces with the same homotopy and n singular homology groupsK asX any· · ·K compactX K polyhedron.K n n KNote that, given a simplicial complexX , we can( apply) the correspondencen of theorem n 3 sequentially to obtainn-th( barycentric( ( ( )))) subdivision = theX n -th barycentricK X · · ·K X subdivisionK of . Similarly,X given any A-space , we can apply the correspondence( ) times to obtain what we will call the = ( ( ( ( )))) of the A-space .

1.4 Persistent homology In the last years, the fields of Computational Topology and Applied Algebraic Topology have had a great and successful development. The deep and abstract mathematical concepts and theorems of (Algebraic) Topology have been shown as a very useful tool in real world problems, so the interest of other areas of science in them, is becoming bigger and bigger. As general references for these topics we give the books, [63, 25, 32]. We are interested in the more specific field of Topological Data Analysis. This consists of the

13 study and management of (maybe belonging to real world) data sets using topological constructions and techniques. The excellent surveys [17] by G. Carlsson and [31] by R. Ghrist are strongly recommended for this topic. In particular, we recall the powefull tool of persistent homology. Persistence is an algebraic topological technique used to detect topological features in contexts where we have not all the information about the space or the information we have is somehow noisy. We recommend, besides the general references quoted, the surveys [24, 61]. It is usually agreed that the concept of persistence born in three different ways: Frosini and Ferri’s group, studying the persistence of 0-dimensional homology of functions (using the concept of size function) [28], Vanessa Robins introducing the concept of persistent Betti numbers in a shape theory context to understand the evolution of homology in fractals [57] and Edelsbrunner group [26]. In this paper, we deal with Robin’s approach to persistence with a shape theory perspective, with the use of inverse sequences. X The idea of persistence We illustrate the notion of persistent homology throughX a very schematic example. Consider we have a finite setX of pointsX (and we know the distances between them), possibly as a noisy sample of an unknown topological space . If we want to detect some topological properties of from , one wayVietoris-Rips could be to construct complex a simplicial complex based on this set of points and study its topologicalproperties. For example, in figure 2, we have the Vietoris-Rips complexes (the

Vε XFigure 2: The Vietoris-RipsX complexes ofε a point cloud with two parameters.

X σ ⊂ X Vε X σ < ε ( ) ofX a finite set of points with parameter in a metricX d, is theS simplicial complex 0 with vertex set and< ε < ε is a simplex of ( ) iff diam( ) X 1 ) of a finite set of points , which is a noisy sample of an underlying space = , with two different real parameters 0X . Both detect the main feature of , the central hole or 1-cycle. But we have that none of them really determine the first homology group of ∼ ∼ ∼ the actual spaceH ,V becauseε0 X Z H Vε X Z Z ⊕ Z Z H X Z :

1 1 1 ( ( ); ) = ( ( ); ) = = ( ; )

14 Vε0 X ,→ Vε X

The persistent homology idea is just to consider the inclusion ( ) ( ) and the ∼ ∼ image of the inducedH mapVε on0 X theZ ﬁrst,→ homologyH Vε X Z groups,Z thatH is,X Z

1 1 1 Im ( ( ); ) ( ( ); ) = = ( ; ) X which really captures only the desired feature,ﬁltration ignoring the noise of .

Filtrations In general, suppose∅ weK ,→ haveK a,→ :::,→,K i.e.,s: a ﬁnite sequence of nested simplicial complexes 0 1 = p ∈ N G p We are interested in the topological evolution of the sequence of the homology groups, so, for every and every abelian group , we can consider the induced -th homology ﬁnite sequence{ } Hp K G ,→ Hp K G ,→ :::,→ Hp Ks G :

0 1 0 = ( ; ) ( ; ) ( ; ) p-th persistent homologyAs we move groups forward in the sequence, new homology classes can appear and some could merge or vanish. We collect the homology classes as follows. The ij are the imagesHp of theHp K homomorphismsi G ,→ Hp Kj G induced by inclusion 6 i < j 6 s = Im p(-th; persistent) ( Betti; ) numbers ij ij βp Hp for 0 . Similarly, the persistenceare the diagram ranks of these groups = rank . We can do the same definitions with reduced homology. The collection of persistent Betti numbers can be visualized in a . Given a filtration of simplicial complexes, there are several algorithms determining these numbers and the evolution of the homology classes. See the quoted references for more details. There are several ways of arriving to a filtration of simplicial complexes. We mention • X the main two of them. point cloud A finite set of points and its distances. Given any finite metric spaceX (asε in > the previous example), called a , we can produce filtrationsX of simplicial complexes taking the Vietoris-Rips, Čech or other complexes of ε for every 0. There will be only a finite number of different complexes since is finite, so we • K f K → obtain a filtration of simplicial complexes along the parameter . R τ σ f τ 6 f σ Consider a simplicial complex and a real−∞ valueda < function a < : :: : < as which is − Kmonotonic,i f −∞ meaning; ai thati if; ;is : : :a ; sface of , then K(i ) ( ). Then, supposingK Ki 0 1 the different values of the function are = , if we set 1 Ki i ; ; : : : ; s − Kas K = ( ] for filtration= 1 2 of the, we function havef that are subcomplexes of , +1 is a subcomplex of , for every = 1 2 1, and = . Thus we have a filtration called the .

15 Structure of persistence One step further in the study of persistence isF to find some structure in thepersistence evolution of module the homologyM classes in a given filtration. InMi this direction,F we recall the Structure'i Mi → TheoremMi byi Carlsson∈ N and Zomorodian [62].6 Let be a field. We define a as' a family of vector spaces over and +1 homomorphisms : M , for .M Fori example, the induced homologyR finite sequence of a filtration,m where the maps'i send a homology classi > m to the one containing it. We will say that is of finite type if is a finitely generated -module and there exists an integer persistencesuch that intervalis an isomorphism for i;. Now j we define6 i < the j i;elements j ∈ Z∪{ for∞} the classification which, in some sense, representsbarcode the beginning and end of an homology class. A is an ordered pair ( ), with 0 , + . A finite set of persistence intervals is called a . The following correspondence is stablished.The isomorphism classes of persistence modules of finite type over a field are in bijective correspondence with barcodes. Theorem 4 (Structure).

The proof of this theorem uses some advanced algebra, including the structure theorem of ﬁnitely generated modules and graded modules over pids, which we do not include here for simplicity. For the algebraic machinery used in the proof, see [22]. The importance of this result, which gives a structure to the persistence modules, is that it allows us to use the barcodes, a very intuitive way of representing the evolution of the homology classes, because they really determine the persistence module, up to isomorphism. So they are a good way to represent persistence. On the other hand, this result enables us to modify the standard reduction algorithm for homology using the properties of the persistence module to derive a rather simple algorithm to compute the barcodes. This is implemented in the Matlab routine Plex and in some functions of the library phat in R.

2 The Main Construction Let us recall the main construction introduced in section 6 of [3]. There, given a compact metric space, it is obtained an inverse sequence of finite approximations of the space and someX; sequences of real numbers thatε allow > us to define continuous maps between theA ⊂ approximations.X finite ε-approximation X x ∈ X Let ( d) abe∈ aA compact metricx; space a < ε and 0 a positive real number. A finite subset is said to be a of if, for every , there is at least Remark . ε ε > one point such that d( ) . F R 3 Compact metric spaces have finite -approximations for every 0. Mi 6 The definition still holds if we replace by a commutative ring with unity, obtaining then modules , but we need this stronger condition for the structure theorem.

16 Point Cloud K ⊂ ::: ⊂ Kn Filtration of complexes 1

Hp K → ::: → Hp Kn Persistence Module Barcode 1 ( ) ( )

Figure 3: The processA of⊂ obtainingX a barcode from a pointX; cloud x ∈ X set of closest points of A AGiven a non-empty ﬁnite subset of a compactx metric space ( d), we consider, for each point , the as the subset of the hyperspace of A X consisting of the pointsA x minimizing{a ∈ A thex; a distancex; toA }: ⊂ u ⊂ u :

A A ( ) = : d( X) = d( ) 2 2 u X u ⊂ Note that is a discrete finite subset of , hencenearby the map topologyX of 2A as a subespace of 2 is a finite space with the relation . It is natural, then, to define a function from A the space to its closest sets. We willq callA X the→ u; from to to the function

qA x A x : 2

X X deﬁned by ( ) = ( ). The extensionrA ofu the→ nearbyu : map will be usually written as

: 2 2

Both will beLet shownX; d tobe be a continuous compact metric maps space because and ofA the⊂ X followinga ﬁnite lemma. subset. For every x ∈ X there exists δ > such that, for every y ∈ B x; δ , A y ⊂ A x . Lemma 2. ( ) − Proof. x ∈ X 0 δ (x; A )> ( ) δ( ) x; A\A x > A \ A x ∅ A x A + Let and consider the distances = d( ) 0 and = d( ( )) δ − δ− 0 (if ( ) = , then ( ) = andδ the result is> obvious,: so we will assume that it is not empty). Now, ﬁx + = 0 2

17 a ∈ A x b ∈ A \ A x y ∈ x; δ

− − δ δ If ( ) and y;( a),6 we seey; x that, forx; a every< δ δ B( ), ; + δ x; b x; y y; b < δ y; b : + 6 d( ) 6 d( ) + d( ) + = + 2 d( ) d( ) + d( ) + d( ) δ δ− y; b > > y; a ; Whence + A y ⊂ A x + X d( ) d( ) 2 so ( ) ( )

As an immediate corollary, we obtain the continuity of the nearby map and its extension. Let X; d be a compact metric space and A ⊂ X a ﬁnite subset. The X nearby map qA X → u is continuous. Hence its extension rA is also continuous. Corollary 1. ( ) Proof. :qA 2 x ∈ X δ >

The map satisﬁes that,q forA everyx; δ ⊂ qA,x there; exists 0 such that

qA X (B( )) ( ) Remark . A ε X; hence is continuous x ∈ X 4 If is a ﬁnite -approximation of a compact metric space ( d), then the A A images of the points U ε Aby the{C nearby∈ map areC < sentε} to ⊂ theu; subspace

2 ( ) = 2 : diam 2 2 qA X →

U ε A because of the triangle inequality. That is, the nearby map may be written as : 2 X; ε > ε ( ). A < ε0 < ε adjusted A Let ( d) be a compact metric space. Given a real number 0, and a ﬁnite - ε − γ approximation , we say that 0 ε0

2 γ { x; A x ∈ X} where X A γ < ε = sup d( ): being the supremum of the distances of points of to which obviously satisfies . The following result is the more important concerning the Main Construction. It says that, given any finite approximation of a compact metric space, we always can find a tighter finite approximation of the space and define nearby continuous maps between some finiteLet spacesX; d basedbe a on compact these metricapproximations. space and consider a real number ε > and a finite ε-approximation A of X. For every < ε0 < ε adjusted to A and every finite Lemma 3. ( ) 0 0

18 0 0 0 ε -approximation A , the map p U ε0 A → U ε A , deﬁned by p C rA C , is well deﬁned and continuous. 2 2 : ( ) ( ) ( ) = ( )

By induction, we can repeat the processFor every indeﬁnitely, compact metricobtaining: space X; d , there exist a decreasing sequence of positive real numbers {εn}n∈N tending to zero, a sequence Theorem 5 (Main Construction). ( ) {An}n∈N of ﬁnite εn-approximations of X with εn adjusted to An for every n ∈ N and continuous maps pn;n U ε An → U ε An for every n ∈ . n n +1 N

+1 2 +1 +1 2 : ( ) ( )

p ; p ; pn− ;n pn;n pn ;n Hence,U ε A we←−−− get anU ε inverseA ←−−− sequence::: ←−−−−− of finiteU spacesεn An and←−−−−− continuousU εn A maps:n ←−−−−−− ::: 1 2 2 3 1 +1 +1 +2 2 1 1 2 2 2 2 2 +1 +1 ( ) ( {U) εn An ; pn;n } {εn(}n∈)N ( ) n ∈ An εn X εn An N 2 +1 An inversefinite sequence approximative( sequence) with X converging to zero, satisfying that +1 for every , is a finite -approximation of with adjusted to will be Remark . called a (fas) of . 5 Observe that, given a compact metric space, this process is completely constructive. We can compute all the real numbers and select finite approximations that satisfy the quoted properties. Being an inductive process, we compute the numbers and Remark . approximations in their strictly necessary order. 6 Strictly speaking, a fas will be the inverse sequence of finite spaces quoted {εn;An; γn; }n∈N above. But we will use fas to make reference also to the approximations and the numbers obtained, , because they determine uniquely the finite spaces Remark . and maps of the inverse sequence. 7 Theorem 5 states that every compact metric space has a fas. In general, fas are not unique. n ∈ N U εn An

We canK U nowε A usen the Alexandroff-McCord correspondenceD ∈ U ε An to obtain an inversehD ;D sequence;:::;Dsi n 0 n2 of polyhedra.D ⊂ D For⊂ every::: ⊂ Ds and finite T space ( ), there exists a simplicial com- 2 2 0 1 plex ( ( )) with vertex set the points ( ) and simplexes 0 1 with such that there is a weak homotopy equivalence between the finite space and the geometricfn |K realizationU εn An | −→ of theU ε simplicialn An ; complex

2 2 :x ∈( |K U( εn))An | ( )

σ hD ;D ;:::;Dsi fn x D 2 defined as follows. Every point ( ( )) is contained in the interior of a unique 0 1 0 simplex = , so7 we can define ( ) = . K pn;n We also have simplicial maps between the polyhedra, defined on the vertices and 7 pn;n +1 Following McCords paper’s notation we should write ( ) for the simplicial maps but we will +1 omit this notation, using the same as for the maps between the finite spaces, , for the sake of simplicity.

19 extendedp asn;n usualK toU simplices:εn An −→ K U εn An

+1 2 +1 +1D 7−→ pn;n 2 D : ( ( )) ( ( )) hD ;D ;:::;Dsi 7−→ hpn;n+1 D ; pn;n D ; : : : ; pn;n Ds i ( ) 0 1 +1 0 +1 1 +1 ( ) ( ) ( )

D ⊂ D ⊂ ::: ⊂ Ds; where, if 0 1

pn;n D ⊂ pn;n D ⊂ ::: ⊂ pn;n Ds : then +1 0 +1 1 +1 n ∈ ( ) ( ) ( ) N

2 +1 2 +1 +1 ( ) ( ) commutes. So we obtain an inverse sequence of polyhedra and a map between the

1 2 +1 2 1 1 2 2 2 2 +1 2 +1 ( ) ( ) ( ) ( ) {|K U εn An ; |pn;n |} Polyhedral Approximative Sequence X 2 +1 Every inverse sequence of polyhedra ( ( )) obtained in this way is called a (or pas) of the space . The analysis of these inverse sequences is proposed in [3] as a reconstruction process of topological properties of a compact metric space. We will show here that every fas is able to recover the homotopy type of the original space. The shape type reconstruction using pas is show in [48]. We propose the use of this sequences as an alternative way of producing persistence modules from a point cloud in a process that will be called inverse persistence.

3 Homotopical reconstruction by fas In this section, we prove the main result concerning the reconstruction of compact metric spaces. It asserts that in the inverse limit of every fas we can ﬁnd the homotopy structure of our space.

20 Let X be a compact metric space. The inverse limit of every

X {U A ; p } Theorem 6. ←− εn n n;n fas

2 +1 contains a subspace X ∗ ⊂ X homeomorphic= lim ( to X) which is a strong deformation retract of X.

{U εn An ; pn;n } X εn γn In order to prove this result, we{ needεn;A somen; γn; } technicaln∈ lemmasn about∈ the aboveεn construc- 2 +1 N N εn−γn tion.ε Considern a fas ( ) εnof; aεn compact< εn metricεn spaceεn εn and the sequences+ of numbers with usual notation . For every , we write = 2 and = For2 . every Theyn clearly < m, we satisfy have and + = .

m Lemma 4. γl < εn: l n X

= εn−γn Proof. n ∈ N εn < γn εn < γn εn < εn n < m m n k; k > +1 +1 +1 For every , we have that 2 , so + + 2 . Now, let be natural numbers, if we write = + 0, we can apply the m k k− k− previous observation inductively to obtain γl γn i < γn i εn k < γn i εn k− < : : : < 1 2 l n i i ! i ! X X + X + + X + + 1 εn − γn εn γn = = =0 < γn =0εn <+ γn =0 + X

+1 + Remark . + + = 2 2 8 The previous lemma gives us a bound in terms of the lower term, so it is ∞ readily seen that the inﬁnite sum converges, and γl < εn: l n X ε = For every n > , εn < n− . 1 1 ε −γ ε LemmaProof. 5. 1 2 n ε < < ε εn−γn εn ε 1 1 1 εn < n− εn < < < n X 2 2 2 1 1 We proceed by induction1 over . The ﬁrst case is clear, . Now, +1 let us suppose thatLet n < m2 be. a Then pair of natural2 numbers.2 Let2 an ∈ An and am ∈ Am be two points of X such that an ∈ pn;m {am} . Then d an; am < εn. Proposition 4. Proof. m n k; k > ( ) ( ) an ∈ An ; : : : ; an k− ∈ Let us write = + 0. The relation between the points means that there +1 +1 + 1 exists a chain of points between them. That is, there exist

21 An k−

+ 1 such that an ∈ pn;n {an } ;

an ∈ pn ;n+1 {an+1} ; ( ) +1 +1 +2 +2::: ( ) an k− ∈ pn k− ;n k {an k } :

+ 1 + 1 + + ( )

k k Using the previous proposition, we can now estimate an; am 6 an l; an i 6 γn i < εn X i i X + + +1 X + d( ) =0 d( ) =0

X X H Before starting the proof of the theorem, let us reinterpret this inverse limit as sequences of points in 2 , with the Haussdorff distance, that is, as sequences of points in 2 . We {Cn}n∈N ∈ X {Cn} n ∈ N Cn ∈ U εn An consider the inversen < m limitpn;m ofC them finiteCn spaces. We will write the points of this limit as 2 sequences ε ( for short), where, for every , ( ), and, for every pair , ( ) = . WeX have to think about this sequences as sets of points of each -approximation, related by a notionX of proximity. It turns out that these sequences converge to points of . To have a notion of measure and see this,X X C;D ∈ H we will use theX Hausdorff distance of the hyperspace 2 of non-emptyC D closedC;D subsets< ε ofC ⊂. It canD; ε by characterized(seeD ⊂ C; ε section 1.2) in the following way: For 2 H closed sets of , we will say that8 the Hausdorff distance of and is d ( ) X if B( ) andEvery pointB( of the) . inverseWe are limit going{C ton} prove ∈ X theis a following Cauchy sequence in H that converges to a singleton {x}, with x ∈ X. Proposition 5. 2 Proof.

{Cn} ∈ X First of all, we see that, in terms of the Hausdorff metric, the diference between two elementsCn of theC sequencem n can < m be boundedCn;C inm terms< εn of: the lower index. Let be a point of the inverse limit.c Then,n ∈ Cn the Hausdorffcm ∈ Cm distance betweencn ∈ pn;m terms{cm} of the H sequencecn; cm and< εn , with , is d ( ) To provecm the∈ firstCm condition of cthen ∈ Hausdorffpn;m Cm distance, consider and such that ( ), and then d( ) by the previous lemma. Analogously, for weX can take {Cn} ∈ X H ε > ( ) and the distancen satisfies∈ N the secondεn < condition. ε n; m > n

Now, the sequenceCn;Cm < of ε closedn < ε: sets is a0 Cauchy sequence in 2 . For any 0 0

0, it suﬃces to consider0 such{ thatCn} ∈ X and then, for every {x} , H Xwe have d ( ) It remains to prove that every sequence converges to a singleton of C; ε ε x ∈ X in the Hausdorﬀ metric. The sequence is Cauchy in the compact metric (and hence 8 c C x; c < ε ε Here,C B( ) is theC; ε generalizedc∈C ballc; ε of radius , i.e., the set of points for which there exists a point of at distance d(S ) or, equivalently, is the union of balls of radius and center any point of , that is, B( ) = B( ).

22 X X H C ∈ complete) space 2 , so there exists a unique limit 2 . The diameter of this point of the hyperspace is C Cn Cn εn n n 6 n

diam( ) = diam(lim ) = lim diam( ) lim 2 = 0 C {x} x ∈ X X because of the continuity of the diameter function regarding to the Hausdorff metric (see Remark . {C } ∈ X {x} ⊂ X [52]). So = , with n ε > n ∈ N n > n {x};Cn < ε Cn ⊂ 9{x}The; ε meaningx ∈ of Cn; ε converging to a set with only onex point∈ c; εis 0 0 H c∈Cn that for every 0 therex exists∈ c;, ε such that, for every , d ( {Cn} ) , i.e., c∈Cn T x B( {)Cnand} −→ x B(x ).H But,{Cn} the first condition, meaning{Cn} {x} B( ), H S implies the second one, B( ). Henceforth, we will say that converges x; Cn {x};Cn to (written or = lim ) for the convergence of to with the H H Hausdorff metric and we will write d ( ) for d ( ), for simplicity. We have the following trivial facts relating the Hausdorff distance on the hyperspace of a metric space and the original distance on the space, for distances between points and closed sets.Let X be a metric space, for every pair of points x; y ∈ X and pair of closed subsets D ⊂ C ⊂ X, we have: Proposition 6. i)d H x; y d x; y .

ii)d H(x; C) = sup( {d) x; c c ∈ C} > inf {d x; c c ∈ C} d x; C .

iii)d H(x; D) =6 dH x; C( but): d x; D > d x; C (. ): = ( )

( ) ( ) ( ) ( )

The last property can be interpreted in some sense as a better behaviour of the Hausdorﬀ Remark . {Cn} ∈ X distance with respect to the upper semiﬁnite topology. x ∈ X n ∈ N

x; Cn 10< εWen can even bound the distances to the limit.m > If n is ax; point Cm of< the εn inverse limit converging to a point in the Hausdorﬀ metric, then, for every , H H d ( ) . This is so because, if we consider an such that d ( ) , then we can write x; Cn 6 x; Cm Cm;Cn < εn εn εn:

H H H d ( ) d ( ) + d ( ) + =

{Cn} ∈ X x ∈ X {Cn} This measure{ allowsx} us to understand this sequences from another point of view: If n ∈is suchx ∈ a sequence,c; then ε we know therex exists an such that N c∈Cn n converges to in the Hausdorﬀ metric. But, from the previous remark, we see that, T for every , B( ). So, we can see as the inﬁnite intersection over x c; εn : all natural numbers: n∈ c∈C ! \N \n = B( )

23 Proof of Theorem 6. ' X → X X

X {Cn} ∈ X

x x Now,H we{Cn can} define a' mapX →: X from{C then} inversex limit to the{ originalCn} ∈ X space . Wex do thisH {C assigningn} to every sequence U xthe uniqueX point in the limit = lim . The map{Cn}: X , sending to is continuous.U Let ε > such that x=∈ lim x; ε ⊂. Then,U consider a neighborhoodn ∈ N of inside . ε Nown n weε wantn < to find a neighborhood of in with image contained in .X There > 0 exists an 0 such that B( ) . Let us consider such that, for every 0 2 C C Cn , V. We claim× that× the::: basic× open× U neighborhoodεn An × of::: the∩ inverse X limit , 1 2 0 2 0+1 0+1 = 2 2 {Cn} X2 {D(n} ∈ V) {Dn} −→ y H is the desiredx; neighborhood y 6 x; Dn of Dinn ; y. So,6 let x; Cn withDn ; y < εn ;. Then we have y ' {HD } ∈ x;H ε ⊂ 0U H 0 H 0 H 0 0 d (n ) d ( ) + d ( ) d ( ) + d ( ) 2 ' X → X x ∈ X so = ( ) B( ) . x ∈ X n ∈ N n Moreover,X the mapx; εn: ∩ An is surjective. For every , we shall construct an nelement∈ N A ofn the inverseε limitn explicitly. To do so, let and consider,n ∈ N for every , the sets = B( ) . These sets are finite and non-empty, because, for every ∗ m , is a finite -approximation.Xn Nowpn;m weX define,; for every , m>n \ = ( )

m m which are non-empty sets,x ∈ asX an intersectionn < m pn;m of aX nested⊂ collectionpn;m X of ﬁnite (hence m m m closed) sets inm a∈ compactpm;m space.X To show⊂ X that itd is∈ indeedpm;m+1 aX nested sequence, we need N +1 m cto∈ proveX that, for everyd ∈ pm;m and{+1c} x;, c < ε(m ) c;+1 d( < ε).m We ﬁrst show +1 +1 that, for+1 every , ( ) . Let ( ). There is an element +1 +1 such that x; d 6 ( x; c), so d(c; d) < εm εandm d(εm; ) and we get

d ∈ X md( ) d( ) + d( ) + =

m m m meaning that pn;m. Now,X it followspn;m directlypm;m thatX ⊂ pn;m X : +1 +1 ∗ ∗+1 +1 X {Xn }( ) = ( ( )) X( ) ∗ ∗ n n ∈ N Xn < εn Xn ⊂ X n < m The sequence∗ =∗ is an element of the inverse limit . This is so because, for pn;m Xm Xn every , diam 2 (by construction,∗ ∗ ) and, for every pair , we n ∈ N pn;n Xn Xn have ( ) = . We just need to prove it for two consecutiven ∈ N terms, i.e., we want +1 +1 to prove that, for every , ( ∗ ) = ,m and the result follows inductively. ∗ n > n m > ∗ n Xn 9pn;m X The last assertionn relies∗ on the following fact : For every there exist an integerm z ∈ X \Xn nz ∈ N m > nz z∈ / pn;m X ( ) such that, for every ( ), = ( ). The proof goes by construction. For every there exists such that, for every , ( ). 9 This is a kind of Mittag-Leﬀer property for these elements of the inverse limit.

24 X n

n ∗ m ∗ Being a∗ ﬁniten set, consider{nz z ∈ X \Xn } {m ∈ N pn;m X Xn } ;

( ) = max : = min∗ N →: N ( ) = m > ∗ n and we have the desired result. The function : is an increasing function. ∗ m m ∗ Considering any pn;n (Xn+ 1) isp elementaryn;n pn ;m toX see thatpn;m X Xn ;

+1 +1 ∗ +1 +1 ( ' )X = x ( ( ε)) > = ( ) =n εn < ε ∗ ∗ ∗ n > n x; X < εn < ε x 0∈ X n 0 n ∗ ∗ ∗ asx; wanted. x < εn We claim thatX ⊂( )x; = εn . Forx every∈ X ; ε0n, consider such that . 0 n H n Then, for every , we have that d ( ) , because, for every , d( ) x ,∈ andX then, B( ) and B( ). The proof of the surjectivity gives us an important element of the inverse limit related∗ {Cn} ∈ X x ' {Cn} Cn ⊂ Xn m with eachn ∈ N . By construction, thism ∈ elementN x; of C them < inverse εm limitCm ⊂ is maximalx; εm ⊂ inX the m ∗ following sense:n ∈ For every m > ∗ n, suchCn thatpn;m C=m ⊂( pn;m),X we haveX that , N H n for every . Indeed,∗ for every , d ( ) so B( ) . Xn Now, given , for every ( ), = ( ) ( ) = . Actually, we ∗ can alternatively define just withXn this propertyC asn; − {Cn}∈[' x = 1 (∗) ' {Xn } x φ X → X because of the maximal∗ property and that ( ) = . φ x {Xn } The previous construction∗ allows us to define a map on the other direction, : V {Xn } X with ( ) = . To prove that this map is continuous in every point, let us consider ∗ ∗ ∗ X X Xr a neighborhood W of ×in .× We::: know× that× U thereεr existsAr × a neighborhood::: ∩ X of the form 1 2 2 +1 +1 W ⊂ V = 2 2 2 ( ) x U ⊂ X φ U ⊂ W such that s . We∗ r need to find points close enought to , that is, an open neighborhood such that ( ) . We do this by the following construction. First of all, consider = ( ). We use the following notation, not to be confused with the s usual topologicalX notation:x; εs ∩ As x; εs {y ∈ X x; y 6 εs} ; s ∂X x; εs \ x; εs ∩ As; s := B( ) (where B( ) = : d( ) ) Xδ x; εs δ ; δ ∈ −εs; ∞ : := B( ) B( ) s := B( + ) forx ( ) As X

s s Let us considerε thes x distance fromx; a toa the∈ A closests \ X pointx; of As \thatX is> not εs: in , + ( ) = min d( ): = d( εs )x εs + If there is not such a point, the proof is easier, just consider ( ) = 2 . In general,

25 we claim the following (see ﬁgure 4):

y x X s s εs z ∈ ∂X Figure 4: For points , close enough to , we do not add exterior points of , when we consider its -neighborhoods. Possibly, some points of the boundary are included, but they are not the image of any point in the next step. s s s i δ < εs x − εs X Xδ c ∈ Xδ x; c < εs δ < εs x c ∈ X s + + ) For every ( ) , = : If then d( ) + ( ), so ii δ < ε y ∈ x; δ p Y s ⊂ Y s \ ∂X s . s s;s s s z ∈ ∂X b ∈ Y +1 +1 ) For every we have that,+1 for every B( ), ( ) : Consider εs andx; z 6 x; y. Then,y; b b; z < εs b; z ;

s s b; z > γs= d( ) zd(∈ / p)s;s + dY( s ) + d( ) 2 ps;s+ d(Y ) ∩ ∂X ∅ s s s ps;s Y ⊂ Y \ ∂X +1 +1 +1 +1 so d( ) ,+1 and then ( ). That means ( ) = , +1 x U x; δ hence ( ) .

The desired neighborhood of δis < = Bε(s x )−withεs; εs : + s s s y ∈ x; δ Y ⊂minXδ X( )

∗ s s s s s ∗ For YBr (⊂ p)r;s, weY have thatpr;s ps;s Y= ⊂andpr;s thatY \ ∂X ⊂ pr;s X Xr : +1 +1 ∗ +1 ∗ +1∗ ∗ ∗ n < r Yn (pn;r Y)r = ⊂ p(n;r Xr ( X))n ( {Yn } ⊂) W ( ) = φ X → X For , = ( ) ( ) = . Then x;and y henceX the map ε : is continuous. ε x; y s ∈ N εs < This map is clearly injective. Suppose we have two diferent points of . Then, they are at distance, let us say, = d( ). Consider such that 2 . Then, for

26 n > s x; εn ∩ y; εn ∅ Xn ∩ Yn ∅ ∗ ∗ ∗ ∗ Xn ∩ Yn ∅ {Xn } 6 {Yn } every , we have that B( ) B( X ∗) =φ ,X that is φ X →= X ∗. So, necesarilly, we have that =∗ , which implies that = , being the map∗ injective.∗ X {Xn }; {Yn } ∈ If we∗ consider the restriction to the image ∗ = ( ), then ∗: is bijective. But, X x 6 y {Xn } φ x {Y } φ y it is easy to see that is Hausdorff. If we consider two different points∗ ∗ s ∈ N n > s Xn ∩ Yn ∅ , then there exist = such that = ( ) and = ( ). Repeating the last proof, we obtain an such that, for every we have = . So, we ∗ ∗ ∗ X Xs Xs ∗ claim that the neighborhoods× ::: × × × U εs As × ::: ∩ X 1 +1 2 +2 +2 2 2 2 ( ) ∗ ∗ ∗ Y Ys Ys ∗ × ::: × × × U εs As × ::: ∩ X and 1 +1 {X ∗} {Y ∗} X ∗ 2 +2 +2 X∗ n n 2 2 2 ( ) of and respectivelyφ X in→ X,∗ are disjoint. Hence is Hausdorff. Then, as a bijective and continuousX ∗ map between a compactX HausdorffX space and a Hausdorff Xspace,∗ we get that the map : X is a homeomorphism. So, we have that is a homeomorphic copy' of· φ insideX → X . Now, we will see that

is a strong deformation retractφ of· '. ToX do → so, X we consider the compositionsX of the maps deﬁned above. It is very easy to see that H X: × ; is→ the X identity map. It is not that easy to see that the map : is homotopic to the identity 1 . We will write the homotopy explicitly: It is{C then} map : t ∈[0 1]; ; given by H {Cn}; t ( φ · ' {Cn} t : if [0 1) ( ) = ( ) if{Cn}=; 1 ∈ X × ; ∗ ∗ φ · ' {Cn} φ x {Xn } V {Xn } X We only need to show the continuity∗ at the points ( 1) [0 1]. Let us write {Xn } ( ) = ( ) = . Consider any neighborhhod of in . We can ∗ ∗ X Xr obtain a neighborhoodW of ×of::: the× form× U εr Ar × ::: ∩ X 1 2 +1 +1 W ⊂ V = 2 2 ( ) s ∗ r s δ εs x x; As \ X δ < εs x − εs; εs t > s εt < such that . As we have done in a previous proof, we consider = ( ), + + ( ) = d( ), and min ( ) . Select such that 2 . We C C Ct claim that the neighborhoodU × × ::: × × U εt At × ::: ∩ X 1 2 2 +1 +1 {Cn}; X ×= ;2 2 H U 2× ; ⊂( W ) {Dn}; t ∈ U × ;

Dn ⊂ Cn n ; : : : ; t t < H {Dn}; t {Dn} ⊂ W of ( 1) in [0 1] satisﬁes∗ ( [0 1]) . Let ( ) [0 1] r < s < t Dn ⊂ Cn ⊂ Xn n ; : : : ; s t where for = 1 . Then,∗ if 1, ( ) =∗ , because H {Dn}; φ · ' {Dn} φ y {Yn } {Yn } ∈ W and for = 1 ∗ . On the other hand, if = 1, then d ∈ Dt ⊂ Ct ⊂ Xt x; y 6 x; d d; y < εt < δ ( 1) = ( t) = t( ) = t . This∗ implies∗ that∗ . To see why, Y ⊂ X ∪ ∂X Yr ⊂ Xr {Yn } ∈ W ﬁrst observe that, for every , d( ) d( ) + d( ) 2 . Then, again as before, and , so , and the homotopy

27 X ∗ X X is then continuous. The space is a strong deformation retract of and the proof of the theorem is ﬁnished

This theoremFor leads every us easily compact to metricthe following space, thereconsequence. exists an inverse sequence of ﬁnite spaces whose inverse limit has the same homotopy type. Corollary 2.

This means that the homotopy type of compact metric spaces is, in some sense, pro-ﬁnite.

X ; 3.1 Example Let = [0 1] be the unit interval with the usual metric on the real line d. In order to perform the1 Main Construction, we are going to use subdivisions ofX theM unit interval in 3 powersε of . The> MconditionsA of the{ } construction willU ε forceA us{ to} take, for each subdivision, an smallγ enough subdivision for the next step. The diameter of is = 1. Let us 1 1 2 1 1 select = 2 and = 0 . Obviously ( ) = 0 . Then, it is easy to see 1 ε − γ that = 1. ε < For the next step, consider 1 1 k 2 1 1 A { ; k ;:::; } = = 3 2 2 2 and = 3 = 0 3 . Then k k U ε A A ∪ ; ; k ;:::; ; 2 2 2 2 + 1 0 ( ) = k k0 = 0 0 2 0 k < k ; <3 3 k − k < k − k 2 X A 3 3 3 because, for , we have d( ) if and only if 2, that is, Ais 2 either γ0 or 1. The largest distance of a point of to a point of the approximation 2 is reached when1 the point lies exactly in the middle of two consecutive points of , 2 6 ε − γ δ hence = . ε < ; · We pick 2 2 2 3 13 1 2 = k = A 3 ; k2 ;:::;2 3 2 and 3 3 = 3 = 0 3 3 k k for the new approximation.U ε A A Then∪ ; ; k ;:::; − ;

3 3 2 3 3 3 +3 1 0 ( ) = k k0 = 0 3 0 1 0 k < k ; 3 < 3 k − k < k k 2 3 3 3 γ · 3 3 3 because, for we have d( ) if and only1 3 if 2, that is and 3 are the same or consecutive integers. Now, = 2 3 (the middle of interval argument holds again).

28 ε − γ For the next approximation, weε would< select : · 3 3 4 15 3 1 2 = = n ∈ ε 3 2 3 2 N n n− 21 3 3 Following this process, we can take,k for an arbitrary n− , = and An n− ; k ;:::; 2 3 = 2 3 = 0 3 εn ; 3 is an -approximation of [0 1]. Observek k that n− U εn An An ∪ n− ; n− ; k ;:::; − : 2 3 2 + 1 ( ) = 2 3 2 3 = 0 3 1 3 3

We can calculate, as in the previous steps, the numbers required to continue the process. The maximum distance of a point of the unit interval to one of the approximation γn · n− is reached in the middle21 3 of any interval formed by two consecutive points of the approximation, so = 2 3 . Next step will consist of taking ε < : n · n−

+1 2 12 3 ε n n− 2 3 n − 2 13+2 2( +1)1 3 +1 3 3 So we are allowed to choose =k = andn − An n − ; k ;:::; 2( +1) 3 +1 = 2( +1) 3 = 0 3 εn ; 3 n ∈ X +1 N is an -approximation of [0 1] and then this construction can be done in this way for every obtaining an explicit fas for the space . x ∈ X Considering this construction done, we analyze' some facts about the proof of the main theoremX in this example.x First, observe that there are points of the interval with onlyx one point∈ in; the preimage by the map , i.e., there; ;::: is only∈ X one point of the inverse limit converging to in the Hausdorﬀ metric. To clarify this, let us see what happens at = 0 [0 1]. It is obvious that the point (0 0 n→∞) n− converges to 0 in the

Hausdorﬀ metric. If we want a diﬀerent element of the inverse21 3 limit converging to 0, it is natural to think that we could use the fact that lim 3 = 0 to obtain it, but it ; ; ;:::; n− ; n − ;::: ∈/ X turns out that 1 13 21 3 2( +1)1 3 n ∈ 0N pn;n n − ∗ 3 3 3 3 X 2( +1)1 3 x n ∈ +1 N because, for every , 3 = 0. If we try to construct the "maximal" element of the inverse limitX withn image; εn ∩=A 0n, we{ obtain} that, for every ,

= B(0 ) = 0

29 pn;m n < m

∗ and (0) = 0, for every X.n Hence,pn;m Xm ; n

Actually, every2 1 point in for some has this property. For any of them, let us say = 3 , we have k k Xn n− ; n− ∩ An n− 1 = B 2 1 2 1 = 2 1 pn;m x x n < m 3pn;n− 3x 3 X x X ∗ ;:::; ; k ; k ;::: n− n− 1 and ( ) = for and ( ) = 0. So, the only element of converging n∈ An 2 1 2 1 N 3 3 to is = 0 0 . The subsetx of the interval consisting of these S points, , is dense in the unit interval. 1 2 If we choose a point not in this subset, for examplek = , we obtain an− diﬀerent result. n ∈ N 1 n− k 2 First of all, we know that is not going1 ton be2 ∈3 aN point of the approximation,2 3 for any 2 3 , because if that was true, then = and1 that implies 3 = 2 which is impossible. Now we claim that, for every , 2 is at the same distance of two consecutive points of the approximation and, because of that, both minimize its distance n− to the approximation.k This is true becausek − n− − − n− ⇐⇒ k : 2 3 + 1 1 1 3 1 2 3 = 2 3 = 3 2 2 3 2 Now, let us constructX the "maximal"; element for this point. We get:

X1 ; ∩ A ; ; = 0 :::2 1 1 2 1 2 = B = n− − n− 2 3 3 3 2 3 2 3 Xn ; n− ∩ An n− ; n− ; ( 3 1 3 +1 ) 2 2 ::: 1 1 = B 2 3 = 2 3 2 3 2 3 3 3 ∗ pn;m Xm Xn n < m Xn Xn n ∈ N x

It is easy1 to see that ( ) = for every so = for every . So, 2 n− − n− for = , ∗ 2 3 2 3 X ; ; ; ; ;:::; n− ; n− ;::: ( 3 1 3 +1 ) ! 2 2 1 2 14 15 = 0 3 3 2 3 2 3 3 3 3 3 3 3

1 pn;m which obviously converges to 2 with the Hausdorff metric. But now, we can see that each term has two elements and the maps are sending the first to the first and the

30 n− − second to the second, so we can consider the sequences 2 3 C ; ; ;:::; n− ;::: ( 3 1 ) ! 2 1 1 14 n− = 0 3 2 3 2 3 C ; 3 ; 3 ;:::; 3 n− ;::: ( 3 +1 ) ! 2 2 2 15 = 0 3 2 3 C ;C ∈ X 3 3 3 ' 1 1 2 and claim that and both converge to 2 . So this point has exactly three points of the inverse limit in its inverse image by , being that map not injective.

' 3.2 Some special cases The same property about the injectivity of in some points observed in the previous example holds inLet general.X be a compact metric space and {εn;An; γn; δn}n∈N a of X. If a point x ∈ X satisﬁes x ∈ An, for every n > n , for some n ∈ N, then the cardinality ofProposition'− x is one. 7. fas 0 0 1 Proof. ( ) x ∈ X An n > n

∗ 0 We are going toX prove that,p ;n ifx ; : : : ; pnbelongs− ;n x ; to x; x; :for : : every: , then

1 0 0 1 0 = C ∈( X) '(C) x X ∗ ∗ X n > n x ∈ Xn x; εn ∩ An ∗ ∗ xSo,∈ thereX are no moren n points pn;msatisfyingx x ( ) = n, appartn < from m (becauseX of n > 0 6 the maximality of ). For , we have that = B( ) , and then, 0 0 for every ∗ , because∗ ( ) = for∗ every∗ ∗ . So has the X p ;n Xn ; : : : ; pn − ;n Xn ;Xn ;Xn ;::: : form 0 0 0 X ∗ 1 {x} 0 n1 n 0 0 y 0+1∈ X y ∈ X ∗ =n ( ) >( ) n n i ∈ yi ∈ An i y0 ∈ pn i;n i yi N 0 0 0

Nowy wei ∈ proveXn that = i ∈ for every .0 Let . Then,0 0 if N + + + +1 +1 and only if there exists, for every , such that ( ) 0+1 and fory everyi ∈ An i . We arei ∈ goingN to see that, if there is a chain of points satisﬁying the ﬁrst condition,0 they cannot satisfy thedi second.x; y So,i leti us∈ suppose + N dthere existsx; y a chain i ∈ N ,y fori every such that one belongs toyi the imagex of the following. For the sake of simplicity, let us write := d( ) for , (and 0 0 +1 = d( )). For every d, i >is closeryi; yi (or< at γn thei: same distance) to than to , so we have d d+1 y ;+1 y 0d+ −d y ; y i 6 i d( i )i i i 6 i i di − di < γn +1 +1 +1 +1 di Moreover, it is obviousi ∈ that di di +d( yi ; yi), i.e., di 0 di dm( m ). Combining N 6 6+1 +1 > this with the previous observation, we get . On the other hand, we +1 +1 +1 + have that, for every , + d( ) 2 , so 2 . We supposed

31 y ∈ Xn εn − d > i ∈ N

0 0 0 0 , so 0. We claim that, forεn every− i d , εn i − di < i : 0 0 0 + 2 (+1+ 2) 2

εn − γn εn − d d We prove it by induction.εn − Thed ﬁrst< case is − d < − 0 0 0 0 1 0+1 1 εn − d d 1 εn − d 6 − : 0 2 0 0 02 0 2 2 3 2 = 2 2 2 2

εn i − γn i εn i − di di Now, supposeεn thei hypothesis− di < of induction is− satisﬁed,di < and we check− 0+ 0+ 0+ +1 0+ +1 +1 εn − i d +1 d εn − i d < i − i i : 0 2 0 0 2 0 2 0 2 (+2+ 2) +2 2 (+2+ 3) i ∈ N i =d > εn i 2 2∗ 2 ∗ εn − di < yi ∈/ Xn i y ∈/ X 0 X {x} n 0 n It is obvious that there exists an suchX that∗ ({+x} 3) 2 n. Forn this , we have 0 0+ 0 n 0 > 0 X that 0, so , and then, . We conclude = and the ' 0 same argument can be applied to show that = , for every

In view of this result, it is natural to look for fas making the map the more "injective" Remark . X An X posible, i.e., injective in the largest possiblen∈N set of points. U ⊂ X x ∈ U ε > x; ε ⊂ U n ∈ S N 11 x;For εn every⊂ fasx; εof , the set a ∈isAn always densex; a in< εn. For each open 0 setx; ε ∩ Anthere6 ∅0 exists and 0 such that B0 ( ) . Let us0 select such that B(0 ) B( ). Then, for any with d( ) , we have that B( ) = . This also shows that every compact metric space has a countable X dense subset. We want to apply Proposition 7 to obtain inyectivity in a dense subset of . The following construction will be useful. X An ⊂ An n ∈ M X ε > M A {x} x ∈ X N +1 Constructionε 1. For every compact metricA space , there exists aAfas withA0 ∩ A forA0 1 1 everyε : If = diamX( ), let us considerA , and = with .n Then,∈ N 2 2 2 2 1 2 consider withA then usualεn rule. Now, for , weX take the unionγn = εn where 2 2 0 0 is a -approximationAn of , then soA is ∩.A Wen can proceedA in thisεn way for every X. n n +1 If we have thatAn is a -approximation of , considerX and takeAn ⊂ An as always. +1 +1 +1 +1 Thenn ∈ consider as the union where is a -approximation of N +1 +1 and, hence, too. In this way, we obtain a fas of such that for every .

The best situations we can have are the following.

Countable spaces

32 Let X be a countable metric space. There exists a of X with inverse limit X homeomorphic to X. Proposition 8. fas Proof. X {x ; x ; : : : ; xn;::: } X xn ∈ An An ⊂ An n ∈ An X 1 2 N n∈N We can write = ' . We just need to ﬁnd a fas for satisfying +1 S and for every . The ﬁrst condition gives us = and the second one will make injective− on the set− ' An ' X X; n∈ ! 1 [N 1 = ( ) = ' X → X

An xn Am mand < then, n : will be a homeomorphism.n ∈ N There are many ways of doing so. We can just do the general construction forcing each to contain and , for every . Forr example,n if wei ∈ consider,N {x ; : for : : ; every xi} εn , the numbers X ;

1 ( )r =n min > r n : is a approximation of it is clear that ( + 1) ( ) andA then we{x } can; write the approximations as

A1 {x1; : : : ; xr }; = 2 ::: 1 (2) = An {x ; : : : ; xr n };

::: 1 ( ) = X X and we are done

Now we face the case of proper dense subsets of . First of all, we observe the following Remark . Y ⊂ X ε > ε A ⊂ Y { x; ε x ∈ X} ε ε 12 For every dense{ subsetx ; ;:::; xkof; a} compact metric space,y ; : :and : ; y everyk ∈ Y 0, 2 ε there existsxi; yi an< -approximationi ; : : : ; k: Let{ usy ; consider : : : ; yk } the coveringε B( ): X 1 1 and a ﬁnite subcovering B( 2 ) B( 2 ) . Now we take such 1 that d( ) 2 for every = 1 , so is an -approximation of . We can state theFor main every result countable in this directiondense subset of a compact metric space, Y ⊂ X, there exists a of X such that there is a dense subset of X which is homeomorphic toPropositionY . 9. fas Proof. Y {y ; y ; : : : ; yn;:::} X

An ⊂ An n ∈ An Y N1 2 n∈N If we write = , it is easy to obtain a fas of such that +1 S , for every , and = . For example, using the previous remark,

33 we can takeA as approximations{y } 0 0 A1 {y1} ∪ A ∪ A A ⊂ Y ε X; = 0 0 A2 {y2} ∪ A1 ∪ A2 A2 ⊂ Y ε2 X; = with an -approximation of 3 ::: 3 2 3 3 3 = with0 0 an -approximation of An {yn} ∪ An− ∪ An An ⊂ Y εn X;

::: 1 = with an -approximation of − ' X → X ⊃ Y n∈N An ' Y 1 S− If we restrict the map : ' |'− Y=' Y −→toY the set ( ) we obtain that

1 1 ( ) : ( )'− − ∗ ' Y ⊂ X ⊂ X 1 − ∗ ∗ is' injectiveY and1 henceX a homeomorphism.V So X is theC desired∈ V set. We have the

Cinclusions1 C ;C ;:::;C( ) n;::: , by construction (recall Proposition 7). Now, to see that ( ) is dense in . Let be any open set of and any point of it, where 1 2 C Cm = ( C ∈ W ). Choose× ::: an× open× neighborhoodU εm Am × from::: the∩ X basis ⊂ V 1 +1 ∗ 2 +1 c ∈ Cm= (2 c :::;c;c;:::2 ( ) An )⊂ An n ∈ N c∗ ∈ W ∩ '− Y ⊂ V ∩ '− Y '− Y X ∗ X+1 and select any 1 . Then, 1 = ( ), because1 for every . Remark . '− Y So ( ) ( ), which implies ( ) = 1 Y n∈N An 13 The inclusion ( ) of last proposition is proper: Recall example 3.1 S where = with k n− An n− k ; ;:::; 2 3 2 3 ∗ = : ∗= 0 1 3 − 3 X ' Y 1 An 1 1 and, while 2 is obviously an element of , it does not belong to ( ), since 2 does not belong to any approximation . X Ultrametric spaces Another structures that will give us a complete topological reconstruction are ultrametric spaces. An ultrametric space is a metric space with an extra property of the distance: Instead of satisfying just the triangle inequality, they satisfy the strong triangle inequality,∀x; y; z ∈ whichX; x; is: y 6 { x; z ; y; z } :

d( ) max d( ) d( )

This inequality gives10 us some properties that make the ultrametric spaces very special ones. For example : 10 See chapter 2 of [56] for more properties and detailed proofs about ultrametric spaces.

34 ·

· Every trianglex; y ∈ is isosceles,X ε > withδ > the nonx; equal ε ∩ sidey; δ smaller6 ∅ than the othery; two. δ ⊂ x; ε For every and 0,B( ) B( ) = implies that B( ) B( ).

We want to show that, for the case of ultrametric spaces, there exists a fas such that they recover the topological type of the space. The key idea here, is that for those spaces thereLet areX verybe a special compact approximations: ultrametric space. For every ε > , there exists an ε-approximation of X, {x ; : : : ; xk }, such that B xi; ε ∩ B xj ; ε ∅ for every i 6 j. Lemma 6. 0 1 Proof. { x; ε (x ∈) X}( X) = = { x ; ε ;:::; xk ; } {x ; : : : ; xk } ε X i 6 j

The coveringxi; ε ∩ byxj ; ε open6 ∅ balls B( ): xi; ε of xj ; εhas a ﬁnite subcover 1 1 X B( ) B( ). So, is an -approximation of . Now for any = such that B( ) B( ) = it turns out that B( ) = B( )

We can stateFor the every main compact theorem ultrametric about ultrametric space X spaces., there exists a with inverse limit X X. Theorem 7. fas Proof. {ε ;A ; γ ; } X = n n n n∈N

x ∈ X n ∈ N qAn x Let us consider any fas of satisfying the property stated in the a ; a ∈ qAn x x; a ; x; a < γn < εn previous lemma. Then,x ∈ fora every; εn ∩ a and; εn every , we have that cardq(A (X))→ =A 1n: 1 2 1 2 n Let us suppose that ( ). Then, d( ) d( ) but,An in that case, 1 2 we will have that B( ) B( ) which is not possible. Then, : +1 is actually a single valued continuous map. Moreover, if we restrict to , we obtain qAn |An pn;n |An An −→ An that +1 +1 +1 +1 = : is a continuous map. So, it makes senseX to write the following diagram, qAn qAn & +1A / A ; n pn ;n n

+1 +1 a ; a ∈ An qAn x a pn;n qA x a x; a < εn n 1 2 which, moreover, is commutative. If not, then there would exist with ( ) = 1 +1 +1 2 1 and x;( a ) =6 . Clearly,x; qAn dx( ) qAn ,x but; p alson;n qAn x < εn − γn 2 < γn γ+1n < εn γ+1n < +1 +1γn < εn: d( ) d( ( )) + d( ( ) ( )) +1 +1 +x ∈ a ; ε+n ∩ a ; εn + qA 2 n X 1 2 and this isX imposible, since thenX B←( An; p)n;n B( ). Adding that is always a surjective map distinguishingC pointsC ;C ;:::;C of (seen;::: corollary∈ X 3 on page 61 ofCn [43]), we +1 have that is the inverse limit = lim ( ). Now, it remains to see that every 1 2 element of the inverse limit = ( ) satisﬁes that card( ) = 1 for

35 n ∈ N a ; a ∈ Cn x; a ; x; a < ε x H {Cn} 1 2 1 2 every . If not, for any pair we would have that d( ) d( ) , with = lim X, which,U again,ε An is; p notn;n possible.A So,n; p wen;n have thatX ← n ← X

2 +1 +1 = lim ( ) = lim( ) =

3.3 Previous work on inverse limits of finite spaces Our construction is a sequence of finite spaces, which, in the limit, are able to reflect topological· properties of the original space. Its main features are:

It is internal: It is constructed from the space itself, without need of external ambient spaces to approximate them. We use the hyperspace, which is constructed · just in terms of the compac metric space.

It is constructive: Given a space explicitly, we can actually select points for each approximation. This is important, since it allows us to perform explictly the construction over the space, and possibly determine some topological structure, up to some error.

There exist previous results on the approximation of topological spaces by ﬁnite spaces. This is an old theme, but nowadays it is becoming more important because of its role in the emerging ﬁeld of computational topology. In this section we will review some of these results and compare them with ours.

Approximation of compact polyhedra There is a paper of E. Clader [19] where the following theorem is proved:Every compact polyhedron is homotopy equivalent to the inverse limit of an inverse sequence of ﬁnite spaces. Theorem 8 (E. Clader).

The proof consists of taking as finite spaces theT vertices of the barycentric subdivisions of the simplicial complexes defining the compact polyhedron.n Given∈ a simplicialn complex, 0 N n n op the McCordK correspondence assignsF an finiteX K space. Cladern assigned the opposite topology to these( ) finite spaces.X K That is, consider for every , the -th barycentric subdivision and the finitepn space |K |= ( )Fn, that is, the -th barycentric|K | subdi- n vision of the finite spaceK ( ) with then opposite > topology of thatqn assignedFn → F byn− McCord.

Then, there is a naturalpn mapp( n) − from to each , because every point of belongs 1 to a unique simplex of . For every 1, there is a map : closing 1 the diagram with and . Then, it is shown that the polyhedron is a retract of the inverse limit of these ﬁnite spaces and maps. Note that every compact polyhedron is a compact metric space (for details of the metric, see, for example, the appendix on polyhedra of [43]). So, this theorem is a particular case of corollary 2.

36 et al. Finite approximations and Hausdorff reflections In a series of papers, R. Kopperman ([35], [36]) proved the following theoremEvery compact about finite Hausdorff approximations. space is the Hausdorff reflection of the inverse limit of an inverse system of finite spaces. Theorem 9 (R. Kopperman, R. Wilson).

The finite spaces involved in this proof are constructed in a very theoretical way. It is considered the set of all possible open coverings of the space and then the spaces are defined with a boolean algebra on the open sets of the coverings. This theorem has the advantage that it is very general: It works for any compact Hausdorff space, with no need for a metric. But it has the desadvantage that it is not always explicitly constructible and that we loose a lot of intuition with the Hausdorff reflection. The idea of a reflection of a topological space is to construct another space, as similarX as posible to the firstT one with an extraµX separationX → XT propertyT reflection and a universalX µ map.X

Somehow it is similarXT to compactiﬁcation.T Concretely, given a topologicalf X space→ Z andZ a separation propertyT , we will say that g :XT → Z is the of if is a continuous map, has the property and every continuous map : with f having property , factors through aX map : 8/ Z , i.e., the diagram µ X g XT

µX commutes. If the map is surjective we will say that the reflection is surjective, too. For some properties,(see [51], the chapter existence 14)Let of reflectorsX be a is topological a well known space. fact. For T being the separation properties Ti; i ; ; ; ; there exists a surjective reflection. Theorem 10. 1 = 0 1 2 3 3 2

It is easy to see that two reflections of the same space are homeomorphic. In many cases, reflections are obtained as quotient spaces (not in every case, as for example the Tychonoff functor -or reflection- which is not obtained as a quotient) for a relation. Nevertheless, it is not allways obtained as the obvious relation. As a matter of fact, in order to obtain the Hausdorff reflection we need to define the following relations (see the reference [59], a short and beautiful paper about reflections, where this• isxR shown):y Ux ;Uy x; y Ux Uy 6 ∅

1 T • xR y iﬀ for every pairx ofz neighborhoods; z ; : : : ; zn y of zresp.,R z R we:::R havezn = .

2 1 2 1 1 2 1 1 • xR y iﬀ there existf X=→ Z Z = such that f x f y .

3 iﬀ for every : X, with Hausdorﬀ, weX haveH X/R( ) = ( ).

3 Then, the Hausdorﬀ reﬂection of is the quotient space = .

37 We want to compare the Hausdorff reflection of a topological space with the space itself in terms of shape type. As a motivation, we can cite [50], where it is shown that the Tychonoff functor indeed induces the identity morphism in shape. So, a topological space and its Tychonoff reflection have the same shape. We will show the same holds for the HausdorffThe Hausdorff reflection. reflection of the product X×I, where I ; , is homeomorphic to XH × I. Lemma 7. = [0 1] Proof.

Consider the continuousf mapX × I −→ XH × I x; t 7−→ µX x ; t ; : ( ) ( XH( ×) I) h X × I H → XH × I which is a quotient map. Moreover, the space is Hausdorﬀ, so there exists a f continuous surjective map :( X × I) such/ XH × thatI the diagram 6 µX×I h X × I H

h ( ) h f µX×I commutes.a ; b ∈ X We× I seeH that ish actuallya h ab homeomorphism.z ; t First of all,µX×I is a quotient map, becausex; t ;andx; t are ([27],µX× pagI x; 91).t Also,a it isµX× anI y; injective t b map. Indeed, let [ ] [ ] ( ) such that ([ ]) = ([ ]) = ([ ] ). Considering that is surjective, 1 2 1 2 there exist ( ) ( ) such that ( ) = [ ] and ( ) = [ ]. Because of the commutativity of thex ; previoust f x; diagram t h µ weX×I havex; t that h a z ; t

y ; t 1 f y; t1 h µX×I y; t1 h b z ; t ([ ] ) = ( ) = ( ( )) = ([ ]) = ([ ] ) 2 2 2 x y z([ ] )t = ( t ) =t ( ( )) = ([t ]) = ([ ] )

1 2 id×t so [ ] = [ ] = [ ] and = =X . For this concrete/ X × I ,; we consider the commutative diagram µX µX×I XH g / X × I H µ ◦ id × t X → X × I X×I ( H ) x; y whichx y exists for being a ( b ):h ( ) a continuous map to a Hausdorff space. We considerX the images of by the two different maps of the diagram. As [ ] = [ ], we obtain that [ ] = [ ], so is injective. A quotient and injective map is a homeomorphismFor every topological space X, the Hausdorff reflection µX X → XH induces the identity morphism in shape. Theorem 11. :

38 Proof.

µX X → XH 11To show this, we are going to use the characterization of identity morphisms in shape :The map : is theXH ;P identity−→ morphismX;P in shape if and only if the map h 7−→ h · f; [ ] [ ] P g X → P P with being anyh metricXH → ANR,P is bijective.g h · µX

It is surjective:h ; Given h X aH map→ P : P , with ANR and then, Hausdorﬀ,h there· µX existsh · aµX map : such that = , that is, whatG X we× wanted.I → P It is 1 2 1 injective:G x; Let h · µ:X x G,x; with hANR,· µX twox continuousP maps such that 2 y are homotopic,F X ×I H i.e.,→ thereP existsG a continuousF ·µX×I map, : such 1 2 that (µX0)×I = µX × id( ) and ( 1) = ( ). Being Hausdorﬀ, there exists a continuous map :( ) such that = . Applying the previous lemma, G we get = , so we haveX that× I the following/7 P diagram: commutes:

µX ×id F XH × I

x ∈ X

Then, for every ,F wex ; have G x; h · µX x h x

F x ; G x; h 1· µX x h 1 x : ([ ] 0) = ( 0) = ( ) = ([ ]) 2 2 h h ([ ] 1) =X ( 1) = ( ) = ([ ])

1 2 So, and Aare topological homotopic space X has the same shape than its Hausdorﬀ reﬂection XH . Corollary 3.

Note that with Theorem 9 and the result just proved here about the shape of the Hausdorff reflectionEvery compact we will get Hausdorff the following space hasgeneralization the same shape of Theorem as the 9. inverse limit of an inverse system of finite spaces. Corollary 4.

In an attempt of understanding better the Hausdorff reflection of an inverse system of spaces, Kopperman and Wilson proved that the original space is not only the Hausdorff reflection but the set of closed points of the inverse limit. We can prove the same in ∗ our construction. For every compact metric space X and every of X, the space X ∗ is just the set of closed points of X. Moreover it is its Hausdorff reflection X XH . Proposition 10. fas

11 = See [38] for this result of shape theory

39 Proof. x ∈ X X ∗ φ x First of all, we are going toX characterize,∗ { forC} every: the point of the inverse limit = ( ). It is the set C∈'− x \ 1 = ( )

∗ ∗ ∗ ∗ We⊂ divide the proof: ' X ' C x X X ;X ;::: X ∗ ∈ {C} X ∗ ∈ V X 1 2 ( ) We show here that if ( ) = ( ) = (notation: = ( )), then . Let an open neighborhood in . Then, there exists an open ∗ ∗ ∗ ∗ X X Xr neighborhoodX ∈ U × × ::: × × U εr Ar × ::: ∩ X: 1 2 2 +1 +1 C ∈=U (2 C 2∈ U ∩ {C}2 6 ∅ ( ) )

⊃ But,D obviously,D ;D ;::: ∈, so C∈'− x {C} = {.Dn} x ∗ D ∈ U 1 U ∩ {X } 6 ∅ 1 2 T ( ) ( ) Let = ( )r ∈ N .Then converges to in the Hausdorf metric. So, for every open neighborhood, we have that = . In D D Dr particular, for every × ×we::: have× neighborhoods× U εr Ar of× the::: form∩ X; 1 2 +1 ∗ 2 +1∗ ∗ X (2 2 r ∈2 N ( Xr ) Dr ) X D

∗ ∗ ∗ where belongs.X So, for every we have = , hence =X ∈. X ∗ X C∈'− x {C} ∗ ∗ ∗ NowC ∈ X to show1' thatC yis theY set∈ of{C closed} { points,C} C firstY observe∈ X that: every is T ( ) = X ∗ , so a closed set. On the other hand,X if there is a closed point α , withX → (Y ) = Ythen = so = C;C 0 ∈ X 'ToC show' thatC 0 xis actually' X ∗ the HausdorffX ∗ ∈ X ∗ reflection of , let us consider a continuous mapX ∗ : withX ∗ ∈a{ HausdorffC} ∩ {C 0} space. Consider two pointsα such that ( ) = ( ) = =Y ( ), with . Then, using the previous characterization of , we have that . Then, applying the map , and using that it is continuous and that is Hausdorff,α X ∗ ∈ weα obtain{C} ∩ α {C 0} ⊂ ⊂ {α C } ∩ {α C 0 } ( ) ( ) ( ) {α C } ∩ {α C 0 }; ( ) ( ) = α X ∗ α C α C 0 = ( ) ( ) φ · ' X → X ∗ ∗ X α |X∗ X → Y so, ( ) = ( ) = ( ). Now, we claim that the map : is actually φ·' the Hausdorff reflection of . This isX so, because/ X ∗ the map : makes the diagram α α|X∗ Y y

α |X∗ φ·' X commutative and is continuous since is a retraction and hence a identiﬁcation

40 4 Inverse Persistence It is clear that a persistence module is nothing but an inverse sequence of vector spaces and homomorphisms reversed, in the sense that the sequence grows in the opposite direction. Moreover, if we "cut" the inverse sequence at some step, we obtain a persistence module of finite type and, hence, the corresponding barcode. In this way, we can obtain persistence modules from inverse sequences of spaces and this makes a connection between shape theory and persistent homology. In this section, we present a different way of persistence in which the persistenceX modules are obtained from inverse sequences of polyhedra. {Kn; pnn } Let us consider a compact metric space and a polyhedral approximative sequence +1 obtained by the main construction as in section 2 . Although all these inverse sequences of polyhedra are the realizations of inverse sequences of simplicial complexes and simplicial maps betweenp ∈ N them, they areF not filtrations of simplicial complexes, even obviating{Hp Kn; pnn the finitenessF } condition, since the maps involved are not the inclusion. But, if we consider, for any , and a field , the induced homology inverse sequences +1 ( ; ) are persistence modules (with maps not induced by the inclusion)X of simplicial complexes, but by means of proximity, as indicated in the quoted section. As a compactX metric space, we can perform this construction to a point cloud . If we consider that this point cloud is a sample, possibly with some noise,X of a compact metric space , the results obtained in section 3 and in [50] aboutX the homotopical and shape properties recovered in theX inverse limit of any fas and pas of make us think that the topological properties obtained applying this method in will represent topological {εn;An; γn; }n∈N X X {εn;An; γn; }n∈N properties of the space . s Considern > s a fas, of . Since is finite, has only a finite number of different approximations:εn < { x; There y x; isy ∈ anX integer} ; such that, for every , A A X U A U A p id n n 2 εn maxn d( εn ):n nn s +1 2 2 +1 +1 +1 and hence = = , ( ) = ( ) and = . So, we have only p ps− s a finite numberUε Aof "changes"←−−− Uε inA the←− sequence,::: ←− U thatεs− A wes− can←−−−− be writtenUεs As as: 12 1 1 1 2 2 1 1 ( ) ( ) ( ) ( )

p ps− s Now, consider the inducedK ←−−− polyhedralK ←− approximative::: ←− Ks− s ←−−−− sequenceKs: of section 2, 12 1 1 2 1 p F

p ps− s Its induced -thH singularp K ←−−− homologyHp K sequence←− ::: ←− (forHp K as ﬁeld− s ←−−−−) Hp Ks 12 1 1 2 1 ( ) ( ) ( ) ( ) BX is indeed a persistence module of ﬁnite type, so it has an associated barcode that we

41 inverse barcode will call . We call this procedure Inverse Persistence. In Figure 4 we represent the Inverse Persistence process. We list some diﬀerences between Persistence and Inverse Persistence.

1. The simplicial complexes used in regular persistence are constructed using all the set of points of the point cloud for every level. In contrast, the simplicial complexes constructed in the inverse persistence are based on subsets of the point cloud. Moreover, we need to add more points to the ﬁnite spaces, in order to make the maps between them continuous.

2. The maps used in the finite sequence of polyhedra constructed from the point cloud are always inclusions in regular persistence, but they are not in inverse persistence. Although they are not inclusions, they are consistent is some sense because they are defined in terms of proximity and, as we have seen, they are constructed in a way that, carried until the infinity, captures the homotopical and shape properties of the space.

For the analysis of the inverse persistence we propose the following steps.

1. Formalize the algorithm outlined here and compare the computational cost with the standard algorithms for persistence on point clouds. interleaving 2. Compare the obtained inverse persistence modules and compare them with the regular persistence modules in terms of the concept of , introduced in [18] by Chazal et al. bottleneck 3. Compare the obtained barcodes from inverse persistence with the ones obtained by regular persistence using the distance on barcodes (see [20] for deﬁnition and main results concerning this distance).

It is expected that the inverse persistence modules have the same behaviour as regular ones in terms of stability (see [20, 18]), because of the shape theoretical framework where they are constructed. We hope this shape approach to persistence to be suitable for real world applications because of its constructibility and its good propertiesX concerning stability. n; m n < m Moreover, we can assign inverse barcodes to every compact space and every pair of integers ( ) with usingHp ﬁniteKn ←− parts::: of← theH polyhedralp Km approximative sequences

( ) ( )

We expect inverse barcodes comming from compact metric spaces and (possibly noisy) samples of them to be quite similar in some yet non deﬁned sense. We compute an inverse barcode in the following section.

42 Point Cloud Polyhedral KApproximative← ::: ← Kn Sequence 1

Hp K ← ::: ← Hp Kn Inverse Persistence Module Barcode 1 ( ) ( )

Figure 5: Process of Inverse Persistence

4.1 Example: The computational Warsaw Circle In this section we will perform the main construction on the Warsaw circle in order to apply the theory previously developed. The Warsaw circle is the paradigmatic example of shape theory and we shall see how inverse persistence works in this case, capturing the shape properties (such as the Čech homology groups)R of this space. See ﬁgure 6. For computational purposes, we are going to deﬁne2 and work with the following12 homeomorphic copy of the Warsaw circle. Consider, in , the following segments : an n− ; − n− ; ; 1 1 b = 2 2; 1 − 2 1 ; 1 ; n 2 n− 2 n− 1 1 1 1 b = 2 2 ; − 2 2; 1 ; n 2 n− 2 2 n− 2 1 1 1 c= 2 1 1; − 2 1 ; : n 2 n− 2 n 2 1 1 1 1 = 2 1 2 2 2 2 2

Then, the computationalW ; − Warsaw; − circle; − is; an bn bn cn: n∈N n∈N\{ } n∈N n∈N [ [ 1 [ 2 [ = (1 0) (0 0) (1 0) (1 1) 1 √ W W M √ √ ε > M A { ; } γ √ √ We nowε perform the< generalε −γ construction on ε. The diameter of Wis = 2. Then, 1 1 1 1 1 we can select 32 = 2 2 ,2 and = (0 0) , so = 2. In the second step, 2 a; b − c; d 2 we take = 2 2 = 2 . To get an -approximation of , we explain the 12 The notation for the segments is ( ) ( ), meaning the segment joining these two points.

43 Figure 6: The Warsaw circle and the computational Warsaw circle. l m − G − ; − ∈ R l; m ∈ Z W process better than31 1 giving just3 the1 coordinates3 1 2 of the points. Consider the intersection 2 of a grid of side 2 , = ( 2 2 ) : with . See ﬁgure 7. Every

G W ε W

2 2 FigureW 7: The intersection of the grid with and the -approximation of . < ε point of , notW in the upper13 left square of the grid, andε the one just below it, are at 2 distance less or equal to 2 . Concerning the two mentioned squares, we see that 2 everyε point of insideW them are at distance less than , except the two centers of the squares, which are exactly at this distance. So, we add these two points and then have 2 an approximationA of G ,∩ W ∪ ; − ; ; − : 2 2 13 13 13 33 √ √ = ( ) 1 1 ε −γ 2 γ 2 2 ε2 < − 2 2 ε W 13 13 62 24 1 2 2 3 2 2 2 FromG the picture,l ; wem can∈ easilyl; msee∈ that = . We pick = W = . − 3− − R Z To61 1 obtain an 6-approximation1 6 1 2 of , we proceed as before.− Consider the grid of side 2 3 2 2 , = ( ) : and its intersection4 with . Then√ add the centers of the upper left square of the grid, and the 15 = 2 1 below it (16 pointsk of 13 k 2 We want some regularity on the epsilon approximations. All of them will be of the form 2 . In this case, there is no lower than 6 in the inequality. This will be proven for the general case, later.

44 W ε W

3 in total),A toG obtain∩ W an∪ approximation; − ; of ; (see− ﬁgure;:::; 8), ; − : 3 3 1 1 1 3 1 31 = ( ) 6 1 6 6 1 6 6 1 6 2 2 2γ 2 2 2

16 3 Now, it is again clear from the picture, that = 2 so we can continue this process to

G W ε W √ 3 3 Figure 8: The intersection of the grid withεn nand− the approximation of n.− l m Gn n− ; n− ∈ R l; m ∈ Z 3 23 W 31 4 2 2 n− the inﬁnity3 in4 the3 4 sameεn way.2 In general, let = . Consider the grid of side , 2 2 3 =5 : . Then, its intersection with and the following 2 points, form an approximation, k − n− An Gn ∩ W ∪ n− ; − n− k ;:::; : 3 5 1 2 1 = ( ) 3 3 1 √ 3 3 : = 1 2 γn n− 2 δn n−2 m n − 31 3 3 23 √ 14 2 2 It is clear that, again, = and εn =− γn . So, writing− = 3 3, we need εn < m :

+1√ 2 +1 1 √ √ ε = k ∈ < − n √ k 2 2 √N k m k− m k− m > 2 > > k >2 m 2+11 +1 2 2 2 We want( +1) tok bem of the form , so we are looking( +1) for , such that, , √ √ √ i.e., 2 2 + 2. We can estimate 2 2 + 2 2 so + 2. k > m εn m n n −

But, actually,εn = + 2 does not satisfy+32 the ﬁrst32 inequality,3( +1)2 3 so we cann − take anyn− +1 2 2 2 + 3, andl hence,m we choose = = = n .− It is clear,n− that we can Gn n− ; n− ∈ R l; m ∈ Z W 3( +1)1 4 31 1 +1 2 2 consider an3 1 -approximation3 1 2 as before, intersecting the3( grid+1) of5 side3 2 = , +1 2 2 = : with k and− add 2 = 2n− points: An Gn ∩ W ∪ n ; − n k ;:::; : 3 2 +1 +1 1 2 1 = ( ) 3 1 3 : = 1 2 γn n − n 2 2

3( +1)1 3 13 +1 n − ε m Then = 2 = 2 and the process is proved to work by induction. 14 The term 3 3 relates the exponent of the denominator with the subindex of each . We use the notation for a moment to understand how the denominator is increased in each step without perturbations of another notations.

45 A Now, we focus on the Alexandrov-McCord sequence related to this finite approximative 1 sequence. The finite space is just a point, so its associated simplicial complex is just a vertex. In the second step, we have a more interesting case. In figure 9, we have depicted

V ε A

2 2 2 Figure 9: The realization of the simplicial complex ( ) in two perspectives: Lateral V A and Aerial. ε 0 2 K U ε A V A 2 2 ε the polyhedron ( ) in two different perspectives. The barycentric subdivision2 of this 2 2 2 2 2 polyhedron isU exactlyε \A the realization of the simplical complex ( ( )) = ( ). Actually, the vertices that are not depicted but belong to the subdivision are the points 2 2 2 of the space . The 1-simplices of this polyhedron are clear from the picture. But Wthere is more structure. First of all there are two empty squares. At their left, there are two piramids whose cusps represent the points added to the intersection of the grid and . Between the two piramids, there is a tetrahedron sharing one face with each one of them. The four points ofε the tetrahedron are the two points added and the two points in common of the two squares (the base of each piramid), which, in the approximation, 2 have diameter less than 2 , so this tetrahedron is “filled”. We have to point outV thatε theA 0 piramids are empty, that is, their four faces areK simplicesU ε A thatV are inA the polyhedron,3 but ε 2 3 there is no “solid” base. For the third step, we also3 depicted the polyhedron ( ) 2 3 2 3 3 (figure 10), whose barycentric subdivision is ( ( )) = ( ). The structure of

V ε A

2 3 3 Figure 10: The realization of the simplicial complex ( ) in two perspectives: Lateral and Aerial. − this polyhedron is the same as the4 previous one. The4 diference is that it has more4 1- simplices, more empty squares (2 ) more piramids (2 ) and more tetrahedrons (2 1).

46 n− εn n− − 3 5

In general, for any 3 approximation5 we will have the same structure, with 2 squares and piramids and 2 1 tetrahedrons. Concerning the maps, we can use pictures to see where they send the points of the approximations, and the sets of those points, but we will focus our attention on the induced maps in homology which actually will tell us the behavior of the maps. Z H K H K Z We now study the previous sequence at the homological level. We will compute the 1 1 first homology group with coefficients in (with notation ( ) := V( ε; A)) of each polyhedron of the sequence and how theK inducedU ε A homology maps work. For2 the first 2 2 approximation, everything is trivial. For the second one, we know that ( ) (in the 2 2 2 figure) has the same homotopy type as ( ( )). It has three 1-cycles: The “big” one and the two littleZ squares. There is noH moreK U 1-homologyε A ' Z on this complex. This is clear from the aerial perspective in figure 9. So, the homology group3 of this polyhedron 1 2 2 2 is just three copies of ,Z whichH weK denoteU ε A ('(Z ( ))) 4. In the third step, as 4 we can see4 in picture 10, there is again one “big” 1-cycle,2 +1 and 2 small squares. I.e., a 1 2 3 3 total of 2 + 1 copies of , so ( ( ( ))) . We are interested in the map induced in homology by thep ; mapK U ε A −→ K U ε A :

2 3 2 3 3 2 2 2 : ( ( )) ( ( )) 15

KWeU needε A to study, for each 1-cycle, where the verticesK U ε areA sent by the map An easy reasoning3 shows that every vertex (included the non drawn2 ones) in the smallK U 1-cyclesε A of 2 3 2 2

( ( )) are sent to null homologousK U ε A cycles in ( ( )) (let us say, they “fall”3 into 2 3 the shaded part which is the contratible part). However, the big 1-cicle of ( ( )) is 2 2 2 mapped into the “big” one of ( ( )) (actually, it is mapped into something bigger which retracts into thisp cycle).; ∗ H So,K U it isε clear,A −→ thatH theK inducedU ε A map; in homology,

2 3 1 2 3 3 1 2 2 2 ( ) : ( ( ( ))) ( ( ( ))) 4 sendsp ; the∗ 2 Zgenerators corresponding to the little squares to zero, and the generator n− of the “big” 1-cycle to the generatorK ofU theε A “big”n one of the target. So, we get that 2 3 n Im(( ) ) = . It is readily seen that, if we consider the nextn step,− it will happen the H K U εn An 3 '5 Z 2 3 5 same. In general, the realization of ( ( )) has 2 1-cycles2 +1 corresponding to 1 2 little squares and one “big” 1-cycle. So, ( ( ( ))) . The map induced pn;n K U εn An −→ K U εn An by +1 2 +1 +1 2 : ( ( )) ( ( )) pn;n ∗ H K U εn An −→ H K U εn An ; in homology, n− +1 1 2 +1 +1 1 K2 U A ( ) : ( ( ( ))) ( ( ( εn))) n 3 2 +1 K U ε An 2 +1 n sends the 2 1-cycles corresponding to little squares of ( ( )) to zero and 2 the 1-cycle corresponding to the “big” one to the “big” one in the image ( ( )). 15 We can visualize the performance of the map by overlying the pictures of the two consecutive approximations, since the map acts in terms of proximity.

47 pn;n ∗ Z

+1 So, again, the image of the map is im(( ) ) = . So, we see that in each step, the “big” 1-cycle is the only non-trivial homology that comes from the image of the previous polyhedron. We could say that the “big” cycle is the only one that survives (or persists) in the whole sequence. In terms of the inverse16 limit, it is clear that the inverse limit of p ; ∗ p ; ∗ pn− ;n ∗ pn;n ∗ pn ;n ∗ the inverseH K sequence←−−− H K induced←−−− in::: homology←−−−− H Kn ←−−−− H Kn ←−−−−− :::: ( 0 1) ( 1 2) ( 1 ) ( +1) ( +1 +2) 1 1 1 2 1 1 +1 ( ) ( ) ( ) ( )

{Kn; pn;n ∗}' : ← Z is +1 lim ( ) W

Hence, the finite approximations are capturing the Čechn < homology m of n; nin the; : : limit. : ; m But, more important, the inverse persistence is able to capture it in a finite number of steps. In this particular example, we consider, for any the steps + 1 . Then, inverse persistence generate an inverse barcode as in figure 11.::: The largest :::

:::

| n n m [ ] + 1 BW

Figure 11: The barcode . W line correspond to theW generator of the “big” hole of , which is present in every step of the construction and preserved by the maps. Hence, it represents the ﬁrstK Čechn homology group of . Ini− the other hand, the shorti linesn; n correspond; : : : ; m to the generators n − of the “fake” and small3 homology5 groupsKn that are created in theK constructionn of at each step. They3( +1) are5 2 generators for each = + 1 as we computed +1 before. The 2 generators of are mapped to zero in and hence they do not generate any long line in theW inverse barcode. Whence, the only feature detected by the inverse persistence is the only that actually exists. We hope that experiments carried out in noisy samples of could reveal a good behavior of this construction in detecting diﬃcult features of spaces.

Kn K U εn An 16 2 Notation: := ( ( ))

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