aa r X i v : . [ m a t h . L O ] J un LOCALLY O-MINIMAL STRUCTURES WITH TAMETOPOLOGICAL PROPERTIES
MASATO FUJITA
Abstract.
We consider locally o-minimal structures possessing tame topo-logical properties shared by models of DCTC and uniformly locally o-minimalexpansions of the second kind of densely linearly ordered abelian groups. Wederive basic properties of dimension of a set definable in the structures includ-ing the addition property, which is the dimension equality for definable mapswhose fibers are equi-dimensional. A decomposition theorem into special sub-manifolds is also demonstrated. Introduction
An o-minimal structure enjoys many tame topological properties such as mono-tonicity and definable cell decomposition [13]. A locally o-minimal structure wasfirst introduced in [12] as a local counterpart of an o-minimal structure. In spite ofits similarity to an o-minimal structure in its definition, a locally o-minimal struc-ture does not enjoy the localized tame properties enjoyed by o-minimal structuressuch as the local monotonicity theorem and the local definable cell decompositiontheorem. Lack of tame topological properties prevents us to establish a tame dimen-sion theory for sets definable in the structures. We expect that discrete definableset is of dimension zero. We also hope that the projection image of a definableset is of dimension not greater than the dimension of the original set. However,the projection image of a discrete definable set is not necessarily discrete in somelocally o-minimal structure as in [6, Example 12].We can recover tame topological properties if we employ additional assumptionson locally o-minimal structures. We can also establish a tame dimension theoryusing such tame topological properties.For instance, the author proposed uniformly locally o-minimal structures of thesecond kind in [3]. Local definable cell decomposition theorem [3, Theorem 4.2]holds true when they are definably complete. We can derive several natural di-mension formulae [3, Section 5] and [4, Theorem 1.1, Corollary 1.2] for a definablycomplete uniformly locally o-minimal expansion of the second kind of a denselylinearly ordered abelian group using the tame topological properties. A definablycomplete uniformly locally o-minimal expansion of the second kind of a denselylinearly ordered abelian group is called a model of DCULOAS in this paper.Another example is a model of DCTC. Schoutens tried to figure out the commonfeatures of the models of the theory of all o-minimal structures in his challengingwork [10]. A model of DCTC was introduced in it. It enjoys tame topological
Mathematics Subject Classification.
Primary 03C64.
Key words and phrases. uniform local o-minimality, Special submanifold, Type complete. and dimensional properties as partially given in [10] and also demonstrated in thispaper.The purpose of this paper is to develop dimension formulae when locally o-minimal structures are definably complete and enjoy the tame topological prop-erties given in the following definition. The previous two examples possess theseproperties.
Definition 1.1.
Consider a locally o-minimal structure M = ( M, <, . . . ). Weconsider the following properties on M .(a) The image of a nonempty definable discrete set under a coordinate projec-tion is again discrete.(b) Let X and X be definable subsets of M m . Set X = X ∪ X . Assumethat X has a nonempty interior. At least one of X and X has a nonemptyinterior.(c) Let A be a definable subset of M m with a nonempty interior and f : A → M n be a definable map. There exists a definable open subset U of A suchthat the restriction of f to U is continuous.(d) Let X be a definable subset of M n and π : M n → M d be a coordinateprojection such that the the fibers X ∩ π − ( x ) are discrete for all x ∈ π ( X ).Then, there exists a definable map τ : π ( X ) → X such that π ( τ ( x )) = x for all x ∈ π ( X ).The following formulae on dimensions are demonstrated in this paper under theassumption that definably complete locally o-minimal structures enjoy the proper-ties (a) through (d) in Definition 1.1.(1) The inequality on the dimensions of the domain of definition and the imageof a definable map (Theorem 3.8(5));(2) The inequality on the dimension of the set of points at which a definablefunction is discontinuous (Theorem 3.8(6));(3) The inequality on the dimensions of a definable set and its frontier (Theo-rem 3.8(7));(4) Addition property. The dimension equality for definable maps whose fibersare equi-dimensional (Theorem 3.14).In o-minimal structures, definable sets are partitioned into finite number of nicelyshaped definable subsets called cells [13, Chapter 3 (2.11)]. Partitions into finitecells are unavailable in locally o-minimal structures. We provide alternative par-titions into finite number of another nicely shaped definable subsets called specialsubmanifolds. The definition of special submanifolds is found in Definition 4.1.Partitions into special submanifolds are available in locally o-minimal structuresenjoying the properties (a) through (d) in Definition 1.1 (Theorem 4.4 and Theo-rem 4.5). It is already known that decomposition theorems into special subman-ifolds also hold true for locally o-minimal expansions of fields [1] and d-minimalexpansions of the real field [9, 11].A model of DCULOAS and a model of DCTC possess the properties (a) through(d) in Definition 1.1. Therefore, the above dimension formulae and the decom-position theorem into special submanifolds also hold true for them. Some of theassertions were presented in the previous studies. In the case of a model of DCU-LOAS, the dimension inequalities (1) through (3) were demonstrated in [3, 4]. Theaddition property (4) and the decomposition theorem first appear in this paper. OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 3
As to a model of DCTC, the inequalities in the planar case were proved in [10]. Adefinably complete locally o-minimal expansion of a field is a model of DCTC andthe decomposition theorem for it is already known [1]. The author could not findthe dimension formulae for higher dimensions and the decomposition theorem fora general model of DCTC in the previous studies.This paper is organized as follows. We first derive several basic lemmas in Section2. Section 3 is devoted for the derivation of the dimension formulae (1) through(4). We also prove the decomposition theorem into special submanifolds in Section4. In Section 5, we demonstrate that a model of DCULOAS is a locally o-minimalstructure enjoying the properties (a) through (d) in Definition 1.1. We prove thata model of DCTC possess the same properties in Section 6.We introduce the terms and notations used in this paper. The term ‘definable’means ‘definable in the given structure with parameters’ in this paper. For anyset X ⊂ M m + n definable in a structure M = ( M, . . . ) and for any x ∈ M m , thenotation X x denotes the fiber defined as { y ∈ M n | ( x, y ) ∈ X } unless anotherdefinition is explicitly given. For a linearly ordered structure M = ( M, <, . . . ), anopen interval is a definable set of the form { x ∈ R | 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.2. Preliminary
We first review the definitions of local o-minimality and definably completeness.
Definition 2.1 ([12]) . A densely linearly ordered structure without endpoints M =( M, <, . . . ) is locally o-minimal if, for every definable subset X of M and for everypoint 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. Definition 2.2 ([8]) . An expansion of a densely linearly ordered set without end-points M = ( M, <, . . . ) is definably complete if any definable subset X of M hasthe supremum and infimum in M ∪ {±∞} .We give an equivalence condition for a definably complete structure being locallyo-minimal. Lemma 2.3.
Consider a definably complete structure M = ( M, <, . . . ) . The fol-lowing conditions are equivalent:(1) The structure M is a locally o-minimal structure.(2) Any definable set in M has a nonempty interior or it is closed and discrete.Proof. The implication (1) ⇒ (2) is obvious by the definition of local o-miniality.We demonstrate the opposite implication. Let X be a definable subset in M .Consider the boundary Y = X \ int( X ). Let J be an arbitrary open interval in M . We have Y ∩ J = ∅ if and only if J ⊂ int( X ) or J ⊂ M \ X by [8, Corollary1.5]. For any arbitrary point a ∈ M , there exists an open interval I containing thepoint a such that I ∩ Y is an empty set or a singleton { a } because Y is closed anddiscrete by the assumption. The open intervals { x ∈ I | x > a } and { x ∈ I | x < a } are contained in int( X ) or M \ X . Hence, I ∩ X is a finite union of points and openintervals. We have demonstrated that the structure M is locally o-minimal. (cid:3) M. FUJITA
We introduce two consequences of the property (a) in Definition 1.1.
Lemma 2.4.
Consider a definably complete locally o-minimal structure with theproperty (a) in Definition 1.1. A definable discrete set is closed.Proof.
Let M = ( M, <, . . . ) be the structure in consideration. Let X be a nonemptydiscrete definable subset of M n . Let π k : M n → M be the coordinate projectiononto the k -th coordinate for all 1 ≤ k ≤ n . The images π k ( X ) are discrete by theproperty (a). They are closed by Lemma 2.3. Let x be an accumulation point of X .We have π k ( x ) ∈ π k ( X ) for all 1 ≤ k ≤ n because π k ( x ) are accumulation pointsof π k ( X ) and π k ( X ) are closed. It means that x ∈ X . (cid:3) Lemma 2.5.
Consider a definably complete locally o-minimal structure M =( M, <, . . . ) with the property (a) in Definition 1.1. Let f : X → M be a defin-able map. If the image f ( X ) and all fibers of f are discrete, then so is X .Proof. We first reduce to the case in which f is the restriction of a coordinate pro-jection. Let X be a definable subset of M n and π : M n +1 → M be the coordinateprojection onto the last coordinate. Consider the graph Γ( f ) of the definable map f . The image π (Γ( f )) = f ( X ) and all the fibers Γ( f ) ∩ π − ( x ) are discrete by theassumption. If the graph Γ( f ) is discrete, the definable set X is also discrete bythe property (a) because X is the projection image of the discrete set Γ( f ). Wehave reduced to the case in which f is the restriction of the coordinate projectiononto the last coordinate π : M n +1 → M to a definable subset Y of M n +1 .Take an arbitrary point x ∈ Y . Since π ( Y ) is discrete by the assumption, we cantake an open interval I containing the point π ( x ) such that π ( Y ) ∩ I is a singleton.Since the fiber π − ( π ( x )) ∩ Y is discrete, there exists an open box B containing thepoint x such that Y ∩ ( B × { π ( x ) } ) is a singleton. The open box B × I contains thepoint x and the intersection of Y with B ∩ I is a singleton. We have demonstratedthat Y is discrete. (cid:3) We introduce the following notations for simplicity.
Notation . Consider a locally o-minimal structure M = ( M, <, . . . ). A definablefunction f : X → M ∪ {∞} denotes a pair of disjoint definable subsets X o and X ∞ with X = X o ∪ X ∞ and a definable function defined on X o . We consider thatthe function f is constantly ∞ on X ∞ . The function f : X → M ∪ {∞} is calledcontinuous if X = X ∞ or X = X o and the function f is continuous. If the structure M enjoys the properties (b) and (c) in Definition 1.1, the restriction of f to somedefinable open set is continuous when the domain of definition X has a nonemptyinterior. We define g : X → M ∪ {−∞} similarly.The following lemma is a consequence of the properties (b) and (c). Lemma 2.7.
Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (b) and (c) in Definition 1.1. Let X be a defin-able subset of M m + n . Set S = { x ∈ M m | the fiber X x has a nonempty interior } .If S has a nonempty interior, X also has a nonempty interior.Proof. We first consider the case in which n = 1. Consider the definable function f : S → M ∪ {−∞} given by f ( x ) = inf { y ∈ M | y is contained in the interior of the fiber X x } . OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 5
Define the definable function g : S → M ∪ {∞} by g ( x ) = sup { y ∈ M | X x contains an interval ( f ( x ) , y ) } .There exists an open box V contained in S such that the restrictions of f and g to V are continuous by the properties (b) and (c) in Definition 1.1. The set X contains an open set { ( x, y ) ∈ V × M | f ( x ) < y < g ( x ) } . We have demonstratedthe lemma for n = 1.We next consider the case in which n >
1. Consider the projection π : M m + n → M m + n − forgetting the last coordinate and the projection π : M m + n − → M m onto the first m coordinates. Set π = π ◦ π , T = { t ∈ π ( X ) | the fiber X t contains a nonempty open interval } and U = { u ∈ π ( X ) | the fiber T u has a nonempty interior } .The definable set S is contained in U . In particular, U has a nonempty interior.Applying the lemma to the pair of T and the restriction of π to T , we haveint( T ) = ∅ by the induction hypothesis. We get int( X ) = ∅ by the lemma for n = 1. (cid:3) We do not use the following proposition in this paper, but it is worth to bementioned. It is a stronger version of definable Baire property discussed in [2, 5].
Proposition 2.8 (Strong definable Baire property) . Consider a definably completelocally o-minimal structure M = ( M, <, . . . ) enjoying the properties (a), (b) and(c) in Definition 1.1. Take c ∈ M . Let { X h r i} r>c be a parameterized increasingfamily of definable sets of M n ; that is, there exists a definable subset X of M n +1 such that X h r i coincides with the fiber X r for any r > c and we have X h r i ⊂ X h s i if r < s . Set X = S r>c X h r i . The definable set X h r i has a nonempty interior forsome r > c if X has a nonempty interior.Proof. We prove the proposition by the induction on n . We first consider the case inwhich n = 1. Assume that X h r i have empty interiors for all r > c . They are closedand discrete by Lemma 2.3. Set Y = { ( r, x ) ∈ M | r = inf { s ∈ M | x ∈ X h s i}} .The set Y is discrete. In fact, consider the fiber Y r of Y at r . Take r ′ ∈ M with r ′ > r . We have Y r ⊂ X h r ′ i because { X h r i} r>c is a parameterized increasingfamily. For any x ∈ M , there exists an open interval I containing the point x suchthat X h r ′ i ∩ I consists of at most one point because X h r ′ i is discrete and closed.Since Y r ⊂ X h r ′ i whenever r < r ′ , the intersection Y ∩ (( c, r ′ ) × I ) consists of atmost one point. We have shown that Y is discrete. Since X is the projection imageof Y , X is also discrete by the property (a). We have demonstrated that X has anempty interior.We next consider the case in which n >
1. Assume that X has a nonemptyinterior. An open box B is contained in X . We may assume that X = B considering X h r i ∩ B instead of X h r i . We lead to a contradiction assuming that X h r i haveempty interiors for all r > c . Take an open box B in M n − and an open interval I with B = B × I . Set Y h r i = { x ∈ B | ( X h r i ) x contains an open interval } .The set Y h r i has an empty interior by Lemma 2.7. We have B = S r>c Y h r i bythe induction hypothesis. Take x ∈ B \ (cid:0)S r>c Y h r i (cid:1) . The union S r>c ( X h r i ) x hasan empty interior because the fibers ( X h r i ) x have empty interiors. It contradictsthe equality S r>c ( X h r i ) x = I . (cid:3) M. FUJITA Dimension theory
We develop a dimension theory for locally o-minimal structures possessing theproperties in Definition 1.1.
Definition 3.1 (Dimension) . Consider an expansion of a densely linearly orderwithout endpoints M = ( M, <, . . . ). Let X be a nonempty definable subset of M n . The dimension of X is the maximal nonnegative integer d such that π ( X )has a nonempty interior for some coordinate projection π : M n → M d . We setdim( X ) = −∞ when X is an empty set.A definable set of dimension zero is always closed and discrete. Proposition 3.2.
Consider a locally o-minimal structure satisfying the property(a) in Definition 1.1. A definable set is of dimension zero if and only if it is discrete.When it is of dimension zero, it is also closed.Proof.
Let X be a definable subset of M n . The definable set X is discrete if andonly if the projection image π ( X ) has an empty interior for all the coordinateprojections π : M n → M by the property (a). Therefore, X is discrete if and onlyif dim X = 0. A discrete definable set is always closed by Lemma 2.4. (cid:3) The following two lemmas are key lemmas of this paper.
Lemma 3.3.
Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (b) and (c) in Definition 1.1. Let X be a defin-able subset of M n of dimension d and π : M n → M d be a coordinate projection suchthat the projection image π ( X ) has a nonempty interior. There exists a definableopen subset U of M d contained in π ( X ) such that the fibers X ∩ π − ( x ) are discretefor all x ∈ U .Proof. Permuting the coordinates if necessary, we may assume that π is the pro-jection onto the first d coordinates. Set S = { x ∈ π ( X ) | the fiber X ∩ π − ( x ) is not discrete } .We have S = { x ∈ π ( X ) | dim( X ∩ π − ( x )) > } by Proposition 3.2. We want toshow that S has an empty interior. Assume the contrary. Let ρ j : M n → M be thecoordinate projections onto the j -th coordinate for all d < j ≤ n . Set S j = { x ∈ π ( X ) | ρ j ( X ∩ π − ( x )) contains an open interval } .We have S = S d Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. Let X ⊂ Y bedefinable subsets of M n . Assume that there exist a coordinate projection π : M n → OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 7 M d and a definable open subset U of M d contained in π ( X ) such that the fibers Y x are discrete for all x ∈ U . Then, there exist • a definable open subset V of U , • a definable open subset W of M n and • a definable continuous map f : V → X such that • π ( W ) = V , • Y ∩ W = f ( V ) and • the composition π ◦ f is the identity map on V .Proof. Permuting the coordinates if necessary, we may assume that π is the projec-tion onto the first d coordinates. Let ρ j : M n → M be the coordinate projectionsonto the j -th coordinate for all d < j ≤ n . The fiber Y x is discrete for any x ∈ U by the assumption. Since X x is a definable subset of Y x , X x is also a discrete set.There exists a definable map g : U → X such that the composition π ◦ g is the iden-tity map on U by the property (d). Note that ρ j ( Y x ) is discrete and closed by theproperty (a) and Lemma 2.4. Consider the definable functions κ + j : U → M ∪{ + ∞} defined by κ + j ( x ) = (cid:26) inf { t ∈ ρ j ( Y x ) | t > ρ j ( g ( x )) } if { t ∈ ρ j ( Y x ) | t > ρ j ( g ( x )) } 6 = ∅ ,+ ∞ otherwisefor all d < j ≤ n . We define κ − j : U → M ∪ {−∞} similarly. Then, we have π − ( x ) ∩ Y ∩ ( { x } × ( κ − d +1 ( x ) , κ + d +1 ( x )) × · · · × ( κ − n ( x ) , κ + n ( x ))) = { g ( x ) } for all x ∈ U . There exists a definable open subset V of U such that the restriction f of g to V and the restrictions of κ − j and κ + j to V are all continuous by theproperties (b) and (c). Set W = { ( x, y d +1 , . . . , y n ) ∈ V × M n − d | κ − j ( x ) < y j <κ + j ( x ) for all d < j ≤ n } . The definable sets V and W and a definable continuousmap f satisfy the requirements. (cid:3) Summarizing the above two lemmas, we get the following lemma. Lemma 3.5. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. Let X ⊂ Y bedefinable subsets of M n of dimension d . There exist • a coordinate projection π : M n → M d , • a definable open subset V of π ( X ) , • a definable open subset W of M n and • a definable continuous map f : V → X such that • π ( W ) = V , • Y ∩ W = f ( V ) and • the composition π ◦ f is the identity map on V .Proof. Immediate from the definition of dimension, Lemma 3.3 and Lemma 3.4. (cid:3) We also need the following lemma and its corollary. Lemma 3.6. Let M = ( M, <, . . . ) be as in Lemma 3.5. Let C ⊂ M n be a definableopen subset and f : C → M n be a definable injective continuous map. The image f ( C ) has a nonempty interior. M. FUJITA Proof. We may assume that C is an open box without loss of generality. The lemmais obvious when n = 0. We assume that n > 0. We lead to a contradiction assumingthat f ( C ) has an empty interior. Set d = dim f ( C ). We have 0 ≤ d < n . When d = 0, the set f ( C ) is discrete by Proposition 3.2. The image f ( C ) is a singleton by[8, Proposition 1.6] because the open box C is definably connected. Contradictionto the assumption that f is injective.We next consider the case in which d = 0. Applying Lemma 3.5, we can takea coordinate projection π : M n → M d and a definable open set W of M n suchthat the restriction of π to f ( C ) ∩ W is injective and its image is a definable openset. We may assume that the restriction of π to f ( C ) is injective by considering f − ( W ) instead of C . Since f is injective and continuous by the assumption, thecomposition π ◦ f is also injective and continuous.Take an open box B contained in C . Let B and B be the open boxes in M d and M n − d with B = B × B , respectively. Take c ∈ B . Consider the definablemap g : B → M d given by g ( x ) = π ( f ( x, c )). It is injective and continuous. Thereexists an open box D in M d with D ⊂ g ( B ) by the induction hypothesis. Takea point x ∈ B with g ( x ) ∈ D and a point c ′ ∈ B sufficiently close to c with c ′ = c . We have π ( f ( x , c ′ )) ∈ D because π ◦ f is continuous. There exists a point x ∈ B with π ( f ( x , c ′ )) = g ( x ) = π ( f ( x , c )) because D ⊂ g ( B ). It contradictsthe fact that π ◦ f is injective. (cid:3) Corollary 3.7. Let M = ( M, <, . . . ) be as in Lemma 3.5. Let B and C be openboxes in M m and M n , respectively. If there exists a definable continuous injectivemap from B to C , we have m ≤ n .Proof. We lead to a contradiction assuming that m > n . Take a definable con-tinuous injective map f : B → C and c ∈ M m − n . Consider the definable map g : B → C × M m − n given by g ( x ) = ( f ( x ) , c ). It is obviously continuous and injec-tive. The image g ( B ) has a nonempty interior by Lemma 3.6. Contradiction. (cid:3) The following theorem is one of the main theorems of this paper. Theorem 3.8. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. The followingassertions hold true:(1) Let X ⊂ Y be definable sets. Then, the inequality dim( X ) ≤ dim( Y ) holdstrue.(2) Let σ be a permutation of the set { , . . . , n } . The definable map σ : M n → M n is defined by σ ( x , . . . , x n ) = ( x σ (1) , . . . , x σ ( n ) ) . Then, we have dim( X ) =dim( σ ( X )) for any definable subset X of M n .(3) Let X and Y be definable sets. We have dim( X × Y ) = dim( X ) + dim( Y ) .(4) Let X and Y be definable subsets of M n . We have dim( X ∪ Y ) = max { dim( X ) , dim( Y ) } .(5) Let f : X → M n be a definable map. We have dim( f ( X )) ≤ dim X .(6) Let f : X → M n be a definable map. The notation D ( f ) denotes the set ofpoints at which the map f is discontinuous. The inequality dim( D ( f )) < dim X holds true.(7) Let X be a definable set. The notation ∂X denotes the frontier of X definedby ∂X = X \ X . We have dim( ∂X ) < dim X . OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 9 Proof. The assertions (1) and (2) are obvious. We omit the proofs.We demonstrate the assertion (3). Assume that X and Y are definable subsetsof M m and M n , respectively. Set d = dim( X ), e = dim( Y ) and f = dim( X × Y ).We first show that d + e ≤ f . In fact, let π : M m → M d and ρ : M n → M e becoordinate projections such that both π ( X ) and ρ ( Y ) have nonempty interiors. Thedefinable set ( π × ρ )( X × Y ) has a nonempty interior. Therefore, we have d + e ≤ f .We show the opposite inequality. Let Π : M m + n → M f be a coordinate projectionwith int(Π( X × Y )) = ∅ . There exist coordinate projections π : M m → M f and π : M n → M f with Π = π × π . In particular, we get f = f + f . SinceΠ( X × Y ) has a nonempty interior, there exist open boxes C ⊂ M f and D ⊂ M f with C × D ⊂ Π( X × Y ). We get C ⊂ π ( X ) and D ⊂ π ( Y ). Hence, we have d ≥ f and e ≥ f . We finally obtain d + e ≥ f + f = f .We next show the assertion (4). The inequality dim( X ∪ Y ) ≥ max { dim( X ) , dim( Y ) } is obvious by the assertion (1). We show the opposite inequality. Set d = dim( X ∪ Y ). There exists a coordinate projection π : M n → M d such that π ( X ∪ Y ) has anonempty interior by the definition of dimension. At least one of π ( X ) and π ( Y )has a nonempty interior by the property (b) because π ( X ∪ Y ) = π ( X ) ∪ π ( Y ).We may assume that π ( X ) has a nonempty interior without loss of generality. Wehave d ≤ dim( X ) by the definition of dimension. We have demonstrated thatdim( X ∪ Y ) ≤ max { dim( X ) , dim( Y ) } .The next target is the assertion (5). Let X be a definable subset of M m .The notation Γ( f ) denotes the graph of the map f . We first demonstrate thatdim(Γ( f )) = dim( X ). In fact, the inequality dim( X ) ≤ dim(Γ( f )) is obvious be-cause X is the projection image of Γ( f ). Set d = dim(Γ( f )) and e = dim( X ).Applying Lemma 3.5 to the graph Γ( f ), we can take a coordinate projection π : M m + n → M d , an open box V contained in π (Γ( f )) and a definable continuousmap τ : V → Γ( f ) such that the composition π ◦ τ is the identity map on V . Inparticular, the map τ is injective.Let Π : M m + n → M m be the projection onto the first m -coordinate. Therestriction of Π to the graph Γ( f ) is obviously injective. Applying Lemma 3.5 tothe set X , we can take a coordinate projection ρ : M m → M e and a definable opensubset W of M m such that the restriction of ρ to W ∩ X is injective. The inverseimage (Π ◦ τ ) − ( W ) contains an open box because Π ◦ τ is continuous. Replacing V with the open box, we may assume that the restriction of ρ to Π( τ ( V )) is injective.We finally get the definable continuous injective map ρ ◦ Π ◦ τ : V → M e . We have d ≤ e by Corollary 3.7. We have shown that dim X = dim Γ( f ).It is now obvious that dim f ( X ) ≤ dim Γ( f ) = dim X because f ( X ) is theprojection image of Γ( f ).We demonstrate the assertion (6). Let X be a definable subset of M m . We leadto a contradiction assuming that d = dim X = dim D ( f ). By Lemma 3.5, there exista coordinate projection π : M m → M d , definable open subsets V ⊂ π ( D ( f )) and W ⊂ M m and a definable continuous function g : V → D ( f ) such that π ( W ) = V , X ∩ W = g ( V ) and π ◦ g is the identity map on V . Shrinking V and replacing W with W ∩ π − ( V ) if necessary, we may assume that f ◦ g is continuous by theproperty (c). Since g is a definable homeomorphism onto its image, the function f is continuous on g ( V ) = X ∩ W . On the other hand, f is discontinuous everywhere on X ∩ W because X ∩ W is open in X and X ∩ W = g ( V ) is contained in D ( f ).Contradiction. We have demonstrated the assertion (6).The remaining task is to demonstrate the assertion (7). Take distinct elements c, d ∈ M . Consider the definable function f : X → M given by f ( x ) = (cid:26) c if x ∈ X and d otherwise.It is obvious that D ( f ) contains ∂X . The assertion (7) follows from the assertions(1) and (6). (cid:3) Remark . Theorem 3.8 (1) through (3) hold true for any expansion of a denselylinearly order without endpoints. Theorem 3.8 (4) is valid for any locally o-minimalstructure with the property (b).A constructible set is a finite boolean combination of open sets. We get thefollowing corollary: Corollary 3.10. Consider a definably complete locally o-minimal structure en-joying the properties (a) through (d) in Definition 1.1. Any definable set is con-structible.Proof. Let X be a definable set of dimension d . We prove that X is constructibleby the induction on d . When d = 0, the definable set X is discrete and closed byProposition 3.2. In particular, it is constructible. When d > 0, the frontier ∂X isof dimension smaller than d by Theorem 3.8(7). It is constructible by the inductionhypothesis. Therefore, X = X \ ∂X is also constructible. (cid:3) The following theorem gives an alternative definition of dimension. The alter-native definition is the same as the definition of dimension given in [3, Definition5.1]. Theorem 3.11. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. A definableset X is of dimension d if and only if the nonnegative integer d is the maximum ofnonnegative integers e such that there exist an open box B in M e and a definableinjective continuous map ϕ : B → X homeomorphic onto its image.Proof. Let d ′ be the maximum of nonnegative integers e satisfying the conditiongiven in the theorem. We first demonstrate d ′ ≤ d . In fact, let B be an openbox contained in M d ′ and ϕ : B → X be a definable injective continuous maphomeomorphic onto its image. We have dim ϕ ( B ) = dim B = d ′ by Theorem3.8(5). We get d = dim X ≥ dim( ϕ ( B )) = d ′ by Theorem 3.8(1).We next demonstrate d ≤ d ′ . Applying Lemma 3.5 to the definable set X , wecan get a coordinate projection π : M n → M d , a definable open box U in π ( X )and a definable continuous map τ : U → X such that π ◦ τ is the identity map on U . In particular, τ is a definable continuous injective map homeomorohic onto itsimage. Therefore, we have d ≤ d ′ by the definition of d ′ . (cid:3) We get the following corollary: Corollary 3.12. Let M = ( M, <, . . . ) be as in Theorem 3.11. Let X be a definablesubset of R n . There exists a point x ∈ M n such that we have dim( X ∩ B ) = dim( X ) for any open box B containing the point x . OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 11 Proof. Set d = dim( X ). There exists an open box U in M d and a definable con-tinuous injective map ϕ : U → X homeomorphic onto its image by Theorem 3.11.Take an arbitrary point t ∈ U and set x = ϕ ( t ). For any open box B containingthe point x , the inverse image ϕ − ( B ) is a definable open set. Take a open box V with t ∈ V ⊂ ϕ − ( B ). The restriction ϕ | V V → X ∩ B is a definable continuousinjective map homeomorphic onto its image. Hence, we have dim( X ∩ B ) ≥ d byTheorem 3.11. The opposite inequality follows from Theorem 3.8(1). (cid:3) We begin to demonstrate the addition property of dimension for definably com-plete locally o-minimal structures enjoying the properties (a) through (d) in Defi-nition 1.1. It is a counterpart of [13, Chapter 4, Proposition 1.5] in the o-minimalcase, that of [14, Theorem 4.2] in the weakly o-minimal case and that of [3, Lemma5.4] in the case of local o-minimal structure admitting local definable cell decom-position. We first treat a special case. Lemma 3.13. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. Let ϕ : X → Y be a definable surjective map whose fibers ϕ − ( y ) are discrete for all y ∈ Y . Wehave dim X = dim Y .Proof. Let X and Y be definable subsets of M m and M n , respectively. Set d =dim( X ) and e = dim( Y ).We first assume that ϕ is continuous. We have d ≥ e by Theorem 3.8(5). Wedemonstrate the opposite inequality. We first reduce to the case in which X is adefinable open subset of M m . There exist a definable open subset U of R d anda definable continuous injective map σ : U → X homeomorphic onto its imageby Theorem 3.11. If the lemma holds true for the composition ϕ ◦ σ , we havedim X = d = dim U = dim ϕ ◦ σ ( U ) ≤ dim Y = e by Theorem 3.8(1). The lemmais also true for the original ϕ . Hence, we may assume that X is open in M m . Inparticular, we have m = d .Let Π : M m + n → M n be the projection onto the last n coordinates. Considerthe graph Γ( ϕ ) of ϕ . Note that Π − ( y ) ∩ Γ( ϕ ) are discrete for all y ∈ Y . Take acoordinate projection π : M n → M e such that π ( Y ) has a nonempty interior. Thedefinable set ( π ◦ Π) − ( z ) ∩ Γ( ϕ ) is discrete and closed if π − ( z ) ∩ Y is discrete for z ∈ M e by Lemma 2.5. By Lemma 3.3 and Lemma 3.4, there exist definable opensubsets V ⊂ π ( Y ) and W ⊂ M m + n and a definable continuous map τ : V → Γ( ϕ )such that π ◦ Π( W ) = V , W ∩ Γ( ϕ ) = τ ( V ) and π ◦ Π ◦ τ is the identity map on V .In particular, the restriction of π ◦ Π to W ∩ Γ( ϕ ) is injective.Let ι : X → Γ( ϕ ) be the natural injection. The map ι is continuous because ϕ is continuous. We may assume that π ◦ Π ◦ ι is injective replacing X with an openbox contained in the definable open set ι − ( W ). We finally obtain the definablecontinuous injective map from an open box in M d to M e . We get d ≤ e by Corollary3.7.We next demonstrate the lemma when ϕ is not necessarily continuous by theinduction on d . When d = 0, the definable set X is discrete and closed by [10,Theorem 4.1, Corollary 4.2]. In particular, the definable map ϕ is continuous.Therefore, the lemma holds true in this case. We next consider the case in which d > 0. Let D ( ϕ ) be the set of points at which ϕ is discontinuous. We have dim D ( ϕ ) < dim X by Theorem 3.8(6). We get dim ϕ ( D ( ϕ )) = dim D ( ϕ ) by the inductionhypothesis. We obtain dim( X \ D ( ϕ )) = dim ϕ ( X \ D ( ϕ )) because ϕ is continuous on X \ D ( ϕ ). We finally get dim ϕ ( X ) = max { dim ϕ ( X \ D ( ϕ )) , dim ϕ ( D ( ϕ )) } =max { dim( X \ D ( ϕ )) , dim( D ( ϕ )) } = dim( X ) by Theorem 3.8(4). (cid:3) The following theorem is the second main theorem of this paper. Theorem 3.14. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. Let ϕ : X → Y be a definable surjective map whose fibers are equi-dimensional; that is, the dimen-sions of the fibers ϕ − ( y ) are constant. We have dim X = dim Y + dim ϕ − ( y ) forall y ∈ Y .Proof. Let X and Y be definable subsets of M m and M n , respectively. Set d =dim( ϕ − ( y )), e = dim( Y ) and f = dim( X ). We first reduce to the case in whichthere exists a coordinate projection π : M m → M d such that π ( ϕ − ( y )) havenonempty interiors for all y ∈ Y . In fact, consider the set Π m,d of all the coordinateprojections of M m onto M d . Set Y π = { y ∈ Y | π ( ϕ − ( y )) has a nonempty interior } .We get Y = S π ∈ Π m,d Y π by the assumption. Assume that the theorem is true forthe restrictions of ϕ to ϕ − ( Y π ) for all π ∈ Π m,d . We havedim X = max π ∈ Π m,d dim ϕ − ( Y π ) = d + max π ∈ Π m,d dim Y π = d + dim( Y )by Theorem 3.8(4). The theorem holds true for the original ϕ . We may assumethat there exists a coordinate projection π : M m → M d such that π ( ϕ − ( y )) havenonempty interiors for all y ∈ Y . We fix such a π through the proof.We next show that d + e ≤ f . By Lemma 3.5, we can get a coordinate projection p : M n → M e , a definable open subset W of M e contained in p ( Y ) and a definablecontinuous injective map τ : W → Y which is homeomorphic onto its image suchthat p ◦ τ is the identity map and p − ( w ) ∩ Y is discrete for any w ∈ W . Considerthe definable set T = { ( w, v ) ∈ W × M d | v ∈ π ( ϕ − ( τ ( w ))) and π − ( v ) ∩ ϕ − ( τ ( w )) is discrete } .The fiber T w has a nonempty interior for any w ∈ W by Lemma 3.3. Therefore, theset T has a nonempty interior by Lemma 2.7. In particular, we have dim( T ) = d + e .Consider the definable subset S = ( p × π ) − ( T ) ∩ Γ ′ ( ϕ ) ∩ ( τ ( W ) × M m ) of M m × M n , where Γ ′ ( ϕ ) denotes the reversed graph of the definable map ϕ givenby Γ ′ ( ϕ ) = { ( y, x ) ∈ Y × X | y = ϕ ( x ) } . It is obvious that ( p × π )( S ) = T and S ∩ ( p × π ) − ( w, v ) are discrete for all ( w, v ) ∈ T . Apply the property (d) to S andthe projection p × π . We can get a definable map ψ ′ : T → S such that ( p × π ) ◦ ψ ′ is the identity map on T . Set ψ = π ◦ ψ ′ : T → X . It is obviously injective. Wehave d + e = dim( T ) = dim ψ ( T ) ≤ f by Lemma 3.13 and Theorem 3.8(1).We next demonstrate the opposite inequality d + e ≥ f . There exist a coordinateprojection q : M m → M f , a definable open subset U of M f contained in q ( X ) anda definable continuous injective map σ : U → X by Lemma 3.5. The notation D ( ϕ )denotes the set of points at which ϕ is discontinuous. Since dim D ( ϕ ) < dim X = f by Theorem 3.8(6), the projection image q ( D ( ϕ )) has an empty interior. Thedifference U \ q ( D ( ϕ )) has a nonempty interior by the property (b). Shrinking U if necessary, we may assume that ϕ is continuous on σ ( U ). Take a coordinateprojection p : M n → M e and a definable set W as in the proof of the inequality d + e ≤ f . Set Z = { ( v, w ) ∈ M d × W | π − ( v ) ∩ ( p ◦ ϕ ) − ( w ) is discrete } . OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 13 We demonstrate that the set Z has a nonempty interior. Fix a point w ∈ W . Wehave only to demonstrate that Z w = { v ∈ M d | π − ( v ) ∩ ( p ◦ ϕ ) − ( w ) is discrete } has a nonempty interior for any w ∈ W by Lemma 2.7.For any z ∈ p − ( w ) ∩ Y , set B ( z ) = { v ∈ M d | π − ( v ) ∩ ϕ − ( z ) is not discrete } .We have dim B ( z ) < d for any z ∈ p − ( w ) by Lemma 2.7. Consider the set D = { ( v, z ) ∈ M d × M n | v ∈ B ( z ) and z ∈ p − ( w ) ∩ Y } . We get dim D =sup z ∈ p − ( w ) ∩ Y dim B ( z ) < d by Theorem 3.11 because p − ( w ) ∩ Y is discrete. Thedefinable set S z ∈ p − ( w ) ∩ Y B ( z ) is the projection image of D , and it is of dimensionsmaller than d by Theorem 3.8(5). In particular, it has an empty interior. Considerthe definable set Z ′ w = S z ∈ p − ( w ) ∩ Y π ( ϕ − ( z )) \ (cid:16)S z ∈ p − ( w ) ∩ Y B ( z ) (cid:17) . The set Z ′ w has a nonempty interior by the property (b) because S z ∈ p − ( w ) ∩ Y π ( ϕ − ( z ))has a nonempty interior by the definition of π . On the other hand, the set Z w contains the set Z ′ w . In fact, take a point v ∈ Z ′ w . Consider the restriction of ϕ to π − ( v ) ∩ ( p ◦ ϕ ) − ( w ). The image is contained in p − ( w ) ∩ Y , and it is discrete. Thefiber at z ∈ p − ( w ) ∩ Y is π − ( v ) ∩ ϕ − ( z ) and it is also discrete by the definitionof B ( z ) and Z ′ w . Finally, the definable set π − ( v ) ∩ ( p ◦ ϕ ) − ( w ) is discrete byapplying Lemma 2.5 to the restriction of ϕ . We have demonstrated that Z w has anonempty interior. Therefore, the definable set Z has a nonempty interior.Take an open box V contained in Z . Consider the definable continuous mapΦ : U → M d × M e given by Φ( x ) = ( π ◦ σ ( x ) , p ◦ ϕ ◦ σ ( x )). Replacing the opendefinable set U with the definable open set Φ − ( V ) if necessary, we may assumethat Φ( U ) ⊂ Z . By the definition of Z , the fiber Φ − ( v, w ) is discrete for any( v, w ) ∈ Z . Therefore, we have f = dim U = dim(Φ( U )) ≤ d + e by Lemma 3.13and Theorem 3.8(1). We have finished the proof of the theorem. (cid:3) The following corollary is the addition property theorem for definably completelocally o-minimal structures enjoying the properties (a) through (d) in Definition1.1. Corollary 3.15 (Addition property) . Let M = ( M, <, . . . ) be as in Theorem 3.14.Let X be a definable subset of M m × M n . Set X ( d ) = { x ∈ M m | dim X x = d } forany nonnegative integer d . The set X ( d ) is definable and we have dim [ x ∈ X ( d ) { x } × X x = dim X ( d ) + d .Proof. It is easy to prove that X ( d ) is definable. We omit the proof. Apply Theorem3.14 to the restriction of the projection Π : M m + n → M m to the set S x ∈ X ( d ) { x } × X x , then we get the corollary. (cid:3) The following corollary also holds true: Corollary 3.16. Let M = ( M, <, . . . ) be as in Theorem 3.14. Let X be a definablesubset of M m + n and π : M m + n → M m be a coordinate projection. Fix a non-negative integer d . Assume that, for any x ∈ M m + n , there exists an open box U containing the point x satisfying the inequality dim( π ( X ∩ U )) ≤ d . Then, we have dim( π ( X )) ≤ d .Proof. We first reduce to the case in which the fibers X ∩ π − ( x ) are equi-dimensionalfor all x ∈ π ( X ). In fact, set Y k = { x ∈ π ( X ) | dim( X ∩ π − ( x )) = k } and X k = X ∩ π − ( Y k ) for all 1 ≤ k ≤ n . They are definable because of the definitionof dimension. Since we have dim( π ( X k ∩ U )) ≤ dim( π ( X ∩ U )) for any open box U by Theorem 3.8(1), the conditions in the corollary are satisfied for X k . Assumethat the corollary holds true for X k . We have dim( Y k ) = dim π ( X k ) ≤ d . Weobtain dim( π ( X )) = max ≤ k ≤ n dim( Y k ) ≤ d by Theorem 3.8(4). The corollary isalso true for X . We have succeeded in reducing to the case in which the fibers areequi-dimensional.Set e = dim( π ( X )) and f = dim( X ∩ π − ( x )) for x ∈ π ( X ). We have dim( X ) = e + f by Theorem 3.14. We can take a point b in R m + n such that dim( X ∩ V ) = e + f for any open box V containing the point b by Corollary 3.12. Choose an open box U containing the point b so that dim( π ( X ∩ U )) ≤ d , which exists by the assumption.Set X ′ = X ∩ U . It is obvious that the fibers X ′ ∩ π − ( x ) are of dimension not greaterthan f for all x ∈ π ( X ∩ U ) = π ( X ′ ). Set Y ′ k = { x ∈ π ( X ′ ) | dim( X ′ ∩ π − ( x )) = k } and X ′ k = X ′ ∩ π − ( Y ′ k ) for 1 ≤ k ≤ f . Since we have X ′ = S fk =1 X ′ k , we getdim( X ′ l ) = dim( X ′ ) = e + f for some 1 ≤ l ≤ f by Theorem 3.8(4). Again byTheorem 3.14 and Theorem 3.8(1), we get e + f = dim π ( X ′ l ) + l ≤ dim( π ( X ∩ U )) + l ≤ d + l . We finally obtain e ≤ d because 0 ≤ l ≤ f . (cid:3) Decomposition into special submanifolds A decomposition theorem into special submanifolds is discussed in this section.We first define special submanifolds. Definition 4.1. Consider an expansion of a densely linearly order without end-points M = ( M, <, . . . ). Let X be a definable subset of M n and π : M n → M d bea coordinate projection. A point x ∈ X is ( X, π )-normal if there exists an openbox B in M n containing the point x such that B ∩ X is the graph of a continuousmap defined on π ( B ) after permuting the coordinates so that π is the projectiononto the first d coordinates.A definable subset is a π -special submanifold or simply a special submanifold if, π ( X ) is a definable open set and, for every point x ∈ π ( X ), there exists an openbox U in M d containing the point x satisfying the following condition: For any y ∈ X ∩ π − ( x ), there exist an open box V in M n and a definable continuous map τ : U → M n such that π ( V ) = U , τ ( U ) = X ∩ V and the composition π ◦ τ is theidentity map on U .Let { X i } mi =1 be a finite family of definable subsets of M n . A decompositionof M n into special submanifolds partitioning { X i } mi =1 is a finite family of specialsubmanifolds { C i } Ni =1 such that S Ni =1 C i = M n , C i ∩ C j = ∅ when i = j and C i has an empty intersection with X j or is contained in X j for any 1 ≤ i ≤ m and1 ≤ j ≤ N . A decomposition { C i } Ni =1 of M n into special submanifolds satisfiesthe frontier condition if the closure of any special manifold C i is the union of asubfamily of the decomposition.The following lemma guarantees that a definable set X in which all the pointsare ( X, π )-normal is always a π -special submanifold in our setting. This propertymakes the proof of the decomposition theorem easy. Lemma 4.2. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (c) in Definition 1.1. Let X bea definable subset of M n and π : M n → M d be a coordinate projection. Assumethat all the points x ∈ X are ( X, π )-normal. Then, X is a π -special submanifold. OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 15 Proof. We may assume that π is the projection onto the first d coordinates withoutloss of generality. It is obvious that π ( X ) is open because X is locally the graph ofa continuous map. We fix a point c ∈ π ( X ). Note that the fiber X c = X ∩ π − ( c )is discrete by the assumption. The fiber X c is also closed by Lemma 2.4. Let p e : M d → M e be the projection onto the first e coordinates for all 0 ≤ e ≤ d . Wedemonstrate the following claim. The lemma is obvious from the claim for e = d . Claim. Let e be a nonnegative integer with 0 ≤ e ≤ d . There exists anopen box U e in M e containing the point p e ( c ) such that, for any y ∈ X c , thereexist an open box V e,y in M d − e and an open box W e,y in M n such that y ∈ W e,y , π ( W e,y ) = U e × V e,y and the intersection of X with W e,y is the graph of a continuousmap defined on U e × V e,y .We prove the claim by the induction on e . The claim follows from the assumptionthat all the points x ∈ X are ( X, π )-normal when e = 0. Consider the case in which e > 0. Let c e be the e -th coordinate of the point c . Take an element d + ,e ∈ M with c e < d + ,e . For any y ∈ X c , let ϕ + ( y ) be the supremum of the point x ∈ M satisfying • c e < x < d + ,e , and • that there exist a ∈ M with a < c e , an open box V e,y in M d − e and an openbox W e,y in M n such that – y ∈ W e,y , – π ( W e,y ) = U e − × ( a, x ) × V e,y and – the intersection of X with W e,y is the graph of a continuous mapdefined on π ( W e,y ).The value ϕ + ( y ) is lager than c e by the induction hypothesis. We get a definablefunction ϕ + : X c → M . The image ϕ + ( X c ) is discrete by the property (a) becausethe fiber X c is discrete. It is closed by Lemma 2.4. Set b e, + = inf { z ∈ ϕ + ( X c ) } .We have b e, + > c e because ϕ + ( X c ) > c e .Take an element d − ,e ∈ M with c e > d − ,e . For any y ∈ X c , we define ϕ − ( y ) asthe infimum of the point x ∈ M satisfying • c e > x > d − ,e , and • that there exist an open box V e,y in M d − e and an open box W e,y in M n such that – y ∈ W e,y , – π ( W e,y ) = U e − × ( x, b e, + ) × V e,y and – the intersection of X with W e,y is the graph of a continuous mapdefined on π ( W e,y ).In the same way as above, the supremum b e, − = sup { z ∈ ϕ − ( X c ) } satisfies theinequality b e, − < c e . Set U e = U e − × ( b e, − , b e, + ). It is now obvious that U e satisfiesthe claim. We have finished the proofs of both the claim and the lemma. (cid:3) We next construct a decomposition of a single definable set. Lemma 4.3. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. Let X bea definable subset of M n . There exists a family { C i } Ni =1 of mutually disjoint specialsubmanifolds with X = S Ni =1 C i and N ≤ n .Proof. We first define the full dimension of a definable subset X of M n . Set d =dim X . The notation Π n,d denotes the set of all the coordinate projections of M n onto M d . The set Π n,d is a finite set. The full dimension fdim( X ) is ( d, e ) bydefinition if d = dim( X ) and e is the number of elements in Π n,d under which theprojection image of X has a nonempty interior. The pairs ( d, e ) are ordered by thelexicographic order.We prove the the theorem by the induction on fdim( X ). When dim( X ) = 0, X is closed and discrete by Proposition 3.2. The definable set X is obviously a specialsubmanifold in this case.We consider the case in which dim( X ) > 0. Set ( d, e ) = fdim( X ). Take acoordinate projection π : M n → M d such that π ( X ) has a nonempty interior. Set G = { x ∈ X | x is ( X, π )-normal } and B = X \ G . It is obvious that any point x ∈ G is ( G, π )-normal. The definable set G is π -special submanifold by Lemma4.2.We demonstrate that π ( B ) has an empty interior. Assume the contrary. Thereexists an open box U such that the fibers B x = π − ( x ) ∩ B are discrete for all x ∈ U by Lemma 3.3. We can take a definable map τ : U → B with π ( τ ( x )) = x for all x ∈ U because the structure M possesses the property (d) in Definition1.1. The dimension of points at which the map τ is discontinuous is of dimensionsmaller than d by Theorem 3.8(6). We may assume that the restriction of τ to U is continuous shrinking U if necessary.Set Z = ∂ ( X \ τ ( U )). We get dim Z = dim ∂ ( X \ τ ( U )) < dim( X \ τ ( U )) ≤ dim X = d by Theorem 3.8(1), (7). We have dim Z = dim Z < d again by Theorem3.8(4), (7). On the other hand, we have d = dim U = dim π ( τ ( U )) ≤ dim τ ( U ) ≤ dim X = d by Theorem 3.8(1), (5). We get dim( τ ( U )) = d . It means that τ ( U ) Z by Theorem 3.8(1).Take a point p in τ ( U ) \ Z . Take a sufficiently small open box V containingthe point p . We have X ∩ V = τ ( U ) ∩ V by the definition of Z and p . Sincethe restriction of τ to U is continuous, there exists an open box U ′ containedin U ∩ τ − ( V ). Consider the open box V ′ = V ∩ π − ( U ′ ). It is obvious that X ∩ V ′ = τ ( U ) ∩ V ′ is the graph of the restriction of τ to U ′ by the definition.Any point τ ( U ) ∩ V ′ is ( X, π )-normal, but it contradicts to the definition of B and the inclusion τ ( U ) ⊂ B . We have shown that π ( B ) has an empty interior. Inparticular, we get fdim( B ) < fdim( X ).There exists a decomposition B = C ∪ . . . ∪ C k of B satisfying the conditions inthe lemma by the induction hypothesis. The decomposition X = G ∪ C ∪ . . . ∪ C k is the desired decomposition of X .It is obvious that the number of special submanifolds N is not greater than n X d =0 (the cardinality of Π n,d ) = n X d =0 (cid:18) nd (cid:19) = 2 n . (cid:3) We finally get the following two decomposition theorems: Theorem 4.4. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. Let { X i } mi =1 be a finite family of definable subsets of M n . There exists a decomposition { C i } Ni =1 of M n into special submanifolds partitioning { X i } mi =1 with N ≤ m + n .Proof. Set X i = X i and X i = M n \ X i for all 1 ≤ i ≤ m . For any σ ∈ { , } m ,the notation σ ( i ) denotes the i -th component of σ . Set X σ = T mi =1 X σ ( i ) i for any OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 17 σ ∈ { , } m . The family { X σ } σ ∈{ , } m is mutually disjoint and satisfies the equality M n = S σ ∈{ , } m X σ . For all σ ∈ { , } m , there exist families { C σ,j } N σ j =1 of mutuallydisjoint special submanifolds with X σ = S N σ j =1 C σ,j and N σ ≤ n by Lemma 4.3.The family S σ ∈{ , } m { C σ,j } N σ j =1 gives the decomposition we are looking for. (cid:3) Theorem 4.5. Consider a definably complete locally o-minimal structure M =( M, <, . . . ) enjoying the properties (a) through (d) in Definition 1.1. Let { X i } mi =1 be a finite family of definable subsets of M n . There exists a decomposition { C i } Ni =1 of M n into special submanifolds partitioning { X i } mi =1 and satisfying the frontiercondition. Furthermore, the number N of special submanifolds is not greater thanthe number uniquely determined only by m and n .Proof. By reverse induction on d , we construct a decomposition { C λ } λ ∈ Λ d of M n into special submanifolds partitioning { X i } mi =1 such that the closures of all thespecial submanifolds of dimension not smaller than d are the unions of subfamiliesof the decomposition.When d = n , take a decomposition { D λ } λ ∈ Λ of M n into special submanifoldspartitioning { X i } mi =1 by Theorem 4.4. Set Λ ′ n = { λ ∈ Λ | dim( D λ ) = n } . Get adecomposition { E λ } λ ∈ f Λ n of M n into special submanifolds partitioning the family { D λ } λ ∈ Λ ∪ { D λ \ D λ } λ ∈ Λ ′ n . Consider the set f Λ n ′ = { λ ∈ f Λ n | E λ is not contained in any D λ ′ with λ ′ ∈ Λ ′ n } .We always have dim( E λ ) < n for all λ ∈ f Λ n ′ by Theorem 3.8(7). Hence, the family { D λ } λ ∈ Λ ′ n ∪{ E λ } λ ∈ f Λ n ′ is trivially a decomposition of M n into special submanifoldspartitioning { X i } mi =1 we are looking for.We next consider the case in which d < n . Let { D λ } λ ∈ Λ d +1 be a decompositionof M n into special submanifolds partitioning { X i } mi =1 such that the closures of allthe special submanifolds of dimension not smaller than d + 1 are the unions of sub-families of the decomposition. It exists by the induction hypothesis. Set Λ ′ d = { λ ∈ Λ d +1 | dim( D λ ) = d } and Λ ′′ d = { λ ∈ Λ d +1 | dim( D λ ) ≥ d } . Get a decomposition { E dλ } λ ∈ f Λ d of M n into special submanifolds partitioning the family { D λ } λ ∈ Λ d +1 ∪{ D λ \ D λ } λ ∈ Λ ′ d . Set f Λ d ′ = { λ ∈ f Λ d | E λ is not contained in any D λ ′ with λ ′ ∈ Λ ′′ d } .The family { D λ } λ ∈ Λ ′′ d ∪ { E λ } λ ∈ f Λ d ′ is a decomposition of M n into special subman-ifolds partitioning { X i } mi =1 we want to construct.The ‘furthermore’ part of the theorem is obvious from the proof. (cid:3) Uniformly locally o-minimal structure of the second kind We consider models of DCULOAS in this section. They were first introducedin [3] and their properties were also investigated in [4]. Their significant feature isthat local definable cell decomposition for them is available. We first review thedefinition of a locally o-minimal structure of the second kind. Definition 5.1 ([3]) . A locally o-minimal structure M = ( M, <, . . . ) is a uniformlylocally o-minimal structure of the second kind if, for any positive integer n , anydefinable 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 X y ∩ I are finite unions of points and open intervals for all y ∈ B . We want to demonstrate that a model of DCULOAS enjoys the properties (a)through (d) in Definition 1.1. We first prove the following definable choice lemma: Lemma 5.2 (Definable choice lemma) . Consider a model of DCULOAS M =( M, <, , + . . . ) . Let π : M m + n → M m be a coordinate projection. Let X and Y bedefinable subsets of M m and M m + n , respectively, satisfying the equality π ( Y ) = X .There exists a definable map ϕ : X → Y such that π ( ϕ ( x )) = x for all x ∈ X .Proof. We may assume that π is the coordinate projection onto the first m coor-dinate without loss of generality. We fix a positive element c ∈ M . We prove thelemma by induction on m .When m = 1, we define ϕ : X → Y as follows: Fix a point x ∈ X . Considerthe fiber Y x = { y ∈ M | ( x, y ) ∈ Y } . Set Y + x = { y ∈ Y x | y ≥ } and Y − x = { y ∈ Y x | y ≤ } . When the definable set Y + x is not an empty set, considerthe element y = inf( Y + x ). If y ∈ Y x , set ϕ ( x ) = ( x, y ). Otherwise, consider y = sup { y > y | y ′ ∈ Y x for all y < y ′ < y } ∈ M ∪ {∞} . When y = ∞ , put ϕ ( x ) = ( x, y + c ). Otherwise, set ϕ ( x ) = ( x, ( y + y ) / ϕ : X → Y in the same manner considering Y − x instead of Y + x when Y + x is an empty set.We next consider the case in which n > 1. Let π : M m + n → M m + n − and π : M m + n − → M m be the coordinate projections forgetting the last coordinateand forgetting the last n − Z = π ( Y ). We have π ( Z ) = X Applying the induction hypothesis to the pair of X and Z , we get adefinable map ϕ : X → Z such that the composition π ◦ ϕ is the identity map.Applying the lemma for n = 1 to the pair of Z and Y , we obtain a definable map ϕ : Z → Y with π ( ϕ ( z )) = z for all z ∈ Z . The composition ϕ = ϕ ◦ ϕ is thedefinable map we are looking for. (cid:3) The following proposition claims that a model of DCULOAS possesses the prop-erties in Definition 1.1. As a consequence of the proposition, the assertions inSection 2 through Section 4 hold true for it.In the proof of the proposition, the claim that the structure in considerationpossesses the property (a) is simply called the property (a). Proposition 5.3. A model of DCULOAS enjoys the properties (a) through (d) inDefinition 1.1.Proof. We first demonstrate the property (a). We temporarily employ a definitionof dimension different from Definition 3.1. The dimension considered here is thatgiven in [3, Definition 5.1]. The above two definitions coincide by Theorem 3.11once we obtain this proposition.A discrete definable set is of dimension zero by [3, Lemma 5.2]. The projectionimage of the set of dimension zero is again of dimension zero by [4, Theorem 1.1].It is discrete by [3, Corollary 5.3]. We have demonstrated the property (a).The property (b) follows from [3, Theorem 3.3]. The property (c) is a directcorollary of [3, Corollary 5.3] and [4, Corollary 1.2]. The property (d) follows fromLemma 5.2. (cid:3) Model of DCTC Schoutens tried to figure out the common features of the models of the theoryof all o-minimal structures [10]. A model of DCTC was introduced in his study. OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 19 He demonstrated tame topological properties enjoyed by it in [10]. The followingis the definition of a model of DCTC. Definition 6.1 ([10]) . A structure M = ( M, <, . . . ) is a model of DCTC if itis a definably complete expansion of a densely linearly ordered structure withoutendpoints with type completeness property. A structure enjoys type completenessproperty by definition if the types a + and a − are complete for any a ∈ M ∪ {±∞} .Here, a definable set Y ⊂ M belongs to a + if there exists b ∈ M with b > a and( a, b ) ⊂ Y . We define a − similarly. For instance, any definably complete locallyo-minimal expansion of an ordered field, which is investigated in [1], is a model ofDCTC.We demonstrate that a model of DCTC enjoys the properties in Definition 1.1.We use the following technical definition in the proof. Definition 6.2. Consider an expansion of densely linearly ordered structure with-out endpoints M = ( M, <, . . . ). Let A be a definable subset of M m and f : A → M be a definable function. Let 1 ≤ i ≤ m . The function f is i -constant if, for any a , . . . , a i − , a i +1 , . . . , a n ∈ M , the univariate function f ( a , . . . , a i − , x, a i +1 , . . . , a n )is constant. We define that the function is i -strictly increasing and i -strictly de-creasing in the same way. The function is i -strictly monotone if it is i -constant, i -strictly increasing or i -strictly decreasing. The function f is i -continuous if, for any a , . . . , a i − , a i +1 , . . . , a n ∈ M , the univariate function f ( a , . . . , a i − , x, a i +1 , . . . , a n )is continuous.Now, we get the following proposition. Proposition 6.3. A model of DCTC is a definably complete locally o-minimalstructure enjoying the properties (a) through (d) in Definition 1.1.Proof. A model of DCTC is definably complete by the definition. It is also a locallyo-minimal structure by Lemma 2.3 and [10, Proposition 2.6]. The property (a) is[10, Corollary 4.3].Recall the properties (b) and (c):(b) Let X and X be definable subsets of M m . Set X = X ∪ X . Assumethat X has a nonempty interior. At least one of X and X has a nonemptyinterior.(c) Let A be a definable subset of M m with a nonempty interior and f : A → M n be a definable map. There exists a definable open subset U of A suchthat the restriction of f to U is continuous.We demonstrate them by induction on m simultaneously. The former is the sameas [10, Corollary 5.3], which is the corollary of [10, Proposition 5.1]. The proof of[10, Proposition 5.1] is given only in the planer case. The proof in the case of m > m = 1, the property (b) follows from [10, Proposition 2.6(iv), Corollary2.9]. The property (c) follows from the monotonicity theorem [10, Theorem 3.2].We consider the case in which m > 1. We first prove the property (b). Assumethat X has a nonempty interior. Take a bounded open box B contained in X . Wemay assume that X = B considering X ∩ B and X ∩ B instead of X and X ,respectively. We have B = B × I for some open interval I and an open box B in M m − . Set Y i = { x ∈ B | the fiber ( X i ) x contains an open interval } for i = 1 , Applying the property (b) in the case of m = 1, we obtain B = Y ∪ Y . Applyingthe property (b) for m − B = Y ∪ Y , Y or Y has a nonempty interior. Wemay assume that int( Y ) = ∅ without loss of generality. We may further assumethat Y = B shrinking B if necessary.Consider the function f : B → I given by f ( x ) = inf { y ∈ I | ∃ α ∈ ( X ) x , ∃ β ∈ ( X ) x such that α < y < β and ∀ y ′ with α < y ′ < β , we have y ′ ∈ ( X ) x } .Since ( X ) x contains an open interval and M is definably complete, the function f is well-defined. We next define the function g : B → I by g ( x ) = sup { y ∈ I | y > f ( x ) and ∀ y ′ with f ( x ) < y ′ < y , we have y ′ ∈ ( X ) x } .The function g is also well-defined for the same reason. We have f ( x ) < g ( x ) forall x ∈ B . Apply the property (c) for m − f and g . There exists an open box V such that the restrictions of f and g to V are continuous. The definable set X contains the open set { ( x, y ) ∈ V × M | f ( x ) < y < g ( x ) } . We have proven theproperty (b).We next demonstrate the property (c). We can prove the property (c) for arbi-trary n by an easy induction on n when the property (c) holds true for n = 1. Wemay assume that n = 1. We may further assume that the domain of definition of f is a bounded open box B without loss of generality. We define I and B in thesame way as above. Set X + = { ( x, x ′ ) ∈ I × B | the univariate function f ( · , x ′ ) isstrictly increasing and continuous on a neighborhood of x } , X − = { ( x, x ′ ) ∈ I × B | the univariate function f ( · , x ′ ) isstrictly decreasing and continuous on a neighborhood of x } , X c = { ( x, x ′ ) ∈ I × B | the univariate function f ( · , x ′ ) isconstant on a neighborhood of x } and X p = B \ ( X + ∪ X − ∪ X c ).The fibers ( X p ) x are discrete for all x ∈ B by the monotonicity theorem [10,Theorem 3.2]. In particular, X p has an empty interior. At least one of X + , X − and X c has a nonempty interior by the property (b) we have just proven. Therefore,we may assume that f is 1-strictly monotone and 1-continuous by considering anopen box contained in one of them instead of B . Applying the same argument( m − f is i -strictly monotone and i -continuous forall 1 ≤ i ≤ m . The function f is continuous on B by [13, Lemma 3.2.16]. We haveproven the property (c).The final task is to demonstrate the property (d). Let X be a definable subsetof M n and π : M n → M d be a coordinate projection such that the the fibers X ∩ π − ( x ) are discrete for all x ∈ π ( X ). Since discrete definable sets are alwaysclosed and bounded by [10, Corollary 4.2], the map τ : π ( X ) → X given by τ ( x ) = lexmin ( X ∩ π − ( x )) is a well-defined definable map satisfying π ( τ ( x )) = x for all x ∈ π ( X ). Here, the notation lexmin denotes the lexicographic minimum definedin [8]. We have demonstrated the property (d). (cid:3) OCALLY O-MINIMAL STRUCTURES WITH TAME PROPERTIES 21 Corollary 6.4. A definably complete locally o-minimal expansion of a field pos-sesses the properties (a) through (d) in Definition 1.1.Proof. The corollary follows from Proposition 6.3 because a definably completelocally o-minimal expansion of a field is a model of DCTC. (cid:3) References [1] A. 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