# Definable separability and second-countability in o-minimal structures

Pablo Andújar Guerrero  University of Leeds<EMAIL_ADDRESS>
###### Abstract.

We show that separability and second-countability are first-order properties
among topological spaces definable in o-minimal expansions of
$(\mathbb{R},<)$. We do so by introducing first-order characterizations
–definable separability and definable second-countability– which make sense in
a wider model-theoretic context. We prove that within o-minimality these
notions have the desired properties, including their equivalence among
definable metric spaces, and conjecture a definable version of Urysohn’s
Metrization Theorem.

_Mathematics Subject Classification 2020._ 03C64 (Primary); 54A05, 54D65,
54D70 (Secondary).
_Key words._ tame topology, o-minimality, separability, second-countability,
definable topological spaces.

##  1\. Introduction

In this paper we deepen the area of tame topology by introducing first-order
notions of separability and second-countability.

O-minimality was birthed as a tame topological setting in the context of the
o-minimal Euclidean topology [PS86]. Since then o-minimal topology has been
developed extensively beyond this seminal context. The first example might be
Pillay’s proof that definable groups have a definable manifold group topology
[Pil88], with the subsequent investigation by van den Dries of definable
manifold spaces [vdD98], which in recent years have served as a framework for
the development of the theory of definable analytic spaces [BBT23]. On the
other hand the study of topological families of definable functions,
inevitably linked to the development of o-minimal analysis [PS01, AF11], has
long been another central topic in o-minimality. The influential research into
parametrizations in [PW06], which led to the celebrated Pila-Wilkie theorem,
motivated the abstraction of the notion of definable normed space [Tho12].
This was later generalized to the notion of definable metric space [Wal15a],
and subsequently by the author to the study of o-minimal (explicitly)
definable topological spaces [AG21], providing thus a usable framework for the
development of o-minimal point-set topology and functional analysis.

In this paper we introduce and investigate definable analogues of separability
and second-countability in the context of definable topological spaces in
o-minimal structures. We observe that these provide non-trivial dividing lines
in our context. Specifically, every definable set with the o-minimal Euclidean
topology is definably second-countable (in particular definably separable),
the definable Sorgenfrey line (Example 3.5) is definably separable but not
definably second-countable, and any infinite definable set with the discrete
topology lacks either property. Our main results (Theorems 4.7 and 4.11) state
that, in o-minimal expansions of $(\mathbb{R},<)$, definable separability and
definable second-countability are equivalent to their classical counterparts,
equating them to properties such as definable connectedness and definable
compactness. We conjecture that, under minimal assumptions, definable second-
countability characterizes o-minimal definable topological spaces which are,
up to definable homeomorphism, Euclidean (Conjecture 6.1).

Our definition of definable separability applies to any structure (regardless
of o-minimality) and our definition of definable second-countability invokes
o-minimal dimension. Our analysis depends largely on basic o-minimality,
including the fact that o-minimal dimension coincides with (naive) topological
dimension, and on the Fiber Lemma described in Fact 2.2. Recently there have
been various axiomatic explorations of tame topology [SW19, DG22] in settings
which generalize o-minimality, as well as non-strongly-minimal dp-minimal
expansions of fields. This literature has yielded in particular the existence
of a well-behaved notion of dimension in these settings. Hence it seems
plausible that the definitions proposed here extend fruitfully to more general
frameworks.

The structure of the paper is as follows. In Section 2 we present our notation
and o-minimal machinery. Section 3 includes our definitions of definable
separability and second-countability, as well as some preliminary results
showing that the properties behave as expected. In Section 4 we prove our main
results, namely the equivalence with the classical properties in o-minimal
expansions of $(\mathbb{R},<)$. We also achieve a deeper characterization of
the introduced properties, and derive that they are definable in families. In
Section 5 we investigate definable separability and second-countability in the
context of o-minimal definable metric spaces, proving their equivalence in
this context. We also contrast our definitions with a different definition of
definable separability for definable metric spaces presented by Walsberg in
[Wal15a]. Finally in Section 6 we conjecture an o-minimal definable version of
Urysohn’s metrization theorem.

In [AGTW21] Thomas, Walsberg and the author proved various results on
o-minimal definable topological spaces which can be seen to capture “definable
first-countability”. In particular it is shown that every definable
topological space in an o-minimal expansion of $(\mathbb{R},<)$ is first-
countable. For this reason the present paper does not dwell into this
property.

### Acknowledgments

The author thanks Dr. Margaret Thomas for her supervision of research that
lead to much of the contents of this paper, which was undertaken during the
author’s doctoral degree, and Erik Walsberg, who informed the author of the
proof of Proposition 5.4. The author also thanks Dr. Pantelis Eleftheriou for
comments on first drafts and him, Mervyn Tong, and Ben De Smet, for helpful
discussions on the topic, which finally lead to the correct definition of
definable second-countability.

### Funding

During the writing of this paper and the undertaking of the research into its
contents the author was supported by the UK Engineering and Physical Sciences
Research Council (EPSRC) Grant EP/V003291/1, and by the Graduate School and
the Department of Mathematics at Purdue University.

##  2\. Preliminaries

###  2.1. Conventions and terminology

Throughout we fix a first-order structure $\mathcal{M}=(M,\ldots)$.
“Definable” always means “definable in $\mathcal{M}$, possibly with
parameters”. Every formula we consider is in the language of $\mathcal{M}$. We
use $n$, $m$ and $k$ to denote positive natural numbers. We use $x$, $y$, $a$,
$b$, $\ldots$ to denote tuples of variables or parameters in $M$. Given a
tuple $x$ we denote its length by $|x|$.

Recall that a family of definable sets $\mathcal{A}=\\{A_{b}:b\in B\\}$ is
(uniformly) definable if there exists a (partitioned) formula $\varphi(x,y)$
such that $B\subseteq M^{|y|}$ is a definable set and $A_{b}=\\{a\in
M^{|x|}:\mathcal{M}\models\varphi(a,b)\\}$ for every $b\in B$.

For any given $n\geq 2$ we denote by $\pi:M^{n}\rightarrow M^{n-1}$ the
projection to the first $n-1$ coordinates, where $n$ will often be clear from
context. Given a set $X\subseteq M^{n}$ and a tuple $a\in M^{m}$, where $m<n$,
let $X_{a}=\\{b\in M^{n-m}:(a,b)\in X\\}$.

We work mostly in the case where $\mathcal{M}$ is an o-minimal expansion of a
dense linear order without endpoints $(M,<)$. In this case, given a definable
set $X$ we denote its o-minimal dimension by $\dim X$. By the _o-minimal
Euclidean topology_ we mean the order topology on $M$ and induced product
topology on $M^{n}$. For background o-minimality we direct the reader to
[vdD98]. In particular we will use o-minimal cell decomposition and uniform
finiteness.

Consider a finite collection of definable families of sets
$\mathcal{B}_{i}=\\{U^{i}_{b}:b\in B_{i}\\}$ for $i\leq k$. We define their
(definable) disjoint union as follows. For each $a=(a_{1},\ldots,a_{k})\in
M^{k}$, let $i(a)$ denote the largest $i\leq k$ with $a_{i}\geq a_{1}$. For
each $\mathbf{b}=(b_{1},\ldots,b_{k})\in\prod_{i=1}^{k}B_{i}$, let
$V_{(a,\mathbf{b})}=U^{i(a)}_{b_{i(a)}}$. The disjoint union of the families
$\mathcal{B}_{i}$ is the definable family
$\\{V_{(a,\mathbf{b})}:(a,\mathbf{b})\in M^{k}\times\prod_{i=1}^{k}B_{i}\\}$.
Note that this construction does not invoke new parameters.

###  2.2. O-minimal preliminaries

In this subsection we present our main tools from o-minimality. They are
mostly connected to the Fiber Lemma stated in Fact 2.2.

In general we will not assume that our structure $\mathcal{M}$ has elimination
of imaginaries. The purpose of the following technical lemma is to circumvent
this fact.

######  Lemma 2.1.

Let $\mathcal{M}$ be an o-minimal structure. Let $\mathcal{A}=\\{A_{b}:b\in
B\\}$ be an infinite definable family of sets. There exists another infinite
definable family $\mathcal{A}^{\prime}=\\{A^{\prime}_{t}:t\in I\\}$, where
$I\subseteq M$ is an interval, with the following properties.

  1. (1)

$\mathcal{A}^{\prime}\subseteq\mathcal{A}$.

  2. (2)

$A^{\prime}_{t}\neq A^{\prime}_{s}$ for every distinct pair $t,s\in I$.

###### Proof.

Let $\mathcal{A}$ be as in the lemma. We begin by proving the case where
$B\subseteq M$.

Let $E\subseteq B\times B$ be the definable equivalence relation on $B$ given
by $(b,c)\in E$ whenever $A_{b}=A_{c}$. For each $b\in B$, let $E(b,M)$ denote
the equivalence class of $b$ by $E$. Let $B^{(0)}=\\{b\in B:\inf E(b,M)\in
E(b,M)\\}$ and $B^{(1)}=B\setminus B^{(0)}=\\{b\in B:\inf E(b,M)\notin
E(b,M)\\}$. For each $i\in\\{0,1\\}$ let $\mathcal{A}^{(i)}=\\{A_{b}:b\in
B^{(i)}\\}$. Clearly $\mathcal{A}=\mathcal{A}^{(0)}\cup\mathcal{A}^{(1)}$.
Since $\mathcal{A}$ is infinite there must be some fixed $i\in\\{0,1\\}$ such
that $\mathcal{A}^{(i)}$ is infinite.

By o-minimality note that, for every $b,c\in B^{(i)}$, if $\inf E(b,M)=\inf
E(c,M)$ then $A_{b}=A_{c}$. In particular the definable set $C=\\{\inf
E(b,M):b\in B^{(i)}\\}\setminus\\{-\infty\\}$ is infinite. For every $c\in C$,
let $A^{\prime}_{c}$ be the set $A_{b}$ where $b\in B^{(i)}$ is any element
satisfying that $c=\inf E(b,M)$. Observe that the family
$\\{A^{\prime}_{b}:b\in C\\}$ is definable, infinite, and it satisfies that
$A^{\prime}_{b}\neq A^{\prime}_{c}$ for every distinct pair $b,c\in C$.
Finally, again by o-minimality there exists an interval $I\subseteq C$ such
that $\\{A^{\prime}_{c}:c\in I\\}$ is infinite. This completes the proof of
the case $B\subseteq M$.

Now let $B\subseteq M^{n}$. We complete the proof of the lemma by showing the
existence of an infinite definable subfamily of $\mathcal{A}$ indexed
definably by a subset of $M$. We proceed by induction on $n$. The base case
$n=1$ is trivial. We assume that $n>1$.

Consider the projection $\pi(B)$ of $B$ to the first $n-1$ coordinates.
Suppose that there exists some $d\in\pi(B)$ such that the definable family
$\mathcal{A}_{d}=\\{A_{(d,t)}:t\in B_{d}\\}$ is infinite. Then it suffices to
pass to this family. Hence onwards suppose that, for every $d\in\pi(B)$, the
family $\mathcal{A}_{d}$ is finite. For each $d\in\pi(B)$ and $t\in B_{d}$ let
$E_{d}(t,M)=\\{s\in B_{d}:A_{(d,t)}=A_{(d,s)}\\}$, and set $C_{d}=\\{\inf
E_{d}(t,M):t\in B_{d}\\}\setminus\\{-\infty\\}$. Observe that the sets $C_{d}$
are definable uniformly in $d$. Moreover by o-minimality for every
$d\in\pi(B)$ we get that $|C_{d}|\leq|\mathcal{A}_{d}|<\infty$ and
$|\mathcal{A}_{d}|\leq 1+2|C_{d}|$. By uniform finiteness we derive that there
exists a fixed $m$ such that $|\mathcal{A}_{d}|\leq m$ for every $d\in\pi(B)$.

By o-minimality for every $d\in\pi(B)$ the sets in $\mathcal{A}_{d}$ are
linearly ordered by the relation $<_{d}$ given by $A_{(d,s)}<_{d}A_{(d,t)}$
whenever there is $t^{\prime}\in B_{d}$ with $A_{(d,t^{\prime})}=A_{(d,t)}$
and $s^{\prime}<t^{\prime}$ for every $s^{\prime}\in B_{d}$ with
$A_{(d,s^{\prime})}=A_{(d,s)}$. For each $1\leq i\leq m$ let
$A^{\prime}_{d}(i)\in\mathcal{A}_{d}$ denote either the $i$-th set in
$\mathcal{A}_{d}$ with respect to the order $<_{d}$ or, in the case where
$|\mathcal{A}_{d}|<i$, the maximum one. Observe that, for each $1\leq i\leq
m$, the family of sets $\mathcal{A}(i)=\\{A^{\prime}_{d}(i):d\in\pi(B)\\}$ is
definable. By choice of $m$ we also have that $\cup_{i\leq
m}\mathcal{A}(i)=\mathcal{A}$. Since $\mathcal{A}$ is infinite it follows that
that $\mathcal{A}(i)$ is infinite for some fixed $i$. We apply the induction
hypothesis to $\mathcal{A}(i)$ to complete the proof. ∎

In Lemma 2.1 above observe that, whenever the sets in $\mathcal{A}$ are not
unary, it might be that $\mathcal{A}^{\prime}$ is not definable over the same
parameters as $\mathcal{A}$.

We now present the Fiber Lemma for o-minimal dimension.

######  Fact 2.2 ([vdD98] Proposition 1.5 and Corollary 1.6).

Let $\mathcal{M}$ be an o-minimal structure. Let $X\subseteq M^{n}\times
M^{m}$ be a definable set and, for any $d\in\\{-\infty,0,1,\ldots,m\\}$, let
$X(d)=\\{a\in M^{n}:\dim X_{a}=d\\}$. Then the sets $X(d)$ are definable and

$\dim\left(\bigcup_{a\in X(d)}\\{a\\}\times X_{a}\right)=\dim(X(d))+d.$

In particular $\dim X=\max_{0\leq d\leq m}\dim(X(d))+d$.

We derive some easy consequences of the above fact in the next two lemmas.

######  Lemma 2.3.

Let $\mathcal{M}$ be an o-minimal structure. Let $\\{A_{b}:b\in B\\}$ be an
infinite definable family of pairwise disjoint sets with $\dim A_{b}\geq n$
for every $b\in B$. Then $\dim(\cup_{b\in B}A_{b})>n.$

###### Proof.

By Lemma 2.1 after passing to a subfamily if necessary we may assume that $B$
is an interval and that $A_{b}\cap A_{c}=\emptyset$ for every distinct pair
$b,c\in B$. In particular the set $X=\cup_{b\in B}\\{b\\}\times A_{b}$ is in
definable bijection with $A=\cup_{b\in B}A_{b}$ by means of the projection
$(b,a)\mapsto a$. So $\dim A=\dim X$. By Fact 2.2 note that $\dim X\geq 1+n$,
and the result follows. ∎

######  Lemma 2.4.

Let $\mathcal{M}$ be an o-minimal structure. Let $X\subseteq M^{n}\times
M^{m}$ be a non-empty definable set. For every $a\in M^{n}$ and $c\in N^{m}$
let $X(a,M)=X_{a}=\\{c^{\prime}\in M^{m}:(a,c^{\prime})\in X\\}$ and
$X(M,c)=\\{a^{\prime}\in M^{n}:(a^{\prime},c)\in X\\}$. Let $A$ and $C$ denote
the projections of $X$ to the first $n$ and last $m$ coordinates respectively.

Suppose that $\dim X(M,c)=\dim A$ for every $c\in C$. Then

$\dim C=\max_{a\in A}\dim X(a,M).$

###### Proof.

Let $f:M^{n}\times M^{m}\rightarrow M^{m}\times M^{n}$ be the permutation map
given by $f(a,c)=(c,a)$. By Fact 2.2, applied to the set
$X^{\text{opp}}=f(X)$, and the fact that $\dim X(M,c)=\dim A$ for every $c\in
C$, we have that $\dim X=\dim A+\dim C$. On the other hand, by Fact 2.2
applied to the set $X$, we have that

$\dim X\leq\dim A+\max_{a\in A}\dim X(a,M)\leq\dim A+\dim C,$

and the lemma follows. ∎

##  3\. Topological definitions and basic results

In this section we introduce our definitions of definable separability and
definable second-countability. The former was inspired by a similar notion
introduced by Walsberg in [Wal15a] in the o-minimal definable metric setting.
(We explain the connection between both definitions in Section 5.) We also
prove some preliminary results. Recall that a topological space is separable
if it has a countable dense subset, and second-countable if the topology has a
countable basis.

Our setting will be that of definable (explicitly in the sense of Ziegler)
topological spaces.

######  Definition 3.1.

A _definable topological space_ $(X,\tau)$, $X\subseteq M^{n}$, is a
topological space such that there exists a basis for $\tau$ that is definable.

Let $\mathcal{M}$ be o-minimal. Any definable subset of $M^{n}$ is a definable
topological space with o-minimal Euclidean topology. Further examples within
o-minimality include the definable manifold spaces studied by Pillay [Pil88]
and van den Dries [vdD98, Chapter 10], the definable Euclidean quotient spaces
of van den Dries [vdD98, Chapter 10] and Johnson [Joh18], the definable normed
spaces of Thomas [Tho12], and the definable metric spaces of Walsberg
[Wal15b]. Peterzil and Rosel [PR20] studied one-dimensional definable
topologies. See the author’s doctoral dissertation [AG21] for an exhaustive
exploration of o-minimal definable topological spaces.

Given a topological space $(X,\tau)$ and a subset $Y\subseteq X$ we denote its
closure in the topology $\tau$ by $cl(X)$. We say that a definable topological
space $(X,\tau)$ has _small boundaries_ if every definable set $Y\subseteq X$
satisfies that $\dim(cl(Y)\setminus Y)<\dim Y$. Any o-minimal Euclidean space
has small boundaries [vdD98, Chapter 4, Theorem 1.8].

We now present our definition of definably separability. Observe that the
notion makes sense in any structure, regardless of o-minimality.

######  Definition 3.2 (Definable separability).

A definable topological space $(X,\tau)$ is _definably separable_ if there
exists no infinite definable family of open pairwise disjoint sets in $\tau$.

The reader will note the similarity between Definition 3.2 and the countable
chain condition (ccc, or Suslin’s condition) for topological spaces, i.e. the
condition that a space does not contain an uncountable family of pairwise
disjoint open sets. Every separable topological space has the ccc, but the
converse is not true. Our main result Theorem 4.7 shows that both being
separable and having the ccc are equivalent to being definably separable among
topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$, and
so “definable separability” could also be labelled the “definable countable
chain condition”.

Clearly any finite definable topological space is definably separable. On the
other hand any infinite definable set $X$ with the discrete topology is not
definably separable, since $\\{\\{x\\}:x\in X\\}$ would be an infinite
definable family of pairwise disjoint open sets. The fact that any definable
set in an o-minimal structure with the o-minimal Euclidean topology is
definably separable (Remark 3.12 below) will follow from showing that these
spaces are definably second-countable and that this property implies definable
separability. We leave it to the interested reader to check that this can also
be derived using o-minimal cell decomposition and Lemma 2.3.

The following easy lemma will be used in Section 4.

######  Lemma 3.3.

Let $(X,\tau)$ be a definably separable definable topological space. Let
$Y\subseteq X$ be a definable open subset. Then the subspace $(Y,\tau|_{Y})$
is definably separable.

###### Proof.

Suppose that $(Y,\tau|_{Y})$ is not definably separable, witnessed by an
infinite definable family of pairwise disjoint open sets $\mathcal{A}$. Since
$Y$ is open we have that $\tau|_{Y}\subseteq\tau$ and so $\mathcal{A}$ also
witnesses that $(X,\tau)$ is not definably separable. ∎

In Section 5 we discuss the fact that Lemma 3.3 does not hold in general if we
drop the assumption that $Y$ is open. Namely the condition of being definably
separable is not hereditary.

We now move on to our notion of definable second-countability.

######  Definition 3.4 (Definable second-countability).

Let $\mathcal{M}$ be an o-minimal structure and let $(X,\tau)$ be a definable
topological space. We say that $(X,\tau)$ is _definably second-countable_ if
there exists a definable basis $\\{U_{b}:b\in B\\}$ for the topology $\tau$
satisfying that, for $x\in X$ and $b\in B$, if $x\in U_{b}$ then

$\dim\\{c\in B:x\in U_{c}\subseteq U_{b}\\}=\dim B.$

We write that $\\{U_{b}:b\in B\\}$ _witnesses_ that $(X,\tau)$ is definably
second-countable.

It is easy to see that any infinite definable discrete set not definably
second-countable. We present a less obvious non-example.

######  Example 3.5.

Let $\mathcal{M}$ be o-minimal and consider the _definable Sorgenfrey line_
$(M,\tau_{\text{sorg}})$, given by the set $M$ with the topology with basis of
right half-open intervals $[x,y)$, for $x,y\in M$. When $(M,<)=(\mathbb{R},<)$
this is the classical Sorgenfrey line.

Since every non-empty definable open set in $(M,\tau_{\text{sorg}})$ is
infinite it is easy to see using Lemma 2.3 that $(M,\tau_{\text{sorg}})$ is
definably separable. On the other hand $(M,\tau_{\text{sorg}})$ is not
definably second-countable. To see this suppose towards a contradiction that
there exists a definable basis $\mathcal{B}=\\{U_{b}:b\in B\\}$ witnessing
that $(M,\tau_{\text{sorg}})$ is definably second-countable. Then, for every
$x\in M$, the definable set $B(x)=\\{b\in B:x\in U_{b}\subseteq[0,\infty)\\}$
satisfies that $\dim B(x)=\dim B$. Moreover the definable family $\\{B(x):x\in
B\\}$ satisfies that $B(x)\cap B(y)=\emptyset$ for every distinct $x,y\in M$.
A contradiction follows from Lemma 2.3.

Following the same arguments as above, the Sorgenfrey line topology restricted
to any interval is still definably separable and not definably second-
countable.

The space $(M,\tau_{\text{sorg}})$, as well as other spaces that fail to be
definably second-countable, arise naturally when considering the pointwise
convergence topology among definable families of functions. For example
$(M,\tau_{\text{sorg}})$ is the topology induced on $M$ by the collection of
characteristic functions $\\{\mathbbm{1}_{(x,\infty)}:x\in M\\}$ with the
pointwise convergence topology, if we identity each characteristic function
$\mathbbm{1}_{(x,\infty)}$ with its index parameter $x$.

In order to work with Definition 3.4 we require a generalization of the
introduced notion that applies to definable subsets of a definable topological
space and which is maintained after passing to finite unions. Sadly this is
not the case with the property of being definably second-countable in the
subspace topology. This generalization will involve the following definition.

######  Definition 3.6.

Let $(X,\tau)$ be a topological space and $Y\subseteq X$. A _$\tau$ -basis for
$Y$_ is a family of open sets $\mathcal{B}$ satisfying that, for every $y\in
Y$ and open set $A$ with $y\in A$, there exists some $U\in\mathcal{B}$ such
that $x\in U\subseteq A$. In other words, $\mathcal{B}$ is a family of open
sets which contains, for every $y\in Y$, a basis of open neighborhoods of $y$.

It is easy to see that if a topological space $(X,\tau)$ admits a countable
cover of subsets which admit each a countable $\tau$-basis then $(X,\tau)$ is
second-countable. This is in contrast with the fact that there exist non-
second-countable topological spaces which can be partitioned into finitely
many second-countable subspaces (see Example A.16 in [AG21]). This key
difference makes the notion of $\tau$-basis useful in proving that a
topological space is second-countable, by proving the existence of one such
partition.

In the definable context the landscape is analogous. In particular the
aforementioned Example A.16 in [AG21] is definable in $(\mathbb{R},+,\cdot,<)$
and (by Theorem 4.11) it fails to be definably second-countable, while
admiting a partition into two definably second-countable subspaces. We
introduce a definable analogue of the property of having a countable
$\tau$-basis. We will rely on this property to prove that definable
topological spaces are definably second-countable, by induction arguments
relying on o-minimal dimension.

######  Definition 3.7.

Let $\mathcal{M}$ be an o-minimal structure. Let $(X,\tau)$ be a definable
topological space and $Y\subseteq X$ be a subset. We say that $Y$ is
_$\text{DSC}_{\tau}$_ 111A more descriptive name for being $\text{DSC}_{\tau}$
would be being _definably $\tau$-second-countable_, however the chosen more
succinct terminology seems to make the text more readable. if it is definable
and there exists a definable $\tau$-basis $\\{U_{b}:b\in B\\}$ for $Y$
satisfying that, for every $y\in Y$ and $b\in B$ with $y\in U_{b}$,

$\dim\\{c\in B:y\in U_{c}\subseteq U_{b}\\}=\dim B.$

We write that $\\{U_{b}:b\in B\\}$ _witnesses_ that $Y$ is
$\text{DSC}_{\tau}$.

Clearly a definable topological space $(X,\tau)$ in an o-minimal structure is
definably second-countable if and only if $X$ is $\text{DSC}_{\tau}$. The use
of Definition 3.7 in this paper relies on the characterization provided by
Lemma 3.9 below. We first present an easy remark.

######  Remark 3.8.

Let $\mathcal{M}$ be an o-minimal structure, $(X,\tau)$ be a definable
topological space, $Y\subseteq X$ be definable subset and
$\mathcal{B}=\\{U_{b}:b\in B\\}$ be a definable $\tau$-basis for $Y$. By
definition of $\tau$-basis observe that $\mathcal{B}$ witnesses that $Y$ is
$\text{DSC}_{\tau}$ if an only if, for every $y\in Y$ and every definable
neighborhood (in $\tau$) $A$ of $y$, it holds that $\dim\\{b\in B:y\in
U_{b}\subseteq A\\}=\dim B$.

######  Lemma 3.9.

Let $\mathcal{M}$ be an o-minimal structure. Let $(X,\tau)$ be a definable
topological space. Any finite union of $\text{DSC}_{\tau}$ subsets of $X$ is
$\text{DSC}_{\tau}$. In particular $(X,\tau)$ is definably second-countable if
and only if it admits a finite covering by $\text{DSC}_{\tau}$ sets.

###### Proof.

By following an inductive argument in the general case it suffices to prove
that the union of two $\text{DSC}_{\tau}$ sets is $\text{DSC}_{\tau}$. Hence
let $X_{1}$ and $X_{2}$ denote two $\text{DSC}_{\tau}$ subsets of $X$. For
each $i\in\\{1,2\\}$, let $\mathcal{B}_{i}=\\{U^{i}_{b}:b\in B_{i}\\}$ be a
definable $\tau$-basis for $X_{i}$ witnessing that $X_{i}$ is
$\text{DSC}_{\tau}$.

Let $m=\max\\{\dim B_{1},\dim B_{2}\\}$. If there exists some fixed
$i\in\\{1,2\\}$ with $\dim B_{i}<m$ let $B^{\prime}_{i}=B_{i}\times M^{m-\dim
B_{i}}$ and let $V^{i}_{(b,c)}=U^{i}_{b}$ for each $(b,c)\in B^{\prime}_{i}$
with $b\in B_{i}$. Applying Fact 2.2, it is easy to see that $\dim
B^{\prime}_{i}=m$, and furthermore that $\\{V^{i}_{b^{\prime}}:b^{\prime}\in
B^{\prime}_{i}\\}$ still witnesses that $X_{i}$ is $\text{DSC}_{\tau}$. Hence,
by passing from $\mathcal{B}_{i}$ to $\\{V^{i}_{b^{\prime}}:b^{\prime}\in
B^{\prime}_{i}\\}$ if necessary, we may assume that $\dim B_{1}=\dim B_{2}$.

Now observe that the disjoint union (described in Section 2.1) of
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ is a $\tau$-basis for $X_{1}\cup
X_{2}$, and furthermore by Remark 3.8 (and using for example Fact 2.2) it
witnesses that $X_{1}\cup X_{2}$ is $\text{DSC}_{\tau}$. ∎

We now present some simple facts about definable second-countability.

######  Proposition 3.10.

Let $\mathcal{M}$ be an o-minimal structure. The following hold.

  1. (1)

A definable subspace of a definably second-countable definable topological
space is also definably second-countable.

  2. (2)

Any finite subset of a definable topological space $(X,\tau)$ is
$\text{DSC}_{\tau}$. In particular any finite definable topological space is
definably second-countable.

  3. (3)

Any definable set with the o-minimal Euclidean topology is definably second-
countable.

  4. (4)

If there exists a finite covering $\\{X_{1},\ldots,X_{n}\\}$ of $X$ by
definable open subsets each of which is definably second-countable in the
subspace topology then $(X,\tau)$ is definably second-countable.

###### Proof.

Let $(X,\tau)$ be a definable topological space.

We begin by proving 1. Hence suppose that $(X,\tau)$ is definably second-
countable, witnessed by a definable basis $\mathcal{B}=\\{U_{b}:b\in B\\}$.
Let $Y$ be a definable subset of $X$. Observe that the family $\\{Y\cap
U_{b}:b\in B\\}$ is a basis for the subspace topology $\tau|_{Y}$ that
witnesses that $(Y,\tau|_{Y})$ is definably second-countable.

We now prove 2. By Lemma 3.9 it suffices to prove that every singleton subset
of $(X,\tau)$ is $\text{DSC}_{\tau}$. Hence fix $x\in X$ and let
$\\{U_{b}:b\in B_{x}\\}$ be a definable basis of open neighborhoods of $x$ (in
particular a definable $\tau$-basis for $\\{x\\}$). For each $b\in B_{x}$ let
$B_{x,b}=\\{c\in B_{x}:x\in U_{c}\subseteq U_{b}\\}.$ Pick some $b_{x}\in
B_{x}$ satisfying that the set $B_{x,b_{x}}$ is of minimum dimension among all
sets $B_{x,b}$ with $b\in B_{x}$. Clearly the definable family $\\{U_{b}:b\in
B_{x,b_{x}}\\}$ is a basis of neighborhoods of $x$. Moreover by choice of
$b_{x}$ we have that, for every $b\in B_{x,b_{x}}$, it holds that

$\dim\\{c\in B_{x,b_{x}}:x\in U_{c}\subseteq U_{b}\\}=\dim B_{x,b}=\dim
B_{x,b_{x}}.$

So $\\{x\\}$ is $\text{DSC}_{\tau}$ as desired.

We prove 3. By 1 it suffices to show that, for any $n$, the space $M^{n}$ with
the o-minimal Euclidean topology is definably second-countable.

Consider the definable basis for the o-minimal Euclidean topology on $M^{n}$
given by boxes, namely
$\mathcal{B}=\\{(a_{1},b_{1})\times\cdots\times(a_{n},b_{n}):a_{i},b_{i}\in
M,\,a_{i}<b_{i},\text{ for }i\leq n\\}$. Let
$B=\\{(a_{1},b_{1},\ldots,a_{n},b_{n}):a_{i},b_{i}\in M,\,a_{i}<b_{i},\text{
for }i\leq n\\}$ and, for any $\bm{b}=(a_{1},b_{1},\ldots,a_{n},b_{n})\in B$,
let $U(\bm{b})=(a_{1},b_{1})\times\cdots\times(a_{n},b_{n})$. Note that $\dim
B=2n$. Now observe that, for every $x=(x_{1},\ldots,x_{n})\in M^{n}$ and
$\bm{b}=(a_{1},b_{1},\ldots,a_{n},b_{n})\in B$, it holds that

$\displaystyle\\{\bm{c}\in B:x\in U(\bm{c})\subseteq U(\bm{b})\\}=$
$\displaystyle\\{(a^{\prime}_{1},b^{\prime}_{1},\ldots,a^{\prime}_{n},b^{\prime}_{n}):a^{\prime}_{i},b^{\prime}_{i}\in
M,$ ($\dagger$) $\displaystyle\,a_{i}\leq
a^{\prime}_{i}<x_{i}<b^{\prime}_{i}\leq b_{i},\text{ for }i\leq n\\},$

and so whenever $x\in U(\bm{b})$ the set in (3) has dimension $2n=\dim B$. It
follows that $M^{n}$ with the o-minimal Euclidean topology is definably
second-countable.

Finally, we prove 4. Let $\\{X_{i}:i\leq n\\}$ be a finite covering of $X$ by
definable open subsets and, for each $i\leq n$, let $\mathcal{B}_{i}$ be a
definable basis for the subspace topology $\tau|_{X_{i}}$ that witnesses that
$(X_{i},\tau|_{X_{i}})$ is definably second-countable. For any $i\leq n$,
since $X_{i}$ is open, the family $\mathcal{B}_{i}$ is a $\tau$-basis for
$X_{i}$, which furthermore witnesses that $X_{i}$ is $\text{DSC}_{\tau}$. By
Lemma 3.9 we conclude that $(X,\tau)$ is definably second-countable. ∎

Observe that 3 and 4 in Proposition 3.10 above imply that the definable
manifold spaces of van den Dries [vdD98, Chapter 10] are definably second-
countable.

We now show that definable second-countability implies definable separability.

######  Proposition 3.11.

Let $\mathcal{M}$ be an o-minimal structure. Let $(X,\tau)$ be a definable
topological space. If $(X,\tau)$ is definably second-countable then it is
definably separable.

###### Proof.

We assume that $(X,\tau)$ is definably second-countable and not definably
separable and reach a contradiction. Hence let $\\{U_{c}:c\in C\\}$ be a
definable basis that witnesses that $(X,\tau)$ is definably second-countable,
and let $\mathcal{A}=\\{A_{b}:b\in B_{\mathcal{A}}\\}$ be an infinite
definable family of pairwise disjoint open sets. We may clearly assume that
$\emptyset\notin\mathcal{A}$.

For each $b\in B_{\mathcal{A}}$, let $C(b)=\\{c\in C:\emptyset\neq
U_{c}\subseteq A_{b}\\}$. Observe that the definable family $\\{C(b):b\in
B_{\mathcal{A}}\\}$ is infinite and contains pairwise disjoint sets. By Lemma
2.3 we may fix some $b\in B_{\mathcal{A}}$ with $\dim C(b)<\dim C$. Since
$\emptyset\notin\mathcal{A}$ then $A_{b}\neq\emptyset$. Let us fix some $x\in
A_{b}$ and, applying the definition of basis, some $c\in C$ with $x\in
U_{c}\subseteq A_{b}$. Then it holds that

$0\leq\dim\\{c^{\prime}\in C:y\in U_{c^{\prime}}\subseteq U_{c}\\}\leq\dim
C(b)<\dim C,$

however this contradicts that $\\{U_{c}:c\in C\\}$ witnesses that $(X,\tau)$
is definably second-countable. ∎

######  Remark 3.12.

It follows from Propositions 3.103 and 3.11 that any definable set in an
o-minimal structure with the o-minimal Euclidean topology is definably
separable. By the sentence below Proposition 3.10 the same is true for
definable manifold spaces.

##  4\. Equivalence with the classical properties

In this section we prove our main results. Primarily, we show that definable
separability and definable second-countability are equivalent to their
classical counterparts among definable topological spaces in o-minimal
expansions of $(\mathbb{R},<)$. We divide this result into Theorems 4.7 and
4.11. We also show in Proposition 4.14 that these properties are definable in
families.

The following lemma is our main tool in proving Theorem 4.7.

######  Lemma 4.1.

Let $(X,\tau)$ be a definably separable definable topological space. Let
$\mathcal{A}=\\{A_{b}:b\in B\\}$ be a definable family of open subsets of $X$
and let $n$ be such that $\dim A_{b}\leq n$ for every $b\in B$. Then there
exists a definable open set $Z\subseteq\cup\mathcal{A}$ such that $\dim Z\leq
n$ and $\cup\mathcal{A}\subseteq cl(Z)$. (In particular if $(X,\tau)$ has
small boundaries then $\dim\cup\mathcal{A}\leq n$.)

Moreover, $Z$ can be chosen definable over the same parameters as the family
$\mathcal{A}$ and the topology $\tau$.

###### Proof.

We fix $(X,\tau)$ and $\mathcal{A}=\\{A_{b}:b\in B\\}$ as in the lemma. We
proceed by induction on $k$, where $B\subseteq M^{k}$. The bulk of the proof
deals with the case $k=1$. The fact that $Z$ can be chosen definable over the
same parameters as $\mathcal{A}$ and $\tau$ will follow immediately from the
proof.

Case $k=1$. For every $x\in\cup\mathcal{A}$ let $B(x)=\\{b\in B:x\in
A_{b}\\}$. We partition $\cup\mathcal{A}$ into two sets as follows. Let
$A^{\text{inf}}=\\{x\in\cup\mathcal{A}:B(x)\text{ is infinite}\\}$ and
$A^{\text{fin}}=\cup\mathcal{A}\setminus
A^{\text{inf}}=\\{x\in\cup\mathcal{A}:B(x)\text{ is finite}\\}$. By
o-minimality these sets are definable.

######  Claim 4.2.

The set $A^{\text{inf}}$ is open and $\dim A^{\text{inf}}\leq n$.

###### Proof.

The fact that $\dim A^{\text{inf}}\leq n$ follows from Fact 2.4 applied to the
set $\\{(b,x)\in B\times A^{\text{inf}}:x\in A_{b}\\}$. We show that
$A^{\text{inf}}$ is open.

By o-minimal uniform finiteness there exits $m=\max\\{|B(x)|:x\in
A^{\text{fin}}\\}<\infty$. Fix $x\in A^{\text{inf}}$. Since $B(x)$ is infinite
we may pick $m+1$ parameters $b_{0},\ldots,b_{m}\in B(x)$. Consider the open
set $U=\cap_{0\leq i\leq m}A_{b_{i}}$. Then we have that $x\in U$ and moreover
every $y\in U$ satisfies that $|B(y)|>m$, which, by definition of $m$, implies
that $B(y)$ is infinite and so $y\in A^{\text{inf}}$. So $x\in U\subseteq
A^{\text{inf}}$. Hence $A^{\text{inf}}$ is open. ∎

Now for every $b\in B$ let $A^{\prime}_{b}=\\{x\in A_{b}\cap
A^{\text{fin}}:b=\min B(x)\\}$. We define

($\dagger$) $Z=A^{\text{inf}}\cup\cup\\{int(A^{\prime}_{b}):b\in B\\}.$

######  Claim 4.3.

The set $Z$ is open, definable, and satisfies that $\dim Z\leq n$.

###### Proof.

It is clear that $Z$ is definable and, by Claim 4.2, open.

Observe that the family of sets $\\{A^{\prime}_{b}:b\in B\\}$ is definable and
pairwise disjoint. Hence the same is clearly true of the family
$\\{int(A^{\prime}_{b}):b\in B\\}$ and so, by definable separability of
$(X,\tau)$, this latter family is finite. In particular since $\dim A_{t}\leq
n$ for every $b\in B$ we have that $\dim\cup\\{int(A^{\prime}_{b}):b\in
B\\}\leq n$. Applying Claim 4.2 we conclude that $\dim Z\leq n$. ∎

We conclude the proof of the case $k=1$ by showing that
$\cup\mathcal{A}\subseteq cl(Z)$. Towards a contradiction suppose that
$\cup\mathcal{A}\setminus cl(Z)\neq\emptyset$. Since $A^{\text{inf}}\subseteq
Z$ note that $\cup\mathcal{A}\setminus cl(Z)\subseteq A^{\text{fin}}$. Hence
by uniform finiteness we may pick and fix $x\in\cup\mathcal{A}\setminus cl(Z)$
satisfying that $|B(x)|=\max\\{|B(y)|:y\in\cup\mathcal{A}\setminus cl(Z)\\}$.
Now, since $x\notin cl(Z)$, there exists an open neighborhood $U$ of $x$ such
that $U\cap Z=\emptyset$. Since $x\in A^{\text{fin}}$ we may pick $U$ so that
it also satisfies that $x\in U\subseteq\cap_{b\in B(x)}A_{b}$. Note that every
$y\in U$ satisfies that $y\notin cl(Z)$ and $B(x)\subseteq B(y)$, and so by
maximality of $|B(x)|$ we have that $B(y)=B(x)$. Finally, let $b=\min B(x)$.
Since $x\notin int(A^{\prime}_{b})\subseteq Z$ there must exists $y\in U$ such
that $y\in A_{b}\setminus A^{\prime}_{b}$. However by definition of the set
$A^{\prime}_{b}$ this means that $\min B(y)<b$, which contradicts that
$B(y)=B(x)$.

Case $k>1$. Consider the projection $\pi(B)$ of $B$ to the first $k-1$
coordinates. For every $c\in\pi(B)$ consider the definable family
$\mathcal{A}_{c}=\\{A_{c,t}:t\in B_{c}\\}$. By the case $k=1$ there exists a
definable open set $Z_{c}\subseteq\cup\mathcal{A}_{c}$ such that $\dim
Z_{c}\leq n$ and $\cup\mathcal{A}_{c}\subseteq cl(Z_{c})$. By the definition
of the set $Z$ in the proof of the case $k=1$ note that the sets $Z_{c}$ may
be chosen to be definable uniformly in $c\in\pi(B)$ (over the same parameters
as the family $\mathcal{A}$ and the topology $\tau$).

We now apply the induction hypothesis to the definable family
$\\{Z_{c}:c\in\pi(B)\\}$ and derive that there exists a definable open set
$Z\subseteq\cup_{c\in\pi(B)}Z_{c}$ with $\dim Z\leq n$ and
$\cup_{c\in\pi(B)}Z_{c}\subseteq cl(Z)$. In particular

$\cup\mathcal{A}\subseteq\bigcup_{c\in\pi(B)}cl(Z_{c})\subseteq
cl\left(\bigcup_{c\in\pi(B)}Z_{c}\right)\subseteq cl(Z).$

∎

We illustrate Lemma 4.1 through the following example.

######  Example 4.4.

For any $n>1$ consider the following non-Hausdorff definable topology on
$M^{n}$. Fix a parameter $a\in M^{n-1}$. For every $x\in M^{n}$, basic open
neighborhoods of $x$ are sets of the form
$\\{x\\}\cup((b,\infty)\times\\{a\\})\subseteq M^{n}$, for $b\in M$. Since any
two non-empty open sets have non-empty intersection, it follows easily that
this space is definably separable.

For every $x\in M^{n}$, consider the open set
$A_{x}=\\{x\\}\cup(M\times\\{a\\})\subseteq M^{n}$. The family $\\{A_{x}:x\in
M^{n}\\}$ is definable and $\dim A_{x}=1$ for every $x\in M^{n}$. On the other
hand $\cup_{x\in M^{n}}A_{x}=M^{n}$, so $\dim(\cup_{x\in M^{n}}A_{x})=n$.
Nevertheless, the definable open subspace $M\times\\{a\\}$ is dense and one-
dimensional.

Example 4.4 shows that we cannot strengthen Lemma 4.1 by changing its
conclusion to $\dim\cup\mathcal{A}\leq n$ (unless the space has small
boundaries). On the other hand, this example is non-Hausdorff. Since there
exist Hausdorff definable topological spaces without small boundaries (e.g.
the Split interval), it is open whether this strengthening of the lemma holds
for all Hausdorff spaces.

Through the next lemma we point out that the construction in the proof of
Lemma 4.1 yields that the property of being definably separable can be
expressed with a single first-order sentence. We will use this fact in
Proposition 4.14 to show that definable separability is definable in families.

######  Lemma 4.5.

Let $\mathcal{M}$ be an o-minimal structure. Let $(X,\tau)$ be a definable
topological space, for which we fix a definable basis $\\{U_{b}:b\in B\\}$.
For every $n$, let $\mathcal{B}(n)\subseteq\mathcal{B}$ be the definable
family of basic open sets of dimension at most $n$, and set
$X(n)=\cup\mathcal{B}(n)$.

The space $(X,\tau)$ is definably separable if and only if, for every
$n\leq\dim X$, there exists a definable open set $Z(n)\subseteq X(n)$ such
that $X(n)\subseteq cl(Z(n))$ and $\dim Z(n)\leq n$.

###### Proof.

The “only if” direction is given by Lemma 4.1, applied to the families
$\mathcal{B}(n)$. We prove the “if” implication.

Suppose that $(X,\tau)$ is not definably separable, and let
$\mathcal{A}=\\{A_{b}:b\in B_{\mathcal{A}}\\}$ be an infinite family of
pairwise dijoint open sets. Let us fix the minimum $n$ such that the family
$\\{X(n)\cap A_{b}:b\in B_{\mathcal{A}}\\}$ is infinite (clearly $0\leq
n\leq\dim X$). Let $Z(n)$ be as described in the lemma. We reach a
contradiction by showing that $\dim Z(n)>n$.

Since $X(n)\subseteq cl(Z(n))$ observe that the family of pairwise disjoint
open sets $\mathcal{Z}=\\{Z(n)\cap A_{b}:b\in B_{\mathcal{A}}\\}$ is infinite.
Now, for any $b\in B_{\mathcal{A}}$ note that, if $\dim(Z(n)\cap A_{b})<n$,
then either $n=0$ and $Z(n)\cap A_{b}=\emptyset$ or otherwise $n>0$ and
$Z(n)\cap A_{b}\subseteq X(n-1)$. By choice of $n$ it follows that the
subfamily of sets in $\mathcal{Z}$ of dimension less than $n$ is finite, since
otherwise we would have $n>0$ and the family $\\{X(n-1)\cap A_{b}:b\in
B_{\mathcal{A}}\\}$ would be infinite. Hence the family $\\{Z(n)\cap
A_{b}:b\in B_{\mathcal{A}},\,\dim(Z(n)\cap A_{b})\geq n\\}$ is an infinite
definable family of pairwise disjoint subsets of $Z(n)$ of dimension at least
$n$. By Lemma 2.3 we conclude that $\dim Z(n)>n$. ∎

In the case where $(X,\tau)$ has small boundaries Lemma 4.1 can be simplified
to provide a more straightforward characterization of definable separability.

######  Proposition 4.6.

Let $\mathcal{M}$ be an o-minimal structure. Let $(X,\tau)$ be a definable
topological space with small boundaries. That is, $\dim(cl(Y)\setminus Y)<\dim
Y$ for every definable subset $Y\subseteq X$. We fix a definable basis for
$\tau$ and, for every $0\leq n\leq\dim X$, let $X(n)$ be the union of all
basic open sets of dimension at most $n$. Then $(X,\tau)$ is definably
separable if and only if $\dim X(n)\leq n$ for every $n$.

###### Proof.

Having small boundaries implies that, for every $n$, if $Z(n)$ is a definable
dense subset of $X(n)$ then $\dim Z(n)=\dim cl(Z(n))=\dim X(n)$. So the
proposition follows from Lemma 4.5. ∎

We now prove our main theorem about definable separability.

######  Theorem 4.7.

Let $\mathcal{M}$ be an o-minimal expansion of $(\mathbb{R},<)$. Let
$(X,\tau)$ be a definable topological space. The following are equivalent.

  1. (1)

$(X,\tau)$ is definably separable.

  2. (2)

$(X,\tau)$ is separable.

  3. (3)

$(X,\tau)$ has the countable chain condition (ccc).

###### Proof.

The implication 2$\Rightarrow$3 is a simple known fact in general topology.
The implication 3$\Rightarrow$1 (or rather its contrapositive) follows easily
from Lemma 2.1. We prove 1$\Rightarrow$2.

Suppose that $(X,\tau)$ is definably separable. We proceed by induction on
$n=\dim X$. The base case $n=0$ is immediate. We assume that $n>0$. Let us fix
a definable basis $\\{U_{b}:b\in B\\}$ for $\tau$. Let $C=\\{b\in B:\dim
U_{b}<n\\}$ and consider the definable family of sets
$\mathcal{A}=\\{U_{c}:c\in C\\}$. Let $Y=\cup\mathcal{A}$. By Lemma 4.1 there
exists a definable open set $Z\subseteq Y$ with $\dim Z<n$ and $Y\subseteq
cl(Z)$. Since $Z$ is open, by Lemma 3.3 the subspace $(Z,\tau)$ is definably
separable. By induction hypothesis we derive that $(Z,\tau)$ is separable. Let
$D_{1}$ denote a countable dense subset of $(Z,\tau)$. Since $Y\subseteq
cl(Z)$ note that $Y\subseteq cl(D_{1})$.

Now recall the classical fact that $X$ with the Euclidean topology is
separable. Let $D_{2}$ denote a countable subset of $X$ that is dense in the
Euclidean topology. Set $D=D_{1}\cup D_{2}$. This is a countable set. We show
that it is dense in $(X,\tau)$.

Let $A$ be a non-empty open set in $(X,\tau)$. By passing to a subset if
necessary we may assume that $A$ is definable. If $A\cap Y\neq\emptyset$ then
$A\cap D_{1}\neq\emptyset$. If $A\cap Y=\emptyset$ then, by definition of $Y$,
it must be that $\dim A=n=\dim X$. But then, by o-minimality (the Euclidean
topology has small boundaries), the set $A$ has non-empty interior in the
Euclidean topology on $X$, and consequently $A\cap D_{2}\neq\emptyset$. ∎

We now turn our attention to proving that a definable topological space in an
o-minimal expansion of $(\mathbb{R},<)$ is definably second-countable if and
only if it is second-countable (Theorem 4.11). In doing so we will use the
next three lemmas.

######  Lemma 4.8.

Let $\mathcal{M}$ be an o-minimal expansion of $(\mathbb{R},<)$. Let
$X\subseteq\mathbb{R}^{n}$ be a non-empty definable set and $\mathcal{C}$ be
countable family (not necessarily definable) of definable subsets of $X$, each
of dimension less than $\dim X$. Then $\mathcal{C}$ is not a cover of $X$,
i.e. $\cup\mathcal{C}\subsetneq X$.

###### Proof.

By o-minimal cell decomposition we can assume that both $X$ and every set in
$\mathcal{C}$ is a cell. We may also assume that $\dim X\geq 1$, since
otherwise $\mathcal{C}$ contains only the empty set and the result is trivial.
We proceed by induction on $n$, where $X\subseteq\mathbb{R}^{n}$. In the case
$n=1$ the sets in $\mathcal{C}$ are either the empty set or singletons and $Y$
is an interval so the result follows. We assume that $n>1$.

Consider the projection $\pi(X)$ of $X$ to the first $n-1$ coordinates. Let
$\mathcal{C}(0)\subseteq\mathcal{C}$ we the subfamily of cells
$C\in\mathcal{C}$ that are graphs of partial functions on $\pi(X)$, and let
$\mathcal{C}(1)=\mathcal{C}\setminus\mathcal{C}(0)$. We first consider the
case where $X$ is the graph of a function on $\pi(X)$. In this case we have
that $\mathcal{C}(1)=\emptyset$ and $\dim\pi(C)=\dim C<\dim X=\dim\pi(X)$ for
all $C\in\mathcal{C}$. By induction hypothesis it holds that
$\cup\\{\pi(C):C\in\mathcal{C}\\}\subsetneq\pi(X)$, and consequently
$\cup\mathcal{C}\subsetneq X$.

Now suppose that $X$ is not a cell given by the graph of a function on
$\pi(X)$, in particular we have that $\dim\pi(X)=\dim(X)-1$. For every
$C\in\mathcal{C}(1)$ it holds that
$\dim\pi(C)=\dim(C)-1<\dim(X)-1=\dim\pi(X)$. Hence by induction hypothesis we
may fix a point $d\in\pi(X)\setminus\cup\\{\pi(C):C\in\mathcal{C}(1)\\}$. Note
that, by assumptions on $X$, the fiber $X_{d}$ is an interval. Moreover for
every $C\in\mathcal{C}(0)$ the fiber $C_{d}$ in a singleton. Hence there
exists $e\in X_{d}\setminus\cup\\{C_{d}:C\in\mathcal{C}(0)\\}$, and we
conclude that $(d,e)\in X\setminus\cup\mathcal{C}$. ∎

######  Lemma 4.9.

Let $(X,\tau)$ be a topological space and $Y\subseteq X$ be a subset with a
countable $\tau$-basis. For any $\tau$-basis $\mathcal{B}$ for $Y$ there
exists a countable subfamily $\mathcal{C}\subseteq\mathcal{B}$ that is also a
$\tau$-basis for $Y$.

###### Proof.

Let $\mathcal{B}_{\omega}$ be a countable $\tau$-basis for $Y$. Let
$\mathcal{H}\subseteq\mathcal{B}_{\omega}\times\mathcal{B}_{\omega}$ be the
set of pairs $(U,U^{\prime})\in\mathcal{B}_{\omega}\times\mathcal{B}_{\omega}$
such that there exists some $A\in\mathcal{B}$ satisfying that $U\subseteq
A\subseteq U^{\prime}$. We fix one such set
$A=A(U^{\prime},U^{\prime\prime})\in\mathcal{B}$ for every pair
$(U,U^{\prime})\in\mathcal{H}$. Let
$\mathcal{C}=\\{A(U,U^{\prime}):(U,U^{\prime})\in\mathcal{H}\\}\subseteq\mathcal{B}$.
The family $\mathcal{C}$ is clearly countable. We show that it is a
$\tau$-basis for $Y$.

Let $x\in Y$ and consider an open set $U^{\prime}\ni x$. By passing to a
smaller neighborhood if necessary we may assume that
$U^{\prime}\in\mathcal{B}_{\omega}$. By definition of $\mathcal{B}$ there
exists some $A\in\mathcal{B}$ satisfying that $x\in A\subseteq U^{\prime}$. By
definition of $\mathcal{B}_{\omega}$ there exists some
$U\in\mathcal{B}_{\omega}$ such that $x\in U\subseteq A$. Hence $U\subseteq
A\subseteq U^{\prime}$. In particular we have that
$(U,U^{\prime})\in\mathcal{H}$. Observe that $x\in A(U,U^{\prime})\subseteq
U^{\prime}$. ∎

The following is the easy direction of Theorem 4.11 below.

######  Lemma 4.10.

Let $\mathcal{M}$ be an o-minimal expansion of $(\mathbb{R},<)$. Let
$(X,\tau)$ be a definable topological space. If $(X,\tau)$ is definably
second-countable then it is second-countable.

###### Proof.

We prove that every $\text{DSC}_{\tau}$ subset of $X$ admits a countable
$\tau$-basis. Let $Y\subseteq X$ be $\text{DSC}_{\tau}$, witnessed by a
$\tau$-basis $\mathcal{B}=\\{U_{b}:b\in B\\}$. That is, for every $y\in Y$ and
$b\in B$ with $y\in U_{b}$ it holds that

(1) $\dim\\{c\in B:x\in U_{c}\subseteq U_{b}\\}=\dim B.$

Let $D$ be a countable subset of $B$ that is dense in $B$ in the Euclidean
topology. For every $y\in Y$ and $b\in B$ with $y\in U_{b}$, equation (1)
implies that the set $\\{c\in B:x\in U_{c}\subseteq U_{b}\\}$ has non-empty
interior in $B$ in the Euclidean topology and so, by density of $D$, there
exists some $c\in D$ such that $y\in U_{c}\subseteq U_{b}$. We have shown that
$\\{U_{c}:c\in D\\}$ is a $\tau$-basis for $Y$. ∎

We may now present our main result on definable second-countability.

######  Theorem 4.11.

Let $\mathcal{M}$ be an o-minimal expansion of $(\mathbb{R},<)$. Let
$(X,\tau)$ definable topological space. Then $(X,\tau)$ is definably second-
countable if and only if it is second-countable.

###### Proof.

The “only if” implication is given by Lemma 4.10. We prove the “if” direction.
Specifically, we prove that any definable subset of $X$ that has a countable
$\tau$-basis is $\text{DSC}_{\tau}$.

Let us fix a definable basis $\\{U_{b}:b\in B\\}$ for $\tau$, and definable
sets $X^{\prime}\subseteq X$ and $B^{\prime}\subseteq B$ such that
$\\{U_{b}:b\in B^{\prime}\\}$ is a $\tau$-basis for $X^{\prime}$. We assume
that $X^{\prime}$ has a countable $\tau$-basis and prove that it is
$\text{DSC}_{\tau}$ by induction on $\dim X^{\prime}+\dim B^{\prime}$. If
$\dim X^{\prime}=0$ then the result is given by Proposition 3.102, and if
$\dim B^{\prime}=0$ then the result is trivial, so we assume that $\dim
X^{\prime}>0$ and $\dim B^{\prime}>0$. In particular this covers the base case
of the induction. To ease notation we assume that $X^{\prime}=X$ and
$B^{\prime}=B$.

Let $H\subseteq X\times B\times B$ denote the relation where $(x,b,c)\in H$
whenever $x\in U_{c}\subseteq U_{b}$. For any $x\in X$, $b\in B$ and set
$C^{\prime}\subseteq B$ let $H(x,b;C^{\prime})=\\{c\in C^{\prime}:(x,b,c)\in
H\\}$. For each $x\in X$ let $B(x)=\\{b\in B:0\leq H(x,b;B)<\dim B\\}$. Note
that the sets $B(x)$ are definable uniformly in $x\in X$. Let $Y=\\{x\in
X:B(x)\neq\emptyset\\}$. Observe that the set $X\setminus Y$ is
$\text{DSC}_{\tau}$, witnessed by $\\{U_{b}:b\in B\\}$. We assume that
$Y\neq\emptyset$ and describe a finite partition of $Y$ into
$\text{DSC}_{\tau}$ sets, hence proving, by Lemma 3.9, that $X$ is
$\text{DSC}_{\tau}$. By passing if necessary to an arbitrary set in a finite
partition of $Y$, we may assume that $\dim B(x)=\dim B(y)$ for every $x,y\in
Y$. The idea of the proof by induction is to find a definable subset
$Z\subseteq Y$ such that $\dim Z<\dim Y$ and moreover $Y\setminus Z$ admits a
definable $\tau$-basis of the form $\\{U_{b}:b\in C\subseteq B\\}$ where $\dim
C<\dim B$.

Onwards for any set $F^{\prime}\subseteq X\times B$ and $c\in B$ let
$H(F^{\prime};c)=\\{(x,b)\in F^{\prime}:(x,b,c)\in H\\}$. Let $F=\cup_{y\in
Y}\\{y\\}\times B(y)$. Let $C$ be the set of all $c\in B$ such that $\dim
H(F;c)=\dim F.$ By Fact 2.4, applied to the set $H\cap(F\times C)$, we derive
that

$\dim C=\max_{(y,b)\in F}\dim H(y,b;C).$

On the other hand by definition of $F$ every $(y,b)\in F$ satisfies that $\dim
H(y,b;B)<\dim B$. It follows that $\dim C<\dim B$.

Now let $Z\subseteq Y$ denote the set of points $z\in Y$ such that
$\\{U_{b}:b\in C\\}$ is not a $\tau$-basis for $\\{z\\}$. Note that the family
$\\{U_{b}:b\in B\setminus C\\}$ is a $\tau$-basis for $Z$ and $\\{U_{b}:b\in
C\\}$ is a $\tau$-basis for $Y\setminus Z$. By the latter, and since $\dim
C<\dim B$, we derive from the induction hypothesis that $Y\setminus Z$ is
$\text{DSC}_{\tau}$.

We complete the proof by showing that $\dim Z<\dim Y$. We may then apply the
induction hypothesis to derive that $Z$ is also $\text{DSC}_{\tau}$,
completing the proof of the theorem. We show that $\dim Z<\dim Y$ by assuming
that $\dim Z=\dim Y$ and deriving a contradiction using the fact that $X$ has
a countable $\tau$-basis.

Let $G=\cup_{z\in Z}\\{z\\}\times B(z)\subseteq F$. Recall that by assumption
on $Y$ it holds that $\dim B(x)=\dim B(y)$ for every $x,y\in Y$. Since by
assumption $\dim Z=\dim Y$ it follows from Fact 2.2 that $\dim G=\dim F$.
Applying the definition of $C$ we reach that every $b\in B\setminus C$
satisfies that

(2) $\dim H(G;b)\leq\dim H(F;b)<\dim F=\dim G.$

Finally, since $\\{U_{b}:b\in B\setminus C\\}$ is a $\tau$-basis for
$Z\subseteq X$, by Lemma 4.9 there exists a countable set $D\subseteq
B\setminus C$ such that $\\{U_{b}:b\in D\\}$ is a $\tau$-basis for $Z$.
Observe that, for any $z\in Z$ and $b\in B(z)$, by definition of $\tau$-basis
there must exists some $c\in D$ such that $z\in U_{c}\subseteq U_{b}$, meaning
that $(z,b,c)\in H$. Hence the sets $\\{H(G;b):b\in D\\}$ cover $G$. By Lemma
4.8 and equation (2) we reach a contradiction. ∎

In the following lemma we extract from the proof of Theorem 4.11 that the
property of not being definably second-countable can be expressed with a
single first-order formula. We use this later in Proposition 4.14 to show to
definable second-countability is definable in families.

######  Lemma 4.12.

A definable topological space $(X,\tau)$ is definably second-countable if and
only if there does not exists a non-empty definable set $Z\subseteq X$, two
definable families of open sets $\\{U_{b}:b\in B\\}$ and $\\{V_{b}:b\in
B^{\prime}\\}$, and a definable set $G\subseteq Z\times B$, with the following
three properties.

  1. (1)

For every $(z,b)\in G$ it holds that $z\in U_{b}$.

  2. (2)

$\\{V_{b}:b\in B^{\prime}\\}$ is a $\tau$-basis for $Z$.

  3. (3)

For every $b^{\prime}\in B^{\prime}$ it holds that

$\dim\\{(z,b)\in G:z\in V_{b^{\prime}}\subseteq U_{b}\\}<\dim G.$

###### Proof.

For the “if” implication, suppose that $(X,\tau)$ does not admit a
construction as described in the lemma. We may then show that $(X,\tau)$ is
definably second-countable by following the proof of Theorem 4.11 up to
equation (2), and then reaching a contradiction using the fact that, by
assumption, $Z$, $G$, $\\{U_{b}:b\in B\\}$, and $\\{U_{b}:b\in B\setminus
C\\}$ as described in said proof cannot exist.

We now prove the “only if” implication. Hence let $Z\subseteq X$,
$\\{U_{b}:b\in B\\}$, $\\{V_{b}:b\in B^{\prime}\\}$, and $G\subseteq Z\times
B$ be as described in the lemma. Towards a contradiction suppose that
$(X,\tau)$ is definably second-countable, witnessed by a definable basis
$\\{A_{c}:c\in C\\}$ for $\tau$.

Let us define $E\subseteq Z\times B\times C\times C$ to be the relation given
by $(z,b,c,c^{\prime})\in E$ whenever $z\in A_{c}\subseteq
A_{c^{\prime}}\subseteq U_{b}.$ For any $c,c^{\prime}\in C$, let
$E(G;c,c^{\prime})=\\{(z,b)\in G:(z,b,c,c^{\prime})\in E\\}$. Let
$D=\\{(c,c^{\prime})\in C\times C:\dim E(G;c,c^{\prime})<\dim G\\}$.
Furthermore, for each $(z,b)\in G$ let $E(z,b;D)=\\{(c,c^{\prime})\in
D:(z,b,c,c^{\prime})\in E\\}$. We prove that, for every $(z,b)\in G$, it holds
that $0\leq\dim E(z,b;D)=\dim D$. By Fact 2.4 it then follows that there
exists some $(c,c^{\prime})\in D$ such that $\dim E(G;c,c^{\prime})=\dim G$,
which contradicts the definition of $D$.

Hence let us fix a pair $(\mathbf{z},\mathbf{b})\in G$. Let
$C^{\prime}=\\{c^{\prime}\in C:\mathbf{z}\in A_{c^{\prime}}\subseteq
U_{\mathbf{b}}\\}$. Observe that, since $\\{A_{c}:c\in C\\}$ witnesses that
$(X,\tau)$ is definably second-countable (see Remark 3.8), it holds that $\dim
C^{\prime}=\dim C$.

######  Claim 4.13.

For each $c^{\prime}\in C^{\prime}$, it holds that $\dim\\{c\in
C:(c,c^{\prime})\in E(\mathbf{z},\mathbf{b};D)\\}=\dim C$.

###### Proof.

Let us fix $c^{\prime}\in C^{\prime}$. Since $\\{V_{b^{\prime}}:b^{\prime}\in
B^{\prime}\\}$ is a $\tau$-basis for $Z$ there exists some $b^{\prime}\in B$
such that $\mathbf{z}\in V_{b^{\prime}}\subseteq A_{c^{\prime}}$. Let
$C^{\prime\prime}=\\{c\in C:\mathbf{z}\in A_{c}\subseteq V_{b^{\prime}}\\}$.
Since $\\{A_{c}:c\in C\\}$ witnesses that $(X,\tau)$ is definably second-
countable (see Remark 3.8) note that $\dim C^{\prime\prime}=\dim C$. We show
that $C^{\prime\prime}\times\\{c^{\prime}\\}\subseteq
E(\mathbf{z},\mathbf{b};D)$.

For any $c\in C^{\prime\prime}$, since $A_{c}\subseteq V_{b^{\prime}}\subseteq
A_{c^{\prime}}$, observe that $E(G;c,c^{\prime})\subseteq\\{(z,b)\in G:z\in
V_{b^{\prime}}\subseteq U_{b}\\}$. Condition 3 in the lemma states that the
latter set has dimension less than $\dim G$, and so we derive that
$(c,c^{\prime})\in D$. We have shown that $C^{\prime\prime}\subseteq\\{c\in
C:(c,c^{\prime})\in E(\mathbf{z},\mathbf{b};D)\\}\subseteq C.$ Since $\dim
C^{\prime\prime}=\dim C$ the claim follows. ∎

Using the fact that $\dim C^{\prime}=\dim C$ and Claim 4.13, together with
Fact 2.2, we derive that

$2\dim C=\dim\bigcup_{c^{\prime}\in C^{\prime}}\\{c\in C:(c,c^{\prime})\in
E(\mathbf{z},\mathbf{b};D)\\}\times\\{c^{\prime}\\}\leq\dim
E(\mathbf{z},\mathbf{b};D).$

Since $E(\mathbf{z},\mathbf{b};D)\subseteq D\subseteq C\times C$, we conclude
that $\dim E(\mathbf{z},\mathbf{b};D)=\dim D=2\dim C\geq 0$. ∎

A family of topological spaces $\\{(X_{c},\tau_{c}):c\in C\\}$ is (uniformly)
definable if it is defined by a partitioned formula $\varphi(x,y,z)$ and a
definable set $B^{\text{opp}}\subseteq M^{|y|+|z|}$ with the following
properties. The set $C$ is the projection of $B^{\text{opp}}$ to the last
$|z|$ coordinates and, for every $c\in C$, the family of sets
$\\{\varphi(M,b,c):b\in B_{c}\\}$, where $B_{c}=\\{b\in M^{|y|}:(b,c)\in
B^{\text{opp}}\\}$ and $\varphi(M,b,c)=\\{a\in
M^{|x|}:\mathcal{M}\models\varphi(a,b,c)\\}$, is a basis for the topology
$\tau_{c}$. An example of a definable family of topological spaces would be
given by any definable family of subsets of a given definable topological
space with the subspace topology.

We show that the properties of definable separability and definable second-
countability are definable in families.

######  Proposition 4.14.

Let $\mathcal{M}$ be an o-minimal structure. Let
$\mathcal{C}=\\{(X_{c},\tau_{c}):c\in C\\}$ be a definable family of
topological spaces. There exists definable subsets $C_{\text{sep}}$ and
$C_{\text{sc}}$ of $C$ such that, for every $c\in C$, the space
$(X_{c},\tau_{c})$ is

  1. (1)

definably separable if and only if $c\in C_{\text{sep}}$,

  2. (2)

definably second-countable if and only if $c\in C_{\text{sc}}$.

###### Proof.

Let $C_{\text{sep}}$ denote the set of elements $c\in C$ such that the space
$(X_{c},\tau_{c})$ is definably separable. By Definition 3.2 (applying uniform
finiteness and Lemma 2.1) and Lemma 4.5 respectively, the sets
$C_{\text{sep}}$ and $C\setminus C_{\text{sep}}$ are each an intersection of
sets definable over the same parameters that define $\mathcal{C}$ (i.e. if
$\mathcal{M}$ were a monster model they would be type-definable). By a
standard first-order logic compactness argument it follows that
$C_{\text{sep}}$ is definable.

The proof that $C_{\text{sc}}$, the set of all $c\in C$ such that
$(X_{c},\tau_{c})$ is definably second-countable, is definable, is analogous,
using Definition 3.4 and Lemma 4.12. ∎

######  Remark 4.15.

Let $\mathcal{M}$ be an o-minimal structure and $(X,\tau)$ be a definable
topological space. Let $\mathcal{A}=\\{A_{b}:b\in B\\}$ be a definable family
of open subsets of $X$ of dimension at most $n$. Observe that the proof of
Lemma 4.1 provides a explicit construction for a definable set
$Z\subseteq\cup\mathcal{A}$, and shows that $Z$ is open in $X$ and dense in
$\cup\mathcal{A}$. Furthermore, it applies the assumption that $(X,\tau)$ is
definably separable to derive that $\dim Z\leq n$. Specifically, if
$B\subseteq M$ then $Z$ is simply given by Equation ($\dagger$ ‣ 4), and in
the general case the same idea is applied recursively. Note that the
construction of $Z$ depends uniformly on the family $\mathcal{A}$.

It follows that in Lemma 4.5 one may fix the sets $Z(n)$ for every $n\leq\dim
X$ to be the unique sets described in the proof of Lemma 4.1 (for
$\mathcal{A}=\mathcal{B}(n)$), and then $(X,\tau)$ is definably separable if
and only if $\dim Z(n)\leq n$ for every $n<\dim X$. Furthermore, if
$\\{(X_{c},\tau_{c}):c\in C\\}$ is a definable family of topological spaces,
then, for each $n$, the corresponding sets
$Z_{c}(n)\subseteq\mathcal{B}_{c}(n)$ are definable uniformly in $c\in C$.
This can be used to yield a new proof of Proposition 4.141, which provides an
explicit description of a formula defining the set $C_{\text{sep}}$.

A similar approach, using the proof of Theorem 4.11 and Lemma 4.12, can be
used to give an explicit description of a formula defining the set
$C_{\text{sc}}$ in Proposition 4.14.

##  5\. Definable metric spaces

In this section we explore definable separability and second-countability in
the context of o-minimal definable metric spaces. Definable metric spaces were
introduced and studied by Walsberg in [Wal15a]. Although Walsberg works under
the assumption that $\mathcal{M}$ is an o-minimal expansion of an ordered
field, any notion that we borrow from [Wal15a], including the definition of
definable metric space below, still makes sense in the ordered group setting.

######  Definition 5.1.

Let $\mathcal{M}=(M,0,+,<,\ldots)$ be an expansion of an ordered group. A
_definable metric space_ is a tuple $(X,d)$, where $X$ is a definable set and
$d:X\times X\rightarrow M^{\geq 0}$ is a definable map that satisfies the
metric axioms, namely identity of indiscernibles, symmetry and subadditivity.

Given a definable metric space $(X,d)$, $x\in X$, and $t\in M^{>0}$, we denote
the open ball of center $x$ and radius $t$ by $B_{d}(x,t)=\\{y\in
X:d(x,y)<t\\}$. A definable metric space $(X,d)$ is a definable topological
space with the topology generated by open balls.

In [Wal15a] Walsberg defined _definable separability_ among o-minimal
definable metric spaces to be the property of not containing an infinite
definable discrete subspace. His main result [Wal15a, Theorem 9.0.1] implies
that a metric space definable in an o-minimal expansion of the field of reals
is definably separable if and only if it is separable. On the other hand,
Walsberg’s definition of definable separability is not suitable for general
o-minimal (non-definably-metrizable) definable topological spaces, since these
include for example the Moore plane [AG21, Example A.12], which is definable
in $(\mathbb{R},+,\cdot,<)$ and separable but nevertheless the subspace
$\mathbb{R}\times\\{0\\}$ is discrete. Another example, this time one-
dimensional, is given by [AG21, Example A.9].

We describe the precise relationship between Walsberg’s definition of
definable separability (generalized to all definable topological spaces) and
ours by means of the next definition, lemma and proposition.

######  Definition 5.2.

A definable topological space $(X,\tau)$ is _hereditarily definably separable_
if every definable subspace of $(X,\tau)$ is definably separable.

The next lemma states that, as long as $\mathcal{M}$ has definable choice,
Walsberg’s definition of definable separability is equivalent, in our
terminology, to being hereditary definably separable.

######  Lemma 5.3.

Suppose that $\mathcal{M}$ has definable choice. A definable topological space
$(X,\tau)$ is hereditarily definably separable if and only if it does not
contain an infinite definable discrete subspace.

###### Proof.

Any infinite discrete space is clearly not definably separable and so the
“only if” implication follows.

For the “if” implication, let $Y\subseteq X$ be a definable subset such that
the subspace $(Y,\tau|_{Y})$ is not definably separable. Let
$\mathcal{A}=\\{A_{b}:b\in B\\}$ be an infinite definable family of pairwise
disjoint open sets in $(Y,\tau|_{Y})$. Using definable choice let
$f:B\rightarrow\cup\mathcal{A}$ be a definable map such that $f(b)\in A_{b}$
for every $b\in B$, and satisfying moreover that $f(b)=f(c)$ whenever
$A_{b}=A_{c}$. Then the image subspace $(f(B),\tau|_{f(B)})$ is an infinite
definable discrete subspace. ∎

Observe that, by Propositions 3.101 and 3.11, any definably second-countable
definable topological space in an o-minimal structure is hereditarily
definably separable.

Recall that any separable metric space is hereditarily separable. We show that
the same holds in the definable setting whenever the underlying structure has
definable choice.

######  Proposition 5.4.

Let $\mathcal{M}$ be an expansion of an ordered group with definable choice
(e.g. $\mathcal{M}$ is an o-minimal expansion of an ordered group). Let
$(X,d)$ be a definable metric space. The following are equivalent.

  1. (1)

$(X,d)$ is definably separable.

  2. (2)

$(X,d)$ is hereditarily definably separable.

  3. (3)

$(X,d)$ does not contain an infinite definable discrete subspace (namely it is
definably separable in the sense of Walsberg [Wal15a]).

###### Proof.

The equivalence 2$\Leftrightarrow$3 is given by Lemma 5.3. The implication
2$\Rightarrow$1 is trivial. We prove 1$\Rightarrow$3 by contraposition.

If the ordered group structure on $\mathcal{M}$ is discrete then every
definable metric space is discrete and so definably separable if and only if
finite. So we may assume that the ordered group is dense, i.e. $M^{>0}$ does
not have a minimum. In particular, since for every $0<s<t$ it holds that
either $2s\leq t$ or $2(t-s)\leq t$, note that, for every $t>0$, there exists
$s>0$ with $2s<t$.

Let $(X,d)$ be a definable metric space and $Y$ be an infinite definable
discrete subspace. By definable choice one may choose definably, for each
$x\in Y$, some $\varepsilon_{x}>0$ such that $2\varepsilon_{x}<d(x,y)$ for
every $y\in Y\setminus\\{x\\}$. We prove that the infinite definable family of
open balls $\\{B_{d}(x,\varepsilon_{x}):x\in Y\\}$ is pairwise disjoint, and
so $(X,d)$ is not definably separable.

Towards a contradiction suppose that there exists $x,y\in Y$ and some $z\in
B_{d}(x,\varepsilon_{x})\cap B_{d}(y,\varepsilon_{y})$. Then by the triangle
inequality $d(x,y)\leq d(x,z)+d(z,y)\leq\varepsilon_{x}+\varepsilon_{y}\leq
2\max\\{\varepsilon_{x},\varepsilon_{y}\\}.$ Without loss of generality
suppose that $\varepsilon_{x}=\max\\{\varepsilon_{x},\varepsilon_{y}\\}$. Then
this contradicts the fact that $2\varepsilon_{x}<d(x,y)$. ∎

Since definable metric spaces are, by definition, defined only in expansions
of ordered groups, Proposition 5.4 addresses all definable metric spaces in
o-minimal structures.

Onwards we restrict our scope exclusively to the o-minimal setting. Recall
that in general topology a metric space is separable if and only if it is
second-countable (if and only if it is hereditarily separable). We prove that
this equivalence also holds in the o-minimal definable context. Joining this
equivalence with the one in Proposition 5.4 we reach the main result of this
section.

######  Theorem 5.5.

Let $\mathcal{M}$ be an o-minimal expansion of an ordered group. Let $(X,d)$
be a definable metric space. The following are equivalent.

  1. (1)

$(X,d)$ is definably separable.

  2. (2)

$(X,d)$ is hereditarily definably separable.

  3. (3)

$(X,d)$ does not contain an infinite definable discrete subspace (namely it is
definably separable in the sense of Walsberg [Wal15a]).

  4. (4)

$(X,\tau)$ is definably second-countable.

The equivalence between 1, 2 and 3 in Theorem 5.5 is provided by Proposition
5.4. Furthermore, the implication $\ref{itm:thm-msp_4}\Rightarrow\ref{itm:thm-
msp_1}$ is given by Lemma 3.11. Hence it remains to show that definable
separability implies definable second-countability. We prove this in Lemma 5.7
below.

In proving Lemma 5.7 we will use the next simple ad hoc lemma. It follows from
an easy application of the triangle inequality, and so we leave the proof to
the reader. Note that it does not necessitate that the metric map $d$ be
definable.

######  Lemma 5.6.

Let $\mathcal{M}$ be an expansion of an ordered divisible group and $(X,d)$ be
a definable metric space. Let $X^{\prime},Y\subseteq X$ be subsets with
$X^{\prime}\subseteq cl(Y)$. Then $\\{B_{d}(y,t):y\in Y,\,t>0\\}$ is a
$\tau$-basis for $X^{\prime}$. Specifically, for every $x\in X^{\prime}$ and
$t>0$ we have that $x\in B_{d}(y,s)\subseteq B_{d}(x,t)$ for every $y\in
B_{d}(x,t/3)\cap Y$ and $t/3<s<2t/3$.

In the proof of Lemma 5.7 below recall that in every o-minimal expansion of an
ordered group the group is necessarily divisible.

######  Lemma 5.7.

Let $\mathcal{M}$ be an o-minimal expansion of an ordered group and $(X,d)$ be
a definable metric space. If $(X,\tau)$ is definably separable then it is
definably second-countable.

###### Proof.

Let $(X,d)$ be a definably separable definable metric space. Our proof
proceeds by showing that every definable subset $X^{\prime}\subseteq X$ is
$\text{DSC}_{\tau}$ by induction on the dimension of a definable set
$Y\subseteq X$ such that $X^{\prime}\subseteq cl(Y)$. If $\dim Y=0$ then, by
Hausdorffness of definable metric spaces, $Y$ is also closed and so
$X^{\prime}\subseteq Y$ is finite and the result is given by Proposition
3.102. This covers the base case. Onwards suppose that $n=\dim Y>0$. To ease
notation we consider $X^{\prime}=X$.

Let $Y^{\prime}$ denote the union of all (basic) open definable sets of
dimension at most $n-1$ in the subspace topology $\tau|_{Y}$. Let $X_{0}=X\cap
cl(Y\setminus Y^{\prime})$ and $X_{1}=X\setminus X_{0}$. Since $X\subseteq
cl(Y)$ we have that $X_{1}\subseteq cl(Y^{\prime})$. Observe that, by Lemma
4.1, there exists a definable set $Z\subseteq Y^{\prime}$ with $\dim Z\leq
n-1$ and such that $Y^{\prime}\subseteq cl(Z)$. In particular $X_{1}\subseteq
cl(Z)$. By induction hypothesis we reach that $X_{1}$ is $\text{DSC}_{\tau}$.
We end the proof by showing that $X_{0}$ is $\text{DSC}_{\tau}$, which
completes the proof by application of Lemma 3.9.

Let $\mathcal{B}_{0}=\\{B_{d}(y,t):y\in Y,\,t>0\\}$. By Lemma 5.6 note that
$\mathcal{B}_{0}$ is a $\tau$-basis for $X$ (in particular for $X_{0}$). We
prove that $\mathcal{B}_{0}$ witnesses that $X_{0}$ is $\text{DSC}_{\tau}$ by
showing that, for every $x\in X_{0}$ and $t>0$, it holds that

$\dim\\{(y,s)\in Y\times M^{>0}:x\in B_{d}(y,s)\subseteq
B_{d}(x,t)\\}=\dim(Y\times M^{>0}).$

Hence let us fix $x\in X_{0}$ and $t>0$. By Lemma 5.6 we have that

$\\{(y,s):y\in Y\cap B_{d}(x,t/3),\,t/3<s<2t/3)\\}\subseteq\\{(y,s)\in Y\times
M^{>0}:x\in B_{d}(y,s)\subseteq B_{d}(x,t)\\}.$

Now notice that, by definition of the set $Y^{\prime}$, any definable open set
$A$ with $A\cap Y\setminus Y^{\prime}\neq\emptyset$ satisfies that $\dim(A\cap
Y)=n=\dim Y$. In particular, since $X_{0}=X\cap cl(Y\setminus Y^{\prime})$ and
$x\in X_{0}$, we have that $\dim(Y\cap B_{d}(x,t/3))=\dim Y$. Applying Fact
2.2, we conclude that $\dim\\{(y,s):y\in Y\cap
B_{d}(x,t/3),\,t/3<s<2t/3\\}=\dim(Y)+1=\dim(Y\times M^{>0})$, as desired. ∎

The above proof can be streamlined if one assumes that $(X,\tau)$ has small
boundaries. This property was proved for definable metric spaces in o-minimal
expansions of ordered fields in [Wal15a, Lemma 9.2.11]. Although it is likely
that it holds for metric spaces definable in o-minimal expansions of groups
too, the author is not aware of any proof of this.

The Sorgenfrey line (Example 3.5 defined in $(\mathbb{R},<)$) is a classical
example of a (non-metrizable) topological space that is hereditary separable
but not second-countable. The Moore plane and the Sorgenfrey plane are
examples of (non-metrizable) topological spaces that are separable but not
hereditarily separable, and in particular not second-countable. An analogous
landscape is present in the o-minimal definable setting. That is, the three
aforementioned examples are definable in the ordered field of reals, and
furthermore have definable analogues in any o-minimal expansion of an ordered
field (see e.g. Example A.12 in [AG21]) which display definable topological
properties analogous to the ones described for the classical spaces.

##  6\. Towards a Definable Urysohn Metrization Theorem

We say that a definable topological space is _affine_ if it is definably
homeomorphic to a set with the o-minimal Euclidean topology. The
characterization of o-minimal affine definable topological spaces has been the
subject of ongoing research in [vdD98, Chapter 10] (manifold and quotient
spaces), [Wal15b] (metric spaces), [Joh18] (quotient spaces) and [PR20] and
[AGT23] (one-dimensional spaces).

We end this paper by conjecturing that, in o-minimal expansions of ordered
fields, definable second-countability characterizes affine definable
topological spaces. This conjecture is informed by the contents of Chapter 7
and Example A.16 in [AG21], the latter which describes a Hausdorff and regular
definable topological space in $(\mathbb{R},+,\cdot,<)$ that can be
partitioned into two definable subsets where the subspace topology is
Euclidean, but nevertheless the space is not definably second-countable (in
particular not affine). We conjecture that this condition is the only obstacle
to achieve affineness.

######  Conjecture 6.1 (Definable Urysohn Metrization Conjecture).

Let $(X,\tau)$ be definable topological space in an o-minimal expansion of an
ordered field. Then $(X,\tau)$ is definably homeomorphic to a set with the
o-minimal Euclidean topology if and only if it is Hausdorff, regular, and
definably second-countable.

The classical Urysohn Metrization Theorem states that every Hausdorff regular
second-countable topological space is metrizable. The main result (Theorem
9.0.1) in [Wal15b] states that a definable metric space in an o-minimal
expansion of an ordered field is affine if and only if it is definably
separable. By Lemma 3.11 it follows that Conjecture 6.1 has a positive answer
among definable metric spaces. Furthermore, in order to prove the conjecture
it would suffice to show that any Hausdorff, regular, definably second-
countable space $(X,\tau)$ is definably metrizable, hence its name.

Theorem 9.1 in [AGT23] states that a one-dimensional Hausdorff definable
topological space in an o-minimal expansion of an ordered field is affine if
and only if it does not contain a subspace definably homeomorphic to an
interval with the discrete or Sorgenfrey line topologies. Since these
topologies are not definably second-countable (see Example 3.5 addressing the
latter), it follows (applying Proposition 3.101) that Conjecture 6.1 also has
a positive answer among one-dimensional spaces.

## References

  * [AF11] Matthias Aschenbrenner and Andreas Fischer.  Definable versions of theorems by Kirszbraun and Helly.  Proc. Lond. Math. Soc. (3), 102(3):468–502, 2011.
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