e-mail: olegb@bilgi.edu.tr

Original Paper

# The space of minimal structures

Article first published online: 29 JAN 2014

DOI: 10.1002/malq.201300012

© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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#### How to Cite

Belegradek, O. (2014), The space of minimal structures. Mathematical Logic Quarterly, 60: 40–53. doi: 10.1002/malq.201300012

#### Publication History

- Issue published online: 8 FEB 2014
- Article first published online: 29 JAN 2014
- Manuscript Accepted: 1 JUL 2013
- Manuscript Revised: 9 JUN 2013
- Manuscript Received: 5 MAR 2013

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### Abstract

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

For a signature *L* with at least one constant symbol, an *L*-structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal *L*-structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier-free *L*-sentences, and therefore one can define a Stone topology on . This topology on generalizes the topology of the space of *n*-marked groups. We introduce a natural ultrametric on , and show that the Stone topology on coincides with the topology of the ultrametric space iff the ultrametric space is compact iff *L* is locally finite (that is, *L* contains finitely many *n*-ary symbols for any ). As one of the applications of compactness of the Stone topology on , 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

- Top of page
- Abstract
- 1 Introduction
- 2 Minimal structures
- 3 The Stone space of minimal structures
- 4 The ultrametric space of minimal structures
- 5 Gromov-Hausdorff spaces and compactness
- 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 , there exists a positive integer

msuch that anyn-generated group, in which for all the size of the ball of radiusrcentered at the identity is at most , has a subgroup of index and nilpotency class at mostm.

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 of all such *S* can be equipped with a topology τ, a basis of which consists of the sets where φ is a quantifier-free *L*-sentence. The topological space 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 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 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 .

For any universally axiomatizable class of *L*-structures, the set of isomorphism types of minimal *L*-structures in is closed in (Proposition 3.5). Let be a variety of *L*-structures and its subvariety. We show that is clopen in iff the -free minimal *L*-structure is finitely presentable in (Proposition 3.7). For example, for any group variety , the set of isomorphism types of *n*-marked -groups is clopen in the space of isomorphism types of *n*-marked groups iff the -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 is metrizable iff *L* is at most countable. For an arbitrary *L*, we define a natural ultrametric on as follows. For two minimal *L*-structures *M* and *N*, the distance between their quantifier-free theories is defined to be equal to , 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 . We show that the topology of the ultrametric space is finer or equal than the Stone topology on ; 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 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.

### 2 Minimal structures

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

Let *L* be a signature containing at least one constant symbol; in this case the set 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 .

Let *L*_{0} be an arbitrary signature, *C* a nonempty set of constant symbols disjoint with *L*_{0}, and . Clearly, an *L*-structure *M* is minimal if and only if the set generates its *L*_{0}-reduct *M*_{0}. Thus, any structure becomes minimal after naming its generators. We call minimal *L*-structures *C**-marked* *L*_{0}*-structures*. For any *L*-structure *M* its core is a *C*-marked *L*_{0}-structure—it is the minimal substructure generated by the set .

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 , 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, , then *C*-marked groups are called *n*-marked groups.

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

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

- 1.;
- 2.;
- 3..

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

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

Proposition 2.2. For the following are equivalent:

- 1., for some minimal
*L*-structure*M*, - 2.
*S*is a maximal finitely satisfiable subset of QF_{L}; - 3.
*S*is finitely satisfiable, and for any either or .

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

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

Corollary 2.3. (Herbrand's theorem) Any finitely satisfiable subset of QF_{L} 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 the set of all maximal finitely satisfiable subsets of QF_{L}. By Proposition 2.2, this is the set of isomorphism types of minimal *L*-structures.

### 3 The Stone space of minimal structures

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

#### 3.1 Topology on

It is easy to see that, for any and ,

- 1.
- 2.
- 3..

In other words, if , we have

- (1)

- (2)

- (3)

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

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

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

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

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

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 , for some .

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

For , denote by [φ] the set of all 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 , and its Stone space by .

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

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

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

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

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

Clearly, for ,

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

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

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

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

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

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

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

Proof. Since this follows from Proposition 3.4.

Similarly to the Stone topology on , one can define a topology on the class a basis of which consists of the classes , where . We call that topology the *Stone topology on* . The class equipped with that topology is called the *Stone space of minimal* *L**-structures*, or, for short, the Stone space . Obviously, analogues of Propositions 3.1–3.5 hold for it, with one exception: the Stone space is not Hausdorff (and so not totally disconnected), because any isomorphic but different members of cannot be separated by open sets. Note that compactness of the Stone space is based on Herbrand's theorem.

If , we call the Stone space the *Stone space of isomorphism types of* *C**-marked* *L*_{0}*-structures*. Let , and γ be the universal *L*_{0}-sentence that axiomatizes the class of groups. Then is a closed subspace of the Stone space , 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 . The space is compact and totally disconnected.

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

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

Proposition 3.6. The set is clopen in iff *C* is finite.

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

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

Now we show that if *C* is infinite then is not clopen. Suppose not, and , where . Let be the finite set of all that occurs in ϑ. Consider any -marked group . It is easy to construct a minimal *L*-structure *N* such that is a substructure of its -reduct, and the *L*_{0}-reduct of *N* is not a group. Since any *L*-expansion of belongs to , we have . Therefore , and hence . Contradiction.

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* . This fact is slightly weaker than Proposition 3.4: for example, it does not imply closedness of the set 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 is closed in , because for any finite *C* the class of *C*-marked centerless groups is axiomatizable by the universal sentence

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

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

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

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

Proof. (2)(1). Suppose *N* is finitely presented in by atomic *L*-sentences . Then

where . So is clopen in .

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

where all 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 by the relations . Then there is a homomorphism from *M* onto *N*. Hence all hold in *M*. Therefore φ holds in *M*, and so . Since *N* is -free there is a homomorphism from *N* onto *M*. Hence this homomorphism is an isomorphism. Thus *N* is finitely presented in .

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

For example, if is any nilpotent group variety then the class of *n*-marked -groups is clopen in . Since, by mel'kin's theorem [11], for 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 . 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 and Gromov's theorem

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

Fix *n*, *k*, and *d*. Let , and , where .

It is easy to construct which says about a *C*-marked group that for all the size of the ball of radius *r* centered at the identity is . Also, there is which says about a *C*-marked group that it has a nilpotent subgroup of class and index . We explain how to construct such 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 , say, ; each of them is finitely generated. Let be a finite set of group words over *C* generating , and be the conjunction of all equalities , where . We show that can be taken as .

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

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

Let denote . Then is clopen in . Thus is an open cover of . It has a finite subcover because is compact. Since for all *m*, we have for some *m*. Thus there is *m* such that every *C*-marked group satisfies , 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 with . If then because implies . Then, by Łoś's theorem, for any non-principal ultraproduct *M* of the *C*-marked groups we have , for all *j*. Then all fail in the *C*-marked group core(*M*), contrary to Gromov's theorem.

#### 3.3 Topology on 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 , and . Then the following are equivalent:

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

Proof. Obviously, (2)(3).

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

Hence , and so contains an element of *X*.

(1)(2). Let . For denote

then , because , by (1). The set

is closed under finite intersections, because if then

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

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

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

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

Proof. Obviously, (2)(3).

(3)(1). Suppose *M* is embeddable into , where 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 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,

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

(1)(2). Let be a family of representatives of all isomorphism types of structures in *X*, which are not isomorphic to *M*. For any , the set

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

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

### 4 The ultrametric space of minimal structures

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

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

for any atomic *L*-sentence ϑ of length .

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

For minimal *L*-structures *M* and *N* we define , the distance between *M* and *N*, as follows. If , put . Otherwise is defined to be , 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 , that is, for any

- 1., and ;
- 2.;
- 3..

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

Clearly, for any , we have

- (i), and
- (ii)means that iff , for any atomic
*L*-sentence ϑ of length .

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

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

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

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

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 by *U*. For any , if then and if then So *U* is clopen.

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

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

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

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

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

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

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

Proof. We prove and .

(1)(2) because every open ball in is closed.

(2)(3). Due to (2), every open set in the ultrametric space is open in the Stone topology on . Together with Corollary 4.2, this gives (3).

(3)(4) because the Stone space is compact, which is a consequence of Herbrand's theorem.

(4)(5) because any compact metric space is separable.

(5)(6). Suppose there are infinitely many *n*-ary symbols in *L*. We show that is not separable.

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

holds in some minimal *L*-structure .

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

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

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

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

Let *U* be a clopen set in the ultrametric space . 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 , by compactness of *U*. By (2), each is , for some . Then , where .

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

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

### 5 Gromov-Hausdorff spaces and compactness

- Top of page
- Abstract
- 1 Introduction
- 2 Minimal structures
- 3 The Stone space of minimal structures
- 4 The ultrametric space of minimal structures
- 5 Gromov-Hausdorff spaces and compactness
- 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 is called a *semi-metric* on *X* if *d* is nonnegative, symmetric, satisfies the triange inequality, and for every . A semi-metric is called a *metric* if for different .

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 .

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 be surjective and distance-preserving. Then if *A* is a compact subset of *X* then is a compact subset of *Y*, and if *B* is a compact subset of *Y* then is a compact subset of *X*.

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

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

The map *d*_{H} is a semi-metric on the set of all subspaces of *Z*. In general, it is not a metric: for example, , for any subspace *X* of *Z* and its closure in *Z*. However, *d*_{H} 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 . An equivalent, often more convenient, definition [[2], Theorem 7.3.25]:

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

The map *d*_{GH} is a semi-metric on the class of all metric spaces; we denote the corresponding semi-metric space by .

The semi-metric *d*_{GH} can be extended to a semi-metric on the class of all semi-metric spaces: for semi-metric spaces *X* and *Y* put

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 is called *uniformly totally bounded* if for every there is a positive integer such that

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

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

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

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

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

We postpone the proof until Subsection 'Compactness of ', 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 of ν-bounded compact metric spaces is compact in .

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

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

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

Clearly, ; therefore the map preserves *d*_{GH}.

#### 5.3 Semi-metric structures

Now we introduce some relational signature *L*_{0}, and associate with any semi-metric space *X* a set of certain *L*_{0}-structures with universe *X*; we call them *X**-structures*. An *L*_{0}-structure, which is an *X*-structure for some semi-metric space *X*, will be called a *semi-metric structure*. We call *L*_{0} the *signature of semi-metric structures*.

The signature *L*_{0} consists of binary relational symbols , where . An *L*_{0}-structure *M* with a universe *X* is called an *X*-*structure* if for any and any

An example of *X*-structure is the *L*_{0}-structure on *X* in which for any ε

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

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

For an *L*_{0}-structure *M* and denote

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

Proof. Let . For any with we have ; so

Suppose . Choose ε with

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

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

Proof. Let Γ be the set of universal *L*_{0}-sentences

- (a), for ;
- (b), for ;
- (c),
- (d), for ;

where run over .

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 is a semi-metric on the universe of *M*. Obviously, is non-negative. The axiom (a) implies that is symmetric. Indeed, suppose, say, . Choose ε with

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

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

Choose reals α and β such that

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

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

#### 5.4 Compactness of

In this subsection we give a proof of Theorem 5.1.

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

For that we define: an extention *L* of the signature *L*_{0} of semi-metric structures by some constants, a universally axiomatizable subclass of , and a continuous surjective map from that subclass onto . Since the subclass is compact in the Stone topology on , due to results of Section 'The Stone space of minimal structures', this implies compactness of .

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

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

- (e);
- (f)

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

Lemma 5.5. (1) For any there exists such that the *L*_{0}-reduct of *M* is an *X*-structure;

(2) for any there is a unique such that the *L*_{0}-reduct of *M* is an *X*-structure.

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

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

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

Consider the *L*-expansion *M* of such that for all . Then *M* satisfies (e) and (f), by the choice of *f*. Since , the *L*-structure *M* is minimal. Thus , and its *L*_{0}-reduct is the *X*-structure .

(2) As *M* satisfies Γ, the *L*_{0}-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, , and so . As *M* satisfies (e), the diameter of *X* is . Since *M* satisfies (f), *X* is covered by the close balls of radius ε centered at with . Thus .

For let be the unique such that the *L*_{0}-reduct of *M* is an *X*-structure, which exists by Lemma 5.5(2). The map

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

Lemma 5.6. The map χ is continuous.

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

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

then we can take as ψ the conjunction of all sentences from that hold in *M*_{0}.

Choose ε with and . Let *m* be the integer with

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

Let . Denote and ; so the *L*_{0}-reduct of *M* is an *X*-structure, and the *L*_{0}-reduct of *N* is a *Y*-structure. Let

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

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

because this implies

To prove it suffices to show that for any

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

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

then and . The lemma is proven.

The proof of Theorem 5.1 is completed.

### References

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

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