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Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Minimal structures
  5. 3 The Stone space of minimal structures
  6. 4 The ultrametric space of minimal structures
  7. 5 Gromov-Hausdorff spaces and compactness
  8. References

For a signature L with at least one constant symbol, an L-structure is called minimal if it has no proper substructures. Let inline image be the set of isomorphism types of minimal L-structures. The elements of inline image can be identified with ultrafilters of the Boolean algebra of quantifier-free L-sentences, and therefore one can define a Stone topology on inline image. This topology on inline image generalizes the topology of the space of n-marked groups. We introduce a natural ultrametric on inline image, and show that the Stone topology on inline image coincides with the topology of the ultrametric space inline image iff the ultrametric space inline image is compact iff L is locally finite (that is, L contains finitely many n-ary symbols for any inline image). As one of the applications of compactness of the Stone topology on inline image, we prove compactness of certain classes of metric spaces in the Gromov-Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Minimal structures
  5. 3 The Stone space of minimal structures
  6. 4 The ultrametric space of minimal structures
  7. 5 Gromov-Hausdorff spaces and compactness
  8. References

In the final remarks in his paper [7], Gromov explained how to deduce from the main result—virtual nilpotency of any finitely generated group of polynomial growth—the following more precise version of the result:

For any positive integers inline image, there exists a positive integer m such that any n-generated group, in which for all inline image the size of the ball of radius r centered at the identity is at most inline image, has a subgroup of index and nilpotency class at most m.

For a proof of that version, he introduced and used a notion of limit of a sequence of groups with distinguished n generators. Implicitely, he defined a topology on the class of such groups, and used its compactness, as well as closedness of a certain subclass. Van den Dries and Wilkie gave a new proof of the result above by means of model-theoretic compactness theorem instead of Gromov's topological compactness argument [12].

Formalizing Gromov's idea, Grigorchuk suggested a precise definition of the topology used by M. Gromov, and showed that the defined topological space is metrizable, separable, compact, and has a base consisting of clopen sets [6]. That topological space, the space of n-marked groups, has been the subject of papers [3-5].

In the present paper we look at the space of marked groups from a model-theoretic point of view, and introduce a more general space of minimal structures.

For a signature L containing at least one constant symbol, an L-structure is called minimal if it has no proper substructures. For example, any n-marked group is a minimal L-structure, where L is the language of groups with added n constant symbols.

It is easy to show that the isomorphism type of a minimal L-structure is completely determined by its quantifier-free theory (Proposition 2.1), and a set S of quantifier-free L-sentences is the quantifier-free theory of a minimal L-structure iff S is a maximal finitely satisfiable set of quantifier-free L-sentences (Proposition 2.2). The set inline image of all such S can be equipped with a topology τ, a basis of which consists of the sets inline image where φ is a quantifier-free L-sentence. The topological space inline image is naturally homeomorphic to the Stone space of the Boolean algebra of quantifier-free L-sentences; therefore it is compact and totally disconnected. Therefore we call inline image the Stone space of isomorphism types of minimal L-structures. The space of isomorphism types of n-marked groups is just a clopen set in the Stone space inline image for a certain L.

We show that the ‘bounded’ version of Gromov's theorem formulated above can be deduced from its standard version using not model-theoretic compactness theorem as it was done in [12], but only compactness of the Stone space inline image.

For any universally axiomatizable class inline image of L-structures, the set inline image of isomorphism types of minimal L-structures in inline image is closed in inline image (Proposition 3.5). Let inline image be a variety of L-structures and inline image its subvariety. We show that inline image is clopen in inline image iff the inline image-free minimal L-structure is finitely presentable in inline image (Proposition 3.7). For example, for any group variety inline image, the set of isomorphism types of n-marked inline image-groups is clopen in the space of isomorphism types of n-marked groups iff the inline image-free group of rank n is finitely presentable.

For an arbitrary set X of minimal L-structures, we characterize in terms of ultraproducts the limit points of X in the Stone topology (Proposition 3.10).

As the Stone space of a Boolean algebra is metrizable iff the Boolean algebra is at most countable, the space inline image is metrizable iff L is at most countable. For an arbitrary L, we define a natural ultrametric on inline image as follows. For two minimal L-structures M and N, the distance between their quantifier-free theories is defined to be equal to inline image, where m is maximal with the property that M and N satisfy the same atomic L-sentences of length at most m. We study the properties of that ultrametric and its relation with the Stone topology on inline image. We show that the topology of the ultrametric space inline image is finer or equal than the Stone topology on inline image; the two topologies coincide iff the signature L is locally finite. (We call L locally finite if L contains finitely many n-ary symbols for any n.) In particular, the ultrametric space inline image is compact iff L is locally finite (Theorem 4.4).

As an application of compactness of the Stone space of minimal structures we give a proof of compactness of certain subclasses in the Gromov-Hausdorff space of metric spaces (Theorem 5.1 & Corollary 5.2). This refines the known result based on Gromov's ideas [2], [8] that any uniformly totally bounded class of compact metric spaces is precompact in the Gromov-Hausdorff topology. For the proof, we associate with every semi-metric space certain relational structures with the same universe called semi-metric structures; the class of such structures is shown to be universally axiomatizable.

For basics of model theory, cf. [9]. The facts and notions of metric geometry we need can be found in [2].

2 Minimal structures

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Minimal structures
  5. 3 The Stone space of minimal structures
  6. 4 The ultrametric space of minimal structures
  7. 5 Gromov-Hausdorff spaces and compactness
  8. References

Let L be a signature containing at least one constant symbol; in this case the set inline image of ground L-terms (that is, the terms without free variables) is not empty. We call an L-structure minimal if it has no proper substructures, or, equivalently, is generated by the empty set. Clearly, an L-structure is minimal iff any its element is the value of some ground L-term in the structure. For any L-structure M the substructure generated by the empty set is a unique minimal substructure; we call it the core of M and denote by core(M). We denote the class of all minimal L-structures by inline image.

Let L0 be an arbitrary signature, C a nonempty set of constant symbols disjoint with L0, and inline image. Clearly, an L-structure M is minimal if and only if the set inline image generates its L0-reduct M0. Thus, any structure becomes minimal after naming its generators. We call minimal L-structures C-marked L0-structures. For any L-structure M its core is a C-marked L0-structure—it is the minimal substructure generated by the set inline image.

The notion of marked structure generalizes the notion of marked group [4], which is defined to be a group with distinguished generators (not necessarily all distinct). In this case inline image, and C consists of names of generators of the group. Note that here we do not assume that the group is finitely generated, and C is finite. If C is finite, inline image, then C-marked groups are called n-marked groups.

Let QFL be the set of all quantifier-free L-sentences. For an L-structure M we denote by inline image the quantifier-free theory of M, that is, the set of sentences in QFL that hold in M, and by inline image the set consisting of all atomic or negated atomic L-sentences from inline image. We shall need the following essentially known facts:

Proposition 2.1. For minimal L-structures M and N the following are equivalent:

  1. 1.
    inline image;
  2. 2.
    inline image;
  3. 3.
    inline image.

Proof. inline image is obvious. If (3) then the map inline image is a well-defined isomorphism from M onto N, and so (1). inline image

Due to this fact, we call inline image the isomorphism type of a minimal L-structure M.

Proposition 2.2. For inline image the following are equivalent:

  1. 1.
    inline image, for some minimal L-structure M,
  2. 2.
    S is a maximal finitely satisfiable subset of QFL;
  3. 3.
    S is finitely satisfiable, and for any inline image either inline image or inline image.

Proof. inline image is easy; we prove (3)[RIGHTWARDS DOUBLE ARROW](1). By (3), inline image is an equivalence relation on inline image. Denote by [t] the equivalence class of inline image. Let M be the L-structure whose universe is inline image, and inline imageinline image, for any function L-symbol f and relation L-symbol R of arity n. Due to (3), inline image and inline image are well-defined. By induction, inline image, for any inline image. Then inline image iff inline image, for any inline image. Using (3), it is easy to show by induction that inline image iff inline image, for any inline image. Thus inline image. Since [t] is inline image for any inline image, the structure M is minimal. inline image

Since, by Zorn's lemma, any finitely satisfiable subset of QFL can be completed to a maximal such subset, we have:

Corollary 2.3. (Herbrand's theorem) Any finitely satisfiable subset of QFL has a minimal model.

Herbrand's theorem is a weak version of model-theoretic compactness theorem. This version admits a simple proof given above, and in the present paper we need only this version of compactness theorem.

Denote by inline image the set of all maximal finitely satisfiable subsets of QFL. By Proposition 2.2, this is the set of isomorphism types of minimal L-structures.

3 The Stone space of minimal structures

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Minimal structures
  5. 3 The Stone space of minimal structures
  6. 4 The ultrametric space of minimal structures
  7. 5 Gromov-Hausdorff spaces and compactness
  8. References

3.1 Topology on inline image

It is easy to see that, for any inline image and inline image,

  1. 1.
    inline image
  2. 2.
    inline image
  3. 3.
    inline image.

In other words, if inline image, we have

  • display math(1)
  • display math(2)
  • display math(3)

Due to (1), inline image is a basis of a topology on inline image; we denote the topology by τ. Due to (3), the sets inline image are clopen in τ. It is easy to show that inline image iff φ and ψ are equivalent.

Let T be the set of finite conjunctions of atomic or negated atomic L-sentences. Since any inline image is equivalent to a finite disjunction of sentences from T then, due to (2), inline image is a basis of τ as well.

Proposition 3.1. The topological space inline image is totally disconnected and compact.

Proof. (totally disconnected) Let S and P be different elements of inline image. Let, say, inline image and inline image. Then inline image, and inline image. Since inline image, by (3), both inline image and inline image are open, and the result follows.

(compact) Suppose inline image covers inline image, where inline image. Then inline image, that is, there is no inline image with inline image. Then, for some finite inline image, there is no inline image with inline image; otherwise inline image would be finitely satisfiable, and so could be completed to a member of inline image, by Zorn's lemma. Hence inline image, and so inline image covers inline image. inline image

The proof of compactness of the topology τ did not use the model-theoretic compactness theorem (nor its weaker version, Herbrand's theorem).

Proposition 3.2. Any set clopen in τ is inline image, for some inline image.

Proof. Any set U open in τ is inline image, for some inline image. If U is closed, it is compact, by Proposition 3.1(ii), and hence inline image, for some finite inline image. By (2), inline image, where inline image. inline image

For inline image, denote by [φ] the set of all inline image equivalent to φ. The sets [φ] form a Boolean algebra with the operations induced by the logical operators ∧, ∨, and ¬. We denote that Boolean algebra by inline image, and its Stone space by inline image.

Recall that for a Boolean algebra inline image its Stone space inline image is defined to be the topological space whose points are ultrafilters of inline image, and a basis of topology is inline image, where

  • display math

It is known that inline image is compact and totally disconnected [[10], Section 8]; it is metrizable iff it has a countable basis iff inline image; its clopen sets are exactly the sets inline image. Any closed subspace X of inline image is a compact, totally disconnected space; its clopen sets are exactly the sets inline image, and they form a basis of X.

For inline image denote inline image by [T]. It is not difficult to show that inline image is a bijection between inline image and the set of ultrafilters of inline image. Moreover, inline image, for any inline image. Therefore inline image is a natural homeomorphism between the topological space inline image and the Stone space inline image. Because of that, we call τ the Stone topology on inline image. Since inline image is the set of isomorphism types of minimal L-structures, we call the topological space inline image the Stone space of isomorphism types of minimal L-structures, or, for short, the Stone space inline image.

As inline image iff inline image, the Stone space inline image is metrizable iff inline image. Since any compact metric space is separable, inline image is separable if inline image. Thus, the Stone space of minimal L-structures is metrizable and separable if inline image.

For an L-sentense φ denote by inline image the class of minimal models of φ, and by inline image the set of isomorphism types of minimal models of φ. In other words,

  • display math

Clearly, for inline image,

  • display math

Thus for any inline image the set inline image is a clopen subspace of the Stone space inline image.

Proposition 3.3. If φ is an existential L-sentence then inline image is open.

Proof. Let φ be inline image where ψ is quantifier-free. Clearly, φ holds in a minimal L-structure M iff inline image for some inline image. Therefore inline image is the union of all sets inline image, where inline image. Since all inline image are clopen, inline image is open. inline image

Proposition 3.4. If φ is a universal L-sentence then inline image is closed.

Proof. The sentence φ is equivalent to inline image for some existential L-sentence ϑ. Then the complement of inline image in inline image is the set inline image, which is open by Proposition 3.3. inline image

For an L-theory T, denote by inline image the set of isomorphism types of minimal models of T.

Proposition 3.5. If T is a universal L-theory then inline image is closed.

Proof. Since inline image this follows from Proposition 3.4. inline image

Similarly to the Stone topology on inline image, one can define a topology on the class inline image a basis of which consists of the classes inline image, where inline image. We call that topology the Stone topology on inline image. The class inline image equipped with that topology is called the Stone space of minimal L-structures, or, for short, the Stone space inline image. Obviously, analogues of Propositions 3.13.5 hold for it, with one exception: the Stone space inline image is not Hausdorff (and so not totally disconnected), because any isomorphic but different members of inline image cannot be separated by open sets. Note that compactness of the Stone space inline image is based on Herbrand's theorem.

If inline image, we call the Stone space inline image the Stone space of isomorphism types of C-marked L0-structures. Let inline image, and γ be the universal L0-sentence that axiomatizes the class of groups. Then inline image is a closed subspace of the Stone space inline image, by Proposition 3.5. Its points are isomorphism types of groups with generators marked by elements of C. We call this topological space the space of isomorphism types of C-marked groups and denote it by inline image. The space inline image is compact and totally disconnected.

For inline image, the set inline image is the set of isomorphism types of C-marked groups satisfying ψ; it is clopen in inline image. Any clopen set in inline image is of that form, and the sets inline image form a basis of inline image. Moreover, for the set Ψ of finite conjunctions of L-sentences of the form inline image or inline image, where w is a group word over C, the set inline image is a basis of the space inline image.

For a finite set of constant symbols C with inline image, the space inline image is exactly the space of isomorphism types of n-marked groups introduced in [6]; we denote it by inline image.

Proposition 3.6. The set inline image is clopen in inline image iff C is finite.

Proof. Suppose C is finite. Let Θ be the set of quantifier-free L-sentences

  • display math

for all constant symbols inline image in C. Clearly, Θ is finite. It is easy to show that inline image where inline image. Therefore inline image is clopen.

Now we show that if C is infinite then inline image is not clopen. Suppose not, and inline image, where inline image. Let inline image be the finite set of all inline image that occurs in ϑ. Consider any inline image-marked group inline image. It is easy to construct a minimal L-structure N such that inline image is a substructure of its inline image-reduct, and the L0-reduct of N is not a group. Since any L-expansion of inline image belongs to inline image, we have inline image. Therefore inline image, and hence inline image. Contradiction. inline image

A special case of Proposition 3.4 was proven in [[4], Section 'Uniform boundedness and compactness']: for any universal sentence ϑ in the group language, the set of isomorphism types of n-marked groups satisfying ϑ is closed in inline image. This fact is slightly weaker than Proposition 3.4: for example, it does not imply closedness of the set inline image of isomorphism types of n-marked centerless groups, because the class of centerless groups is not closed under subgroups and therefore is not universally axiomatizable. However, Proposition 3.4 implies that inline image is closed in inline image, because for any finite C the class of C-marked centerless groups is axiomatizable by the universal sentence

  • display math

Note that inline image is not open in inline image if inline image. Indeed, let G be a free group of rank n, and inline image a free k-nilpotent group of rank n; then inline image, where inline image is the k-th member of the lower central series of G. Consider G and inline image as groups with marked free generators. Then G is a limit of the sequence inline image; this follows from the well-known fact that inline image. But G is centerless, and all inline image are not.

For a variety inline image of L-structures, we call a inline image-free structure generated by the empty set a inline image-free minimal structure. Denote by inline image the set of isomorphism types of minimal L-structures from inline image.

Proposition 3.7. Let inline image and inline image be varieties of L-structures, and inline image. The following are equivalent:

  1. 1.
    inline image is clopen in inline image;
  2. 2.
    the inline image-free minimal structure N is finitely presentable in inline image.

Proof. (2)[RIGHTWARDS DOUBLE ARROW](1). Suppose N is finitely presented in inline image by atomic L-sentences inline image. Then

  • display math

where inline image. So inline image is clopen in inline image.

(1)[RIGHTWARDS DOUBLE ARROW](2). Suppose inline image is clopen in inline image. Since inline image is closed by Proposition 3.5, inline image for some inline image. We may assume that φ is a finite disjunction of sentences of the form

  • display math

where all inline image are atomic L-sentences. Then one of these disjuncts—say, the disjunct written above—holds in N. Let M be the minimal L-structure presented in inline image by the relations inline image. Then there is a homomorphism from M onto N. Hence all inline image hold in M. Therefore φ holds in M, and so inline image. Since N is inline image-free there is a homomorphism from N onto M. Hence this homomorphism is an isomorphism. Thus N is finitely presented in inline image. inline image

Corollary 3.8. Let inline image be a group variety, and inline image. Then the set of isomorphism types of n-marked inline image-groups is clopen in inline image iff the inline image-free group of rank n is finitely presented.

For example, if inline image is any nilpotent group variety then the class of n-marked inline image-groups is clopen in inline image. Since, by inline imagemel'kin's theorem [11], for inline image the n-generated free m-solvable group is not finitely presented (for a more general result, cf. [1]), the class of n-marked m-solvable groups is not open in inline image. The latter fact was explained in [[4], Section 2.6] in a completely different way based on some D. V. Osin's result. Note that there is an open question posed by A. Yu. Olshanski whether any finitely presented relatively free group is virtually nilpotent.

3.2 Compactness of inline image and Gromov's theorem

Now we explain how one can use compactness of inline image to deduce from Gromov's theorem its ‘bounded’ version formulated at the beginning of the present paper.

Fix n, k, and d. Let inline image, and inline image, where inline image.

It is easy to construct inline image which says about a C-marked group that for all inline image the size of the ball of radius r centered at the identity is inline image. Also, there is inline image which says about a C-marked group that it has a nilpotent subgroup of class inline image and index inline image. We explain how to construct such inline image using the idea suggested in  [[12], Section 7].

Let F be the free group with basis C. It is known that F has only finitely many subgroups of index inline image, say, inline image; each of them is finitely generated. Let inline image be a finite set of group words over C generating inline image, and inline image be the conjunction of all equalities inline image, where inline image. We show that inline image can be taken as inline image.

Let M be a C-marked group; then there is a unique epimorphism inline image of C-marked groups. Suppose M has a nilpotent subgroup N of class inline image and index inline image. Then inline image, and hence inline image for some i. Since N is nilpotent of class inline image, we have inline image, and so inline image. Now suppose inline image; then inline image for some i. Therefore the subgroup inline image of M is nilpotent of class inline image and index inline image. Indeed, since inline image generates inline image, and inline image for all inline image, the subgroup inline image is nilpotent of class inline image. This subgroup is of index inline image in M because inline image.

Let inline image denote inline image. It is easy to see that if inline image then inline image implies inline image. Every C-marked group M satisfies inline image for some m (possibly, depending on M). Indeed, if M is virtually nilpotent then inline image for some m; if M is not virtually nilpotent then, by Gromov's theorem, M is not of polynomial growth, and therefore inline image, for some m.

Let inline image denote inline image. Then inline image is clopen in inline image. Thus inline image is an open cover of inline image. It has a finite subcover because inline image is compact. Since inline image for all m, we have inline image for some m. Thus there is m such that every C-marked group satisfies inline image, and the result follows.

Note that the proof above is based on compactness of the Stone space of C-marked groups, which follows from a general fact on compactness of Stone spaces of Boolean algebras. For a proof of the latter fact one needs only Zorn's lemma but not model-theoretic compactness theorem. The proof of the result given in [[12], Section 7] is based on model-theoretic compactness theorem; so our proof is different, even though uses the same idea.

Another way to realize that idea is to use ultraproducts. Towards a contradiction, suppose for every i there is a C-marked group inline image with inline image. If inline image then inline image because inline image implies inline image. Then, by Łoś's theorem, for any non-principal ultraproduct M of the C-marked groups inline image we have inline image, for all j. Then all inline image fail in the C-marked group core(M), contrary to Gromov's theorem.

3.3 Topology on inline image and ultraproducts

In general, there is a link between ultraproducts and the Stone topology on the class of minimal structures (cf. [[4], Proposition 6.4], where a link between ultraproducts and convergence of groups in the space of marked groups had been demonstrated).

Proposition 3.9. Let X be a subset of inline image, and inline image. Then the following are equivalent:

  1. 1.
    M belongs to the closure of X in the Stone space inline image;
  2. 2.
    M is isomorphic to the core of an ultraproduct of structures from X;
  3. 3.
    M is embeddable into an ultraproduct of structures from X.

Proof. Obviously, (2)[RIGHTWARDS DOUBLE ARROW](3).

(3)[RIGHTWARDS DOUBLE ARROW](1). Suppose M is embeddable into an ultraproduct inline image of structures from X. We show that any basic neighbourhood inline image of M, where inline image, contains an element of X. Since φ is quantifier-free and holds in M, it holds in the ultraproduct. Therefore, by Łoś's theorem,

  • display math

Hence inline image, and so inline image contains an element of X.

(1)[RIGHTWARDS DOUBLE ARROW](2). Let inline image. For inline image denote

  • display math

then inline image, because inline image, by (1). The set

  • display math

is closed under finite intersections, because if inline image then

  • display math

Therefore P has the finite intersection property, and hence can be completed to an ultrafilter D on I. For any inline image we have inline image, and hence inline image, by Łoś's theorem. It follows that any inline image holds in the core of the ultraproduct. Therefore M is isomorphic to the core, by Proposition 2.1. inline image

A point M of the Stone space inline image is called a limit point of a subset X of inline image if every open neighbourhood of M in inline image contains a member of X which is non-isomorphic to M.

Proposition 3.10. Let X be a subset of inline image, and M a structure in inline image, which is non-isomorphic to any member of X. Then the following are equivalent:

  1. 1.
    M is a limit point of X in the Stone space inline image;
  2. 2.
    M is isomorphic to the core of a non-principal ultraproduct of pairwise non-isomorphic structures from X;
  3. 3.
    M is embeddable into a non-principal ultraproduct of pairwise non-isomorphic structures from X.

Proof. Obviously, (2)[RIGHTWARDS DOUBLE ARROW](3).

(3)[RIGHTWARDS DOUBLE ARROW](1). Suppose M is embeddable into inline image, where inline image is a family of pairwise non-isomorphic structures from X, and D is a non-principal ultrafilter on I. We need to show that any basic neighbourhood inline image of M contains an element of X non-isomorphic to M. Since φ is quantifier-free and holds in M, it holds in the ultraproduct. Therefore, by Łoś's theorem,

  • display math

Since the ultrafilter D is non-principal, inline image. Since all inline image are pairwise non-isomorphic, there is inline image such that inline image is not isomorphic to M. Then inline image, and inline image.

(1)[RIGHTWARDS DOUBLE ARROW](2). Let inline image be a family of representatives of all isomorphism types of structures in X, which are not isomorphic to M. For any inline image, the set

  • display math

is infinite. Indeed, suppose not. By Proposition 2.1, for each i there is inline image such that inline image. Since M is a limit point of X, there is inline image which is non-isomorphic to M and such that inline image. Then none of inline image is isomorphic to N. Contradiction. The set

  • display math

is closed under finite intersections, as in the proof of (1)[RIGHTWARDS DOUBLE ARROW](2) at Proposition 3.9. Let F be the Fréchet filter on I. The set inline image has the finite intersection property: otherwise, for some inline image the set inline image is disjoint with a set from F, and hence is finite. Hence inline image is contained in an ultrafilter D on I. The ultrafilter D is non-principal because it contains F. For any inline image we have inline image, and therefore inline image, by Łoś's theorem. It follows that any inline image holds in the core of the ultraproduct. Therefore M is isomorphic to the core, by Proposition 2.1. inline image

4 The ultrametric space of minimal structures

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Minimal structures
  5. 3 The Stone space of minimal structures
  6. 4 The ultrametric space of minimal structures
  7. 5 Gromov-Hausdorff spaces and compactness
  8. References

For inline image, we say that L-structures M and N are m-close if

  • display math

for any atomic L-sentence ϑ of length inline image.

Note that minimal L-structures M and N are m-close for arbitrary large m iff inline image iff inline image, by Proposition 2.1.

For minimal L-structures M and N we define inline image, the distance between M and N, as follows. If inline image, put inline image. Otherwise inline image is defined to be inline image, where m be the maximal positive integer such that M and N are m-close.

It is easy to see that d is semi-ultrametric on inline image, that is, for any inline image

  1. 1.
    inline image, and inline image;
  2. 2.
    inline image;
  3. 3.
    inline image.

Since inline image iff inline image, the semi-ultrametric d induces an ultrametric on the set of isomorphism types of minimal L-structures, that is, on inline image. We denote the induced ultrametric on inline image by the same letter d; so for any inline image, we have inline image, where inline image and inline image.

Clearly, for any inline image, we have

  1. (i)
    inline image, and
  2. (ii)
    inline image means that inline image iff inline image, for any atomic L-sentence ϑ of length inline image.

We call inline image and inline image the semi-ultrametric and ultrametric space of minimal L-structures, respectively.

Clearly, in inline image and inline image for any point x the open ball inline image is the whole space if inline image. If for a positive integer m

  • display math

then the open ball inline image is equal to the closed ball inline image. Thus in the spaces inline image and inline image any open ball is a closed set. It follows that the metric space inline image is totally disconnected.

Proposition 4.1. For any inline image, the set inline image is clopen in the ultrametric space inline image.

Proof. Since a boolean combination of clopen sets is clopen, we may assume that φ is atomic. Let m be the length of φ. Denote inline image by U. For any inline image, if inline image then inline image and if inline image then inline image So U is clopen. inline image

Since inline image is a basis of the Stone topology on inline image, we have

Corollary 4.2. The ultrametric topology is equal to or finer than the Stone topology on inline image.

In general, the two topologies do not coincide: in the Stone space inline image the clopen sets are exactly inline image, where inline image, but in the ultrametric space inline image it is not always so. For example, in Proposition 3.6 we proved that if C is infinite then inline image, for any inline image. However,

Proposition 4.3. The set inline image is clopen in the ultrametric space inline image, for any C.

Proof. Let Θ be defined as in the proof of Proposition 3.6; then

  • display math

Since all inline image are clopen, inline image is closed. Also, inline image is open because if inline image, and m is the maximal length of sentences in Θ, then inline image. inline image

We call a signature L locally finite if for every n the set of n-ary symbols in L is finite. Clearly, any locally finite signature is finite or countable.

Theorem 4.4. The following are equivalent:

  1. 1.
    any clopen set in the ultrametric space inline image is inline image for some inline image;
  2. 2.
    any open ball in the ultrametric space inline image is inline image for some inline image;
  3. 3.
    the Stone and ultrametric topologies on inline image coincide;
  4. 4.
    the ultrametric space inline image is compact;
  5. 5.
    the ultrametric space inline image is separable;
  6. 6.
    the signature L is locally finite.

Proof. We prove inline image and inline image.

(1)[RIGHTWARDS DOUBLE ARROW](2) because every open ball in inline image is closed.

(2)[RIGHTWARDS DOUBLE ARROW](3). Due to (2), every open set in the ultrametric space inline image is open in the Stone topology on inline image. Together with Corollary 4.2, this gives (3).

(3)[RIGHTWARDS DOUBLE ARROW](4) because the Stone space inline image is compact, which is a consequence of Herbrand's theorem.

(4)[RIGHTWARDS DOUBLE ARROW](5) because any compact metric space is separable.

(5)[RIGHTWARDS DOUBLE ARROW](6). Suppose there are infinitely many n-ary symbols in L. We show that inline image is not separable.

First we show that there is a family inline image of atomic L-sentences of the same length m such that for any inline image the set of sentences

  • display math

holds in some minimal L-structure inline image.

Let c be a constant symbol in L. If L contains infinitely many distinct and different from c constant symbols inline image, one can take the sentence inline image as inline image. If L contains infinitely many distinct n-ary function symbols inline image, where inline image, one can take the sentence inline image as inline image. If L contains infinitely many distinct n-ary relation symbols inline image one can take the sentence inline image as inline image. Clearly, for such choice of inline image the set inline image holds in some L-structure, and hence in its core inline image.

We prove that no countable subset is dense in inline image. To show that, we construct for any sequence inline image in inline image a member of inline image which is not m-close to inline image for every inline image. Let

  • display math

Then for any i the structure inline image is not m-close to inline image. Indeed, if inline image then inline image and inline image; if inline image then inline image and inline image.

(6)[RIGHTWARDS DOUBLE ARROW](2). Let inline image, and inline image. We show that inline image for some inline image. Since L is locally finite, the set of atomic L-sentences of length inline image is finite. Let φ be the conjunction of all sentences from inline image of length inline image. Then inline image means exactly that P and S are inline image-close, that is, inline image.

inline image Let U be a clopen set in the ultrametric space inline image. Since U is closed, it is compact, by (4). Since U is open, it is a union of open balls, and so a union of finitely many open balls inline image, by compactness of U. By (2), each inline image is inline image, for some inline image. Then inline image, where inline image. inline image

Corollary 4.5. If L is locally finite then all subspaces of the ultrametric space inline image are separable.

Proof. For metric spaces separability is equivalent to existence of a countable base, which is a hereditary property. inline image

5 Gromov-Hausdorff spaces and compactness

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Minimal structures
  5. 3 The Stone space of minimal structures
  6. 4 The ultrametric space of minimal structures
  7. 5 Gromov-Hausdorff spaces and compactness
  8. References

5.1 Gromov-Hausdorff distance

First we recall some notions and facts of metric geometry [[2], Chapter 7]. We already used above the notion of semi-metric; we shall need a bit more general definition of semi-metric, in which distances between points can be infinite.

A map inline image is called a semi-metric on X if d is nonnegative, symmetric, satisfies the triange inequality, and inline image for every inline image. A semi-metric is called a metric if inline image for different inline image.

A set (or, more generally, a class) equipped with a (semi-)metric is said to be a (semi-)metric space. Usually, the set and the space are denoted with the same letter, and the (semi-)metric of the space X is denoted by inline image.

Like a metric, any semi-metric d on X defines a topology on X in a usual way; this topology is Hausdorff iff d is a metric.

We shall use the following easy observations. Let X and Y be semi-metric spaces, and inline image be surjective and distance-preserving. Then if A is a compact subset of X then inline image is a compact subset of Y, and if B is a compact subset of Y then inline image is a compact subset of X.

For a semi-metric d on X, the relation inline image is an equivalence relation on X. Denote by [x] the equivalence class of inline image, and by inline image the set of all equivalence classes. Clearly, inline image is a well-defined metric on inline image; thus inline image is a surjective and distance-preserving map from the semi-metric space X to the metric space inline image.

The Hausdorff distance inline image between subspaces X and Y of a metric space Z is defined to be the infimum of all inline image such that for any inline image there is inline image with inline image, and for any inline image there is inline image with inline image. If there is no such r then inline image. Clearly, inline image for bounded X and Y.

The map dH is a semi-metric on the set of all subspaces of Z. In general, it is not a metric: for example, inline image, for any subspace X of Z and its closure inline image in Z. However, dH is a metric on the set of closed subsets of Z.

Any two metric spaces X and Y are isometrically embeddable into a third metric space Z; for each such embeddings the Hausdorff distance between the isometric images of X and Y is defined. The infimum of Hausdorff distances between X and Y for all such embeddings is called the Gromov-Hausdorff distance between X and Y (cf. [2, 8]); it is denoted by inline image. An equivalent, often more convenient, definition [[2], Theorem 7.3.25]:

  • display math

where the infimum is taken over all maps inline image, inline image from sets I onto X, Y.

The map dGH is a semi-metric on the class of all metric spaces; we denote the corresponding semi-metric space by inline image.

The semi-metric dGH can be extended to a semi-metric on the class of all semi-metric spaces: for semi-metric spaces X and Y put

  • display math

It is easy to show that (⋆) holds for semi-metric spaces X and Y as well.

5.2 Uniform boundedness and compactness

It is known that any uniformly totally bounded class of compact metric spaces is precompact in the Gromov-Hausdorff topology (that is, its closure is compact) [[2], Theorem 7.4.15]. Here a class of metric spaces inline image is called uniformly totally bounded if for every inline image there is a positive integer inline image such that

  1. 1.
    the diameter of every space in inline image is inline image;
  2. 2.
    for any inline image any space in inline image can be covered by inline image closed balls of radius ε.

Our goal is to prove compactness of certain subclasses of inline image using compactness of the Stone space inline image for a certain L.

We call semi-metric spaces satisfying (1) and (2) ν-bounded, where

  • display math

We denote the class of ν-bounded metric spaces by inline image. So a class inline image of metric spaces is uniformly totally bounded if inline image, for some ν.

Theorem 5.1. For any ν, the class inline image is compact in inline image.

We postpone the proof until Subsection 'Compactness of inline image', because for that we need a certain correspondence between semi-metric spaces and structures, which requires some preparatory work.

Theorem 5.1 has a corollary which is a refinement of the result on precompactness of any uniformly totally bounded class of compact metric spaces in the Gromov-Hausdorff topology.

Corollary 5.2. For any ν, the class inline image of ν-bounded compact metric spaces is compact in inline image.

Proof. It suffices to show that the map inline image, where inline image is a completion of X, is a surjective distance-preserving map from inline image to inline image.

First we note that if inline image then inline image. If inline image is a dense subspace of a metric space Y, then inline image. (Indeed, first, X and Y have the same diameter, and, second, if for some inline image and inline image the closed balls inline image cover X then the closed balls inline image cover Y because otherwise the complement of inline image in Y is open and nonempty but does not meet X, contrary to density of X in Y.) So inline image. Since a metric space is compact iff it is complete and totally bounded, inline image is compact. So inline image.

Now we show that inline image maps inline image onto inline image. For inline image and inline image choose inline image closed balls inline image which cover Y. Let X be the subspace of all inline image. Then inline image and inline image.

Clearly, inline image; therefore the map inline image preserves dGH. inline image

5.3 Semi-metric structures

Now we introduce some relational signature L0, and associate with any semi-metric space X a set of certain L0-structures with universe X; we call them X-structures. An L0-structure, which is an X-structure for some semi-metric space X, will be called a semi-metric structure. We call L0 the signature of semi-metric structures.

The signature L0 consists of binary relational symbols inline image, where inline image. An L0-structure M with a universe X is called an X-structure if for any inline image and any inline image

  • display math

An example of X-structure is the L0-structure inline image on X in which for any ε

  • display math

We shall use the structure inline image in the proof of Theorem 5.1. This example is not unique: replacing ⩽ with < in the definition of inline image, we obtain another example of X-structure.

Now we show that any X-structure completely determines the space X.

For an L0-structure M and inline image denote

  • display math

Proposition 5.3. If M is an X-structure then inline image. In particular, if M is an X-structure and Y-structure then the semi-metric spaces X and Y coincide.

Proof. Let inline image. For any inline image with inline image we have inline image; so

  • display math

Suppose inline image. Choose ε with

  • display math

Since inline image, we have inline image, and hence inline image, contrary to the choice of ε. inline image

Proposition 5.4. The class of semi-metric structures is universally axiomatizable.

Proof. Let Γ be the set of universal L0-sentences

  1. (a)
    inline image, for inline image;
  2. (b)
    inline image, for inline image;
  3. (c)
    inline image,
  4. (d)
    inline image, for inline image;

where inline image run over inline image.

It is easy to check that any X-structure is a model of Γ. We show that any model M of Γ is an X-structure for some semi-metric space X.

Let M be a model of Γ. We show that inline image is a semi-metric on the universe of M. Obviously, inline image is non-negative. The axiom (a) implies that inline image is symmetric. Indeed, suppose, say, inline image. Choose ε with

  • display math

By definition of inline image, there is inline image with inline image. By (a), inline image. Hence inline image. Contradiction.

The axiom (b) implies that inline image satisfies the triangle inequality. Towards a contradiction, suppose

  • display math

Choose reals α and β such that

  • display math

By definition of inline image, there are ε and δ such that inline image and inline image. By (b), inline image. Then inline image. Contradiction.

By (c), inline image for any inline image. Let X be the semi-metric space which is the universe of M equipped with inline image. We show that M is an X-structure. By definition of inline image, if inline image then inline image; if inline image then inline image for some inline image, and therefore inline image, by (d). inline image

5.4 Compactness of inline image

In this subsection we give a proof of Theorem 5.1.

Let inline image be the class of ν-bounded semi-metric spaces of cardinality at most inline image. Then inline image is a distance-preserving map from inline image to inline image. Since any ν-bounded metric space is of cardinality at most inline image, the map is surjective. Therefore inline image is compact iff inline image is compact. We shall prove compactness of inline image.

For that we define: an extention L of the signature L0 of semi-metric structures by some constants, a universally axiomatizable subclass of inline image, and a continuous surjective map from that subclass onto inline image. Since the subclass is compact in the Stone topology on inline image, due to results of Section 'The Stone space of minimal structures', this implies compactness of inline image.

Let inline image, where C is the union of a family of pairwise disjoint sets of constant symbols inline image, with inline image, and inline image for inline image.

Let inline image be the union of Γ and the set of universal L-sentences

  1. (e)
    inline image;
  2. (f)
    inline image

Denote by inline image the class inline image. By results of Section 'The Stone space of minimal structures', the class inline image is compact in Stone topology on inline image.

Lemma 5.5. (1) For any inline image there exists inline image such that the L0-reduct of M is an X-structure;

(2) for any inline image there is a unique inline image such that the L0-reduct of M is an X-structure.

Proof. (1) Let inline image. Since inline image and X is ν-bounded, there is a inline image such that inline image, and for every inline image the closed balls of radius ε centered at inline image, where inline image, cover X.

The X-structure inline image defined in Subsection 'Semi-metric structures' has the following property:

  • display math

for all inline image and all inline image. By Proposition 5.4, inline image is a model of Γ.

Consider the L-expansion M of inline image such that inline image for all inline image. Then M satisfies (e) and (f), by the choice of f. Since inline image, the L-structure M is minimal. Thus inline image, and its L0-reduct is the X-structure inline image.

(2) As M satisfies Γ, the L0-reduct of M is an X-structure for some semi-metric space X, which is unique, by Proposition 5.3. Since M is a minimal L-structure, inline image, and so inline image. As M satisfies (e), the diameter of X is inline image. Since M satisfies (f), X is covered by the close balls of radius ε centered at inline image with inline image. Thus inline image. inline image

For inline image let inline image be the unique inline image such that the L0-reduct of M is an X-structure, which exists by Lemma 5.5(2). The map

  • display math

is surjective, by Lemma 5.5(1). Now, to complete the proof of compactness of inline image, it suffices to prove

Lemma 5.6. The map χ is continuous.

Proof. To prove that χ is continuous at inline image, we need to show that for any inline image there is inline image with inline image such that for any inline image with inline image

  • display math

For any inline image we construct a finite inline image such that, for any inline image,

  • display math

then we can take as ψ the conjunction of all sentences from inline image that hold in M0.

Choose ε with inline image and inline image. Let m be the integer with

  • display math

Let Φ be the set of all sentences inline image, where inline image and inline image. We show that the finite set Φ satisfies the required conditions.

Let inline image. Denote inline image and inline image; so the L0-reduct of M is an X-structure, and the L0-reduct of N is a Y-structure. Let

  • display math

Since X is covered by the closed balls centered at points of the set inline image, we have inline image. Similarly, inline image. Therefore by the triangle inequality,

  • display math

Hence it suffices to show that if inline image iff inline image for all inline image, then

  • display math

because this implies

  • display math

To prove inline image it suffices to show that for any inline image

  • display math

by (⋆) from Subsection 'Gromov-Hausdorff distance'. The latter inequality holds because, first, the diameters of M and N are inline image, and so

  • display math

and, second, none of the numbers inline image can be strictly between inline image and inline image: if, say,

  • display math

then inline image and inline image. The lemma is proven. inline image

The proof of Theorem 5.1 is completed. inline image

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Minimal structures
  5. 3 The Stone space of minimal structures
  6. 4 The ultrametric space of minimal structures
  7. 5 Gromov-Hausdorff spaces and compactness
  8. References
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