# On definable Skolem functions and trichotomy

Bruno Dinis  Departamento de Matemática
Escola de Ciências e Tecnologia, Universidade de Évora
Rua Romão Ramalho, 59
7000–671 Évora, Portugal<EMAIL_ADDRESS>and  Mário J. Edmundo
Departamento de Matemática
Faculdade de Ciências, Universidade de Lisboa
Campo Grande, Edifício C6
P-1749-016 Lisboa, Portugal<EMAIL_ADDRESS>
###### Abstract.

In this paper we give an explicit characterization of o-minimal structures
with definable Skolem functions/definable choice. Such structures are, after
naming finitely many elements from the prime model, a union of finitely many
trivial points each defined over $\emptyset$ and finitely many open intervals
each a union of a $\emptyset$-definable family of group-intervals with fixed
positive elements.

###### 2010 Mathematics Subject Classification:

03C64

Both authors were supported by FCT - Fundação para a Ciência e a Tecnologia,
under the projects UIDB/04561/2020 and UIDP/04561/2020 and the research center
CMAFcIO - Centro de Matemática, Aplicações Fundamentais e Investigação
Operacional. The first author was also supported by CIMA - Centro de
Investigação em Matemática e Aplicações – UIDP/04674/2020.
Keywords and phrases: O-minimal structures, definable Skolem functions,
definable choice, trichotomy.

##  1\. Introduction

In several papers topological invariants of definable sets, such as the
fundamental group and cohomology, were developed in the setting of o-minimal
structures ${\mathbb{M}}=(M,<,(c)_{c\in{\mathcal{C}}},$
$(f)_{f\in{\mathcal{F}}},(R)_{R\in{\mathcal{R}}})$ with definable Skolem
functions/definable choice ( [7], [11], [13], [14], [15]), extending similar
work in o-minimal expansions of ordered fields ([1], [2], [3], [4], [5], [6],
[12], [16]) and in o-minimal expansions of ordered groups ([8], [9], [10],
[17], [19]). A referee of a previous version of [7], challenged us to properly
discuss the assumption of definable Skolem functions/definable choice and how
it differs from the setting of products of finitely many definable group-
intervals which is used in [11]. In fact, the referee asks the following
questions: (i) are all examples of such structures, which are not obtained
from finitely many group-intervals, similar to the expansion of the real field
by the function $f(x,y)=x+y$ with $|y|<1$? (ii) is there a definable family of
local group intervals covering all points of $M$? (iii) are all but finitely
many points in $\mathbb{M}$ non-trivial?

Taking into account the questions above we tried to understand the
relationship between definable Skolem functions/definable choice and the
trichotomy theorem ([22]). The outcome of this research is the following:

######  Theorem 1.1.

Let $\mathbb{M}$ be an o-minimal structure and let
$\mathbb{P}\preceq\mathbb{M}$ be the prime model. Then the following are
equivalent:

  1. (1)

$\mathbb{M}$ has definable Skolem functions.

  2. (2)

After naming finitely many elements of $\mathbb{P}$, there is a finite
collection $\mathcal{Y}$ of $\emptyset$-definable open subintervals of $M$
such that:

     * -

$M\setminus\bigcup\mathcal{Y}$ is a finite set of trivial points each defined
over $\emptyset$.

     * -

each $Y\in\mathcal{Y}$ is the union of a uniformly $\emptyset$-definable
family of group-intervals, each with a fixed positive element, parametrized by
the end points of the intervals.

     * -

each $Y\in\mathcal{Y}$ has a fixed $\emptyset$-definable element.

  3. (3)

$\mathbb{M}$ has definable choice.

The proof of our main theorem relies on geometrical arguments and
constructions. This justifies the development of some technical tools (Section
4) and the careful analysis, in Section 5, of certain definable families of
curves obtained from the existence of definable Skolem functions.

Our main theorem gives a full characterization of o-minimal structures with
definable Skolem functions/definable choice. This is particularly relevant
because so far only loose examples were known such as o-minimal expansions of
ordered groups ([26]), structures obtained from a product of finitely many
definable group-intervals ([22]) or the example mentioned in question (i)
above. One can ask for a similar characterization of o-minimal structures with
(parametric) elimination of imaginaries. We return to this question again in
Section 7.

The structure of the paper is the following. For the readers convenience, we
include, in the first two sections, basic definitions and properties. Most of
these are well known or can be found in the literature but are included for
completeness. In Section 2 we recall the definition of definable Skolem
functions/definable choice as well as some properties of these notions. In
Section 3 we recall the trichotomy theorem ([22]), the definition of definable
group-intervals ([18]) as well as some properties of trivial/non-trivial
o-minimal structures.

In Section 4 we prove some auxiliary lemmas. Section 5 is devoted to proving
the main technical lemma (Lemma 5.9) required to obtain the main theorem
(Theorem 6.4), proved in Section 6. Some final remarks, including an
unparametric version of the main theorem, are left to Section 7.

Acknowledgements: We thank the anonymous referee of a previous version of [7]
who’s questions challenged us to do this paper!

##  2\. Preliminaries on definable Skolem functions

Recall that an o-minimal structure ${\mathbb{M}}=(M,<,\cdots)$ has definable
Skolem functions if and only if for every uniformly definable family
$\\{X_{t}\\}_{t\in T}$ of nonempty definable subsets of some $M^{k}$, there is
a definable function $h:T\to M^{k}$ such that:

  * -

$h(t)\in X_{t}$ for all $t\in T$.

One says that ${\mathbb{M}}$ has definable choice if in addition the definable
function $h$ can be chosen so that:

  * -

for all $t,t^{\prime}\in T,$ if $X_{t}=X_{t^{\prime}},$ then
$h(t)=h(t^{\prime})$.

Obviously definable choice implies the existence of definable Skolem
functions. In this section we collect a couple of other well known
observations about these two notions.

######  Lemma 2.1.

For ${\mathbb{M}}$ to have definable Skolem functions (resp. definable choice)
it is enough to consider $\emptyset$-definable families $\\{Y_{s}\\}_{s\in S}$
of nonempty definable subsets of $M$.

Proof. Assume that ${\mathbb{M}}$ has definable Skolem functions (resp.
definable choice) for $\emptyset$-definable families of nonempty definable
subsets of $M^{k}$. Let $\\{X_{t}\\}_{t\in T}$ be a definable family of
nonempty definable subsets of $M^{k}$ defined over $b\in M^{l}$. By varying
the parameter $b$ we get a $\emptyset$-definable family
$\\{Y_{(t,z)}\\}_{(t,z)\in S}$ of nonempty subsets of $M^{k}$ such that
$T=\\{t:(t,b)\in S\\}$ and $Y_{(t,b)}=X_{t}$ for all $t\in T$. So there is a
definable function $g:S\to M^{k}$ such that for all $(t,z)$ we have $g(t,z)\in
Y_{(t,z)}$. Therefore, $h:T\to M^{k}$ given by $h(t)=g(t,b)$ is a definable
function such that for all $t\in T$ we have $h(t)\in X_{t}=Y_{(t,b)}$.
Furthermore, if $g$ satisfies the definable choice property, then $h$ will
also satisfy it.

Now assume that ${\mathbb{M}}$ has definable Skolem functions (resp. definable
choice) for $\emptyset$-definable families of nonempty definable subsets of
$M$. We prove by induction on $k$ that the same holds for
$\emptyset$-definable families of nonempty definable subsets of $M^{k}$. Let
$\\{Y_{s}\\}_{s\in S}$ be such a $\emptyset$-definable family. By the
induction hypothesis, there is $g:T\to M^{k-1}$ a definable Skolem function
for $\\{\pi(Y_{s})\\}_{s\in S}$ where $\pi:M^{k}\to M^{k-1}$ is the projection
onto the first $k-1$ coordinates. On the other hand, by the first part of the
proof, there is $f:S\to M$ a definable Skolem function for the definable
family $\\{Z_{s}\\}_{s\in S}$ where $Z_{s}=\\{z\in M:(g(s),z)\in Y_{s}\\}$. It
follows that $h:S\to M^{k}$ given by $h(s)=(g(s),f(s))$ is a definable Skolem
function for $\\{Y_{s}\\}_{s\in S}.$ Moreover, if $g$ and $f$ satisfy the
definable choice property, then $h$ will also satisfy it. ∎

######  Proposition 2.2.

If ${\mathbb{M}}$ has definable Skolem functions (resp. definable choice),
then every ${\mathbb{K}}\prec{\mathbb{M}}$ has definable Skolem functions
(resp. definable choice).

Proof. Let ${\mathbb{K}}\prec{\mathbb{M}}$. Let $\\{Y_{s}\\}_{s\in S}$ be a
uniformly $\emptyset$-definable family of nonempty definable subsets of $K.$
Then there is a definable function $h:S(M)\to M$ such that for all $z\in S(M)$
we have $h(z)\in Y_{z}.$ Let $b$ be the parameter in $M^{q}$ over which $h$ is
defined. Then there is a $\emptyset$-definable function $g:S\times D\to K$
such that $b\in D(M)$ and $h(z)=g(z,b)$ for all $z\in S(M).$ Let

$D^{\prime}=\\{d\in D:\,\,\,g(s,d)\in Y_{s}\,\,\,\textrm{for all $s\in
S$}\\}.$

Then $D^{\prime}$ is $\emptyset$-definable and, since $b\in D^{\prime}(M)$ and
$\mathbb{K}\prec\mathbb{M}$, there is $c\in D^{\prime}.$ It follows that if
$f:S\to K$ is given by $f(s)=g(s,c)$ for all $s\in S,$ then $f$ is
${\mathbb{K}}$-definable and $f(s)\in Y_{s}$ for all $s\in S.$

If $h$ satisfies the definable choice property, then we can choose $g$
satisfying that property and so $f$ will also satisfy it. ∎

The following is similar:

######  Proposition 2.3.

If ${\mathbb{M}}$ has definable Skolem functions (resp. definable choice),
then every ${\mathbb{M}}\prec{\mathbb{K}}$ has definable Skolem functions
(resp. definable choice).

Recall that for $C\subseteq M$, the definable closure over $C$ is defined as

${\rm dcl}(C)=\\{u\in M:\\{u\\}\,\,\,\textrm{is definable over $C$}\\}.$

Equivalently, $u\in\mathrm{dcl}(C)$ if and only if there is a
$\emptyset$-definable function $f:Z\subseteq M^{l}\to M$ and $c\in C^{l}\cap
Z$ such that $f(c)=u.$ Recall also that, since we have an order in $M$,
definable closure over $C$ coincides with the algebraic closure over $C$,
defined as

$u\in{\rm acl}(C)\,\,\ \textrm{if and only if $u$ belongs to a finite set
definable over $C$}.$

As usual if $a=(a_{1},\ldots,a_{l})\in M^{l}$ and $C\subseteq M,$ then $aC$
denotes $\\{a_{1},\ldots,a_{l}\\}\cup C.$

From now on we let ${\mathbb{P}}\prec{\mathbb{M}}$ be the prime model over the
empty set and we let $P$ be its domain. See [25, Theorem 5.1] for its
existence in the o-minimal context.

The following is implicit in the observations before the proof of [26, Chapter
6, (1.2)]:

######  Proposition 2.4.

Assume that ${\mathbb{M}}$ is sufficiently saturated. Then ${\mathbb{M}}$ has
definable Skolem functions if and only if every nonempty definable subset
$X\subseteq M$ defined with parameters $a\in M^{l}$ has an element in
$\mathrm{dcl}(a\,\mathbb{P})$.

Proof. ($\Rightarrow$) Let $X$ be a nonempty definable subset of $M$ defined
with parameters $a\in M^{l}.$ Then there is a uniformly $\emptyset$-definable
family $\\{X_{t}\\}_{t\in T}$ of nonempty definable subsets of $P$ such that
$a\in T(M)$ and $X_{a}=X.$ By Proposition 2.2, there is a
${\mathbb{P}}$-definable Skolem function $f:T\to P$ for the family
$\\{X_{t}\\}_{t\in T}$ of the form $f=g(-,c)$ for some $\emptyset$-definable
function $g:T\times D\to P$ and some $c\in P^{q}$. Then $f(a)\in X\cap{\rm
dcl}(a{\mathbb{P}})$.

($\Leftarrow$) Let $\\{X_{t}\\}_{t\in T}$ be a uniformly $\emptyset$-definable
family of nonempty definable subsets of $M$. Suppose that $T\subseteq M^{l}$
and fix $v\in T$. Since $X_{v}$ is defined over $v$, there is an element in
$X_{v}\cap{\rm dcl}(v\,{\mathbb{P}})$. So there is a $\emptyset$-definable
function $g_{v}:Z_{v}\subseteq T\times M^{q_{v}}\to M$ and $c_{v}\in
P^{q_{v}}$ such that $(v,c_{v})\in Z_{v}$ and $g_{v}(v,c_{v})\in X_{v}$. Let
$B_{v}=\\{t\in T:(t,c_{v})\in Z_{v}\\}$. Then each $B_{v}$ is defined over
${\mathbb{P}}$.

Since $T=\bigcup_{v}B_{v}$, by saturation, there are finitely many
$v_{0},\ldots,v_{m}$ such that $T=B_{v_{0}}\cup\ldots\cup B_{v_{m}}$. For
$i=0,\ldots,m$, let $f_{i}:B_{v_{i}}\to M$ be given by
$f_{i}(t)=g_{v_{i}}(t,c_{v_{i}})$. Then, by recursion on $m$, there is a
${\mathbb{P}}$-definable function $f:T\to M$ such that $f(t)\in X_{t}$ for all
$t\in T$. ∎

######  Corollary 2.5.

${\mathbb{M}}$ has definable Skolem functions if and only if every $C\subseteq
M$, we have ${\rm dcl}(C\,\mathbb{P})\preceq\mathbb{M}$.

Proof. It is clear that ${\rm dcl}(C\,\mathbb{P})$ is a substructure of
$\mathbb{M}$. To conclude we may assume that ${\mathbb{M}}$ is sufficiently
saturated and use Proposition 2.4 and Tarski’s lemma for elementary
substructure. ∎

There is an analogue of Proposition 2.4 for definable choice and definable
closure in ${\mathbb{M}}^{{\rm eq}}$. To this end, let us recall a couple of
definitions from [18, Section 7]. Given definable sets $X$ and $Y$ and a
definable equivalence relation $E$ on $X,$ one says that a function $f:X/E\to
Y$ is $A$-definable if and only if the set $\\{(x,y)\in X\times Y:f([x])=y\\}$
is $A$-definable. Equivalently, $f:X/E\to Y$ is $A$-definable if and only if
there is an $A$-definable function $g:X\to Y$ such that $f([x])=g(x)$ for all
$x\in X$ i.e. making the following diagram commutative:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$$\textstyle{X/E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$

One says that a subset $U$ of $X/E$ is definable if it is the image of a
definable subset of $X$ under the quotient map $X\to X/E.$

With these definitions, for $a\in X/E$ and $C\subseteq M$ one sets

$\mathrm{dcl}(aC)=M\cap\mathrm{dcl}^{\textrm{eq}}(aC).$

Equivalently, $u\in\mathrm{dcl}(aC)$ if and only if there is a
$\emptyset$-definable function $f:Z\subseteq X/E\times M^{q}\to M$ and $c\in
C^{q}\subseteq M^{q}$ such that $f(a,c)=u.$

If $X$ is a nonempty definable subset of $M$ defined with parameters in $a\in
M^{l},$ then there is a uniformly $\emptyset$-definable family
$\\{X_{t}\\}_{t\in T}$ of nonempty definable subsets of $P$ such that $a\in
T(M)$ and $X_{a}=X.$ We consider on $T$ the $\emptyset$-definable equivalence
relation given by $t\sim t^{\prime}$ if and only if $X_{t}=X_{t^{\prime}}$
and, we denote by $[t]$ the $\sim$-equivalence class of $t\in T$ and
$[T]=\\{[t]:t\in T\\}.$ In particular, below we consider $[a],$ a code for
$X.$

######  Proposition 2.6.

Assume that ${\mathbb{M}}$ is sufficiently saturated. Then ${\mathbb{M}}$ has
definable choice if and only if every nonempty definable subset $X\subseteq M$
defined with parameters $a\in M^{l}$ has an element in
$\mathrm{dcl}([a]\,\mathbb{P})$.

Proof. ($\Rightarrow$) Let $X$ be a nonempty definable subset of $M$ defined
with parameters $a\in M^{l}$. Then there is a uniformly $\emptyset$-definable
family $\\{X_{t}\\}_{t\in T}$ of nonempty definable subsets of $P$ such that
$a\in T(M)$ and $X_{a}=X$. By Proposition 2.2, there is a
${\mathbb{P}}$-definable choice function $h:T\to P$ for the family
$\\{X_{t}\\}_{t\in T}$ of the form $h=g(-,c)$ for some $\emptyset$-definable
function $g:T\times D\to P$ and some $c\in P^{q}$. Since $g$ has the choice
property, it induces a $\emptyset$-definable function $\overline{g}:[T]\times
D\to M$. Then $\overline{g}([a],c)\in X\cap{\rm dcl}([a]\,{\mathbb{P}})$.

($\Leftarrow$) Let $\\{X_{t}\\}_{t\in T}$ be a uniformly $\emptyset$-definable
family of nonempty definable subsets of $M$. Suppose that $T\subseteq M^{l}$
and fix $v\in T$. Since $X_{v}$ is defined over $v$, there is an element in
$X_{v}\cap{\rm dcl}([v]\,{\mathbb{P}})$. So there is a $\emptyset$-definable
function $g_{v}:Z_{v}\subseteq[T]\times M^{q_{v}}\to M$ and $c_{v}\in
P^{q_{v}}$ such that $([v],c_{v})\in Z_{v}$ and $g_{v}([v],c_{v})\in X_{v}$.
Let $B_{v}=\\{[t]\in[T]:([t],c_{v})\in Z_{v}\\}$. Then each $B_{v}$ is defined
over ${\mathbb{P}}$.

Since $[T]=\bigcup_{v}B_{v}$, by saturation, there are finitely many
$v_{0},\ldots,v_{m}$ such that $[T]=B_{v_{0}}\cup\ldots\cup B_{v_{m}}$. For
$i=0,\ldots,m$, let $f_{i}:B_{v_{i}}\to M$ be given by
$f_{i}([t])=g_{v_{i}}([t],c_{v_{i}})$. Then, by recursion on $m$, there is a
${\mathbb{P}}$-definable function $f:[T]\to M$ such that $f([t])\in X_{t}$ for
all $t\in T$. Hence, the ${\mathbb{P}}$-definable function $h:T\to M$ given by
$h(t)=f([t])$ is a definable choice function for the family $\\{X_{t}\\}_{t\in
T}$. ∎

##  3\. Preliminaries on the trichotomy theorem

Recall that, in an o-minimal structure ${\mathbb{M}}=(M,<,\cdots)$, a point
$a\in M$ is non-trivial over $A\subseteq M$ if and only if there is an
infinite open interval $I\subseteq M$ containing $a$ and a definable
continuous function $F:I\times I\to M$ defined over $A$ such that $F$ is
strictly monotone in each variable (separately). We say that $a\in M$ is non-
trivial if it is non-trivial over some $A$. A point which is not non-trivial
is called trivial. Finally, let us say that an o-minimal structure
${\mathbb{M}}$ is trivial if and only if every $a\in M$ is trivial.

In this section we recall the trichotomy theorem in an arbitrary o-minimal
structure ${\mathbb{M}}$ and present several well known characterizations of
trivial/non-trivial that we require later.

First note the following ([21, Lemma 2.2]):

######  Fact 3.1.

${\mathbb{M}}$ is non-trivial if and only if there are infinite open intervals
$J,K\subseteq M$ and a definable continuous function $G:J\times K\to M$ such
that $G$ is strictly monotone in each variable.

The main theorem of Peterzil and Starchenko in [22] is the following:

######  Theorem 3.2 (Trichotomy).

Let ${\mathbb{M}}$ be a sufficiently saturated o-minimal structure structure.
Given $a\in M$, then $a$ is non-trivial if and only if either

  1. (T2)

the structure that ${\mathbb{M}}$ induces on some convex neighbourhood of $a$
is an ordered vector space over an ordered division ring, or

  2. (T3)

the structure that ${\mathbb{M}}$ induces on some open interval around $a$ is
an o-minimal expansion of a real closed field.

So, the Trichotomy theorem implies that if $a\in M$ is non-trivial, then $a$
is contained in a definable group-interval. Recall that a definable group-
interval $J=\langle(-b,b),0,+,<\rangle$ is an open interval $(-b,b)\subseteq
M$, with $-b<b$ in $M\cup\\{-\infty,+\infty\\}$, together with a binary
partial continuous definable operation $+:J^{2}\to J$ and an element $0\in J$,
such that:

  * (i)

$x+y=y+x$ when defined; $(x+y)+z=x+(y+z)$ when defined; if $x<y$ and $x+z$ and
$y+z$ are defined then $x+z<y+z$;

  * (ii)

for every $x\in J$, if $x>0$, then the set $\\{y\in J:\,\,x+y\,\,\textrm{is
defined}\\}$ is an interval of the form $(-b,r(x))$;

  * (iii)

for every $x\in J$, we have $\lim_{z\to 0}(z+x)=x$ and if $x>0$ we also have
$\lim_{z\to r(x)^{-}}(x+z)=b$;

  * (iv)

for every $x\in J$ there exists $-x\in J$ such that $x+(-x)=0$.

Let us give other characterizations of trivial/non-trivial that will be useful
later. First recall that an o-minimal structure ${\mathbb{M}}$ is a _geometric
structure_ , i.e.:

  * -

$\mathrm{acl}(-)$ satisfies the Exchange Principle: if $a,b\in M$, $A\subseteq
M$ and $b\in\mathrm{acl}(aA)$, then either $b\in\mathrm{acl}(A)$ or
$a\in\mathrm{acl}(bA)$ and,

  * -

for any formula $\varphi(x,\overline{y})$, there exists $n\in\mathbb{N}$ such
that for any $\overline{b}\in M^{r}$, either $\varphi(x,\overline{b})$ has
fewer than $n$ solutions in ${\mathbb{M}}$ or it has infinitely many.

See [22, Theorem 4.1]. So as in any geometric structure, given $A\subseteq M$,
for $\overline{a}\in M^{n}$ we defined $\dim(\overline{a}/A)$ to be the
minimal cardinality of $\overline{a}^{\prime}\subseteq\overline{a}$ such that
${\rm acl}(\overline{a}^{\prime}A)={\rm acl}(\overline{a}A)$, and for
$U\subseteq M^{n}$ a definable subset over $A$, we define
$\dim(U)=\max\\{\dim(\overline{a}/A):\overline{a}\in U\\}$. Furthermore, we
say that $\overline{a}\in U$ is generic in $U$ over $A$ if and only if
$\dim(\overline{a}/A)=\dim(U)$. Note that in the o-minimal case, dimension of
definable subsets over $A$ defined in this way coincides with the o-minimal
dimension defined using the cell decomposition theorem (see for example [24,
Lemma 1.4] and [26, Chapter 3, (2.7)]).

Here is another characterization of trivial o-minimal structure related with
the notion of trivial pregeometry (see [21, Lemmas 1.8, 2.1 and 2.2]). Recall
that $\\{a_{1},\ldots,a_{m}\\}\subseteq M$ is independent if there is no
$1\leq i\leq m$ such that $a_{i}$ is algebraic over
$\\{a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{m}\\}$.

######  Fact 3.3.

${\mathbb{M}}$ is trivial if and only if any finite subset of $M$ whose
elements are pairwise independent is independent.

Recall that a _curve_ is a definable one-dimensional subset of $M^{2}$. A
family of curves $\mathcal{F}$ is said to be _definable_ if there exist
definable sets $U\subseteq M^{k}$ and $F\subseteq U\times M^{2}$ such that
$\mathcal{F}=\\{C_{\overline{u}}:\overline{u}\in U\\}$ where for each
$\overline{u}\in U$ we let $C_{\overline{u}}:=\\{\langle
x,y\rangle:\langle\overline{u},x,y\rangle\in F\\}$. We say that
$C_{\overline{u}}$ is _generic_ in $F$ if $\overline{u}$ is generic in $U$
over the parameters defining $F$. The family $F$ is said to be _interpretable_
if $U$ is replaced by $U/E$, where $E$ is a definable equivalence relation on
$U$. A definable (or interpretable) family of curves
$\mathcal{F}=\\{C_{\overline{u}}:\overline{u}\in U\\}$ is _normal_ of
dimension $n$ if $\dim(U)=n$ (or $\dim(U/E)=n$) and for
$\overline{u}\neq\overline{v}$ from $U$ (or $U/E$), $C_{\overline{u}}$ and
$C_{\overline{v}}$ intersect in at most finitely many points.

We have the following characterization given in [22]:

######  Theorem 3.4.

Let ${\mathbb{M}}$ be a sufficiently saturated o-minimal structure. Then
${\mathbb{M}}$ is trivial if and only if the following holds:

  1. $(Z_{1})$

for every interpretable infinite normal family of curves $\mathcal{F}$, if
$\mathcal{C}$ is a generic curve in $\mathcal{F}$ and $\langle a,b\rangle$ is
generic in $\mathcal{C}$, then either $\dim(\mathcal{C}\cap(\\{a\\}\times
M))=1$ or $\dim(\mathcal{C}\cap(M\times\\{b\\}))=1$.

We conclude with an observation that is somewhat implicit in [21] and follows
from [26, Chapter 2, (2.19) Exercises 1 and 2]. First let us recall the
material from the later source. We say that an open cell $C\subseteq M^{n}$ is
regular if for each $i\in\\{1,\ldots,n\\}$ and each two points $x,y\in C$ that
differ only on the $i^{{\rm th}}$ coordinate and each point $z\in M^{n}$ that
differs from $x$ and $y$ only in the $i^{{\rm th}}$ coordinate, if
$x_{i}<z_{i}<y_{i}$ then $z\in C$.

We say that a definable function $f:C\subseteq M^{n}\to M$ on a regular open
cell is strictly increasing in the $i^{{\rm th}}$ coordinate (resp. strictly
decreasing in the $i^{{\rm th}}$ coordinate or constant in the $i^{{\rm th}}$
coordinate) if for each two points $x,y\in C$ that differ only on the $i^{{\rm
th}}$ coordinate, if $x_{i}<y_{i}$ then $f(x)<f(y)$ (resp. $f(y)<f(x)$ and
$(f(x)=f(y)$).

We say that a definable function $f:C\subseteq M^{n}\to M$ on a regular open
cell is regular if it is continuous, and for each $i\in\\{1,\ldots,n\\}$, $f$
is (depending on $i$) either strictly increasing or strictly decreasing or
constant in the $i^{{\rm th}}$ coordinate.

Let us call a definable function $f:C\subseteq M^{n}\to M$ on a regular open
cell unary if it is regular and is constant in the $i^{{\rm th}}$ coordinate
for all except for at most one $i\in\\{1,\ldots,m\\}$.

We have the following unary version of the Cell Decomposition Theorem for
trivial o-minimal structures. We refer the reader to [26, Chapter 3, (2.10)]
for the definition of a _decomposition_ of $M^{n}$.

######  Theorem 3.5 (Unary CDT).

Suppose that ${\mathbb{M}}$ is a trivial o-minimal structure. Then

  1. (1)

Given any definable sets $A_{1},\dots,A_{k}\subseteq M^{n}$, there is a
decomposition $\mathcal{C}$ of $M^{n}$ that partitions each $A_{i}$, all of
whose open cells are regular.

  2. (2)

Given a definable function $f:A\subseteq M^{n}\to M$, there is a decomposition
$\mathcal{C}$ of $M^{n}$ that partitions $A$ all of whose open cells are
regular, and such that for each open cell $B\in\mathcal{C}$ with $B\subseteq
A$ the restriction $f_{|B}$ is unary.

Proof. [26, Chapter 2, (2.19) Exercise 2] shows that the regular cell
decomposition theorem holds i.e., (1) above holds and (2) holds with the
restrictions $f_{|B}$ being regular. Since ${\mathbb{M}}$ is a trivial
o-minimal structure, using Fact 3.1, the restrictions $f_{|B}$ from (2) must
be unary. ∎

##  4\. Auxiliary lemmas

Here we let ${\mathbb{M}}$ be an arbitrary o-minimal structure and we prove a
couple of auxiliary lemmas needed in the next section.

######  Lemma 4.1.

Let $Z\subseteq M$ be a finite union of open intervals defined over $s$.
Suppose that we have a definable function $\mu:Z\to M$ defined over $s$ such
that $\mu(u)<u$ for all $u\in Z$. Then $Z$ is a finite union of points and
open subintervals, all defined over $s$, such that on each such open
subinterval $X$, if $u\in X$, then for all $v,w\in(\mu(u),u)$ with $v<w$ we
have $v\in(\mu(w),w)$.

Proof. By the monotonicity theorem ([26, Chapter 3, (1.2)]), $Z$ is a finite
union of points and open subintervals such that on each open subinterval $X$,
the restriction $\mu:X\to Y_{s}$ is either constant or strictly monotone and
continuous. Let $u\in X$ and suppose that $v,w\in(\mu(u),u)$ with $v<w$.

Observe that $\mu(u)<v<w$. It follows that: If $\mu$ is constant, then
$\mu(u)=\mu(w)=\mu(v)$ and so $v\in(\mu(w),w)$. If $\mu$ is strictly
increasing, then $\mu(v)<\mu(w)<\mu(u)$ and therefore $v\in(\mu(w),w)$. If
$\mu$ is strictly decreasing, then $\mu(w)<\mu(v)$ and hence $v\in(\mu(w),w)$.
∎

Similarly:

######  Lemma 4.2.

Let $Z\subseteq M$ be a finite union of open intervals defined over $s$.
Suppose that we have a definable function $\mu:Z\to M$ defined over $s$ such
that $u<\mu(u)$ for all $u\in Z$. Then $Z$ is a finite union of points and
open subintervals, all defined over $s$, such that on each such open
subinterval $X$, if $u\in X$, then for all $v,w\in(u,\mu(u))$ with $v<w$ we
have $w\in(v,\mu(v))$.

The following lemma is obtained combining Lemma 4.1 and the Unary CDT (Theorem
3.5):

######  Lemma 4.3.

Let $Z\subseteq M$ be a finite union of open intervals and $b:Z\to M$ a
definable function defined over $s$ such that $b(u)<u$ for all $u\in Z$. Let
$\Omega:(b,\mathrm{id}_{Z})_{Z}\to M$ be a definable function over $s$ such
that for each $u\in Z$ we have:

  * -

$\Omega(u,-)$ is strictly monotone and continuous.

Suppose that $Z$ has no non-trivial points. Then $Z$ is a finite union of
points and open subintervals, all defined over $s$, such that on each such
open subinterval $X$:

  * -

there is a definable function $d_{X}:X\to M$ defined over $s$ such that
$b_{|X}(u)\leq d_{X}(u)<u$ for all $u\in X$;

  * -

for every $u\in X$, if $v,w\in(d_{X}(u),u)\cap X$ are such that $v<w$, then we
have $v\in(d_{X}(w),w)$ and $\Omega(u,v)=\Omega(w,v)$.

Proof. By Theorem 3.5, take $\mathcal{D}$ a regular cell decomposition of
$M^{2}$ that partitions $C:=(b,\mathrm{id}_{Z})_{Z}$ such that for each open
cell $D\in\mathcal{D}$ contained in $C$ the restriction $\Omega_{|D}$ is
unary.

Let $Z_{1},\ldots,Z_{k}$ be the open cells of $\pi(\mathcal{D})$ contained in
$Z$. For each $i$ let $D_{i}$ be the open cell of $\mathcal{D}$ of the form
$(d_{i},\mathrm{id}_{Z_{i}})_{Z_{i}}$ where $d_{i}:Z_{i}\to M$ is the maximal
element of the set of continuous definable function with domain $Z_{i}$, which
are below $\mathrm{id}_{Z_{i}}$ and whose graph is a cell of $\mathcal{D}$. Of
course $b_{|Z_{i}}\leq d_{i}$.

Fix $i$. Since for all $u\in Z_{i}$ we have $d_{i}(u)<u$, by Lemma 4.1,
$Z_{i}$ is a finite union of points and open subintervals
$Z_{i,1},\ldots,Z_{i,l_{i}}$, all defined over $s$, such that for each
$j\in\\{1,\ldots,l_{i}\\}$, if $u\in Z_{i,j}$, then for all
$v,w\in(d_{i}(u),u)$ with $v<w$ we have $v\in(d_{i}(w),w)$.

Let $u\in Z_{i,j}$ and let $v,w\in(d_{i}(u),u)\cap Z_{i,j}$ with $v<w$. Then
we have $v\in(d_{i}(w),w)$ and therefore $(u,v),(u,w),(w,v)\in D_{i}$. Since
$D_{i}\in\mathcal{D}$ and $\Omega_{|D}$ is unary we have
$\Omega(u,v)=\Omega(w,v)$ as required. ∎

Similarly, using Lemma 4.2 instead of Lemma 4.1 we have:

######  Lemma 4.4.

Let $Z\subseteq M$ be a finite union of open intervals and $b:Z\to M$ a
definable function defined over $s$ such that $u<b(u)$ for all $u\in Z$. Let
$\Omega:(\mathrm{id}_{Z},b)_{Z}\to M$ be a definable function over $s$ such
that for each $u\in Z$ we have:

  * -

$\Omega(u,-)$ is strictly monotone and continuous.

Suppose that $Z$ has no non-trivial points. Then $Z$ is a finite union of
points and open subintervals, all defined over $s$, such that on each such
open subinterval $X$:

  * -

there is a definable function $d_{X}:X\to M$ defined over $s$ such that
$u<d_{X}(u)\leq b_{|X}(u)$ for all $u\in X$;

  * -

for every $u\in X$, if $v,w\in(u,d_{X}(u))\cap X$ are such that $v<w$, then we
have $w\in(v,d_{X}(v))$ and $\Omega(u,w)=\Omega(v,w)$.

We end with the following lemma which will be used later to “remove”
parameters from certain uniformly definable families:

######  Lemma 4.5.

Assume that $\mathbb{M}$ has definable Skolem functions. Consider the
following data $(Z,T,\\{Z_{t}\\}_{t\in T},\\{\mathcal{U}_{t}\\}_{t\in T})$
where: $Z\subseteq M$ is a $\emptyset$-definable open subinterval, $T\subseteq
Z$ is a $\emptyset$-definable subset such that $Z\setminus T$ is finite,
$\\{Z_{t}\\}_{t\in T}$ is a uniformly $\emptyset$-definable family of open
subintervals of $Z$ such that $Z\setminus\bigcup Z_{t}$ is finite and
$\\{\mathcal{U}_{t}\\}_{t\in T}$ is a uniformly $\emptyset$-definable family
of finitely many two by two disjoint open subintervals of each $Z_{t}$ such
that $Z_{t}\setminus\bigcup\mathcal{U}_{t}$ is finite. Then after naming
finitely many elements of $\mathbb{P}$, and after removing finitely many
$\emptyset$-definable points from $Z$ and replacing $Z$ by one of the
remaining open subintervals, we may assume that there are
$\emptyset$-definable functions $q^{-},q^{+}:Z\to M$ such that $q^{-}<q^{+}$
and $t:Z\to T$ such that, for each $z\in Z$, $(q^{-}(z),q^{+}(z))$ is one of
the open subintervals of $Z_{t(z)}$ in $\mathcal{U}_{t(z)}$ and
$z\in(q^{-}(z),q^{+}(z))$.

Proof. First we show that

######  Claim 4.6.

The set $Z^{\prime}:=\\{z\in Z:z\notin\bigcup\mathcal{U}_{t}\,\,\textrm{for
all}\,\,t\in T\\}$ is finite and defined over $\emptyset$.

Proof. Suppose that $Z^{\prime}$ is infinite. By the monotonicity theorem and
the finiteness theorem (see [26, Chapter 3, (1.8)]) there is a partition of
$T$ into finitely many points and open intervals, all defined over
$\emptyset$, such that on each such open subinterval $J$, there are $n_{J}$
continuous definable functions $f_{J,i}:J\to M\cup\\{-\infty,+\infty\\}$ such
that $-\infty\leq f_{J,1}<\ldots<f_{J,n_{J}}\leq+\infty$ and for each $t\in J$
the end points of the open subintervals in $\mathcal{U}_{t}$ are exactly
$f_{J,1}(t)<\ldots<f_{J,n_{J}}(t)$.

Fix such $J$ and let $Z^{\prime}_{J}:=\\{z\in
Z:z\notin\bigcup\mathcal{U}_{t}\,\,\textrm{for all}\,\,t\in J\\}$. Since
$Z^{\prime}\subseteq Z^{\prime}_{J}$, we have that $Z^{\prime}_{J}$ is
infinite. On the other hand, for each $z\in Z^{\prime}_{J}$ and each $t\in J$,
there is a unique index $i(z,t)$ such that $z=f_{J,i(z,t)}(t)$. So we have a
definable function $i:Z^{\prime}_{J}\times J\to\\{1,\ldots,n_{J}\\}$ and, by
the cell decomposition theorem, there is a finite partition $\mathcal{C}$ of
$Z^{\prime}_{J}\times J$ into finitely many cells such that on each such cell
$C$ the function is constant, say with value $i_{C}$.

Since $J$ is an open subinterval and $Z^{\prime}_{J}$ is infinite, there is an
open cell $C\in\mathcal{C}$ of the form $(f,g)_{B}$ with $B\subseteq
Z^{\prime}_{J}$ an open subinterval and $f,g:B\to M\cup\\{-\infty,+\infty\\}$
continuous definable functions such that $-\infty\leq f<g\leq+\infty$. Fix
such a cell $C$ and note that for all $z\in B$ the definable function
$f_{J,i_{C}}:(f(z),g(z))\to M\cup\\{-\infty,+\infty\\}$ is constant with value
$z$.

Now fix $z^{\prime}\in B$ and let $I$ be a closed infinite interval contained
in $(f(z^{\prime}),g(z^{\prime}))$. Then by continuity there is an open
subinterval $O\subseteq B$ such that for all $z\in O$ we have
$I\subseteq(f(z),g(z))$. This implies that for each $z\in O$, the definable
function $f_{J,i_{C}}:I\to M\cup\\{-\infty,+\infty\\}$ is constant with value
$z$, which is absurd since $O$ is infinite. ∎

It follows that for each $z\in Z\setminus Z^{\prime}$, the set $T_{z}=\\{t\in
T:z\in\bigcup\mathcal{U}_{t}\\}$ is a nonempty definable set. So
$\\{T_{z}\\}_{z\in Z\setminus Z^{\prime}}$ is a uniformly definable family of
nonempty definable sets, defined over $\emptyset$. By definable Skolem
functions, there is a definable function $t:Z\setminus Z^{\prime}\to T$ such
that for each $z\in Z\setminus Z^{\prime}$, we have
$z\in\bigcup\mathcal{U}_{t(z)}$. Varying the parameter over which
$t:Z\setminus Z^{\prime}\to T$ is defined, we get a uniformly
$\emptyset$-definable family of such functions. By Proposition 2.4 we can
choose a parameter in ${\rm dcl}(p)$ for some $p\in P^{k}$. After naming the
elements in $p$ we see that we assume that $t:Z\setminus Z^{\prime}\to T$ is
defined over $\emptyset$.

Now $\\{Z_{t(z)}\setminus\mathcal{U}_{t(z)}\\}_{z\in Z\setminus Z^{\prime}}$
is a uniformly $\emptyset$-definable family of finite sets. By Finiteness
lemma ([26, Chapter 3 (1.7)]), the cardinality of such finite sets are
uniformly bounded. Hence there is a finite definable subset
$Z^{\prime\prime}\subseteq Z$ such that $Z\setminus(Z^{\prime}\cup
Z^{\prime\prime})$ is a finite union of open subintervals, all defined over
$\emptyset$, such that on each such open subinterval the cardinality of all
the finite sets is fixed. So replacing $Z$ by one of the remaining open
subintervals of $Z\setminus(Z^{\prime}\cup Z^{\prime\prime})$, we can assume
that there is $n_{Z}$ such that
$\\#(Z_{t(z)}\setminus\mathcal{U}_{t(z)})=n_{Z}$ for every $z\in Z$.

Consider the $\emptyset$-definable functions $q_{i}:Z\to M$
($i=1,\ldots,n_{Z}$) such that $q_{1}<\ldots<q_{n_{Z}}$ and, for each $z\in
Z$, $(q_{i}(z),q_{i+1}(z))$ are the finitely many open subintervals of
$Z_{t(z)}$ in $\mathcal{U}_{t(z)}$. Since the open subintervals of
$\mathcal{U}_{t(z)}$ are two by two disjoint, the subsets $\\{z\in
Z:z\in(q_{i}(z),q_{i+1}(z))\\}$ with $i=1,\ldots,n_{Z}$, determine a partition
of $Z$ into finitely many $\emptyset$-definable subsets. These subsets are
each a finite union of open intervals and points all defined over $\emptyset$,
so removing all those finitely many points and replacing $Z$ by one of the
remaining open subintervals, we may assume that there are
$\emptyset$-definable functions $q^{-},q^{+}:Z\to M$ such that $q^{-}<q^{+}$
and, for each $z\in Z$, $(q^{-}(z),q^{+}(z))$ is one of the open subintervals
of $Z_{t(z)}$ in $\mathcal{U}_{t(z)}$ and $z\in(q^{-}(z),q^{+}(z))$. ∎

##  5\. Technical lemmas

Now we assume that ${\mathbb{M}}$ is a sufficiently saturated o-minimal
structure with definable Skolem functions.

First, using existence of definable Skolem functions, we try to find finitely
many open subintervals $I$ of $M$ obtained by removing from $M$ finitely many
points, such that on each one of them we have a definable function $H:I\times
I\to M$ which is strictly monotone in each variable. This goal will only be
partially achieved in Lemmas 5.1 and 5.2. We will get instead on each such
subinterval a uniformly definable family $\\{H_{s}:Y_{s}\times I_{s}\to
M\\}_{s\in S}$ of functions which are strictly monotone in the second variable
(when the first variable is fixed).

For each $u\in M$ let

$L_{u}=\\{(x,y)\in M^{2}:x<y<u\\}$

and let

$R_{u}=\\{(x,y)\in M^{2}:u<y<x\\}.$

######  Lemma 5.1.

After naming finitely many elements of $\mathbb{P}$, there is a finite
collection $\mathcal{I}$ of $\emptyset$-definable open intervals of $M$ such
that:

  * (1)

$M\setminus\bigcup\mathcal{I}$ is finite.

  * (2)

For each $I\in{\mathcal{I}}$ there are $\emptyset$-definable functions
$l,r:I\to M$, with $l<r$, which are either constant or strictly monotone and
continuous and, there is a uniformly $\emptyset$-definable family

$\\{h_{I,u}:(l(u),r(u))\to M\\}_{u\in I}$

of definable, continuous and strictly increasing functions such that for each
$u\in I:$

    * -

$(t,h_{I,u}(t))\in L_{u}$ for all $l(u)<t<u$ and

    * -

$(t,h_{I,u}(t))\in R_{u}$ for all $u<t<r(u)$.

Proof. Fix $u\in M$. By definable Skolem functions there is a definable
function $f_{u}:(-\infty,u)\to(-\infty,u)$ such that for each
$t\in(-\infty,u)$ we have $(t,f_{u}(t))\in L_{u}$ and a definable function
$g_{u}:(u,+\infty)\to(u,+\infty)$ such that for each $t\in(u,+\infty)$ we have
$(t,g_{u}(t))\in R_{u}$. Varying the parameter over which $f_{u}$ (resp.
$g_{u}$) is defined, we get a uniformly definable family of such functions
defined over $u$. By Proposition 2.4 we can choose a parameter in ${\rm
dcl}(u\,p)$ for some $p\in P^{k}$. After naming the elements in $p$ we see
that we assume that $f_{u}$ (resp. $g_{u}$) is defined over $u$.

By the monotonicity theorem ([26, Chapter 3, (1.2)]), there is
$l_{u}\in(-\infty,u)$ such that $f_{u|(l_{u},u)}$ is either constant, or
strictly monotone and continuous and, there is $r_{u}\in(u,+\infty)$ such that
$g_{u|(u,r_{u})}$ is either constant or strictly monotone. Note that $f_{u}$
on $(l_{u},u)$ cannot be constant, for if $f_{u}(t)=c$ for all
$t\in(l_{u},u),$ then $l_{u}<c<u$ and so for $s\in(c,u)$ we would have
$c<s<f_{u}(s)=c.$ Note also that $f_{u}$ on $(l_{u},u)$ cannot be strictly
decreasing, otherwise, taking $d\in(l_{u},u),$ then for all $t\in(d,u)$ we
would have $f_{u}(t)<f_{u}(d)<u,$ consequently for
$s\in(f_{u}(d),u)\subseteq(d,u)$ we would have $f_{u}(d)<s<f_{u}(s)<f_{u}(d).$
So $f_{u}$ must be strictly increasing and continuous on $(l_{u},u)$.
Similarly, $g_{u}$ must be strictly increasing and continuous on $(u,r_{u})$.

Figure 1. The functions $f_{u}$ and
$g_{u}$$\,\,\,\,\,\,g_{u}$$f_{u}\,\,\,\,$$u$$u$$l_{u}$$r_{u}$

Note also that $\lim_{t\to u^{-}}f_{u}(t)=u=\lim_{t\to u^{+}}g_{u}(t)$. Let
$h_{u}:(l_{u},r_{u})\to M$ given by

$\displaystyle h_{u}(t)=\begin{cases}f_{u}(t)\qquad\textrm{if}\quad
t\in(l_{u},u)\\\ \\\ u\qquad\quad\,\,\,\,\,\textrm{if}\quad t=u\\\ \\\
g_{u}(t)\qquad\,\,\textrm{if}\quad t\in(u,r_{u})\end{cases}$

Then, for each $u\in M,$ there is a definable function $h_{u}:(l_{u},r_{u})\to
M$ such that $h_{u}$ is continuous and strictly increasing, $(t,h_{u}(t))\in
L_{u}$ for all $t\in(l_{u},u)$, $(t,h_{u}(t))\in R_{u}$ for all
$t\in(u,r_{u})$ and $\lim_{t\to u^{-}}h_{u}(t)=u=\lim_{t\to u^{+}}h_{u}(t)$.
Furthermore, $h_{u}$ is defined over $u$.

Consider the collection of all first-order formulas $\varphi(x,y,v)$ over
$\emptyset$ such that for some $a\in M$, $\varphi(x,y,a)$ defines the graph of
a function on some open interval containing $a$ with the same properties as
$h_{a}$ above. Since all of the above properties of $h_{u}$ are first-order on
$u$, for each such $\varphi$, the subset $I_{\varphi}$ of $M$ of all $a\in M$
such that $\varphi(x,y,a)$ defines the graph of a function on some open
interval containing $a$ with the same properties as $h_{a}$ above is
$\emptyset$-definable. Let ${\mathcal{I}}$ be the collection of all those
$\emptyset$-definable subsets of $M$. Then $\bigcup{\mathcal{I}}=M$ and for
each $I\in{\mathcal{I}}$ there is a uniformly $\emptyset$-definable family

$\\{h_{I,u}:(l_{u},r_{u})\to M\\}_{u\in I}$

of definable, continuous and strictly increasing functions such that for each
$u\in I:$

  * -

$(t,h_{I,u}(t))\in L_{u}$ for all $t\in(l_{u},u)$,

  * -

$(t,h_{I,u}(t))\in R_{u}$ for all $t\in(u,r_{u})$ and,

  * -

$\lim_{t\to u^{-}}h_{I,u}(t)=u=\lim_{t\to u^{+}}h_{I,u}(t)$.

By saturation we may assume that ${\mathcal{I}}$ is finite. By o-minimality,
we may also assume, after ignoring finitely many elements of $M$, that each
$I\in{\mathcal{I}}$ is a $\emptyset$-definable open interval of $M$.

Fix $I\in{\mathcal{I}}$ and consider the $\emptyset$-definable functions
$l,r:I\to M$ given by $l(v)=l_{v}$ and $r(v)=r_{v}$. By the monotonicity
theorem ([26, Chapter 3, (1.2)]), after removing finitely many points from
each $I$ we can assume that on each $I\in{\mathcal{I}}$ the
$\emptyset$-definable functions $l,r:I\to M$ are constant or strictly monotone
and continuous. ∎

Given $Z\subseteq M$ let us set $\Delta_{Z}=\\{(x,x):x\in Z\\}$.

######  Lemma 5.2.

Let $\mathcal{I}$ and for each $I\in\mathcal{I}$ let
$\\{h_{I,w}:(l(w),r(w))\to M\\}_{w\in I}$ be as in Lemma 5.1. Then after
naming more finitely elements of $\mathbb{P}$, for each $I\in\mathcal{I}$
there is a finite collection $\mathcal{Y}^{I}$ of $\emptyset$-definable open
subintervals of $I$ such that:

  * -

$\bigcup\mathcal{Y}^{I}=I$.

  * -

For each $Y\in\mathcal{Y}^{I}$ there is a $\emptyset$-definable $e_{Y}\in Y$
and a uniformly $\emptyset$-definable family of definable functions

$\\{H_{s}:Y_{s}\times I_{s}\to M\\}_{s\in S}$

given by $H_{s}(w,t)=h_{I,w}(t)$ where, $S=Y\setminus\\{e_{Y}\\}$, for each
$s\in S$,

$\displaystyle
Y_{s}=\begin{cases}Y\cap(s,e_{Y})\qquad\textrm{if}\,\,\,s<e_{Y}\\\ \\\
Y\cap(e_{Y},s)\qquad\textrm{otherwise}\end{cases}$

and $I_{s}=(l_{s},r_{s})\subseteq I$ with $l_{s}=\max\\{l(s),l(e_{Y})\\}$ and
$r_{s}=\min\\{r(s),r(e_{Y})\\}$. Moreover, $\Delta_{Y_{s}}\subseteq
Y_{s}\times I_{s}$.

Proof. For each $u\in I$, since $l(u)<u<r(u)$, by continuity, there is an open
interval $Y_{u}\subseteq I$ containing $u$ such that for all $v\in Y_{u}$ we
have $l(v)<u<r(v)$. Since the property of $Y_{u}$ is first-order on $u$, there
is a collection $\mathcal{Y}^{I}$ of $\emptyset$-definable subsets of $I$ such
that $\bigcup\mathcal{Y}^{I}=I$ and for each $Y\in\mathcal{Y}^{I}$, $Y$ is a
$\emptyset$-definable open subinterval of $I$ and for all $x,x^{\prime}\in Y$
we have $l(x^{\prime})<x<r(x^{\prime})$. By saturation, $\mathcal{Y}^{I}$ is
finite.

Fix $Y\in\mathcal{Y}^{I}$. By Proposition 2.4, we can choose an element of $Y$
in ${\rm dcl}(p)$ for some $p\in P^{k}$. After naming the elements in $p$ we
see that we can choose a $\emptyset$-definable $e_{Y}\in Y$.

For $s\in Y\setminus\\{e_{Y}\\}$, let $l_{s}$, $r_{s}$, $I_{s}$ and $Y_{s}$ be
as in the statement of the lemma. Let $w\in Y_{s}\subseteq Y\subseteq I$.
Since the $\emptyset$-definable functions $l,r:I\to M$ are constant or
strictly monotone and continuous and $w$ is between $s$ and $e_{Y}$, we have
$(l_{s},r_{s})\subseteq(l(w),r(w))$. Moreover, since $s,e_{Y}\in Y$ we also
have $l_{s}<w<r_{s}$. It follows that $\Delta_{Y_{s}}\subseteq Y_{s}\times
I_{s}$ and we can define uniformly the definable functions

$H_{s}:Y_{s}\times I_{s}\to M$

by $H_{s}(w,t)=h_{I,w}(t)$. ∎

Each $H_{s}$ of Lemma 5.2 is continuous and strictly increasing in the second
coordinate when we fix the first coordinate. We need to investigate what
happens in the first coordinate when we fix the second coordinate. Following
this line of thought does not seem to produce any conclusive results, so we
try to see what happens to the restrictions of the uniformly definable family
of curves

$C_{w}=\\{(t,f_{w}(t)):t\in I_{s,w}^{-}\\},$

with $w\in Y_{s}$ and $I_{s,w}^{-}=\\{t\in I_{s}:t<w\\}$, induced by
$\\{H_{s}\\}_{s\in S}$.

In Lemmas 5.3 and 5.6 below, we will find finitely many definable open
subintervals of each $Y_{s}$, obtained by removing finitely many points,
uniformly in $s$, such that on each one of them the family $\\{C_{w}\\}_{w\in
Y_{s}}$ induces one-dimensional definable normal families of curves or if not
then we have density of non-trivial points on the subinterval.

Below for each $u\in Y_{s}$ we let $Y_{s,u}^{-}=\\{v\in Y_{s}:v<u\\}$.

######  Lemma 5.3.

For each $Y_{s}$ there is a finite collection $\mathcal{X}_{s}$, uniformly
defined over $s$, of two by two disjoint open intervals uniformly defined over
$s$ such that $Y_{s}\setminus\bigcup\mathcal{X}_{s}$ is finite and for each
$X\in\mathcal{X}_{s}$ either the following holds

  * (N1)

for every $u\in X$ there is an open subinterval $X_{s,u}^{-}\subseteq
Y_{s,u}^{-}$ with right endpoint $u$ such that for every $v\in X_{s,u}^{-}$
there is an open subinterval $J_{s,u,v}^{-}\subseteq I_{s,v}^{-}$ with right
endpoint $v$ such that $\\{C_{v|J_{s,u,v}^{-}}\\}_{v\in X_{s,u}^{-}}$ is a
one-dimensional definable normal family of curves,

or if not, then the following holds

  * (L)

for every $u\in X$ there is an open subinterval $X_{s,u}^{-}\subseteq
Y_{s,u}^{-}$ with right endpoint $u$ such that for every $v\in X_{s,u}^{-}$ we
have $\overline{C}_{v|J_{s,u,v}^{-}}\cap C_{u|J_{s,u}^{-}}$ is finite where
$J_{s,u,v}^{-}$ (resp. $J_{s,u}^{-}$) is an open subinterval of $I_{s,v}^{-}$
(resp. $I_{s,u}^{-}$) with right endpoint $v$ (resp. $u$),

or else

  * (D1)

every open subinterval of $X$ defined over $s$ has a non-trivial point.

###### Proof.

We are going to construct definable subsets

$\mathcal{T}_{s}\subseteq\mathcal{B}_{s}\subseteq Y_{s},$

all defined over $s$, so that

$Y_{s}=\mathcal{T}_{s}\sqcup(\mathcal{B}_{s}\setminus\mathcal{T}_{s})\sqcup(Y_{s}\setminus\mathcal{B}_{s}).$

Each one of these pieces of $Y_{s}$ is a finite union of points and open
intervals. We let $\mathcal{X}_{s}$ be the collection of all such open
subintervals. Let $X\in\mathcal{X}_{s}$. We show that:

  * -

if $X$ is an open subinterval of $Y_{s}\setminus\mathcal{B}_{s}$ then (N1) of
the lemma holds on $X$;

  * -

if $X$ is an open subinterval of $\mathcal{B}_{s}\setminus\mathcal{T}_{s}$
then (L) of the lemma holds on $X$;

  * -

if $X$ is an open subinterval of $\mathcal{T}_{s}$ then (D1) of the lemma
holds on $X$.

Fix $u\in Y_{s}$. Consider the definable subset

$\mathcal{B}_{s,u}=\\{v\in Y_{s,u}^{-}:\,\textrm{for
some}\,\,t<v\,\,\textrm{we have}\,\,f_{u}(t)\leq f_{v}(t)\\}$

and the definable function $\tau_{s,u}:\mathcal{B}_{s,u}\to I_{s,u}^{-}$ given
by

$\tau_{s,u}(v)=\sup\\{t<v:f_{u}(t)\leq f_{v}(t)\\}.$

Note that the family $\\{\tau_{s,u}:\mathcal{B}_{s,u}\to I_{s,u}^{-}\\}$ is
uniformly defined over $s$ and $u$.

Figure 2. The functions $f_{u}$, $g_{u}$, $f_{v}$ and
$g_{v}$$\,\,\,\,\,\,g_{u}$$f_{u}\,\,\,\,$$\,\,\,\,\,\,\,\,\,g_{v}$$f_{v}\,\,\,\,\,\,\,$$u$$u$$v$$v$

######  Claim 5.4.

For all $v\in\mathcal{B}_{s,u}$, we have $\tau_{s,u}(v)<v$ and
$f_{u}(\tau_{s,u}(v))=f_{v}(\tau_{s,u}(v))$.

Proof. Note that for all $v<u$, since $f_{u}(v)\in L_{u}$, we have
$v<f_{u}(v)$. So by continuity there is an open definable neighbourhood $V$ of
$v$ such that for all $t\in V$ we have $v<f_{u}(t)<u$. In particular, there is
$t^{\prime}<v$ an element of $V$ such that $\\{t<v:f_{u}(t)\leq f_{v}(t)\\}$
is below $t^{\prime}$, since for $t<v$ we have $f_{v}(t)<v$. It follows that
$\tau_{s,u}(v)<v$.

Suppose that $f_{v}(\tau_{s,u}(v))<f_{u}(\tau_{s,u}(v))$. Then by continuity
there is an open definable neighbourhood $U$ of $\tau_{s,u}(v)$ such that for
all $t\in U$ with $t<\tau_{s,u}(v)$ we have $f_{v}(t)<f_{u}(t)$, contradicting
the definition of $\tau_{s,u}(v)$. Similarly, if
$f_{u}(\tau_{s,u}(v))<f_{v}(\tau_{s,u}(v))$, then by continuity there is an
open definable neighbourhood $U$ of $\tau_{s,u}(v)$ such that for all $t\in U$
with $\tau_{s,u}(v)<t<v$ we have $f_{u}(t)<f_{v}(t)$, contradicting again the
definition of $\tau_{s,u}(v)$. Therefore,
$f_{u}(\tau_{s,u}(v))=f_{v}(\tau_{s,u}(v))$.

∎

For each $u\in Y_{s}$, $\mathcal{B}_{s,u}$ is a finite union of points and
open subintervals of $Y_{s,u}^{-}$. The collection of these finitely many
points and open subintervals is uniformly defined over $s$ and $u$. Therefore,
either $\mathcal{B}_{s,u}$ or its complement in $Y_{s,u}^{-}$, but not both,
contains an open subinterval with right endpoint $u$. Let

$\mathcal{B}_{s}=\\{u\in Y_{s}:\mathcal{B}_{s,u}\,\,\textrm{contains an open
subinterval with right endpoint}\,\,u\\}.$

Note that $\mathcal{B}_{s}$ is uniformly defined over $s$.

Inside $Y_{s}\setminus\mathcal{B}_{s}$. We have that
$Y_{s}\setminus\mathcal{B}_{s}$ is a finite union of points and open
subintervals of $Y_{s}$, all uniformly defined over $s$. Removing these
finitely many points we may assume that $Y_{s}\setminus\mathcal{B}_{s}$ is a
finite union of open subintervals of $Y_{s}$ uniformly defined over $s$.

Let $u\in Y_{s}\setminus\mathcal{B}_{s}$. Then there is a maximal open
subinterval $X_{s,u}^{-}$ with right endpoint $u$ such that
$\mathcal{B}_{s,u}\cap X_{s,u}^{-}=\emptyset$.

Then for all $v\in X_{s,u}^{-}$ we have $f_{v}(t)<f_{u}(t)$ for all $t\in
I_{s,v}^{-}$. It follows that, for all $v\in X_{s,u}^{-}$ we have $C_{v}\cap
C_{u}=\emptyset$.

Since $\\{X_{s,u}^{-}\\}_{u\in Y_{s}\setminus\mathcal{B}_{s}}$ is uniformly
definable, we have a definable map $\mu:Y_{s}\setminus\mathcal{B}_{s}\to
Y_{s}$ given by $\mu(u)=\inf X_{s,u}^{-}$. So we have $X_{s,u}^{-}=(\mu(u),u)$
and $\mu(u)<u$ for all $u\in Y_{s}\setminus\mathcal{B}_{s}$.

Now by Lemma 4.1, the set $Y_{s}\setminus\mathcal{B}_{s}$ is a finite union of
open subintervals $X$ all defined over $s$ such that, if $u\in X$ and $v,w\in
X_{s,u}^{-}$ are such that $v<w$, then $v\in X_{s,w}^{-}$. But by definition
of $X_{s,w}^{-}$, we have $C_{v}\cap C_{w}=\emptyset$. Hence
$\\{C_{v}\\}_{v\in X_{s,u}^{-}}$ is a one-dimensional definable normal family
of curves, and (N1) holds on $Y_{s}\setminus\mathcal{B}_{s}$.

We have that $\mathcal{B}_{s}\subseteq Y_{s}$ is a finite union of points and
open subintervals of $Y_{s}$, all uniformly defined over $s$. Removing these
finitely many points we may assume that $\mathcal{B}_{s}$ is a finite union of
open subintervals of $Y_{s}$ uniformly defined over $s$.

Let $u\in\mathcal{B}_{s}$. Then $\mathcal{B}_{s,u}$ contains a maximal open
subinterval with right endpoint $u$. By abuse of notation we assume that
$\mathcal{B}_{s,u}$ is that subinterval. These open subinterval are uniformly
defined over $s$. Let

$\mathcal{T}_{s}=\\{u\in\mathcal{B}_{s}:\sup\tau_{s,u}=u\\}.$

Note that $\mathcal{T}_{s}$ is uniformly defined over $s$.

Inside $\mathcal{B}_{s}\setminus\mathcal{T}_{s}$. If
$u\in\mathcal{B}_{s}\setminus\mathcal{T}_{s}$, then $\sup\tau_{s,u}<u$. Let
$X_{s,u}^{-}=\mathcal{B}_{s,u}\cap(\sup\tau_{s,u},u)$ (recall we are assuming
that $\mathcal{B}_{s,u}$ is an open subinterval with right endpoint $u$). So,
for all $v\in X_{s,u}^{-}$, since $\tau_{s,u}(v)\leq\sup\tau_{s,u}$, we have
$f_{v}(t)<f_{u}(t)$ for all $t\in(\sup\tau_{s,u},v)$. From here we conclude
that, for all $v\in X_{s,u}^{-}$ we have $C_{v|(\sup\tau_{s,u},v)}\cap
C_{u}=\emptyset$. Hence (L) holds on
$\mathcal{B}_{s}\setminus\mathcal{T}_{s}$.

Inside $\mathcal{T}_{s}$. Without loss of generality, after removing finitely
many points we may assume that $\mathcal{T}_{s}$ is a finite union of open
subintervals of $Y_{s}$ uniformly defined over $s$.

Let $u\in\mathcal{T}_{s}$. Then

######  Claim 5.5.

There is a maximal open subinterval of $\mathcal{B}_{s,u}$ with right endpoint
$u$ on which $\tau_{s,u}$ is strictly increasing and continuous.

Proof. By the monotonicity theorem ([26, Chapter 3, (1.2)]), there is a
maximal open subinterval of $\mathcal{B}_{s,u}$ with right endpoint $u$ on
which $\tau_{s,u}$ is constant or strictly monotone and continuous. Now:

  * -

If $\tau_{u}$ is constant with value $t_{0}$, then $t_{0}=\tau_{s,u}(v)<v<u$
(Claim 5.4), which contradicts $\sup\tau_{s,u}=u$.

  * -

If $\tau_{u}$ is strictly decreasing, then for $v<w$ we have
$\tau_{s,u}(w)<\tau_{s,u}(v)<v<u$ and the supremum of $\tau_{s,u}$ is the
limit going to the left which leads again to a contradiction.

∎

By abuse of notation, we may assume that $\mathcal{B}_{s,u}$ is the maximal
open subinterval with right endpoint $u$ on which $\tau_{s,u}$ is strictly
increasing and continuous. Since the family
$\\{\mathcal{B}_{s,u}\\}_{u\in\mathcal{T}_{s}}$ is uniformly definable, we
have a definable function $b:\mathcal{T}_{s}\to Y_{s}$, given by
$b(u)=\inf\mathcal{B}_{s,u}$, such that $(b(u),u)=\mathcal{B}_{s,u}$. Consider
the partition of $\mathcal{T}_{s}$ into finitely many points and open
intervals given by Lemma 4.1. Replace $\mathcal{T}_{s}$ by the collection of
those finitely many open subintervals. It follows that given
$u\in\mathcal{T}_{s}$, if $w,v\in\mathcal{B}_{s,u}\cap\mathcal{T}_{s}$ are
such that $v<w$, then $v\in\mathcal{B}_{s,w}$. Moreover we have the following
three possibilities:

  * -

$\tau_{s,w}(v)\leq\tau_{s,u}(v)<\tau_{s,u}(w)$;

  * -

$\tau_{s,u}(v)<\tau_{s,w}(v)<\tau_{s,u}(w)$;

  * -

$\tau_{s,u}(v)<\tau_{s,u}(w)\leq\tau_{s,w}(v)$.

Suppose that there is an open subinterval $O$ of $\mathcal{T}_{s}$ defined
over $s$ with no non-trivial points. Applying Lemma 4.3 to $O$, the definable
function $b:O\to M$ and the definable function
$\Omega:(b,\mathrm{id}_{O})_{O}\to M$ given by $\Omega(u,t)=\tau_{s,u}(t)$, we
obtain that $O$ is a finite union of points and open subintervals, all defined
over $s$, such that on each such open subinterval $X$:

  * -

there is a definable function $d_{X}:X\to M$ defined over $s$ such that
$b_{|X}(u)\leq d_{X}(u)<u$ for all $u\in X$;

  * -

for every $u\in X$, if $v,w\in(d_{X}(u),u)\cap X$ are such that $v<w$, then we
have $v\in(d_{X}(w),w)$ and $\Omega(u,v)=\Omega(w,v)$.

Given $u\in O$ let $X_{s,u}^{-}=(d_{X}(u),u)\cap X$ where $X$ as above is such
that $u\in X$. Then for all $v,w\in X_{s,u}^{-}$, if $v<w$, then
$\tau_{s,w}(v)\leq\tau_{s,u}(v)<\tau_{s,u}(w)$. Now let $C_{v}$ (resp.
$C_{w}$) denote the restriction $C_{v|(\tau_{s,u}(v),v)}$ (resp.
$C_{w|(\tau_{s,u}(w),w)}$). Since
$f_{v|(\tau_{s,w}(v),v]}<f_{w|(\tau_{s,w}(v),v]}$, it follows that we have
$C_{v}\cap C_{w}=\emptyset$ for all $v,w\in X_{s,u}^{-}$ with $v<w$. Therefore
$\\{C_{v}\\}_{v\in X_{s,u}^{-}}$ is a one-dimensional definable normal family
of curves. Let $u^{\prime}\in X$ be such that $u^{\prime}<u$ and consider the
restriction of the family $\\{C_{v}\\}_{v\in X_{s,u}^{-}}$ to
$[u^{\prime},u]\times[u^{\prime},u]$. By [22, Lemma 2.3], the restriction of
$\\{C_{v}\\}_{v\in X_{s,u}^{-}}$ to $[u^{\prime},u]\times[u^{\prime},u]$ is
definable over $[u^{\prime},u]$, hence over $O$. So by Theorem 3.4 applied to
${\mathbb{M}}_{|O}$ there is a non-trivial point in $O$, which is a
contradiction, since we assumed that $O$ had no non-trivial points. ∎

In the next lemma, although the condition (N2) is exactly the same as the
condition (N1) from Lemma 5.3, the curves appearing in (N2) are constructed in
a different way than those appearing in (N1) and it is convenient for now to
keep track of this information. A similar remark applies to condition (D1)
from Lemma 5.3 and (D2) below.

######  Lemma 5.6.

Let $X\in\mathcal{X}_{s}$ be of type (L). Then after removing finitely many
points from $X$, all uniformly defined over $s$, and replacing $X$ by one of
the remaining open subintervals we may assume that the following holds

  * (N2)

for every $u\in X$ there is an open subinterval $X_{s,u}^{-}\subseteq
Y_{s,u}^{-}$ with right endpoint $u$ such that for every $v\in X_{s,u}^{-}$
there is an open subinterval $J_{s,u,v}^{-}\subseteq I_{s,v}^{-}$ with right
endpoint $v$ such that $\\{C_{v|J_{s,u,v}^{-}}\\}_{v\in X_{s,u}^{-}}$ is a
one-dimensional definable normal family of curves,

or else

  * (D2)

every open subinterval of $X$ defined over $s$ has a non-trivial point.

Proof. Let $X\in\mathcal{X}_{s}$ be of type (L). Since the family
$\\{X_{s,u}^{-}\\}_{u\in X}$ of open subintervals of $X$, each with right end
point $u$, is uniformly definable, there is a definable function $\mu:X\to
Y_{s}$, given by $\mu(u)=\inf X_{s,u}^{-}$, such that
$(\mu(u),u)=X_{s,u}^{-}$. Now consider the partition of $X$ into finitely
points and open intervals given by Lemma 4.1. Note that, since the family
$\\{\mu:X\to Y_{s}\\}$ is uniformly defined over $s$ and the conclusion of
Lemma 4.1 is first-order in $s$, these finitely many points and open intervals
are uniformly defined over $s$. Replace $X$ by one such open subinterval. It
follows that if $u\in X$, then for all $v,w\in X_{s,u}^{-}$ with $v<w$ we have
$v\in X_{s,w}^{-}$.

Fix $u\in X$. If $v,w\in X_{s,u}^{-}$ are such that $v<w$, then, since $v\in
X_{s,w}^{-}$, there are open subintervals $J_{s,w,v}^{-}$ of $I_{s,v}^{-}$ and
$J_{s,w}^{-}$ of $I_{s,w}^{-}$, uniformly in $v$ and $w$, with right endpoint
$v$ and $w$ respectively such that $C_{v|J_{s,w,v}^{-}}\cap C_{w|J_{s,w}^{-}}$
is finite, equivalently $\dim(C_{v|J_{s,w,v}^{-}}\cap C_{w|J_{s,w}^{-}})=0$.
So there is a uniformly definable family of definable functions

$\\{j_{s,v}:(v,u)\to I_{s,v}^{-}\\}_{v\in X_{s,u}^{-}}$

given by $j_{s,v}(w)=\inf J_{s,w,v}^{-}$ for $w\in(v,u)$.

By construction we have $J_{s,w,v}^{-}=(j_{s,v}(w),v)$, in particular
$j_{s,v}(w)<v$, for all $w\in(v,u)$ and all $v\in X_{s,u}^{-}$.

Let

$\mathcal{V}_{s,u}=\\{v\in X_{s,u}^{-}:\sup j_{s,v}=v\\}.$

Then for each $u\in X$ either $\mathcal{V}_{s,u}$ or
$X_{s,u}^{-}\setminus\mathcal{V}_{s,u}$, but not both, contains an open
subinterval of $X_{s,u}^{-}$ with right endpoint $u$. Let

$\mathcal{V}_{s}=\\{u\in X:\mathcal{V}_{s,u}\,\,\textrm{contains an open
subinterval with right endpoint}\,\,u\\}.$

Note that $\mathcal{V}_{s}$ is uniformly defined over $s$.

Inside $X\setminus\mathcal{V}_{s}$. We have that
$X\setminus\mathcal{V}_{s}\subseteq Y_{s}$ is a finite union of points and
open subintervals of $Y_{s}$, all uniformly defined over $s$. Removing these
finitely many points we may assume that $X\setminus\mathcal{V}_{s}$ is a
finite union of open subintervals of $Y_{s}$ uniformly defined over $s$.

If $u\in X\setminus\mathcal{V}_{s}$, then
$X_{s,u}^{-}\setminus\mathcal{V}_{s,u}$ contains an open subinterval with
right endpoint $u$. Let us assume, with abuse of notation, that $X_{s,u}^{-}$
is the maximal open such subinterval. If $v\in X_{s,u}^{-}$, then $\sup
j_{s,v}<v$. Note also that $v\in X$. Consider the open subinterval $(\sup
j_{s,v},v)\cap J^{-}_{s,v}$, which by abuse of notation we still denote by
$J_{s,u,v}^{-}$. Then for all $w\in(v,u)$ we have
$\dim(C_{v|J_{s,u,v}^{-}}\cap C_{w|J_{s,w}^{-}})=0$ since $j_{s,v}(w)<\sup
j_{s,v}$. Now if $w\in X_{s,u}^{-}$ is such that $v<w$, then since $(\sup
j_{s,w},w)\cap J^{-}_{s,w}\subseteq J^{-}_{s,w}$ we have
$\dim(C_{v|J_{s,u,v}^{-}}\cap C_{w|J_{s,u,w}^{-}})=0$. Hence
$\\{C_{v|J_{s,u,v}^{-}}\\}_{v\in X_{s,u}^{-}}$ is a one-dimensional definable
normal family of curves, and (N2) holds on $X\setminus\mathcal{V}_{s}$.

Step 2: Inside $\mathcal{V}_{s}$. If $u\in\mathcal{V}_{s}$, then
$\mathcal{V}_{s,u}$ contains an open subinterval with right endpoint $u$. Let
us assume, with abuse of notation, that $\mathcal{V}_{s,u}$ is the maximal
open such subinterval.

Let $v\in\mathcal{V}_{s,u}$. Then the following holds.

######  Claim 5.7.

There is a maximal open subinterval $D_{s,v}$ of $(v,u)$ such that
$j_{s,v|D_{s,v}}:D_{s,v}\to I_{s,v}^{-}$ is strictly decreasing and $D_{s,v}$
has left endpoint $v$.

Proof. By the monotonicity theorem ([26, Chapter 3, (1.2)]), $(v,u)$ is a
finite union of points and open subintervals such that on each such open
subinterval the restriction of $j_{s,v}$ to that interval is either constant
or strictly monotone and continuous. Since $\sup j_{s,v}=v$, there is an open
subinterval $D_{s,v}$ of $(v,u)$ such that $j_{s,v|D_{s,v}}:D_{s,v}\to
I_{s,v}^{-}$ is strictly monotone and continuous and $j_{s,v}(D_{s,v})$ is an
open subinterval of $I_{s,v}^{-}$ with left endpoint $v$.

Let $a^{-}$ (resp. $a^{+}$) be the left (resp. right) endpoint of $D_{s,v}$.
Note that there are open subintervals of $D_{s,v}$ with right endpoint $a^{+}$
(resp. left endpoint $a^{-}$) such that the restriction of the definable
function $a\mapsto H_{s}(a,j_{s,v}(a))$ of Lemma 5.2 is continuous on them.
Note also that for all $a\in D_{s,v}$ we have
$f_{v}(j_{s,v}(a))=f_{a}(j_{s,v}(a))$ since, by definition of $j_{s,v}(a)$,
the curves $C_{v}$ and $C_{a}$ coincide in a subinterval with right endpoint
$j_{s,v}(a)$.

Suppose that $j_{s,v|D_{s,v}}$ is strictly increasing, in particular,
$\lim_{a\to a^{+}}j_{s,v}(a)=\sup j_{s,v}=v$. Then by continuity, we have

$\displaystyle v$ $\displaystyle=$ $\displaystyle\lim_{a\to
a^{+}}f_{v}(j_{s,v}(a))$ $\displaystyle=$ $\displaystyle\lim_{a\to
a^{+}}f_{a}(j_{s,v}(a))$ $\displaystyle=$ $\displaystyle\lim_{a\to
a^{+}}H_{s}(a,j_{s,v}(a))$ $\displaystyle=$ $\displaystyle f_{a^{+}}(v)$

contradicting the fact that $v<f_{a^{+}}(v)$ since $v<a^{+}$.

Suppose that $j_{s,v|D_{s,v}}$ is strictly decreasing but $v<a^{-}$. Then by
continuity again, we have $v=\lim_{a\to a^{-}}f_{v}(j_{s,v}(a))=\lim_{a\to
a^{-}}f_{a}(j_{s,v}(a))=f_{a^{-}}(v)$ contradicting the fact that
$v<f_{a^{-}}(v)$ since $v<a^{-}$. ∎

Since the family $\\{D_{s,v}\\}_{v\in\mathcal{V}_{s,u}}$ is uniformly
definable, we have a definable function $b:\mathcal{V}_{s,u}\to Y_{s}$, given
by $b(v)=\sup D_{s,v}$, such that $(v,b(v))=D_{s,v}$. Consider the partition
of $\mathcal{V}_{s,u}$ into finitely many points and open intervals given by
Lemma 4.2. Replace $\mathcal{V}_{s,u}$ by the maximal rightmost such open
subinterval. It follows that given $v\in\mathcal{V}_{s,u}$, if $x,y\in
D_{s,v}\cap\mathcal{V}_{s,u}$ are such that $x<y$, then $x\in D_{s,y}$.
Moreover we have the following three possibilities:

  * -

$j_{s,x}(y)\leq j_{s,v}(y)<j_{s,v}(x)$;

  * -

$j_{s,v}(y)<j_{s,x}(y)<j_{s,v}(x)$;

  * -

$j_{s,v}(y)<j_{s,v}(x)\leq j_{s,x}(y)$.

Suppose that there is an open subinterval $O$ of $\mathcal{V}_{s}$ defined
over $s$ with no non-trivial points. Then for every $u\in O$,
$\mathcal{V}_{s,u}$ has no non-trivial points. Applying Lemma 4.4 to
$\mathcal{V}_{s,u}$, the definable function $b:\mathcal{V}_{s,u}\to M$ and the
definable function
$\Omega:(\mathrm{id}_{\mathcal{V}_{s,u}},b)_{\mathcal{V}_{s,u}}\to M$ given by
$\Omega(v,t)=j_{s,v}(t)$, we obtain that $\mathcal{V}_{s,u}$ is a finite union
of points and open subintervals, all defined over $s$, such that on each such
open subinterval $X$:

  * -

there is a definable function $d_{X}:X\to M$ defined over $s$ such that
$v<d_{X}(v)\leq b_{|X}(v)$ for all $v\in X$;

  * -

for every $v\in X$, if $x,y\in(v,d_{X}(v))\cap X$ are such that $x<y$, then we
have $y\in(x,d_{X}(x))$ and $\Omega(v,y)=\Omega(x,y)$.

Let $X_{s,u}^{-}$ be the open subinterval $X$ of $\mathcal{V}_{u}$ with right
endpoint $u$. Given $v\in X_{s,u}^{-}$ let
$X_{s,u,v}^{+}=(v,d_{X}(v))\cap\mathcal{V}_{u}$. Then for all $x,y\in
X_{s,u,v}^{+}$, if $x<y$, then $j_{s,x}(y)\leq j_{s,v}(y)<j_{s,v}(x)$. Now let
$C_{x}$ (resp. $C_{y}$) denote the restriction $C_{x|(j_{s,v}(x),x)}$ (resp.
$C_{y|(j_{s,v}(y),y)}$). Since $C_{x|(j_{s,x}(y),x)}\cap C_{y|(j_{s,x}(y),x)}$
is finite, it follows that $C_{x}\cap C_{y}$ is finite for all $x,y\in
X_{s,u,v}^{+}$ with $x<y$. Therefore $\\{C_{x}\\}_{x\in X_{s,u,v}^{+}}$ is a
one-dimensional definable normal family of curves. Let $v^{\prime}\in X$ be
such that $v<v^{\prime}$ and consider the restriction of the family
$\\{C_{x}\\}_{x\in X_{s,u,v}^{+}}$ to $[v,v^{\prime}]\times[v,v^{\prime}]$. By
[22, Lemma 2.3], the restriction of $\\{C_{x}\\}_{x\in X_{s,u,v}^{+}}$ to
$[v,v^{\prime}]\times[v,v^{\prime}]$ is definable over $[v,v^{\prime}]$, hence
over $O$. So by Theorem 3.4 applied to ${\mathbb{M}}_{|O}$ there is a non-
trivial point in $O$, which is a contradiction since we assumed that $O$ had
no non-trivial points. ∎

The cost for obtaining Lemmas 5.3 and 5.6 was the introduction of extra
parameters. To proceed we will apply Lemma 4.5 to remove first the parameter
$s$ (Lemma 5.8) and then the extra parameters (Lemma 5.9).

######  Lemma 5.8.

Consider the $\emptyset$-definable open intervals $I\in\mathcal{I}$ and
$Y\in\mathcal{Y}^{I}$ of Lemmas 5.1 and 5.2. Then after naming finitely many
elements of $\mathbb{P}$, and after removing finitely many
$\emptyset$-definable points and replacing each $Y$ by one of the remaining
open subintervals we may assume that for each $Y\in\mathcal{Y}^{I}$ there are
$\emptyset$-definable functions $q^{-},q^{+}:Y\to M$ with $q^{-}<q^{+}$, such
that $y\in(q^{-}(y),q^{+}(y))$ for all $y\in Y$. Moreover, letting
$U_{y}=(q^{-}(y),q^{+}(y))$, then for all $y\in Y$ the following holds

  * (N′)

for every $u\in U_{y}$ there is an open subinterval $X_{y,u}^{-}\subseteq
U_{y}$ with right endpoint $u$, uniformly definable in $y$, such that for
every $v\in X_{y,u}^{-}$ there is an open subinterval $J_{y,u,v}^{-}\subseteq
U_{y}$ with right endpoint $v$, uniformly definable in $y$, such that
$\\{C_{v|J_{y,u,v}^{-}}\\}_{v\in X_{y,u}^{-}}$ is a one-dimensional definable
normal family of curves,

or else for all $y\in Y$

  * (D′)

every open subinterval of $U_{y}$ defined over $y$ has a non-trivial point.

Proof. Consider the $\emptyset$-definable open intervals $I\in\mathcal{I}$,
$Y\in\mathcal{Y}^{I}$ and the definable intervals

$\displaystyle
Y_{s}=\begin{cases}Y\cap(s,e_{Y})\qquad\textrm{if}\,\,\,s<e_{Y}\\\ \\\
Y\cap(e_{Y},s)\qquad\textrm{otherwise}\end{cases}$

with $s\in S$, introduced in Lemmas 5.1 and 5.2. Consider also the finite
collection $\mathcal{X}_{s}$, uniformly defined over $s$, of two by two
disjoint open intervals defined over $s$, given by Lemmas 5.3 and 5.6, such
that $Y_{s}\setminus\bigcup\mathcal{X}_{s}$ is finite and for each
$X\in\mathcal{X}_{s}$ either $X$ is of one of the types (N1), (D1), (N2) or
(D2).

If consider the data $(Y,S,\\{Y_{s}\\}_{s\in S},\\{\mathcal{X}_{s}\\}_{s\in
S})$ then we are in the set up of Lemma 4.5. So after naming finitely many
elements of $\mathbb{P}$, and after removing finitely many
$\emptyset$-definable points from $Y$ and replacing $Y$ by one of the
remaining open subintervals, we may assume that there are
$\emptyset$-definable functions $q^{-},q^{+}:Y\to M$ such that $q^{-}<q^{+}$
and $s:Y\to S$ such that, for each $y\in Y$, $(q^{-}(y),q^{+}(y))$ is one of
the open subintervals of $Y_{s(y)}$ in $\mathcal{X}_{s(y)}^{-}$ and
$y\in(q^{-}(y),q^{+}(y))$.

By the remarks about uniform definability over $s(y)$ throughout the proof of
Lemma 5.3, we see that the open subinterval $(q^{-}(y),q^{+}(y))$ being of
type (N1), not being of type (N1) but being of type (L) and not being of type
(N1) neither of type (L) but of type (D1) are first-order properties of $y$.
Indeed, these correspond to: the open subinterval $(q^{-}(y),q^{+}(y))$ is
contained in $Y_{s(y)}\setminus\mathcal{B}_{s(y)}$ (resp. in
$\mathcal{B}_{s(y)}\setminus\mathcal{T}_{s(y)}$ or in $\mathcal{T}_{s(y)}$).
So one can partition $Y$ into one, two or three $\emptyset$-definable subsets
such that on each one of them, all the open subintervals $(q^{-}(y),q^{+}(y))$
are of type (N1), or all are of type (L) but not of type (N1) or none is of
type (N1) or type (L) but are of type (D1).

Similarly, using the remarks about uniform definability over $s(y)$ throughout
the proof of Lemma 5.6, we see that the part of $Y$ corresponding to type (L)
but not type (N1) can be further partitioned into one or two
$\emptyset$-definable subsets such that on each one of them, all the open
subintervals $(q^{-}(y),q^{+}(y))$ are of type (N2) or none is of type (N2)
and all are of type (D2).

So removing finitely many $\emptyset$-definable points and replacing $Y$ by
one of the remaining open subintervals, we can assume that the open intervals
$(q^{-}(y),q^{+}(y))$ are either all of type (N1), or all of type (D1), or all
of type (N2) or all of type (D2).

Using the fact that for each $y\in Y$, $(q^{-}(y),q^{+}(y))$ is an open
subinterval of $Y_{s(y)}$, $I_{s(y),v}^{-}$ is an open subinterval with right
endpoint $v$ and $v\in(q^{-}(v),q^{+}(v))$ for each $v\in(q^{-}(y),q^{+}(y))$,
the result now follows at once from Lemmas 5.3 and 5.6. ∎

We can apply Lemma 4.5 once more to remove the parameter $u$ from Lemma 5.8
and reach our main technical lemma:

######  Lemma 5.9.

After naming finitely many elements of $\mathbb{P}$, there is a collection
$\mathcal{Y}$ of finitely many $\emptyset$-definable open subinterval of $M$
such that $M\setminus\bigcup\mathcal{Y}$ is finite and for each
$Y\in\mathcal{Y}$ there is a uniformly $\emptyset$-definable family
$\\{X_{y}\\}_{y\in Y}$ of open subintervals such that $y\in X_{y}$ for all
$y\in Y$ and moreover, for all $y\in Y$ the following holds

  * (N)

for every $v\in X_{y}$ there is an open subinterval $J_{y,v}^{-}$ of $X_{y}$
with right endpoint $v$, uniformly definable in $y$, such that
$\\{C_{v|J_{y,v}^{-}}\\}_{v\in X_{y}}$ is a one-dimensional definable normal
family of curves,

or else for all $y\in Y$

  * (D)

every open subinterval of $X_{y}$ defined over $y$ has a non-trivial point.

Proof. We start with $I\in\mathcal{I}$, $Y\in\mathcal{Y}^{I}$ and $U_{y}$ as
in Lemma 5.8.

Case (D′): Suppose that for all $y\in Y$, $U_{y}$ is of type (D′). Then set
$X_{y}=U_{y}$.

Case (N′): Suppose that for all $y\in Y$, $U_{y}$ is of type (N′). Let $Z=Y$,
$T=Y$ and for $t\in T$ let $Z_{t}=U_{t}$. Note that
$F_{t}:=Z_{t}\setminus\bigcup\\{X_{t,u}^{-}:u\in Z_{t}\\}$ is finite.
Otherwise, this definable subset of $M$ would contain an open subinterval $W$.
But then, for $u\in W$, we would have $W\cap X_{t,u}^{-}\neq\emptyset$ since
$W\subseteq X$ is an open subinterval and, on the other hand, $W\cap
X_{t,u}^{-}=\emptyset$ since $W\subseteq F_{t}$.

Let $\mathcal{U}_{t}$ be the collection of two by two disjoint open
subintervals obtained by removing $F_{t}$ from $Z_{t}$. If we consider the
data $(Z,T,\\{Z_{t}\\}_{t\in T},\\{\mathcal{U}_{t}\\}_{t\in T})$, then we are
in the set up of Lemma 4.5. Therefore, after naming finitely many elements of
$\mathbb{P}$, and after removing finitely many $\emptyset$-definable points
from $Z$ and replacing $Z$ by one of the remaining open subintervals, we may
assume that there are $\emptyset$-definable functions $p^{-},p^{+}:Z\to M$
such that $p^{-}<p^{+}$ and $t:Z\to T$ such that, for each $z\in Z$,
$(p^{-}(z),p^{+}(z))$ is one of the open subintervals of $Z_{t(z)}$ in
$\mathcal{U}_{t(z)}$ and $z\in(p^{-}(z),p^{+}(z))$. Hence, for each $z\in Z$,
there is $u\in Z_{t(z)}$ such that $z\in X_{t(z),u}^{-}$. By definable Skolem
functions there is another definable function $t^{\prime}:Z\to T$ such that,
for each $z\in Z$, we have $t^{\prime}(z)\in Z_{t(z)}\subseteq T$ and $z\in
X_{t(z),t^{\prime}(z)}^{-}\subseteq Z_{t(z)}\subseteq Z$. Varying the
parameter over which $t^{\prime}:Z\to T$ is defined, we get a uniformly
$\emptyset$-definable family of such functions. By Proposition 2.4 we can
choose a parameter in ${\rm dcl}(p^{\prime})$ for some $p^{\prime}\in P^{k}$.
After naming also the elements in $p^{\prime}$ we see that we assume that
$t^{\prime}:Z\to T$ is defined over $\emptyset$.

For $z\in Z$, take $X_{z}=X_{t(z),t^{\prime}(z)}^{-}$ and for $v\in X_{z}$
take $J_{z,v}^{-}=J_{t(z),t^{\prime}(z),v}^{-}$.

It is now clear that if we let $\mathcal{Y}$ be the collection of all those
finitely many $Y$’s in $\mathcal{Y}^{I}$ from Case (D′) together with those
finitely many $Z$’s obtained in Case (N′), the result follows. ∎

##  6\. The main results

Here if we don’t say otherwise, $\mathbb{M}$ is a sufficiently saturated
o-minimal structure with definable Skolem functions.

The first consequence of the main technical lemma (Lemma 5.9) is the density
of non-trivial points in $M$:

######  Proposition 6.1.

Every open subinterval of $M$ defined over $s$ has, after naming finitely many
elements of $\mathbb{P}$, a non-trivial point over $s$. In particular, the set
of non-trivial points is dense in $M$.

Proof. Let $O$ be an open subinterval defined over $s$. Note if we add the
parameters $s$ to the language, the structure ${\mathbb{M}}$ in the new
language still has definable Skolem functions. So it is enough to proceed
assuming that $O$ is defined over $\emptyset$.

Let $\mathcal{Y}$ be as in Lemma 5.9. Then it is enough to show that for each
$Y\in\mathcal{Y}$, $O\cap Y$ has a non-trivial point over $\emptyset$. Fix
$Y\in\mathcal{Y}$. By Proposition 2.4, after naming finitely many elements of
$\mathbb{P}$, we can choose $y\in O\cap Y$ defined over $\emptyset$.

Case (N): Suppose that $Y$ is of type (N). Take $u^{\prime}<u$ both in $O\cap
Y$ such that $y\in(u^{\prime},u)$. By [22, Lemma 2.3], the restriction of
$\\{C_{v|J^{-}_{y,v}}\\}_{v\in X_{y}}$ to $[u^{\prime},u]\times[u^{\prime},u]$
is definable over $[u^{\prime},u]$, hence over $O\cap Y$. So by Theorem 3.4
applied to ${\mathbb{M}}_{|O\cap Y}$ there is a non-trivial point in $O\cap
Y$.

Case (D): Suppose that $Y$ is of type (D). Then $O\cap Y\cap X_{y}$ is an open
sub-interval of $X_{y}$ defined over $y$. So there is a non-trivial point in
$O\cap Y$.

Now we conclude using:

######  Claim 6.2.

If $O$ is an open subinterval defined over $s$ with a non-trivial point, then
after naming finitely many elements of $\mathbb{P}$ it has a non-trivial point
over $s$.

Proof. Let $a$ be a non-trivial point in $O$. Since $O$ is an open interval,
by [22, Lemma 2.5], $a$ is non-trivial over $O$. Therefore, there is an
${\mathbb{M}}_{|O}$-definable map $G:I\times I\to J$ with $I,J\subseteq O$
open subintervals, which is continuous and strictly monotone in each variable.
Let $c\in O^{m}$ be the parameter over which $G$ is defined. Then there is a
uniformly definable family $\\{G_{u}:I_{u}\times I_{u}\to J_{u}\\}_{u\in U}$,
defined over $s$, of definable maps such that for each $u\in U\subseteq
O^{m}$, $I_{u},J_{u}\subseteq O$ are open subintervals, $G_{u}:I_{u}\times
I_{u}\to J_{u}$ is continuous and strictly monotone in each variable, $c\in U$
and $G_{c}=G$. By Proposition 2.4 and after naming finitely many elements of
$\mathbb{P}$ we can choose $u\in U$ defined over $s$ and then choose $b\in
I_{u}$ defined over $s$. This $b$ is therefore a non-trivial point of $O$ over
$s$. ∎

∎

Density of non-trivial points in $M$ implies the following:

######  Lemma 6.3.

Every trivial point of $M$ is $\mathbb{P}$-definable.

Proof. Let $t$ be a trivial point of $M$ and assume that it is defined over
$c\,\mathbb{P}$ for some $c=(c_{1},\ldots,c_{l})\in M^{l}$. Then there is a
$\mathbb{P}$-definable function $f:Z\subseteq M^{l}\to M$ such that $c\in Z$
and $f(c)=t$. We can assume that there is no $c^{\prime}\subset c$ for which
there is a $\mathbb{P}$-definable $f^{\prime}:Z^{\prime}\subseteq M^{k}\to M$
such that $c^{\prime}\subseteq Z^{\prime}$ and $f^{\prime}(c^{\prime})=t$.
Note that if some $c_{i}$ is defined over $c^{\prime}\,\mathbb{P}$ for
$c^{\prime}\subset c$ consisting of the remaining elements, then we may
replace $f:Z\to M$ by another $\mathbb{P}$-definable
$f^{\prime}:Z^{\prime}\subseteq M^{l-1}\to M$ such that $c^{\prime}\subseteq
Z^{\prime}$ and $f^{\prime}(c^{\prime})=t$. So no $c_{i}$ is defined over
$c^{\prime}\,\mathbb{P}$ for $c^{\prime}\subset c$ consisting of the remaining
elements.

Since ${\rm dim}(c/\mathbb{P})=\dim Z$, by the cell decomposition theorem, we
can furthermore assume that $Z$ is an open cell in $M^{l}$.

Let $\pi:M^{l}\to M^{l-1}$ be the projection onto the first $l-1$ coordinates.
Let $W=\\{w^{\prime}\in\pi(Z):(w^{\prime},c_{l})\in Z\\}$. Since $Z$ is open,
$W$ is an open definable subset of $\pi(Z)$, defined over $c_{l}\,\mathbb{P}$.
For each $w^{\prime}\in W$, let $I_{w^{\prime}}=\\{x\in M:(w^{\prime},x)\in
Z\\}$ and let $g_{w^{\prime}}:I_{w^{\prime}}\to M$ be the definable function
given by $g_{w^{\prime}}(x)=f(w^{\prime},x)$. Then each $I_{w^{\prime}}$ is an
open subinterval of $M$ with $c_{l}\in I_{w^{\prime}}$ and
$g_{w^{\prime}}:I_{w^{\prime}}\to M$ is uniformly defined over
$w^{\prime}\,\mathbb{P}$.

Let $c^{\prime}=(c_{1},\ldots,c_{l-1})\in W$. Then by the monotonicity
theorem, $I_{c^{\prime}}$ is a finite union of points and open subintervals
all defined over $c^{\prime}\,\mathbb{P}$ such that on each such open
subinterval $g_{c^{\prime}}$ is either constant or strictly monotone and
continuous. Note that $c_{l}$ cannot be one the these points, since otherwise
it would be defined over $c^{\prime}\,\mathbb{P}$. So $c_{l}$ is in one of the
open subintervals, say $J$. Then $g_{c^{\prime}|J}$ must be strictly monotone
and continuous, otherwise it would be constant and, by Proposition 2.4, we
could choose $b\in J$ defined over $c^{\prime}\,\mathbb{P}$ and obtain another
$\mathbb{P}$-definable function $f^{\prime}:Z^{\prime}\subseteq M^{l-1}\to M$
such that $c^{\prime}\in Z^{\prime}$ and $f^{\prime}(c^{\prime})=t$,
contradicting our assumption on $c$ and $t$.

Now the properties $J\subseteq I_{c^{\prime}}$ is an open subinterval,
$c_{l}\in J$ and $g_{c^{\prime}|J}$ is strictly monotone and continuous are
first-order in $c^{\prime}$. So there is a definable subset $U\subseteq W$,
defined over $c_{l}\,\mathbb{P}$, and a uniformly definable family of open
subintervals $J_{u^{\prime}}\subseteq I_{u^{\prime}}$ such that: (i)
$c^{\prime}\in U$; (ii) $J_{c^{\prime}}=J$; (iii) for each $u^{\prime}\in U$
we have $c_{l}\in J_{u^{\prime}}$ and $g_{u^{\prime}|J_{u^{\prime}}}$ is
strictly monotone and continuous. Moreover, since $c^{\prime}\in U\subseteq W$
and ${\rm dim}(c^{\prime}/\mathbb{P})=\dim W$, by the cell decomposition
theorem we may assume that $U$ is an open cell in $M^{l-1}$.

Consider the definable function $U\to M:u^{\prime}\mapsto
g_{u^{\prime}|J_{u^{\prime}}}(c_{l})$, defined over $c_{l}\,\mathbb{P}$. Its
image, which is defined over $c_{l}\,\mathbb{P}$, must be infinite, otherwise
$t$, which is in the image, would be defined over $c_{l}\,\mathbb{P}$. Take
one of the maximal open subintervals contained in this image, say $K$. Since
$g_{c^{\prime}|J}$ is strictly monotone and $g_{c^{\prime}|J}(c_{l})=t$,
$c_{l}$ is a trivial point of $M$. So similarly, every point in $K$ must be a
trivial point, contradicting Proposition 6.1. ∎

We are ready to show the main result:

######  Theorem 6.4.

Let $\mathbb{M}$ be an o-minimal structure. Then the following are equivalent:

  1. (1)

$\mathbb{M}$ has definable Skolem functions.

  2. (2)

After naming finitely many elements of $\mathbb{P}$, there is a finite
collection $\mathcal{Y}$ of $\emptyset$-definable open subintervals of $M$
such that:

     * -

$M\setminus\bigcup\mathcal{Y}$ is a finite set of trivial points each defined
over $\emptyset$.

     * -

each $Y\in\mathcal{Y}$ is the union of a uniformly $\emptyset$-definable
family of group-intervals, each with a fixed positive element, parametrized by
the end points of the intervals.

     * -

each $Y\in\mathcal{Y}$ has a fixed $\emptyset$-definable element.

  3. (3)

$\mathbb{M}$ has definable choice.

Proof. First note that if the result holds in a sufficiently saturated
elementary extension of $\mathbb{M}$, then it holds for $\mathbb{M}$. So we
may assume that $\mathbb{M}$ sufficiently saturated.

(1) $\Rightarrow$ (2): If $x\in M$ is non-trivial, then by the Trichotomy
theorem $x$ is contained in a definable group-interval $(I,0,+,<)$. As
observed in [18, Note 3.2], by varying the parameter over which $+$ and $0$
are defined but requiring that the domain remains $I$, there is a uniformly
definable family of such $+$’s and corresponding $0$’s defined over the end
points of $I$. By Proposition 2.4, we can choose a parameter in the family
defined over the end points of $I$ and some $p\in P^{k}$. Therefore, after
naming the elements in $p$, we may assume that $+$ and $0$ are defined over
those end points. Applying again Proposition 2.4 and naming finitely many the
elements of $\mathbb{P}$ we can choose a positive element $1$ in $(I,0,+,<)$
defined over the end points of $I$.

Suppose that $I=(c_{1},c_{2})$ and let $c=(c_{1},c_{2})$. Varying $c$ we
obtain a $\emptyset$-definable family $\\{(I_{w},0_{w},1_{w},+_{w},<)\\}_{w\in
W}$ of definable group-intervals, each with a fixed positive element, such
that $x\in I_{W}$ where $I_{W}=\bigcup_{w\in W}I_{w}$.

Let $\mathcal{W}$ be the set of all such $I_{W}$’s and let $\mathcal{T}$ be
the set of all trivial points of $M$. Then
$\mathcal{T}\cup\bigcup\mathcal{W}=M$. Since every element of $\mathcal{T}$ is
defined over $\mathbb{P}$ (Lemma 6.3), by saturation, there is a finite subset
$T\subseteq\mathcal{T}$ and finitely many
$I_{W_{1}},\ldots,I_{W_{l}}\in\mathcal{W}$ such that $T\cup
I_{W_{1}}\cup\ldots\cup I_{W_{l}}=M$. Since
$\mathcal{T}\cap(I_{W_{1}}\cup\ldots\cup I_{W_{l}})=\emptyset$ and, by
Proposition 2.4, for each $i$, after naming finitely many the elements of
$\mathbb{P}$, we can choose a $\emptyset$-definable element in the
$\emptyset$-definable set $I_{W}$, the result follows.

(2) $\Rightarrow$ (3): Suppose that (2) holds. Then we have $T\sqcup
I_{W_{1}}\sqcup\ldots\sqcup I_{W_{l}}=M$ where $T$ is a finite set of trivial
points of $M$ all defined over $\mathbb{P}$ and each $I_{W}$ is of the form
$I_{W}=\bigcup_{w\in W}I_{w}$ with $\\{(I_{w},0_{w},1_{w},+_{w},<)\\}_{w\in
W}$ a $\mathbb{P}$-definable family of definable group-intervals, each with a
fixed positive element, with $w$ the pair of left and right end point of
$I_{w}$ in $M\cup\\{\pm\infty\\}$. Moreover, for each $i$, let $e_{i}\in
I_{W_{i}}$ be a fixed $\mathbb{P}$-definable element.

Let $e=(e_{1},\ldots,e_{l})$. Let $X\subseteq M$ be a non empty definable
subset of $M$ defined over $a=(a_{1},\ldots,a_{n})$. By Proposition 2.6 we
need to show that there is an element of $X$ in ${\rm
dcl}([a]e\,\mathbb{P})={\rm dcl}([a]\,\mathbb{P})$.

Take a cell decomposition of $M$ respecting $X$ and the $\mathbb{P}$-definable
finite set $X\cap T$. Then $X$ is the disjoint union of points
$d_{1}<\ldots<d_{k}$ containing $X\cap T$ and open intervals
$J_{1}<\ldots<J_{m}$ with the $d_{i}$’s and the end points of the $J_{j}$’s in
${\rm dcl}(a\,\mathbb{P})$. By varying $a$ we have a uniformly
$\mathbb{P}$-definable family $\\{X_{s}\\}_{s\in S}$ of subsets of $M$ such
that $a\in S$ and $X_{a}=X$, together with a finite collection $\mathcal{G}$
of $\mathbb{P}$-definable functions from $S$ to $M$ such that: (i) for each
$s\in S$, $X_{s}$ is the disjoint union of $k$ many points and $m$ many open
intervals; (ii) for each $g\in\mathcal{G}$ and $s\in S$, $g(s)$ is one of the
$k$ many points of $X_{s}$ exactly when $g(a)$ is one of the $k$ many points
of $X$ or $g(s)$ is one of end points of the $m$ many of open subintervals of
$X_{s}$ exactly when $g(a)$ is an end point of one of the $J_{j}$’s; (iii) for
every $g,g^{\prime}\in\mathcal{G}$ and $s\in S$ we have $g(s)<g^{\prime}(s)$
if and only if $g(a)<g^{\prime}(a)$.

Note that if $s\in[a]$, then $X_{s}=X$ and so for each $g\in\mathcal{G}$ we
have $g(s)=g(a)$. Therefore, the $g$’s show that each of the $d_{i}$’s and
each of the end points of the open subintervals $J_{j}$’s are in ${\rm
dcl}([a]\,\mathbb{P})$.

If $k\geq 1$ then we are done. So we may assume that $X$ is disjoint union of
open intervals $J_{1}<\ldots<J_{m}$ and $X\cap T=\emptyset$. Let $J=J_{1}$ and
take $W\in\\{W_{1},\ldots,W_{l}\\}$ the unique element such that $J\subseteq
I_{W}$. Then $J$ has end points in $\mathrm{cl}(I_{W})$ and either $J$ has an
end point in $I_{W}$ or $J=I_{W}$. If $J=I_{W}$, then one of the elements of
$\\{e_{1},\ldots,e_{l}\\}$ is in $X\cap{\rm dcl}([a]e\,\mathbb{P})$.

Suppose $J$ has an end point $c$ in $I_{W}$ (which is in ${\rm
dcl}([a]\,\mathbb{P})$). Let $g\in\mathcal{G}$ be the unique element such that
$g(a)=c$. Let $U=\\{u\in W:c\in I_{u}\\}$,

$v_{1}=\inf\\{x:x<c\,\,\textrm{and}\,\,(x,y)\in U\,\,\textrm{for
some}\,\,c<y\\},$ $v_{2}=\sup\\{y:c<y,\textrm{and}\,\,(v_{1},y)\in U\\}$

and $v=(v_{1},v_{2})$. Then $U$ is definable over $c\,\mathbb{P}$, $v\in U$
and is also defined over $c\,\mathbb{P}$. So there are $\mathbb{P}$-definable
functions $h_{1},h_{2}:C\to M$ such that $c\in C$ and $h_{i}(c)=v_{i}$ for
$i=1,2$. Then the $\mathbb{P}$-definable functions $h_{i|g(S)\cap C}\circ
g:S\to M$ ($i=1,2$) show that $v\in{\rm dcl}([a]\,\mathbb{P})$.

Since $c\in I_{v}$, we have that $J\cap I_{v}$ is an open subinterval of
$I_{v}$. Let $c^{\prime}$ be the other end point of $J\cap I_{v}$ in
$\mathrm{cl}(I_{v})$. Then either $c^{\prime}\in I_{v}$ and in this case
$J\subseteq I_{v}$ or $c^{\prime}$ is one of the end points of $I_{v}$. In
both cases we have $c^{\prime}\in{\rm dcl}([a]\,\mathbb{P})$.

Suppose that $c^{\prime}\in I_{v}$. In $\mathbb{M}_{|I_{v}}$, the set $J\cap
I_{v}$ is definable over $c\,c^{\prime}$. Since $\mathbb{M}_{|I_{v}}$ is an
o-minimal expansion of a group-interval with a fixed positive element, it has
definable choice (as in [18, Fact 4.5] or [26, Chapter 6 (1.1)]). Therefore,
by Proposition 2.6 in the o-minimal structure $\mathbb{M}_{|I_{v}}$, there is
an element of $J$ in ${\rm
dcl}_{\mathbb{M}_{|I_{v}}}([c\,c^{\prime}]\,\mathbb{P}_{|I_{v}})$. But
$[c\,c^{\prime}]=\\{(c,c^{\prime})\\}$ and $c,c^{\prime},v\in{\rm
dcl}([a]\,\mathbb{P})$. It follows that the element of $J$ in ${\rm
dcl}_{\mathbb{M}_{|I_{v}}}([c\,c^{\prime}]\,\mathbb{P}_{|I_{v}})$ is in ${\rm
dcl}([a]\,\mathbb{P})$.

In the case $c^{\prime}$ is one of the end points of $I_{v}$, by a similar
argument, we get an element of $J$ in ${\rm
dcl}_{\mathbb{M}_{|I_{v}}}([c]\,\mathbb{P}_{|I_{v}})$ which will also be in
${\rm dcl}([a]\,\mathbb{P})$. So in both case there is an element of $X$ in
${\rm dcl}([a]\,\mathbb{P})$.

∎

##  7\. Concluding remarks

Note that some authors might define definable Skolem functions/definable
choice in a slightly different way, explained below. The way we defined these
notions in Section 2 corresponds to the way they were used in the papers
developing definable analogues of algebraic topology tools mentioned in the
Introduction.

One may (and often people choose to do that) define definable Skolem
functions/definable choice in an “unparametric” way: $\mathbb{M}$ has
unparametric definable Skolem functions if for every $A$-definable family
$\\{X_{t}\\}_{t\in T}$ of nonempty definable subsets of some $M^{k}$, there is
a $A$-definable function $f:T\to M^{k}$ such that $f(t)\in X_{t}$ for all
$t\in T$; $\mathbb{M}$ has unparametric definable choice, if furthermore, we
can choose $f:T\to M^{k}$ such that $f(t)=f(t^{\prime})$ whenever
$X_{t}=X_{t^{\prime}}$.

With these definitions it is straightforward to obtain the following analogues
of Propositions 2.4 and 2.6.

######  Fact 7.1.

If $\mathbb{M}$ is sufficiently saturated, then:

  * $\bullet$

${\mathbb{M}}$ has unparametric definable Skolem functions if and only if
every definable subset $\emptyset\neq X\subseteq M$ defined with parameters
$a\in M^{l}$ has an element in $\mathrm{dcl}(a)$.

  * $\bullet$

${\mathbb{M}}$ has unparametric definable choice if and only if every
definable subset $\emptyset\neq X\subseteq M$ defined with parameters $a\in
M^{l}$ has an element in $\mathrm{dcl}([a])$.

Replacing the use of Propositions 2.4 and 2.6 throughout the paper by the two
unparametric versions above, we obtain the following unparametric version of
our main theorem:

######  Theorem 7.2.

Let $\mathbb{M}$ be an o-minimal structure. Then the following are equivalent:

  1. (1)

$\mathbb{M}$ has unparametric definable Skolem functions.

  2. (2)

There is a finite collection $\mathcal{Y}$ of $\emptyset$-definable open
subintervals of $M$ such that:

     * -

$M\setminus\bigcup\mathcal{Y}$ is a finite set of trivial points each defined
over $\emptyset$.

     * -

each $Y\in\mathcal{Y}$ is the union of a uniformly $\emptyset$-definable
family of group-intervals, each with a fixed positive element, parametrized by
the end points of the intervals.

     * -

each $Y\in\mathcal{Y}$ has a fixed $\emptyset$-definable element.

  3. (3)

$\mathbb{M}$ has unparametric definable choice.

Finally note also that our main results characterize (unparametric) definable
Skolem functions/definable choice in a rather explicit way. An obvious
question is whether there is such a characterization for (parametric)
elimination of imaginaries. We wonder if the technics used here and the
examples from [23, Proposition 3.2] and counter-example from [20] might help
attack this question.

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