Definably complete and Baire structures

Version 4.1

A. Fornasiero & T. Servi

March 24, 2008

Abstract

We consider definably complete and Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain can not be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire. So is every o-minimal expansion of a field. The converse is clearly not true. However, unlike the o-minimal case, the structures considered form an elementary class. In this context we prove a version of Kuratowski-Ulam’s Theorem and some restricted version of Sard’s Lemma.

1 Introduction

We recall that a subset A of a topological space X is said to be meager if there exists a collection {Y_{i} : i ∈ ℕ} of nowhere dense subsets of X such that A ⊆ ⋃_{i∈ℕ} Y_{i}. The Baire Category Theorem states that every open subset of ℝ (with the usual topology) is not meager, i.e. ℝ is a Baire space.

The notion of Baire space is clearly not first order. Here we consider a similar (definable) notion, which instead is preserved under elementary equivalence, and which coincides with the classical notion over the real numbers (this is made precise in Section 2).

Def. complete and Baire, v. 4.1  1 Introduction

The (first order) structures we consider are definably complete expansions of ordered fields. Definable completeness (see Definition 1.3) is a weak version of Dedekind completeness, which is preserved under elementary equivalence. The study of definably complete structures, which is mainly due to C. Miller, follows that of o-minimal structures; the aim is to develop an analogue to the theory of o-minimality in a more general situation, where the assumptions have been considerably weakened, but, unlike o-minimality, they are first order. It is shown in [Miller01], [Servi07], [Fratarc06] that, as in the ominimal case, (a definable version of) most results of elementary real analysis can be proved in every definably complete expansion of an ordered field.

However, to obtain less elementary results one would need some more sophisticated machinery, in the direction of Sard’s Lemma and Fubini’s Theorem. Both of the quoted classical results refer to a notion of smallness (having measure zero), which has no natural translation in our context. We consider instead a topological notion of smallness (being meager), propose a definable version of this notion and carry out a theory of definably complete and Baire structures. In this context we prove an analogue to Fubini’s Theorem (the Kuratowski-Ulam Theorem 4.1) and a very restricted form of an analogue to Sard’s Lemma (Theorem 6.17). Notice that it is not known whether every definably complete structure is definably Baire.

The main motivation for the study of definably complete and Baire structures is to generalize the o-minimality results present in [Wilkie99] and [Kar–Mac99], to the situation where the base field is not necessarily ℝ. In [Wilkie99], the author proves that, given an expansion R of the real field with a family of C^{∞} functions, if there are bounds (uniform in the parameters) on the number of connected components of quantifier free definable sets, then R is o-minimal. In particular, thanks to a well known finiteness result in [Khov91], the structure generated by all real Pfaffian functions is o-minimal. In [Kar–Mac99], the authors generalize Wilkie’s theorem (by weakening the smoothness assumption) in a way which allows them to derive the following result (originally due to Speissegger, see [Speiss99]): the Pfaffian closure of an o-minimal expansion of the real field is o-minimal.

In a subsequent paper we use the results obtained in this paper to prove that, given a definably complete and Baire expansion 𝕂 of an ordered field with a family of C^{∞} functions, if there are bounds (uniform in the parameters) on the number of connected components of quantifier free definable sets, then 𝕂 is o-minimal. By using our restricted version 6.17 of Sard’s Lemma, we then proceed to prove the analogue to Khovanskii’s finiteness result in the context of definably complete and Baire structures. We derive the o-minimality of every definably complete and Baire expansion of an ordered field with any family of definable Pfaffian functions. Finally, we prove that the relative Pfaffian closure of an o-minimal structure 𝕂_{0} into a definably complete and Baire expansion 𝕂 of 𝕂_{0} is o-minimal. This latter result, whose proof is shaped on the one present in [Kar–Mac99], can be compared with the main result in [Fratarc06], where instead Speissegger’s method was followed.

This work contributes to the study of ordered structures which satisfy properties implied by (but not equivalent to) o-minimality; hence it has a natural collocation in the framework depicted in [DMS08].

1.1 Preliminaries and notation

Throughout this paper, 𝕂 is a (first-order) structure expanding an ordered field. We use the word “definable” as a shorthand for “definable in 𝕂 with parameters from 𝕂”.

For convenience, on 𝕂^{m} instead of the usual Euclidean distance we will use the equivalent distance

d : (x, y) ↦ max_{i=1,...,m}|x_{i} − y_{i}|.

For every δ > 0 and x ∈ 𝕂^{m}, we define

B^{m}(x; δ) := {y ∈ 𝕂^{m} : d(x, y) < δ},

the open “ball” of center x and “radius” δ; we will drop the superscript m if it is clear from the context.

Notation 1.1. Let X ⊆ Y ⊆ 𝕂^{n}, with Y definable. We write cl_{Y} (X) (or simply \overline{X} if Y is clear from the context) for the topological closure of X in Y, int_{Y} (X) (or simply X̊) for the interior part of X in Y, and bd_{Y} (X) := \overline{X} \X̊ for the boundary of X (in Y).

Notation 1.2. We define Π_{n}^{m+n} : 𝕂^{m+n} → 𝕂^{m} as the projection onto the first m coordinates. If A ⊂ 𝕂^{m+n} and x ∈ 𝕂^{m}, we denote by A_{x} the fibre of A over x, i.e. the set {y ∈ 𝕂^{n} : (x, y) ∈ A}.

1.2 Definably complete structures

Definition 1.3. An expansion 𝕂 of an ordered field is called definably complete if every definable subset of 𝕂 has a supremum in 𝕂 ∪ {±∞}.

Generalities on definably complete structures can be found in [Servi07] and [Miller01].

Definition 1.4. X ⊆ 𝕂^{m} is definably compact (d-compact for short) if it is definable, closed in 𝕂^{m} and bounded.

We order 𝕂^{m} lexicographically. We will denote by N a definable subset of 𝕂^{m} which is cofinal in the lexicographic ordering.

Lemma 1.5 (Miller). X is definably compact iff for every (Y (y))_{y∈N} definable decreasing family of closed non empty subsets of X, we have ⋂_{y} Y (y) ≠ ∅.

Definition 1.6. Let f : N → 𝕂^{n} be definable. Define acc_{y→∞} f(y) (and write for simplicity acc f) to be the set of accumulation points of f; that is, x ∈ acc f iff

(∀r ∈ 𝕂^{m})(∀ε > 0)(∃y > r) y ∈ N & d(f(y), x) < ε.

Lemma 1.7. If X is definably compact, then for all definable N and for all f : N → X definable we have acc f ≠ ∅.

It is not clear if the converse of the above lemma is true.

Definition 1.8. Let (A(y))_{y∈N} be a definable family of non-empty subsets of 𝕂^{n}. Define acc_{y→∞} A(y) (and write for simplicity acc A) to be the set of accumulation points of A; that is, x ∈ acc A iff (∀r ∈ 𝕂^{m}) (∀ε > 0) (∃y > r) y ∈ N and d(A(y), x) < ε.

Note that acc A = ⋂_{y} \overline{(⋃_{z≥y} A(z))}.

Remark 1.9. Let (A(t))_{0<t∈𝕂} be a definable family of subsets of 𝕂^{m}, and G := ⋃_{t>0} A(t) × {t}. Then, acc_{t→0} A(t) = (cl_{𝕂^{m+1}}(G))_{0} := {x ∈ 𝕂^{m} : (x, 0) ∈ \overline{G}\}.

Lemma 1.10. X is definably compact iff for all A definable family of nonempty subsets of X we have X ∩ acc A ≠ ∅.

Proof. First assume that X is d-compact. Let Y (y) := \overline{(⋃_{z≥y} A(y))}. Then (X ∩ Y (y)) is a definable decreasing family of closed subsets of X. By Lemma 1.5, ⋂_{y} Y (y) ≠ ∅, and we are done.

Conversely, assume that X is not d-compact. By Lemma 1.5, there exists a definable decreasing family Y := (Y (y))_{y∈𝕂^{m}} of closed subsets of X such that ⋂_{y} Y (y) = ∅. However, since Y is decreasing, X ∩ acc Y = ⋂_{y} Y (y), and we are done.

Proof of Lemma 1.7. Define A(y) := {f(y)}. By Lemma 1.10, acc A is nonempty. Note that acc A = acc f.

Lemma 1.11. Let C ⊂ 𝕂^{n} be a nonempty d-compact set, and let V := {V (t) : t ∈ I} be a definable open cover of C. Then, there exists δ_{0} ∈ 𝕂^{+} (a Lebesgue number for V and C) such that, for every subset X ⊆ C of diameter smaller than δ_{0}, there exists t ∈ I such that X ⊆ V (t).

Def. complete and Baire, v. 4.1  2 Meager sets

Proof. Suppose for a contradiction that

∀δ > 0 ∃y ∈ C ∀t ∈ I B(y; δ) ⊈ V (t).

For every δ > 0, define

Y (δ) := {y ∈ C : ∀t ∈ I B(y; δ) ⊈ V (t)}.

Note that (Y (δ))_{δ>0} is a definable family of subsets of C, increasing as δ decreases. Let y_{0} be an accumulation point for the family (Y (δ))_{δ>0}, as δ → 0 (which exists by Lemma 1.10).

Let t_{0} ∈ I and δ_{0} > 0 such that B(y_{0}; 2δ_{0}) ⊆ V (t_{0}). Let y ∈ Y (δ_{0}) such that |y − y_{0}| < δ_{0}. Therefore, B(y; δ_{0}) ⊆ B(y_{0}; 2δ_{0}) ⊆ V (t_{0}), contradicting the fact that y ∈ Y (δ_{0}).

We will often use without further comment the following result:

Lemma 1.12 (Miller). Let f : 𝕂^{n} → 𝕂^{m} be a definable continuous function and let C ⊂ 𝕂^{n} be d-compact. Then f(C) is d-compact.

Definition 1.13. A n-dimensional definable embedded C^{∞} 𝕂-manifold V ⊆ 𝕂^{d} (which we will simply call n-dimensional 𝕂-manifold) is a definable subset V of 𝕂^{d}, such that for every x ∈ V there exists a definable neighbourhood U(x) of x (in 𝕂^{d}), and a definable C^{∞} diffeomorphism f_{x} : U(x) ≃ 𝕂^{d}, such that U(x) ∩ V = f_{x}^{−1}(𝕂^{n} × {0}).

Remark 1.14. Note that a 𝕂-manifold V can always be written as the intersection of a definable closed set and a definable open set. In fact, let δ : V → 𝕂^{+} be a definable map, such that, for every x ∈ V, B(x, δ(x)) ⊆ U(x). Let U := ⋃_{x∈V} B(x, δ(x)); then, V = \overline{V} ∩ U.

Note moreover that the dimension n of a 𝕂-manifold V is uniquely determined by V, because 𝕂^{n} and 𝕂^{n′} are locally diffeomorphic iff n = n′.

2 Meager sets

Let X ⊆ Y ⊆ 𝕂^{n}, with Y definable.

Definition 2.1. X is nowhere dense (in Y) if int_{Y} (cl_{Y} (X)) = ∅. X is definably meager (in Y) if there exists a definable increasing family (A(t))_{t∈K} of nowhere dense subsets of Y, such that X ⊆ ⋃_{t} A(t). We will call such a family (cl_{Y} (A(t)))_{t∈𝕂} a witness of the fact that X is definably meager. X is definably residual (in Y) if Y \ X is definably meager. If Y is clear from the context, we will simply say that X is nowhere dense (resp. definably meager, definably residual).

Notice that, if (A(t))_{t∈𝕂} is a witness of the fact that X is meager in 𝕂^{n}, then also the family

(\overline{A(t)} ∩ [−|t|, |t|]^{n})_{t∈𝕂}

is a witness, hence we may always assume that each A(t) is d-compact. Notice also that we do not require that a meager set is definable. The subsets of Y, with the operations Δ (symmetric difference) and ∩,

form a commutative ring; the definably meager subsets of Y form an ideal of this ring.

Definition 2.2. Y is definably Baire if every non-empty open definable subset of Y is not meager (in Y).

Note that if 𝕂 has countable cofinality, then X is definably meager (Baire, respectively) in 𝕂^{n} if X is meager (Baire, respectively) in the usual topological sense. In general, the converse is not true: for instance, if 𝕂 is a countable o-minimal structure, then it is definably Baire, but not Baire in the topological sense. If 𝕂 is the expansion of the real field with a predicate for every subset of ℝ^{n} (n ∈ ℕ), then the two notions coincide. From now on, we will write “meager” for “definably meager”, and “topologically meager” for the usual topological notion, and similarly for “residual” and “Baire”.

Proposition 2.3. Let Y be definable, and ∅ ≠ U ⊆ Y be definable and open. Then, U is meager in Y iff it is meager in itself.

Proof. Suppose U is meager in Y and let (Y (t))_{t∈𝕂} be a witness of this fact. For every t ∈ 𝕂, define X(t) := Y (t) ∩ U. Since U is open, int_{U}(X(t)) = int_{Y} (Y (t)) ∩ U = ∅. Hence, (X(t))_{t∈𝕂} is a witness of the fact that U is meager in itself.

Viceversa, let (X(t))_{t∈𝕂} be a witness of the fact that U is meager in itself, and Y (t) := cl_{Y} (X(t)). We claim that int_{Y} (Y (t)) = ∅. In fact, int_{Y} (Y (t)) = int_{Y} ((cl_{Y} (X(t)) ∩ U)) = int_{U}(Y (t)) = int_{U}(X(t)) = ∅. Hence, (Y (t))_{t∈𝕂} is a witness of the fact that U is meager in Y.

Corollary 2.4. Let Y be definable, and ∅ ≠ U ⊆ Y be definable and open. Then,

1. If U is of not meager in itself, then Y is also not meager in itself.

2. If Y is Baire, then U is also Baire.

Proof. For 1, note that every subset of a meager set is meager, and in particular U is meager.

Regarding 2, if Y is Baire, let V ⊆ U be non-empty, definable and open in U. Since U is open in Y, V is also open in V. Hence, by 2.3, V is not meager in itself, and, again by 2.3, V is not meager in U. Therefore, U is Baire.

Lemma 2.5. Let Y ⊆ 𝕂^{m} be definable. The following are equivalent:

1. Y is Baire;

2. for all X ⊆ Y, if X is meager, then X̊ = ∅;

3. every x ∈ Y has a definable neighbourhood which is Baire;

4. every residual subset of Y is dense;

5. every open definable non-empty subset of Y is not meager in itself;

6. every meager closed definable subset of Y has empty interior.

Proof.

(2 ⇒ 1) is obvious.

(1 ⇒ 3) is obvious, because Y itself is a Baire neighbourhood of each point.

(3 ⇒ 4) Let X ⊆ Y be meager. Suppose, for a contradiction, that U is a nonempty definable subset of X open in Y, and let x ∈ U. Let V be a definable Baire neighbourhood of x, and W := V ∩ U. By Proposition 2.3, W is Baire, and therefore it is not meager in Y (by the same lemma), which is not possible.

(4 ⇒ 2) Let X ⊆ Y be meager. Hence, Y \ X is dense, and therefore X̊ = ∅.

(1 ⇔ 5) Use Proposition 2.3.

(1 ⇒ 6) Let C ⊆ Y be definable, closed and meager. If C̊ ≠ ∅, then C̊ is not meager, and thus C is not meager.

(6 ⇒ 1) Let U ⊆ Y be open, definable and meager in Y. Then, \overline{U} is also meager, because \overline{U} = U ⊔ bd U, and bd U is nowhere dense. Therefore, \overline{U} has empty interior, and therefore U is empty.

Remark 2.6. 𝕂^{n} is Baire iff it is not meager in itself.

Proof. One implication is obvious.

For the other implication, assume that 𝕂^{n} is not meager in itself, and let U ⊆ 𝕂^{n} be an open definable subset. If, for a contradiction, U were meager in itself, then we could find an open non empty box B ⊆ U. By Proposition 2.3, B is also meager in itself. However, B is definably homeomorphic to 𝕂^{n}, because 𝕂 expands a field, contradicting the hypothesis.

Def. complete and Baire, v. 4.1  3 F_{σ}-sets

The following result is not trivial and will be proved in Section 4.

Proposition 2.7. If 𝕂 is Baire, then for every m ≥ 1, 𝕂^{m} is Baire.

3 F_{σ}-sets

We now consider a class of sets for which it is easy to determine whether they are meager or not.

Definition 3.1. Let X ⊆ Y ⊆ 𝕂^{n}, with Y definable. X is in F_{σ} (in Y) if X is the union of a definable increasing family of closed subsets of Y, indexed by 𝕂. X is in G_{δ} if its complement is an F_{σ}.

Lemma 3.2. Let A be either the family of F_{σ} or the family of G_{δ} subsets of some 𝕂^{n}, for n ∈ ℕ. Then, each A ∈ A is definable. Moreover, A is closed under finite unions, finite intersections, Cartesian products, and preimages under definable continuous functions. Besides, the following are in A

1. definable closed subsets of 𝕂^{n};

2. definable open subsets of 𝕂^{n};

3. finite boolean combinations of definable open subsets of 𝕂^{n}.

The family of F_{σ} subsets is also closed under images under definable continuous functions.

Proof. Let A and B be in F_{σ}. The fact that A ∪ B and A × B are also in F_{σ} is obvious.

Let A = ⋃_{t} A(t) and B = ⋃_{t} B(t), where (A(t))_{t∈𝕂} and (B(t))_{t∈𝕂} are two definable increasing families of closed (and, we may assume, d-compact) sets.

Then, A ∩ B = ⋃_{t}(A(t) ∩ B(t)), because (A(t))_{t∈𝕂} and (B(t))_{t∈𝕂} are increasing families. Hence, A ∩ B is also in F_{σ}.

If f : 𝕂^{n} → 𝕂^{n′} is continuous, then f(A) = ⋃_{t} f(A(t)). For every t ∈ 𝕂, f(A(t)) is d-compact, because A(t) is d-compact, and therefore f(A) is in F_{σ}. A similar proof works for preimages.

Let U ⊆ 𝕂^{n} be open and definable, and C := 𝕂 \ U. For every r ∈ 𝕂^{+}, define U(r) := {x ∈ 𝕂^{n} : d(x, C) ≥ r}. Note that each U(r) is closed. Since U is open, U = ⋃_{r>0} U(r), and therefore U is in F_{σ}.

If D is a finite boolean combination of open definable subsets of 𝕂^{n}, then it is a finite union of sets of the form C_{i} ∩ U_{i}, for some definable sets C_{i} and U_{i}, such that each C_{i} is closed and each U_{i} is open. Hence, D is in F_{σ}.

The corresponding results for G_{δ} follow immediately by considering the complements.

If Y ⊆ 𝕂^{n} is definable, X ⊆ Y is an F_{σ}-subset of Y, and f : Y → Y is definable and continuous, it might not be true that f(X) is an F_{σ}. The point where the above proof breaks down for Y ≠ 𝕂^{n} is the fact that it is not necessarily true that every F_{σ} set X is an increasing definable union of d-compact sets.

Notice that, by Remark 1.14, every 𝕂-manifold is an F_{σ}-set.

Remark 3.3. If X ⊆ 𝕂^{n} is meager, then there exists a meager F_{σ}-set containing X.

Lemma 3.4. Let Y be definable and Baire, and D ⊆ Y. Assume that D is in F_{σ}. Then, D is meager iff D̊ = ∅.^{1}

Proof. If D̊ ≠ ∅, then, since Y is Baire, D cannot be meager. Conversely, assume that D is not meager. If D is in F_{σ}, then D = ⋃_{t} D(t), for some definable increasing family of closed subsets. Since D is not meager, at least one of the D(t), say D(t_{0}), is not meager. Hence, D̊(t_{0}) ≠ ∅ (otherwise, D(t_{0}) would be nowhere dense), and therefore D̊ ≠ ∅.

Note that if X ⊆ ℝ^{n} is in F_{σ} and of measure zero, then X is meager, but the converse is not true.

Remark 3.5. Let X ⊆ 𝕂^{n}. X is an F_{σ} iff X is of the form Π_{n}^{n+m}(Z) for some Z ⊆ 𝕂^{n+m} closed and definable.

Proof. The “if” direction follows from Lemma 3.2. For the other direction, let (X(t))_{t∈𝕂} be a definable increasing family of closed subsets of 𝕂^{n}, such that X = ⋃_{t∈𝕂} X(t). Define Z := ⨆_{t∈𝕂}(X(t) × {t}).

Notice that, if 𝕂 is o-minimal, then every X definable subset of 𝕂 is a finite Boolean combination of definable closed sets (because X is a finite union of cells), and therefore X is an F_{σ}.

We now give a local condition which is sufficient to prove that the image of an F_{σ}-set under a continuous definable function is meager.

Proposition 3.6. Let C ⊆ 𝕂^{m} × 𝕂^{n} be in F_{σ}, f : C → 𝕂^{d} be definable and continuous. Assume that for every y ∈ Π_{m}^{m+n}(C) there exists a neighbourhood V_{y} ⊆ 𝕂^{m} of y, such that f((V_{y} × 𝕂^{n}) ∩ C) is meager. Then, f(C) is meager.

1This is not true for G_{δ} sets: for instance, the set of irrational numbers in ℝ is a G_{δ} which is not meager (it is even residual), but has empty interior.

Proof. If 𝕂 is meager in itself, then by Proposition 2.7 there is nothing to prove. Hence, we may assume that 𝕂 is Baire.

We proceed by induction on m. The case m = 0 is clear, because if m = 0, then V_{0} = 𝕂^{0}.

Assume that we have already proved the conclusion for m − 1 (and every n). We want to prove it for m. First, we consider the case when C is d-compact. W.l.o.g., 0 ∈ C. For every r > 0 and y ∈ 𝕂^{m}, let T^{m}(y; r) ⊂ 𝕂^{m} be the closed hypercube of side 2r and center y, and S^{m}(y; r) be its boundary. Moreover, define D(r) := f(C ∩ (T^{m}(0; r) × 𝕂^{n})).

Note that f(C) = ⋃_{r} D(r), and that each D(r) is d-compact. Therefore, to prove that f(C) is meager, it suffices to prove that each D(r) has empty interior. Suppose, for a contradiction, that f(C) is not meager, and let

r_{0} := inf{r > 0 : int(D(r)) ≠ ∅}.

Since the D(r) are closed, r_{0} = inf{r > 0 : D(r) is not meager}. We have that 0 < r_{0} by hypothesis, and r_{0} < +∞ because f(C) is not meager.

Let P := Π_{m}^{n+m}(C). Since P is d-compact, if 𝕂 = ℝ, we could find y_{1}, …, y_{k} ∈ P such that P ⊆ V_{y_{1}} ∪ ⋯ ∪ V_{y_{k}}. In the general situation, we need another argument. Let 5δ_{0} be a Lebesgue number for the open cover {V_{y} : y ∈ P } of P (we may assume that δ_{0} is small in comparison with r_{0}); δ_{0} > 0 exists by Lemma 1.11.

Note that

T^{m}(0; r_{0} + δ_{0}/2) ⊆ T^{m}(0; r_{0} − δ_{0}/2) ∪ ⋃_{y∈S^{m}(0;r_{0})} T^{m}(y; δ_{0}),

hence

D(r_{0} + δ_{0}/2) ⊆ D(r_{0} − δ_{0}/2) ∪ ⋃_{y∈S^{m}(0;r_{0})} f(C ∩ (T^{m}(y; δ_{0}) × 𝕂^{n})).

By definition of r_{0}, we know that D(r_{0} + δ_{0}/2) is not meager, while D(r_{0} − δ_{0}/2) is meager. Hence, to obtain a contradiction, it suffices to show that ⋃_{y∈S^{m}(0;r_{0})} f(C ∩ (T^{m}(y; δ_{0}) ∩ 𝕂^{n})) is meager.

Note that S^{m}(0; r_{0}) is the finite union of the faces of the closed hypercube T^{m}(0; r_{0}): hence, we only need to show that for each face S of S^{m}(0; r_{0}) the set D := ⋃_{y∈S} f(C ∩ (T^{m}(y; δ_{0}) × 𝕂^{n})) is meager. W.l.o.g., we can assume that S is the “top” face {y ∈ T^{m}(0; r_{0}) : y_{m} = r_{0}} and we may identify S with T^{m−1}(0; r_{0}) × {r_{0}}.

Define

\begin{aligned}C′ & := C ∩ ⋃_{y∈S}(T^{m}(y; δ_{0}) × 𝕂^{n}), \\ f′ & := f ↾ C′. \end{aligned}

Claim. C′ and f′ satisfy the hypothesis of the proposition, with n′ = n + 1, m′ = m − 1, and V′_{z} = B(z; δ_{0}).

C′ is d-compact, and therefore it is in F_{σ}. Let P′ ⊆ 𝕂^{m−1} be the projection of C′ onto 𝕂^{m−1}; note that P′ is d-compact. Fix z ∈ P′; by definition, there exists t ∈ [r_{0} − δ_{0}, r_{0} + δ_{0}] such that y := (z, t) ∈ P. Notice that

C′ ∩(V′_{z} ×𝕂×𝕂^{n}) ⊆ C ∩(V′_{z} ×[r_{0} −δ_{0}, r_{0} +δ_{0}]×𝕂^{n}) ⊆ C ∩(T^{m}(y; 2δ_{0})×𝕂^{n}).

Since 5δ_{0} is a Lebesgue number for the cover {V_{y} : y ∈ P } of P, it follows that there exists y′ ∈ P such that T^{m}(y; 2δ_{0}) ⊂ V_{y′}. Putting everything together, we have that C′ ∩ (V′_{z} × 𝕂^{n+1}) ⊂ C ∩ (V_{y′} × 𝕂^{n}) and thus f′(C′ ∩ (V′_{z} × 𝕂^{n+1})) is meager, which proves the claim.

Therefore, by inductive hypothesis, f′(C′) is meager. However, D ⊆ f′(C′), and we reached a contradiction.

We now treat the general case when C is in F_{σ}. Note that C is an increasing union of d-compact sets C(t). For each t ∈ 𝕂, define D(t) := f(C(t)): note that each D(t) is d-compact. By the d-compact case, we can conclude that each D(t) is meager, and therefore nowhere dense. Thus, D = ⋃_{t} D(t) is meager.

Corollary 3.7. Let C ⊆ 𝕂^{m} be in F_{σ}, and f : C → 𝕂^{d} be definable and continuous. Assume that for every x ∈ C there exists V_{x} ⊆ C neighbourhood of x, such that f(C ∩ V_{x}) is meager. Then, f(C) is meager.

Proof. Apply the proposition to the case n = 0.

Corollary 3.8. Let W ⊆ 𝕂^{m} be a definable 𝕂-manifold, C ⊆ W be an F_{σ}- set (in W), and f : C → 𝕂^{d} be definable and continuous. Assume that for every x ∈ C there exists V_{x} ⊆ C neighbourhood of x (in C), such that f(V_{x}) is meager. Then, f(C) is meager.

Proof. Since W is a 𝕂-manifold, it is in F_{σ}. Since C is an F_{σ}-set in W, it is also an F_{σ}-set in 𝕂^{m}. Apply the previous corollary.

Corollary 3.9. Let C ⊆ 𝕂^{m} be an F_{σ}. If every x ∈ C has a neighbourhood V_{x} such that C ∩ V_{x} is meager, then C is meager.

Let ℝ̃ be the structure on the reals numbers, with a predicate for every subset of ℝ^{n}. Proposition 3.6 and the following corollaries are trivial for 𝕂 = ℝ̃, because an F_{σ}-subset C of ℝ̃^{n} has a countable basis of open sets; for instance, the hypothesis in Proposition 3.6 on f and C imply that f(C) is a countable union of meager sets, and hence meager.

Def. complete and Baire, v. 4.1  4 The Kuratowski-Ulam Theorem

4 The Kuratowski-Ulam Theorem

The main result of this section is the following theorem.

Theorem 4.1. Let D ⊆ 𝕂^{m+n}. For every x ∈ 𝕂^{m}, let D_{x} := {y ∈ 𝕂^{n} : (x, y) ∈ D} be the corresponding section of D. Let T := T^{m}(D) := {x ∈ 𝕂^{m} : D_{x} is meager in 𝕂^{n}}.

If D is meager (in 𝕂^{m+n}), then T is residual.

This is a definable version of Kuratowski-Ulam’s Theorem [Oxtoby80, Thm. 15.1], which in turn is an analogue of Fubini’s Theorem: they both imply that if D is negligible, then D_{y} is negligible for almost every y; in Kuratowski-Ulam’s Theorem negligible means “meager”, while in Fubini’s Theorem negligible means “of measure zero”.

It is not clear whether in the above theorem D definable implies that T is definable. Note that if 𝕂 is o-minimal and D is definable, then T is also definable.

As a corollary, we obtain Proposition 2.7.

Proof of Proposition 2.7. By induction on m. The case m = 1 is our assumption on 𝕂. Assume that we already proved that 𝕂^{m} is Baire: we want to prove that 𝕂^{m+1} is Baire. Suppose not; then 𝕂^{m+1} is meager in itself. If we apply Theorem 4.1 with n = 1, we obtain that either 𝕂^{m} or 𝕂 is meager in itself, a contradiction.

Definition 4.2. A definable function f : Y → 𝕂 is lower semi-continuous if, for every x ∈ Y, either x is an isolated point of Y, or

lim inf_{x′∈Y^{x′→x}} f(x′) ≥ f(x).

Remark 4.3. Let C ⊆ 𝕂^{n+1} be d-compact. For every x ∈ D := Π_{n}^{n+1}(C), let f(x) := min D_{x}. Then, f : D → 𝕂 is lower semi-continuous.

Lemma 4.4. Let Y ⊆ 𝕂^{n} be definable, f : Y → 𝕂 be lower semi-continuous and definable, and Δ ⊆ Y be the set of points of discontinuity of f. Then, Δ is meager (in Y).

Proof. By a change of variable in the co-domain, w.l.o.g. we can assume that f is bounded. For every ε > 0, let

Δ(ε) := {x ∈ Y : x is not isolated in Y & lim sup_{x′→x} f(x′) ≥ f(x) + ε}.

Since f is lower semi-continuous, we have that Δ = ⋃_{ε>0} Δ(ε). Hence, to prove the lemma it suffices to show that each Δ(ε) is nowhere dense. Fix ε > 0. Claim 1. Δ(ε) is closed.

In fact, let x_{0} ∈ \overline{Δ(ε)}. We have to prove that x_{0} ∈ Δ(ε). Assume not. It is clear that x_{0} is not an isolated point of Y. Let

s := lim sup_{x→x_{0}} f(x),
i := lim inf_{x→x_{0}} f(x).

Note that s and i are in 𝕂, because 𝕂 is definably complete. Since x_{0} ∉ Δ(ε), we have that s − ε < f(x_{0}). Hence, since f is lower semi-continuous,

f(x_{0}) ≤ i ≤ s < f(x_{0}) + ε ≤ i + ε;

let ρ := i + ε − s > 0.

Therefore, if x and x′ are sufficiently near x_{0}, then f(x) < s+ρ/2 (because s is the lim sup), and f(x) > s − ε + ρ/2 (because i = s − ε + ρ and i is the lim inf), and similarly for x′. Thus,

|f(x) − f(x′)| < (s + ρ/2) − (s − ε + ρ/2) = ε.  (1)

On the other hand, since x_{0} ∈ \overline{Δ(ε)}, we have that there exist x ∈ Δ(ε) near x_{0}. By definition of Δ(ε), there exists x′ near x (and therefore near x_{0}) such that f(x′) ≥ f(x) + ε, contradicting (1). Claim 2. int(Δ(ε)) = ∅.

Assume, for a contradiction that Δ(ε) contains a nonempty open subset U of Y. Let m := sup_{x∈U} f(x). Note that m ∈ 𝕂, because f is bounded.

Let x ∈ U be such that f(x) > m − ε/4. Since x ∈ Δ(ε), there exists x′ ∈ Y near x (and therefore in U) such that f(x′) > f(x) + ε/2. Hence, f(x′) > m + ε/4 > m, contradicting the definition of m.

The two claims imply that Δ(ε) is nowhere dense, and we are done.

In the above lemma, if Y = 𝕂 = ℝ, we can not conclude that Δ has measure zero. In fact, let C ⊆ ℝ be closed, with empty interior, and of positive measure, and f be the characteristic function of ℝ\C. Then, Δ = C, and therefore it is of positive measure.

On the other hand, it is always true that if f : 𝕂^{m} → 𝕂 is definable, then Δ is in F_{σ}. In fact, for every ε > 0 define

Λ(ε) := {x ∈ 𝕂^{m} : ∀δ > 0 ∃y ∈ B(x; δ) |f(y) − f(x)| ≥ ε}.

Note that Δ = ⋃_{ε>0} Λ(ε). While Λ(ε) might not be closed, we have that \overline{Λ(ε)} ⊆ Λ(ε/2). Therefore, Δ = ⋃_{ε>0} \overline{Λ(ε)}.

Proof of Theorem 4.1. If 𝕂^{m} is meager in itself, then the conclusion is trivially true, because then every subset of 𝕂^{m} is meager. Hence, we can assume that 𝕂^{m} is Baire.

Case 1: n = 1 and D is d-compact. Hence, D has empty interior, and each D_{x} is also d-compact. Therefore, by Lemma 3.4, T = {x ∈ 𝕂^{m} : D̊_{x} = ∅}. Let E := 𝕂^{m} \ T. We have to prove that E is meager.

For every ε > 0 let

X(ε) := {(x, y) ∈ 𝕂^{m} × 𝕂 : B^{1}(y; ε) ⊆ D_{x}}.

Let

E(ε) := π(X(ε)) = {x ∈ 𝕂^{m} : D_{x} contains a ball of radius ε}.

Note that X(ε) is d-compact, since its complement is the projection of an open set, therefore so is E(ε). Note that E = ⋃_{ε>0} E(ε); hence, to prove that E is meager, it suffices to prove that each E(ε) is nowhere dense. Since each E(ε) is d-compact, it suffices to prove the following claim. Claim 1. For every ε > 0, int(E(ε)) = ∅.

Assume, for a contradiction, that there exists a nonempty open box U ⊆ E(ε). Define

\begin{aligned}f : & U → 𝕂 \\ & x ↦ min\{y ∈ 𝕂 : (x, y) ∈ X(ε)\}. \end{aligned}

Note that f is lower semi-continuous and definable. By Lemma 4.4, f is continuous outside a meager set Δ ⊆ U. Since 𝕂^{m} is Baire, Δ ≠ U, and therefore there exists x_{0} ∈ U such that f is continuous at x_{0}. It is now easy to show that a neighbourhood of (x_{0}, f(x_{0})) is contained in D, contradicting the fact that D̊ = ∅.

Case 2: n = 1 and D arbitrary meager subset of 𝕂^{m}. Let (D(p))_{p∈𝕂} be an increasing definable family of d-compact subsets of 𝕂^{m+1} with empty interior, such that D ⊆ ⋃_{p} D(p). For each p ∈ 𝕂, let E(p) := {x ∈ 𝕂^{m} : D(p)_{x} is not meager in 𝕂}. By what we have seen above, E(p) = ⋃_{ε>0} E(p, ε), where (E(p, ε))_{p∈𝕂^{ε∈𝕂^{+}}} is a definable family of subsets

of 𝕂, increasing in p and decreasing in ε, such that each E(p, ε) is closed and nowhere dense. Let

E′ := ⋃_{ε,p} E(p, ε) = ⋃_{p} E(p).

Def. complete and Baire, v. 4.1  5 Almost open sets

Claim 2. 𝕂^{m} \ T ⊆ E′.

In fact, let x ∉ T. Thus, D_{x} is not meager. However, D_{x} ⊆ ⋃_{p} D(p)_{x}. Since (D(p)_{x})_{p∈𝕂} is an increasing definable family of closed subsets of 𝕂, we obtain that there exists p_{0} such that D(p_{0})_{x} has non-empty interior. Thus, x ∈ E(p_{0}) ⊆ E′.

Therefore, it suffices to prove that E′ is meager to obtain that T is residual. However, E′ = ⋃_{p>0} E(p, 1/p), and we are done.

Case 3: n > 1 and D arbitrary meager subset of 𝕂^{m}. We argue by induction on n. Suppose that we have already proved the conclusion for n (and for every m). We want to prove the conclusion for n + 1. First, we will assume that D is in F_{σ}. We want to prove that the set T := T^{m}(D) := {x ∈ 𝕂^{m} : D_{x} is meager} is residual. Define

S := 𝕂^{m+1} \ T^{m+1}(D) := {(x, y_{n+1}) ∈ 𝕂^{m} × 𝕂 : D_{(x,y_{n+1})} is not meager},
R := T^{m}(S) = {x ∈ 𝕂^{m} : S_{x} is meager}.

By Lemma 3.4, D_{(x,y_{n+1})} is meager iff its interior is empty, and therefore S = {(x, y_{n+1}) ∈ 𝕂^{m} × 𝕂 : int(D_{(x,y_{n+1})}) ≠ ∅} (and in particular S is definable). Claim 3. S is meager.

By inductive hypothesis. Claim 4. R is residual.

By the case n = 1 and the previous claim. Claim 5. R ⊆ T.

Fix x ∈ 𝕂^{m}. Assume that x ∉ T. We have to prove that x ∉ R. Define F := D_{x} ⊆ 𝕂^{n+1}. Note that F is in F_{σ}: therefore, since x ∉ T, F̊ ≠ ∅. Let U := U_{1} × U_{2} be a non-empty open box contained in F, U_{1} ⊆ 𝕂, U_{2} ⊆ 𝕂^{m}. For every y_{n+1} ∈ U_{1}, D_{(x,y_{n+1})} = F_{y_{n+1}} ⊇ U_{2}, and therefore (x, y_{n+1}) ∈ S. Thus, U_{1} ⊆ S_{x}, and x ∉ R.

Hence, T contains a residual set, and therefore it is residual.

For D arbitrary, let D′ ⊆ 𝕂^{m+n} be a meager F_{σ} containing D. By the previous case, the corresponding set T′ := T^{m}(D′) is residual. Since T′ ⊆ T, we are done.

5 Almost open sets

In this section we will assume that 𝕂 is definably complete and Baire.

Let Y ⊆ 𝕂^{m} be definable. We have seen that the family of meager subsets of Y is an ideal, hence it defines an equivalence relation on the family of subsets of Y, given by X ∼ X′ iff X Δ X′ is meager.

Remark 5.1. X ∼ X′ iff there exists Z meager such that X Δ Z = X′

Proof. Set Z := X Δ X′.

Definition 5.2. X ⊆ Y is almost open, or a.o. for short, if X is equivalent to a definable open set.^{2}

Lemma 5.3. Let Y ⊆ 𝕂^{m} be definable, and A and B be a.o. subsets of Y. Then, A ∩ B, A ∪ B and Y \ A are also a.o.. Moreover, F_{σ} and G_{δ} subsets of Y are a.o..

Finally, if Y_{1} and Y_{2} are definable, and A_{i} ⊆ Y_{i} are a.o. for i = 1, 2, then A_{1} × A_{2} is a.o. in Y_{1} × Y_{2}.

Proof. It is trivial to see that A ∩ B, A ∪ B and A_{1} × A_{2} are a.o..

Let A = U Δ E, where U is open and definable, and E is meager. Then, Y \ A = (Y \ U) Δ E. Hence, to prove that Y \ A is a.o. it suffices to prove that C := Y \ U is a.o.. However, C = C̊ ∪ bd(C). Since C is closed, bd(C) is nowhere dense, and a fortiori meager, and we are done.

Let (D(t))_{t∈𝕂} be a definable increasing sequence of closed subsets of Y. We have to prove that D := ⋃_{t} D(t) is a.o.. Let U := D̊ and E := D \ U. It is enough to prove that E is meager. For every t, let E(t) := E ∩ D(t). Note that D̊(t) ⊆ U; therefore, E(t) ⊆ bd(D(t)) is nowhere dense, and we are done.

Consequently, X ⊆ Y is a.o. iff it is equivalent to a definable closed subset of Y.

Remark 5.4. Every meager set is a.o., being equivalent to the empty set. By Lemma 5.3, every residual set is a.o..

Corollary 5.5. Let A ⊆ Y. The following are equivalent:

1. A is a.o.;

2. A is of the form E Δ F, for some meager set E and some set F in F_{σ};

3. A is of the form G ⊔ E, for some G in G_{δ} and E meager.

Proof. Cf.[Oxtoby80, Thm. 4.4]. (1 ⇔ 2) and (3 ⇒ 1) are obvious. For (1 ⇒ 3), let A = U Δ E for some U open and E meager. Let Q be a meager set in F_{σ} containing E, and G := U \ Q. Note that G is in G_{δ}, and

U Δ E = [(U \ Q) Δ(U ∩ Q)] Δ(E ∩ Q) = G Δ[(U Δ E) ∩ Q] = G ⊔ E′,

where E′ := (U Δ E) ∩ Q is meager.

2Almost open sets are called sets with the “property of Baire” in [Oxtoby80].

Def. complete and Baire, v. 4.1  6 Further results and open problems

The following is a partial converse of Theorem 4.1. Here it is important that 𝕂 be Baire.

Lemma 5.6. Let D be an a.o. subset of 𝕂^{m+n}, and T (D) := {x ∈ 𝕂^{m} : D_{x} is meager}. Then, D is meager iff T (D) is residual.

Proof. The “only if” direction is Theorem 4.1. For the other direction, let U be an open set such that E := D Δ U is meager. By Theorem 4.1, T (E) is residual. Moreover, since U_{x} = D_{x} Δ E_{x}, we have T (U) ⊇ T (D) ∩ T (E), and therefore T (U) is also residual. However, U is open and 𝕂^{n} is Baire: therefore, T (U) is the complement of the projection of U on 𝕂^{m}. Since U is open, T (U) is closed. Therefore, T (U) is closed and residual; since 𝕂^{m} is Baire, T (U) = 𝕂^{m}. Thus, U is empty, and we are done.

The hypothesis that D is a.o. in the above lemma is necessary: [Oxtoby80, Thm. 15.5] gives an example of a set E ⊆ ℝ^{2} that is not meager, such that no three points of E are collinear.

Note that, if D ⊆ 𝕂^{m+n} is in F_{σ}, then D is meager iff 𝕂^{m} \ T^{m}(D) has non-empty interior.

6 Further results and open problems

Remark 6.1. It follows from Remark 2.6 that the fact that 𝕂 is Baire can be expressed by a recursive set of first-order sentences: that is, every 𝕂′ elementary equivalent to 𝕂 also satisfies the hypothesis.

Notice that an ultra-product of definably complete (resp. Baire) structures is also definably complete (resp. Baire); the same cannot be said for “o-minimal” instead of “definably complete”.

Examples 6.2. • Every expansion of ℝ is definably Baire (because ℝ is topologically Baire).

• Every o-minimal expansion of a field is definably Baire (the union of a definable increasing family of nowhere dense sets is finite, and hence can not coincide with the whole structure).

• Let B be an o-minimal expansion of a field, let A ≼ B be a dense substructure. Then the structure B_{A}, generated by adding a unary predicate symbol for A, is definably Baire. This follows from the fact that if X ⊆ B is B_{A}-definable, then its topological closure \overline{X} is Bdefinable (see [vdDries98, Theorem 4]). Hence, as above, the union of a definable increasing family of closed nowhere dense sets is finite.

Notice that the structures considered in all of the above examples are also definably complete.

Open problem 6.3. It is not known to the authors if there exists a definably complete structure which is not Baire.

Assume that 𝕂 is definably complete and Baire.

Open problem 6.4. Let (Y (t))_{t∈𝕂} be a definable increasing family of meager subsets of 𝕂^{m}, and let Y := ⋃_{t} Y_{t}. Is Y necessarily meager? In particular, is it necessarily Y ≠ 𝕂^{m}?

Notice that, if the Y (t) are closed, then Y is meager, whereas the same conclusion does not necessarily hold if the Y (t) are in F_{σ} (actually, since every meager set is contained in a meager F_{σ}-set, it is enough to reduce to this situation).

Moreover, the above question has positive answer if 𝕂 is o-minimal, because then each Y_{t} has (o-minimal) dimension less than m, and therefore Y has dimension less than m. In fact, if 𝕂 is o-minimal, and Y ⊆ 𝕂^{m} is definable, then Y is meager iff dim Y < m; moreover, if (Y (t))_{0<t∈𝕂} is a definable family, decreasing in t, then ⋃_{t} Y (t) ⊆ acc_{t→0} Y (t). Thus, the following lemma proves what we want.

Lemma 6.5. Let 𝕂 be an o-minimal structure, n ≤ m ∈ ℕ, and (Y (t))_{t>0} be a definable family of subsets of 𝕂^{m}, and Z := acc_{t→0} Y (t). If, for every t > 0, dim(Y (t)) ≤ n, then dim(Z) ≤ n.

Proof. Define W := ⋃_{t>0} Y (t) × \{t\} ⊆ 𝕂^{m+1}. Note that Z = (\overline{W})_{0} := \{x ∈ 𝕂^{m} : (x, 0) ∈ \overline{W} \}. Moreover, since dim Y (t) ≤ n, we have dim W ≤ n + 1. Since Z × \{0\} ⊆ ∂W := \overline{W} \ W, we have dim Z < dim W ≤ n + 1.

The following is a partial result for the case of a.o. sets.

Lemma 6.6. Let Y ⊆ 𝕂^{n} be definable and Baire, D ⊆ Y be a.o. (in Y), and (Y (t))_{t∈𝕂} be a definable increasing family of closed subsets of Y, such that Y = ⋃_{t} Y (t). Then, D is meager in Y iff each D ∩ Y (t) is meager (in Y).

Proof. The “only if” direction is clear.

For the other direction, let C ⊆ Y be closed, such that E := C Δ D is meager. It suffices to prove that C is meager. For every t ∈ 𝕂, define

\begin{aligned}C(t) & := C ∩ Y (t), \\ D(t) & := D ∩ Y (t). \end{aligned}

Then, D(t) Δ C(t) ⊆ E. Therefore, since D(t) and E are meager, C(t) is meager and closed. Since Y is Baire, C(t) is nowhere dense, and thus C is meager.

Let C ⊆ 𝕂^{n} be meager, and f : 𝕂^{n} → 𝕂^{n} be definable and C^{1}. We want to investigate in which circumstances f(C) is meager. When 𝕂 = ℝ, Sard’s Lemma implies that f(C) is meager. This suggests the following definition.

Definition 6.7. Fix d, r, m ∈ ℕ. Let V ⊆ 𝕂^{d} be a 𝕂-manifold of dimension n. Let f : V → 𝕂^{m} be a definable C^{r} function and Δ_{f} be the set of singular points of f. If Σ_{f} := f(Δ_{f}) is meager in 𝕂^{m}, then we say that f has the Sard property.

Open problem 6.8. Does every C^{r} definable function f : V → 𝕂^{m} as above (with r > max{0, n − m}) have the Sard property?

Remark 6.9. If 𝕂 is o-minimal, then every C^{1} definable function f : V → 𝕂^{m} has the Sard property [Ber–Ot01, Thm. 3.5].

Lemma 6.10. If 𝕂 = ℝ and f : V → 𝕂^{m} is as in the above definition, with r > max{0, n − m}, then f has the Sard property.

Proof. By Sard’s Lemma, Σ_{f} has Lebesgue measure zero, and therefore it has empty interior. Since Σ_{f} is in F_{σ}, it is also meager.

Proposition 6.11. Suppose f : 𝕂^{n} → 𝕂^{m} has the Sard property, and let C ⊂ 𝕂^{n} be meager. Then f(C) is meager.

Proof. We may assume that C ∈ F_{σ}, since C is contained in a meager F_{σ}-set. Let Λ := 𝕂^{n} \ Δ_{f} be the set of regular points of f. Note that Λ is open.

By the Sard property, f(C ∩ Δ_{f}) is meager. Hence, it suffices to show that f(C ∩ Λ) is meager. Let x ∈ C ∩ Λ. Since x is a regular point for f, by the Implicit Function Theorem there exists a neighbourhood V of x such that f is a diffeomorphism on V; therefore, f(C ∩ V) is meager, and, by Corollary 3.7, f(C ∩ Λ) is meager.

The following lemma is an analogue of the Co-area Formula.

Lemma 6.12. Let f : 𝕂^{n} → 𝕂 be a C^{∞} definable function with the Sard property. Let Λ be the set of the regular points of f, and C ⊆ 𝕂^{n} be in F_{σ}. For every t ∈ 𝕂, let F_{t} := f^{−1}(t), C_{t} := F_{t} ∩ C, and T := {t ∈ 𝕂 : C_{t} is meager in F_{t}}. Then, T is residual iff C ∩ Λ is meager.

Proof. If n = 0 we have a tautology. Assume that n ≥ 1.

Let x ∈ C ∩ Λ. Since x is a regular point for f, there exists V open neighbourhood of x such that, up to a change of coordinates, f is the projection on the last coordinate x_{n}. For every x_{n} ∈ T, the set C_{x_{n}} ∩ V is meager in 𝕂^{n−1} × {x_{n}}.

Hence, if T is residual, then, by Kuratowski-Ulam’s Theorem 4.1, C ∩ V is meager; therefore, by Corollary 3.7, C ∩ Λ is meager.

Conversely, assume that C∩Λ is meager; we must prove that T is residual. Since f(Δ_{f}) is meager, it suffices to prove that T (C∩Λ) is residual. Therefore, w.l.o.g. we can assume that C ⊆ Λ. Again by Kuratowski-Ulam’s Theorem, the set T (V ∩ C) := {x_{n} ∈ 𝕂 : C_{x_{n}} ∩ V is meager in 𝕂^{n−1}} is residual. Therefore, T is residual.

In the following subsection we produce examples of classes of functions in definably complete and Baire structures, which have the Sard property.

6.1 The Sard property and Noetherian Differential Rings

Notation 6.13. Fix n ∈ ℕ \ {0} and a definably connected definable open set U ⊆ 𝕂^{n}. Let C^{∞}(U, 𝕂) be the ring of definable C^{∞} functions from U to 𝕂.

Definition 6.14. A ring M with the following properties

• M ⊆ C^{∞}(U, 𝕂);

• M is noetherian;

• M is closed under partial differentiation;

• M ⊇ ℤ[x_{1}, …, x_{n}].

is called a Noetherian differential ring.

If G := (g_{1}, …, g_{k}) ∈ M^{k}, we denote by V (G) the set of zeroes of G, and by V^{reg}(G) the set of regular zeroes of G.

Generalities on Noetherian differential rings of functions over definably complete structures can be found in [Servi07]. In particular, we will need the following result, which states that in a Noetherian differential ring there are no flat functions.

Proposition 6.15. Let M ⊆ C^{∞}(U, 𝕂) be a Noetherian differential ring and let 0 ≢ g ∈ M. Then for every x ∈ U such that g(x) = 0, there exist k ∈ ℕ and a derivative θ of order k such that θg(x) ≠ 0.

Fix a Noetherian differential ring M ⊆ C^{∞}(U, 𝕂).

Remark 6.16. For g_{1}, …, g_{k} ∈ M, the set V := V^{reg}(g_{1}, …, g_{k}) is in F_{σ}; in fact consider the following closed definable subset of U × 𝕂:

C := ⋃_{E(x)}{(x, y) ∈ U × 𝕂 : ⋀_{i=1}^{k} g_{i}(x) = 0 ∧ det(E(x))y − 1 = 0},

where E(x) ranges over all maximal rank minors of the Jacobian matrix of (g_{1}, …, g_{k}) in x. Now, V = Π_{n}^{n+1}(C); since C is an F_{σ} of 𝕂^{n+1} and Π_{n}^{n+1} is continuous, V is also an F_{σ}.

In this subsection we prove the following version of Sard’s Lemma:

Theorem 6.17. Fix k, m ∈ ℕ, k ≤ n. Let

• H = (h_{1}, …, h_{n−k}) ∈ M^{n−k} and V := V^{reg}(H) ≠ ∅;

• F = (F_{1}, …, F_{m}) ∈ M^{m} and f := F ↾ V : V → 𝕂^{m};

• Δ_{f} ⊆ V be the set of singular points of f, and Σ_{f} := f(Δ_{f}) be the set of singular values of f.

Then, Σ_{f} is a meager set (in 𝕂^{m}).

Proof. We proceed by induction on dim V and m. If m = 0, there are no singular points. If dim V = 0, then V is discrete. In particular, for every a ∈ Δ_{f} there exists U_{a} neighbourhood of a such that Δ_{f} ∩ U_{a} = {a}. Hence we can apply Corollary 3.7 and we are done.

Consider now the general case. Claim 1. We can restrict to the case V = 𝕂^{k}. By Corollary 3.7, it suffice to prove that for every a ∈ Δ_{f} there exists a neighbourhood U_{a} of a such that f(U_{a} ∩ Δ_{f}) is meager. Fix a ∈ Δ_{f}. Using the Implicit Function Theorem, it is easy to check that there is a neighbourhood U_{a} of a and a definable diffeomorphism Φ : 𝕂^{k} → V ∩ U_{a} such that H ∘ Φ ≡ 0 and each F_{i} ∘ Φ belong to a Noetherian differential ring M′ ⊆ C^{∞}(𝕂^{k}, 𝕂) (see [Servi07] for the details). Hence Claim 1 is proved and we may assume that f : 𝕂^{k} → 𝕂^{m}, and f ∈ M ⊆ C^{∞}(𝕂^{k}, 𝕂).

Let X_{0} := {a ∈ Δ_{f} : Df(a) ≠ 0}, where Df is the Jacobian matrix of f. We first prove that f(X_{0}) is meager.

Again by Corollary 3.7, it suffice to prove that for every a ∈ X_{0} there exists a neighbourhood U_{a} of a such that f(U_{a} ∩ X_{0}) is meager.

Fix a ∈ X_{0}. Claim 2. We may assume that f(x) = (x_{1}, f_{2}(x), …, f_{m}(x)). In fact, since Df(a) ≠ 0, w.l.o.g. we can assume that ∂f_{1}(a)/∂x_{1} ≠ 0 and a = 0.

Def. complete and Baire, v. 4.1  References

Consider definable neighbourhoods O and Õ ⊂ 𝕂^{k} of 0, where the following map is a diffeomorphism:

\begin{aligned}G : & O & → & Õ \\ & x & ↦ & (f_{1}(x), x_{2}, …, x_{k}). \end{aligned}

The ring M̃ := {g ∘ G^{−1}| g ∈ M} ⊂ C^{∞}(Õ, 𝕂) is clearly Noetherian and differentially closed and f̃:= f ∘ G^{−1} ∈ M̃. Since G is a diffeomorphism, it is enough to prove the statement for M̃ and f̃, and Claim 2 is proved.

For every t ∈ 𝕂, consider the Noetherian differential ring

N_{t} := {g_{t} := g(t, x_{2}, …, x_{k})| g ∈ M} ⊂ C^{∞}(Õ ∩ 𝕂^{k−1}, 𝕂).

Let f_{t} : 𝕂^{k−1} → 𝕂^{m−1} be the map ((f_{2})_{t}, …, (f_{m})_{t}). By inductive hypothesis, the set Σ_{f_{t}} is meager in 𝕂^{m−1}. Moreover, f(X_{0}∩Õ)∩({t}×𝕂^{m−1}) ⊆ {t}×Σ_{f_{t}}. Hence f(X_{0} ∩ Õ) ⊆ D := {(t, y) ∈ 𝕂 × 𝕂^{k−1}| y ∈ Σ_{f_{t}}}. By what we have just observed, T (D) := {t ∈ 𝕂 : D_{t} is meager} is residual, because D_{t} = Σ_{f_{t}}, hence by Lemma 5.6, D is meager. It follows by Corollary 3.7 that f(X_{0}) is meager.

Now, let a ∈ Δ_{f} such that Df(a) = 0, and let P be the least natural number such that there exists i ≤ m and a derivative θ of order P such that, if g_{θ} := θf_{i}, then g_{θ}(a) = 0 and Dg_{θ}(a) ≠ 0. Such a P exists by Proposition 6.15. Let W_{θ} := V^{reg}(g_{θ}) ⊂ 𝕂^{k} (notice that the inclusion is strict, hence dim W_{θ} < k). Then there is a definable open neighbourhood O of a such that

Δ_{f} ∩ O ⊆ ⋃_{ord(θ)≤P} W_{θ}.

Hence it is enough to prove that f(Δ_{f} ∩ W_{θ}) is meager. Let h_{θ} := f ↾ W_{θ}. By inductive hypothesis, Σ_{h_{θ}} is meager. Note that if x ∈ W_{θ} is a singular point for f, then x is also a singular point for h_{θ}; that is, Δ_{f} ∩ W_{θ} ⊆ Δ_{h_{θ}}, and we are done.

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