Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition

Annals of Pure and Applied Logic 171 (2020) 102756

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Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition Masato Fujita Department of Liberal Arts, Japan Coast Guard Academy, 5-1 Wakaba-cho, Kure, Hiroshima 737-8512, Japan

a r t i c l e

i n f o

Article history: Received 12 June 2019 Received in revised form 2 November 2019 Accepted 4 November 2019 Available online 13 November 2019 MSC: primary 03C64

a b s t r a c t We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which admits local definable cell decomposition. © 2019 Elsevier B.V. All rights reserved.

Keywords: Uniformly locally o-minimal structure Local monotonicity theorem Local definable cell decomposition

1. Introduction An o-minimal structure enjoys tame properties such as monotonicity and definable cell decomposition [5,9,12]. Structures similar to o-minimal structures have been proposed and investigated such as weakly o-minimal structures [6,13], locally o-minimal structures [11] and structures having (locally) o-minimal open cores [2,3]. However, the tame properties enjoyed by o-minimal structures are generally unavailable in them. For instance, a weakly o-minimal structure has monotonicity [6,1], but definable cell decomposition is not available. Local o-minimal structure localizes the definition of o-minimal structures, but local monotonicity and local definable cell decomposition are unavailable. A geometric counterpart to a locally o-minimal structure [10] enjoys the tame properties, but the projection image of a definable set is required to be definable only when their fibers are compact. Toffalori and Vozoris proposed more stronger condition than local o-minimality and called it strongly local o-minimality [11]. As demonstrated in [4], strongly locally o-minimal structures have local monotonicity and

E-mail address: [email protected]. https://doi.org/10.1016/j.apal.2019.102756 0168-0072/© 2019 Elsevier B.V. All rights reserved.

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local definable cell decomposition. However, the definition of strongly local o-minimality seems to be too restrictive. Local monotonicity and local definable cell decomposition are available in a locally o-minimal structure with more relaxed condition than strongly local o-minimality. This is the reason why we propose a new structure, a uniformly locally o-minimal structure of the second kind. A uniformly locally o-minimal structure of the second kind has local monotonicity, and it admits local definable cell decomposition if it is definably complete. Kawakami et al. already used the term ‘uniformly locally o-minimal’ in [4], but it is not identical to the definition of uniformly local o-minimality in this paper. The definition proposed in this paper is called ‘of the second kind’ so as to distinguish ours from Kawakami’s definition. We also investigate locally o-minimal structures which admit local definable cell decomposition. Dimensions of definable sets are defined and we deduce their basic properties. This paper is organized as follows: We first give the definition and examples of uniformly locally o-minimal structures of the second kind in Section 2. Section 3 is devoted to local monotonicity theorems, Theorem 3.2 and Corollary 3.1. A local definable cell decomposition theorem, Theorem 4.2, is proved in Section 4 for definably complete uniformly locally o-minimal structures of the second kind. We define dimensions of definable sets and investigate their basic properties when the given structure is a locally o-minimal structure which admits local definable cell decomposition theorem in Section 5. We introduce the terms and notations used in this paper. The term ‘definable’ means ‘definable in the given structure with parameters’ in this paper. The notation ThL (M) denotes the set of all sentences which are valid in a structure M. For any set X ⊂ M n+1 definable in a structure M = (M, . . .) and for any y ∈ M n , the notation Xy denotes the fiber defined as {x ∈ M | (x, y) ∈ X} unless another definition is explicitly given. For a linearly ordered structure M = (M, <, . . .), an open interval is a definable set of the form {x ∈ M | a < x < b} for some a, b ∈ M . It is denoted by ]a, b[ in this paper. An open box in M n is the direct product of n open intervals. Let A be a subset of a topological space. The notations int(A) and A denote the interior and the closure of the set A, respectively. The boundary bd(A) of A is given by A \ int(A). The frontier ∂A of A is defined by A \ A. The notation |S| denotes the cardinality of a set S. 2. Definitions We review definitions on local o-minimality and introduce uniformly local o-miniality of the second kind. Definition 2.1. A densely linearly ordered structure M = (M, <, . . .) is definably complete if every definable subset of M has both a supremum and an infimum in M ∪ {±∞} [8]. A definable set is definably connected if it is not a disjoint union of two nonempty definable open subsets. A densely linearly ordered structure M = (M, <, . . .) is locally o-minimal if, for every definable subset X of M and for every point a ∈ M , there exists an open interval I containing the point a such that X ∩ I is a finite union of points and open intervals. A locally o-minimal structure M = (M, <, . . .) is strongly locally o-minimal if, for every point a ∈ M , there exists an open interval I containing the point a such that X ∩ I is a finite union of points and open intervals for every definable subset X of M . A locally o-minimal structure M = (M, <, . . .) is a uniformly locally o-minimal structure of the first kind if, for any positive integer n, any definable set X ⊂ M n+1 and a ∈ M , there exists an open interval I containing the point a such that the definable sets Xy ∩ I are finite unions of points and open intervals for all y ∈ M n . A uniformly locally o-minimal structure of the first kind is called a uniformly locally o-minimal structure in [4]. A locally o-minimal structure M = (M, <, . . .) is a uniformly locally o-minimal structure of the second kind if, for any positive integer n, any definable set X ⊂ M n+1 , a ∈ M and b ∈ M n , there exist an open interval I containing the point a and an open box B containing b such that the definable sets Xy ∩ I are finite unions of points and open intervals for all y ∈ B.

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It is obvious that a uniformly locally o-minimal structure of the first kind is a uniformly locally o-minimal structure of the second kind. The converse is not generally true as illustrated in the following example. Example 2.2. For each positive q ∈ Q, we prepare a binary predicate Pq (x, y). The language is L = {<, Pq }q∈Q+ , where Q+ denotes the set of all positive rational numbers. We define an L-structure M = (Q,
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{xi = xj }, {xi > π k · xj }, {xi < π k · xj }, {xi = q}, {xi > π k · q} and {xi < π k · q}, where 1 ≤ i, j ≤ n, k are integers and q are rational numbers. The notation {xi = xj } denotes the set {(x1 , . . . , xn ) ∈ Qn | xi = xj }. The other notations are similar. Proof. The claim is obviously satisfied when X is defined by a quantifier-free formula. It is also obvious that, if the definable sets X and Y satisfy the claim, the intersection X ∩ Y and the complement Qn \ X satisfy the claim. It suffices to show that the projection image Π(X) satisfies the claim when the definable set X ⊂ Qn satisfies the claim. The notation Π denotes the projection of Qn onto Qn−1 forgetting the last coordinate. We prove it by the induction on n. It is obvious when n = 1. We consider the case in which n > 1. Let {qj }N j=1 be the rational numbers appearing in the formula defining X. Define definable subsets Yij , Zij and X  as follows: Yij = X ∩ {xi = xj }, Zij = X ∩ {xi = qj } and ⎛   X  = X \ ⎝ Yij ∪ i
⎞ Zij ⎠ .

1≤i≤n,1≤j≤N

  We have only to show the claim for Π(Yij ), Π(Zij ) and Π(X  ) because X = X  ∪ i π k · xj , xn < π k · xj , xn > π k · q and xn < π k · q from the inequalities defining the definable set X. Let K be the number of such inequalities and rl (x1 , . . . , xn−1 ) be the right hand of the l-th inequality. The functions rl (x) are of the forms π k · xj or π k · qj for some integers k. Set Wσ = X  ∩ {(x, xn ) ∈ M n−1 × M | rσ(1) (x) < rσ(2) (x) < · · · < rσ(K) (x)} for all the permutations σ of the set {1, . . . , K}. The projection image Π(Wσ ) is empty or defined by the inequalities used in the definition of X such that the variable xn is not involved and by the inequalities rσ(1) (x) < rσ(2) (x) < · · · < rσ(K) (x). Hence, the claim is valid for Π(Wσ ). The claim holds true also for the definable set Π(X  ) because the set Π(X  ) is a finite union of definable sets of the form Π(Wσ ). 2 We demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind in Section 4. A local definable cell decomposition theorem for strongly locally o-minimal structures is already available in [4]. We give an example of a definably complete uniformly locally o-minimal structure of the first kind (and trivially of the second kind) which is not strongly locally o-minimal. Example 2.4. Let N denote the set of positive integers. Consider an ultrafilter U of N containing the sets {n ∈ N | n > N } for all N ∈ N. The notation F is the family of all maps from N to Q+ . We prepare a family of unary predicate symbols {Pf }f ∈F . Set L = {<, {Pf }f ∈F } and define L-structures Sn = (R,
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• γ}, {x ∈ R | x = γ} and {x ∈ R | x < γ}, where γ is one of b1 , . . . , bd , c, αj,u and βj,u . In particular, (Yj )b ∩]αj,u , βj,u [ is the union of at most (d + 1) points and at most (d + 2) open intervals. lj For all 1 ≤ j ≤ k, there exist finite points uj,1 , . . . , uj,lj in [0, 1] such that {Uj,up }p=1 are open coverings



max αj,uj,p and β = min min αj,uj,p . Then, they satisfy of the compact set [0, 1]d . Set α = max 1≤j≤k

1≤p≤lj

1≤j≤k

1≤p≤lj

the condition in Claim 2. It is obvious from the following claim that M is uniformly locally o-minimal of the first kind. Claim 3. Claim 2 remains true if we replace R and S in the statement with M and M, respectively. There are an L-formula ϕ(x, y) and parameters a ∈ M p such that X is defined by the formula ϕ(x, a). It is obvious that, if the claim is true for ϕ(x, y), the claim also holds true for ϕ(x, a). We may assume

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that X is defined by a formula without parameters. Claim 3 immediately follows from Claim 2 by the Łoś’s theorem because the statement of Claim 2 is represented by a first-order sentence. We finally demonstrate that M is not strongly o-minimal. Fix arbitrary α, β ∈ M with α < 0 < β. The notation 0 denotes the equivalence class of the series (0, 0, . . .) in the ultraproduct M . We have to find a definable set whose intersection with ]α, β[ is not a finite union of points and open intervals. The elements α and β are equivalence classes of some sequences (a1 , a2 , . . .) and (b1 , b2 , . . .), respectively. Set U = {n ∈ N | an < 0 < bn }, then we have U ∈ U. Take a positive rational number qn with an < −qn < 0 < qn < bn for any n ∈ U . Consider the map f : N → Q+ given by f(n) =

n/qn if n ∈ U and 1

elsewhere.

At least (2n + 1) points x in R satisfy the conditions that an < x < bn and Sn |= P S n (x) for any n ∈ U . f

It is obvious that the set {x ∈ M | M |= PfM  (x)} is discrete using the Łoś’s theorem. The remaining

task is to demonstrate that the definable set C = {x ∈ M | α < x < β, M |= PfM  (x)} is infinite. Assume the contrary and that C consists of K points for some positive integer K. The set  

   V = n ∈ N | an < 0 < bn , {x ∈]an , bn [ | Sn |= P S n (x)} ≤ K f

is an element of the ultrafilter U by the Łoś’s theorem. On the other hand, we have U ∩ V ∩ {n ∈ N | n > K} = ∅. It is a contradiction to the definition of U. We have finished the proof. 2 We demonstrate that a uniformly locally o-minimal ordered field of the second kind is o-minimal. Proposition 2.1. A uniformly locally o-minimal ordered field of the second kind is o-minimal and real closed. Proof. Let M = (M, <, +, ·, 0, 1, . . .) be a uniformly locally o-minimal ordered field of the second kind. Let X be a definable subset of M . We show that X is a finite union of points and open intervals. We first consider the case in which X is bounded. Consider the set Y = {(x, r) ∈ M 2 | r > 0, x/r ∈ X}. The fiber Yr = {x ∈ M | x/r ∈ X} is a finite union of points and open intervals for any sufficiently small r > 0 because M is uniformly locally o-minimal of the second kind and X is bounded. The set X is also a finite union of points and open intervals. When X is unbounded, set X1 = X ∩ [−1, 1] and X2 = X \ X1 . The set X1 is a finite union of points and open intervals because X1 is bounded. Consider the set Z = {x ∈ M | 1/x ∈ X2 }. It is bounded and a finite union of points and open intervals. Therefore, X2 is also a finite union of points and open intervals. The ordered field M is a real closed field by [9, Theorem 2.3]. 2 According to [11, Lemma 6.3], a strongly locally o-minimal ordered field is real closed. We can get a stronger result using Proposition 2.1. A strongly locally o-minimal ordered field is o-minimal because a strongly locally o-minimal structure is a uniformly locally o-minimal structure of the second kind. 3. Local monotonicity theorem The purpose of this section is to prove the parameterized local monotonicity theorem given in Theorem 3.2. We review the definition of local monotonicity. Definition 3.1 (Local monotonicity). A function f defined on an open interval I is locally constant if, for any x ∈ I, there exists an open interval J such that x ∈ J ⊂ I and the restriction f |J of f to J is constant.

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A function f defined on an open interval I is locally strictly increasing if, for any x ∈ I, there exists an open interval J such that x ∈ J ⊂ I and f is strictly increasing on the interval J. We define a locally strictly decreasing function similarly. A locally strictly monotone function is a locally strictly increasing function or a locally strictly decreasing function. A locally monotone function is locally strictly monotone or locally constant. The following three lemmas are necessary for the proof of the parameterized local monotonicity theorem. Lemma 3.1. Let M = (M, <, . . .) be a uniformly locally o-minimal structure of the second kind. Let f : I → M be a definable function on an open interval I. Assume that, for any a ∈ I, there exists an open interval Ia such that a ∈ Ia ⊂ I, f (x) < f (a) for all x ∈ Ia with x < a and f (x) > f (a) for all x ∈ Ia with x > a. Then, f is locally strictly increasing. Proof. Let a ∈ I be an arbitrary point. We consider the definable set X = {(s, x) ∈ I 2 | s > x and f (x ) > f (x) for all x ∈ I with x < x ≤ s}. Since M is a uniformly locally o-minimal structure of the second kind, there exists an open interval J such that a ∈ J ⊂ I and Xx ∩ J is a finite union of points and open intervals. We may assume that the closure of J is contained in I by shrinking J if necessary. Set s(x) = sup{s ∈ J | s ∈ Xx ∩ J}. The function s(x) is well-defined without assuming that M is definably complete because Xx ∩ J is a finite union of points and open intervals. We show that the value s(x) coincides with the right endpoint of the interval J. It means that f is strictly increasing on J. Assume that s(x) is not the right endpoint of J. By the assumption of the lemma, we have f (x1) < f (s(x)) < f (x2 ) for all x1 < s(x) and x2 > s(x) sufficiently close to s(x). Fix points x1 and x2 sufficiently close to s(x) with x1 < s(x) < x2 . We have f (x) < f (x1 ) by the definition of s(x). We have f (x) < f (x ) for all x with x < x < x2 . It contradicts the definition of s(x). We have shown that f is strictly increasing on the interval J. 2 Lemma 3.2. Let M = (M, <, . . .) be a uniformly locally o-minimal structure of the second kind. No injective definable functions defined on open intervals have the local minimum throughout the intervals. Proof. We prove the lemma in the same way as weakly o-minimal structures [1]. We lead a contradiction assuming that f : I → M is an injective definable function on an open interval I which have the local minimum throughout I. For any a ∈ I, set Ua as follows: Ua = {x ∈ I | x > a and f (y) > f (a) for all a < y ≤ x} ∪ {a}∪ {x ∈ M | x < a and f (y) > f (a) for all x ≤ y < a}. The definable set Ua is convex and contains a neighborhood of the point a because f is injective and locally minimal at the point a. The notation a ≺ b denotes the relation Ua  Ub . Since f is injective, we have Ua = Ub if a = b. The relation ≺ is a partial order. We first show the following equivalence: a ≺ b ⇔ b ∈ Ua .

(1)

The implication ⇒ is obvious. We demonstrate the opposite implication. We have only to show that Ub ⊂ Ua . We have f (a) < f (b) by the definition of Ua . Assume that a < b. We can show in the same way when b < a.

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Take an arbitrary point c ∈ Ub . If c < a, we have f (d) > f (b) for any c ≤ d < a. We have f (d) > f (b) > f (a). It means that c ∈ Ua . We can show that c ∈ Ua when c > b similarly. Since Ua is convex, c ∈ Ua when a ≤ c ≤ b. We have proven the equivalence (1). Secondly, we consider the following claim: Ua ∩ Ub = ∅ ⇒ a ≺ b or b ≺ a.

(2)

We may assume that a < b by the symmetry. We consider the case in which f (a) < f (b). We can prove in the same way when f (a) > f (b). We have a ∈ / Ub . Let c ∈ Ua ∩ Ub . We have only to consider the case in which a < c < b because we easily get a ∈ Ub or b ∈ Ua in the other cases using the fact that Ua and Ub are convex. We can demonstrate that an element d ∈ Ub is contained in Ua elementally considering the cases in which d < a, a ≤ d ≤ c and d > c, separately. We skip the proof. We have Ub  Ua ; that is, that a ≺ b. As a direct corollary of the claim (2), we get the following implication: b ≺ a and c ≺ a ⇒ b ≺ c or c ≺ b.

(3)

a ≺ b ≺ c and a < c ⇒ a ≤ b,

(4)

a ≺ b ≺ c and a > c ⇒ a ≥ b and

(5)

a ≺ b ≺ c and a < b ⇒ a < c.

(6)

It is also easy to prove that

We next show that, shrinking the interval I if necessary, the definable set Ca = {x ∈ I | x ≺ a} is a finite set for any a ∈ I. Consider the definable set C = {(a, x) ∈ I 2 | x ∈ Ca }. Shrinking the interval I if necessary, we may assume that Ca is a finite union of points and open intervals because M is a uniformly locally o-minimal structure of the second kind. We lead to a contradiction assuming that Ca contains an open interval J for some a ∈ I. Let b ∈ J. Take c, d ∈ Ub ∩ J with c < b < d. Since c, d ∈ Ca , we have c ≺ a and d ≺ a; hence, we get c ≺ d or d ≺ c by the claim (3). We only consider the case in which c ≺ d because the proof is similar when d ≺ c. The point b is an element of Uc because Uc is convex, c < b < d and d ∈ Uc . It is a contradiction to the conditions that c ∈ Ub and f is injective. Consider the following sets: K = {x ∈ I | y ⊀ x for all y ∈ I} and  a = {x ∈ I | a ≺ x and y ∈ I a ≺ y ≺ x}, where a is an element of I. Using the claim (3) and the fact that Ca is finite, we can easily show the following equality: I \K =

 ˙

 a.

(7)

a∈I

 The symbol ˙ represents disjoint union. We can also demonstrate that neither K nor  a contain an open interval without difficulty. The definable sets K and  a are locally finite because M is locally o-minimal. Since M is a uniformly locally o-minimal structure of the second kind, shrinking I if necessary, we may assume that  a is a finite set. In fact, consider the definable set

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X = {(a, x, α, β) ∈ I 4 | α < β, a ≺ x, α < a < β, α < x < β, y α < y < β and a ≺ y ≺ x}. Taking a sufficiently small subinterval I  of I, the definable set X(a,α,β) ∩ I  is a finite union of points and open intervals for any a, α , β  ∈ I  , where X(a,α,β) = {x ∈ M | (a, x, α, β) ∈ X}. Consequently, X(a,α,β) ∩ I  consists of finite points because  a does not contain an open interval. Take α, β ∈ I  with α < β and set I =]α, β[. Then,  a is a finite set for any a ∈ I. By the equality (7), the family { a}a∈I is infinite because I \ K is infinite and the set  a is finite for any a ∈ I. We define a definable relation E on I \ K by E(a, b) ⇔ M |= ∃c (a ∈  c∧b∈ c). It is an equivalence relation by the equality (7). Set Y = {x ∈ I \ K | M |= ∀y ∈ I \ K (E(x, y) → x ≤ y)}. The smallest element of  a belongs to the set Y for all a ∈ I. Therefore, the definable set Y is an infinite set because the family { a}a∈I is infinite. We may assume that Y is a finite union of points and open intervals, shrinking the interval I if necessary, in the same way as we proved that  a is a finite set. Let Z be the largest convex subset of Y with t < Z for all t ∈ Y \ Z. Take a ∈ Z and b1 , b2 ∈ Ua with b1 < a < b2 . We have a ≺ b1 and a ≺ b2 by the equivalence (1). Since Cb1 and Cb2 are finite sets, there are b1 and b2 with b1 ∈  a, b2 ∈  a, b1  b1 and b2  b2 . E(b1 , b2 ) holds true. We get b1 ≤ a < b2 by the relations (4) and (5). We have b2 ∈ / Y by the definition of Y . Take an element c ∈ Ub2 with b2 < c. There exists a d ∈  c with d ∈ Y because the smallest element of  c is contained in the set Y . We obtain b2 < d by the claim (6) because b2 ≺ c ≺ d and b2 < c. Since a < b2 < d, a ∈ Z, d ∈ Z and Z is convex, we finally get b2 ∈ Z ⊂ Y . It is a contradiction. 2 Lemma 3.3. Let M = (M, <, . . .) be a locally o-minimal structure. Strictly monotone definable functions defined on open intervals which are discontinuous everywhere have discrete images. Proof. Let f be a strictly increasing definable function on an interval I which is discontinuous everywhere. Assume that f (I) is not discrete. Since f (I) is a definable set and f is strictly increasing, the definable set f (I) contains an open interval J. Let r, s ∈ J with r < s. There exist unique elements a, b with r = f (a) and s = f (b) because f is injective. Since f is monotone, the restriction g of f to the interval ]a, b[ gives a bijection onto the interval ]r, s[. It is easy to show that g is continuous. Contradiction to the assumption that f is discontinuous at all points in I. The proof for strictly decreasing functions is similar. 2 We are now ready to prove the following parameterized local monotonicity theorem. Theorem 3.2 (Parameterized local monotonicity theorem). Consider a uniformly locally o-minimal structure of the second kind M = (M, <, . . .). Let A ⊂ M and P ⊂ M n be definable subsets. Let f : A × P → M be a definable function. For any (a, b, p) ∈ M × M × M n , any sufficiently small open intervals I and J with a ∈ I and b ∈ J and any sufficiently small open box B with p ∈ B, the following assertions hold true: (1) The set f −1 (J) ∩ (I × {z}) is a finite union of points and open intervals for all z ∈ B. (2) There exists a mutually disjoint definable partition {Xf , X− , X+ , Xc } of f −1 (J) ∩ (I × B) satisfying the following conditions for any z ∈ B: (i) the definable set Xf ∩ (f −1 (J) ∩ (I × {z})) is a finite set;

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(ii) the definable set Xc ∩ (f −1 (J) ∩ (I × {z})) is a finite union of open intervals and f is locally constant on the set; (iii) the definable set X− ∩ (f −1 (J) ∩ (I × {z})) is a finite union of open intervals and f is locally strictly decreasing and continuous on the set; (iv) the definable set X+ ∩ (f −1 (J) ∩ (I × {z})) is a finite union of open intervals and f is locally strictly increasing and continuous on the set. Proof. Consider the definable set X = {(x, y1 , y2 , z) ∈ M 3 × P | y1 < b, y2 > b, y1 < f (x, z) < y2 }. Since M is a uniformly locally o-minimal structure of the second kind, there exist an open interval I with a ∈ I, open intervals J1 , J2 containing b and an open box B with p ∈ B such that the definable set X(y1 ,y2 ,z) ∩ I is a finite union of points and open intervals for any y1 ∈ J1 , y2 ∈ J2 and z ∈ B. Here, the notation X(y1 ,y2 ,z) denotes the fiber of X at the point (y1 , y2 , z). Take b1 ∈ J1 and b2 ∈ J2 with b1 < b and b2 > b, and set J =]b1 , b2 [. Then, the set f −1 (J) ∩ (I × {z}) is a finite union of points and open intervals for all z ∈ B. We have shown the existence of I, J and B satisfying the condition (1) of the theorem. Note that, even if we shrink I, J and B, the condition (1) remains true. Claim 1. Shrinking I, J and B if necessary, there exists a partition f −1 (J) ∩ (I × B) = Xf ∪ Xc ∪ Xn such that, for any z ∈ B, (a) the definable set Xf ∩ (f −1 (J) ∩ (I × {z})) is a finite set; (b) the definable set Xc satisfies the condition (ii) of the theorem; (c) the definable set Xn ∩ (f −1 (J) ∩ (I × {z})) is a finite union of open intervals and f is locally injective on the open intervals. Here, a function g : I → M is called locally injective if, for any x ∈ I, there exists an open interval I  such that x ∈ I  ⊂ I and the restriction of g to I  is injective. We show Claim 1. Set Φ = {(x, y, z) ∈ I × J × B | y = f (x, z)}. Since M is a uniformly locally o-minimal structure of the second kind, we may assume that the fiber Φ(y,z) is a finite union of points and open intervals for any y ∈ J and z ∈ B. The notation fz denotes the function given by f (·, z). Set Xc = {(x, z) ∈ I × B | x ∈ I ∩ fz−1 (J) and ∃x1 , x2 ∈ I, x1 < x < x2 and (x ∈ I ∩ fz−1 (J)) ∧ (f (x) = f (x )) for all x with x1 < x < x2 }. The set Xc clearly satisfies the condition (ii) of the theorem by shrinking I, J and B if necessary. Set E = {(x, z) ∈ I × B | x is an endpoint of the fiber (Xc )z } and Y = (f −1 (J) ∩ (I × B)) \ (E ∪ Xc ). Note that the fiber Yz is a finite union of open intervals for all z ∈ B. We consider the definable set F = {(x, y, z) ∈ I × J × B | y = f (x, z) and (x, z) ∈ Y }. Shrinking I, J and B if necessary, we may assume that the fiber F(y,z) is a finite union of points and open intervals for any y ∈ J and z ∈ B. If the fiber F(y,z) contains an interval, the function fz is constant on the interval. It contradicts the definition of Y . The fiber F(y,z) is a finite set; that is, the function fz is finite-to-one on the fiber Yz for any z ∈ B. Set Xn = {(x, z) ∈ Y | ∃x1 < x, ∃x2 > x, fz is injective on the interval ]x1 , x2 [}. We show that (Xn )z is dense in Yz for any z ∈ B. Fix arbitrary points z ∈ B and x ∈ Yz . There exists x2 ∈ Yz such that the open interval Iz =]x , x2 [ is contained in Yz . Consider a definable map gz : fz (Iz ) → Iz given by gz (y) = inf{x ∈ Iz | f (x) = y}. We can take the infimum without assuming that M is definably complete

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because the set fz is finite-to-one on Yz . The image of gz contains an open interval one of whose endpoints is x . Otherwise, there exists an open interval ]x , x2 [⊂ Iz with ]x , x2 [∩gz (fz (Iz )) = ∅. Take a point u1 ∈]x , x2 [. There exists a point u2 ∈]x , x2 [ with u2 < u1 and f (u1 ) = f (u2 ) by the definition of gz . We can take a point u3 ∈]x , x2 [ such that u3 < u2 and f (u2 ) = f (u3 ) similarly. We can get infinite number of points in Iz at which the value of fz is identical in this way. Contradiction to the fact that fz is finite-to-one. Shrinking Iz if necessary, fz is injective on the open interval Iz . We have shown that (Xn )z is dense in Yz . We may assume that (Xn )z is a finite union of open intervals for any z ∈ B by shrinking I and B if necessary. The definable set Xn satisfies the condition (c) of Claim 1. Set Xf = E ∪ (Y \ Xn ), then the set Xf satisfies the condition (a) of Claim 1. We have finished the proof of Claim 1.   ∪ X− such that, for any Claim 2. Shrinking I and B if necessary, there exists a partition Xn = Xf ∪ X+ z ∈ B,

(a) the definable set Xf ∩ (f −1 (J) ∩ (I × {z})) is a finite set;  ∩ (f −1 (J) ∩ (I × {z})) is a finite union of open intervals and f is locally strictly (b) the definable set X− decreasing on the open intervals;  ∩ (f −1 (J) ∩ (I × {z})) is a finite union of open intervals and f is locally strictly (c) the definable set X+ increasing on the open intervals.   We demonstrate Claim 2. Define the definable subsets X− , X+ , Xmax and Xmin of Xn as follows:  X− = {(x, z) ∈ Xn | ∃x1 < x ∃x2 > x (x1 , z) ∈ Xn , (x2 , z) ∈ Xn ,

∀x ((x1 < x < x) → (f (x , z) > f (x, z)) ∧ (x , z) ∈ Xn ), ∀x ((x < x < x2 ) → (f (x, z) > f (x , z)) ∧ (x , z) ∈ Xn )}  = {(x, z) ∈ Xn | ∃x1 < x ∃x2 > x (x1 , z) ∈ Xn , (x2 , z) ∈ Xn , X+

∀x ((x1 < x < x) → (f (x , z) < f (x, z)) ∧ (x , z) ∈ Xn ), ∀x ((x < x < x2 ) → (f (x, z) < f (x , z)) ∧ (x , z) ∈ Xn )} Xmax = {(x, z) ∈ Xn | ∃x1 < x ∃x2 > x (x1 , z) ∈ Xn , (x2 , z) ∈ Xn , ∀x ((x1 < x < x) → (f (x , z) < f (x, z)) ∧ (x , z) ∈ Xn ), ∀x ((x < x < x2 ) → (f (x, z) > f (x , z)) ∧ (x , z) ∈ Xn )} Xmin = {(x, z) ∈ Xn | ∃x1 < x ∃x2 > x (x1 , z) ∈ Xn , (x2 , z) ∈ Xn , ∀x ((x1 < x < x) → (f (x , z) > f (x, z)) ∧ (x , z) ∈ Xn ), ∀x ((x < x < x2 ) → (f (x, z) < f (x , z)) ∧ (x , z) ∈ Xn )} Since M is locally o-minimal and f is locally injective on Xn , we have   Xn = X− ∪ X+ ∪ Xmax ∪ Xmin .   Set E  = {(x, z) | x is an isolated point or an endpoint of X− or X+ }. Shrinking I and B if necessary, the    definable set X+ = X+ \ E satisfies the condition (c) of Claim 2 by Lemma 3.1. In the same way, the   definable set X− = X− \ E  satisfies the condition (b) of Claim 2. Shrinking I and B if necessary, the fiber (Xmin )z is a finite set for any z ∈ B by Lemma 3.2. In the same way, we may assume that (Xmax )z is a finite set. The definable set Xf = E  ∪ Xmin ∪ Xmax satisfies the condition (a) of Claim 2. We have proven Claim 2.

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We return to the proof of the parameterized local monotonicity theorem. Consider the definable sets  Z+ = {(x, z) ∈ X+ | fz is not continuous at x} and  | fz is not continuous at x}. Z− = {(x, z) ∈ X−

Set g+ = f |Z+ . The notation (g+ )z denotes the restriction of g+ to (Z+ )z . The image (g+ )z (I  ) of an open subinterval I  of (Z+ )z is a finite union of points and open intervals for any z ∈ B by shrinking I, J and B if necessary. In fact, consider the set V = {(y, z, a1 , a2 ) | y = (g+ )z (x) for some a1 < x < a2 }. The definable set V(z,a1 ,a2 ) ∩ J is a finite union of points and open intervals for any z ∈ B and a1 , a2 ∈ I by shrinking I, J and B if necessary. It means that the image (g+ )z (I  ) of an open subinterval I  of (Z+ )z is a finite union of points and open intervals. We next demonstrate that (Z+ )z is a finite set for any z ∈ B. Shrinking I, J and B if necessary, we may assume that (Z+ )z is a finite union of points and open intervals for any z ∈ B. Assume that the definable set (Z+ )z contains an open interval I  for some z ∈ B. We may assume that (g+ )z is strictly increasing on I  by shrinking I  if necessary because (g+ )z is locally strictly increasing. The image (g+ )z (I  ) is a finite union of points and open intervals by the claim proved in the previous paragraph. The image (g+ )z (I  ) is a discrete set by Lemma 3.3 because (g+ )z is strictly increasing on I  and discontinuous everywhere on I  . Hence, the image (g+ )z (I  ) is a finite set. On the other hand, the image (g+ )z (I  ) is simultaneously an infinite set because (g+ )z is injective and I  is an infinite set. Contradiction. We have shown that (Z+ )z is a finite set. We can prove that (Z− )z is a finite set similarly.   \ Z+ , X− = X− \ Z− and Xf = Xf ∪ Xf ∪ Z+ ∪ Z− . Then, the definable sets Xc , X+ , X− Set X+ = X+ and Xf satisfy the conditions of the theorem. 2 Corollary 3.1 (Local monotonicity theorem). Consider a uniformly locally o-minimal structure of the second kind M = (M, <, . . .). Let A ⊂ M be a definable subset. Let f : A → M be a definable function. For any (a, b) ∈ M 2 , there exist an open interval I containing the point a, an open interval J containing the point b and a mutually disjoint definable partition {I− , I+ , Ic } of f −1 (J) ∩ I satisfying the following conditions: (1) the function f is locally strictly decreasing and continuous on I− , (2) the function f is locally strictly increasing and continuous on I+ , and (3) the function f is locally constant on Ic . Proof. Immediate from Theorem 3.2.

2

We restricted the function f to the definable set f −1 (J) ∩ I in Corollary 3.1. The restriction of the function f to the interval I alone is not sufficient to satisfy the monotonicity condition described in (1) to (3) of Corollary 3.1. A counterexample is found in [4, Example 12]. A locally strictly monotone function is strictly monotone when the structure M is definably complete. Proposition 3.1. Let M = (M, <, . . .) be a definably complete local o-minimal structure. A locally strictly monotone definable function defined on an open interval is strictly monotone. Proof. Let f : I → M be a locally strictly increasing definable function. Set s(x) = sup{s ∈ I | s > x, f (x ) > f (x) for all x with x < x ≤ s}. Since f is locally strictly increasing, s(x) is the right endpoint of the interval I in the same way as Lemma 3.1. It means that f is strictly increasing. We can show the lemma similarly when f is a locally strictly decreasing definable function. 2

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We show the following theorems simultaneously. Theorem 3.3. Consider a uniformly locally o-minimal structure of the second kind M = (M, <, . . .). Let X be a definable set with a nonempty interior and X = X1 ∪ · · · ∪ Xm be a finite partition into definable subsets. Then, at least one definable subset Xi has a nonempty interior. Theorem 3.4. Consider a uniformly locally o-minimal structure of the second kind M = (M, <, . . .). Let f : B → M be a definable function defined on an open box B and b ∈ M . For any sufficiently small open interval J with b ∈ J, the preimage f −1 (J) has an empty interior or there exists an open box C such that C ⊂ f −1 (J) and f is continuous on C. Proof. Assume that X and B in the theorems are subsets of M n . We prove Theorem 3.3 and Theorem 3.4 by the induction on n. Theorem 3.3 is obvious when n = 1 because M is locally o-minimal. Theorem 3.4 follows from Corollary 3.1. We consider the case in which n > 1. We first demonstrate Theorem 3.3 for n assuming that Theorem 3.3 and Theorem 3.4 hold true for n − 1. Since X has a nonempty interior, there exists an open box B contained in X. If B ∩ Xi has a nonempty interior, Xi also has a nonempty interior. Therefore, we may assume that X is an open box. Let π : M n → M n−1 be the projection forgetting the first coordinate. Set X = B = I × B  , where I is an open interval and B  is an open box in M n−1 . Take an arbitrary point a = (a1 , a ) ∈ I × B  . Set Yi = {y ∈ B  | Xi ∩ (J × {y}) contain open intervals for all open interval J with a1 ∈ J and a1 < j for any j ∈ J}  for all 1 ≤ i ≤ m. The sets Yi are definable and B  = 1≤i≤m Yi because the family {Xi }m i=1 covers X. At least one Yi has a nonempty interior by the inductive hypothesis. We may assume that Y1 contains an open box B  . Replacing B  with B  , we may assume that Y1 = B  . Since M is a uniformly locally o-minimal structure of the second kind, taking a smaller open box X, we may assume that (X1 )y ∩ I is a finite union of points and open intervals for all y ∈ B  . Define a definable function f : B  → M as follows: f (y) = sup{x ∈ I | ]a1 , x[×{y} is contained in X1 }. It is well-defined without assuming that M is definably complete because (X1 )y ∩ I is a finite union of points and open intervals. Take a sufficiently small open interval J containing a1 . We first consider the case in which f −1 (J) has an empty interior. There exists an open box C ⊂ B  with f −1 (J) ∩ C = ∅ by the inductive hypothesis. Let c be the right endpoint of the open interval J. We have c < f (y) for all y ∈ C by the definition of f . It means that ]a1 , c[×B  is contained in X1 . The definable set X1 has a nonempty interior. We next consider the case in which f −1 (J) has a nonempty interior. Taking an open box C ⊂ B  , the definable function f is continuous on C by Theorem 3.4. The definable set {(x, y) ∈ I × C | a1 < x < f (y)} ⊂ X1 obviously contains an open box because f is continuous on C. We have proven Theorem 3.3. We next demonstrate Theorem 3.4 for n assuming that Theorem 3.3 and Theorem 3.4 hold true for n and n − 1, respectively. Set B = I1 × · · · × In , where Ii are open intervals for all 1 ≤ i ≤ n, and set Bi = I1 × · · · × Ii−1 × Ii+1 × · · · × In for all 1 ≤ i ≤ n. Take a sufficiently small open interval J with b ∈ J and shrink the open box B if necessary. By Theorem 3.2, the following assertions hold true for all 1 ≤ i ≤ n. 1. The set f −1 (J) ∩ ({(z1 , . . . , zi−1 )} × Ij × {(zi+1 , . . . , zn )}) is a finite union of points and open intervals for all z = (z1 , . . . , zi−1 , zi+1 , . . . , zn ) ∈ Bi .

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i i 2. There exists a mutually disjoint definable partition {Xfi , X− , X+ , Xci } of fi−1 (J) ∩ B satisfying the  following conditions for any z = (z1 , . . . , zi−1 , zi+1 , . . . , zn ) ∈ Bi : (i) the definable set Xfi ∩ (f −1 (J) ∩ ({(z1 , . . . , zi−1 )} × Ii × {(zi+1 , . . . , zn )})) is a finite set; (ii) the definable set Xci ∩ (f −1 (J) ∩ ({(z1 , . . . , zi−1 )} × Ii × {(zi+1 , . . . , zn )})) is a finite union of open intervals and f is locally constant on the set; i (iii) the definable set X− ∩ (f −1 (J) ∩ ({(z1 , . . . , zi−1 )} × Ii × {(zi+1 , . . . , zn )})) is a finite union of open intervals and f is locally strictly decreasing and continuous on the set; i (iv) the definable set X+ ∩ (f −1 (J) ∩ ({(z1 , . . . , zi−1 )} × Ii × {(zi+1 , . . . , zn )})) is a finite union of open intervals and f is locally strictly increasing and continuous on the set. 1 1 , X+ contains an open box C by Theorem 3.3. If f −1 (J) has a nonempty interior, at least one of Xc1 , X− We set B = C. The restriction of f to I1 × {y} is continuous and locally monotone for any y ∈ B1 . We assume that the restriction is locally strictly increasing. We can demonstrate Theorem 3.4 in the same way in the other cases. Take an arbitrary point a = (a1 , a ) ∈ I1 × B1 . Consider the set

Z = {(x, y) ∈ I1 × B1 | x < a1 and fy (x) < fy (x ) < fy (a1 ) for all x < x < a1 } ∪ ({a1 } × B1 ) ∪ {(x, y) ∈ I1 × B1 | x > a1 and fy (x) > fy (x ) > fy (a1 ) for all x > x > a1 }, where fy (·) denotes the function f (·, y). Since M is a uniformly locally o-minimal structure of the second kind, we may assume that Zy is a finite union of points and open interval for any y ∈ B1 , shrinking B if necessary. The fiber Zy is an open interval because f (·, y) are locally strictly increasing on I1 for all y ∈ B1 . By the definition of Zy , the function f (·, y) is continuous and strictly increasing on Zy . Since Zy is an open interval for any y ∈ B  , there exist definable functions g, h : B  → M with ]g(y), h(y)[×{y} = Zy for all y ∈ B1 . We may assume that both g and h are continuous by taking a smaller open box B1 by the induction hypothesis. There exists an open interval I  with I  ⊂ I1 and an open box B  ⊂ B1 with I  × B  ⊂ {(x, y) ∈ I1 × B1 | g(y) < x < h(y)} ⊂ Z. Therefore, taking a smaller open box B, we may assume that f (·, y) are continuous and monotone on I1 for all y ∈ B  . Applying the same argument to I2 , . . . , In and shrinking the open box, we may assume that the restrictions of f to {(y1 , . . . , yi−1 )} × Ii × {(yi+1 , . . . , yn )} are continuous and monotone for all (y1 , . . . , yi−1 , yi+1 , . . . , yn ) ∈ Bi and for all 1 ≤ i ≤ n. We can easily show that the function f is continuous on B by applying [12, Lemma 3.2.16] inductively. 2 Corollary 3.2. Consider a uniformly locally o-minimal structure of the second kind M = (M, <, . . .). Let f : X → M be a definable function and b ∈ M . For any sufficiently small open interval J with b ∈ J, the preimage f −1 (J) has an empty interior or there exists an open box C such that C ⊂ f −1 (J) and f is continuous on C. Proof. Assume that X is a subset of M n . Take c ∈ M with c = b. We consider the extension g : M n → M of f defined by g(x) =

f (x) if x ∈ X, c

otherwise.

Take a sufficiently small open interval J with b ∈ J and c ∈ / J. We have g −1 (J) = f −1 (J). The corollary follows from Theorem 3.4. 2

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4. Local definable cell decomposition theorem We demonstrate that a definably complete uniformly locally o-minimal structure of the second kind admits local definable cell decomposition in this section. We need the assumption that the structure is definably complete because we use the fact that an open cell is definably connected in our proof. We review the definitions used in this section. Definition 4.1 (Definable cell decomposition). Consider a densely linearly ordered structure M = (M, <, . . .). Let (i1 , . . . , in ) be a sequence of zeros and ones of length n. (i1 , . . . , in )-cells are definable subsets of M n defined inductively as follows: • A (0)-cell is a point in M and a (1)-cell is an open interval in M . • An (i1 , . . . , in , 0)-cell is the graph of a continuous definable function defined on an (i1 , . . . , in )-cell. An (i1 , . . . , in , 1)-cell is a definable set of the form {(x, y) ∈ C × M | f (x) < y < g(x)}, where C is an (i1 , . . . , in )-cell and f and g are definable continuous functions defined on C with f < g. A cell is an (i1 , . . . , in )-cell for some sequence (i1 , . . . , in ) of zeros and ones. An open cell is a (1, 1, . . . , 1)-cell. We inductively define a definable cell decomposition of an open box B ⊂ M n . For n = 1, a definable cell m decomposition of B is a partition B = i=1 Ci into finite cells. For n > 1, a definable cell decomposition of m m B is a partition B = i=1 Ci into finite cells such that π(B) = i=1 π(Ci ) is a definable cell decomposition of π(B), where π : M n → M n−1 is the projection forgetting the last coordinate. Consider a finite family {Aλ }λ∈Λ of definable subsets of B. A definable cell decomposition of B partitioning {Aλ }λ∈Λ is a definable cell decomposition of B such that the definable sets Aλ are unions of cells for all λ ∈ Λ. Lemma 4.1. Consider a uniformly locally o-minimal structure of the second kind M = (M, <, . . .). The followings are equivalent: (1) Let X be a definable subset of M n+1 . For any a ∈ M and b ∈ M n , there exist an open interval I containing the point a, an open box B with b ∈ B and a positive integer N such that the definable set Xy ∩ I contains an interval or |Xy ∩ I| ≤ N for any y ∈ B. (2) Let X be a definable subset of M n+1 such that the fiber Xy is empty or a discrete set for any y ∈ M n . For any a ∈ M and b ∈ M n , there exist an open interval I containing the point a, an open box B with b ∈ B and a positive integer N such that |Xy ∩ I| ≤ N for any y ∈ B. Proof. The assertion (2) obviously follows from the assertion (1). We demonstrate the opposite implication. Let X be a definable subset of M n+1 . Fix a ∈ M and b ∈ M n . Since M is a locally o-minimal structure of the second kind, there exist an open interval I and an open box B such that a ∈ I, b ∈ B and the definable set Xy ∩ I is a finite union of points and open intervals for any y ∈ B. Set Y = {(x, y) ∈ I × B | x ∈ / Xy and Xy ∩ J = ∅ or x ∈ Xy and Xyc ∩ J = ∅ for all open interval J with x ∈ J}. Here, Xyc denotes the complement of the set Xy in M n × {y}. The fiber Yy is the set of the isolated points in Xy ∩ I and the endpoints of maximal intervals contained in Xy ∩ I. In particular, Yy is empty or a discrete set. Applying the assertion (2) to Y , we may assume that |Yy ∩ I| ≤ N for some positive integer N for any y ∈ B, by shrinking I and B if necessary. By the definition of Yy , Xy ∩ I is the union of at most N points and open intervals for any y ∈ B. We get the assertion (1). 2

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The following theorem is a main theorem of this paper. Theorem 4.2 (Local definable cell decomposition theorem). Consider a definably complete uniformly locally o-minimal structure of the second kind M = (M, <, . . .). For any positive integer n, the following assertions hold true: (D)n

(C)n

(U )n

Let {Aλ }λ∈Λ be a finite family of definable subsets of M n . For any point a ∈ M n , there exist an open box B containing the point a and a definable cell decomposition of B partitioning the finite family {B ∩ Aλ | λ ∈ Λ and B ∩ Aλ = ∅}. Let A ⊂ M n be a definable subset and f : A → M be a definable function. For any a ∈ M n , b ∈ M and any sufficiently small open interval J with b ∈ J, there exist an open box B containing the point a and a definable cell decomposition of B partitioning f −1 (J) ∩ B such that the function f is continuous on any cell contained in f −1 (J) ∩ B. Let X be a definable subset of M n+1 . For any a ∈ M and b ∈ M n , there exist an open interval I containing the point a, an open box B with b ∈ B and a positive integer N such that, for any y ∈ B, the definable set Xy ∩ I contains an interval or |Xy ∩ I| ≤ N for any y ∈ B.

Proof. We show the assertions (D)n , (C)n and (U )n by the induction on n. We first consider the case in which n = 1. The assertion (D)1 follows from the local o-minimality of the structure M. The assertion (C)1 follows from Corollary 3.1. We next show that the assertions (U )n hold true for all positive integers n assuming the assertions (C)n , (D)n and (U )k for all k < n. Fix a ∈ M and b ∈ M n . We take a sufficiently small open interval I with a ∈ I. We may assume that the fiber Xx is a discrete set for any x ∈ M n by Lemma 4.1. In the proof, we call a point (y, x) ∈ I × B normal if there exist an open subinterval Iy of I and an open box Bx such that y ∈ Iy , x ∈ Bx and (Iy ×Bx ) ∩X = ∅ or (Iy ×Bx ) ∩X is of the form Γ (f ) for some definable continuous function f : Bx → Iy . Here, the notation Γ (f ) is the set given by {(y  , x ) ∈ Iy ×Bx | y  = f (x )}. We consider the definable sets AI = {x ∈ M n | (y, x) is not normal for some y ∈ I} and NI = M n \ AI . Claim 1. For any definably connected definable subset C of NI , there exists a finite family {fi : C → I}ki=1 of k definable continuous functions defined on C such that f1 < f2 < · · · < fk on C and X ∩(I ×C) = i=1 Γ (fi ). We begin to prove Claim 1. Take a point c ∈ C. Set |Xc ∩ I| = k. Let Xc ∩ I = {y1 , . . . , yk } with yi < yi+1 for all 1 ≤ i ≤ k − 1. For any 1 ≤ i ≤ k, there exists a definable continuous function fi defined on a neighborhood of c with yi = fi (c) by the definition of NI . We first show that fi is extendable to the entire C. Assume otherwise. Let Di be the largest domain of definition of the function fi . Let x ∈ ∂Di ∩ C. Take a point (y, x) ∈ Γ (fi ) ∩ (I × {x}). Since x ∈ NI , there exists a definable continuous function g : B  → J  such that X ∩ (J  × B  ) = Γ (g), where B  is an open neighborhood of x in C and J  is an open neighborhood of y in I. Therefore, the definable function fi is extendable to Di ∪ B  . It is a contradiction. We next show that f1 < f2 < · · · < fk on C. Assume otherwise. By the assumption, there exist 1 ≤ i < j ≤ k and a point q ∈ C such that fi (q) ≥ fj (q). Note that the definable set {x ∈ C | fi (x) = fj (x)} is open by the definition of the set NI . The definably connected set C is covered by the mutually disjoint definable open subsets {x ∈ C | fi (x) = fj (x)}, {x ∈ C | fi (x) < fj (x)} and {x ∈ C | fi (x) > fj (x)}. At least two of them are not empty. Contradiction.

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 We finally show that X ∩ (I × C) = i=1 Γ (fi ). Assume otherwise. Take a point (y  , x ) ∈ X ∩ (I ×  C) \ ( i=1 Γ (fi )). As we have shown previously, there exists a definable continuous function g on C such that Γ (g) ⊂ X ∩ (I × C) and y  = g(x ). We have g(c) = fi (c) for some 1 ≤ i ≤ k by the definition of f1 , . . . , fk . We can show that g = fi on C in the same way using the assumption that C is definably connected. Contradiction. We have finished the proof of Claim 1. Claim 2. The interior of AI ∩ B is empty for some interval I with a ∈ I and some open box B with b ∈ B. Assume the contrary. Fix a sufficiently small open interval I containing the point a. We consider the sets n A− I = {x ∈ M | (y, x) is not normal for some y ∈ I with y ≤ a} and n A+ I = {x ∈ M | (y, x) is not normal for some y ∈ I with y > a}. + We have AI = A− I ∪ AI . It is easy to verify that, for any open intervals I1 and I2 with a ∈ I1 ⊂ I2 , we have − + + A− I1 ⊂ AI2 and AI1 ⊂ AI2 .

(8)

+ ∗ Since AI = A− I ∪ AI , there exists ∗ ∈ {−, +} such that AI  have nonempty interiors for all open intervals I  with a ∈ I  ⊂ I by Theorem 3.3 by shrinking I if necessary. We only consider the case in which A− I  have   nonempty interiors for all open intervals I with a ∈ I ⊂ I. We can show the claim in the same way in the other case. Consider the definable function βI : A− I → M given by

βI (x) = sup{y ∈ I | y ≤ a and (y, x) is not normal}. Note that, for any open intervals I1 and I2 with a ∈ I1 ⊂ I2 , we have βI1 = βI2 |A− .

(9)

I1

We also consider the following definable sets: BI− = {x ∈ A− I | ∃y y ∈ I, y < βI (x) and (y, x) ∈ X}, BI+ = {x ∈ A− I | ∃y y ∈ I, y > βI (x) and (y, x) ∈ X}. The definable functions γI− : BI− → M and γI+ : BI+ → M are defined by γI− (x) = sup{y ∈ I | y < βI (x) and (y, x) ∈ X} and γI+ (x) = inf{y ∈ I | y > βI (x) and (y, x) ∈ X}, respectively. These functions are well-defined because the definable set Xx ∩ I is finite for any x ∈ B if we take sufficiently small I and B. We have BI−1 ⊂ BI−2 , BI+1 ⊂ BI+2 , γI−1 = γI−2 |B− and γI+1 = γI+2 |B+ I1

I1

(10)

for all open intervals I1 and I2 with a ∈ I1 ⊂ I2 . Apply the assertion (C)n to βI , γI− and γI+ . Apply the assertion (D)n and get a definable cell decomposition partitioning the obtained cells. Using the relations (8) to (10), we finally obtain a definable cell decomposition of B such that, for each cell, the functions among βI , γI− and γI+ defined on the cell are continuous on the cell by shrinking I and B if necessary. Since A− I has a nonempty interior, there exists an open cell C contained in A− I . We have the following four cases:

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− − − − + + C ⊂ BI− ∩ BI+ , C ⊂ A− I \ BI , C ⊂ AI \ BI and C ⊂ AI \ (BI ∩ BI ).

In each case, the functions among βI , γI− and γI+ defined on C are continuous on C. We consider the case in which C is contained in BI− ∩ BI+ . Consider the two definable sets {x ∈ / X}. We may assume that at least one of the above deC | (βI (x), x) ∈ X} and {x ∈ C | (βI (x), x) ∈ finable sets contains an open box by Theorem 3.3. If the former set contains an open box D, the points (βI (x), x) are normal for all x ∈ D. It is a contradiction to the definition of βI . If the latter set contains an open box, a sufficiently small open box containing the point (βI (x), x) has an empty intersection with X. It means that (βI (x), x) is normal, which is also a contradiction to the definition of βI . We can lead to contradictions in the same way in the other cases. We have demonstrated Claim 2. We are now ready to prove the assertion (U )n . Applying the assertion (D)n , we get a definable cell decomposition partitioning π(X) and NI . For non-open cells D, there exist positive integers ND such that |Xx | ≤ ND for all x ∈ D by the induction hypothesis. For open cells D, D is contained in NI by Claim 2. There exist positive integers ND such that |Xx | ≤ ND for all x ∈ D by Claim 1 because D is definably connected by [2, 2.9]. Set N = max{ND | D is a cell of the decomposition}, then N satisfies the requirement of the assertion (U )n . We show that the assertion (C)n follows from the assertions (D)n and (C)n−1 when n > 1. The set D denotes the set of points at which f is discontinuous. It is clearly a definable set. Consider the restriction g of f to D. The preimage g −1 (J) has an empty interior for a sufficiently small open interval b ∈ J. Otherwise, it contradicts to Corollary 3.2. We first apply the assertion (D)n to the definable sets A and D. There exist an open box B with a ∈ B and a definable cell decomposition of B partitioning A ∩ B. If the assertion (C)n is true for any restriction of f to a cell C, the assertion (C)n is true for the original f . We consider the case in which C is not an open cell. There exists a coordinate projection π : M n → M n−1 whose restriction to C is a definable homeomorphism onto its image. The assertion (C)n follows from (C)n−1 in this case. The remaining case is the case in which C is an open cell. We have C ∩ D = ∅ because the interior of D is empty. It means that f is continuous on C. We finally demonstrate the assertion (D)n+1 assuming the assertions (C)n , (D)n and (U )n when n > 1. Fix a ∈ M n+1 . We have B = C × I for some interval I and some open box C in M n . For any definable set S ⊂ M n+1 , the notation bdn (S) denotes the set {(x, y) ∈ M n × M | (x, y) ∈ B and y ∈ bd(Sx )}. Here, Sx = {y ∈ M | (x, y) ∈ S}. The similar notations such as Yx and (Aλ )x are defined similarly in the rest of  the proof. Set Y = λ∈Λ bdn (Aλ ∩ B). Taking a smaller open box B if necessary, the sets Yx are finite sets for all x ∈ C. There exists a positive integer N with |Yx ∩ I| ≤ N for all x ∈ C by the assertion (U )n . Set Ci = {x ∈ C | |Yx ∩ I| = i} for all 0 ≤ i ≤ N . For all 0 ≤ i ≤ N , there exist definable functions fi1 , . . . , fii : Ci → I such that fi1 (x) < fi2 (x) < · · · < fii (x) and Yx ∩ I = {fi1 (x), . . . , fii (x)} for all x ∈ Ci . We define definable sets Cλij and Dλij as follows: Cλij = {x ∈ Ci | fij (x) ∈ (Aλ )x } and Dλij = {x ∈ Ci | ]fij (x), fij+1 (x)[ ⊂ (Aλ )x }. Applying the assertions (C)n and (D)n , we get a definable cell decomposition of C such that the restrictions of fij to the cells are continuous for all i and j by shrinking C and I if necessary. Let b1 and b2 be the left and right endpoints of the open interval I, respectively. For any cell E, set DE = {]b1 |E , fi1 |E [, . . . , ]fij |E , fij+1 |E [, . . . , ]fii |E , b2 |E [, Γ(fi1 ), . . . , Γ(fii )}. Here, ]f, g[= {(x, y) ∈ E × M | f (x) < y < g(x)} and Γ(f ) = {(x, y) ∈ E × M | y = f (x)} for all definable functions f, g : E → M . The notations b1 |E and b2 |E denote the restrictions of the constant functions b1

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 and b2 to E, respectively. The family D = E is a cell. DE is a definable cell decomposition of B partitioning {B ∩ Aλ | λ ∈ Λ and B ∩ Aλ = ∅}. We have demonstrated the assertion (D)n+1 . 2 Remark 4.3. We did not use the assumption that M is definably complete in the proof of the assertions (C)n and (D)n . The assertions (C)n and (D)n in Theorem 4.2 hold true for any uniformly locally o-minimal structure of the second kind satisfying the conditions (U )n in Theorem 4.2 for all positive integers n. For instance, the non-definably complete structure given in Example 2.2 satisfies the conditions (U )n in Theorem 4.2 for all positive integers n and satisfies the statements in Theorem 4.2. Definition 4.4. A densely linearly ordered structure admits local definable cell decomposition if the assertions (D)n hold true for all positive integers n. Note that a locally o-minimal structure which admits local definable cell decomposition is always a uniformly locally o-minimal structure of the second kind. A strongly locally o-minimal structure also admits local definable cell decomposition by [4, Proposition 13]. Corollary 4.1. A definably complete locally o-minimal structure admits local definable cell decomposition if and only if it is a uniformly locally o-minimal structure of the second kind. Proof. A definably complete uniformly locally o-minimal structure of the second kind admits a definable cell decomposition by Theorem 4.2. The converse is obvious. 2 5. Dimension In this section, we define dimension of a definable set and investigate its basic properties when the given structure admits local definable cell decomposition. Definition 5.1 (Dimension of a definable set). Consider a densely linearly ordered structure M = (M, <, . . .). A definable set X ⊂ M n is of dim(X) ≥ m if there exists an open box B ⊂ M m and a definable continuous injective map f : B → X which is homeomorphic onto its image. A definable set X ⊂ M n is of dim(X) = m if it is of dim(X) ≥ m and it is not of dim(X) ≥ m + 1. The empty set is defined to be of dimension −∞. Lemma 5.1. Consider a densely linearly ordered structure M = (M, <, . . .). Let X ⊂ Y be definable sets. Then, the inequality dim(X) ≤ dim(Y ) holds true. Proof. Obvious. 2 Lemma 5.2. Consider a densely linearly ordered structure. A discrete definable set is of dimension zero. Proof. Let X be a discrete definable set. Assume that X is of dimension greater than zero. There exists a definable continuous injection f : I → X defined on an interval I. Take b ∈ f (I). Since X is discrete, the singleton {b} is open in X. The definable set f −1 (b) is open because f is continuous. The open definable set f −1 (b) is an infinite set because the considered structure is densely ordered. It is a contradiction to the assumption that f is injective. 2 Example 5.2. Consider an o-minimal structure M = (M, <, · · · ) and a definable injective map f : X → M n . We have dim(X) = dim(f (X)) by [12, Proposition 4.1.3(ii)]. The same equality holds true for weakly o-minimal structures by [13, Theorem 2.13]. We next consider the strongly o-minimal structure N discussed in [4, Example 12]. Let M be any o-minimal structure and a ∈ M . Let f : {a} × M → M 2 be the function defined by f (a, b) = (b, a). The M -definable structure N is given by (M 2 ,
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strongly o-minimal structure. In this case, the definable set {a} × M is of dimension greater than zero and the image of f is discrete and of dimension zero by Lemma 5.2. We have dim({a} × M ) = dim(f ({a} × M )). A strongly locally o-minimal structure admits local definable cell decomposition by [4, Proposition 13]. The example demonstrates that the equality dim(X) = dim(f (X)) is not satisfied even for a locally ominimal structure which admits a local definable cell decomposition if f is a definable bijection. We will show that the equality holds true when f is a definable homeomorphism in Corollary 5.2 and Theorem 5.4. Definition 5.3 (Alternative definition of dimension). Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. Let X ⊂ M n be a definable set. We define dim (X) as follows: dim (X) = max{i1 + · · · + in | X contains an (i1 , . . . , in )-cell}. The notation πi1 ,...,in denotes the projection from M n to M d , where d = i1 + · · · + in , forgetting all the j-th coordinates with ij = 0. This definition is employed as the definition of sets definable in o-minimal structures in [12]. The condition in the following proposition is similar to the definition of a set definable in weakly o-minimal structures in [6]. We show that our definition of dimension coincides with the definition in o-minimal structures and the definition similar to that in weakly o-minimal structures in Corollary 5.3 when the structure is a locally o-minimal structure which admits local definable cell decomposition. Proposition 5.1. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. Let X ⊂ M n be a definable set. We have dim (X) ≥ m if and only if there exist a coordinate projection π : M n → M m and a point a ∈ M n such that the definable set π(B ∩ X) has a nonempty interior for any open box B containing the point a. Proof. We first assume that dim (X) ≥ m. There exists an (i1 , . . . , in )-cell C with m = i1 + · · · + in and C ⊂ X. Let π = πi1 ,...,in and a ∈ C. The definable set π(B ∩ X) is an open neighborhood of π(a) for any open box containing the point a. In particular, it has a nonempty interior. We next consider the converse implication. We consider the coordinate projection π and the point a given in the proposition. The set π(B ∩ X) contains an open box by the assumption. The set B ∩ X contains an (i1 , . . . , in )-cell with i1 + · · · + in ≥ m. We have dim (X) ≥ i1 + · · · + in ≥ m. 2 We prove a lemma and its corollaries necessary for proving Corollary 5.3. Lemma 5.3. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. Let C ⊂ M n be an open cell and f : C → M n be a definable injective continuous map. Then, the definable set f (C) contains an open cell. Proof. We demonstrate the lemma by the induction on n. We can show the lemma in the same way as Lemma 5.2 when n = 1. We consider the case in which n > 1. Let a ∈ C and b = f (a). Since M admits local definable cell k decomposition, there exist an open box B with b ∈ B and a decomposition f (C) ∩ B = i=1 Ci into cells. The preimage f −1 (B) is a definable open set because B is open and f is continuous. One of f −1 (Ci ) has a nonempty interior by Theorem 3.3. We may assume that f −1 (C1 ) has a nonempty interior without loss of generality. We lead to a contradiction assuming that C1 is not an open cell. Let C1 be an (i1 , . . . , in )-cell and let π = πi1 ,...,in : M n → M d , where d = i1 + · · · + in . The restriction of π to C1 is a definable homeomorphism

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onto its image. Apply definable cell decomposition partitioning f −1 (C1 ), then we get an open cell C  contained in f −1 (C1 ). Consider the definable continuous injective map g : C  → M d given by g(x) = π(f (x)). Taking an open box contained in C  , we may assume that C  is an open box. We have C  = B1 × B2 for some open boxes B1 in M d and B2 in M n−d . Take c ∈ B2 . Define a definable continuous injective map h : B1 → M d given by h(x) = g(x, c). We have an open box D ⊂ h(B1 ) by the induction hypothesis. Take y ∈ D and choose x ∈ B1 with y = h(x). Get c ∈ M sufficiently close to c with c = c. We have g(x, c ) ∈ D because g is continuous. Since D ⊂ h(B1 ), we obtain x ∈ B1 with h(x ) = g(x, c ). It means that g(x, c ) = g(x , c), which contradicts that g is injective. 2 Corollary 5.1. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. Let m > n. There are no definable injective continuous maps from open boxes in M m to open boxes in M n . Proof. Let f : B1 → B2 be a definable injective continuous map from an open box in M m to an open box in M n . Take a point c ∈ M n−m . Consider a definable injective continuous map g : B1 → B2 × M n−m given by g(x) = (f (x), c). The image g(B1 ) contains an open cell by Lemma 5.3. Contradiction. 2 Corollary 5.2. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. If there exists a definable homeomorphism between definable sets X and Y , then we have dim (X) = dim (Y ). Proof. Let X and Y be definable subsets of M m and M n , respectively. Let f : X → Y be a definable homeomorphism. Set d = dim (X) and e = dim (Y ). We have only to show the inequality d ≤ e by the m symmetry. Let C be an (i1 , . . . , im )-cell in X with j=1 ij = d. Set π = πi1 ,...,im , then the restriction π to C is a definable homeomorphism onto its image. We may assume that X is an open cell by setting X = π(C) of M d and f = f ◦ (π|C )−1 . Take y ∈ f (X) and a sufficiently small open box B with y ∈ B. There exists a decomposition B ∩ f (X) = k −1 (B) is open j=1 Dj into cells because M admits local definable cell decomposition. The preimage f  k −1 −1 because B is open and f is continuous. We get the decomposition f (B) = j=1 f (Dj ). One of the preimages f −1 (Dj ) has a nonempty interior by Theorem 3.3. We may assume that j = 1 without loss of n generality. We assume that D1 is an (j1 , . . . , jn )-cell with e = k=1 jk . We have only to show that d ≤ e .  Assume the contrary. Let B  be an open box contained in f −1 (D1 ). Set π  = πj1 ,...,jn : M n → M e . Then,  D = π  (D1 ) is an open cell, and the map g : B  → M e given by g(x) = π  (f (x)) is a definable injective continuous map. It contradicts Corollary 5.1. 2 Theorem 5.4. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. For any definable set X, the equality dim (X) = dim(X) is satisfied. Proof. Let X be a definable subset of M n . Set m = dim(X). There exist an open box B in M m and a definable injective continuous map f : B → X homeomorphic onto its image. It is obvious that dim (B) = m. We have m = dim (B) = dim (f (B)) ≤ dim (X) by Corollary 5.2. We next lead to a contradiction assuming that dim (X) ≥ m + 1. There exists an (i1 , . . . , in )-cell C with n j=1 ij ≥ m + 1 and C ⊂ X. Set π = πi1 ,...,in , then the restriction of π to C is definable homeomorphism onto its image. Let B be an open box contained in π(C). Consider a definable map f : B → X given by

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f (x) = (π|C )−1 (x). It is an injective continuous map homeomorphic onto its image. In particular, we have dim(X) ≥ m + 1. Contradiction. 2 Corollary 5.3. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. The following conditions are equivalent for any definable subset X ⊂ M n : • dim(X) ≥ m; n • the definable set X contains an (i1 , . . . , in )-cell with j=1 ij ≥ m, and • there exist a coordinate projection π : M n → M m and a point a ∈ M n such that the definable set π(B ∩ X) has a nonempty interior for any open box B containing the point a. Proof. Immediate from Proposition 5.1 and Theorem 5.4.

2

The following basic properties of dimension follows from Corollary 5.3. Corollary 5.4. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. The following assertions hold true: (i) Let σ be a permutation of the set {1, . . . , n}. The definable map σ : M n → M n is defined by σ(x1 , . . . , xn ) = (xσ(1) , . . . , xσ(n) ). Then, we have dim(X) = dim(σ(X)) for any definable subset X of M n . (ii) Let X and Y be definable subsets of M n . We have dim(X ∪ Y ) = max{dim(X), dim(Y )}. (iii) Let X be a cell in M n and π : M n → M m be a coordinate projection. The inequality dim(X) ≥ dim(π(X)) holds true. (iv) Let X and Y be definable sets. We have dim(X × Y ) = dim(X) + dim(Y ). Proof. The assertions (i), (ii) and (iii) are obvious by Corollary 5.3. We prove the assertion (iv). Let X and Y be definable subsets of M m and M n , respectively. Set d = m dim(X) and e = dim(Y ). There exist an (i1 , . . . , im )-cell C contained in X with k=1 ik = d and a n (j1 , . . . , jn )-cell D contained in Y with k=1 jk = e by Corollary 5.3. We have dim(X × Y ) ≥ d + e because C × D is an (i1 , . . . , im , j1 , . . . , jn )-cell contained in X × Y . n m Let E be an (i1 , . . . , im , j1 , . . . , jn )-cell contained in X × Y with k=1 ik + k=1 jk = dim(X × Y ). Let π : M m × M n → M m be the projection forgetting the last n coordinates. The definable set π(E) is an m (i1 , . . . , im )-cell contained in X. We have k=1 ik ≤ d. Since the definable set Ex = {y ∈ M n | (x, y) ∈ E} n is a (j1 , . . . , jn )-cell contained in Y for all x ∈ π(E), we get k=1 jk ≤ e. We finally obtain dim(X × Y ) ≤ d + e. 2 Remark 5.5. The inequality in Corollary 5.4 (iii) does not necessarily hold true if X is a definable set which is not a cell. In fact, let M be an o-minimal abelian group and consider the M -definable structure N = (N = M 2 ,
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Corollary 5.5. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. Let S1 and S2 be definable subsets of M n . Assume that, for any point a ∈ M n , we have dim(S1 ∩ B) < dim(S2 ∩ B) or S1 ∩ B = S2 ∩ B = ∅ for all sufficiently small open boxes B with a ∈ B. Then, we have dim(S1 ) < dim(S2 ). n Proof. Set d = dim(S2 ). Assume that dim(S1 ) ≥ dim(S2 ). Let C be an (i1 , . . . , in )-cell with e = j=1 ij ≥ d and C ⊂ S1 . Let π = πi1 ,...,in . The set π(C) is an open cell. Take a ∈ C and a sufficiently small open box B with a ∈ B. We have dim(B ∩ C) ≥ e by Corollary 5.3. Consider a definable cell decomposition partitioning B ∩ C. There exists a cell D such that π(D) has a nonempty interior by applying Theorem 3.3 to π(C). We have dim(D) ≥ e by Corollary 5.4(iii). We get dim(S1 ∩ B) ≥ e by Lemma 5.1 because D ⊂ S1 ∩ B. We have dim(S2 ∩ B) > dim(S1 ∩ B) ≥ e ≥ d = dim S2 . It contradicts Lemma 5.1. 2 Our next goal is to demonstrate that dim ∂S < dim S for all definable sets S. We prepare two lemmas. Lemma 5.4. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. Let S ⊂ M m × M n be a definable set. For any (a, b) ∈ M m × M n , there exist open boxes B1 ⊂ M m and B2 ⊂ M n with a ∈ B1 and b ∈ B2 such that, for any 0 ≤ d ≤ n, the definable set SB1 (d) is a definable set and the following equality holds true: ⎛ dim ⎝



⎞ {x} × Sx,B2 ⎠ = dim (SB1 (d)) + d.

x∈SB1 (d)

Here, the definable sets Sx,B2 and SB1 (d) are defined as follows for all x ∈ B1 : Sx,B2 = {y ∈ B2 | (x, y) ∈ S} and SB1 (d) = {x ∈ B1 | dim (Sx,B2 ) = d}. Proof. The set SB1 (d) is a definable set because the last condition of Corollary 5.3 is expressed by a first-order formula. Fix (a, b) ∈ M m ×M n . There exist open boxes B1 ⊂ M m and B2 ⊂ M n and a definable cell decomposition {Ci }N i=1 of B1 ×B2 partitioning S because M admits local definable cell decomposition. Let C be an arbitrary (i1 , . . . , im+n )-cell and π : M m ×M n → M m be the projection forgetting the last n coordinates. The definable set π(C) is an (i1 , . . . , im )-cell and the fibers Cx = {y ∈ M n | (x, y) ∈ C} are (im+1 , . . . , im+n )-cells for all x ∈ π(C). In particular, we have dim(C) = dim(π(C)) + dim(Cx ) for all x ∈ π(C).

(11)

N The family {π(Ci )}N i=1 is a definable cell decomposition of B1 . Take an arbitrary cell D in {π(Ci )}i=1 with D ⊂ π(S). We may assume that {i | π(Ci ) = D and Ci ⊂ S} = {1, 2, . . . , K} without loss of generality. For any x ∈ D, we have Sx,B2 = (C1 )x ∪ · · · ∪ (CK )x for all x ∈ D. We have

dim(Sx,B2 ) = sup dim(Ci )x = sup dim(Ci ) − dim D 1≤i≤K

1≤i≤K

by the equality (11) and Corollary 5.4(ii). In particular, dim(Sx,B2 ) is independent of the choice of x ∈ D. Set d = dim(Sx,B2 ) for some x ∈ D, then we have D ⊂ SB1 (d). We get d = sup dim(Ci ) − dim D = dim 1≤i≤K

K  i=1

 Ci

 − dim D = dim



x∈D

 {x} × Sx,B2

− dim D

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by Corollary 5.4(ii). Taking the supremum of all cells contained in SB1 (d), we finally obtain ⎛



dim ⎝

⎞



sup

dim

D⊂SB1 (d),D is a cell





{x} × Sx,B2 ⎠ = dim ⎝

x∈SB1 (d)

=

⎛

D⊂SB1 (d):cell





⎞ {x} × Sx,B2 ⎠

x∈D



{x} × Sx,B2

x∈D

=

sup

dim(D) + d

D⊂SB1 (d),D is a cell

= dim(SB1 (d)) + d by Corollary 5.4(ii). 2 Lemma 5.5. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. Let X ⊂ M n be a definable set, where n is a positive integer. Let a ∈ M and b ∈ M n−1 . There exists an open interval I with a ∈ I such that the inequality dim(Y ) ≤ 0 holds true, where Y = {x ∈ I | ∂(Xx ) ∩ B = (∂X)x ∩ B for any open box B ⊂ M n−1 with b ∈ B}. Proof. Fix an arbitrary open interval I with a ∈ I. Note that ∂(Xx ) ⊂ (∂X)x for any x ∈ M . The notation Dx denotes the definable set (∂X)x \ ∂(Xx ). For any open box B in M n−1 , we set IB = {x ∈ I | Dx ∩ B = ∅ and Xx ∩ B = ∅}. Define the definable sets G, B(α, β) and C as follows: G = {(α, β) = (α1 , . . . , αn−1 , β1 , . . . , βn−1 ) ∈ M n−1 × M n−1 | αi < βi for all i}, B(α, β) =]α1 , β1 [× · · · ×]αn−1 , βn−1 [ (∀(α, β) ∈ G) and  C= IB(α,β) × {(α, β)} ⊂ M 2n−1 . (α,β)∈G

Taking a sufficiently small open interval I and a sufficiently small open box B, we may assume that the equality in Lemma 5.4 holds true for C. Fix such I and B. We first demonstrate that dim C∩(I×B 2 ) ≤ 2(n−1). This inequality follows from Lemma 5.4 immediately if we prove that dim IB(α,β) ≤ 0 for any (α, β) ∈ G ∩ B 2 . We assume the contrary. There exists an open interval I  contained in IB(α,β) for some (α, β) ∈ G ∩ B 2 . We have X ∩ (I  × B(α, β)) = ∅ by the definition of IB(α,β) . We have X ∩ (I  × B(α, β)) = ∅ because I  × B(α, β) is an open set. On the other hand, for any x ∈ I  , we get Dx ∩ B(α, β) = ∅. Take a point y ∈ Dx ∩ B(α, β), then we have (x, y) ∈ X ∩ (I  × B(α, β)). It is a contradiction. We next show that dim C ∩ (I × B 2 ) ≥ 2(n − 1) + 1 assuming that dim Y ≥ 1. It contradicts the inequality proved in the previous paragraph. We therefore get dim Y ≤ 0. The remaining task is to show dim C ∩ (I × B 2 ) ≥ 2(n − 1) + 1. The inequality dim C ∩ (I × B 2 ) ≥ dim C ∩ (Y × B 2 ) holds true by Lemma 5.1. Therefore, we demonstrate that dim C ∩ (Y × B 2 ) ≥ 2(n − 1) + 1. This inequality follows from Lemma 5.4 immediately if we prove that dim Cx ∩ B 2 ≥ 2(n − 1) for any x ∈ Y . We have only to show that Cx ∩ B 2 contains an open box for any x ∈ Y . Fix x ∈ Y . There exists (α, β) ∈ G with B(α, β) ⊂ B and x ∈ IB(α,β) by the definition of Y . Take y ∈ Dx ∩B(α, β). For any (α , β  ) ∈ G with y ∈ B(α , β  ) ⊂ B(α, β), we have y ∈ Dx ∩ B(α , β  ) and x ∈ IB(α ,β  ) . We get (α , β  ) ∈ Cx by the definition of the set C. It means that Cx contains an open box. 2

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Theorem 5.6. Consider a locally o-minimal structure M = (M, <, . . .) which admits local definable cell decomposition. The inequality dim(∂S) < dim(S) is satisfied for any definable set S. Proof. Let S be a definable subset of M n . We prove the theorem by the induction on dim(S). It is obvious when dim(S) = 0. We consider the case in which dim(S) > 0. Take an arbitrary point a ∈ M n . We have only to show that dim(∂S∩B) < dim(S∩B) for all sufficiently small open boxes B with a ∈ B and ∂S∩B = ∅ by Corollary 5.5. Take a sufficiently small open box B with a ∈ B. Let σi : M n → M n be a definable map given by σi (x1 , . . . , xn ) = (xi , x1 , . . . , xi−1 , xi+1 , . . . , xn ). Take open intervals I1 , . . . , In such that B = I1 ×I2 ×· · ·×In . Set Ji = I1 × · · · × Ii−1 × Ii+1 × · · · × In for all 1 ≤ i ≤ n. We consider the definable sets Fi defined by Fi = {xi ∈ Ii | ∂(σi (S))xi ∩ Ji = ∂(σi (S)xi ) ∩ Ji } n for all 1 ≤ i ≤ n. We have dim(Fi ) ≤ 0 by Lemma 5.5. Set Hi = M i−1 × Fi × M n−i and H = i=1 Hi = n F1 × · · · × Fn . We get dim H ≤ 0 by Corollary 5.4(iv). We have ∂S ⊂ H ∪ (∂S \ H) = H ∪ i=1 (∂S \ Hi ). Therefore, we have only to show that dim((∂S \ Hi ) ∩ B) < dim(S ∩ B) for all 1 ≤ i ≤ n. It is equivalent to the inequality dim (σi (∂S \ Hi ) ∩ (Ii × Ji )) < dim (σi (S) ∩ (Ii × Ji )) by Corollary 5.4(i). We may assume that i = 1 without loss of generality. We omit the suffixes of F1 , H1 , I1 and J1 in order to simplify the notation. Consequently, we have only to show the following inequality: dim ((∂S \ H) ∩ (I × J)) < dim (S ∩ (I × J)) .

(12)

We have (∂S \ H) ∩ (I × J) = S ∩ (I × J) ⊃





{x} × (∂(Sx ) ∩ J) and

{x} × (Sx ∩ J).

x∈I\F

For any d ∈ {−∞, 0, 1, . . . , n}, set Td = {x ∈ I \ F | dim(∂(Sx ) ∩ J) = d} and Td = {x ∈ I \ F | dim(Sx ∩ J) = d}. We obtain 



dim

 {x} × (∂(Sx ) ∩ J)

= dim(Td ) + d and

x∈Td

⎛ dim ⎝

 x∈Td

(13)

x∈I\F

⎞ {x} × (Sx ∩ J)⎠ = dim(Td ) + d

(14)

M. Fujita / Annals of Pure and Applied Logic 171 (2020) 102756

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for all d ∈ {−∞, 0, 1, . . . , n} by Lemma 5.4. Set Kk = {d | dim(Td ) = k} and Kk = {d | dim(Td ) = k} for k = 0, 1. We have dim ((∂S \ H) ∩ (I × J)) = max{max d, max d + 1} and

(15)

dim (S ∩ (I × J)) ≥ max{max d, max d + 1}

(16)

d∈K0

d∈K0

d∈K1

d∈K1

by the relations (13), (14), Lemma 5.1 and Corollary 5.4(ii). We get the following inequality by the induction hypothesis: dim(∂(Sx ) ∩ J) < dim(Sx ∩ J).

(17)

max{max d, max d + 1} < max{max d, max d + 1}.

(18)

We demonstrate the inequality d∈K0

d∈K1

d∈K0

d∈K1

When max{maxd∈K0 d, maxd∈K1 d + 1} = maxd∈K0 d, there exists d ∈ K0 ∪ K1 with maxd∈K0 d < d by the inequality (17). We get max{maxd∈K0 d, maxd∈K1 d +1} = maxd∈K0 d < d ≤ max{maxd∈K0 d, maxd∈K1 d + 1}. When max{maxd∈K0 d, maxd∈K1 d + 1} = maxd∈K1 d + 1, there exists d ∈ K1 with maxd∈K1 d <  d by the inequality (17). We get max{maxd∈K0 d, maxd∈K1 d + 1} = maxd∈K1 d + 1 < d + 1 ≤ max{maxd∈K0 d, maxd∈K1 d + 1}. The inequality (12) follows from the equalities (15), (16) and (18). 2 Declaration of competing interest The author has no conflict of interest, financial or otherwise. Acknowledgement The author is grateful to the referees for their insightful comments, especially, for proposing a much shorter proof of Proposition 2.1 than the proof initially given by the author. References [1] R. Arefiev, On the property of monotonicity for weakly o-minimal structures, in: A.G. Pinus, K.N. Ponomarev (Eds.), Algebra and Model Theory II, Novosibirsk, 1997, pp. 8–15. [2] A. Dolich, C. Miller, C. Steinhorn, Structure having o-minimal open core, Trans. Amer. Math. Soc. 362 (2010) 1371–1411. [3] A. Fornasiero, Locally o-minimal structures and structures with locally o-minimal open core, Ann. Pure Appl. Logic 164 (2013) 211–229. [4] T. Kawakami, K. Takeuchi, H. Tanaka, A. Tsuboi, Locally o-minimal structures, J. Math. Soc. Japan 64 (2012) 783–797. [5] J. Knight, A. Pillay, C. Steinhorn, Definable sets in ordered structure II, Trans. Amer. Math. Soc. 295 (1986) 593–605. [6] D. Macpherson, D. Marker, C. Steinhorn, Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 (2000) 5435–5483. [7] D. Marker, Model Theory: An Introduction, Graduate Texts in Mathematics, vol. 217, Springer Science+Business Media, New York, 2002. [8] C. Miller, Expansions of dense linear orders with the intermediate value property, J. Symbolic Logic 66 (2001) 1783–1790. [9] A. Pillay, C. Steinhorn, Definable sets in ordered structure I, Trans. Amer. Math. Soc. 295 (1986) 565–592. [10] M. Shiota, Geometry of Subanalytic and Semialgebraic Sets, Progress in Mathematics, vol. 150, Springer Science+Business Media, LLC, New York, 1997. [11] C. Toffalori, K. Vozoris, Notes on local o-minimality, MLQ Math. Log. Q. 55 (2009) 617–632. [12] L. van den Dries, Tame Topology and o-Minimal Structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. [13] R. Wencel, Topological properties of sets definable in weakly o-minimal structures, J. Symbolic Logic 75 (2010) 841–867.