A PRESENTATION THEOREM FOR CONTINUOUS LOGIC AND METRIC ABSTRACT ELEMENTARY CLASSES

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1 A PRESENTATION THEOREM FOR CONTINUOUS LOGIC AND METRIC ABSTRACT ELEMENTARY CLASSES WILL BONEY Cotets 1. Itroductio 1 2. Models ad Theories 3 3. Elemetary Substructure Types ad Saturatio T dese as a Abstract Elemetary Class Metric Abstract Elemetary Classes 18 Refereces Itroductio I the spirit of Chag ad Shelah s presetatio results (from [Cha68] ad [Sh88], respectively), we prove a presetatio theorem for classes of cotiuous structures, both those axiomatized by first-order ad beyod, i terms of a class of discrete structures. The thrust of this presetatio theorem is the basic aalytic fact that the behavior of cotiuous fuctios is determied by their values o a dese subset of their domai. Focusig o dese subsets is key because it allows us to drop the requiremet that structures be complete, which is ot a property expressible by discrete (classical) logic, eve i the broader cotexts of L λ,ω or Abstract Elemetary Classes. The specific statemets of the presetatio theorems appear below (see Theorem 2.1 for cotiuous first-order logic ad Theorem 6.1 for Metric Abstract Elemetary Classes), but the geeral idea is the same i both cases: give a cotiuous laguage L, we defie a discrete laguage L + that allows us to approximate the values of the fuctios ad relatios by a coutable dese subset of values, amely Q [0, 1]. Note that the specificatio that this dese set (ad its completio) is stadard already requires a L + ω 1,ω setece, eve if we are workig i cotiuous first-order logic. The, give a cotiuous L-structure M ad a icely dese (see Defiitio 1.1 below) subset of it A, we ca form a discrete L + -structure A with uiverse A that ecodes all of M. Date: August 15,

2 2 WILL BONEY Give a class of cotiuous L-structures, we represet it by all dese approximatios (i the way described above) to members of the class. I each presetatio theorem, the way the cotiuous class is defied helps defie the class of approximatios. If the class is axiomatized by a cotiuous first-order theory, the the uiform cotiuity of the fuctios ad relatios (see Be-Yaacov, Berestei, Heso, ad Usvyatsov [BBHU08]) allow a L + ω 1,ω axiomatizatio of the dese approximatios. If the cotiuous class is o-elemetary ad forms a Metric Abstract Elemetary Class (see Hirvoe ad Hyttie [HH09] or Villaveces ad Zambrao [ViZa]), the the class of dese approximatios has o explicit axiomatizatio, but forms a Abstract Elemetary Class. We carry the relatio betwee the class of cotiuous models ad discrete approximatios further by cosiderig various model-theoretic properties ad fidig aalogues for them i the class of approximatios. We cosider types, saturatio, (ad for Metric Abstract Elemetary Classes), amalgamatio, joit embeddig, ad d-tameess. Throughout we assume that the reader is familiar with the basics of the cotiuous cotexts either first-order or Metric Abstract Elemetary Classes but try to provide refereces ad remiders whe discussig the cocepts. Parts of this paper come from the author s Ph.D. thesis uder the directio of Rami Grossberg at Caregie Mello Uiversity ad I would like to thak Professor Grossberg for his guidace ad assistace i my research i geeral ad i this work specifically. This material is based upo work while the author was supported by the Natioal Sciece Foudatio uder Grat No. DMS This paper represets a i progress draft. May of the argumets i the two cases are similar. I order to avoid repeatig the same argumets twice i slightly differet cotexts, we provide the details oly oce. We have chose to provide the details for cotiuous first-order logic because this cotext ofte allows more specific formulatios of the correspodece. Dese sets are ot quite the right cotext because they eed ot be substructures of the larger structure. Istead, we itroduce icely dese sets to require them to be closed uder fuctios. Defiitio 1.1. Give a cotiuous model M ad a set A M, we say that A is icely dese iff A is dese i the metric structure ( M, d M ) ad A is closed uder the fuctios of M. I the what follows, we will ofte wat to prove similar results for both greater tha ad less tha. I order to avoid writig everythig twice, we ofte use to stad i for both ad. Thus, assertig a statemet for r s meas that that statemet is true both for r s ad for r s. Our goal is to traslate the real-valued formulas of L ito classic, true/false formulas of L +. We do this by ecodig relatios ito L + that are iteded to specify the value of φ by decidig if it is above or below each possible value.

3 T dese AND K dese 3 To esure that the size of the laguage does t grow, we take advatage of the separability or R ad oly compare each φ to the ratioals i [0, 1]. For otatioal ease, we set Q := [0, 1] Q. 2. Models ad Theories The mai thesis of the presetatio of cotiuous first-order logic is that modeltheoretic properties of cotiuous first order structures ca be traslated to modeltheoretic (but typically quatifier free) properties of discrete structures that model a specific theory i a expaded laguage. We use F ml c L to deote the cotiuous formulas of the laguage L. The mai theorem about this presetatio is the followig: Theorem 2.1. Let L be a cotiuous laguage. The there is (a) a discrete laguage L + ; (b) a L + ω 1,ω theory T dese ; (c) a map that takes cotiuous L-structures M ad icely dese subsets A to discrete L + -structures M A that model T dese ; (d) a map that takes discrete L + structures A that model T dese to cotiuous L-structures A with the properties that (1) M A T dese has uiverse A ad, for ay a A, φ(x) F ml c L, r Q, ad stadig for ad, we have M A R φ r [a] φ M (a) r (2) A is a dese subset of A ad, for ay a A, φ(x) F ml c L, r Q, ad stadig for ad, we have A R φ r [a] φ A (a) r (3) these maps are (essetially) each other s iverse. That is, give ay icely dese A M, we have M = A M A ad, give ay L + -structure A T dese, we have (A) A = A. The essetially i the last clause comes from the fact that completios are ot techically uique as the objects selected as limits ca vary, but this fairly pedatic poit is the oly obstacle. Restrictig to dese subsets ad their completios have already bee cosidered i cotiuous first-order logic, where it goes by the ame prestructure (see [BBHU08]. 3). The key differece here is that, while prestructures are still cotiuous objects with uiformly cotiuous fuctios ad relatios, M A is a discrete object with the relatios o it beig either true or false. Proof: Our proof is log, but straightforward. First, we will defie L + ad T dese. The, we will itroduce the map (M, A) M A ad prove it satisfies (1). After this, we will itroduce the other map A A ad prove (2). Fially, we will

4 4 WILL BONEY prove that they satisfy (3). Defiig the ew laguage ad theory We defie the laguage L + to be F + i, R φ(x) r, R φ(x) r i<f,φ(x) F ml c (L),r Q with the arity of F + i matchig the arity of F i ad the arity of R φ(x) r matchig l(x). Sice we oly use a full (dese) set of coectives (see [BBHU08].6.1), we have esured that L + = L + ℵ 0. We defie T dese L + ω 1,ω to be the uiversal closure of all of the followig formulas ragig over all cotiuous formulas φ(z) ad ψ(z ), all terms τ(z, z ), ad all r, s Q ad t Q {0}. We have divided them ito headigs so that their meaig is (hopefully) more clear. Whe we refer to specific seteces of T dese later, we referece the orderig i this list. As always, a i a formula meas that it should be icluded with both a ad a replacig the. (1) The ordered structure of R (a) R φ(z) r (x) = R φ(z) r (x) (b) R φ(z) r (x) = R φ(z) r (x) (c) If r > s, the iclude R φ(z) r (x) R φ(z) s (x) (d) If r s, the iclude R φ(z) s (x) = R φ(z) r (x); ad R φ(z) r (x) = R φ(z) s (x) (e) R φ(z) r (x) R φ(z) r (x) (f) <ω r,s Q, r s < 1 R φ r (x) R φ s (x) (g) ( <ω R φ(z) r 1 (x)) = R φ(z) r (x) (h) ( <ω R φ(z) r+ 1 (x)) = R φ(z) r (x) (2) Costructio of formulas (a) R φ(z) 0 (x) R φ(z) 1 (x) (b) R 0 t (x) R 1 1 t (x); (c) R φ(z) r(x) R φ(z) 2r(x) 2 (d) R φ(z) r(x) R φ(z) 2r(x); 2 (e) R φ(z) ψ(z ) r(x, x ) s Q (R ψ(z ) s(x ) R φ(z) r+s (x)) (f) R φ(z) ψ(z ) r(x, x ) s Q ( R ψ(z ) s(x ) R φ(z) r+s (x)); (g) R supy τ(y,y) r(x) xr τ(y,y) r (x, x) (h) R supy τ(y,y) r(x) <ω xr τ(y,y) r 1 (i) R ify τ(y,y) r(x) <ω xr τ(y,y) r+ 1 (j) R ify τ(y,y) r(x) xr φ(y,y) r (x, x); (k) R φ(y,y) r (τ(x ), x) R φ(τ(y ),x) r(x, x) (3) Metric structure (a) R d(y,y ) 0(x, x ) x = x ; (x, x) ; (x, x)

5 T dese AND K dese 5 (b) R d(y,y ) r(x, x ) R d(y,y ) r(x, x); (c) r Q (R d(y,y ) r(x, x ) = x s Q [0,r]R d(y,y ) s(x, x ) R d(y,y ) r s(x, x )) (4) Uiform Cotiuity (a) For each r, s Q ad i < L F such that s < Fi (r), we iclude the setece i< R d(z,z ) s(x i, y i ) = R d(z,z ) r(f i (x), F i (y)) (b) For each r, s Q ad j < L R such that s < Rj (r), we iclude the setece i< R d(z,z ) s(x i, y i ) = (R Rj (z) R j (z ) r(x, y) R Rj (z) R j (z ) r(y, x)) We have bee careful about the specific eumeratio of these axioms for a reaso. If the origial cotiuous laguage is coutable, the T dese is coutable. I particular, we could take the cojuctio of it ad make it a sigle L + ω 1,ω setece. This meas that it is expressible i a coutable fragmet of L ω1,ω. Coutable fragmets are the most well-studied ifiitary laguages ad may of the results i, say, Keisler [Kei71] use these fragmets. I geeral, T dese is expressible i a L + ℵ 0 sized fragmet of L + ω 1,ω. From cotiuous to discrete... This is the easier of the directios. We defie the structure M A so that all of the iteded correspodeces hold ad everythig works out well. Suppose we have a cotiuous L-structure M ad a icely dese subset A. Now we defie a L + structure M A by (1) the uiverse of M A is A; (2) (F + i ) M A = Fi M A for i < F ; ad (3) for r Q ad φ(x) F ml c (L), set R M A φ r = {a A : φm (a) r} This is a L + -structure sice it is closed uder fuctios. The real meat of this part is the followig claim, which is (1) from the theorem. Claim: M A T dese ad, for ay a A, φ(x) F ml c L, r Q, ad =,, we have M A R φ r [a] φ M (a) r Proof of Claim: This is all straightforward. From the defiitio, we kow that, for ay a A ad formula φ(x) F ml c (L) ad {, }, we have M A R φ r [a] φ M (a) r

6 6 WILL BONEY This gives a easy proof of the fact that M A T dese because they are all just true facts if R φ r (a) is replaced by φ(a) r. Claim...ad back agai This is the harder directio. We wat to read out the L-structure that A is a dese subset of from the L + structure. First, we use the axioms of T dese to show that we ca read out the metric ad relatios of L from the relatios of L + ad that these are well-defied. The we complete A ad use the uiform cotiuity of the derived relatios to expad them to the whole structure. I the first directio, T dese could have bee ay collectio of true seteces about cotiuous structures ad the real lie, but this directio makes it clear that the axioms chose are sufficiet. Suppose that we have a L + -structure A that models T dese. The followig claim is a importat step i readig out the relatios of the completio of A from A. Claim 2.2. For ay φ(x) F ml c (L) ad a A, we have sup{t Q : A = R φ(x) t (a)} = if{t Q : A = R φ(x) t (a)} Proof: We show this equality by showig two iequalities. Let r {t Q : A = R φ(x) t (a)} ad s {t Q : A = R φ(x) t (a)}. The A = R φ(x) r (a) R φ(x) s (a) The, sice M + satisfies 1c, we must have r s. Thus sup{t Q : A = R φ(x) t (a)} if{t Q : A = R φ(x) t (a)}. By 1f, we have A = <ω r,s Q ; r s < 1 R φ(x) r (a) R φ(x) s (a) Let ɛ > 0. The there is 0 < ω such that ɛ > 1 0. By the above, there are r, s Q such that r s < 1 0 ad M + = R φ(x) r (a) R φ(x) s (a) As above, 1c implies r s, so we have r s < 1 0 < ɛ. Thus r < s + ɛ ad s {t Q : A = R φ(x) t (a)} ad r {t Q : A = R φ(x) t (a)}. The, if{t Q : A = R φ(x) t (a)} sup{t Q : A = R φ(x) t (a)}. Claim The first relatio that we eed is the metric. Give a, b A, we set D(a, b) := sup{r Q : A R d(x,y) r [a, b]} = if{r Q : A R d(x,y) r [a, b]}

7 T dese AND K dese 7 These defiitios are equivalet by Claim 2.2. We show that this is ideed a metric o A. Claim 2.3. ( A, D) is a metric space. Proof: We go through the metric space axioms. Let a, b A. (1) D(a, b) = 0 = if{r Q : A R d(x,y) r (a, b)} = 0 (2) = < ω r Q so A = R d(x,y) r (a, b) ad r 1 = 1d < ω, A = R d(x,y) 1 (a, b) = 1h A = R d(x,y) 0 (a, b) = a = b a = b = A = R d(x,y) 0 (a, b) = if{r Q : A = R d(x,y) r (a, b)} = 0 = D(a, b) = 0 D(a, b) = sup{r Q : A = R d(x,y) r (a, b)} = 3b sup{r Q : A = R d(x,y) r (b, a)} = D(b, a) (3) Let c A. We wat to show D(a, c) D(a, b) + D(b, c). It is eough to show r Q (D(a, c) r = D(a, b) + D(b, c) r) Thus, let r Q ad suppose D(a, c) r. The sup{s Q : A = R d(x,y) s (a, c)} r. By 3c, this meas sup{s Q : A = t Q [0,s]R d(x,y) t (a, b) R d(x,y) s t (b, c)} r Fix < ω. There is some s Q such that s r 1 ad A = t Q [0,s ]R d(x,y) t (a, b) R d(x,y) s t(b, c) Thus, there is some t Q such that 0 t s ad A = R d(x,y) t (a, b) R d(x,y) s t (b, c) By the defiitio of D, this meas that D(a, b) t ad D(b, c) s t ; thus, D(a, b) + D(b, c) s. Sice this is true for all < ω, we get that D(a, b) + D(b, c) r as desired. Thus, D is a metric o A. Claim Now we defie partial fuctios ad relatios o ( A, D) such that they are uiformly cotiuous. I particular, (1) for i < L F, set f i := Fi A with modulus fi (r) = sup{s Q : A x 0,..., x (Fi ) 1; y 0,..., y (Fi ) 1 ( i<(fi )R d(z,z ) s(x i, y i ) R d(z,z ) r(f i (x), F i (y))}.

8 8 WILL BONEY (2) for j < L R, set r j (a) := sup{r Q : A R Rj (z) r[a]} with modulus rj (r) = sup{s Q : A = x y( i<(rj )R d(z,z ) s(x i, y i ) (R Rj (z) R j (z ) r(x, y) R Rj (z) R j (z ) r(y, x)))}. These fuctios are ot defied o the desired structure (ie the completio of A), but they already fulfill our goal i terms of agreeig with the discrete relatios i the followig sese. Claim: For all a A ad all formulas φ(x) built up from these fuctios ad D, we have that φ(a) r A R φ(z) r [a] Proof: We proceed by iductio o the costructio of φ(x). We assume that is i our proofs, but the proofs for are the same. If φ is atomic, the it falls ito oe of the followig cases. Suppose φ(x) R j (τ(x)) for some term τ. The R M j (τ(a)) r if{s Q : A = R Rj (x) s[τ(a)]} r 1d 1g 2k < ω, s Q so s r 1 ad A = R R j (x) s [τ(a)] < ω, A = R Rj (x) r 1 [τ(a)] A = R Rj (x) r[τ(a)] A = R Rj (τ(y)) r[a] Suppose that φ(x, y) d(τ 1 (x), τ 2 (y)) for terms τ 1 ad τ 2. The detail are essetially as above: D M (τ 1 (a), τ 2 (b)) iff (by 1d, the defiitio of sup, ad 1h ad 1g) M + = R d(x,y) r [τ 1 (a), τ 2 (b)] iff (by 2k) M + = R d(τ1 (x),τ 2 (y)) r(a, b). For the iductive step, we deal with each coective (from our full set) i tur. The iductio steps for x 0, x 1, ad x x are clear. 2 Suppose φ ψ τ, where τ is a formula ad ot a term. Note if r = 0, the this is obvious. So assume r 0. Recall that φ M (a) = ψ M (a) τ M (a) = { ψ M (a) τ M (a) if ψ M (a) 0 0 otherwise Thus, we ca assume we are i the case that ψ M (a) > τ M (a). First, suppose ψ M (a) τ M (a) r. Sice ψ M (a) > τ M (a), there is some s Q such that ψ M (a) > s > τ M (a). The τ M (a) s ad ψ M (a) s + r. By iductio, we have that M + = R τ(x) s [a] R ψ M (x) s+r[a] The, by 2e, we have that M + = R ψ τ r[a] as desired.

9 T dese AND K dese 9 Now, suppose M + = R ψ τ r[a]. Agai, 2e implies there is is some s Q such that M + = R τ s [a] R ψ r+s [a] By iductio, we get τ M (a) s ad ψ M (a) r + s. The φ M (a) = ψ M (a) τ M (a) (r + s) s = r as desired. Suppose φ(x) sup x ψ(x, x). We will cosider both sides of the iequality sice they re ot symmetrically axiomatized (see 2g ad 2h), but we wo t worry about if. Suppose that sup x φ M (x, a) r. The for ay < ω, there is some a M such that φ M (a, a) > r 1 2. Sice φm is uiformly cotiuous, there is some δ > 0 such that, if d(a, b) < δ, the φ M (a, a) φ M (b, a) < 1 2. Sice M + is dese i M, there is some a M + such that d(a, a ) < δ. Thus, φ M (a, a) > r 1. By iductio, we have that M + = <ω xr φ(y,y) r 1 (x, a) The 2h says that M + = R supy φ(y,y) r(a). Suppose that M + = R supy φ(y,y) r[a]. The, by 2h, M + = <ω xr φ(y,y) r 1 (x, a). So, for each < ω, there is some a M + such that M + = R φ(y,y) r 1 [a, a]. By iductio, we have that φ M (a, a) r 1. Sice this is true for each < ω, we get sup y φ M (y, a) r. The other directio is easier ad we ca combie the two parts sup φ M (x, a) r a Mφ M (a, a) r x a M + φ(a, a) r Iductio M + = xr φ r (x, a) 2g M + = R supx φ(x,x)[a] We have give these fuctios moduli, but do ot kow they are uiformly cotiuous. We show this ow. It is also worth otig that these moduli might ot be the same moduli i the origial sigature L. Istead, these are the optimal moduli, while the origial laguage might have moduli that could be improved. Claim: The fuctios f i ad r j are cotiuous. Proof: We do each of these cases separately.

10 10 WILL BONEY Sub-Claim 1: F M + i is uiformly cotiuous o ( M +, D) with modulus Fi. Let r Q ad let a, b M + such that max i< D(a i, b i ) < Fi (r). Thus, for each i <, D(a i, b i ) = if{s Q : M + = R d(x,y) s (a i, b i )} < Fi (r). Sice this is strict, there is some s i Q such that M + = R d(x,y) si (a i, b i ). Note that 1d implies that the set Fi (r) is supremumig over is dowward closed. Thus, s = max i< s i is i it. Thus, we ca coclude M + = R d(x,y) r [F i (a), F i (b)] This meas that D(F i (a), F i (b)) r, as desired. Sub-Claim 2: R M + j is uiformly cotiuous o ([0, 1], ) with modulus Rj. Let r Q ad a, b M + such that i< D(a i, b i ) < Rj (r). From the ifimum defiitio of D, for each i <, there is s i Q such that s i < Rj (r) ad M + = R d(z,z ) s i [a i, b i ]. Thus, M + = R Rj (z) R j (z ) r(a, b) R Rj (z) R j (z ) r(b, a) For this ext part, we eed some of the future proofs, but essetially we have eough to show that this implies R M + j (a) R M + j (b) r ad R M + j (b) R M + j (a) r This implies R M + j (a) R M + j (b) r, so R M + j is uiformly cotiuous. Now we have a prestructure, see [BBHU08]. 3. Now we complete A to A i the stadard way; see Mukries [Mu00] for a referece for the topological facts. I particular, we defie the cotiuous L structure A by A is the completio of ( A, D); the metric d A is the extesio of D to A ; for i < F, Fi A is the uique extesio of f i to A ; ad for j < R, Rj A is the uique extesio of r j to A. Essetial iverses Propositio 2.4. Give ay cotiuous L-structure M ad dese subset A, we have that M = A M A ad, give ay L + structure A that models T dese, we have that (A) A = A. Proof: First, let M be a cotiuous L-structure ad A M be icely dese. We defie a map f : M (M A ) as follows: if a A, the f(a) = a. For a M A, fix some (ay) sequece a A : < ω such that lim a = a (this limit computed i M). We kow that a : < ω is Cauchy i M, so it s

11 T dese AND K dese 11 Cauchy i (M A ). The set f(a) = lim a, where that limit is computed i (M A ). This is well-defied ad a bijectio because A is dese i both sets. That this is a L-isomorphism follows from applyig the correspodece twice: for all a A ad φ(x) F ml c (L) φ M (a) r M A R φ(x) r [a] φ (M A) (a) r ad the fact that the values of φ o A determies its values o M ad (M A ). Secod, let A be a L + structure that models T dese. Clearly, the uiverses are the same, ie, (A) A = A. For ay relatio R φ r ad a A, we have Give a fuctio F + i A R φ r [a] φ A (a) r (A) A R φ r [a] ad a, a A, we have that (F + i ) A (a) = a A R d(f + i (x),x) 0[a, a] + (A) A R d(f + i (x),x) 0[a, a] (F i ) (A) A (a) = a We ca exted this correspodece to theories. Suppose that T is a cotiuous theory i L. Followig [BBHU08].4.1, theories are sets of closed L-coditios; that is, a set of φ = 0, where φ is a formula with o free variables. The followig is immediate from Theorem 2.1. Corollary 2.5. If φ = 0 is a closed L-coditio, the φ M = 0 M A R φ 0 With our fixed theory T, set T to be T dese {R φ 0 : φ = 0 T }. The our represetatio of cotiuous L-structures as discrete L + -structures modelig T dese ca be exteded to a represetatio of cotiuous models of T ad discrete models of T. 3. Elemetary Substructure We ow discuss traslatig the otio of elemetary substructure betwee our two cotexts. Depedig o the geerality eeded, this is either easy or difficult. For the easy case, we have the followig. Theorem 3.1. Let M, N be cotiuous L structures. The M L N iff, for every icely dese A M ad B N such that A B, we have that M A L + N B. Note that the relatio betwee M A ad N B is just substructure. So eve though they are models of ifiitary theories, their relatio just cocers atomic formulas. This is because we have built the quatifiers of L ito the relatios of L +. Proof: : Let A = M ad B = N. The M N, so M M L + N N by assumptio. Thus they agree o all relatios cocerig elemets of M. Now we

12 12 WILL BONEY wat to show that M c L N. Let φ(x) F mlc L ad a M. From the theorems proved last sectio, we have, for each r Q, φ M (a) r M M R φ(x) r [a] Theorem 2.1 N N R φ(x) r [a] M M L + N N φ N (a) r Theorem 2.1 Thus φ M (a) = φ N (a) ad M L N as desired. : Let A M ad B N be icely dese so A B. We wat to show that M A L + N B. Let F + L + ad a A. The, by defiitio of the structures, (F + ) M A (a) = F M (a) = F N (a) = (F + ) N B (a) Let R φ r (x) L + ad a A. M M R φ(x) r [a] φ M (a) r Theorem 2.1 Similarly, we have the followig. φ N (a) r M N N N R φ(x) r [a] Theorem 2.1 Theorem 3.2. Give A, B T dese, if A L + B, the A L B. However, these are ot the best theorems possible. I particular, the requiremet that A B limits the scope of this theorem. We would like to kow whe L + structures complete to L-elemetary substructures eve whe the dese substructures are ot subsets or each other; for istace, the completios of Q [0, 1] ad Q + 2 are icely related, but the previous theorem does ot see that. We would like to develop a criterio for L + structures A, B T dese that is equivalet to A L B. Our first attempt is the followig. Theorem 3.3. Suppose M N are cotiuous L-structures ad A M ad B N are icely dese. The (1) M A L + N C, where C is the closure of A B uder the fuctios of N. (2) There is a icely dese B N such that M A L + N B ad B = A + dc(n) + L. Proof: (1) Note that A C ad C is icely dese i N. By Theorem 3.1, M A L + N C. (2) Let B N be dese of size dc(n) ad let B be the closure of B A uder the fuctios of N. By Theorem 3.1, M A L + N B.

13 T dese AND K dese 13 While this is a improvemet, it is still ot the best desireable. I particular, it still makes referece to the cotiuous structures. We would prefer a correspodece that oly ivolved L + structures. To that ed, we give the followig defiitio of iessetial extesios. Defiitio 3.4. Give A L + B, we say that B is a iessetial extesio of A A iff for every b B ad < ω, there is some a A such that B R d(x,y)< 1 [b, a]. Propositio 3.5. If B is a iessetial extesio of A, the A = B. This gives us the followig theorem. Theorem 3.6. Let A, B T dese. The TFAE (1) A L B. (2) There is a L + structure C such that A, B L + C ad C is a iessetial A extesio of B. (3) There is a L + structure C T dese such that A, B L + C ad C is a iessetial extesio of B. (4) There is a extesio of the fuctios ad relatios of L + to A B such that A B T dese that are still uiformly cotiuous ad such that for every a A B ad < ω, there is some b B such that A B R d(x,y)< 1 [b, a]. Proof: (1) = (2) Take C = (B) A B. (2) = (3) Immediate. (3) = (4) Take the extesio iherited from C. (4) = (1) We have A, B A B from the first coditio ad B = A B form the secod. 4. Types ad Saturatio I this sectio, we will coect types i the cotiuous logic sese to types i the discrete sese. However, just as elemets i the complete structure are represeted by sequeces of the discrete structure, we represet types by sequece types. Recall from [BBHU08].8.1 that a type over B is a cosistet collectio of coditios of the form φ(x, b) = 0 with b B. Defiitio 4.1. We say that r : < ω is a sequece l-type over B iff r 0 (x) is a l-type over B ad r +1 (x, y) is a 2l-type over B such that there is some idex set I, (possibly repeatig) formulas φ i : i I ; ad (possibly repeatig) Cauchy sequeces b i B <ω : i I so d(b i, b i +1) 1 2 such that r 0 (x) = {R φi (z,z ) w i(2)(x, b i 0) : i I}; ad φ

14 14 WILL BONEY r +1 (x, y) = {R φi (z,z ) w φ i( 1 2 ) (x, bi +1) : i I} {R d(z,z ) 1 2 (x k, y k ) : k < l} A realizatio of a sequece type r : < ω is a : < ω such that a 0 realizes r 0 ; ad a +1 a realizes r +1. Note that the use of 1 is ot ecessary; this could be replaced by ay summable 2 1 sequece for a equivalet defiitio (also replacig by the trailig sums). 2 1 However, we fix 1 for computatioal ease. The fudametal coectio betwee 2 cotiuous types ad sequece types is the followig. Theorem 4.2. Let A M be icely dese. (1) If B M ad r(x) is a partial l-type over B, the for ay B A such that B B, there is a sequece l type r : < ω over B such that M realizes r iff M A realizes r : < ω (2) If B A ad r : < ω is a partial sequece l-type over B, the there is a uique l-type r over B such that M realizes r iff M A realizes r : < ω We ca deote the type i (2) by lim r. I each case, we have that a M A : < ω realizes r : < ω implies lim a realizes lim r. Proof: (1) Recall that r(x) cotais coditios of the form φ(x, b) = 0 for φ F ml c (L) ad b B. For < ω ad b B, set B (b) = {b B : d M (b, b) < 1 }. To make the cardiality work out icer, fix a choice 2 fuctio G, ie G(B (b)) B (b). The B (b) ad G(B (b)) have the obvious meaigs. Defie r + (x) := {R φ(z;y)<w φ 1 ( (b))) : φ(x; b) = 0 r} 2 1 r 0 (x) := r 0 + (x) r +1 (x, y) := r + +1(x) {R d(z,z )< 1 2 (x i, y i ) : i < l(x)} The r : < ω is a sequece type over B ; we ca see this by takig r as the idex set, φ i = φ, ad b i = G(B (b)) for i = φ(x; b) = 0 r. To show it has the desired property, first suppose that a : < ω from M A realizes r : < ω. We kow that a : < ω is a Cauchy sequece; i particular, for m >, d M (a, a m ) m i= 1 2 = 2m+1 1 i 2 m+1

15 T dese AND K dese 15 Sice M A = M is complete, there is a MA such that lim a = a. We claim that a r. Let φ(x; b) = 0 r. The d M A (ab; a G(B (b))) = max{d M (a, a ), d M (b, G(B (b))} 1 max{ 2, 1 i 2 } = Thus, φ M (a; b) φ M (a ; G(B (b))) < w φ 1 ( 2 ) 1 Lettig, we have that φ M A (a; b) = lim φ M (a ; G(B (b))) lim w φ 1 ( 2 ) = 0 1 as desired. Now suppose that a M realizes r. Sice A is dese, we ca fid a A : < ω such that d M (a, a +1 ) < 1 ad a 2 a. We wat to show that r + (a ) holds. Let φ(x, b) r. We kow that d M (a G(B (b)), ab) = 1 2 1, so we get φ M A (a, G(B (b))) = φ M (a, b) φ M (a, G(B (b))) < w φ 1 ( 2 ) 1 as desired. (2) Let r : < ω be a partial sequece l-type give by I, φ i : i I, ad b i <ω : i I. The set r(x) := {φ i (x, lim b i ) = 0 : i I} First, suppose that a M A : < ω realizes r : < ω. The, sice a +1 a r +1, we have d M A (a +1, a ) < 1 ad, thus, the sequece is 2 Cauchy. Sice M is complete, let a = lim a M. The, by uiform cotiuity, we have φ M i (a, b i ) = φ M i ( lim a, lim b i ) = lim φ M i (a, b i ) lim w φ i 1 ( 2 ) 1 = 0 So a r. Now suppose that a M realizes t. The, by deseess, we ca fid a Cauchy sequece a A : < ω such that d(a +1, a ) 1. The 2 d(ab i, a b i ) 1. The we ca coclude 2 1 i=

16 16 WILL BONEY φ M i (a, b i ) φ M i (a, b i ) w φ i 1 ( 2 ) 1 φ M A (a, a i ) w φ i 1 ( 2 ) 1 So a : < ω realizes r : < ω. We ow coect type-theoretic cocepts i cotiuous logic (e.g. saturatio ad stability) with cocepts i our discrete aalogue. Recall (see [BBHU08].7.5) that a cotiuous structure M is κ-saturated iff, for ay A M of size < κ ad ay cotiuous type r(x) over A, if every fiite subset of r(x) is satisfiable i M, the so is r(x). Defiitio 4.3. If r : < ω is a sequece type defied by a idex set I ad I 0 I, the r : < ω I 0 is the sequece type defied by I 0. We say that M A T dese is κ-saturated for sequece types iff, for all B A ad sequece type r : < ω over B that is defied by I, if r : < ω I 0 is realized i M A for all fiite I 0 I, the r : < ω is realized i M A. Theorem 4.4. (1) If M is κ-saturated ad λ ℵ 0 < κ, the M A is λ + saturated for sequece types. (2) If M A is κ saturated, the M is κ saturated. Proof: (1) Let M be κ-saturated ad A M be icely dese. Let B M A of size λ ad let r : < ω be a sequece type over B that is fiitely satisfiable i M A. Set r = lim r from Theorem 4.2; this is a type over B where B λ ℵ 0 < κ. We claim that r is fiitely satisfiable i M. Ay fiite subset of r of r = {φ i (x, lim b i ) : i I} correspods to a fiite I 0 I. The, by Theorem 4.2, r 0 is realized i M iff r : < ω I 0 is realized i M A. The, sice each r : < ω I 0 is realized i M A by assumptio, we have that r is fiitely satisfiable i M. By the κ-saturatio of M, r is realized i M. By Theorem 4.2, r : < ω is realized i M A. So M A is λ + -saturated. (2) Let M A be κ-saturated for sequece types. Let B M of size < κ ad r be a type over B that is fiitely satisfied i B. Fid B A such that B B; this ca be doe with B B + ℵ 0 < κ. The form the sequece type r : < ω over B that coverges to r as i Theorem 4.2. As before, sice r is fiitely satisfiable i M, so is r : < ω i M A. So r : < ω is realized i M A by saturatio. Thus, r is realized i M. We immediately get the followig corollary.

17 T dese AND K dese 17 Corollary 4.5. If κ = (λ ℵ 0 ) + or, more geerally, κ = sup λ<κ (λ ℵ 0 ) + ad M is of size κ, the M is saturated iff M A is saturated for some icely dese A M of size κ. 5. T dese as a Abstract Elemetary Class I this sectio we view the discrete side of thigs as a Abstract Elemetary Class; see Baldwi [Bal09] or Grossberg [Gro1X]. Theorem 5.1. Let T be a complete, cotiuous first order L-theory. The let L + ad T dese be from Theorem 2.1. Set K = (Mod (T dese T, L +). The (1) K is a AEC; (2) K has amalgamatio, joit embeddig, ad o maximal models; ad (3) Galois types i K correspod to sequece types (Defiitio 4.1). Note that if T were ot complete, the amalgamatio would ot hold. However, the other properties will cotiue to hold, icludig the correspodece betwee Galois types ad sequece types. Proof: T dese T + is a L ω1,ω theory, so all of the examples hold except perhaps the chai axioms. For those, cosider a L +-icreasig chai M Ai : i < α. The, by Theorem 2.1 ad Theorem 3.2, the sequece M Ai : i < α is L -icreasig chai that each model T. The by the chai axiom for cotiuous logic, there is M = i<α (M Ai ) that models T. Additioally, A := i<α A i is icely dese i M. Thus, M A = i<α M Ai is as desired. Additioally, if M Ai L + M B for some B, the M L M B, so M A L + M B. These properties all follow from the correspodig properties of cotiuous firstorder logic. For istace, cosiderig amalgamatio, suppose M A L + M B, M C. The we have M A L M B, M C. By amalgamatio for cotiuous first-order logic, there is some N L M B ad elemetary f : M C M A N. Let D N be icely dese that cotais B C. The we have M B L + N D ad f M C : M C MA N D ; this is a amalgamatio of the origial system. Fially, we wish to show that Galois types are sequece types ad vice versa. Note that there are moster models i each class. Further more, we may assume that, if C is the moster model of T, that there is some icely dese U C such that the moster model of K is M U ; i fact, we could take U = C. Let cotiuous M T ad A M be icely dese. If we have tuples a ad b, the gtp K (a/m A ) = gtp K (b/m A ) f Aut MA M U.f(a) = b f Aut MA C.f(a) = b tp(a/m) = tp(b/m) lim r a = lim r b

18 18 WILL BONEY where r x : < ω is the sequece type derived from tp(x/m) as i Theorem 4.2 usig A as the dese subset. 6. Metric Abstract Elemetary Classes I this sectio, we exted the above represetatio to Metric Abstract Elemetary Classes. Recall from Hirvoe ad Hyttie [HH09] or that a Metric Abstract Elemetary Class (MAEC) is a class of cotiuous L-structures K ad a strog substructure relatio K satisfyig the followig axioms: (1) K is a partial order o K; (2) for every M, N K, if M K N, the M L N; (3) (K, K ) respects L(K) isomorphisms, if f : N N is a L(K) isomorphism ad N K, the N K ad if we also have M K with M K N, the f(m) K ad f(m) K N ; (4) (Coherece) if M 0, M 1, M 2 K with M 0 K M 2 ; M 1 K M 2 ; ad M 0 M 1, the M 0 M 1 ; (5) (Tarski-Vaught chai axioms) suppose M i K : i < α is a K -icreasig cotiuous chai, the (a) i<α M i K ad, for all i < α, we have M i K i<α M i ; ad (b) if there is some N K such that, for all i < α, we have M i K N, the we also have i<α M i K N; ad (6) (Löweheim-Skolem umber) There is a ifiite cardial λ L(K) such that for ay M K ad A M, there is some N K M such that A N ad N A + λ. We deote the miimum such cardial by LS(K). These axioms were first give i Hirvoe ad Hyttie [HH09]. A key differece is that the fuctios ad relatios L are o loger required to be uiformly cotiuous, but just cotiuous. This is due to the lack of compactess i the MAEC cotext. This iitially seems problematic because fuctios must be uiformly cotiuous o a set to be guarateed a extesio to its closure. However, we get aroud this by simply defiig K dese to be all the structures that happe to complete to a member of K, the use the MAEC axioms to show that K dese satisfies the AEC axioms. For this reaso, whe we refer to cotiuous laguages, structures, etc. i this sectio, we will ot mea that they are uiformly cotiuous. Theorem 6.1. Let L be a cotiuous laguage. The there is a discrete laguage L + such that, for every MAEC K with LS(K) = L, there is (1) a AEC K dese with L(K dese ) = L + ad LS(K dese ) = LS(K); (2) a map from M K ad icely dese subsets A of M to M A K dese ; ad (3) a map from A K dese to A K with the properties that

19 T dese AND K dese 19 (1) M A has uiverse A ad for each a A, r Q, ad {, }, we have that M A R Rj (z) r[a] R M j (a) r (2) A has uiverse that is the completio of A with respect to the derived metric ad for each a A, r Q, ad {, }, we have that R A j (a) r A R Rj (z) r[a] (3) The maps above are essetially iverses, i the sice of Theorem (4) Give M l K ad A l icely dese i M l for l = 0, 1, if f : M 0 M 1 is a K-embeddig such that f(a 0 ) A 1, the f A 0 is a K dese - embeddig from (M 0 ) A0 to (M 1 ) A1. Give A, B K dese ad a K dese -embeddig f : A B, this lifts caoically to a K-embeddig f : A B. Proof: The proof proceeds similar to the first-order versio, Theorem 2.1. I particulalr, may of the defiitios of cotiuous structures from discrete approximatios (such as gettig the metric ad relatios from their approximatios) did ot use compactess ad oly used uiform cotiuity to esure that a completio existed, which will be guarateed by the defiitio of K dese i this case. Give cotiuous L = F i, R j i<f,j< r, defie L + := F + i, R Rj (z) r, R Rj (z) r i<f,j< r,r Q Give a MAEC K, we defie the AEC K dese as follows: L(K dese ) = L + ; Give a L + structure A, we use the followig procedure to determie membership i K dese : defie D o A A by D(a, b) := sup{r Q : A R d(x,y) r [a, b]} = if{r Q : A R d(x,y) r [a, b]} The proof that this is a well-defied ad is a metric proceeds exactly as i the previous case. We ca similarly defie the relatios R j o A ad complete the uiverse (A, D) to A. We call the structure A completable iff (1) for every a A ad every Cauchy sequece a A : < ω covergig to to a, the value of lim R j (a ) is idepedet of the choice of the sequece; ad similarly (2) for every a A ad every Cauchy sequece a A : < ω covergig to to a, the value of lim F + i (a ) is idepedet of the choice of the sequece. If A is completable, the we defie A to be the L + -structure where F i ad R j are defied o A accordig to the idepedet value give above. Fially, we say that A K dese iff (1) A is completable; ad

20 20 WILL BONEY (2) A K. Give A, B K dese, we say that A dese B iff (1) A L + B; ad (2) A K B. Now that we have the defiitio of K dese, we must show that it is i fact a AEC. The verificatio of the axioms are routie; we give the argumets for coherece ad the chai axioms as templates. For coherece, suppose that A, B, C K dese such that A dese C; B dese C; ad A L + B. The, takig completios, we get that A K C; B K C; ad A L B By coherece i K, we the have that A K B. The, by defiitio, A dese B, as desired. For the chai axioms, suppose that A i K dese : i < α is a cotiuous, dese -icreasig chai such that, for all i < α, A i dese B K dese. Agai, takig completios, we get that A i K : i < α is a cotiuous, K -icreasig chai such that, for all i < α, A i K B K. The, by the uio axioms for K, we have that i<α A i K ad i<α A i K B. Note that the existece of i<α A i shows that i<α A i is completable ad that a easy computatio shows that i<α A i = i<α A i. Thus, i<α A i K dese ad, for all j < α, A j dese i<α A i dese B Oce we have defied the maps ad show that K dese is a AEC, the rest of the proof proceeds exactly as i the cotiuous first-order case, i some ways simpler sice L + oly has relatios for each relatio of L, rather tha each formula of L. We ow tur to a applicatio. Both Hirvoe ad Zambrao have proved versios of Shelah s Presetatio Theorem for MAECs i their theses. The more geeral is Zambrao s Theorem from [Zam]: Theorem 6.2. Let K be a MAEC. There is L 1 L ad a cotiuous 1 L 1 -theory T 1 ad a set of T 1 -types Γ such that K = P C(T 1, Γ, L). A immediate corollary to our presetatio theorem is a discrete presetatio theorem. Corollary 6.3 (Discrete Presetatio Theorem for Metric AECs). Let K be a MAEC. The there is a (discrete) laguage L 1 of size LS(K), a L 1 -theory T 1, ad a set of T 1 -types Γ such that K = {M 1 L(K) : M 1 T 1 ad omits Γ}, where the completio is take with respect to a caoically defiable metric. Proof: Apply Theorem 6.1 to represet K as a discrete AEC K dese ad the apply Shelah s Presetatio Theorem from Shelah [Sh88]. 1 But ot uiformly cotiuous

21 T dese AND K dese 21 Additioally, Zambrao asks (as Questio [Zam].1.2.9) if there exists is a Haf umber for model existece i MAECs. Usig our presetatio theorem, we ca aswer this questios i the affirmative. Furthermore, the Haf umber for MAECs is the same as for AECs. Theorem 6.4. If K is a MAEC with LS(K) = κ models of size or desity character cofial i ℶ (2 κ ) +, the K has models with desity character arbitrarily large. Proof: For every, λ < ℶ (2 κ ) +, let M λ K have size λ. M λ is icely dese i itself, so (M λ ) Mλ K dese has size λ. By the defiitio of Haf umber for discrete AECs, this meas that K dese has arbitrarily large models. Takig completios, this meas K has arbitrarily large models. The proof for desity character is the same. Give this represetatio, we ca determie basic structural properties of K by lookig at K dese ad vice versa. The above theorem already shows how to trasfer arbitrary large models ad the other properties trasfer similarly. Propositio 6.5. Suppose K is a MAEC. For P beig amalgamatio, joit embeddig, or o maximal models, K has P iff K dese has P. The fourth clause of the coclusio of Theorem 6.1 is stated as it is to make the proof of this propositio easy to see. We ow look at the otio of type i K dese that correspods to Galois types i K. As before, we pass to a sequece of types represetig a Cauchy sequece for a realizatio of the Galois type we wish to represet. Defiitio 6.6. Give A K dese, r : < ω is a sequece Galois type over A iff (1) r 0 gs l (A); ad (2) r +1 gs l2 (A) such that (a) if ab r +1, the d(a, b) 1 ; ad 2 (b) r {l,...,l2 1} +1 = r, l i. e. the first l coordiates of r are the same as the fial l coordiates of r +1. Give a sequece Galois type r : < ω over A ad A dese B, we say that a B : < ω realizes r : < ω iff (1) a 0 r 0 ; ad (2) a +1 a r +1 Give a sequece Galois type r : < ω over A ad A dese B, we say that a B : < ω weakly realizes r : < ω iff there is some C ad b C : < ω such that (1) B dese C; (2) b : < ω realizes r : < ω ; ad (3) lim <ω a = lim <ω b.

22 22 WILL BONEY Note that realizig a sequece Galois type is differet tha realizig each idividual Galois type i the sequece separately. However, we ca always realize sequece Galois types give amalgamatio. Lemma 6.7. Suppose that K has amalgamatio. Give ay sequece Galois type r : < ω over A K dese, there is A dese B K dese that cotais a realizatio of r : < ω. Proof: By the defiitio of Galois type, we ca write each r as gtp Kdese (a 0 0/A; B 0 ) ad gtp Kdese (a +1 +1a +1 /A; B +1 ). Usig amalgamatio i K dese, which follows from amalgamatio i K, we ca costruct icreasig C : < ω ad icreasig f : B A C such that f 0 is the idetity ad f (a ) = f +1 (a +1 ); this secod part is due to the secod clause i the defiitio of sequece types. Settig C = <ω C, we have that f (a ) : < ω realizes r : < ω. Ufortuately, we do t have the same tight coectio betwee Galois types i K ad sequece Galois types i K dese as exists i Theorem 4.2. This is due to the fact that the Galois versio of sequece types specifies a distace betwee cosecutive members of the Cauchy sequece, rather tha just specifyig a boud o the distace. It is possible that perturbatios (as i Hirvoe ad Hyttie [HH12]) might be used to restore this coectio. Istead, we have itroduced the otio of weakly realizig a sequece Galois type because this is eough to prove a variat of Theorem 4.4 i this cotext. Theorem 6.8. Let M K ad A K dese such that A = M. (1) If M is κ-galois saturated ad λ ℵ 0 < κ, the A is λ + -weakly saturated for sequece Galois types (i.e. give ay A 0 dese A of size < λ +, every sequece Galois type over A 0 is weakly realized i A). (2) If A is κ-weakly saturated for sequece Galois types, the M is κ-galois saturated. Proof: First, suppose that M is κ saturated ad λ ℵ 0 < κ. Let A 0 dese A be of size λ ad let r : < ω be a sequece Galois type over A 0. By Lemma 6.7, there is some B dese A 0 ad a : < ω that realizes r : < ω. After completig the members of K dese, we have A 0 M ad A 0 B K. Sice a B : < ω is a Cauchy sequece, there is some a B such that lim a = a. Sice M is κ-saturated ad A 0 λ ℵ 0 < κ, there is b M that realizes gtp K (a/a 0 ; B). Sice A is dese i M, there is some Cauchy sequece b A : < ω that coverges to b. The b : < ω weakly realizes r : < ω. Secod, suppose that A is κ-weakly saturated for sequece Galois types ad let M 0 M of size < κ ad r gs(m 0 ). Let A 0 A be icely dese i M 0 ; the A 0 := (M 0 ) A0 dese A. I K, we ca write r as gtp K (a/m 0 ; N). The, i K dese, r := gtp Kdese (a/a 0 ; N N ) : < ω is a sequece Galois type over A 0. Sice A 0 < κ, by A s weak saturatio, there is some b : < ω that weakly realizes

23 T dese AND K dese 23 r. This meas that we ca fid a extesio B dese A ad f : N N A0 B such that lim f(a) = lim b Sice b A, this meas that f(a) M. Sice N N is complete, the K dese - embeddig f is i fact a K-embeddig from N ito B ad fixes A 0 = M 0. Thus, we have that f(a) M realizes gtp K (a/m 0 ; N). A crucial property i the study of MAECs is whether the atural otio of distace betwee Galois types defies a metric or ot. This property is called the Pertubatio Property by Hirvoe ad Hyttie [HH09] ad the Cotiuity Type Property by Zambrao [Zam12]. For ease, we assume that K (equivaletly, K dese ) has a moster model C. Defiitio 6.9. Give M K ad p, q S(M), we defie d(p, q) = if{d(a, b) : a p ad b q} K has the Pertubatio Property (PP) iff, give ay Cauchy sequece b C : < ω ad M K, if, for all < m < ω, gtp(b /M) = gtp(b m /M) the, gtp(lim b /M) = gtp(b 0 /M). Although these properties might ot iitially seem related, a little work shows that d is always a pseudometric ad that it is a metric iff PP holds; this is due Hirvoe ad Hyttie. The, similar to tameess from AECs, Zambrao [Zam12].2.9 defies a otio of tameess i MAECs that satisfy PP. Defiitio K is µ-d-tame iff for every ɛ > 0, there is a δ > 0 such that for every M K of desity character µ ad p, q gs(m), if d(p, q) ɛ, the there is some N M of desity character µ such that d(p N, q N) δ ɛ. Agai these properties trasfer to sequece types i K dese. We ca defie a pseudometric o sequece Galois types i K dese by d dese ( r : < ω, s : < ω ) := if{ lim d(a, b ) : a : < ω realizes r : < ω, b : < ω realizes s : < ω } The d dese is a metric iff d is, ad µ-d-tameess trasfers from K to K dese i the obvious way. Refereces [Bal09] Joh Baldwi, Categoricity, Uiversity Lecture Series, America Mathematical Society, [BBHU08] Itai Be Yaacov, Alexader Berestei, C. Ward Heso, Alexader Usvyatsov, Model theory for metric structures, Model theory with applicatios to algebra ad aalysis, vol. 2 (Chatzidakis, Macpherso, Pillay, Wilke eds.). Lodo Math Society Lecture Note Series, Cambridge Uiversity Press, 2008.

24 24 WILL BONEY [Cha68] C. C. Chag, Some remarks o the model theory of ifiitary laguages, The Sytax ad Sematics of Ifiitary Laguages (J. Barwise, ed.), vol. 72, [Gro1X] Rami Grossberg, A Course i Model Theory, I Preparatio, 201X. [HH09] Åsa Hirvoe ad Tapai Hyttie, Categoricity i homogeeous complete metric spaces, Archive of Mathematical Logic 48 (2009), o. 3-4, [HH12] Åsa Hirvoe ad Tapai Hyttie, Metric Abstract Elemetary Classes with pertubatios, Fudameta Mathematica 217 (2012), [Kei71] H. Jerome Keisler, Model Theory for Ifiitary Logic, Studies i Logic ad the Foudatios of Mathematics, vol. 62, North-Hollad Publishig Co., Amsterdam, [Mu00] James Mukres, Topology, 2d ed., Pretice Hall 2000 [Sh88] Saharo Shelah, Classificatio of oelemetary classes, II. Abstract elemetary classes, Classificatio theory (Joh Baldwi, ed.), 1987, pp [ViZa] Adres Villaveces ad Pedro Zambrao, Aroud superstability i Metric Abstract Elemetary Classes: limit models ad r-towers, Accepted, Preprit series of Mittag-Leffler Istitute. [Zam] Pedro Zambrao, Aroud superstability i Metric Abstract Elemetary Classes, Ph. D thesis, [Zam12] Pedro Zambrao, A stability trasfer theorem for d-tame Metric Abstract Elemetary Classes, Math Logic Q (58), p , address: wboey2@uic.edu Mathematics, Statistics, ad Computer Sciece Departmet, Uiversity of Illiois at Chicago, Chicago, Illiois, USA