Type spaces of metric structures and topometric spaces Ita - - PDF document

Type spaces of metric structures and topometric spaces

Ita¨ ı Ben-Yaacov September 2006

1

1 Continuous logic

Origins Many classes of (complete) metric structures arising in analysis are “tame” (e.g., admit well-behaved notions of independence) although not elementary in the classical sense. Continuous logic [BU, BBHU] (B., Berenstein, Henson & Usvyatsov) is an attempt to apply model-theoretic tools to such classes. It was preceded by:

• Henson’s logic for Banach structures (positive bounded formulae, approximate sat-

isfaction).

• Positive logic and cats (compact abstract theories).
• Chang and Keisler’s continuous logic (1966 – too general, and not adequate for

metric structures).

Lukasiewicz’s many-valued logic (similar, although probably devised for other pur- poses).

• . . . ?

1.1 Basic definitions

Intellectual game: replace {T, F} with [0, 1]

• The basic idea is: “replace the space of truth values {T, F} with [0, 1], and see

what happens”. . .

• Things turn out more elegant if we agree that 0 is “True”.
• Greater truth value is falser.

Ingredient I: non-logical symbols

• A signature L consists of function and predicate symbols, as usual.
• n-ary function symbols: interpreted as functions M n → M.
• n-ary predicate symbols: interpreted as functions M n → [0, 1].
• Syntactically: L-terms and atomic L-formulae are as in classical logic.

2

Ingredient II: Connectives

• Any continuous function [0, 1]n → [0, 1] should be admitted as an n-ary connective.
• Problem: uncountable syntax. But a dense subset of C([0, 1]n, [0, 1]) (in uniform

convergence) is good enough.

• The following connectives generate a (countable) dense family of connectives (lattice

Stone-Weierstrass): ¬x = 1 − x;

1 2x = x/2;

x − . y = max{x − y, 0}.

• “ϕ −

. ψ” replaces “ψ → ϕ”. In particular: {ψ, ϕ − . ψ} ϕ (Modus Ponens: if ψ = 0 and ϕ − . ψ = 0 then ϕ = 0). Ingredient III: Quantifiers

• If R ⊆ M n+1 is a predicate on M,
• ∀xR(x,¯

b) M is the falsest among {R(a,¯ b): a ∈ M}.

• By analogy, if R: M n+1 → [0, 1] is a continuous predicate:
• ∀x R(x,¯

b) M = sup

a∈M

R(a,¯ b). We will just use “supx ϕ” instead of “∀xϕ”.

• Similarly, “∃xϕ” becomes “infx ϕ”.
• Prenex normal form exists since the connectives ¬, 1

2, −

. are monotone in each ar- gument: ϕ − . infxψ ≡ supx(ϕ − . ψ), &c. . . Ingredient IV: Equality...? In classical logic the symbol = always satisfies: x = x (x = y) → (x = z) → (y = z) (ER) Replacing “x = y” with “d(x, y)” and “ϕ → ψ” with “ψ − . ϕ”: d(x, x) d(y, z) − . d(x, z) − . d(x, y) 3

i.e., d is a pseudo-metric: d(x, x) = 0 d(y, z) ≤ d(x, z) + d(x, y) (PM) Similarly, = is a congruence relation: (x = y) → P(x, ¯ z) → P(y, ¯ z) (CR) Translates to: P(y, ¯ z) − . P(x, ¯ z) − . d(x, y) I.e., P is 1-Lipschitz: P(y, ¯ z) − . P(x, ¯ z) ≤ d(x, y) (1L) Conclusion: all predicate (and function) symbols must be 1-Lipschitz in d. Structures

• Definition. A set M, equipped with a pseudo-metric dM and 1-Lipschitz interpretations

f M, P M of symbols f, P ∈ L is an L-pre-structure. It is an L-structure if dM is a complete metric.

• Once =M is a congruence relation, classical logic cannot tell whether it is true

equality or not.

• Similarly, once all symbols are 1-Lipschitz, continuous logic cannot tell whether:

– dM is a true metric or a mere pseudo-metric. – A Cauchy sequence has a limit or not.

• A pre-structure M is logically indistinguishable from its completion

M/∼d. (a ∼d b ⇐ ⇒ d(a, b) = 0) Example: probability algebras

• Let (Ω, B, µ) be a probability space.
• Let B0 ≤ B be the null-measure ideal, and ¯

B = B/B0. Then ¯ B is a Boolean algebra and µ induces ¯ µ: ¯ B → [0, 1]. The pair ( ¯ B, ¯ µ) is a probability algebra.

• It admits a complete metric: d(a, b) = ¯

µ(a△b). ¯ µ and the Boolean operations are 1-Lipschitz.

• ( ¯

B, ∧, ∨, ·c, ¯ µ) is a continuous structure. 4

Semantics

• If M is a structure, ϕ(x<n) a formula and ¯

a ∈ M n, then ϕM(¯ a) ∈ [0, 1] is defined by induction on ϕ.

• Elementary equivalence: If M, N are two structures then M ≡ N if ϕM = ϕN ∈

[0, 1] for every sentence ϕ (i.e.: formula without free variables).

• Elementary inclusion: M N if M ⊆ N and ϕM(¯

a) = ϕN(¯ a) for every formula ϕ and ¯ a ∈ M. This implies M ≡ N.

• The elementary chain theorem holds. Attention: we may need to replace the union
• f a countable increasing chain with its completion.

Bottom line

• By replacing {T, F} with [0, 1] we obtained a logic for (bounded) complete metric

1-Lipschitz structures.

• It is fairly easy to replace “1-Lipschitz” with “uniformly continuous”.
• One can also overcome “bounded”, but it’s trickier.
• Since all structures are complete metric structures we do not measure their cardi-

nality, but their density character: (M, d) = min{|A|: A ⊆ M is dense}.

1.2 Theories

Theories

• A theory T is a set of sentences (closed formulae).

M T ⇐ ⇒ ϕM = 0 for all ϕ ∈ T.

• We sometimes write T as a set of statements “ϕ = 0”. For r ∈ [0, 1] we may also

consider ϕ ≤ r as a statement (same as ϕ − . r = 0). Theorem (Compactness). A theory is satisfiable if and only if it is (approximately) finitely satisfiable. Proved using ultraproducts or Henkin’s method. 5

Examples of continuous elementary classes

• Hilbert spaces (infinite dimensional).
• Probability algebras (atomless).
• Lp Banach lattices (atomless).
• Fields with a non-trivial valuation in (R, +) (algebraically closed, in characteristic

(p, q)).

• &c. . .

All these examples are complete and admit QE. Universal theories

• A theory consisting solely of “
• sup¯

x ϕ(¯

x)

• = 0”, where ϕ is quantifier-free, is called
• universal. Universal theories are those stable under sub-models.
• For a formula ϕ, ψ: ∀¯

x

• ϕ = 0
• is shorthand for
• sup¯

xϕ

• = 0.
• More generally, for formulae ϕ, ψ: ∀¯

x

• ϕ = ψ
• is shorthand for
• sup¯

x|ϕ − ψ|

• = 0.
• For terms σ, τ: ∀¯

x(σ = τ) is shorthand for

• sup¯

xd(σ, τ)

• = 0.

The (universal) theory of probability algebras The class of probability algebras is axiomatised by: universal equational theory of Boolean algebras ∀xy d(x, y) = µ(x △ y) ∀xy µ(x) + µ(y) = µ(x ∧ y) + µ(x ∨ y) µ(1) = 1 The model completion is the ∀∃-theory of atomless probability algebras: supxinfy|µ(x ∧ y) − µ(x)/2| = 0. 6

Types

• Definition. Let M be a structure, ¯

a ∈ M n. Then: tpM(¯ a) = {“ϕ(¯ x) = r”: ϕ(¯ x) ∈ L, r = ϕ(¯ a)M}. Sn(T) is the space of n-types in models of T. If p ∈ Sn(T): “ϕ(¯ x) = r” ∈ p ⇐ ⇒ ϕp = r.

• The logic topology on Sn(T) is minimal such that p → ϕp is continuous for all ϕ.
• This is the analogue of the Stone topology in classical logic; it is compact and

Hausdorff (not totally disconnected).

• Types over parameters and Sn(A) are defined similarly. Saturated and homogeneous

models exist. Omitting types Theorem (Omitting types). Assume T is countable and X ⊆ S1(T) is meagre (i.e., contained in a countable union of closed nowhere-dense sets). Then T has a model M such that a dense subset of M omits each type in X. (Similarly with Xn ⊆ Sn(T) meagre for each n.) Proof ([Ben05]). The classical Baire category argument works. What about omitting types in M, and not only in a dense subset? Later. . . Definable predicates

• We identify a formula ϕ(x<n) with the function ϕ: Sn(T) → [0, 1] it induces. By

Stone-Weierstrass these functions are dense in C(Sn(T), [0, 1]).

• An arbitrary continuous function ψ: Sn(T) → [0, 1] is called a definable predicate.

It is a uniform limit of formulae: ψ = limn→∞ ϕn. Its interpretation: ψM(¯ a) = lim

n ϕM n (¯

a). Since each ϕM

n is uniformly continuous, so is ψM.

• Same applies with parameters. Note that a definable predicate may depends on

countably many parameters. 7

Imaginaries and algebraic closure

• In continuous logic imaginary elements are introduced as canonical parameters of

formulae and predicates with parameters. Imaginary sorts are also metric: d(cp(ψ), cp(χ)) = sup

¯ x |ψ(¯

x) − χ(¯ x)|.

• An element a is algebraic over A if the set of its conjugates over A is compact

(replaces “finite”).

• acleq(A) is the set of all imaginaries algebraic over A.

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2 Topometric spaces

2.1 Motivating example

Motivating example: Sn(T) The topological structure of Sn(T) is insufficient. We will also need to consider the distance between types: d(p, q) = inf{d(a, b): a, b ∈ M T & M p(a) ∪ q(b)}. (In case T is incomplete and p, q belong to different completions: d(p, q) = inf ∅ = ∞.) The infimum is always attained as minimum. Indeed, apply compactness to the partial type: p(x) ∪ q(y) ∪ {d(x, y) ≤ d(p, q) + 2−n : n < ω}. Some properties of (Sn(T), d)

• If f : Sn(T) → [0, 1] is topologically continuous (f is a definable predicate) then it

is metrically uniformly continuous.

• Implies: The metric refines the topology.
• If F ⊆ Sn(T) is closed, then so is the set:

¯ B(F, r) = {p ∈ Sn(T): d(p, F) ≤ r}.

• Implies: (Sn(T), d) is complete.
• And: If F ⊆ Sn(T) is closed and p ∈ Sn(T), then there is q ∈ F such that

d(p, q) = d(p, F). All these properties are consequences of compactness + “metric Hausdorff” property:

• Lemma. The distance function d: Sn(T)2 → [0, ∞] is lower semi-continuous. That is

to say that {(p, q): d(p, q) ≤ r} is closed for all r.

• Proof. The projection S2n(T) → Sn(T) × Sn(T) is closed, and [d(¯

x, ¯ y) ≤ r] ⊆ S2n(T) is closed, whereby so is its image {(p, q): d(p, q) ≤ r} ⊆ Sn(T)2. 9

2.2 Topometric spaces

Topometric spaces

• Definition. A compact topometric space is a compact topological space X = (X, TX)

equipped with a metric dX such that dX : X2 → [0, ∞] is lower semi-continuous. It follows that X is Hausdorff.

• Convention. Metric space vocabulary (e.g., “uniformly continuous”) refers to the metric

part; other vocabulary (e.g., “continuous”) refers to the topological part. All properties of (Sn(T), d) hold in general Proof of “continuous = ⇒ uniformly continuous”. Let f : X → [0, 1] be continuous, ε > 0. Let U = {(x, y) ∈ X2 : |f(x) − f(y)| < ε}. Then U is open and contains

• δ>0{(x, y): d(x, y) ≤ δ}. By compactness U ⊇ {(x, y): d(x, y) ≤ δ} for some δ > 0.

Proof of “F closed = ⇒ ¯ B(F, r) closed”. As F is closed so are (F ×X)∩{(x, y): d(x, y) ≤ r} and its projection Fr = {x: (∃y ∈ F)(d(x, y) ≤ r)}. Thus ¯ B(F, r) =

r′>r Fr′ is

closed. Examples

• Example. The motivating example (Sn(T), d) is a compact topometric space.
• Example. Every (compact) topological space is automatically a topometric space via the

discrete metric. We call such a space a maximal topometric space.

• Example. Every (compact) metric space is automatically a topometric space via the

metric topology. We call such a space a minimal topometric space. A non-compact example

• Example. Let M be a complete metric structure. Let:

G = Aut(M) TG = point-wise convergence topology dG = uniform convergence metric. Then (G, TG, dG) is a non-compact topometric group: The metric is complete, lower semi- continuous and refines the topology; All group operations are continuous and uniformly continuous. If M is separable, the topology is Polish, i.e., separable and completely metrisable (but this separable and complete metric is not dG which is almost never separable.) 10

Morphisms

• Definition. Let f : X → Y be a map between compact topometric spaces.
• It is a morphism if it is continuous and uniformly continuous.
• It is an embedding if it is an isomorphism with a subspace.
• It is a quotient map if in addition it is a topological quotient map, and in addition:

(∀ε)(∃δ)dY (f(x), y) < δ = ⇒ dX(x, f −1(y)) ≤ ε. Note that an injective epimorphism is an isomorphism.

2.3 Cantor-Bendixson analysis

Cantor-Bendixson analysis

• Cantor-Bendixson analysis of type spaces is a useful tool of classical model theory.
• E.g.: local stability; ω-stability; small theories. . .
• The classical Cantor-Bendixson derivative is defined as X′ = X {isolated points}.

Unfortunately we do not expect to find (topologically) isolated points in the type space of a continuous theory (or more generally, in “non-trivial” topometric spaces). Metric Cantor-Bendixson derivatives

• Alternative understanding of the Cantor-Bendixson derivative: take out open sets

which are “small”.

• How do we define “small”? Metrically, relative to some ε > 0:

X′

d,ε

= X {open sets of diameter ≤ ε}

• r also:

X′

b,ε

= X {“ε-bounded” open sets} X′

f,ε

= X {“ε-finite” open sets} . . . 11

Derivative sequences and ranks We define X(α)

∗,ε as usual: X(α+1) ∗,ε

= (X(α)

∗,ε )′ ∗,ε, . . . This stabilises: X(∞) ∗,ε = X(|X|+) ∗,ε

. Definition.

• For x ∈ X, A ⊆ X:

CB∗,ε(A) = sup{α: A ∩ X(α)

∗,ε = ∅}

CB∗,ε(x) = CB∗,ε({x})

• X is (∗, ε)-CB-analysable if CB∗,ε(X) < ∞ (⇐

⇒ X(∞)

(∗,ε) = ∅).

• X is ∗-CB-analysable if (∗, ε)-CB-analysable for all ε > 0.

Note that ε > ε′ > 0 = ⇒ CB∗,ε(A) ≤ CB∗,ε′(A).

• Proposition. All notions of ∗-CB-analysability are the same, ∗ ∈ {d, f, b}. Moreover,

the ranks are essentially the same: CBd,2ε(A) ≤ CB∗,ε(A) ≤ CBd,ε(A). If M is a metric space and X a topological space: M = min{|M0|: M0 ⊆ M dense} wt(X) = min{|B|: B is a base for the topology}

• Proposition. If X is CB-analysable then X ≤ wt(X).
• Lemma. A subspace or quotient space of a CB-analysable space is CB-analysable.
• Theorem. (Assuming χdef(X, d) = ℵ0.) A compact topometric X is CB-analysable if

and only if all quotients of X with countable base are metrically separable. Definition complexity of a closed set

• Definition. Let X be compact topometric space.
• Let F ⊆ X be closed. Its definition complexity χdef(F) is the minimal κ such that

there is B ⊆ C(X, [0, 1]), |B| ≤ κ which defines F: F =

f∈B f −1({0}).

• The definition complexity of the metric is defined as:

χdef(X, d) = sup{χdef( ¯ B(F, r)): r ∈ R+, χdef(F) = ℵ0}.

• Fact. For every theory T and set of parameters A: χdef(Sn(A), d) = ℵ0. This is true not
• nly for the standard metric, but for any “reasonable” metric on Sn(A).

12

Measures on CB-analysable spaces Recall that a finite measure on a compact space X is regular if for every measurable S: µ(S) = sup{µ(K): K ⊆ S compact}. Regular Borel measures on X are in bijection with positive integration functionals on C(X, R) (or C(X, [0, 1])). Theorem (Motivated by Pillay’s question.). Let X be a compact, CB-analysable topomet- ric space, µ a regular Borel probability measure on X (Borel – according to the topology.) Then for every ε > 0, 1 − ε of the mass of µ is supported by a metrically compact set. (Obvious in case X = ℵ0.)

• Proof. Fix r > 0; let Uα = X X(α)

d,r .

• Assume µ(U1) > ε. Then there is F ⊆ U1 compact such that µ(F) > µ(U1) − ε,

and F is r-finite.

• By regularity: Uδ = supα<δ µ(Uα) (δ limit).

Thus {α: µ(Uα+1 Uα) > 0} = {αn : n < ω}.

• For each n there is Fn ⊆ Uαn+1 Uαn r-finite, µ(Fn) > µ(Uαn+1 Uαn) − 2−n−1ε.
• For some m: µ(

n<m Fn) > µ(X) − ε, and K = n<m Fn is r-finite.

∴ (∀k)(∃Kk 2−k-finite)(µ(Kk) > µ(X)−2−k−1ε). K = Kk is metrically compact, and µ(K) > µ(X) − ε. Questions

• 1. If X is a compact CB-analysable (classical) topological space then it is totally

disconnected. What is the analogue for topometric spaces? 13

3 Topometric type spaces

3.1 Local stability

Application: local stability [BU] (B. & Usvyatsov)

• Definition. Let ϕ(¯

x, ¯ y) be a formula, ¯ a a tuple, M a model. Then: tpϕ(¯ a/M) = {“ϕ(¯ x,¯ b)” = r: ¯ b ∈ M, r = ϕ(¯ a,¯ b)M}. The space of all ϕ-types over a model M is Sϕ(M). It is equipped with the logic topology, and the metric: d(p, q) = sup{|ϕ(¯ x,¯ b)p − ϕ(¯ x,¯ b)q|: ¯ b ∈ M}.

• Fact. Sϕ(M) is a compact topometric space.
• Proof. It is compact as a quotient of Sn(M).

Then for each ¯ b ∈ M the function ϕ(¯ x,¯ b): Sϕ(M) → [0, 1] is continuous, so the relation |ϕ(¯ x,¯ b)p−ϕ(¯ x,¯ b)q| ≤ r is closed.

• Definition. A formula ϕ(¯

x, ¯ y) is stable if there are no r < s and ¯ ai,¯ bi : i < ω such that: i < j = ⇒ ϕ(¯ ai,¯ bj) ≤ r < s ≤ ϕ(¯ aj,¯ bi). (This definition motivated the introduction of continuous logic.)

• Proposition. If ϕ is stable then every p ∈ Sϕ(M) is definable. That is to say that the

function ¯ b → ϕ(¯ x,¯ b)p is a definable predicate with parameters in M, denoted dpϕ(¯ y). The canonical parameter of dpϕ(¯ y) is an imaginary in M eq.

• Theorem. The following are equivalent:
• 1. ϕ is stable.
• 2. Every ϕ-type is definable.
• 3. M ≤ ℵ0 =

⇒ Sϕ(M) ≤ wt(Sϕ(M)) ≤ ℵ0 (we may assume T countable).

• 4. Sϕ(M) is CB-analysable for all M.
• 5. For all λ: M ≤ λ =

⇒ Sϕ(M) ≤ wt(Sϕ(M)) ≤ λ. If ϕ is stable then: CB-analysability = ⇒ existence of types of maximal rank = ⇒ existence of non-forking extensions. One problem: we have a whole family of ranks CBd,ε! Let p ∈ Sϕ(A), and let P ⊆ Sϕ( ¯ M) be the set of extensions of p to ϕ-types over ¯

• M. What do we mean by “maximal

rank in P”? 14

• Let P0 = P, and let Pn+1 ⊆ Pn be the set of types of maximal CBd,2−n-rank in Pn.
• Each Pn+1 is closed, non-empty, and covered by finitely many balls of radius 2−n.
• Pω = Pn = ∅ is metrically compact, and invariant over A.
• If q ∈ Pω then cp(dqϕ) ∈ acleq(A): q does not fork over A.
• . . . a ϕ-type over acleq(A) has a unique non-forking extension to ¯

M, so Pω = p↾

¯ M

(the set of non-forking extensions). ∴ q ∈ p↾

¯ M iff q has maximal CBd,ε rank in P for all ε.

Local stability = ⇒ global stability Definition.

• We say that T is λ-stable if A ≤ λ =

⇒ Sn(A) ≤ λ.

• It is stable if it is λ-stable for some λ.
• It is superstable if it is λ-stable for all λ big enough.
• Proposition. The following are equivalent:
• T is stable.
• Every ϕ is stable.
• If M ≤ 2|T| then | Sn(M)| ≤ 2|T|.

Assume T stable, A ⊆ M. By “coding” finitely many formulae in one, and compact- ness: every global type p ∈ Sn(acleq(A)) admits a unique acleq(A)-definable extension q = p↾M ∈ Sn(M) (i.e.: q↾ϕ is acleq(A)-definable for all ϕ). Say that ¯ a | ⌣A B if tp(¯ a/acleq(AB))↾

¯ M is acleq(A)-definable.

• Theorem. If T is stable then |

⌣ satisfies invariance, symmetry, transitivity, extension, finite character, local character, and stationarity of types over acleq(A). Conversely, if |′ ⌣ is any notion of independence satisfying the above then T is stable and |′ ⌣ = | ⌣. 15

Examples H = (model completion of the theory of) Hilbert Spaces; PA = probability algebras; LpL = Lp Banach lattices; V FR = fields with valuation in R.

• H, PA, LpL, V FR are stable
• H, PA, LpL are ℵ0-stable (=

⇒ superstable).

• H, PA, LpL are ℵ0-categorical.
• H is uncountably categorical.

Local stability = ⇒ Keisler measures (Pillay, B. – ?) Keisler measure µ over M = regular Borel probability measure µ on Sn(M) = positive integration functional Iµ : {ϕ(x<n) ∈ L(M)} → [0, 1]: Iµ(¬ϕ) = 1 − Iµ(ϕ) Iµ(1

2ϕ)

= 1

2Iµ(ϕ)

Iµ(ϕ) = Iµ(ϕ − . ψ) + Iµ(ϕ ∧ ψ).

• µ given by {µϕ = image of µ on Sϕ(M): ϕ(x<n, . . .) ∈ L}.
• If ϕ is stable: Sϕ(M) is CB-analysable =

⇒ ∀ε µϕ is concentrated up to ε on a metrically compact set.

• Since each ϕ-type is definable, so is µϕ (i.e.: ¯

b → Iµ(ϕ(¯ x,¯ b)) is definable). ∴ If T is stable then µ is definable (or “continuous”). ∴ Fubini: I¯

x(J¯ y(ϕ(¯

x, ¯ y))) = J¯

y(I¯ x(ϕ(¯

x, ¯ y))).

3.2 d-isolated types and ω-stability

d-isolated points

• Definition. A point x ∈ X is d-isolated if for all r > 0: x ∈ B(x, r)◦ (i.e., the metric

and the topology coincide near x). It is weakly d-isolated if we only have B(x, r)◦ = ∅ for all r > 0. The set Xdi ⊆ X of all d-isolated points is Gδ: Xdi =

• n
• x∈Xdi

B(x, 2−n)◦. (It is not usually open, so X Xdi is not closed and would not have served as a Cantor- Bendixson derivative.)

• Fact. A metric limit of weakly d-isolated points is weakly d-isolated.

16

Omitting and realising types in models A type p ∈ (Sn(T), d) is d-isolated ⇐ ⇒ weakly d-isolated. (Henson calls such types principal.) Proposition (Henson). A d-isolated type p is realised in every model of T. If T is countable, then the converse is also true. Proof. = ⇒ As B(p, 2−n)◦ = ∅ for all n, it must be realised in M, say by an. We can furthermore arrange that d(an, an+1) < 2−n−1. Then an → a p. ⇐ = If B(p, r)◦ = ∅ for some r > 0, we can omit it in a dense subset of M. Then M

• mits p.

Existence of d-isolated types

• Proposition. Assume X is CB-analysable. Then the set of d-isolated points is dense.
• Corollary. Assume that T is small, i.e., Sn(T) = ℵ0 for all n. Then T has a d-atomic

(and therefore prime) model. Proof.

• Let An ⊆ Sn(T) be the set of d-isolated types = set of weakly d-isolated

types.

• Sn(T) separable =

⇒ CB-analysable = ⇒ An is topologically dense. An is Gδ = ⇒ Bn = Sn(T) An is meagre.

• We can find M T such that a dense subset Qn ⊆ M n omits Bn for all n.
• Since An is closed in d: all tuples in M realise d-isolated types.

Ryll-Nardzewski Theorem

• Definition. A theory T is λ-categorical if for all M, N T:

M = N = λ = ⇒ M ≃ N. Theorem (Ryll-Nardzewski, Henson). For a complete countable theory T, TFAE:

• T is ℵ0-categorical (unique separable model).
• Every n-type over ∅ is d-isolated for all n.
• The metric and topology coincide on each (Sn(T), d).
• Every automorphism-invariant uniformly continuous predicate on ¯

M is definable. 17

Towers of d-isolated types

• Theorem. Let π: X → Y be a surjective morphism. Let x ∈ X, y = π(x) ∈ Y , and let

Z = π−1(y) be the fibre above Y .

• 1. If π is a quotient map:

y d-isolated in Y and x d-isolated in Z = ⇒ x d-isolated in X.

• 2. If π is open, then the converse holds:

x d-isolated in X = ⇒ y d-isolated in Y and x d-isolated in Z. Proof of 1 (“going up”) Let ∆: (0, ∞) → (0, ∞) be such that dX(x, π−1(y)) < ∆(ε) = ⇒ dY (π(x), y) ≤ ε dY (π(x), y) < ∆(ε) = ⇒ dX(x, π−1(y)) ≤ ε.

• Let r > 0 be given. Then x ∈ U := intZ(B(x, r/2)).
• Find V ⊆op X, U = Z ∩ V .
• Find V ′ ⊆op X, x ∈ V ′ ⊆ ¯

V ′ ⊆ V .

• Find 0 < s ≤ r/2 such that ¯

B(¯ V ′, s) ⊆ V (compactness).

• Now: y ∈ W := intY (B(y, ∆(s))).
• Then:

x ∈ V ′ ∩ π−1(W) ⊆ B(x, r) Application: towers of types in (S(T), d) Fact.

• The natural map π: Sn+m(T) → Sn(T) is a quotient map.
• If p = tp(¯

a) ∈ Sn(T) then θ: Sm(¯ a) → π−1(p) is an open surjective morphism.

• Corollary. Assume tp(¯

a) and tp(¯ b/¯ a) are both d-isolated. Then tp(¯ a,¯ b) is d-isolated.

• Proof. Let p = tp(¯

a), tp(¯ b/¯ a), π and θ as above. Then θ(q) = tp(¯ a,¯ b) is d-isolated in π−1(p), and p is d-isolated, whereby θ(q) is d-isolated. 18

Application: Morley’s theorem [Ben05] Assume T is countable, (i) λ-categorical but (ii) not λ′-categorical λ, λ′ > ℵ0. Then:

• From (i) T is ℵ0-stable =

⇒ Sn(M) CB-analysable for all M = ⇒ T λ-stable for all λ = ⇒ has saturated models at all λ > ℵ0.

• d-isolated types over any set are dense, so prime (constructed) models exist over

every set.

• From (ii) there are A countable, p ∈ Sn(A) and an A-indiscernible sequence (ai : i <

. . .) such that the prime model over A ∪ {ai : i < λ} omits p, so is non-saturated, contradicting λ-categoricity. (Morley ranks defined in [Ben05] coincide with CBb,ε in Sn( ¯ M).) 19

3.3 Perturbations

Perturbations of metric structures

• Changing a classical structure (without changing the underlying set) consists of

changing some predicate from “True” to “False” (or vice versa). = ⇒ Changing by “little” = not changing at all.

• A continuous predicate/function can be changed by a little, e.g., by no more than

ε > 0. = ⇒ Notions of small perturbations of metric structures. Motivating examples

• Consider π: Sn+m(T) → Sn(T), p = tp(¯

a) ∈ Sn(T). Topologically: Sm(¯ a) ≃ π−1(p).

• For q0 = tp(¯

b/¯ a), q1 = tp(¯ c/¯ a) we have two natural distances: Sm(¯ a) : d1(q0, q1) = min{d(¯ b, f(¯ c)): f ∈ Aut( ¯ M/¯ a)} Sn+m(T) : d2(q0, q1) = min{d(¯ b¯ a, f(¯ c¯ a)): f ∈ Aut( ¯ M)}

• d1 ≥ d2, i.e., d2 is coarser: d1 only allows to move the realisation; d2 also allows to

move the parameters.

• d2 allows to perturb the tuple ¯

a in ( ¯ M, ¯ a).

• Banach Mazur distance = perturbations of the norm.
• Perturbations of σ in (M, σ) T + automorphism allow taking into account the

metric on Aut(M). Formal definition [Ben]

• Definition. A perturbation system p for T consists of a family of [0, ∞]-valued metrics

dp,n on Sn(T), such that: TM Each dp,n is lower semi-continuous (i.e., (Sn(T), dp) is a topometric space). INV dp,n is invariant under permutations of n. EXT Let π: Sn+1(T) → Sn(T), p ∈ Sn(T), q ∈ Sn+1(T): dp,n(p, π(q)) = dp,n+1(π−1(p), q) 20

UC If b = c: dp,2(tp(aa), tp(bc)) = ∞.

• Definition. A bijection f : M → N is a p(r)-perturbation if for all ¯

a ∈ M: dp(tp(f(¯ a)), tp(¯ a)) ≤ r. The set of all p(r)-perturbations is denoted: Pertp(r)(M, N). By UC a perturbation is always uniformly continuous.

• Fact. Let ¯

a ∈ M, ¯ b ∈ N. TFAE:

• dp(tp(¯

a), tp(¯ b)) ≤ r.

• ∃
• M ′ M, N ′ N, θ ∈ Pertp(r)(M ′, N ′)
• (θ(¯

a) = ¯ b). This allows us to specify a perturbation system p by specifying Pertp(r)(M, N) for all M, N, r (they must satisfy some conditions. . . ) Examples Example (Trivial perturbation system: id). did(p, q) =

• p = q

∞ p = q ; Pertid(r)(M, N) = Iso(M, N). Example (Banach Mazur distance). T = Banach spaces (with no additional structure). θ ∈ PertBM(r)(E, F) if θ: E → F is a bijection and: ∀v ∈ E ve−r ≤ θ(u) ≤ ver. Example (Perturbation of a new symbol). T = an L-theory, L′ = L ∪ {P(¯ x)}, p a perturbation system for T. pP = p + perturbation of P: θ ∈ PertpP (r)((M, P), (N, P)) ⇐ ⇒      θ ∈ Pertp(r)(M, N), and for all ¯ b ∈ M : |P M(¯ b) − P N(θ(¯ b))| ≤ r Same can be done with a finite tuple ¯ P of new symbols. A function symbol f(¯ x) can be replaces with Gf(¯ x, y) = d(f(¯ x), y). Perturbations of the automorphism in (M, σ). Perturbations of parameters of types.

• When doing “model theory up to p-perturbations” we mustn’t forget the standard

metric d on types. We merge d and dp by allowing to perturb the structure and move the realisations. 21

• We define ˜

dp(p, q) as the minimum r for which there are M p(¯ e), N q( ¯ f) and θ ∈ Pertpr(M, N) such that: (∀a ∈ M)

• |dM(a, ei) − dN(a, fi)| ≤ r
• . This is the

natural common coarsening of d and dp on Sn(T).

• Note that if M p(¯

e), N q( ¯ f) then: ˜ dp,n(p, q) = dp¯

c,0(ThL(¯

c)(M, ¯

e), ThL(¯

c)(N, ¯

f))

• Finally, for p, q ∈ Sn(¯

a): p¯

a allows to move the parameters ¯

a, and ˜ dp¯

a(p, q) is the

minimal distance we need to “travel”, moving parameters and realisations (and perturbing the underlying structure), to get from p to q. Perturbed Ryll-Nardzewski Definition.

• Two structures are p-isomorphic, M ≃p N, if for all ε > 0 there exists

an ε-perturbation θ ∈ Pertp(ε)(M, N).

• T is p-λ-categorical if M, N T and M = N = λ imply M ≃p N.
• Theorem. Let T be a countable complete theory. TFAE:
• T is p-ℵ0-categorical.
• For all finite ¯

a and n < ω, every point in the topometric space (Sn(¯ a), ˜ dp¯

a) is weakly

˜ dp¯

a-isolated.

• For all finite ¯

a, every point in the topometric space (S1(¯ a), ˜ dp¯

a) is weakly ˜

dp¯

a-

isolated. Corollary (Sufficient condition, no parameters). Assume T is countable, complete, and for every n < ω each point in (Sn(T), ˜ dp) is ˜ dp-isolated. Then T is p-ℵ0-categorical. Corollary (Transfer to a reduct). Assume T is countable, complete. Let L′ = L ∪ { ¯ P}, T ′ an L′-completion of T, p′ = p ¯

• P. If for every n < ω each point in (Sn(T ′), ˜

dp′) is ˜ dp′-isolated then T is p-ℵ0-categorical. An anomaly with p-categoricity In the previous corollary, “T ′ p-ℵ0-categorical” would not suffice. Example.

• T0 = Th(Lp(R), ∧, ∨, . . .) theory of Lp Banach lattices [BBH].
• Let a = χ[0,1], b = χ[1,2]; T = T0(a), p = idT; T ′ = T(b) = T0(a, b), p′ = pb (So p′

fixes a and perturbs b). 22

• T is not ℵ0-categorical. Two models:

(Lp[0, 1], χ[0,1]) ≃ (Lp[0, 2], χ[0,1]),

• But T ′ is p′-ℵ0-categorical:

(Lp[0, 2], χ[0,1], χ[0,2]) ≃p′ (Lp[0, 3], χ[0,1], χ[0,2]), Perturbations of types with parameters Say a perturbation system p for T is given, and we wish to work over A ⊆ ¯

• M. How

do we extend dp to Sn(A)? We need a perturbation system for T(A) = ThL(A)( ¯ M, A).

• In case A = ¯

a = a<n is finite, we have already seen one option: use p¯

a.

• In case A is infinite, this doesn’t make sense, or else boils down to p/A which fixes
• A. For M, N T(A) (so A ⊆ M, N):

θ ∈ Pert(p/A)(r)(M, N) ⇐ ⇒ θ ∈ Pertp(r)(M, N)&θ↾A = idA . T is p-λ-stable if M ≤ λ implies (Sn(M), ˜ dp/M) ≤ λ. Expansion by a generic automorphism Theorem (Chatzidakis, Pillay [CP98]). Let T be stable, Tσ = T ∪ {“σ is an automorphism”}. Assuming Tσ has a model-companion TA:

• 1. TA is simple.
• 2. If T is superstable, then TA is supersimple.

The first part generalises to continuous logic. What about the second part? Consider T = PA theory of atomless probability algebras. Then:

• PA is superstable (in fact ℵ0-stable).
• (Berenstein, Henson [BH]) PAA exists and is stable.
• (B.) PAA is not superstable, and therefore not supersimple.

Let p = idT. pσ allows to perturb the automorphism, while fixing the underlying model of T. Proposition (B., Berenstein). PAA is pσ-superstable, and in fact pσ-ℵ0-stable.

• Question. Let T be any superstable theory such that TA exists. Is TA is pσ-supersimple?

23

References

[BBH] Ita¨ ı Ben-Yaacov, Alexander Berenstein, and C. Ward Henson, Model-theoretic independence in the Banach lattices Lp(µ), submitted. [BBHU] Ita¨ ı Ben-Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvy- atsov, Model theory for metric structures, Expanded lecture notes for a workshop given in March/April 2005, Isaac Newton Institute, University of Cambridge. [Ben] Ita¨ ı Ben-Yaacov, On perturbations of continuous structures, submitted. [Ben05] Itay Ben-Yaacov, Uncountable dense categoricity in cats, Journal of Symbolic Logic 70 (2005), no. 3, 829–860. [BH] Alexander Berenstein and C. Ward Henson, Model theory of probability spaces with an automorphism, submitted. [BU] Ita¨ ı Ben-Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, submitted. [CP98] Zo¨ e Chatzidakis and Anand Pillay, Generic structures and simple theories, An- nals of Pure and Applied Logic 95 (1998), 71–92. 24