Nonsensitivity, Stability, NIP, and Non-SOP; Model Theoretic Properties of Formulas in Continuous Logic. Karim Khanaki

Transcription

1 Nonsensitivity, Stability, NIP, and Non-SOP; Model Theoretic Properties of Formulas in Continuous Logic Karim Khanaki Faculty of Fundamental Sciences, Arak University of Technology, P.O. Box , Arak, Iran; School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box , Tehran, Iran; Abstract We characterize SOP, NIP, and stability in terms of topological and measure theoretical properties of classes of functions in continuous logic. We show that a formula φ(x, y) has the strict order property if there are a i s such that the sequence φ(x, a i ) is pointwise convergence but its it is not sequentially continuous. We deduce from this a theorem of Shelah: a theory is unstable iff it has the IP or the SOP. We study a measure theoretic property, Talagrand s stability, and explain the relationship between it and the NIP in continuous logic. This makes clear that Talagrand s stability is the correct counterpart of NIP in integral logic. Then we study forking and independence in stable and NIP theories, and their connections to measure theory. We also study sensitive families of functions and chaotic maps and their connections with stability. Keywords: Sensitivity, chaotic function, integral logic, local stability, independence property, strict order property, continuous logic AMS subject classification: 03B48, 03C45, 28C10, 43A07 Contents 1 Introduction 2 2 Continuous Logic Syntax and semantics Compactness Types Stability 6 4 NIP Talagrand s stability NIP in Continuous Logic Almost definability of types in NIP theories

2 5 Forking and independence relation Non-forking and stability Properties of non-forking Almost non-forking and NIP Properties of almost non-forking The number of coheirs SOP NIP+Non-SOP = Stable Definability of types and Fréchet-Urysohn spaces The Eberlein-Šmulian theorem and NIP/Non-SOP First order logic is angelic Sensitivity and chaotic maps 22 8 Appendix 23 References 25 1 Introduction Shelah s stability theory essentially leads to classifying the special properties of formulas in which the number of types (or models) can be controlled by them. The set-theoretic criteria largely pass over in favor of definitions which mention ranks or combinatorial properties of a particular formula. Up to present time there are a few properties of formulas in which their definitions are in terms of topological (or analysis) concepts. Maybe the discrete nature of formulas in classical model theory is its cause. For example, a formula φ(x, y) has the order property if there exist a i, b j, i, j < ω such that φ(a i, b j ) if and only if i < j. One can assume that φ is a 0-1 valued function, then φ has the order property iff there exist a i, b j such that i j φ(a i, b j ) = 1 0 = j i φ(a i, b j ). Therefore stability is equivalent to the absence of two different double its. The latter is a topological property of a class of functions by Grothendieck s criterion [Gro52]. What happen if we study real valued formulas? One might therefore hope to obtain new classes of functions (formulas) and develop a sharper stability theory by making use of topological properties of Stone spaces instead of only their cardinalities. In Mathematics, more exactly Analysis, the set X R of all real-valued functions on a topological space X can be classified by many classes such as differentiable, continuous, Baire, Borel, measurable and so on. A function belongs to a special class if it satisfies a desirable property. We know that the class of continues function is not closed for pointwise topology; there are sequences of continuous functions such that their poinwise its are not continuous. In classical model theory, we classify not only sets of real-valued functions (or formulas) on a specific space X, but that we study the (first order) properties so that a set A has the property P if (i) the closure A of A is a compact subset of a suitable set A P, (ii) every element of A is a closure of a small subset of A. For example, we study the 2

3 sets of continuous functions such that the pointwise closure of their elements are again continuous, and every point in the closures is a it of a sequence of the elements of those sets. In model theory, the later classes are called stable. Let us explain further. In this paper we work in Continuous Logic which is an extension of classical first order logic, and therefore our results hold for classical case. Let M be an L-structure in this logic. A maximal set of formulas in L(M) which is finitely satisfiable in a model of T h(m) is called a complete type. By Kakutani representation theorem of M-spaces, there is a compact Hausdorff space S(M), is called the space of types, such that every type is realized in S(M). Then for every formula φ, there is a continuous function f φ : S(M) R which is an extension of the interpretation of formula φ(x) in M. By a result of Grothendieck, it is known that a subset A of real-valued functions on on M (or S(M)) is stable if and only if it is relatively compact in C(S(M)) for the topology of poinwise convergence, where C(S(M)) is the space of all continuous functions on S(M). In fact, one can continue this pattern for the NIP. By a result of Bourgain, Fremlin and Talagrand (short BFT), we show that the set A has the NIP if and only if the pointwise closure of A is a set of measurable functions for all Radon measures on S(M). In fact, the role played by the Grothendieck s criterion in stable theories is mirrored in NIP theories by the result of BFT. This is not the end of the story, we give a notation of non-forking extension in NIP theories, we call almost non-forking, such that it satisfy symmetry and transitivity. Moreover, in stable cases, this notation is consistence with the classical notation. We study the strict order property (short SOP) and show that a formula φ(x, y) has the strict order property if there are a i s such that the sequence φ(x, a i ) is pointwise convergence but its it is not sequentially continuous. We deduce from this a theorem of Shelah: a theory is unstable iff it has the IP or the SOP. Then we show that the theorem definability of types in stable theories is equivalent to the Fréchet-Urysohn property, and Shelah s theorem is equivalent to the Eberlein-Šmulian theorem. Finally, we study sensitive families of functions and chaotic maps and their connections with stability. At the end, it is worth stating that many of results in this paper holds in other realvalued logics, such as integral logic, operator logic, and so on. 2 Continuous Logic In this section we give a brief review of continuous logic from [BU10] and [BBHU08]. Results stated without proof can be found there. Readers familiar with continuous logic can go to the next section. 2.1 Syntax and semantics A Language is a set L consisting of constant symbols and function/relation symbols of various arities. To each relation symbol R is assigned a bound R 0 and we assume that its interpretations is bounded by R. It is always assumed that L contains the metric symbol d and d = 1. We use R as value space and its common operations +, and 3

4 scalar products as connectives. Moreover to each relation symbol R (function symbol F ) is assigned a modulus of uniform continuity R ( F ). We also use the symbols sup and inf as quantifiers. Let L be a language. L-terms and their bound are inductively define as follows: Constant symbols and variables are terms If F is a n-ary function symbol and t 1,..., t n are terms, then F (t 1,..., t n ) is a term. All L-terms are constructed in this way. Definition 2.1 L-formulas and their bounds are inductively defined as follows: Every r R is an atomic formula with bound r. If R is a n-ary relation symbol and t 1,..., t n are terms, R(t 1,..., t n ) is an atomic formula with bound R. If φ, ψ are formula and r R then φ + ψ, φ ψ and rφ are formula with bound resp φ + ψ, φ ψ, r φ. If φ is a formula and x is a variable, sup x φ and inf x φ are formulas with the same bound as φ. Definition 2.2 A perstructure in L is pseudo-metric space (M, d) equipped with: for each constant symbol c L, an element c M M for each n-ary function symbol F a function F M : M n M such that d M n ( x, ȳ) F (ɛ) = d M (F M ( x), F M (ȳ)) ɛ for each n-ary relation symbol R a function R M : M n [ R, R ] such that d M n ( x, ȳ) R (ɛ) = R M ( x) R M (ȳ) ɛ. If M is a prestructure, for each formula φ( x) and ā M, φ M (ā) is defined inductively starting from atomic formulas. In particular, (sup y φ) M (ā) = sup b M φ M (ā, b). Similar for inf y φ. Proposition 2.3 Let M be an L-prestructure and φ( x) a formula with x = n. Then φ M ( x) is a real valued function on M n with a modulus of uniform continuity φ and φ M (ā) φ for every ā. Interesting prestructures are those which are complete metric spaces. They are called L-structures. Every prestructure can be easily transformed to a complete L-structure by first taking the quotient metric and then completing the resulting metric space. By uniform continuity, interpretations of function and relation symbols induce well-defined function and relations on the resulting metric space (cf. [BU10]). 4

5 2.2 Compactness Let L be a language. An expression of the form φ ψ, where φ, ψ are formulas, is called a condition. The equality φ = ψ is called a condition again. These conditions are called closed if φ, ψ are sentences. A theory is a set of closed conditions. The notion M T is defined in the obvious way. M is then called a model of T. A theory is satisfiable if has a model. In [BBHU08], a variant of the ultraproduct constructon is defined. The most important application of this construction in logic is to prove the Loś theorem and to deduce the compactness theorem. Theorem 2.4 (Compactness Theorem) Let T be an L-theory and C a class of L- structures. Assume that T is finitely satisfiable in C. Then there exists an ultraproduct of structures from C that is a model of T. 2.3 Types There is a intrinsic connection between some concepts from functional analysis and continuous logic. In fact, types are well-known mathematical objects, Riesz homomorphisms. For showing of this connection, one can use two results in functional analysis, Gelfand representation of C -algebras and Kakutani representation of M-spaces. Here we work in a real-valued logic, therefore we use the later. Suppose that L is an arbitrary language. Let M be an L-structure, A M and T A = T h(m, a) a A. Let p(x) be a set of L(A)-statements in free variable x. We shall say that p(x) is a type over A if p(x) T A is satisfiable. A complete type over A is a maximal type over A. We let S M (A) be the set of all complete types over A. The type of a in M over A, denoted by tp M (a/a), is the set of all L(A)-statements satisfied in M by a. We give a characterization of complete types. First we need some definitions. Let L A be the family of all interpretations φ M in M where φ is an L(A)-formula (not a condition) with a free variable x. L A is an Archimedean Riesz space of measurable functions on M. Let σ A (M) be the set of Riesz homomorphisms I : L A R such that I(1) = 1. The set σ A (M) is called the spectrum of T A. Note that σ A (M) is a weak* compact subset of L A. The next proposition shows that a complete type can be coded by a Riesz homomorphism and gives a characterization of complete types. In fact, by Kakutani representation theorem, the map S M (A) σ A (M), defined by p I p where I p (φ M ) = r if φ(x) = r is in p, is bijection. Proposition 2.5 Assume that M, A and T A are as above. (i) The map S M (A) σ A (M) defined by p I p is bijection. (ii) p S M (A) if and only if there is an elementary extension N of M and a N such that p = tp N (a/a). 5

6 We equip S M (A) = σ A (M) with the related topology induced from L A. Therefore, S M (A) is a compact and Hausdorff space. For a complete type p = tp M (a/a) and a formula φ, we let φ(p) = φ M (a). It is easy to verify that the topology on S M (A) is the weakest topology in which all the functions p φ(p) are continuous. This topology sometimes called the logic topology. 3 Stability Definition 3.1 A formula φ(x, y) is called stable in a structure M if there are no r > s and infinite sequences a n, b n M such that for all i > j: φ(a i, b j ) r and φ(a j, b i ) s. A formula φ is stable in a theory T if it is stable in every model of T. It is easy to verify that φ(x, y) is stable in M if whenever a n, b n M form two sequences we have n m φ(a n, b m ) = m n φ(a n, b m ), provided both its exist. Fact 3.2 (Grothendieck s Criterion, [Gro52]) Let X be an arbitrary topological space, X 0 X a dense subset. Then the following are equivalent for a subset A C b (X): (i) The set A is relatively weakly compact in C b (X). (ii) The set A is bounded, and whenever f n A and x n X 0 form two sequences we have n m f n (x m ) = m n f n (x m ), whenever both its exists. Because we believe that the proof of Grothendieck s criterion in the case of compact spaces which is presented in [KN63, 8.18] is informative and useful we mention it in 8. Fact 3.3 (Fundamental Theorem of Stability) Let φ(x, y) be a formula and T a theory. Then the following are equivalent. (i) The formula φ is stable in T. (ii) For every model M T, every φ-type over M is definable by a φ-predicate over M. (iii) For each cardinal λ = κ ℵ 0 T, and model M T with M λ, S φ (M) λ. (iv) There exists a cardinal λ = κ ℵ 0 then S φ (M) λ. T such that for every model M T, M λ We give some definitions and results that we use later. 6

7 Definition 3.4 (Order property) A bounded family F of real valued functions on a set X has the (strong) order property (or short OP) if there exist real numbers s < r and an (infinite) sequence {f n } n F and {a m } m X such that either min m n f n (a m ) r and max m<n f n (a m ) s, or min m<n f n (a m ) r and max m n f n (a m ) s. F has the weak order property (or short wop) if there exist real numbers s < r and an (infinite) sequence {f n } n F and {a m } m X such that either inf m f n (a m ) r and sup n f n (a m ) s, or inf n f n (a m ) r and sup m f n (a m ) s. A formula φ has the (weak) order property in a model M if the family {φ(x, b), φ(a, y) : a, b M} has the (weak) order property. A formula φ has the (weak) order property in a theory T if it has the (weak) order property in a model of T. We note that if {f n } C(X), f n f pointwise, and {f n } n be ordered, then f is not sequential continuous. For this, we have Definition 3.5 Let X be a topological space. A family F C(X) has sequential continuous property (or short SCP) if for every convergence sequence {f n } n of F, its it is sequential continuous, i.e. if f n f pointwise, then there are not a m, a X such that a m a and m f(a m ) f(a). Later we show that the connection between the SCP and the strict order property. Now we give some simple lemmas that we use later. Lemma 3.6 Let X be compact and A C(X) be bounded. (i) Non-wOP = SCP. (ii) Stable = SCP. (iii) Stable = Non-OP. (iv): The formula φ(x, y) has Non-OP in theory T = φ(x, y) has SCP in T. (v): The formula φ(x, y) has Non-OP in theory T = φ(x, y) is stable in T. Therefore, in a theory T, stable Non-OP. Proof. (i): Assume that Non-SCP. Therefore there are f n A, a m X, such that f n f pointwise, and a m a but m f(a m ) f(a). Note that we can assume m f(a m ) exists, because we can find a subsequence a mk such that mk f(a mk ) exists. Therefore, m n f n (a m ) n m f n (a m ). Therefore, A has wop. (ii) Similar to (i). (iii): Assume that OP. Let f n A and a m X such that min m n f n (a m ) r and max m<n f n (a m ) s where s < r. By a technical result in [Rud64] (Theorem 7.23), since A is bounded and X is compact, we can assume that the its m n f n (a m ) and n m f n (a m ) exist. But they are not equal, and hence A is not stable. (iv): We use the notation f n (y) for φ(b n, y). Assume that φ has Non-SCP. Therefore there are f n A, a m X, such that f n f pointwise, and a m a but m f(a m ) s > r f(a). Then, by a selection argument, one can find subsequences {a mk } and {f nk } such that they are witnesses of the OP. Indeed, by induction, we assume that we find a 1,..., a n and {f k } k N such that max k i n f k (a i ) r and min m n<k f k (a m ) s. First, 7

8 without loss of generality we assume that f k (a) r for all k 0. If f k (a n+1 ) > r for some k n, then we omit a n+1 and take a n+1 := a m for some m > n + 1 such that f k (a m ) r for all k n. If f k (a n+1 ) < s for some k n + 1, then we can find a N > n + 1 such that f k (a n+1 ) s for each k N, and we take f n+1+i := f N+i for all i 0. And we continue in this way. Therefore, for each n, we can find a 1,..., a n and {b k } k N such that max m k φ(b k, a m ) r and min m<k φ(b k, a m ) s. Now, by the compactness theorem of model theory, φ has OP. (v)similar to (iv). Definition 3.7 We say that a (bounded) family F of real valued function on a set X has the convergence subsequence property (or short CSP) if for every sequence in F has a pointwise convergence subsequence in R X. Later we show that the connection between the CSP and the independence property. Lemma 3.8 (i) SCP + CSP = Non-wOP. (ii) Stable = Non-wOP. Proof. (i): Assume that wop and CSP(=every sequence in A has a pointwise convergence subsequence in R X ) hold. Therefore there are f n, a m and s < r such that inf m f n (a m ) r and sup n f n (a m ) s where s < r. We can assume that a m a. By CSP, there is a function f such that f n f pointwise. Therefore, f(a m ) = n f n (a m ) sup n f n (a m ) s. Since f n s are continuous, f n (a) = m f n (a m ) = inf m f n (a m ) r. Since X is compact and f is bounded, we can find a subsequence a mk such that {f(a mk )} mk be convergence, and a mk a. Therefore, mk f(a mk ) f(a). Therefore, A has not SCP. (ii): Similar to (i). 4 NIP In this section, first we study some concepts presented in [GM14] and give a new one that has a meaningful connection to stability, see (1) in below. Then we study Talagrand s stability and its relationship to the NIP in continuous logic. Finally, we give a variant of definability of types in NIP theories. Definition 4.1 ([GM14], Definition 2.1) Let (X, τ) be a topological space and ɛ a positive number. We say that a family of (not necessarily continuous) functions F = {f i : X R} i I is: (1) ɛ-non-sensitive in every point (ɛ-nse in short) if for every x X there exists an open subset O x x in X such that diam(f i (O x )) ɛ for every i I. (2) ɛ-non-sensitive (ɛ-ns in short) if there exists a non-void open subset O in X such that diam(f i (O)) ɛ for every i I. (3) Non-Sensitive (in every point) if F is ɛ-ns (ɛ-nse) for every ɛ > 0. 8

9 (4) ɛ-fragmented if for every nonempty closed subset A of X the restriction F A := {f A : A R} f F is an ɛ-ns family. (5) Fragmented, or Hereditarily Non-Sensitive, if F is ɛ-fr for every ɛ > 0. (6) Eventually Fragmented if for every infinite subfamily L F there exists an infinite fragmented subfamily K L. Lemma 4.2 Assume that X is a compact space, and F = {f i : X R} i I C(X) is bounded. Then (i) (ii). (i) ɛ-non-stability: F is ɛ-non-stable, i.e. there are f m s in F and x n s in X such that m n f n (x m ) n m f n (x m ) > ɛ. (ii) ɛ-sensitivity in a point: F is ɛ-sensitive in a point x X, i.e. for every non-void open subset O x in X, there exist some i I such that diam(f i (O)) > ɛ. Proof. (i)= (ii): Assume that F is ɛ-non-stable, and m n f n (x m ) n m f n (x m ) > ɛ. (By Theorem 2.16 in [GM14], we can assume that I is countable.) Assume that x m x. We claim that F is ɛ-sensitive in x. If not, there exists a non-void open subset O x in X such that diam(f i (O)) ɛ for every n I. Since x m x, we can assume that x m O for all m. Therefore, f n (x m ) f n (x) ɛ for all m, n. This contradicts with the ɛ-non-stability of F. (ii)= (i): See Proposition 3.5 in [Meg03]. We can give an explicit proof. Indeed, assume that F is ɛ-stable but it is ɛ-sensitive in x X. Let O n = {y : d(x, y) > ɛ} for all n. Then there exist some f n F such that diam(f n (O n )) > ɛ. Since O n O n+1, diam(f n (O k )) > ɛ for all k n. (For simplicity, we assume that F is stable.) By NIP, there is a sequential continuous function f such that f n f. Therefore, ɛ < m diam(f m (O n )) = diam(f(o n )) for all n. On that other hand, n diam(f(o n )) = diam(f({x})) = 0. But this is a contradiction. Remark 4.3 In the previous lemma ɛ -stability implies ɛ-non-sensitivity. If not: indeed, by ɛ-sensitivity, ɛ < m diam(f m (O n )) = sup y,z On m d(f m (y), f m (z)). Then by 2 Lemma 5.2 in [BBHU08], there exist y n, z n O n such that n sup y,z On m d(f m (y), f m (z)) = n m d(f m (y n ), f m (z n )). By ɛ -stability, 2 n m f m (y n ) m n f m (y n ) < ɛ 2 and n m f m (z n ) m n f m (z n ) < ɛ. Therefore, 2 n m d(f m (y n ), f m (z n )) = m n d(f m (y n ), f m (z n )) ɛ + ɛ. But this is a contradiction. 2 2 To summarize: ɛ-sensitive in a point ɛ/2-non-stable Definition 4.4 (Independence Property) A family F of real valued functions on a set X is said to be independent or has the independence property if there exist real numbers s < r and a sequence f n F such that for each k 1 and for each I {1,..., k}, there is x X with f i (x) s for i I and f i (x) r for i / I. A formula φ is independent in a model M if the family {φ(x, b), φ(a, y) : a, b M} is independent. A formula φ has the independence property in a theory T if it has dependence property in a model of T. 9

10 Fact 4.5 ([GM14], Theorem 2.11) Let X be a compact space and F C(X) a bounded subset. The following conditions are equivalent: (1) F does not contain an independent subsequence. (2) F does not contain a subsequence equivalent to the unit basis of l 1. (3) Each sequence in F has a pointwise convergent subsequence in R X. (4) F is a Rosenthal family for X. (5) F is an eventually fragmented family. Lemma 4.6 ([GM14], Lemma 7.9) Let {f n } be a bounded sequence of continuous functions on a topological space X. Let Y be a dense subset of X. Then {f n } is an independent sequence on X if and only if the sequence of restrictions {f n Y } is an independent sequence on Y. Proposition 4.7 (i) Non-OP = NIP. (ii) Non-OP = NIP + SCP. Proof. (i): Easy. By (i) and 3.6, the proof is completed. 4.1 Talagrand s stability Historically, Talagrands stability (see 4.8), which we call the almost independence property, arose naturally when Talagrand and Fremlin were studying pointwise compact sets of measurable functions; they found that in many cases a set of functions was relatively pointwise compact because it was stable (Fact 4.9). Later did it appear that the concept was connected with Glivenko-Cantelli classes in the theory of empirical measures, as explained in [Tal87]. In this subsection we study this property and show that it is the correct counterpart of NIP in integral logic (cf. [Kha14]). Then, we give the connection between the NIP in continuous logic and this property. Definition 4.8 (Talagrand stability, [Fre06]) Let A C(X) be a (not necessarily countable) pointwise bounded family of continuous functions from X to R. Suppose that µ is a measure on X. We say that A has the µ-almost non independence property, or short the µ-almost NIP, if A is a stable set of functions in the sense of Definition 465B in [Fre06], that is, whenever E M is measurable, µ(e) > 0 and s < r in R, there is some k 1 such that (µ 2k ) D k (A, E, s, r) < (µe) 2k where D k (A, E, s, r) = f A{w E 2k : f(w 2i ) s, f(w 2i+1 ) r for i < k}. We say that A has the universal almost non independence property, or short the universal almost NIP, if A has the µ-almost NIP for all Radon measure µ on X. Fact 4.9 ([Fre06, Proposition 465D]) Let X be a compact Housdorff space and A C(X) be a (not necessarily countable) pointwise bounded family of continuous functions 10

11 from X to R. Suppose that µ is a Radon measure on X. If A has the µ-almost NIP, then the poinwise closure of A, denoted by cl p (A), has the µ-almost NIP and every element in cl p (A) is µ-measurable. In [Fre75] Fremlin obtained an important result, which we call Fremlin s dichotomy. He proved that sets of measurable functions on perfect measure spaces are either good (relatively countably compact for the pointwise topology and relatively compact for the topology of convergence in measure) or bad (with neither property). (We recall that a subset A of a topological space X is relatively countably compact if every sequence of A has a cluster point in X.) Now, we remember this result: Theorem 4.10 (Fremlin s dichotomy, [Fre75]) Let (X, Σ, µ) be a perfect σ-finite measure space, and {f n } a sequence of real-valued measurable functions on X. Then either {f n } has a subsequence which is convergent almost everywhere or {f n } has a subsequence with no measurable cluster point in R X. Proposition 4.11 Let (X, Σ, µ) be a finite Radon measure on compact a space X, and A L 0 a bounded family of real-valued measurable functions on X. Then (i)= (ii). If A has the property (M) in [Tal87], i.e. for all s < r and all k, the set D k (A, X, r, s) be measurable (particularly A be countable), then (ii)= (i). (i)= (iii), but not(iii)= (i). (i) A has the µ-almost NIP. (ii) The following property, that we call ( ), fails: There exist measurable set E, µ(e) > 0 and s < r in R, such that for each n, and almost all w E n, for each subset I of {1,..., n}, there is f A with f(w i ) < s if i I and f(w i ) > r if i / I. (iii) Every sequence in A has a subsequence which is convergent almost everywhere. Proof. (i)= (ii): Clear. (M) (ii)= (i): Proposition 4 in [Tal87], or part (iii) of case 2 of the proof of 465L in [Fre06]. (i)= (iii): Assume that {f n } A. We take an arbitrary subsequence of it (still denoted by {f n }). Let D be a non-principal ultrafilter on N. Define f(x) = D f i (x) for all x X. (By assumption, there is a real number r such that for each h A, h r. Therefore, f is well defined.) Since A has the µ-almost NIP and f cl p ({f n }), the function f is measurable (cf. 456Db in [Fre06]). Therefore every subsequence of {f n } has a measurable cluster point. Then, by Fremlin s dichotomy, {f n } has a subsequence which is convergent almost everywhere. not(iii)= (i): In [SF93], Shelah and Fremlin find that in a model of set theory there is a separable pointwise compact set of real-valued Lebesgue measurable functions on the unit interval, which it is not stable. Professor Fremlin kindly pointed out to the author that Shelah s model, described in their paper [SF93], in fact deals with the point that there is a countable set of continuous 11

12 functions which is relatively pointwise compact in L 0 but that it is not R-stable. Of course this will not happen if a set be relatively pointwise compact in L 0 (µ) for each Radon measure µ: Proposition 4.12 Let X be compact and Hausdorff, and F C(X) be uniformly bounded. Then the following are equivalent: (i) F is NIP. (ii) Each sequence in F has a subsequence which is pointwise convergence. (iii) Each sequence in F has a subsequence which is µ-almost everywhere convergence for each Radon measure µ on X. (iv) For each Radon measure µ on X, each sequence in F has a subsequence which is µ-almost everywhere convergence. Proof. (i) (ii): This is the equivalence (ii) (vi) in Theorem 2F in [BFT78]. (i) (iii): By either Fremlin s dichotomy and the equivalence (v) (vi) in Theorem 2F in [BFT78], or 465X(a) in [Fre06] and the equivalence (ii) (vi) in Theorem 2F in [BFT78], the proof is completed. (i) (iv): This is the equivalence (v) (vi) in Theorem 2F in [BFT78] and Fremlin s dichotomy. 4.2 NIP in Continuous Logic The following fact shows that why the µ-almost NIP is the correct notation of the NIP for integral logic. First we review a definition in Continuous Logic. Let M be an L-structure in continuous logic, and φ(x; y) a formula. We say φ(x; y) has the non independent property (short the NIP) on M, in the sense of continuous logic, if for each sequence φ(a n, y) in the set A = {φ(a, y) : a M}, and r > s there is some I N such that {y S φ(y) (M) : ( i I φ(a i, y) < s ) ( i/ I φ(a i, y) > r ) } =, where S φ(y) (M) is the space of all complete φ(y)-types on M. Theorem 4.13 (Theorem 2F, [BFT78]) Let M be an L-structure in continuous logic, and φ(x; y) a formula. Let A = {φ(a, y) : a M} and à = {φ(x, b) : b M}. Then the following are equivalent: (i) φ has the NIP on M (in the sense of continuous logic). (ii) à has the µ-almost NIP for all Radon measures on M. Proof. (i)= (ii): By the NIP of φ(x; y) and the compactness theorem of continuous logic, there is some integer n such that no set of size n is shattered by φ(x, y). Therefore if E M, µ(e) > 0, r > s, then for each (a 1,..., a n ) E n there is a set I {1,..., n} such that {y S φ(y) (M) : ( φ(a i, y) < s ) ( φ(a i, y) > r )} =, i I where S φ(y) (M) is the space of all complete φ(y)-types on M. Therefore, the set A = {φ(x, b) : b S φ(y) (M)} (and particularly the set Ã) has the µ-almost NIP for all Radon 12 i/ I

13 measures on X = S φ(y) (M) (equivalently on M). (Note that by Theorem 465T in [Fre06], A has the µ-almost NIP if and only if every countable subset of A has the µ-almost NIP. Therefore, the conditions (i) and (ii) in 4.11 are equivalent.) (ii)= (i): Conversely, assume that à has the µ-almost NIP for all Radon measures on M (or equivalently on X = S φ(x) (M)). Therefore, by Fact 4.9, à is relatively compact in M r ( X) (the space of all µ-measurable functions on X for each Radon measure µ). By the equivalence (iv) (vi) in Theorem 2F in [BFT78], for each sequence φ(x, a n ) in Ã, and r < s, there is some I N such that {x S φ(x) (M) : ( φ(x, a i ) < s ) ( φ(x, a i ) > r )} =. i I Therefore, the set à (and equivalently the dual formula φ(y; x) := φ(x; y)) has the NIP on M (in the sense of continuous logic). So, by applying the direction (i)= (ii) on φ (equivalently on Ã), we conclude that à = A has the µ-almost NIP (for all Radon measures). Then, again by Theorem 2F in [BFT78], we conclude that A has the NIP in the sense of continuous logic. In fact the proof of the previous result says more: Corollary 4.14 (NIP duality) By the above assumptions, φ has the NIP if and only if φ has the NIP. Similarly, A has the µ-almost NIP (for all Radon measures µ) if and only if à has the µ-almost NIP (for all Radon measures µ). Consequently, φ has the NIP if and only if A has the µ-almost NIP (for all Radon measures µ). 4.3 Almost definability of types in NIP theories It is well-known that every type in a stable theories is definable. Here we want to give a counterpart of this fact for NIP theories. In [Kha14] the author showed that if a formula φ (in integral logic) has the µ-almost NIP on a model M, then every type in S φ (M) is µ-almost definable. Here we say a formula ψ is universally measurable on M, if it is µ-measurable for all probability Radon measures on M. We need some definitions from [Kha14]. Let ψ be a measurable function on (S φ (M), µ φ ) where µ φ is the unique Radon measure induced by φ M (x, b) for all b M. Then ψ is called an almost φ-definable relation over M if there is a sequence g n : S φ (M) R, g n φ, of continuous functions such that n g n (b) = ψ(b) for almost all b S φ (M). (We note that by the Stone-Weierstrass theorem every continuous function g n : S φ (M) R can be expressed as a uniform it of algebraic combinations of (at most countably many) functions of the form p φ(p, b), b M.) An almost definable relation ψ(y) over M defines p S φ (M) if φ(p, b) = ψ(b) for almost all b M, and in this case we say that p is almost definable. Assume that every type p in S φ (M) is almost definable by a measurable function ψ p. Then, we say that p is almost equal to q, denoted by p q, if ψ p (b) = ψ q (b) for almost all b M. Define [p] = {q S φ (M) : p q} and [S φ ](M) = {[p] : p S φ (M)}. Now by the above results it is easy to show that: 13 i/ I

14 Theorem 4.15 (Almost definability) Let φ(x, y) be a formula NIP in a structure M (in the sense of continuous logic). Then every p S φ (M) is almost definable by a (unique up to measures) universally measurable and almost φ-definable relation ψ(y) over M, where φ(y, x) = φ(x, y). Proof. See [Kha14]. In fact, by the BFT theorem, the converse of the above theorem holds: Theorem 4.16 Let T be a theory. Then the following are equivalent: (i) T is NIP. (ii) For every formula φ and M T, every type p S φ (M) is almost definable (for all Radon measures on M). 5 Forking and independence relation It is a known fact that in NIP theories, non-forking is not an independence relation, which means in particular that it does not satisfy symmetry and transitivity (see for example [Sim14a]). Historically, H. Jerome Keisler was the first person who discovered the relationship between forking, measures and NIP theories in the frame work of classical model theory. Almost a decade before that, Bourgain, Fremlin and Talagrand (short BFT) were shown the intrinsic connection between measures theory and the dependence property. In fact, the role played by the Grothendieck s criterion in stable theories is mirrored in NIP theories by a result of them. In this section we define non-forking extensions in two levels: stable and NIP theories. Then we show that non-forking is an independence relation in both levels. 5.1 Non-forking and stability A global type, is a type over the monster model U. Let A U and let p S x (U). We say that p is finitely satisfiable (finitely realized) in A if for every formula φ(x, a) = r p there exists c A such that U φ(c, a) = r. Definition 5.1 Let T be stable, A U, and p S x (U). We say that p does not fork over A if p is finitely satisfiable in every model containing A. Construction. Let A = {a i : i < κ} U be a small set and D be an ultrafilter on κ. We define global type p D S(U) by: p D φ(x, b) = r {i : U φ(a i, b) = r} D, for each formula φ(x, b) = r L(U). Then p D is finitely satisfiable in A. Conversely, every global type finitely satisfiable in A is equal to p D for some (not necessarily unique) ultrafilter D. In particular note that if we take D to be a principal ultrafilter, then we obtain a realized type. 14

15 If p 0 S x (M) is a type, then a coheir of p 0 is a global extension p of p 0 which is finitely satisfiable in M, i.e. p 0 p S x (U) and for every formula φ(x; b) = r p, there is a M such that φ(a; b) = r holds. Equivalently, p S x (U) is a coheir of a p 0 S x (M) if there are types p i S x (M) such that i p i = p. Indeed, by the definition of coheir; if φ(x, b) = r p then there is a M such that U φ(a, b) = r. Therefore p [φ(x, b) = r] = {q S x (U) : q φ(x, b) = r} and tp(a) [φ(x, b) = r], i.e. for each neighborhood [φ(x, b) = r] of p, there exists a type tp(a) = tp(a) M S x (M) in [φ(x, b) = r], and this means that p is a cluster point of some elements of S x (M). A type p 0 S x (M) always has at least a coheir: let {a i M : i < M } be an indexed set of elements of M, and D an ultrafilter on M such that if p 0 φ(x, b) = r then {i : U φ(a i, b) = r} D. Then we define p D S x (U) by: p D φ(x, b) = r {i : U φ(a i, b) = r} D, for every formula φ(x, b) = r L(U). Clearly, p D is a coheir of p 0. (Note that the set E = {{i : U φ(a i, b) = r} : p 0 φ(x, b) = r} has the finite intersection property. If D be an ultrafilter on M such that E D. Then p D φ(x, b) = r {i : U φ(a i, b) = r} D.) We remark that (in continuous logic) every type in S x (M) is finitely satisfiable in M. Indeed, if φ(x, a) r p S x (M) then U sup x φ(x, a) r, and hence M sup x φ(x, a) r (because sup x φ(x, a) r T h(m)), i.e. there is a c M such that φ(c, a) r holds. Theorem 5.2 Let T be stable, A U, and p S x (U). Then the following are equivalent: (i) p does not fork over A; (ii) p is definable over every model containing of A. Proof. (i)= (ii): Assume that p S x (U) is a coheir of p M which M is a model containing of A. Then there are p i S x (M), i I such that i,d p i = p which D is an ultrafilter on I. Let φ(x, y) L. Since T is stable, every type in S x (M) is M- definable. Therefore, there is a M-definable function ψ i (=d i φ) such that p i φ(x, b) = r iff ψ i (b) = r, for all b M (and hence for all b U). Again, by stability, the set {ψ i } i is a relatively compact subset of C(S φ (M)). Let ψ := i,d ψ i (pointwise it). Then for all b U p φ(x, b) = r i,d φ(p i, b) = r i,d ψ i (b) = r ψ(b) = r. since φ is continuous (ii)= (i): Clear. Assume that p is definable over every model containing of A and p [φ(x, b) = r] which b is in U. Since A = {φ(a i, y) : a i M} is relatively compact, there are a i M, i I and an ultrafilter D on I such that i,d φ(a i, y) = ψ(y). Now, clearly p φ = i,d tp φ (a i /M). Therefore, i,d tp(a i ) [φ(x, b) = r]. 15

16 Proposition 5.3 Let A U. Every type over A has an extension over U which does not fork over A. proof. See the construction. 5.2 Properties of non-forking In this subsection T is stable. In the next subsection we study forking and its properties in NIP theories. Definition 5.4 Let A B, and p S x (B). We say that p does not fork over A, or that p is a nonforking extension of p A, if p has an extension p over U such that p does not fork over A. Proposition 5.5 (symmetry) tp(a/ab) does not fork over A iff tp(b/aa) does not fork over A. proof. Assume that p S x (U) is a global non-forking extension of tp(a/m) which M is a model containing Ab. For each formula φ(x, y) there is a M-definable formula ψ(y) such that p φ(x, c) = r iff ψ(c) = r for all c U. We can assume that ψ has the following form: ψ(y) = ψ(y, A, b, m) which m M. Now we can define the global type q as the following: q φ(x, c) = r iff ψ(a, A, c, m) = r for all c U. It is easy to verify that q is a definable extension of tp(b/m) and ψ = ψ(a, A, y, m) is its definable framework, where ψ is M -definable which M is a model containing of m, a (or containing of M, a). (We note that if ψ(y) = i ψ(y, A, b, m) which ψ i are M-definable, then ψ(y) = i ψ(a, A, y, m).) ( See Iovino s paper. Also, see Proposition 7.14 in [BU10].) Proposition 5.6 (Transitivity) Let A B C and p S x (C). If p does not fork over B and p B does not fork over A, then p does not fork over A. proof. Assume that p is an extension of p and q is an extension of p B such that p does not fork over B and q does not fork over A. Then p does not fork over B, so, by uniqueness of non-forking extension, p = q. Thus, p does not fork over A. 5.3 Almost non-forking and NIP We give a generalization of the notation of non-forking extension in NIP theories. It is easy to verify that the restriction of the new notation to stable case is consistence with the (first) classical notation. The theory T is NIP if all formulas φ(x; y) L are NIP. Definition 5.7 Let T be NIP, A U, and p S x (U). (i) We say that p almost does not fork over A, or is an almost non-forking over A, if p is almost finitely satisfiable in every model containing A, i.e. there are p i S x (M) such that i p i [p] which [p] is the class of types almost equial to p (cf. [Kha14]). 16

17 (ii) p is almost definable over A if for each formula φ(x, y) and every model M containing A, there is an almost (M-)definable function ψ(y) such that φ(p, b) = ψ(b) for almost all b M (and hence b U) for each Radon measure on U. (see [Kha14].) Similar to the stable case: Definition 5.8 Let A B, and p S x (B). We say that p almost does not fork over A, or that p is an almost nonforking extension of p A, if p has an extension p over U such that p almost does not fork over A. Theorem 5.9 Let T be NIP, A U, and p S x (U). Then the following are equivalent: (i) p almost does not fork over A; (ii) p is almost definable over every model containing of A. Proof. Use The proof is similar to the stable case. 5.4 Properties of almost non-forking Proposition 5.10 Let T be NIP. (i) (symmetry): tp(a/ab) almost does not fork over A iff tp(b/aa) almost does not fork over A. (ii) (Transitivity): Let A B C and p S x (C). If p almost does not fork over B and p B almost does not fork over A, then p almost does not fork over A. proof. The proofs are similar to the stable case. 5.5 The number of coheirs Proposition 5.11 If T has IP, then for each cardinal κ > T there is some model M of cardinality κ and p S(M) having 2 2κ coheirs. Proof. Assume that T has IP. Then there is a set {a i : i < κ}, a formula φ(x; y) and r > s such that for any A κ, we can find some b A such that: φ(a i ; b A ) r if i A, and φ(a i, b A ) s if i / A. Let M be a model of size κ containing all the a i s. For D an ultrafilter over κ, we define p D by p D φ(x, b) = r {i : U φ(a i, b) = r} D. Then we have p D φ(x; b A ) r A D. This shows that for D D, the two types p D and p D are distinct. Furthermore, p D is finitely satisfiable in M. As there are 2 2κ pairwise distinct ultrafilters on κ, we have obtained 2 2κ global types finitely satisfiable in M. Since there are only 2 κ types in variable x over M, there is at least one such type which has 2 2κ global coheirs. 6 SOP In classical model theory a formula φ(x, y) has the strict order property (SOP) if there exists a sequence (a i : i < ω) in the monster model U such that for all i < ω, φ(u, a i ) φ(u, a i+1 ). 17

18 We can assume that φ(x, y) is a 0-1 valued function on U such that φ(a, b) = 1 iff φ(a, b). Then φ(x, y) has the strict order property if there are a i s such that for each b U, the sequence {φ(b, a i )} i is monotone, and therefore the pointwise it {φ(x, a i )} i exists, but its it is not sequentially continuous. Indeed, take b j φ(u, a j+1 ) \ φ(u, a j ). On the other hand, if b i φ(u, a i) then there is a k such that for all i k, b φ(u, a i ), therefore i φ(b, a i ) = 1. Otherwise, b U \ i φ(u, a 1), and therefore i φ(b, a i ) = 0. In the other word, for each b the sequence φ(b, a i ) is monotone and hence the pointwise it ψ(x) := i φ(x, a i ) is well-defined. But j ψ(b j ) = j i φ(b j, a i ) = 1 0 = i j φ(b j, a i ). One can assume that U is equipped with the Stone topology (or the logic topology). Since this topology is compact, there is a subsequence b jk and b U such that b jk b. Therefore, the subsequence b jk converges to b but k ψ(b jk ) ψ(b), i.e. ψ is not sequentially continuous. To summarize, it is natural to conjecture that in the generalized model theory the strict order property is equivalent to the existence of a pointwise convergence sequence of real valued functions such that its it is not sequentially continuous. Our next goal is to convince the reader that this is indeed the case. Definition 6.1 (Strict order property) A bounded family F of real valued functions on a set X has the ɛ-strict order property (or short ɛ-sop) if there exist (i): {a m } m X, and (ii): an (infinite) sequence {f n } n F such that for each x X the sequence {f n (x)} m is monotone, and hence f n is convergence to a function f R X, but m f(a m ) n m f n (a m ) > ɛ. F has strict order property (or short SOP) if for some ɛ > 0 it has ɛ-sop. If X be a compact space and f n s be sequentially continuous then the SOP means that f is not sequentially continuous. A formula φ(x, y) has the (ɛ-)strict order property in a model M if the family {φ(x, a) : a M} has the (ɛ-)strict order property. The formula φ has the (ɛ-)strict order property in a theory T if it has the (ɛ-)strict order property in a model of T. The next result shows that the SCP and the Non-SOP are actually the same. Proposition 6.2 Non-SOP SCP. Proof. = : Assume that f n C(X) and a m X such that f n f but n m f n (a m ) m f(a m ), i.e. T has not SCP. Then, for a countable model M T (or equivalently a separable model) we can find a subsequence f nk such that for each a M the sequence f nk (a) is monotone. (Indeed, let M be a countable model of T such that {a m } M. Let b 1, b 2,... be an enumeration of the elements of M. Then for each b i there is a monotone subsequence of {f n (b i )}, which we shall denote by {f i,n }. By a diagonal argument, we can find a sequence {f n,n } such that it is desirable (cf. Theorem 7.23 in [Rud64]).) Now, by the compactness theorem, since T is complete, this sequence is monotone in every model of T, i.e. if N T then {f nk (b)} is monotone for each b N, and hence T has SOP. =: Clear. 18

19 By the above result, SOP is expresible. Indeed, let ψ n (x) := ( n f i (x) f i+1 (x) ) ( n f i (x) f i+1 (x) ). i=1 Then by the compactness theorem, {ψ n } n expresses that the sequence {f n } n is monotone. Also, for all i < j, f i (b j ) f j (b i ) ɛ expresses the ɛ-non-stability. To summarize, the SOP is expressible. 6.1 NIP+Non-SOP = Stable Now we can give a topological proof of the following classical result. We note that the NIP and the Non-SOP are negative properties but they are equivalent to the CSP and the SCP which are positive. Theorem 6.3 Let X be a compact space and A C(X) be bounded. Then A is stable iff it has CSP and SCP. Proof. We show that cl p (A) C(X) iff every sequence of A has a convergence subsequence in R X and the it of every convergence sequence of A is sequentially continuous. Assume that A has NIP(=CSP) and SCP. Assume that {f n } n A and {a m } m X and the double its m n f n (a m ) and n m f n (a m ) exist. Let a be a closure point of {a m } m. Without loss of generality we can assume that a m a. By NIP, there is a convergence subsequence f nk such that f nk f. Therefore m nk f nk (a m ) = m f(a m ) and nk m f nk (a m ) = nk f nk (a) = f(a). By SCP, f is sequentially continuous and therefore m f(a m ) = f(a). Since the double its exist, it is easy to verify that m n f n (a m ) = m nk f nk (a m ) and n m f n (a m ) = nk m f nk (a m ). Therefore A has the double it property and hence it is stable(=wap). The converse is just 4.7. Corollary 6.4 (Shelah, [She71]) Theory T is unstable iff it has the IP or the SOP. We say a family F C(X) is sequentially close (or has sequentially closeness property), if the pointwise it of each convergence sequence of its elements is in C(X). If X be compact, then it is known that F has SCP iff it is sequentially close (cf. 8.2). To summarize: stable NIP + Non-SOP WAP CSP + SCP Eberlein Rosenthal + Sequentially Closeness i=1 19

20 6.2 Definability of types and Fréchet-Urysohn spaces Almost all the combinatorial content of stability, in classical model theory, is contained in the theorem of definability of types (cf. Lemma 2.2 in [Pil96]). In [Ben1?] Ben Yaacov gave a proof of this theorem using the Grothendieck s criterion. Here we take a closer study of this theorem and show that it is equivalent to that every compact subset of C(X) is a Fréchet-Urysohn space. Proposition 6.5 (i) (ii). (i) Fréchet-Urysohn space: If X is a compact space and A C(X) is relatively compact, then every point of the closure of A is a it of a sequence in A. (ii) Definability of types: If φ is a stable formula in model M, then every type in S φ (M) is definable. Proof. We use the Grothendieck s criterion as a catalyst. (i)= (ii): The set A = {φ(a, y) : a M} is relatively compact in C(S φ (M)) because φ is stable. Let p(x) S φ (M), and let a i M be any net such that i tp φ (a i /M) = p. Since A is relatively pointwise compact, there is a ψ C(X) such that i φ a i (y) = ψ(y). By (i), ψ is the pointwise it of a sequence φ an (y) of the family {φ a i (y)} i. Therefore, there is a subsequence φ an k (y) such that k φ an k (y) = ψ(y). Clearly, ψ(y) is a φ-definable relation over M, and for b M we have φ(p, b) = k φ(a nk, b) = ψ(b). Therefore, p is definable by a φ-definable relation ψ over M. (ii)= (i): Assume that the set A = {φ(a, y) : a M} C(S φ (M)) is relatively compact. Equivalently, φ is stable in M. Let ψ(y) A. Then there are a i M, i I and an ultrafilter D on I such that i,d φ(a i, y) = ψ(y). We define the type p D as the following: p D φ(x, b) = r ɛ > 0 {i I : φ(a i, b) r ɛ} D, for every b M. It is easy to verify that p D is a complete type on M. Indeed, let a := i,d a i, since S φ (M) is compact, a is well-defined. Therefore, if p D φ(x, b) = r then φ(a, b) = i,d φ(a i, b) = r because φ(x, b) is continuous for each b. Thus p D = tp(a) is a complete type in S φ (M). Now, by (ii), p D is definable, i.e. there is a sequence φ an (y) in A such that n φ an = ψ. Therefore every point of the closure of A is a it of a sequence in A. The interested reader can compare the proofs of Theorem 462B in [Fre06] and Lemma 2.2 in [Pil96]. 6.3 The Eberlein-Šmulian theorem and NIP/Non-SOP The well known compactness theorem of Eberlein and Šmulian says that weakly compactness and weakly sequentially compactness are equivalent on a Banach space. By the previous observation (6.5), for the Banach space C(X) which X is compact, this theorem 20

21 is equivalent to the following expression: relatively weakly compactness and relatively weakly sequentially compactness are equivalent on C(X). Now, we show the connection between Shelah s theorem (Stable = NIP+Non-SOP) and the Eberlein-Šmulian theorem for the Banach space C(X) which X is compact. Proposition 6.6 (i) (ii). (i) The Eberlein-Šmulian theorem: If X is a compact space and A C(X), then the following statements are equivalent: (a) The weak closure of A is weakly compact in C(X). (b) Each sequence of elements of A has a subsequence that is weakly convergent in C(X). (ii) NIP+Non-SOP = Stable: If T is a first order theory, then the following statements are equivalent: (a ) T is stable. (b ) T has the NIP and the Non-SOP. Proof. First, we note that by the Grothendieck s criterion, (a) (a ). (i)= (ii): Assume that (a ) holds, then T A is weakly compact and hence by (i), it has the NIP and the Non-SOP. Conversely, assume that (b ) holds, i.e. T has the NIP and the Non-SOP. Then by the NIP, every sequence has a pointwise convergence subsequence. By the Non-SOP, its it is sequentially continuous, and hence it would be continuous (cf. 8.2). (ii)= (i): Assume that (a) holds, i.e. A is relatively weakly compact. By (ii), A has the NIP, and hence each sequence of elements of A has a subsequence that is pointwise convergent in R X. Since A has not the SOP, the pointwise it f of a convergence sequence {f n } A is sequentially continuous and hence it would be continuous. Conversely, if (b) holds, then A has the NIP and the Non-SOP. Therefore, by (ii), A is relatively weakly compact. Of course, the Eberlein-Šmulian theorem is proved for arbitrary Banach spaces, but it follows easily from the case C(X) (cf. Theorem 462D in [Fre06]). On the other hand, we note that for a compact space X, countably compactness and sequentially compactness are equivalent in C(X). 6.4 First order logic is angelic Fremlin s notation of an angelic topological space is as follow: A regular Hausdorff space X is angelic if whenever A is a subset of X which is relatively countably compact in X, then (i) A is relatively compact (ii) every point of A is the it of a sequence in A. Roughly an angelic space is one for which the conclusions of the Eberlein-Šmulian theorem hold. By the previous observations it is easy to show that: 21