Some model theory of Polish structures

Krzysztof Krupiński

Abstract

We define a topological notion of independence in Polish structures and we prove that it has some nice properties. Using this notion, we prove counterparts of some basic results from geometric stability theory in the context of small Polish structures. Examples of small Polish structures are also given.

0 Introduction

We propose a new, model theoretic, approach to study classical descriptive set theoretic objects, like Polish G-spaces or Borel G-spaces. More generally, we are going to study Polish structures which are defined as follows.

Definition 0.1 A Polish structure is a pair (X, G) where G is a Polish group acting faithfully on a set X so that the stabilizers of all singletons are closed subgroups of G. We say that (X, G) is small if for every n ∈ ω, there are only countably many orbits on X^{n} under the action of G.

Notice that the assumption that G acts faithfully on X is purely cosmetic as we can always divide G by the maximal subgroup acting trivially on X.

Profinite structures (X and G are profinite and the action is continuous) introduced by Newelski in [13, 14] and then considered also by Wagner [17] and by myself [4, 5, 6, 8], and, more generally, compact structures [7] (X is a compact metric space, G is a compact group and the action is continuous) are particular cases of Polish structures. More generally, Polish G-spaces (X is Polish and the action is continuous) and Borel G-spaces (X is a standard Borel space and the action is Borel-measurable) are also examples of Polish structures.

In Section 1 we introduce the notion of nm-independence and we prove that it has some nice properties, similarly as forking independence in stable theories. We show that nmindependence generalizes the notion of m-independence introduced by Newelski for profinite structures. However, the proof that nm-independence is transitive is rather complicated and it uses some descriptive set theory, whereas transitivity of m-independence follows immediately from the definition. Similarly as in the case of profinite structures, in order to get the existence of nm-independent extensions we need to assume smallness. In fact, we could just assume the existence of nm-independent extensions (as in [7]) but we prefer to assume smallness since it is more natural and easier to check in concrete examples.

02000 Mathematics Subject Classification: 03C45, 03E15, 54H11, 54F15

0Key words and phrases: Polish group, independence relation

In Section 2 we consider counterparts of some basic notions from geometric stability theory and we prove fundamental results about them.

In Section 3 we give examples of small Polish structures. We end up with a list of open questions.

The general goal of this paper, similarly as in [7] where I was investigating compact structures, is to make an attempt to apply stability theory ideas to various classical mathematical objects. A similar motivation appears in [13, 14] where Newelski was considering small profinite structures. However, Polish structures (particularly Polish G-spaces) seem to be more interesting than profinite structures from the point of view of descriptive set theory or topology. Moreover, in my opinion Polish structures yield a more adequate generalization of profinite structures than compact structures. This is because each small compact structure is profinite. So instead of smallness we should assume here the existence of m-independent extensions; but I do not know any interesting examples of such compact structures which are not profinite. In contrast, we have several natural examples of small Polish structures which are not profinite, and it seems to me that it should be not very difficult to find another ones.

Having all the notions and basic results established in this paper, the natural next step would be to try and prove the counterparts of some deep results from stability theory, e.g. a group configuration theorem. Such results were proved by Newelski for small profinite structures [14]. In [7] I noticed that most of them can be generalized to the case of compact [profinite] structures satisfying the existence of m-independent extensions. In small Polish structures the situation is more complicated and it is even not clear how to formulate the appropriate conjectures.

There are also certain open questions about the existence of small profinite structures with some model theoretic properties (e.g. M-gap conjecture [13, 14, 17]). I think it would be interesting to find counterexamples for them in the wider context of small Polish structures.

The notions introduced in this paper (e.g. nm-independence, N M-rank) may also turn out to be new tools to deal with purely descriptive set theoretic or topological problems.

I am very grateful to Sławomir Solecki for interesting discussions and suggestions, in particular, for pointing me the pseudo-arc as an example of small Polish structure.

1 Independence relation

In this section we define a notion of independence, which we call nm-independence (read non-meager independence), and we study its properties. We also prove that it coincides with Newelski’s m-independence in compact structures.

If (X, G) is a Polish structure and A ⊆ X, then by G_{A} we denote the pointwise stabilizer of A. For a ∈ X^{n} we define o(a/A) = {f(a) : f ∈ G_{A}} (the orbit of a over A).

Let us recall the definition of m-independence.

Definition 1.1 Let (X, G) be a compact structure, a be a finite tuple and A, B finite subsets of X. We say that a is m-independent from B over A (written a⌣ |^{m}_{A}B) if o(a/AB) is open in o(a/A). We say that a is m-dependent on B over A (written a⌣̸ |^{m}_{A}B) if o(a/AB) is nowhere dense in o(a/A).

We cannot use the above definition for a Polish structure (X, G) as there is no topology on X. Even if we assumed that (X, G) is a Polish G-space, orbits could be weird, e.g. meager in their relative topologies, and then m-independence would not have nice properties in this context.

The idea to omit the above obstacle is to define a relation of independence in terms of the Polish group G.

Definition 1.2 Let (X, G) be a Polish structure, a be a finite tuple and A, B finite subsets of X. Let π_{A} : G_{A} → o(a/A) be defined by π_{A}(g) = ga. We say that a is nm-independent from B over A (written a⌣ |^{nm}_{A}B) if π_{A}^{−1}[o(a/AB)] is non-meager in π_{A}^{−1}[o(a/A)]. Otherwise, we say that a is nm-dependent on B over A (written a⌣̸ |^{nm}_{A}B).

One can also define o-independence just replacing the word ’non-meager’ by ’open’ in the above definition. Some of the results will work for both notions of independence. However, the proof of the existence of nm-independent extensions in small Polish structures does not work for o-independence. In Section 3 we will show that the pseudo-arc is an example of a small Polish structure without the existence of o-independent extensions. That is why nm-independence is a more appropriate notion of independence.

Notice that nm and o in Definition 1.2 come from topological properties ’non-meager’ and ’open’, whereas m in Definition 1.1 comes from the word ’multiplicity’.

Notation If T is a topological space and U, V ⊆ T, then U ⊆_{nm} V means that U is a non-meager subset of V and U ⊆_{o} V means that U is an open subset of V. When we write ∗ = nm, it means that ∗ stands for non-meager; similarly, ∗ = o means that ∗ stands for open.

Proposition 1.3 Let (X, G) be a Polish structure, a be a finite tuple and A, B finite subsets of X. Assume ∗ = nm or ∗ = o. Then TFAE:

(1) a⌣ |^{∗}_{B}A,

(2) G_{AB}G_{Aa} ⊆_{∗} G_{A},

(3) G_{AB}/G_{Aa} ⊆_{∗} G_{A}/G_{Aa}.

Proof. (1) ↔ (2). An easy computation shows that π_{A}^{−1}[o(a/AC)] = G_{AC}G_{Aa} for any C ⊆ X. Hence π_{A}^{−1}[o(a/AB)] = G_{AB}G_{Aa} and π_{A}^{−1}[o(a/A)] = G_{A}. Now the desired equivalence follows directly from the definition of ∗-independence.

(2) → (3). Let π : G_{A} → G_{A}/G_{Aa} be the quotient map. Then π is continuous and open. If ∗ = o, the implication follows from the fact that π is open. So consider the case

∗ = nm. Suppose G_{AB}/G_{Aa} is a meager subset of G_{A}/G_{Aa}, i.e. G_{AB}/G_{Aa} is covered by a countable union ⋃_{i∈ω} D_{i} of closed and nowhere dense subsets of G_{A}/G_{Aa}. Then G_{AB}G_{Aa} = π^{−1}[G_{AB}/G_{Aa}] ⊆ ⋃_{i∈ω} π^{−1}[D_{i}]. Since π is continuous and open, we get that π^{−1}[D_{i}], i ∈ ω, are closed and nowhere dense. So G_{AB}G_{Aa} is a meager subset of G_{A}.

(3) → (2). Since π is continuous and G_{AB} is Polish, we see that G_{AB}/G_{Aa} is analytic so it has the Baire property [3, Theorem 21.6], i.e. G_{AB}/G_{Aa} = D△U where D is meager and U is open in G_{A}/G_{Aa}. Assume now that G_{AB}/G_{Aa} ⊆_{nm} G_{A}/G_{Aa}. Then U is nonempty. We have G_{AB}G_{Aa} = π^{−1}[D]△π^{−1}[U]. Since π is continuous and open, we get that π^{−1}[D] is meager and π^{−1}[U] ≠ ∅ is open in G_{A}. Hence G_{AB}G_{Aa} ⊆_{nm} G_{A}.

■

Definition 1.4 Let (X, G) be a Polish structure and A be a finite subset of X. We define the algebraic closure of A (written Acl(A)) as the set of all elements of X with countable orbits over A. The strong algebraic closure of A (denoted by acl(A)) is the set of all elements of X with finite orbits over A. If A is infinite, we define Acl(A) = ⋃{Acl(A_{0}) : A_{0} ⊆ A is finite} and acl(A) = ⋃{acl(A_{0}) : A_{0} ⊆ A is finite}.

In Polish structures Acl plays a similar role as acl in compact [profinite] structures.

Theorem 1.5 In any Polish structure (X, G) ∗-independence, where ∗ = nm or ∗ = o, has the following properties.

(1) (Symmetry) For every finite a, b, C ⊆ X we have that a⌣ |^{∗}_{C}b iff b⌣ |^{∗}_{C}a.

(2) (Transitivity) For every finite A ⊆ B ⊆ C ⊆ X and a ⊆ X we have that a⌣ |^{∗}_{B}C ∧

a⌣ |^{∗}_{A}B iff a⌣ |^{∗}_{A}C.

(3) For every finite A ⊆ X we have that a ∈ Acl(A) iff for all finite B ⊆ X, a⌣ |^{∗}_{A}B .

Proof. Only (2) requires a serious argument, but let us first check (1) and (3) for completeness.

(1) a⌣ |^{∗}_{A}b ⟺ G_{Ab}G_{Aa} ⊆_{∗} G_{A} ⟺ G_{Aa}G_{Ab} ⊆_{∗} G_{A} ⟺ b⌣ |^{∗}_{A}a.

(3) (⇒). a ∈ Acl(A) means that o(a/A) is countable. Hence [G_{A} : G_{Aa}] ≤ ω. So G_{Aa} is non-meager in G_{A} by the Baire category theorem. Since G_{Aa} is a closed subgroup of G_{A},

we get G_{Aa} ⊆_{o} G_{A} ⟹ G_{AB}G_{Aa} ⊆_{o} G_{A} ⟹ a⌣ |^{o}_{A}B ⟹ a⌣ |^{nm}_{A}B for any finite B ⊆ X.

(⇐). Suppose that a⌣ |^{∗}_{A}B for every finite B ⊆ X. Then a⌣ |^{∗}_{A}Aa. Hence G_{Aa} ⊆_{∗} G_{A} ⟹ G_{Aa} ⊆_{o} G_{A} ⟹ [G_{A} : G_{Aa}] ≤ ω ⟹ a ∈ Acl(A).

In order to prove transitivity we need first to prove some purely descriptive set theoretic lemmas, which seem to be interesting on their own rights.

Lemma 1.6 Suppose that H_{1} and H_{2} are closed subgroups of a Polish group H. Let H_{3} = H_{1} ∩ H_{2}. Then H_{1}H_{2} is a Borel subset of H. Moreover, for every A_{1} = A_{1}H_{3} a Borel subset of H_{1} and A_{2} = H_{3}A_{2} a Borel subset of H_{2} we have that A_{1}A_{2} is a Borel subset of H_{1}H_{2}.

Proof. Define f : H_{1} × H_{3} × H_{2} → H by f(h_{1}, h_{3}, h_{2}) = h_{1}h_{3}h_{2}. By [3, Theorem 12.17] we can choose a Borel set S_{1} meeting every coset from H_{1}/H_{3} in exactly one point and a Borel set S_{2} meeting every coset from H_{3}\H_{2} in exactly one point.

Let f_{0} be the function f restricted to S_{1}×H_{3}×S_{2}. Notice that f_{0} is an injection. Indeed, if s_{1}, s′_{1} ∈ S_{1}, s_{2}, s′_{2} ∈ S_{2}, h, h′ ∈ H_{3} and s_{1}hs_{2} = s′_{1}h′s′_{2}, then s_{1}^{′−1}s_{1} = h′s′_{2}s_{2}^{−1}h^{−1} ∈ H_{1} ∩ H_{2} = H_{3}. Hence s_{1} = s′_{1}. Similarly s_{2} = s′_{2}. So h = h′.

Since S_{1} × H_{3} × S_{2} is a Borel subset of the Polish space H_{1} × H_{3} × H_{2}, we get that f_{0}[S_{1} × H_{3} × S_{2}] = S_{1}H_{3}S_{2} = H_{1}H_{2} is a Borel subset of H.

To prove the second part of the lemma, first notice that A′_{1} := A_{1} ∩ S_{1} and A′_{2} := A_{2} ∩ S_{2} are Borel subsets of H_{1} and H_{2}, respectively. Hence f_{0}[A′_{1} × H_{3} × A′_{2}] is a Borel subset of H. On the other hand, f_{0}[A′_{1} × H_{3} × A′_{2}] = A′_{1}H_{3}A′_{2} = A_{1}H_{3}A_{2} = A_{1}A_{2}. Hence A_{1}A_{2} is a Borel subset of H, which implies that it is a Borel subset of H_{1}H_{2}.

■

Lemma 1.7 Suppose that H_{1} and H_{2} are closed subgroups of a Polish group H such that H_{1}H_{2} is non-meager in its relative topology. Let H_{3} = H_{1} ∩ H_{2}. Then for every A_{1} = A_{1}H_{3} ⊆_{o} H_{1} and H_{2} = H_{3}A_{2} ⊆_{o} H_{2} we have A_{1}A_{2} ⊆_{o} H_{1}H_{2}.

Proof. Let us define a function ψ : H_{1}H_{2} → H_{1}/H_{3} × H_{3}\H_{2} by

ψ(h_{1}h_{2}) = (h_{1}H_{3}, H_{3}h_{2})

where h_{1} ∈ H_{1} and h_{2} ∈ H_{2}.

We check that ψ is well-defined. Suppose h_{1}h_{2} = h′_{1}h′_{2} where h_{1}, h′_{1} ∈ H_{1} and h_{2}, h′_{2} ∈ H_{2}. Then h_{1}^{′−1}h_{1} = h′_{2}h_{2}^{−1} ∈ H_{1} ∩ H_{2} = H_{3}. Hence h_{1}H_{3} = h′_{1}H_{3} and H_{3}h′_{2} = H_{3}h_{2}.

To finish the proof it is enough to show that ψ is continuous. The following Claim follows from Lemma 1.6.

Claim 1 ψ is Baire measurable.

By Claim 1 and [3, Theorem 8.38] there is a set C ⊆ H_{1}H_{2} which is a countable intersection of dense open subsets of H_{1}H_{2} such that ψ ↾ C is continuous.

Claim 2 ψ is continuous.

Proof of Claim 2. Suppose for a contradiction that there are h_{1}^{i} ∈ H_{1} and h_{2}^{i} ∈ H_{2}, i ∈ ω, such that

1. h_{1}^{i}h_{2}^{i} ⟶ h_{1}h_{2},

2. (h_{1}^{i}H_{3}, H_{3}h_{2}^{i}) −̸→ (h_{1}H_{3}, H_{3}h_{2}).

Notice that (1) and (2) hold for hh_{1}^{i}, h_{2}^{i}g, hh_{1}, h_{2}g instead of h_{1}^{i}, h_{2}^{i}, h_{1}, h_{2} for arbitrary h ∈ H_{1} and g ∈ H_{2}. Hence to get a contradiction it is enough to find h ∈ H_{1} and g ∈ H_{2} such that hh_{1}h_{2}g ∈ C and hh_{1}^{i}h_{2}^{i}g ∈ C for all i ∈ ω. So we will be done if we prove the following:

Subclaim (∀h_{1}h_{2} ∈ H_{1}H_{2})(∀^{∗}(h, g) ∈ H_{1} × H_{2})(hh_{1}h_{2}g ∈ C).

Proof of Subclaim. Since C^{∗} := {h_{1}h_{2} ∈ H_{1}H_{2} : (∀^{∗}(h, g) ∈ H_{1} × H_{2})(hh_{1}h_{2}g ∈ C)} is invariant under multiplication by H_{1} on the left and by H_{2} on the right, it is enough to show that

(∗) (∃h_{1}h_{2} ∈ H_{1}H_{2})(∀^{∗}(h, g) ∈ H_{1} × H_{2})(hh_{1}h_{2}g ∈ C).

Since C is comeager in H_{1}H_{2}, we have

(∀(h, g) ∈ H_{1} × H_{2})(∀^{∗}h_{1}h_{2} ∈ H_{1}H_{2})(hh_{1}h_{2}g ∈ C).

So by the Kuratowski-Ulam theorem [3, Theorem 8.41] we get

(∀^{∗}h_{1}h_{2} ∈ H_{1}H_{2})(∀^{∗}(h, g) ∈ H_{1} × H_{2})(hh_{1}h_{2}g ∈ C).

Since H_{1}H_{2} is non-meager in its relative topology, we get (∗) and we are done. ■

Lemma 1.8 Suppose that H_{1} and H_{2} are closed subgroups of a Polish group H such that H_{1}H_{2} is non-meager in its relative topology. Let H_{3} = H_{1} ∩ H_{2}. Then for every analytic set A_{1} = A_{1}H_{3} ⊆_{nm} H_{1} and A_{2} = H_{3}A_{2} ⊆_{nm} H_{2} we have A_{1}A_{2} ⊆_{nm} H_{1}H_{2}.

Proof. In the same way as in the proof of Proposition 1.3 we get that A_{1}/H_{3} ⊆_{nm} H_{1}/H_{3} and H_{3}\A_{2} ⊆_{nm} H_{3}\H_{2}. Moreover, since A_{1} and A_{2} are analytic, A_{1}/H_{3} and H_{3}\A_{2} have the Baire property [3, Proposition 14.4, Theorem 21.6]. So A_{1}/H_{3} = D_{1}△U_{1} and H_{3}\A_{2} = D_{2}△U_{2} where D_{1}, D_{2} are meager and U_{1}, U_{2} are nonempty and open in H_{1}/H_{3} and H_{3}\H_{2}, respectively. So there are h_{i}, g_{i}, i ∈ ω, such that R_{1} := ⋃_{i∈ω} h_{i}A_{1}/H_{3} and R_{2} := ⋃_{i∈ω} H_{3}\A_{2}g_{i} are comeager in H_{1}/H_{3} and H_{3}\H_{2}, respectively. So there are U_{i}, V_{i}, i ∈ ω, open and dense in H_{1}/H_{3} and H_{3}\H_{2}, respectively, such that ⋂_{i∈ω} U_{i} ⊆ R_{1} and ⋂_{i∈ω} V_{i} ⊆ R_{2}.

Let π_{1} : H_{1} → H_{1}/H_{3} and π_{2} : H_{2} → H_{3}\H_{2} be the quotient maps. Let U′_{i} = π_{1}^{−1}[U_{i}] and V′_{i} = π_{2}^{−1}[V_{i}]. Then

(∗) U′_{i} is open and dense in H_{1} and V′_{i} is open and dense in H_{2} for all i ∈ ω,

(∗∗) U′_{i} = U′_{i}H_{3} and V′_{i} = H_{3}V′_{i} for all i ∈ ω,

(∗ ∗ ∗) ⋂_{i∈ω} U′_{i} ⊆ π_{1}^{−1}[R_{1}] = ⋃_{i∈ω} h_{i}A_{1} and ⋂_{i∈ω} V′_{i} ⊆ π_{2}^{−1}[R_{2}] = ⋃_{i∈ω} A_{2}g_{i}.

Claim (⋂_{i∈ω} U′_{i})(⋂_{i∈ω} V′_{i}) = ⋂_{i∈ω} U′_{i}V′_{i}.

Proof of Claim. The inclusion ⊆ is obvious. So let us prove ⊇. Consider any a ∈ ⋂_{i∈ω} U′_{i}V′_{i}. Then for every i ∈ ω we can choose u_{i} ∈ U′_{i} and v_{i} ∈ V′_{i} so that a = u_{i}v_{i}. For every i, j ∈ ω we have u_{i}v_{i} = u_{j}v_{j} ⟹ u_{j}^{−1}u_{i} = v_{j}v_{i}^{−1} ∈ H_{1} ∩ H_{2} = H_{3}. So u_{i} ∈ u_{j}H_{3} and v_{i} ∈ H_{3}v_{j}. Hence by (∗∗) we get u_{0} ∈ ⋂_{i∈ω} U′_{i} and v_{0} ∈ ⋂_{i∈ω} V′_{i}. So a = u_{0}v_{0} ∈ (⋂_{i∈ω} U′_{i})(⋂_{i∈ω} V′_{i}).

□

By (∗ ∗ ∗) and Claim we get

⋃_{i,j∈ω} h_{i}A_{1}A_{2}g_{j} = (⋃_{i∈ω} h_{i}A_{1})(⋃_{i∈ω} A_{2}g_{i}) ⊇ (⋂_{i∈ω} U′_{i})(⋂_{i∈ω} V′_{i}) = ⋂_{i∈ω} U′_{i}V′_{i}.

By Lemma 1.7, (∗) and (∗∗), U′_{i}V′_{i} ⊆_{o} H_{1}H_{2}. By (∗), U′_{i}V′_{i} is dense in H_{1}H_{2}. Hence ⋂_{i∈ω} U′_{i}V′_{i} is comeager, and thus non-meager, in H_{1}H_{2}. Therefore ⋃_{i,j∈ω} h_{i}A_{1}A_{2}g_{j} and so A_{1}A_{2} are non-meager in H_{1}H_{2}. ■

To prove (2) in Theorem 1.5 we will need the following corollary of Lemma 1.7 and 1.8.

Corollary 1.9 Suppose that H_{1} and H_{2} are closed subgroups of a Polish group H such that H_{1}H_{2} is non-meager in its relative topology. Then

(i) for every A_{1} ⊆_{o} H_{1} we have A_{1}H_{2} ⊆_{o} H_{1}H_{2}, (ii) for every analytic set A_{1} ⊆_{nm} H_{1} we have A_{1}H_{2} ⊆_{nm} H_{1}H_{2}.

Proof. Apply Lemma 1.7 and 1.8 for A_{2} = H_{2}. ■

Now we are returning to the proof of Theorem 1.5.

Proof of (2) in Theorem 1.5. First consider the case ∗ = o. We need to prove that

G_{C}G_{Ba} ⊆_{o} G_{B} ∧ G_{B}G_{Aa} ⊆_{o} G_{A} ⟺ G_{C}G_{Aa} ⊆_{o} G_{A}.

(⇒). Assume G_{C}G_{Ba} ⊆_{o} G_{B} ∧ G_{B}G_{Aa} ⊆_{o} G_{A}. Define H := G_{A}, H_{1} := G_{B}, H_{2} := G_{Aa}, A_{1} := G_{C}G_{Ba}. Then H_{1}H_{2} = G_{B}G_{Aa} ⊆_{o} G_{A}, so it is non-meager in its relative topology. Moreover, A_{1} ⊆_{o} H_{1}. Hence by Corollary 1.9(i) we get G_{C}G_{Aa} = G_{C}G_{Ba}G_{Aa} = A_{1}H_{2} ⊆_{o} H_{1}H_{2} = G_{B}G_{Aa} ⊆_{o} G_{A}. So G_{C}G_{Aa} ⊆_{o} G_{A}. (⇐). Assume G_{C}G_{Aa} ⊆_{o} G_{A}. Then of course G_{B}G_{Aa} ⊆_{o} G_{A}. On the other hand, taking the intersection with G_{B}, we get that G_{C}G_{Ba} ⊆_{o} G_{B}.

Now consider the case ∗ = nm. We need to prove that

G_{C}G_{Ba} ⊆_{nm} G_{B} ∧ G_{B}G_{Aa} ⊆_{nm} G_{A} ⟺ G_{C}G_{Aa} ⊆_{nm} G_{A}.

(⇒). The proof is similar as in the case ∗ = o. We need only to check the last implication, namely G_{C}G_{Aa} ⊆_{nm} G_{B}G_{Aa} ⊆_{nm} G_{A} implies G_{C}G_{Aa} ⊆_{nm} G_{A}.

Suppose for a contradiction that there are closed and nowhere dense subsets D_{i}, i ∈ ω, of G_{A} such that G_{C}G_{Aa} ⊆ ⋃_{i∈ω} D_{i}. By the assumption that G_{C}G_{Aa} ⊆_{nm} G_{B}G_{Aa}, there is i ∈ ω such that D_{i} ∩ G_{B}G_{Aa} has a non-empty interior in G_{B}G_{Aa}. So countably many translates (by elements of G_{B} on the left and by elements of G_{Aa} on the right) of D_{i} cover G_{B}G_{Aa}. Since D_{i} is nowhere dense in G_{A}, we get that G_{B}G_{Aa} is meager in G_{A}. This is a contradiction. (⇐). Assume G_{C}G_{Aa} ⊆_{nm} G_{A}. Then of course G_{B}G_{Aa} ⊆_{nm} G_{A}. It remains to prove that G_{C}G_{Ba} ⊆_{nm} G_{B}.

By Proposition 1.3 we get G_{C}/G_{Aa} ⊆_{nm} G_{A}/G_{Aa}. Moreover, G_{C}/G_{Aa} is Polish, so it is a G_{δ} subset of G_{A}/G_{Aa}. So G_{C}/G_{Aa} = D△U where D is meager and analytic and U ≠ ∅ is open in G_{A}/G_{Aa}. Let π : G_{A} → G_{A}/G_{Aa} be the quotient map. Define D′ = π^{−1}[D] and U′ = π^{−1}[U]. Then we have that G_{C}G_{Aa} = D′△U′, D′ is meager and analytic [3, Proposition 14.4] and U′ ≠ ∅ is open in G_{A}. Moreover, D′G_{Aa} = D′, U′G_{Aa} = U′ and G_{C} ⊆ G_{B}. Hence U′ ∩ G_{B} ≠ ∅.

Suppose for a contradiction that G_{C}G_{Ba} is meager in G_{B}. Since G_{C}G_{Ba} = (G_{C}G_{Aa}) ∩ G_{B} = (D′△U′) ∩ G_{B} = (D′ ∩ G_{B})△(U′ ∩ G_{B}) and U′ ∩ G_{B} ≠ ∅ is open in G_{B}, we get that D′ ∩G_{B} ⊆_{nm} G_{B}. We also know that D′ ∩G_{B} is analytic. Moreover, G_{B}G_{Aa} ⊆_{nm} G_{A} so it is non-meager in its relative topology. Hence by Corollary 1.9(ii) we get (D′∩G_{B})G_{Aa} ⊆_{nm} G_{B}G_{Aa} ⊆_{nm} G_{A}. So, in the same way as in the proof of (⇒), we get (D′∩G_{B})G_{Aa} ⊆_{nm} G_{A}. Hence D′G_{Aa} ⊆_{nm} G_{A}, which means that D′ ⊆_{nm} G_{A}, a contradiction. ■

In order to get the existence of nm-independent extensions we assume smallness.

Theorem 1.10 (Existence of nm-independent extensions) Let (X, G) be a small Polish structure. Then for all finite a ⊆ X and A ⊆ B ⊆ X, there is b ∈ o(a/A) such that

b⌣ |^{nm}_{A}B.

Before the proof let us show the following remark.

Remark 1.11 A Polish structure (X, G) satisfies the existence of ∗-independent extensions, where ∗ = nm or ∗ = o, iff for all finite a ⊆ X and A ⊆ B ⊆ X there exists f ∈ G_{A} such that G_{B}fG_{Aa} ⊆_{∗} G_{A}.

Proof. The existence of ∗-independent extensions is equivalent to the fact that for every finite a ⊆ X and A ⊆ B ⊆ X there is b ∈ o(a/A) such that G_{B}G_{Ab} ⊆ G_{A}, which in turn is equivalent to the conclusion of the remark. ■

Proof of Theorem 1.10. Consider a, A, B as in the theorem. Let {a_{i} : i ∈ ω} be a set of representatives of all orbits over B contained in o(a/A). Then o(a/A) = ⋃_{i∈ω} o(a_{i}/B).

Take f_{i} ∈ G_{A}, i ∈ ω, such that f_{i}(a) = a_{i}.

Claim G_{A} = ⋃_{i∈ω} G_{B}f_{i}G_{Aa}.

Proof of Claim. Consider any f ∈ G_{A}. Then f(a) ∈ o(a_{i}/B) for some i ∈ ω. Hence there is g ∈ G_{B} such that g(f(a)) = f_{i}(a). Then f_{i}^{−1}gf ∈ G_{Aa}, so f ∈ g^{−1}f_{i}G_{Aa} ⊆ G_{B}f_{i}G_{Aa}. Hence G_{A} ⊆ ⋃_{i∈ω} G_{B}f_{i}G_{Aa}. The opposite inclusion is obvious. □

By Claim and the Baire category theorem there is i ∈ ω such that G_{B}f_{i}G_{Aa} ⊆_{nm} G_{A}. So the proof is completed by Remark 1.11. ■

We see that the above application of the Baire category theorem works only for nmindependence. In Section 3 (Remark 3.5) we will see that the pseudo-arc is an example of a small Polish structure without the existence of o-independence extensions.

One justification for our definition of nm-independence is the fact that it satisfies all the fundamental properties (Theorem 1.5 and 1.10) necessary to develop a counterpart of basic geometric stability theory. Another justification is given by the next corollary, which shows that in compact [profinite] structures nm-independence coincides with m-independence.

Theorem 1.12 Let (X, G) be a Polish structure such that G acts continuously on a separable metrizable space X. Let a, A, B ⊆ X be finite. Assume that o(a/A) is non-meager in its relative topology (e.g. it is Polish). Then a⌣ |^{∗}_{A}B ⟺ o(a/AB) ⊆_{∗} o(a/A) where ∗ = o or ∗ = nm.

Proof. Let π : G_{A}/G_{Aa} → o(a/A) be defined by π(gG_{Aa}) = ga. Since o(a/A) is nonmeager in its relative topology, by Effros’ theorem [1, Theorem 2.2.2] we have that π is a homeomorphism. Hence o(a/AB) ⊆_{∗} o(a/A) ⟺ π^{−1}[o(a/AB)] ⊆_{∗} π^{−1}[o(a/A)] ⟺ G_{AB}/G_{Aa} ⊆_{∗} G_{A}/G_{Aa}. We finish using Proposition 1.3. ■

Corollary 1.13 In every compact structure o-independence, nm-independence and mindependence coincide.

In Section 3 (Remark 3.5) we will see that the pseudo-arc considered with the group of all homeomorphisms is an example of a small Polish structure where o-independence and nm-independence differ. Below we give a simpler example of a Polish structure in which these two notions differ, but this structure is not small.

Example 1 Let X = (S^{1})^{ω} and G = (Homeo(S^{1}))^{ω}. We consider the Polish structure (X, G) where G acts naturally on X on the appropriate coordinates. Then in (X, G) oindependence is different from nm-independence. Moreover, (X, G) is not small, it does not have the existence of o-independent extensions but it satisfies the existence of nmindependent extensions.

Proof. Take any \overline{x} = ⟨x_{0}, x_{1}, …⟩ ∈ X and \overline{y} = ⟨y_{0}, y_{1}, …⟩ ∈ X such that x_{i} ≠ y_{i} for all i ∈ ω. Then o(\overline{x}) = X and o(\overline{x}/\overline{y}) = S^{1} \ \{y_{0}\} × S^{1} \ \{y_{1}\} × . . .. So o(\overline{x}/\overline{y}) is non-meager and not open in o(\overline{x}). Since o(\overline{x}) = X is Polish, by Theorem 1.12 we get that \overline{x}⌣ |^{nm} \overline{y} and \overline{x}⌣̸ |^{o}\overline{y}. The ’moreover’ part is left to the reader. ■

2 Basic model theory

In this section we introduce counterparts of some basic notions from stability theory and we investigate their properties.

In compact [profinite] structures definable sets were defined as the sets which are closed and invariant over finite subsets. Since for a Polish structure (X, G) we do not have any topology on X, we need another definition of definable sets. Moreover, as in model theory we would like to have (imaginary) names for definable sets for which forking calculus works in the same way as for the real elements. So we propose the following definition of definable sets and imaginary elements.

From now on we assume that (X, G) is a Polish structure.

Definition 2.1 The imaginary extension, denoted by X^{eq}, is the union of all sets of the form X^{n}/E with E ranging over all invariant equivalence relations such that for all a ∈ X^{n}, Stab([a]_{E}) <_{c} G. The sets X^{n}/E will be called the sorts of X^{eq}.

Remark 2.2 Let E be an invariant equivalence relation on X^{n} whose classes have closed stabilizers in G. Then G induces a group of permutations of X^{n}/E, denoted by G ↾ X^{n}/E, which is Polish, and (X^{n}/E, G ↾ X^{n}/E) is a Polish structure.

As in model theory, (X^{eq})^{eq} = X^{eq} which means that if E is an invariant equivalence relation on a product of sorts X^{n_{1}}/E_{1} × … × X^{n_{k}}/E_{k} whose classes have closed stabilizers in G, then the set of E-classes can be identified with the sort X^{n_{1}} × … × X^{n_{k}}/E′ where

E′(x_{1}, …, x_{k}; y_{1}, …, y_{k}) ⟺ E(x_{1}/E_{1}, …, x_{k}/E_{k}; y_{1}/E_{1}, …, y_{k}/E_{k}).

Definition 2.3 A subset D of X (or more generally of any sort of X^{eq}) is said to be definable over a finite subset A of X^{eq} if D is invariant over A and Stab(D) <_{c} G. We say that d ∈ X^{eq} is a name for D if for every f ∈ G we have f[D] = D ⟺ f(d) = d.

Proposition 2.4 Each set definable in (X, G) [or in X^{eq}] has a name in X^{eq}.

Proof. The proof is similar to the proof of Proposition 1.9 in [7]. Suppose D is a-definable for some finite a ∈ X^{n}. We define an equivalence relation E on X^{n} by:

E(a_{1}, a_{2}) ⟺ [a_{1} = a_{2} ∨ (a_{1}, a_{2}) ∈ S(a, a)]

where S = {(f, g) ∈ G × G : f[D] = g[D]}. It is easy to check that E is invariant, every class of E has a closed stabilizer and a/E is a name for D. ■

Working in X^{eq}, we can define Acl^{eq} in the same way as in X. Then the results of Section 1 (including Theorem 1.5 and 1.10) are true in X^{eq} (the only exception is the fact that the Polish structure (X, G) considered in Example 1 does not satisfy the existence of nm-independent extensions in X^{eq}). However, in the case of a compact [profinite] structure (X, G) both the family of definable sets and X^{eq} computed according to the definitions given in this paper are larger than those computed according to the definitions from [13, 14, 7].

As usual, having a notion of independence satisfying the properties listed in Theorem 1.5 and 1.10, one can define a rank, which has some nice properties.

Definition 2.5 The rank N M is the unique function from the collection of orbits over finite sets to the ordinals together with ∞, satisfying

N M(a/A) ≥ α + 1 iff there is a finite set B ⊇ A such that

a⌣̸ |^{nm}_{A}B and N M(a/B) ≥ α.

From now on we assume that (X, G) is small. In fact, more generally, one could only assume that (X, G) satisfies the existence of nm-independent extensions and, if we work in X^{eq}, one should also assume that the existence of nm-independent extensions holds in X^{eq}.

The results formulated below follow from a standard forking calculation (e.g. see [16, Lemma 5.1.4 and Theorem 5.1.6]). In the next proposition a, b, A are finite tuples (subsets) of X or X^{eq}.

Proposition 2.6 (1) a⌣ |^{m}_{A}b iff N M(a/Ab) = N M(a/A).

(2) N M(a/bA) + N M(b/A) ≤ N M(ab/A) ≤ N M(a/bA) ⊕ N M(b/A).

(3) Suppose N M(a/Ab) < ∞ and N M(a/A) ≥ N M(a/Ab) ⊕ α. Then N M(b/A) ≥ N M(b/Aa) + α.

(4) Suppose N M(a/Ab) < ∞ and N M(a/A) ≥ N M(a/Ab) + ω^{α}n. Then N M(b/A) ≥ N M(b/Aa) + ω^{α}n.

(5) If a⌣ |^{m}_{A}b, then N M(ab/A) = N M(a/bA) ⊕ N M(b/A).

As in stable or simple theories, the inequalities in the point (2) will be called Lascar inequalities.

An easy induction and Proposition 2.6(1) yield the following remark.

Remark 2.7 Let a, A ⊆ X be finite. Then N M-rank of o(a/A) computed in X is the same as N M-rank of o(a/A) computed in X^{eq}.

Definition 2.8 (X, G) is nm-stable if every 1-orbit has ordinal N M-rank.

Remark 2.9 (X, G) is nm-stable iff there is no infinite sequence A_{0} ⊆ A_{1} ⊆ … of finite subsets of X and a ∈ X such that a⌣̸ |^{nm}_{A_{i}}A_{i+1} for every i ∈ ω.

By Lascar inequalities and Remark 2.7 we easily get:

Remark 2.10 (X, G) is nm-stable iff each n-orbit, n ≥ 1, has ordinal N M-rank iff each n-orbit, n ≥ 1, in X^{eq} has ordinal N M-rank.

Proposition 2.11 TFAE:

(1) (X, G) is nm-stable.

(2) There is no a ∈ X and finite sets A_{0} ⊆ A_{1} ⊆ … ⊆ X such that G_{A_{i+1}}G_{A_{i}a} is meager in G_{A_{i}} for every i ∈ ω.

(3) For every a ∈ X and finite sets A_{0} ⊆ A_{1} ⊆ … X there is n ∈ ω such that G_{A_{n+i+1}}G_{A_{n}a} ⊆_{nm} G_{A_{n+i}}G_{A_{n}a} for every i ∈ ω.

Proof. (1) ⇔ (2) is obvious by Proposition 1.3 and Remark 2.9.

(1) ⇒ (3). By Proposition 1.3, Remark 2.9 and transitivity we can find n ∈ ω such that G_{A_{n+i+1}}G_{A_{n+i}a} ⊆_{nm} G_{A_{n+i}} for every i ∈ ω. So by Corollary 1.9(ii) and induction we easily get that G_{A_{n+i+1}}G_{A_{n}a} ⊆_{nm} G_{A_{n+i}}G_{A_{n}a} for every i ∈ ω.

(3) ⇒ (1). Take any a ∈ X and finite sets A_{0} ⊆ A_{1} ⊆ … ⊆ X. By (3) there is n ∈ ω such that G_{A_{n+i+1}}G_{A_{n}a} ⊆_{nm} G_{A_{n+i}}G_{A_{n}a} for every i ∈ ω. Hence we easily get that

G_{A_{n+i}}G_{A_{n}a} ⊆_{nm} G_{A_{n}} for every i ∈ ω. So a⌣ |^{nm}_{A_{n}}A_{n+i+1}, and hence a⌣ |^{nm}_{A_{n+i}}A_{n+i+1}, for

every i ∈ ω. We are done by Remark 2.9. ■

As it was mentioned in the introduction I hope that nm-independence, N M-rank and maybe some other model theoretic notions may be useful in descriptive set theory or topology. It is worth mentioning here that by Theorem 1.12, if orbits are non-megear in their relative topologies, then N M-rank and nm-stability can be expressed in terms of X instead of G, and N M-rank measures a ’topological size’ of orbits. More precisely, Theorem 1.12 gives us the following remark.

Remark 2.12 Suppose that X is a separable metrizable space and G acts continuously on X. Assume that all orbits over finite sets are non-meager in their relative topologies. Then

(1) N M(a/A) ≥ α + 1 iff there is a finite set B ⊇ A such that o(a/B) is meager in o(a/A) and N M(a/B) ≥ α.

(2) (X, G) is nm-stable iff there is no infinite sequence A_{0} ⊆ A_{1} ⊆ … of finite subsets of X and a ∈ X such that o(a/A_{i+1}) is meager in o(a/A_{i}) for every i ∈ ω.

By Remark 2.12 we see that in compact structures N M-rank and nm-stability coincide with M-rank and m-stability, respectively.

Similarly as in model theory and in profinite structures one can define a natural pregeometry on an orbit of N M-rank 1. To introduce this definition first we need to prove several remarks.

Remark 2.13 For any finite a, A ⊆ X^{eq} we have that N M(a/A) = 0 iff a ∈ Acl^{eq}(A).

Proof. (⇒). Assume N M(a/A) = 0. Then a⌣ |^{nm}_{A}B for every finite B ⊆ X^{eq}. By Theorem 1.5(3) we get a ∈ Acl^{eq}(A). (⇐). Assume a ∈ Acl^{eq}(A). By Theorem 1.5(3) we get that a⌣ |^{nm}_{A}B for every finite B ⊆ X^{eq}. Hence N M(a/A) = 0. ■

Corollary 2.14 If o(a/A) is Polish, then N M(a/A) = 0 iff o(a/A) is countable with the discrete topology.

For a finite set A ⊆ X^{eq} we define the operator Acl_{A}^{eq} by Acl_{A}^{eq}(B) = Acl^{eq}(A ∪ B).

Remark 2.15 Assume N M(a/A) = 1 and B is a finite subset of X^{eq}. Then a ∈ Acl_{A}^{eq}(B)

iff a⌣̸ |^{nm}_{A}B.

Proof. (⇒). Assume a⌣̸ |^{nm}_{A}B. Then N M(a/AB) < N M(a/A) = 1. Hence N M(a/AB) = 0. By Remark 2.13 we get a ∈ Acl_{A}^{eq}(B). (⇐). Assume a ∈ Acl_{A}^{eq}(B). By Remark 2.13, N M(a/AB) = 0 < N M(a/A). Hence by Proposition 2.6(1), a⌣̸ |^{nm}_{A}B. ■

Proposition 2.16 Assume N M(a/A) = 1. Then (o(a/A), Acl_{A}^{eq}) is a pregeometry.

Proof. The proof is the same as in model theory. Let us only check the exchange property. Take any B ⊆ o(a/A) and b ∈ o(a/A). Consider any a′ ∈ Acl_{A}^{eq}(Bb) \ Acl_{A}^{eq}(B). Wlog B is finite. By Remark 2.15, a′⌣̸ |^{nm}_{A}ABb and a′⌣ |^{nm}_{A}AB. Hence by Theorem 1.5, b⌣̸ |^{nm}_{AB}a′. So b⌣̸ |^{nm}_{A}Ba′, which implies that b ∈ Acl_{A}(Ba′). ■

Similarly, one can show that if N M(X) := sup{N M(a) : a ∈ X} = 1, then (X, Acl) is a pregeometry.

3 Examples

In this section we give several examples of small Polish structures. To begin with, notice that all small profinite structures are such examples. This class contains for instance all abelian profinite groups of finite exponent presented as the inverse limit of a system indexed by the natural numbers and considered with the standard structural group [5, Theorem 1.9]. Below we give examples which are not profinite.

Example 2 For every n ≥ 1 the Polish structure (S^{n}, Homeo(S^{n})) where S^{n} is the ndimensional sphere and Homeo(S^{n}) is the group of all homeomorphisms of S^{n} with the compact-open topology is small of N M-rank 1. Similarly, (S^{n}, Diff(S^{n})) where Diff(S^{n}) is the group of all diffeomorphisms of S^{n} is also small of N M-rank 1.

Example 3 The Polish structure (I^{ω}, Homeo(I^{ω})) where I^{ω} is the Hilbert cube is small of N M-rank 1.

Proof. By [11, Section 6.1, Exercise 2] we know that the action is n-transitive for every n ≥ 1. ■

Example 4 The Polish structure (P, Homeo(P)) where P is the pseudo-arc is small and not nm-stable.

Before the proof we recall some notions and results about continua. Recall that the pseudo-arc P is the unique hereditarily indecomposable chainable continuum. By hereditary idecomposability we get that for every A ⊆ P the intersection of all subcontinua of P containing A is the smallest subcontinuum of P containing A.

Let C be any nondegenerate continuum (e.g. P). We say that C is irreducible between subsets A and B if there is no proper subcontinuum of C containing A and B. For p ∈ C we define the composant of p, denoted by κ(p), as the set of the points x ∈ C for which there is a proper subcontinuum A of C such that p, x ∈ A.

For every p ∈ C, κ(p) is the union of countably many proper subcontinua of C containing p [12, Proposition 11.14] and κ(p) is dense and connected [12, Exercise 5.20]. Moreover, by [12, Exercise 6.19] C is indecomposable iff every proper subcontinuum of C is nowhere dense in C. Hence

(∗) If C is indecomposable, then κ(p) is meager and dense in C for every p ∈ C.

If C is indecomposable, we define an equivalence relation E on C by:

E_{C}(x, y) ⟺ y ∈ κ(x).

By (∗) and the Baire category theorem E_{C} has uncountably many classes (they are just composants). We will need the following two facts, which can be found in [9, Theorem 2, Theorem 6].

Fact 3.1 The pseudo-arc is homeomorphic to each of its nondegenerate subcontinua.

Fact 3.2 Suppose H_{1,1}, H_{1,2}, …, H_{1,n} are proper subcontinua of the pseudo-arc P and that P is irreducible between each pair of them. Suppose T is a homeomorphism of H_{1,1} ∪H_{1,2} ∪ … ∪ H_{1,n} onto H_{2,1} ∪ H_{2,2} ∪ … ∪ H_{2,n} where H_{2,1}, H_{2,2}, …, H_{2,n} are proper subcontinua of P such that P is irreducible between each pair of them. Then T can be extended to a homeomorphism of P onto P.

Now we can prove the following Lemma which immediately implies that the Polish structure (P, Homeo(P)) is small.

Lemma 3.3 For every n ≥ 1, there are only finitely many orbits on P^{n} under the action of Homeo(P).

Proof. The proof is by induction on n. The case n = 1 follows from the fact that the pseudo-arc is homogeneous.

Assume that Lemma 3.3 is true for the tuples of length < n. Suppose for a contradiction that there are infinitely many n-tuples t_{0} = (t_{0}^{0}, …, t_{0}^{n−1}), t_{1} = (t_{1}^{0}, …, t_{1}^{n−1}), … lying in different orbits under Homeo(P). Then there is an infinite subsequence t_{i_{0}}, t_{i_{1}}, … such that for every j, k ∈ ω, the tuples t_{i_{j}} and t_{i_{k}} are isomorphic, via f_{jk}, with respect to the relation E_{P}. Wlog i_{j} = j for every j ∈ ω. Then there is m ≤ n−1 such that for every i ∈ ω, {t_{i}^{0}, …, t_{i}^{n−1}} can be partitioned into classes modulo E_{P}, say A_{i}^{0}, …, A_{i}^{m}, so that f_{jk}[A_{j}^{l}] = A_{k}^{l} for every j, k ∈ ω and l ≤ m. Let B_{i}^{0}, …, B_{i}^{m} be the smallest proper subcontinua of P containing A_{i}^{0}, …, A_{i}^{m}, respectively. Then B_{i}^{0}, …, B_{i}^{m} are pairwise disjoint proper subcontinua of P such that P is irreducible between each pair of them.

Case 1 m > 1. By Fact 3.1 for every i ∈ ω we can find homeomorphisms f_{i}^{l} : B_{i}^{l} → B_{0}^{l}, l = 0, …, m. By the inductive hypothesis and Fact 3.1 there are i ≠ j for which there exist homeomorphisms g_{l} : B_{0}^{l} → B_{0}^{l}, l = 0, …, m, such that

T := (f_{j}^{0})^{−1}g_{0}f_{i}^{0} ∪ … ∪ (f_{j}^{m})^{−1}g_{m}f_{i}^{m} : B_{i}^{0} ∪ … ∪ B_{i}^{m} → B_{j}^{0} ∪ … ∪ B_{j}^{m}

is a homeomorphism extending f_{ij}. Since both {B_{i}^{0}, …, B_{i}^{m}} and {B_{j}^{0}, …, B_{j}^{m}} are collections of proper subcontinua of P such that P is irreducible between each pair of them, by Fact 3.2, T can be extendend to an element of Homeo(P). Since T (t_{i}) = t_{j}, we get a contradiction.

Case 2 m = 1. Then B_{i}^{0}, i ∈ ω, are proper subcontinua of P, so by Fact 3.1 we can find homeomorphisms f_{i} : B_{i}^{0} → B_{0}^{0}. Let t′_{i} = f_{i}(t_{i}). If we show that there exist i ≠ j and a homeomorphism f : B_{0}^{0} → B_{0}^{0} with f(t′_{i}) = t′_{j}, then T := f_{j}^{−1}ff_{i} is a homeomorphism from B_{i}^{0} onto B_{j}^{0} such that T (t_{i}) = t_{j}, and we finish using Fact 3.2. By the minimality of B_{i}^{0}, i ∈ ω, working with B_{0}^{0} and t′_{0}, t′_{1}, …, we are in the situation described in Case 1, so the proof is completed. ■

Lemma 3.4 (P, Homeo(P)) is not nm-stable.

Proof. Take any p ∈ P. By (∗) we can find p_{0} ∈ κ(p) such that p_{0} ≠ p. Now choose the smallest proper subcontinuum P_{0} of P containing p and p_{0}. Then P_{0} is also the pseudo-arc and it is nowhere dense in P. We see that P_{0} is definable over {p, p_{0}}. Let q_{0} be a name of P_{0}. By Fact 3.2, o(p/q_{0}) = P_{0}, which is nowhere dense in o(p) = P. Since P is Polish, Theorem 1.12 implies that p⌣̸ |^{nm}q_{0}.

Let us repeat this step within P_{0}. By (∗) we can find p_{1} ∈ κ_{P_{0}}(p) such that p_{1} ≠ p, where κ_{P_{0}} is κ computed within P_{0}. Now choose the smallest proper subcontinuum P_{1} of P_{0} containing p and p_{1}. Then P_{1} is also the pseudo-arc and it is nowhere dense in P_{0}. We see that P_{1} is definable over {p, p_{0}, p_{1}}. Let q_{1} be a name of P_{1}. By Fact 3.2, o(p/q_{0}q_{1}) = P_{1}, which is nowhere dense in o(p/q_{0}) = P_{0}. Since P_{0} is Polish, Theorem 1.12 implies that p⌣̸ |^{nm}_{q_{0}}q_{1}.

We repeat this procedure and obtain an infinite sequence of imaginaries q_{0}, q_{1}, … such that p⌣̸ |^{nm}_{q_{<i}}q_{≤i} for all i ∈ ω. By Remark 2.9 and 2.10 the proof is completed. ■

It is not clear how to repeat the above proof without using imaginaries. More precisely, by Fact 3.1, 3.2 and (∗) we have that o(p/p_{0}) = κ(p) is meager in o(p) = P and P is Polish, so by Theorem 1.12, p⌣̸ |^{nm}p_{0}. However, by (∗) and Effros’ theorem, o(p/p_{0}) = κ(p) is meager in its relative topology, so starting from this point we cannot work just with orbits, but we should look at their preimages in Homeo(S), which is rather complicated.

Remark 3.5 (i) (P, Homeo(P)) does not satisfy the existence of o-independent extensions. (ii) In (P, Homeo(P)) the relations ⌣ |^{o} and ⌣ |^{nm}are different.

Proof. (i) Since P is homogeneous, by Theorem 1.12 it is enough to show that there is p ∈ P such that every orbit over p is not open in P. Take any p ∈ P. Using Fact 3.1 and 3.2 we easily get that the only orbits over p are κ(p) and P \ κ(p), so we are done by (∗). (ii) It follows from (i), Lemma 3.3 and Theorem 1.10. We can also see it directly. Take any p ∈ P and q ∉ κ(p). By Fact 3.2, o(q/p) = P \ κ(p). So by (∗) we get that o(q/p) is non-meager in P, and hence, by Theorem 1.12, q⌣ |^{nm}p.

On the other hand, by (∗) we see that P \κ(p) is not open in P, and hence, by Theorem

1.12, q⌣̸ |^{o}p. ■

Definition 3.6 (i) A Polish group structure is a Polish structure (H, G) such that H is a group and G acts as a group of automorphisms of H. (ii) A Polish group regarded as a Polish structure is a Polish structure (H, G) such that H is a Polish group and G acts continuously as a group of automorphisms of H.

Of course every Polish group regarded as a Polish structure is a Polish group structure. The next example is an example of a small Polish group regarded as a Polish structure.

Example 5 Let us consider the discrete topology on ℚ and the product topology on ℚ^{ω}. We consider the additive group structure on ℚ. Let Aut^{0}(ℚ^{ω}) be the group of all automorphisms of ℚ^{ω} respecting the inverse system ℚ ⟵ ℚ × ℚ ⟵ . . .. Then Aut^{0}(ℚ^{ω}) can be considered as the inverse limit of the system consisting of Aut^{0}(ℚ^{n}), n ∈ ω, where on Aut^{0}(ℚ^{n}) we have the pointwise convergence topology. Then (ℚ^{ω}, Aut^{0}(ℚ^{ω})) is a small of N M-rank 1 Polish group regarded as a Polish structure.

Proof. We leave to the reader checking that (ℚ^{ω}, Aut^{0}(ℚ^{ω})) is a Polish group regarded as a Polish structure. Now we will show that it is small of N M-rank 1.

The following claim is obvious.

Claim If f ∈ Aut^{0}(ℚ^{n}) and {η_{1}, …, η_{n}} is a basis of ℚ^{n} over ℚ, then for all ϵ_{0}, ϵ_{1}, …, ϵ_{n}, ϵ′_{0}, ϵ′_{1}, …, ϵ′_{n} ∈ ℚ such that ϵ_{0}, ϵ′_{0} ≠ 0, there is \overline{f} ∈ Aut^{0}(ℚ^{n+1}) such that \overline{f}(0^{⌢}ϵ_{0}) = 0^{⌢}ϵ′_{0} and \overline{f}(η_{i}^{⌢}ϵ_{i}) = f(η_{i})^{⌢}ϵ′_{i}, i = 1, …, n.

Now consider any finite set A ⊆ ℚ^{ω} and a ∈ ℚ^{ω}. Then either a ∈ Lin(A) or there is the largest natural number n such that a↾n ∈ Lin(A)↾n. So by Claim we get that either a ∈ Lin(A), and then o(a/A) = {a}, or a ∉ Lin(A), and then o(a/A) = {η ∈ ℚ^{ω} : η↾n = a↾ n ∧ η↾(n + 1) ∉ Lin(A)↾(n + 1)} where n is the largest n such that a↾n ∈ Lin(A)↾n. So each orbit over A is either a singleton from Lin(A) or an open set. Hence there are countably many orbits over A, so (ℚ^{ω}, Aut^{0}(ℚ^{ω})) is small; we also see that N M(ℚ^{ω}) = 1. ■

We end up with a remark which follows from Lascar inequalities and yields examples of small Polish structures of arbitrary finite N M-rank.

Remark 3.7 If (X, G) is a small Polish structure of N M-rank 1, then for every natural number n ≥ 1, (X^{n}, G) is a small Polish structure of N M-rank n.

4 Final comments and questions

In model theory there are results, known as group configuration theorems, which say that under some general geometric assumptions one can find a definable group (e.g. [15, Chapter 5, 7]). Such theorems were also proved for small profinite structures [14, Theorem 1.7, Theorem 3.3] or, more generally, for compact structures satisfying the existence of mindependent extensions [7, Theorem 3.15]. Considerations concerning the existence of a definable (in some sense) group structure are the best example illustrating how stability theory ideas may lead to new aspects in the analysis of classical topological objects. In the context of small Polish structures it is not clear how to prove such kind of results. We can ask here the following general questions.

Question 4.1 Suppose (X, G) is a small Polish structure. When is there a function · : X ×X → X definable (or invariant) over a finite subset A of X such that (X, ·) is a group?

Question 4.2 Suppose (X, G) is a small Polish structure. When is there an infinite set Y ⊆ X and a function · : Y × Y → Y, both definable (or invariant) over a finite subset A of X, such that (Y, ·) is a group?

The group configuration theorem for small profinite structures [14, Theorem 1.7] yields a partial answer to Question 4.2. Namely, if we additionally assume that (X, G) is profinite and m-normal, then in every non-trivial orbit of M-rank 1 there is an open definable group.

So one of the possible ways of further research is to prove for small Polish structures counterparts of some advanced results from geometric stability theory, e.g. a variant of the group configuration theorem.

There are also certain open questions about the existence of small profinite structures satisfying some additional assumptions. I think it would be interesting to find counterexamples for these questions in the bigger class of small Polish structures. Let us formulate two such questions.

Conjecture 4.3 (N M-gap conjecture) Let (X, G) be a small Polish structure. Then for every orbit o over a finite subset A of X one has N M(o) ∈ ω ∪ {∞}.

This conjecture is open in the class of small profinite structures; it was proved only for small m-stable profinite groups [17, Theorem 18]. In the context of small Polish structures it is open even for small nm-stable Polish groups.

Question 4.4 Is it true that every small Polish group regarded as a Polish structure or, more generally, every small Polish group structure, is abelian-by-countable?

In the case of small profinite structures the appropriate question asks if every small profinite group is abelian-by-finite. This question is open in general but it was answered positively for small m-stable profinite groups [17, Theorem 1]. Question 4.4 is open even for small nm-stable Polish groups.

Let us notice here that all the structural results about small profinite groups were based on the fact that such groups are locally finite [13, Proposition 2.4], which is not true for small Polish groups [Example 5]. So an interesting general question is what we can prove about small Polish groups regarded as Polish structures.

At the end I would like to formulate several purely descriptive set theoretic (or topological) facts and questions which came up naturally during my considerations on Polish structures.

The following fact is Corollary 2.6.8 of [1].

Fact 4.5 The Polish Homeo(I^{ω})-space (I^{ω}, Homeo(I^{ω})) is universal for Borel G-spaces, i.e. every Borel G-space (X, G) can be embedded into (I^{ω}, Homeo(I^{ω})) in the sense that there is a topological isomorphism ψ : G → ψ[G] <_{c} Homeo(I^{ω}) and a Borel embedding ϕ : X → I^{ω} such that ϕ(gx) = ψ(g)ϕ(x) for every g ∈ G.

We know that this universal Borel G-space is a small Polish structure [Example 3]. The following question is a counterpart of the above fact for groups.

Question 4.6 Does there exist a Polish G-group (U, G) which is universal for Borel Hgroups (i.e. every Borel H-group (G_{0}, H) can be embedded into (U, G), preserving the group structure on G_{0})? If yes, is (U, G) a small Polish group regarded as a Polish structure?

From now on we fix a Polish group G. The following fact comes from [2, Theorem 0.3] and [1, Corollary 2.6.8].

Fact 4.7 There is a Polish G-space which is universal for Polish [Borel] G-spaces.

Question 4.8 Does there exist a Polish G-group which is universal for Polish [Borel] Ggroups?

It is obvious that any universal Polish G-space (or G-group, if it exists) for a fixed G is not small. However, we can ask the following question (or its variants for Polish structures which are additionally Polish or Borel G-spaces or G-groups).

Question 4.9 Does there exist a small Polish structure [group structure] which is universal for small Polish structures [group structures] of the form (X, G)?

At the very end we would like to formulate a question concerning small profinite structures. The following is a well-known fact [10].

Fact 4.10 If G is a locally compact group and (X, G) is a topological G-space, then there is a topological group H ⊇ X such that the topology on X is inherited from H and the action of G on X can be extended to an action on H so that (H, G) is a topological G-group.

Question 4.11 Is it true that for every small profinite structure (X, G) one can find a profinite group H ⊇ X such that X is closed in H, its topology is inherited from H and the action of G on X can be extended to an action on H so that (H, G) is a small profinite group?

It is not difficult to construct H so that (H, G) is not necessarily small. Namely, if

X = lim_{⟵} X_{i},

we can define H as the inverse limit of the system consisting of the linear spaces spanned freely by X_{i} over the two element field F_{2}.

Proposition 4.12 If Question 4.11 has the positive answer, then M-gap conjecture is true for small m-stable profinite structures.

Proof. Suppose for a contradiction that (X, G) is a small m-stable profinite structure with an orbit o over some finite set A such that M(o) ∈ Ord \ ω; hence wlog M(o) = ω. By the assumption we have that X is ∅-definable (i.e. closed and invariant) in a small profinite group (H, G).

By [13, Proposition 2.3] the subgroup ⟨X⟩ generated by X is generated in finitely many steps. This easily implies that (⟨X⟩, G) is a small m-stable profinite group. Hence the M-rank of o computed within (⟨X⟩, G) also equals ω, and we get a contradiction with the fact that M-gap conjecture holds for small m-stable profinite groups [17, Theorem 18]. ■

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Mathematical Institute, University of Wrocław pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. e-mail: kkrup@math.uni.wroc.pl and Mathematics Department University of Illinois at Urbana-Champaign 1409 W. Green Street, Urbana, IL 61801, USA. e-mail: kkrup@math.uiuc.edu