A model theoretic Baire category theorem for simple theories and its applications

Ziv Shami

March 7, 2011

Abstract

We prove a model theoretic Baire category theorem for τ̃_{low}^{f}-sets in a countable simple theory in which the extension property is firstorder and show some of its applications. We also prove a trichotomy for minimal types in countable nfcp theories: either every type that is internal in a minimal type is essentially-1-based by means of the forking topology or T interprets a strongly-minimal formula or T interprets an infinite definable 1-based group of finite D-rank.

1 Introduction

The goal of this paper is to generalize a result from [S1] and to give some applications. In [S1] The first step for proving supersimplicity of countable unidimensional simple theories eliminating hyperimaginaries is to show the existence of an unbounded type-definable τ^{f}-open set (a set defined in terms of forking by formulas, see definition 2.1) of bounded finite SU_{se}-rank (for definition see section 4). In this paper we develop a general framework for this kind of result. It is the new idea of a model theoretic Baire category theorem, namely, one deals with certain ”uniformly-definable” family of generalized closed sets (in complicated ”logic”), roughly speaking, given a partition of a complicated open set into countably many sets, each of which is the intersection of a ”uniformly definable” family of generalized closed sets, one can find a nice (this is the main advantage) open set that is contained in some generalized closed set in one of these families. In particular, it is not just the usual Baire category theorem for a complicated topological space. The proof is quite similar to the proof in [S1] and has some important consequences, e.g. in a countable wnfcp theory if for every non-algebraic element a (even in some fixed non-empty τ̃_{low}^{f}-set) there is a′ ∈ acl(a)\acl(∅) of finite SU-rank, then there exists a weakly-minimal formula. We also prove a trichotomy for countable nfcp theories as indicated in the abstract.

2 Preliminaries

The forking topology is introduced in [S0] and is a variant of Hrushovski’s and Pillay’s topologies from [H0] and [P] respectively. In this section T is assumed to be simple and we work in C.

Definition 2.1 Let A ⊆ C and let x be a finite tuple of variables.

1) An invariant set U over A is said to be a basic τ^{f}-open set over A if there is ϕ(x, y) ∈ L(A) such that

U = {a|ϕ(a, y) forks over A}.

2) An invariant set U over A is said to be a basic τ_{∞}^{f}-open set over A if U is a type-definable τ^{f}-open set over A.

Note that the family of basic τ^{f}-open sets over A is closed under finite intersections, thus form a basis for a unique topology on S_{x}(A). Likewise, we define the τ_{∞}^{f}-topologies.

Recall, the following definition from [S0] whose roots are in [H0].

Definition 2.2 We say that the τ^{f}-topologies over A are closed under projections (T is PCFT over A) if for every τ^{f}-open set U(x, y) over A the set ∃yU(x, y) is a τ^{f}-open set over A. We say that the τ^{f}-topologies are closed under projections (T is PCFT) if they are over every set A.

Recall that a formula ϕ(x, y) ∈ L is low in x if there exists k < ω such that for every ∅-indiscernible sequence (b_{i}|i < ω), the set {ϕ(x, b_{i})|i < ω} is inconsistent iff every subset of it of size k is inconsistent. T is low if every ϕ(x, y) is low in x.

Remark 2.3 Assume ϕ(x, t) ∈ L is low in t and ψ(y, v) ∈ L is low in v (x ∩ y or t ∩ v may not be ∅). Then θ(xy, tv) ≡ ϕ(x, t) ∨ ψ(y, v) is low in tv.

Proof: Let k_{1} < ω be a witness that ϕ(x, t) is low in t and let k_{2} < ω be a witness that ψ(y, v) is low in v. Let k = k_{1} + k_{2} − 1. By adding dummy variables we may assume x = y and t = v (as tuples of variables). Let (a_{i}| i < ω) be indiscernible such that {ϕ(a_{i}, t) ∨ ψ(a_{i}, t)|i < ω} is inconsistent. Thus, every subset of {ϕ(a_{i}, t)|i < ω} of size k_{1} is inconsistent, and every subset of {ψ(a_{i}, t)|i < ω} of size k_{2} is inconsistent. Thus every subset of size k of {ϕ(a_{i}, t) ∨ ψ(a_{i}, t)|i < ω} is inconsistent.

In [BPV, Proposition 4.5] the authors proved the following equivalence which, for convenience, we will use as a definition (their definition involves extension with respect to pairs of models of T).

Definition 2.4 The extension property is first-order in T iff for every formulas ϕ(x, y), ψ(y, z) ∈ L the relation Q_{ϕ,ψ} defined by:

Q_{ϕ,ψ}(a) iff ϕ(x, b) doesn’t fork over a for every b ⊧ ψ(y, a)

is type-definable (here a can be an infinite tuple from C whose sorts are fixed). We say that T has wnfcp if T is low and the extension property is first-order in T.

Fact 2.5 [S1] Suppose the extension property is first-order in T. Then T is PCFT.

We say that an A-invariant set U has finite SU-rank if SU(a/A) < ω for all a ∈ U, and has bounded finite SU-rank if there exists n < ω such that SU(a/A) ≤ n for all a ∈ U. The existence of a τ^{f}-open set of bounded finite SU-rank implies the existence of a weakly-minimal formula:

Fact 2.6 [S0, Proposition 2.13] Let U be an unbounded τ^{f}-open set over some set A. Assume U has bounded finite SU-rank. Then there exists a set B ⊇ A and θ(x) ∈ L(B) of SU-rank 1 such that θ^{C} ⊆ U ∪ acl(B).

In [S1] the class of τ̃^{f}-sets is introduced, this class is much wider than the class of basic τ^{f}-open sets. Here, we look at the class of τ̃_{low}^{f}-sets, instead of the class of τ̃_{st}^{f}-sets from [S1].

Definition 2.7 A relation V (x, z_{1}, ...z_{l}) is said to be a pre-τ̃^{f}-set relation if there are θ(x̃, x, z_{1}, z_{2}, ..., z_{l}) ∈ L and ϕ_{i}(x̃, y_{i}) ∈ L for 0 ≤ i ≤ l such that for all a, d_{1}, ..., d_{l} ∈ C we have

V (a, d_{1}, ..., d_{l}) iff ∃ã [θ(ã, a, d_{1}, d_{2}, ..., d_{l}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(ã, y_{i}) forks over d_{1}d_{2}...d_{i})]

(for i = 0 the sequence d_{1}d_{2}...d_{i} is interpreted as ∅). If each ϕ_{i}(x̃, y_{i}) is assumed to be low in y_{i} , V (x, z_{1}, ...z_{l}) is said to be a pre-τ̃_{low}^{f}-set relation.

Definition 2.8 1) A τ̃^{f}-set (over ∅) is a set of the form

U = {a| ∃d_{1}, d_{2}, ...d_{l} V (a, d_{1}, ..., d_{l})}

for some pre-τ̃^{f}-set relation V (x, z_{1}, ...z_{l}).

2) A τ̃_{low}^{f}-set (over ∅) is a set of the form

U = {a| ∃d_{1}, d_{2}, ...d_{l} V (a, d_{1}, ..., d_{l})}

for some pre-τ̃_{low}^{f}-set relation V (x, z_{1}, ...z_{l}).

3 The Theorem

In this section T is assumed to be a simple theory and we work in C (so, T not necessarily eliminates imaginaries).

Definition 3.1 Let Θ = {θ_{i}(x_{i}, x)}_{i∈I} be a set of L-formulas such that ∃^{<∞}x_{i}θ_{i}(x_{i}, x) for all i ∈ I. Let s be the sort of x. For A ⊆ C^{s}, let acl_{Θ}(A) = {b| θ_{i}(b, a) for some θ_{i} ∈ Θ and a ∈ A}.

Definition 3.2 An invariant set U(x, y_{1}, ...y_{r}) is said to be a generalized uniform family of τ̃_{low}^{f}-sets if there is a formula ρ(x̃, x, y_{1}, ..., y_{r}, z_{1}, z_{2}, ..., z_{k}) ∈ L and there are formulas ψ_{i}(x̃, v_{i}), μ_{j}(x̃, w_{j}) ∈ L for 0 ≤ i ≤ r and 1 ≤ j ≤ k that are low in v_{i} and low in w_{j} respectively, such that for all a, d_{1}, ...d_{r} we have U(a, d_{1}, ...d_{r}) iff ∃ã∃e_{1}...e_{k}

ρ(ã, a, d_{1}, ...d_{r}, e_{1}, ...e_{k})∧[ ⋀_{i=0}^{r} ψ_{i}(ã, v_{i}) forks over d_{1}...d_{i}]∧[ ⋀_{j=1}^{k} μ_{j}(ã, w_{j}) forks over d_{1}...d_{r}e_{1}...e_{j}].

Definition 3.3 An invariant set F(x, y_{1}, ...y_{r}) is said to be a generalized uniform family of τ̃_{low}^{f}-closed sets if F(x, y_{1}, ...y_{r}) = ⋂_{i} ¬U_{i}(x, y_{1}, ...y_{r}), where each U_{i}(x, y_{1}, ...y_{r}) is a generalized uniform family of τ̃_{low}^{f}-sets.

Fact 3.4 Assume the extension property is first-order in T. Then

1) Let U be an unbounded τ̃^{f}-set over ∅. Then there exists an unbounded τ^{f}-open set U^{∗} over some finite set A^{∗} such that U^{∗} ⊆ U. In fact, if V (x, z_{1}, ..., z_{l}) is a pre-τ̃^{f}-set relation such that U = {a|∃d_{1}...d_{l}V (a, d_{1}, ..., d_{l})}, and (d_{1}^{∗}, ..., d_{m}^{∗}) is any maximal sequence (with respect to extension) such that ∃d_{m+1}...d_{l}V (C, d_{1}^{∗}, ..., d_{m}^{∗}, d_{m+1}, ..., d_{l}) is unbounded, then

U^{∗} = ∃d_{m+1}...d_{l}V (C, d_{1}^{∗}, ..., d_{m}^{∗}, d_{m+1}, ..., d_{l})

is a τ^{f}-open set over d_{1}^{∗}...d_{m}^{∗}.

Theorem 3.5 Let T be a countable simple theory in which the extension property is first-order. Assume:

1) Θ = {θ_{i}(x′_{i}, x)}_{i<ω} is a set of L-formulas such that ∃^{<∞}x′_{i}θ_{i}(x′_{i}, x) for all i < ω.

2) U_{0}(x) is a non-empty τ̃_{low}^{f}-set over ∅.

3) {F_{n}(x_{n})}_{n<ω} is a family of ∅-invariant sets such that F_{n}(C) ∩ acl(∅) = ∅ for all n < ω.

4) For every n < ω and every variables ȳ = y_{1}, ...y_{r}, let F ȳ ; n(x_{n}, ȳ) be a generalized uniform family of τ̃_{low}^{f}-closed sets such that F_{n}(C) ⊆ F ȳ ; n(C, d̄) for all d̄. Now, assume for all a ∈ U_{0} there exists b ∈ acl_{Θ}(a) and n < ω such that b ∈ F_{n}(C). Then there is an unbounded τ_{∞}^{f}-open set U^{∗} over a finite tuple d̄^{∗} and variables ȳ^{∗} of the sort of d̄^{∗}, and n^{∗} < ω such that

U^{∗} ⊆ F_{n^{∗}}^{ȳ^{∗}}(C, d̄^{∗}) ∩ acl_{Θ}(U_{0}).

Proof: First, we may assume Θ is closed downwards (i.e. if θ ∈ Θ and θ′ ⊢ θ then θ′ ∈ Θ). Assume the conclusion of the theorem is false. It will be sufficient to show that for every non-empty τ̃_{low}^{f}-set U ⊆ U_{0}, every θ ∈ Θ, and every n < ω there exists a non-empty τ̃_{low}^{f}-set U^{∗} ⊆ U such that either ¬∃x′θ(x′, a) for all a ∈ U^{∗} or for all a ∈ U^{∗} there exists b ⊧ θ(x′, a) with b ∉ F_{n}(C). Indeed, by iterating this for every pair (θ, n) ∈ Θ × ω we get by compactness a^{∗} such that for all θ ∈ Θ and all n < ω either ¬∃x′θ(x′, a^{∗}) or there exists b_{n,θ} ⊧ θ(x′, a^{∗}) such that b_{n,θ} ∉ F_{n}(C). Since we assume Θ is closed downwards, we get contradiction to the assumption that for all a ∈ U_{0} there exists b ∈ acl_{Θ}(a) and n < ω such that b ∈ F_{n}(C) (note that F_{n}(C) is ∅-invariant). To show this, let U, θ and n < ω be given. Let V (x, z_{1}, ...z_{l}) be a pre-τ̃_{low}^{f}-set relation such that

U = {a| ∃d_{1}, d_{2}, ...d_{l} V (a, d_{1}, ..., d_{l})}.

where V is defined by:

V (a, d_{1}, ..., d_{l}) iff ∃ã [σ(ã, a, d_{1}, d_{2}, ..., d_{l}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(ã, t_{i}) forks over d_{1}d_{2}...d_{i})]

for some σ(x̃, x, z_{1}, z_{2}, ..., z_{l}) ∈ L and ϕ_{i}(x̃, t_{i}) ∈ L which are low in t_{i} for 0 ≤ i ≤ l. Let V_{θ} be defined by: for all b, d_{1}, ..., d_{l} ∈ C,

V_{θ}(b, d_{1}, ..., d_{l}) iff ∃a(θ(b, a) ∧ V (a, d_{1}, ..., d_{l})).

and let

U_{θ} = {b| ∃d_{1}, d_{2}, ...d_{l} V_{θ}(b, d_{1}, ..., d_{l})}.

Since by the assumption F_{n}(C) ∩ acl(∅) = ∅, we may assume U_{θ} ∩ acl(∅) = ∅ and U_{θ} is non-empty. Now, let d̄^{∗} = (d_{1}^{∗}, ..., d_{m}^{∗}) be a maximal sequence, with respect to extension (0 ≤ m ≤ l) such that

Ṽ_{θ}(x′) ≡ ∃d_{m+1}, d_{m+2}, ...d_{l}V_{θ}(x′, d_{1}^{∗}, ...d_{m}^{∗}, d_{m+1}, ...d_{l})

is non-algebraic. We may assume m < l (by choosing V appropriately). By Fact 3.4, Ṽ_{θ}(C) is an unbounded basic τ_{∞}^{f}-open set over d̄^{∗}. Since we assume the conclusion of the theorem is false, Ṽ_{θ}(C) ⊈ F_{n}^{ȳ^{∗}}(C, d̄^{∗}) where ȳ^{∗} = y_{1}^{∗}, ..., y_{m}^{∗} has the same sort as d̄^{∗}. Now, let U_{s,n}(x_{n}, ȳ^{∗}) for s < ω be each a generalized uniform family of τ̃_{low}^{f}-sets such that F_{n}(x_{n}, ȳ^{∗}) = ⋂_{s} ¬U_{s,n}(x_{n}, ȳ^{∗}). Let b^{∗} ∈ Ṽ_{θ}(C)\F_{n}^{ȳ^{∗}}(C, d̄^{∗}). So, there exists s^{∗} < ω such that b^{∗} ∈ U_{s^{∗},n}(C, d̄^{∗}). Let ρ(x̃′, x_{n}, y_{1}^{∗}, ..., y_{m}^{∗}, z′_{1}, z′_{2}, ..., z′_{k}) ∈ L and let ψ_{i}(x̃′, v_{i}), μ_{j}(x̃′, w_{j}) ∈ L for 0 ≤ i ≤ m and 1 ≤ j ≤ k be low in v_{i} and low in w_{j} respectively, such that for all b, d_{1}, ...d_{m} we have U_{s^{∗},n}(b, d_{1}, ...d_{m}) iff ∃b̃∃e_{1}...e_{k}

ρ(b̃, b, d_{1}, ...d_{m}, e_{1}, ...e_{k})∧[ ⋀_{i=0}^{m} ψ_{i}(b̃, v_{i}) forks over d_{1}...d_{i}]∧[ ⋀_{j=1}^{k} μ_{j}(b̃, w_{j}) forks over d_{1}...d_{m}e_{1}...e_{j}].

Now, let d_{m+1}^{∗}, ...d_{l}^{∗} and a^{∗}, ã^{∗} and E^{∗} = (e_{1}^{∗}, ..., e_{k}^{∗}) and b̃^{∗} be such that

θ(b^{∗}, a^{∗}) ∧ σ(ã^{∗}, a^{∗}, d_{1}^{∗}, d_{2}^{∗}, ..., d_{l}^{∗}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(ã^{∗}, y_{i}) forks over d_{1}^{∗}d_{2}^{∗}...d_{i}^{∗}) (∗1)

and

ρ(b̃^{∗}, b^{∗}, d_{1}^{∗}, ..d_{m}^{∗}, e_{1}^{∗}, ..e_{k}^{∗}) (∗2)

and

[ ⋀_{i=0}^{m} ψ_{i}(b̃^{∗}, v_{i}) forks over d_{1}^{∗}...d_{i}^{∗}] ∧ [ ⋀_{j=1}^{k} μ_{j}(b̃^{∗}, w_{j}) forks over d_{1}^{∗}...d_{m}^{∗}e_{1}^{∗}...e_{j}^{∗}] (∗3).

By maximality of d̄^{∗}, we know b^{∗} ∈ acl(d̄^{∗}d_{m+1}^{∗}). Thus, by taking a nonforking extension of tp(b̃^{∗}E^{∗}/acl(d̄^{∗}d_{m+1}^{∗})) over acl(d_{1}^{∗}...d_{l}^{∗}a^{∗}ã^{∗}) we may assume E^{∗} is independent from d_{1}^{∗}...d_{l}^{∗}a^{∗}ã^{∗} over d̄^{∗}d_{m+1}^{∗} and (∗1), (∗2) and (∗3) still hold. We conclude that

⋀_{i=m+1}^{l}(ϕ_{i}(ã^{∗}, t_{i}) forks over d_{1}^{∗}d_{2}^{∗}...d_{i}^{∗}E^{∗}).

Now, we define the τ̃_{low}^{f}-set U^{∗}. First, define a relation V^{∗} by:

V^{∗}(a, d_{1}, ...d_{m}, e_{1}, ...e_{k}, d_{m+1}, ..d_{l}) iff ∃ã, b, b̃(θ^{∗} ∧ V_{0}^{∗} ∧ V_{1}^{∗} ∧ V_{2}^{∗}),

where θ^{∗} is defined by: θ^{∗}(ã, b, b̃, a, d_{1}, ..d_{m}, e_{1}, ...e_{k}, d_{m+1}, ..d_{l}) iff

θ(b, a) ∧ σ(ã, a, d_{1}, d_{2}, ..., d_{l}) ∧ ρ(b̃, b, d_{1}, ...d_{m}, e_{1}, ..., e_{k}),

V_{0}^{∗} is defined by: V_{0}^{∗}(ã, b̃, d_{1}, ...d_{m}) iff

⋀_{i=0}^{m}(ϕ_{i}(ã, t_{i}) ∨ ψ_{i}(b̃, v_{i}) forks over d_{1}d_{2}...d_{i}),

V_{1}^{∗} is defined by V_{1}(b̃, d_{1}, ..d_{m}, e_{1}, ...e_{k}) iff

⋀_{j=1}^{k}(μ_{j}(b̃, w_{j}) forks over d_{1}...d_{m}e_{1}...e_{j}), and

V_{2}^{∗} is defined by V_{2}(ã, d_{1}, ...d_{m}, e_{1}, ...e_{k}, d_{m+1}, ..d_{l}) iff

⋀_{i=m+1}^{l}(ϕ_{i}(ã, t_{i}) forks over d_{1}d_{2}...d_{i}e_{1}...e_{k}).

Note that V^{∗} is a pre-τ̃_{low}^{f}-set. Let

U^{∗} = {a|∃d_{1}, ..d_{m}, e_{1}, ...e_{k}, d_{m+1}, ..d_{l} V^{∗}(a, d_{1}, ..d_{m}, e_{1}, ...e_{k}, d_{m+1}, ...d_{l})}.

By the definition of U^{∗}, U^{∗} ⊆ U. U^{∗} is a τ̃_{low}^{f}-set using Remark 2.3. By the construction, U^{∗} ≠ ∅. Now, let a ∈ U^{∗}. By the definition of U^{∗}, there are b̃, b, d_{1}, ...d_{m}, e_{1}, ...e_{k} such that θ(b, a), ρ(b̃, b, d_{1}, ...d_{m}, e_{1}, ..., e_{k}),

⋀_{i=0}^{m}(ψ_{i}(b̃, v_{i}) forks over d_{1}, ...d_{i}) and ⋀_{j=1}^{k}(μ_{j}(b̃, w_{j}) forks over d_{1}, ...d_{m}e_{1}...e_{j}).

Thus U_{s^{∗},n}(b, d_{1}...d_{m}) and therefore ¬F_{n}^{ȳ^{∗}}(b, d_{1}...d_{m}). Hence b ∉ F_{n} as required.

4 Applications

In this section we show some applications of Theorem 3.5. In fact, we will show several instances of this theorem that apparently new even for stable theories. In this section T is assumed to be a simple theory and we work in C.

We start by pointing out that theorem 3.5 generalizes [S1, Theorem 9.4] that is one of the essential steps towards the proof of supersimplicity of countable simple unidimensional theories with elimination of hyperimaginaries. First recall the following definitions from [S1].

Definition 4.1 For a ∈ C, A ⊆ B ⊆ C, \begin{aligned}a & ⌣̸ |s & B \\ & A \end{aligned} if for some stable

ϕ(x, y) ∈ L, there is b over B and a′ ⊧ ϕ(x, b) for some a′ ∈ dcl(Aa) such that ϕ(x, b) forks over A.

Definition 4.2 The SU_{se}-rank of tp(a/A) is defined by induction on α: if

α = β + 1, SU_{se}(a/A) ≥ α if there exist B_{1} ⊇ B_{0} ⊇ A such that \begin{aligned}a & ⌣̸ |s & B_{1} \\ & B_{0} \end{aligned}

and SU_{se}(a/B_{1}) ≥ β. For limit α, SU_{se}(a/A) ≥ α if SU_{se}(a/A) ≥ β for all β < α.

Remark 4.3 First, recall that in a simple theory in which Lstp = stp over sets ⌣ |s is symmetric [Lemma 6.7, S1]. Thus for any finite tuples of sorts s_{0} and s_{1} and n < ω the set F_{n}^{s_{0},s_{1}} defined by

F_{n}^{s_{0},s_{1}} = {(a, A) ∈ C^{s_{0}} × C^{s_{1}}| SU_{se}(a/A) < n}.

is a generalized uniform family of τ̃_{low}^{f}-closed sets.

For an A-invariant set V, let acl_{1}(V) = {a′| a′ ∈ acl(a) for some a ∈ V }. The following corollary generalizes [S1, Theorem 9.4].

Corollary 4.4 Let T be a countable simple theory in which the extension property is first-order and assume Lstp = stp over sets. Let U_{0} be a nonempty τ̃_{low}^{f}-set. Assume for every a ∈ U_{0} there exists a′ ∈ acl(a)\acl(∅) such that SU_{se}(a′) < ω. Then there exists an unbounded τ_{∞}^{f}-open set U ⊆ acl_{1}(U_{0}) over a finite set such that U has bounded finite SU_{se}-rank.

Proof: Let x be the variable of U_{0}, so U_{0} = U_{0}(x). Let

Θ = {θ(x′, x)| ∃^{<∞}x′θ(x′, x), x′ any variable}.

Let S be the set of sorts. Let I : ω → S × ω be a bijection, I_{1}, I_{2} the projections of I to the first and second coordinate respectively. Now, for each n < ω let F_{n} = {a ∈ C^{I_{1}(n)}\acl(∅) |SU_{se}(a) < I_{2}(n)}. Now, for every finite tuple of variables Y and n < ω let s(Y) be the finite sequence of sorts of Y and let

F_{n}^{Y} = {(a, A) ∈ C^{I_{1}(n)} × C^{s(Y)}| SU_{se}(a/A) < I_{2}(n)}.

Now, by the definition of the SU_{se}-rank, F_{n}(C) ⊆ F_{n}^{Y} (C, A) for every n < ω and every Y, A. By Remark 4.3, F_{n}^{Y} is a generalized uniform family of τ̃_{low}^{f}- closed sets for all Y, n. By our assumptions, we see that the assumptions of Theorem 3.5 hold for U_{0}(x), Θ ,{F_{n}}_{n} and {F_{n}^{Y} }_{Y,n} and thus by its corollary we are done.

Corollary 4.5 Let T be a countable theory with wnfcp. Let U_{0} be an unbounded τ̃^{f}-set over ∅ of finite SU-rank. Then there exists a finite set A and a SU-rank 1 formula θ ∈ L(A) such that θ^{C} ⊆ U_{0} ∪ acl(A).

Proof: First, by modifying U_{0}, we may assume U_{0} ∩ acl(∅) = ∅. Let Θ = {x′ = x}, U_{0}(x) = U_{0}. Let s(x) be the sort of x. Now, for each n < ω let

F_{n} = {a ∈ C^{s(x)}\acl(∅) |SU(a) < n}.

For every finite tuple of variables Y and n < ω let s(Y) be the finite sequence of sorts of Y and let

F_{n}^{Y} = {(a, A) ∈ C^{s(x)} × C^{s(Y)}| SU(a/A) < n}.

By symmetry of forking and the assumption that T is low, each F_{n}^{Y} is a generalized uniform family of τ̃_{low}^{f}-closed sets. Clearly, F_{n}(C) ⊆ F_{n}^{Y} (C, A) for every n < ω and every Y, A. By our assumption, the assumptions of Theorem 3.5 are satisfied for U_{0}, Θ, {F_{n}}_{n} and {F_{n}^{Y} }_{Y,n} and thus by its corollary there exists an unbounded τ_{∞}^{f}-open set U^{∗} ⊆ U_{0} over a finite set A_{0} and U^{∗} has bounded finite SU-rank. By Fact 2.6, there exists a finite set A ⊇ A_{0} and there exists a SU-rank 1 formula θ ∈ L(A) such that θ^{C} ⊆ U^{∗} ∪ acl(A).

Corollary 4.6 Let T be a countable theory with wnfcp. Let U_{0} be a nonempty τ̃^{f}-set over ∅. Assume for every a ∈ U_{0} there exists a′ ∈ acl(a)\acl(∅) such that SU(a′) < ω. Then there exists a finite set A and a SU-rank 1 formula θ ∈ L(A) such that θ^{C} ⊆ acl_{1}(U_{0}) ∪ acl(A).

Proof: Just like the proof of Corollary 4.5.

5 Dichotomies for countable theories with the wnfcp

In this section we show the dichotomy [S1, Theorem 5.5] implies a strong dichotomy between essential 1-basedness and supersimplicity in the case T is a countable wnfcp theory that eliminates hyperimaginaries. Before we state the above dichotomy for the special case of the τ^{f}-topologies (simplified version), let us recall the basic definitions. In this section T is assumed to be simple.

Definition 5.1 A type p ∈ S(A) is said to be essentially 1-based by means of the τ^{f}-topologies if for every finite tuple c̄ from p and for every typedefinable τ^{f}-open set U over Ac̄, the set {a ∈ U| Cb(a/Ac̄) ∉ bdd(aA)} is nowhere dense in the Stone-topology of U.

Fact 5.2 Let T be a countable simple theory with PCFT that eliminates hyperimaginaries. Let p_{0} be a partial type over ∅ of SU-rank 1. Then, either there exists an unbounded finite-SU-rank τ^{f}-open set over some countable set, or every type p ∈ S(A), with A countable, that is internal in p_{0} is essentially 1-based by means of the τ^{f}-topologies.

Let V, W be invariant sets (over some small set). We say that V almost contain W if for some A over which W is invariant V ⊇ W \acl(A). We say that V is generated over W if there exists a small set B such that V ⊆ dcl(W ∪ B).

Theorem 5.3 Let T be a countable theory with wnfcp that eliminates hyperimaginaries. Let p be a partial type over ∅ of SU-rank 1 . Then, either

1) every type q ∈ S(A), with A countable, that is internal in p is essentially 1-based by means of the τ^{f}-topologies, or

2) there exists n < ω and g(x, y) ∈ L such that g(x, a) is a definable function for all a, and there exists a small sequence (a_{i}|i < α) such that ⋃_{i<α} g(p^{n}, a_{i}) almost contains a weakly-minimal definable set. In particular, there exists a weakly-minimal definable set that is generated over p(C).

Proof: Assume 1) is false. By the proof of Fact 5.2, there exists an unbounded basic τ^{f}-open set U over some countable set A such that tp(a/A) is almost p-internal for every a ∈ U.

Subclaim 5.4 There exists an unbounded basic τ^{f}-open set U^{∗} ⊆ U over A that is generated over p(C).

Proof: By [WB] or [S2, Corollary 4.9], for every a ∈ U\acl(A) there exists a′ ∈ dcl(aA)\acl(A) such that tp(a′/A) has fundamental system of solutions over p(C), (i.e. tp(a′/A) is generated over p(C) by a set of realizations p.) In particular, there exists a (finite) set A′ of realizations of tp(a′/A) that is independent from A′ and tuple c̄ of realizations of p such that a′ ∈ dcl(A′c̄). For every A-definable functions f, g let

F_{f,g} = {a ∈ U|f(a) = g(b̄, c̄) ∉ acl(A) for some b̄, c̄ with f(a) ⌣ | b̄ , where c̄is a tuple of realizations of p, and b̄ is a tuple of realizations of tp(f(a)/A)}.

Note that each F_{f,g} is τ^{f}-closed over A. Thus, by Baire category theorem for the τ^{f}-topology and Fact 2.5 we conclude that there exists an unbounded τ^{f}-open set U^{∗} over A and A-definable function g^{∗} such that for every a ∈ U^{∗} there exists a tuple b̄ of realizations of tp(a) that is independent from a such that a = g^{∗}(b̄, c̄) for tuple c̄ of realizations of p. Now, the subclaim follows directly from fact [S2, Theorem 3.7]:

Fact 5.5 Let p ∈ S(∅) and let R be ∅-invariant. Suppose the internality of p in R is witnessed by a generic parameter whose type q is R-internal. Then p is generated over R by a set of realizations of q.

Moreover, we may assume A = acl(A), so it is not hard to see that there exists a definable family of functions g^{∗∗}(x, y) ∈ L (constructed from g^{∗}) such that for every a ∈ U^{∗} there exists a′ such that if n = l(x), then g^{∗∗}(p^{n}, a′) contains tp(a)^{C} (see the method in [S3, Theorem 5.6]). Now, as U^{∗} has bounded finite SU-rank (the bound is determined by g^{∗}), by Fact 2.6, there exists a SU-rank 1 formula θ(x, b) such that θ(C, b) ⊆ U^{∗} ∪ acl(Ab). Thus 2) follows.

5.1 A trichotomy for countable theories with the nfcp

Work in C = C^{eq}.

Theorem 5.6 Let T be a countable theory with nfcp. Let p ∈ S(∅) be minimal. Then, either

1) every type q ∈ S(A), with A countable, that is internal in p is essentially 1-based by means of the τ^{f}-topologies, or

2) there exists a strongly minimal definable set that is p-internal, or

3) there is an infinite definable 1-based group of finite D-rank that is pinternal.

Proof: Assume 1) is false. By Theorem 5.3, there exists a weakly-minimal formula θ(x, b) that is p-internal. First, assume θ(C, b) ⊆ acl(p(C) ∪ b). By Baire category theorem, there exists a b-definable θ^{∗}(x, b) ⊢ θ(x, b) and bdefinable functions f, g and n < ω such that g[p^{n}(C)] ⊇ f[θ^{∗}] ≡ f[θ^{∗}(C, b)] and f[θ^{∗}] is non-algebraic. Since p is minimal, f[θ^{∗}] must have Morley rank and thus contains a strongly minimal formula. Thus, we may assume θ(C, b) ⊈ acl(p^{C} ∪ b). Let a ∈ θ(C, b)\acl(p^{C} ∪ b). Let q = tp(a/acl(b)) and let Γ = Aut(q^{C}/p^{C} ∪ acl(b)). Then there is an infinite type-definable group G over acl(b) that is isomorphic to Γ. Now, G can be realized as acl(b)-definable projections of certain type-definable set over acl(b) of finite tuples from q. As q ⊢ θ(x, b), and G is intersection of definable groups over acl(b), we conclude that there an infinite acl(b)-definable group G_{0} that is p-internal and has finite D-rank. By Buechler’s dichotomy, every minimal type r ⊢ G_{0} is either 1-based or of Morley rank 1. Thus if 2) fails, then any such r is 1-based. As G_{0} has finite SU-rank, every non-algebraic type of G_{0} is non-orthogonal to a minimal type in G_{0} and therefore analyzable in it. By [W], G_{0} is 1-based.

References

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[S1] Z.Shami, Countable hypersimple unidimensional theories, to appear.

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