Dimension inequality for a definably complete uniformly locally o-minimal structure of the second kind
aa r X i v : . [ m a t h . L O ] F e b DIMENSION INEQUALITY FOR A DEFINABLY COMPLETEUNIFORMLY LOCALLY O-MINIMAL STRUCTURE OF THESECOND KIND
MASATO FUJITA
Abstract.
Consider a definably complete uniformly locally o-minimal ex-pansion of the second kind of a densely linearly ordered abelian group. Let f : X → R n be a definable map, where X is a definable set and R is the uni-verse of the structure. We demonstrate the inequality dim( f ( X )) ≤ dim( X )in this paper. As a corollary, we get that the set of the points at which f isdiscontinuous is of dimension smaller than dim( X ). We also show that thestructure is defiably Baire in the course of the proof of the inequality. Introduction
A uniformly locally o-minimal structure of the second kind was first defined andinvestigated in the author’s previous work [5]. It enjoys several tame properties suchas local monotonicity. In addition, it admits local definable cell decomposition whenit is definably complete.In [5], the author defined dimension of a set definable in a locally o-minimal struc-ture admitting local definable cell decomposition. Many assertions on dimensionknown in o-minimal structures [2] also hold true for locally o-minimal structuresadmitting local definable cell decomposition which are not necessarily definablycomplete [5, Section 5.5]. An exception is the inequality dim( f ( X )) ≤ dim( X ),where f : X → R n is a definable map. The author gave an example which doesnot satisfy the above dimension inequality in [5, Remark 5.5]. The structure in theexample is not definably complete. A question is whether the dimension inequalityholds true when the structure is definably complete. This paper gives an affirmativeanswer to this question. Our main theorem is as follows: Theorem 1.1.
Let R = ( R, <, + , , . . . ) be a definably complete uniformly locallyo-minimal expansion of the second kind of a densely linearly ordered abelian group.The inequality dim( f ( X )) ≤ dim( X ) holds true for any definable map f : X → R n . We get the following corollary:
Corollary 1.2.
Let R = ( R, <, + , , . . . ) be the same structure as Theorem 1.1. Let f : X → R be a definable function. The set of the points at which f is discontinuousis of dimension smaller than dim( X ) . Mathematics Subject Classification.
Primary 03C64; Secondary 54F45, 54E52.
Key words and phrases. uniformly locally o-minimal structure of the second kind, definablyBaire structure, definably complete.
The author proved the dimension inequality in [6, Theorem 2.4] when the uni-verse of the structure is the set of reals. This fact is not a direct corollary of theabove theorem because the structure should be an expansion of an abelian groupin the theorem.The paper is organized as follows. In Section 2, we first review definitions used inthe paper. We prove several basic facts in Section 3. Satisfaction of the dimensioninequality is relevant to defiably Baire property introduced in [3]. Section 4 treatsthe definably Baire property. We show that a definably complete uniformly locallyo-minimal expansion of the second kind of a densely linearly ordered abelian groupis definably Baire in the section. We finally demonstrate Theorem 1.1 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. A
CBD set is a closed, bounded and definable set. For any set X ⊂ R m + n definable ina structure R = ( R, . . . ) and for any x ∈ R m , the notation X x denotes the fiberdefined as { y ∈ R n | ( x, y ) ∈ X } . For a linearly ordered structure R = ( R, <, . . . ),an open interval is a definable set of the form { x ∈ R | a < x < b } for some a, b ∈ R .It is denoted by ( a, b ) in this paper. We define a closed interval in the same mannerand it is denoted by [ a, b ]. An open box in R n is the direct product of n openintervals. A closed box is defined similarly. Let A be a subset of a topologicalspace. The notations int( A ) and A denote the interior and the closure of the set A , respectively. The notation | S | denotes the cardinality of a set S .2. Definitions
We review the definitions given in the previous works. The definition of a de-finably complete structure is found in [8] and [1]. A locally o-minimal structure isdefined and investigated in [9]. Readers can find the definitions of uniformly locallyo-minimal structures of the second kind and locally o-minimal structures admittinglocal definable cell decomposition in [5]. We use D Σ -sets introduced in [1]. Definition 2.1 (D Σ -sets) . Consider an expansion of a linearly ordered structure R = ( R, <, , . . . ). A parameterized family of definable sets is the family of thefibers of a definable set. A parameterized family { X r,s } r> ,s> of CBD subsetsof R n is called a D Σ -family if X r,s ⊂ X r ′ ,s and X r,s ′ ⊂ X r,s whenever r ≤ r ′ and s ≤ s ′ . A definable subset X of R n is a D Σ -set if X = [ r> ,s> X r,s for someD Σ -family { X r,s } r> ,s> .A parameterized family of definable sets { X s } s> is a definable decreasing familyof CBD sets if we have X s = X r,s for some D Σ -family { X r,s } r> ,s> with X r ,s = X r ,s for all r , r and s .We next review definably Baire property introduced in [3]. Definition 2.2.
Consider an expansion of a densely linearly ordered structure.A parameterized family of definable sets { X r } r> is called a definable increasingfamily if X r ⊂ X r ′ whenever 0 < r < r ′ . A definably complete expansion of adensely linearly ordered structure is definably Baire if the union S r> X r of anydefinable increasing family { X r } r> with int (cid:0) X r (cid:1) = ∅ has an empty interior.The following proposition is a direct corollary of the local definable cell decom-position theorem [5, Theorem 4.2]. IMENSION INEQUALITY FOR UNIFORMLY LOCALLY O-MINIMAL STRUCTURE 3
Proposition 2.3.
Consider a definably complete uniformly locally o-minimal struc-ture of the second kind. It is definably Baire if and only if the union S r> X r ofany definable increasing family { X r } r> with int( X r ) = ∅ has an empty interior.Proof. Because int (cid:0) X r (cid:1) = ∅ iff int( X r ) = ∅ iff X r contains an open cell in this caseby [5, Theorem 4.2]. (cid:3) The dimension of a set definable in a locally o-minimal structure admitting localdefinable cell decomposition is defined in [5, Section 5]. We get the following lemmaon the dimension of the projection image. A lemma similar to it is found in [6],but we give a complete proof here.
Lemma 2.4.
Consider a locally o-minimal structure R = ( R, <, . . . ) admitting localdefinable cell decomposition. Let X be a definable subset of R m + n and π : R m + n → R m be a coordinate projection. Assume that the fibers X x are of dimension ≤ forall x ∈ R m . Then, we have dim X ≤ dim π ( X ) .Proof. For any ( a, b ) ∈ R m × R n , there exist open boxes B a ⊂ R m and B b ⊂ R n with ( a, b ) ∈ B a × B b and dim( X ∩ ( B a × B b )) = dim π ( X ∩ ( B a × B b )) by [5, Lemma5.4]. We have dim π ( X ∩ ( B a × B b )) ≤ dim π ( X ) by [5, Lemma 5.1]. On the otherhand, we have dim( X ) = sup ( a,b ) ∈ R m × R n dim( X ∩ ( B a × B b )) by [5, Corollary 5.3]. Wehave finished the proof. (cid:3) Preliminaries
From now on, we consider a definably complete uniformly locally o-minimalexpansion of the second kind of a densely linearly ordered abelian group R = ( R, <, + , , . . . ). We demonstrate several basic facts in this section. Lemma 3.1.
Let X be a bounded definable set. There exists a definable decreasingfamily of CBD sets { X s } s> with X = S s> X s .Proof. We demonstrate the lemma by the induction on d = dim( X ). When d = 0, X is discrete and closed by [5, Corollary 5.3]. We have only to set X s = X for all s > d >
0. Let ∂X denote the frontier of X .We have dim ∂X < d by [5, Theorem 5.6]. We get dim( X ∩ ∂X ) < d by [5,Proposition 5.1]. There exists a definable decreasing family of CBD sets { Y s } s> with X ∩ ∂X = S s> Y s by the induction hypothesis. Set Z s = { x ∈ X | d ( x, ∂X ) ≥ s } for all s >
0, where the notation d ( x, ∂X ) denotes the distance of the point x tothe set ∂X . They are CBD. It is obvious that S s> Z s = X \ ∂X = X \ ∂X . Set X s = Y s ∪ Z s . The family { X s } s> is a definable decreasing family we are lookingfor. (cid:3) Lemma 3.2.
Any definable set X is a D Σ -set. That is, there exists a D Σ -family { X r,s } r> ,s> with X = S r> ,s> X r,s .Proof. Let X be a definable subset of R n . Set X r = X ∩ [ − r, r ] n . We can constructsubsets X r,s of X r satisfying the condition in the same manner as the proof ofLemma 3.2. We omit the details. (cid:3) Lemma 3.3.
Let X be a bounded definable set and { X s } s> be a definable de-creasing family of CBD sets with X = S s> X s . The CBD set X s has a nonemptyinterior for some s > if X has a nonempty interior. M. FUJITA
Proof.
We prove the lemma following the same strategy as the proof of [1, 3.1]. Let X be a definable subset of R n . We prove the lemma by the induction on n . Wefirst consider the case in which n = 1. Assume that int( X s ) = ∅ for all s >
0. Fixan arbitrary point a ∈ R . There exist a positive integer N , an interval I with a ∈ I and t > < s < t , I ∩ X s contains an open interval or consistsof at most N points by [5, Theorem 4.2]. The sets I ∩ X s consist of at most N points because int( X s ) = ∅ . We get | X ∩ I | = (cid:12)(cid:12)S s> ( I ∩ X s ) (cid:12)(cid:12) ≤ N . In particular, X has an empty interior.We next consider the case in which n >
1. Assume that X has a nonemptyinterior. We show that the definable set X s has a nonempty interior for some s > B = C × I ⊂ R n − × R is contained in X . We have B = S s> ( B ∩ X s ).Hence, we may assume that X is a closed box B without loss of generality.Shrinking B if necessary, we may assume that the fiber ( X s ) x consists of at most M points and N closed intervals for some M > N > s > x ∈ C by [5, Theorem 4.2]. Set I = [ c , c ]. Take 2 N distinct points inthe open interval ( c , c ), say b , . . . , b N . We may assume that b i < b j whenever i < j . Set b = c and b N +1 = c . Put I j = [ b j − , b j ] for all 1 ≤ j ≤ N + 1.Consider the sets Y ks = { x ∈ C | I k ⊂ ( X s ) x } for all s > ≤ k ≤ N + 1.They are CBD. Therefore, nS N +1 k =1 Y ks o s> is a definable decreasing family of CBDsets. We demonstrate that C = S s S N +1 k =1 Y ks . Let x ∈ C be fixed. We have onlyto show that I k ⊂ ( X s ) x for some k and s . For any k , there exists s k > I k ∩ ( X s k ) x ) = ∅ by the induction hypothesis because { I k ∩ ( X s ) x } s> isa decreasing family of CBD sets with I k = S s I k ∩ ( X s ) x . Take s = min { s k | ≤ k ≤ N + 1 } . We have int( I k ∩ ( X s ) x ) = ∅ for all 1 ≤ k ≤ N + 1. Assume that I k ( X s ) x for all k . A maximal closed interval in ( X s ) x should be contained in I k , I k ∪ I k +1 or I k − ∪ I k for some k . Therefore, int( I j ∩ ( X s ) x ) is empty for some1 ≤ j ≤ N + 1. Contradiction. We have proven that I k ⊂ ( X s ) x for some k and s .Apply the induction hypothesis to C = S s> S N +1 k =1 Y ks . The set S N +1 k =1 Y ks hasa nonempty interior for some s >
0. The CBD set Y ks has a nonempty interior forsome k by [5, Theorem 3.3]. The CBD set X s has a nonempty interior because I k × Y ks is contained in X s . (cid:3) Lemma 3.4.
Assume that R is definably Baire. Let X be a definable set and { X r,s } r> ,s> be a D Σ -family with X = S r> ,s> X r,s . The CBD set X r,s has anonempty interior for some r > and s > if X has a nonempty interior.Proof. Let X be a definable subset of R n . Set X ′ r,s = X r,s ∩ [ − r, r ] n . We have X = S r> ,s> X ′ r,s . We may assume that X r = S r> ,s> X r,s is bounded considering X ′ r,s instead of X r,s . The lemma is now immediate from Proposition 2.3 and Lemma3.3. (cid:3) On definably Baire property
We demonstrate that the structure R is definably Baire. Lemma 4.1.
Let X be a bounded definable subset of R n +1 . Set S = { x ∈ R n | X x contains an open interval } .The set S has an empty interior if X has an empty interior. IMENSION INEQUALITY FOR UNIFORMLY LOCALLY O-MINIMAL STRUCTURE 5
Proof.
Assume that S has a nonempty interior. There exists a definable decreasingfamily of CBD sets { X s } s> with X = S s> X s by Lemma 3.1. Set S s = { x ∈ R n | ∃ t ∈ R, [ t − s, t + s ] ⊂ ( X s ) x } for all s >
0. They are CBD by [8, Lemma1.7] because they are the projection images of the CBD sets S s = { ( x, t ) ∈ R n × R | [ t − s, t + s ] ⊂ ( X s ) x } . We have S = S s> S s . In fact, it is obvious that S s> S s ⊂ S by the definition. Take a point x ∈ S . There exist t ∈ R and s > t − s , t + s ] ⊂ X x . In particular, we have int( X x ) = ∅ . We have int( X s ) x = ∅ for some s > t − s , t + s ] ⊂ ( X s ) x bytaking new s and t again. Set s = min { s , s } , then we have x ∈ S s . We havedemonstrated that S = S s> S s .Again by Lemma 3.3, we have int( S s ) = ∅ for some s >
0. We obtain int( X s ) = ∅ by [1, 2.8(2)]. We get int( X ) = ∅ . (cid:3) We reduce to the one-dimensional case.
Lemma 4.2.
The structure R is definably Baire if the union S r> S r of any de-finable increasing family { S r } r> of subsets of R has an empty interior whenever S r have empty interiors for all r > .Proof. Let { X r } r> be a definable increasing family of subsets of R n . Set X = S r> X r . We have only to show that the definable set X r has a nonempty interiorfor some r > X has a nonempty interior. The definable set X contains abounded open box B . We may assume that X is a bounded open box B withoutloss of generality by considering B and { X r ∩ B } r> in place of X and { X r } r> ,respectively.We prove the lemma by the induction on n . The lemma is obvious when n = 0. We next consider the case in which n >
0. We lead to a contradic-tion assuming that X r have empty interiors for all r >
0. Let π : R n → R n − be the projection forgetting the last coordinate. We have B = B × I for someopen box B in R n − and some open interval I . Consider the set Y r = { x ∈ B | the fiber ( X r ) x contains an open interval } for all r >
0. They have emptyinteriors by Lemma 4.1. The union S r> Y r has an empty interior by the induc-tion hypothesis. In particular, we have B = S r> Y r and we can take a point x ∈ B \ (cid:0)S r> Y r (cid:1) . Since x S r> Y r , the fiber ( X r ) x does not contain an openinterval for any r >
0. Therefore, the union S r> ( X r ) x has an empty interior by theassumption. On the other hand, we have I = S r> ( X r ) x because B = S r> X r .It is a contradiction. (cid:3) We prove that R is definably Baire now. Theorem 4.3.
A definably complete uniformly locally o-minimal expansion of thesecond kind of a densely linearly ordered abelian group is definably Baire.Proof.
Let R = ( R, <, + , , . . . ) be the considered structure. Let { X r } r> be adefinable increasing family of subsets of R . Set X = S r> X r . We have only toshow that the definable set X has an empty interior if X r have empty interiors forall r > X r are discrete and closed because the structureis locally o-minimal.Assume that X has a nonempty interior. The definable set X contains an openinterval. Take a point a contained in the open interval. Consider the definablefunction f : { r ∈ R | r > } → { x ∈ R | x > a } defined by f ( r ) = inf { x >a | x ∈ X r } . It is obvious that f is a decreasing function because { X r } r> is a M. FUJITA definable increasing family. We demonstrate that lim r →∞ f ( r ) = a . Let b be anarbitrary point sufficiently close to a with b > a . Since X = S r> X r contains aneighborhood of a , there exists a positive element r ∈ R with b ∈ X r . We have a < f ( r ) ≤ b by the definition of f . We have shown that lim r →∞ f ( r ) = a .Consider the image Im( f ) of the function f . Take a sufficiently small openinterval I containing the point a with I ⊂ X . The intersection I ∩ Im( f ) is a finiteunion of points and open intervals because it is definable in the locally o-minimalstructure R . Take an arbitrary point b ∈ Im( f ) and a point r > b = f ( r ).Since X r is closed, we have b ∈ X r . Any point b ′ ∈ Im( f ) with b ′ > b is alsocontained in X r . In fact, take a point r ′ > b ′ = f ( r ′ ). If r ′ > r , the set X r ′ contains the point b because X r ⊂ X r ′ . We have b ′ = f ( r ′ ) ≤ b by the definition ofthe function f . It is a contradiction. If r ′ < r , we have b ′ ∈ X r ′ ⊂ X r .Set b = inf { b ′ ∈ Im( f ) | b ′ > b } . We have b ∈ X r and b > b because { b ′ ∈ Im( f ) | b ′ > b } ⊂ X r and X r is closed and discrete. The open interval( b, b ) has an empty intersection with Im( f ). We have shown that I ∩ Im( f ) doesnot contain an open interval. The set I ∩ Im( f ) consists of finite points. It is acontradiction to the fact that lim r →∞ f ( r ) = a . (cid:3) Remark . It is already known that a definably complete expansion of an orderedfield is definably Baire [7]. Our research target is a uniformly locally o-minimalstructure of the second kind. A uniformly locally o-minimal expansion of the secondkind of an ordered field is o-minimal by [5, Proposition 2.1]. In this case, it istrivially definably Baire by the definable cell decomposition theorem [2, Chapter 3,(2,11)]. We have more interest in the case in which the structure is not an expansionof an ordered field. 5.
Proof of Theorem 1.1
We demonstrate Theorem 1.1 in this section. We first show that a definable mapis continuous on an open subset of the domain of definition.
Lemma 5.1.
A definable map f : U → R n defined on an open set U is continuouson a nonempty definable open subset of U .Proof. The structure R is definably Baire by Theorem 4.3. We may use Lemma3.4 in the proof.Let U be a definable open subset of R m . Consider the projection π : R m + n → R m onto the first m coordinates. The notation Γ( f ) denotes the graph of f . There existsa D Σ -family { X r,s } r,s with Γ( f ) = S r,s X r,s by Lemma 3.2. Note that π ( X r,s ) isCBD by [8, Lemma 1.7]. We have U = S r,s π ( X r,s ) and the fiber π − ( x ) ∩ Γ( f ) is asingleton for any x ∈ U . Therefore, we obtain X r,s = Γ( f | π ( X r,s ) ), where f | π ( X r,s ) is the restriction of f to π ( X r,s ). Take a closed box B contained in U . The family { π ( X r,s ) ∩ B } is a D Σ -family and B = S r,s π ( X r,s ) ∩ B . The CBD set π ( X r,s ) ∩ B has a nonempty interior for some r and s by Lemma 3.4. Take a closed box B ′ contained in π ( X r,s ) ∩ B . The set X r,s ∩ ( B ′ × R n ) = Γ( f | B ′ ) is closed. Therefore, f is continuous on int( B ′ ). (cid:3) We finally prove Theorem 1.1.
Proof of Theorem 1.1.
We prove the following assertion:
IMENSION INEQUALITY FOR UNIFORMLY LOCALLY O-MINIMAL STRUCTURE 7 ( ∗ ) : The inequality dim( f ( X )) ≤ dim( X ) holds true for any definable map f : X → R n .Lemma 3.4 is available as in the proof of Lemma 5.1 for the same reason.Set d = dim( f ( X )). We demonstrate that dim( X ) ≥ d . We can reduce to thecase in which the image f ( X ) is an open box B of dimension d . In fact, thereexist an open box B in R d and a definable map g : B → f ( X ) such that the map g is a definable homeomorphism onto its image by the definition of dimension [5,Definition 5.1]. Set Y = f − ( g ( B )) and h = g − ◦ f | Y : Y → B . When dim( Y ) ≥ d ,we get dim( X ) ≥ d by [5, Lemma 5.1] because Y is a subset of X . We may assumethat f ( X ) = B by considering Y and h instead of X and f , respectively.We next reduce to the case in which the map f is the restriction of a coordinateprojection. Consider the graph G = Γ( f ) ⊂ R m + d of the definable map f . Let π : R m + d → R d be the projection onto the last d coordinates. We have dim( X ) ≥ dim( G ) ≥ d by Lemma 2.4 when dim( G ) ≥ d . We may assume that f : X → B isthe restriction of the projection π : R m + d → R d to X .We have a D Σ -family { X r,s } r> ,s> with X = S r,s X r,s by Lemma 3.2. Thefamily { f ( X r,s ) } r> ,s> is also a D Σ -family by [8, Lemma 1.7] because f is therestriction of a projection. We have B = S r,s f ( X r,s ). The CBD set f ( X r,s ) hasa nonempty interior for some r > s > r > s >
0. Take an open box U contained in f ( X r,s ). Note that the inverse image { y ∈ X r,s | f ( y ) = x } of x ∈ U is CBD because f is continuous. Consider adefinable function ϕ : U → X r,s given by ϕ ( x ) = lexmin { y ∈ X r,s | f ( y ) = x } ,where the notation lexmin denotes the lexicographic minimum defined in [8]. Wecan get an open box V contained in U such that the restriction ϕ | V of ϕ to V is continuous by Lemma 5.1. The definable set X r,s is of dimension ≥ d by thedefinition of dimension because it contains the graph of the definable continuousmap ϕ | V defined on the open box V in R d . We have dim X ≥ dim( X r,s ) ≥ d by [5,Lemma 5.1]. We have proven Theorem 1.1. (cid:3) The proof of Corollary 1.2 is the same as that of [6, Corollary 2.6]. We give aproof here because it is brief.
Proof of Corollary 1.2.
Let D be the set of points at which the definable function f is discontinuous. Assume that the domain of definition X is a definable subsetof R m . Let G be the graph of f . We have dim( G ) = dim( X ) by Lemma 2.4 andTheorem 1.1. Set E = { ( x, y ) ∈ X × R | y = f ( x ) and f is discontinuous at x } . Weget dim( E ) < dim( G ) by [5, Theorem 4.2, Corollary 5.3]. Let π : R m +1 → R m bethe projection forgetting the last coordinate. We have D = π ( E ) by the definitionsof D and E . We finally obtain dim( D ) = dim( π ( E )) ≤ dim( E ) < dim( G ) = dim( X )by Theorem 1.1. (cid:3) References [1] A. Dolich, C. Miller and C. Steinhorn, Structure having o-minimal open core, Trans. Amer.Math. Soc., 362 (2010), 1371-1411.[2] L. van den Dries, Tame topology and o-minimal structures, London Mathematical SocietyLecture Note Series, Vol. 248. Cambridge University Press, Cambridge, 1998.[3] A. Fornasiero and T. Servi, Definably complete Baire structure, Fund. Math., 209 (2010),215-241.[4] A. Fornasiero, Locally o-minimal structures and structures with locally o-minimal open core,Ann. Pure Appl. Logic, 164 (2013), 211-229.
M. FUJITA [5] M. Fujita, Uniformly locally o-minimal structures and locally o-minimal structures admittinglocal definable cell decomposition, Ann. Pure Appl. Logic, 171 (2020), 102756.[6] M. Fujita, Uniform local definable cell decomposition for locally o-minimal expansion of thegroup of reals, preprint, 2020.[7] P. Hieronymi, An analogue of the Baire category theorem, J. Symbolic Logic, 78 (2013),207-213.[8] C. Miller, Expansions of dense linear orders with the intermediate value property, J. SymbolicLogic, 66 (2001), 1783-1790.[9] C. Toffalori and K. Vozoris, Notes on local o-minimality, MLQ Math. Log. Q., 55 (2009),617-632.
Department of Liberal Arts, Japan Coast Guard Academy, 5-1 Wakaba-cho, Kure,Hiroshima 737-8512, Japan
E-mail address ::