# The product of Lindelöf groups and $\mathbb{R}$-factorizability

Evgeny Reznichenko<EMAIL_ADDRESS>Department of General Topology and
Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State
University, Leninskie Gory 1, Moscow, 199991 Russia

###### Abstract

Lindelöf topological groups $\mathsf{{G}_{1}}$, $\mathsf{{H}_{1}}$,
$\mathsf{{G}_{2}}$, $\mathsf{{H}_{2}}$ are constructed in such a way that the
products of $\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ and
$\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ are not $\mathbb{R}$-factorizable
groups and (1) the group $\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ is not
pseudo-$\aleph_{1}$-compact; (2) the group
$\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ is a separable not normal group and
contains a discrete closed subset of the cardinality $2^{\omega}$.

###### keywords:

product of groups , Lindelöf groups , $\mathbb{R}$-factorizability ,
pseudo-$\aleph_{1}$-compact groups

##  1 Introduction

A topological group $G$ is said to be $\mathbb{R}$-factorizable if for any
continuous function $f:G\to\mathbb{R}$ there exists a topological separable
metrizable group $G$, a continuous homomorphism $\varphi:G\to H$, and a
continuous function $h:H\to\mathbb{R}$, such that $f=h\circ\varphi$, i.e. the
following diagram is commutative

${G}$${H}$${H}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{h}$

$\mathbb{R}$-factorizable groups play a big role in topological group theory,
see Chapter 8 in [1]. One of the main questions in the theory of
$\mathbb{R}$-factorizable groups: is the product of two
$\mathbb{R}$-factorizable groups a $\mathbb{R}$-factorizable group [2, 3]?
Lindelöf topological groups are $\mathbb{R}$-factorizable [1, Theorem 8.1.6.].
In this note, Lindelöf groups are constructed whose product is not
$\mathbb{R}$-factorizable. These examples also answer Problems 8.5.3, 8.5.4
and 8.5.9 from [1].

Lindelöf topological groups $\mathsf{{G}_{1}}$, $\mathsf{{H}_{1}}$,
$\mathsf{{G}_{2}}$, $\mathsf{{H}_{2}}$ are constructed in such a way that the
products of $\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ and
$\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ are not $\mathbb{R}$-factorizable
groups and (1) the group $\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ is not
pseudo-$\aleph_{1}$-compact; (2) the group
$\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ is a separable not normal group and
contains a discrete closed subset of the cardinality $2^{\omega}$ (Theorem 5).
The group $\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ is a separable
non-$\mathbb{R}$-factorizable group. The first such group was constructed in
[4].

Recently O.V. Sipacheva [5, 6] also independently found a
non-$\mathbb{R}$-factorizable product of Lindelöf groups. Moreover, one of
these groups is metrizable separable. In Section 4, products of Lindelöf
groups from [5, 6] are studied and it is proved that the product of Lindelöf
groups from [7] is also not $\mathbb{R}$-factorizable.

##  2 Notation and definition

Let $X$ be a set and $\tau_{1}$, $\tau_{2}$, …, $\tau_{n}$ be a family of
subsets of $X$. Denote by $\mathcal{B}(\tau_{1},\tau_{2},...,\tau_{n})$ the
topology on $X$ whose prebase is formed by the family
$\bigcup_{i=1}^{n}\tau_{i}$. If $\tau_{i}$ is a topology on $X$ for
$i=1,2,...,n$, then the topology $\mathcal{B}(\tau_{1},\tau_{2},...,\tau_{n})$
is the topology of the diagonal $\\{(x,x,...,x)\,:\,x\in X\\}$ in the product
$\prod_{i=1}^{n}(X,\tau_{i}).$

Denote by $B(X)$ the free Boolean topological group over $X$.

A family ${\mathcal{F}}$ is called non-Archimedean if for any
$A,B\in{\mathcal{F}}$ one of the following conditions is true: $A\cap
B=\varnothing$; $A\subset B$; $B\subset A$. Spaces that have a non-Archimedean
base are called non-Archimedean spaces. Separable metrizable zero-dimensional
spaces are non-Archimedean.

We call a space $X$ co-non-Archimedean if there is a non-Archimedean topology
$\tau$ on $X$ and a family ${\mathcal{F}}$ of $\tau$-closed subsets such that
the topology $X$ coincides with $\mathcal{B}(\tau,{\mathcal{F}})$.

######  Theorem 1 ([8]).

If $X$ is a co-non-Archimedean space, then $X$ is a retract $B(X)$.

The group with the topology is called

  1. (1)

paratopological if the multiplication in the group is (jointly) continuous;

  2. (2)

semitopological if the multiplication in the group is separately continuous.

Consider the following classes of spaces and groups with topology:

  1. ($\mathsf{TS}_{sm}$)

separable metrizable spaces;

  2. ($\mathsf{TS}_{cnw}$)

spaces with a countable network;

  3. ($\mathsf{TG}_{sm}$)

separable metrizable topological groups;

  4. ($\mathsf{PG}_{sm}$)

separable metrizable paratopological groups;

  5. ($\mathsf{SG}_{cnw}$)

semitopological groups with a countable network.

Let $\mathsf{G}$ be some class of groups with topology. A group with the
topology $G$ is called $\mathsf{G}$-factorizable if for any continuous
function $f:G\to\mathbb{R}$ there exists a group $H\in\mathsf{G}$, a
continuous homomorphism $\varphi:G\to H$ and a continuous function
$h:H\to\mathbb{R}$, such that $f=h\circ\varphi$.

A topological group $G$ is called $\mathbb{R}$-factorizable if $G$ is
$\mathsf{TG}_{sm}$-factorizable. A paratopological group $G$ is called
$\mathbb{R}$-factorizable if $G$ is $\mathsf{PG}_{sm}$-factorizable.

Let $\mathsf{S}$ be some class of spaces. We call a function on the product of
spaces $f:X_{1}\times X_{2}\times...\times X_{n}\to\mathbb{R}$ (separately)
$\mathsf{S}$-factorizable if there exist spaces $Y_{1}$, $Y_{2}$, …, $Y_{n}$
from $\mathsf{S}$, continuous mappings $g_{i}:X_{i}\to Y_{i}$ for
$i=1,2,...,n$ and a (separately) continuous function $h:Y=Y_{1}\times
Y_{2}\times...\times Y_{n}\to\mathbb{R}$, so $f=h\circ\prod_{i=1}^{n}g_{i}$,
that is, the following diagram is commutative.

${X_{1}\times X_{2}\times...\times X_{n}}$${\mathbb{R}}$${Y_{1}\times
Y_{2}\times...\times
Y_{n}}$$\scriptstyle{f}$$\scriptstyle{\prod\limits_{i=1}^{n}g_{i}}$$\scriptstyle{h}$

We call a function $f$ (separately) $\mathbb{R}$-factorizable if $f$ is
(separately) $\mathsf{TS}_{sm}$-factorizable.

A product of spaces $X=X_{1}\times X_{2}\times...\times X_{n}$ is called
(separately) $\mathsf{S}$-factorizable if any continuous function on
$f:X\to\mathbb{R}$ is (separately) $\mathsf{S}$-factorizable.

A product of spaces $X=X_{1}\times X_{2}\times...\times X_{n}$ is called
(separately) $\mathbb{R}$-factorizable if $X$ is (separately)
$\mathsf{TS}_{sm}$\- factorizable.

The discrete Suslin number $dc(X)$ of the space $X$ is called the supremum of
cardinals $\tau$, such that there exists a discrete family of open sets of
cardinality $\tau$. Spaces with a countable discrete Suslin number are also
called pseudo-$\aleph_{1}$-compact.

######  Theorem 2 ([9]).

If $X$ is a normal separable space then the space does not contain a discrete
closed subset of the cardinality of the continuum.

There are two definitions of covering dimension, in the sense of Čech and in
the sense of Katětov; following [10], we denote the Katětov dimension by
$\operatorname{dim}_{0}$. Recall that, given a topological space $X$,
$\operatorname{dim}_{0}X$ is the least integer $n>-1$ such that any finite
cozero cover of $X$ has a finite cozero refinement of order $n$, provided that
such an integer exists. If it does not exist, then
$\operatorname{dim}_{0}X=\infty$. A space $X$ for which
$\operatorname{dim}_{0}X=0$ is said to be strongly zero-dimensional.

In [11], as in many other places, Katětov dimension is denoted as
$\operatorname{dim}$.

######  Theorem 3.

Let the topological group $H$ be $\mathbb{R}$-factorizable.

  1. (1)

If $H$ is embedded in a topological group $G$, then
$\operatorname{dim}_{0}H\leq\operatorname{dim}_{0}G$ [2, Theorem 2.7].

  2. (2)

If $H$ is zero-dimensional then $H$ is strongly zero-dimensional [12, Theorem
3.3].

######  Definition 1 (Definition 2.28 [13]).

A product space $X\times Y$ is said to be rectangular if any ﬁnite cozero-
cover ${\mathcal{U}}$ of the space $X\times Y$ has a $\sigma$-locally ﬁnite
reﬁnement ${\mathcal{V}}$ consisting of sets of the form $U\times V$, where
$U$ is cozero subset of $X$ and $V$ is cozero subset of $Y$.

######  Theorem 4 (Theorem 2.32 [13]).

If a product space $X\times Y$ is rectangular, then

$\operatorname{dim}_{0}X\times
Y\leq\operatorname{dim}_{0}X+\operatorname{dim}_{0}Y.$

##  3 Product of groups

Let $X\subset\mathbb{R}$ and $A\subset X$, $\tau$ be the topology of $X$
induced from $\mathbb{R}$. Denote

$\displaystyle\tau_{[A]}$ $\displaystyle=\mathcal{B}(\tau,\\{\\{x\\}\,:\,x\in
X\setminus A\\}),$ $\displaystyle X_{[A]}$ $\displaystyle=(X,\tau_{[A]}),$
$\displaystyle\tau_{[A)}$
$\displaystyle=\mathcal{B}(\tau,\\{X\cap[x,+\infty)\,:\,x\in X\setminus
A\\}),$ $\displaystyle X_{[A)}$ $\displaystyle=(X,\tau_{[A)}),$
$\displaystyle\tau_{(A]}$
$\displaystyle=\mathcal{B}(\tau,\\{X\cap(-\infty,x]\,:\,x\in X\setminus
A\\}),$ $\displaystyle X_{(A]}$ $\displaystyle=(X,\tau_{(A]}).$

The following assertions follow from the definition.

######  Statement 1.

The spaces $X_{(A]}$ and $X_{[A)}$ are continuous images of $X_{[A]}$.

######  Statement 2.

If $X\setminus A$ is dense in $\mathbb{R}$, then the spaces $X_{[A]}$,
$X_{(A]}$, $X_{[A)}$ are co-non-Archimedean.

There exists a partition $\\{M_{\alpha}\,:\,\alpha<2^{\omega}\\}$ of the line
$\mathbb{R}$, such that $\left(\mathbb{R}_{[M_{\alpha}]}\right)^{\omega}$ is
Lindelöf for $\alpha<2^{\omega}$ [14].

Let $A=M_{0}$, $B=M_{1}$, $C=\mathbb{R}\setminus(A\cup B)$, $\mathsf{L}=A\cup
C$, $\mathsf{R}=B\cup C$.

The construction and Statements 1 and 2 imply the following statement.

######  Proposition 1.

The spaces $\mathsf{L}_{[A]}$, $\mathsf{L}_{(A]}$, $\mathsf{L}_{[A)}$,
$\mathsf{R}_{[B]}$, $\mathsf{R}_{(B]}$, $\mathsf{R}_{[B)}$ are co-non-
Archimedean and Lindelöf to countable degree.

######  Lemma 1.

If a separately continuous function $f:X\times Y\to\mathbb{R}$ is separately
$\mathsf{TS}_{cnw}$-factorizable, then $\varphi(X)$ has a countable network,
where

$\varphi:X\to C_{p}(Y),\ \varphi(x)(y)=f(x,y).$

###### Proof.

Since $f$ is a separately continuous $\mathsf{TS}_{cnw}$-factorizable
function, there exist spaces $X_{*}$ and $Y_{*}$ with a countable network,
continuous mappings $g_{X}:X\to X_{*}$, $g_{Y}:Y\to Y_{*}$ and a separately
continuous function $h:X_{*}\times Y_{*}\to\mathbb{R}$, such that
$f=h\circ(g_{X}\times g_{Y})$. Then

$\psi:X_{*}\to C_{p}(Y),\ \psi(x_{*},y)=h(x_{*},g_{Y}(y))$

is continuous and $\varphi=g_{X}\circ\psi$. Then $\varphi(X)$ is a continuous
image of $X_{*}$ and hence $\varphi(X)$ has a countable network. ∎

######  Proposition 2.

(1) The product $\mathsf{L}_{[A]}\times\mathsf{R}_{[B]}$ is not Lindelöf, has
a discrete Suslin number $2^{\omega}$, and is not separately
$\mathsf{TS}_{cnw}$-factorizable. (2) The product
$\mathsf{L}_{(A]}\times\mathsf{R}_{[B)}$ is separable, contains a discrete
closed subset of the cardinality $2^{\omega}$, is not normal, and is not
separately $\mathsf{TS}_{cnw}$-factorizable.

###### Proof.

Denote $D=\\{(x,x)\,:\,(x,x)\in\mathsf{L}\times\mathsf{R}\\}=\\{(x,x)\,:\,x\in
C\\}$.

(1) The family $\\{\\{p\\}\,:\,p\in D\\}$ is a discrete family of open sets in
$\mathsf{L}_{[A]}\times\mathsf{R}_{[B]}$. Let us define the function
$f:\mathsf{L}\times\mathsf{R}\to\mathbb{R}$,

$f(x,y)=\begin{cases}0,&\text{if }y\geq x,\\\ 1,&\text{if }y<x.\end{cases}$

Let us put

$\varphi:\mathsf{L}\to C_{p}(\mathsf{R}),\ \varphi(x)(y)=f(x,y).$

Then

$\varphi(x)(y)=\begin{cases}0,&\text{if }y\geq x,\\\ 1,&\text{if
}y<x.\end{cases}$

The uncountable space $\varphi(C)$ embeds in the Songefrey line and,
therefore, $\varphi(C)$ and $\varphi(\mathsf{L})$ do not have a countable
network.

The function $f$ is continuous with respect to the topology
$\mathsf{L}_{[A]}\times\mathsf{R}_{[B]}$. Since $\varphi(\mathsf{L})$ does not
have a countable network, it follows from the Lemma 1 that the function $f$ is
not separately $\mathsf{TS}_{cnw}$-factorizable.

(2) The topologies $\mathsf{L}_{(A]}$ and $\mathsf{R}_{[B)}$ are weaker than
the Songegfrey topology, and the Songegfrey line is a hereditarily separable
space. Hence the spaces $\mathsf{L}_{(A]}$, $\mathsf{R}_{[B)}$ and
$\mathsf{L}_{(A]}\times\mathsf{R}_{[B)}$ are separable.

The set $D$ is a discrete closed subset of
$\mathsf{L}_{(A]}\times\mathsf{R}_{[B)}$ and $|D|=2^{\omega}$. Theorem 2
implies that the product $\mathsf{L}_{(A]}\times\mathsf{R}_{[B)}$ is not
normal. The function $f$ is continuous with respect to the topology
$\mathsf{L}_{(A]}\times\mathsf{R}_{[B)}$. Since $\varphi(\mathsf{L})$ does not
have a countable network, it follows from the Lemma 1 that the function $f$ is
not separately $\mathsf{TS}_{cnw}$-factorizable. ∎

######  Proposition 3.

Let $\mathsf{T}$ be a class of spaces closed with respect to taking subspaces,
$\mathsf{PT}$ is the class of paratopological groups in $\mathsf{T}$ and
$\mathsf{ST}$ is the class of semitopological groups in $\mathsf{T}$. Let $G$
and $H$ be groups with topology.

  1. (1)

If the product of groups $G\times H$ is $\mathsf{PT}$-factorizable, then the
product of spaces $G\times H$ is $\mathsf{T}$-factorizable.

  2. (2)

If the product of groups $G\times H$ is $\mathsf{ST}$-factorizable, then the
product of spaces $G\times H$ is separately $\mathsf{T}$-factorizable.

###### Proof.

Let $f:G\times H\to\mathbb{R}$ be a continuous function.

(1) Since the group $G\times H$ is $\mathsf{PT}$-factorizable, then there
exists a paratopological group $S\in\mathsf{T}$, a continuous homomorphism
$g:G\times H\to S$, and a continuous function $h:S\to\mathbb{R}$, so that
$f=h\circ g$. Let $G_{*}=g(G)$, $H_{*}=g(H)$, $g_{G}=\left.g\right|_{G}$,
$g_{H}=\left.g\right|_{H}$,

$h_{*}:G_{*}\times H_{*}\to\mathbb{R},\ (x_{*},y_{*})\mapsto h(x_{*}y_{*}).$

Then the function $h$ is continuous and $f=h_{*}\circ(g_{G}\times g_{H})$.

(2) Since the group $G\times H$ is $\mathsf{ST}$-factorizable, then there
exists a semitopological group $S\in\mathsf{T}$, a continuous homomorphism
$g:G\times H\to S$, and a continuous function $h:S\to\mathbb{R}$, so that
$f=h\circ g$. Let $G_{*}=g(G)$, $H_{*}=g(H)$, $g_{G}=\left.g\right|_{G}$,
$g_{H}=\left.g\right|_{H}$,

$h_{*}:G_{*}\times H_{*}\to\mathbb{R},\ (x_{*},y_{*})\mapsto h(x_{*}y_{*}).$

Then the function $h$ is separately continuous and $f=h_{*}\circ(g_{G}\times
g_{H})$. ∎

######  Corollary 1.

Let $G$ and $H$ be groups with topology. If the product of groups $G\times H$
is $\mathsf{SG}_{cnw}$-factorizable, then the product of spaces $G\times H$ is
separately $\mathsf{TS}_{cnw}$-factorizable.

######  Corollary 2.

Let $G$ and $H$ be groups with topology. Assume that one of the following
conditions is true:

  1. (1)

the product of groups $G\times H$ is $\mathsf{PG}_{sm}$-factorizable;

  2. (2)

the groups $G$ and $H$ are paratopological groups, and the group $G\times H$
is $\mathbb{R}$-factorizable.

Then the product of spaces $G\times H$ is $\mathbb{R}$-factorizable.

######  Proposition 4.

Let $\mathsf{T}$ be a class of spaces. Let $X$ and $Y$ be topological spaces,
$X^{\prime}$ retract $X$ and $Y^{\prime}$ retract $Y$. If the product $X\times
Y$ is (separately) $\mathsf{T}$-factorizable, then the product
$X^{\prime}\times Y^{\prime}$ is (separately) $\mathsf{T}$-factorizable.

###### Proof.

Let $r_{X}:X\to X^{\prime}$ and $r_{Y}:Y\to Y^{\prime}$ be retractions, the
function $f^{\prime}:X^{\prime}\times Y^{\prime}\to\mathbb{R}$ is continuous.
We put $f=f^{\prime}\circ(r_{X}\times r_{Y})$. Since the product $X\times Y$
is (separately) $\mathsf{T}$-factorizable, there exist spaces
$X_{*},Y_{*}\in\mathsf{T}$, continuous mappings $g_{X}:X\to X_{*}$,
$g_{Y}:Y\to Y_{*}$ and (separately) a continuous function $h:X_{*}\times
Y_{*}\to\mathbb{R}$, such that $f=h\circ(g_{X}\times g_{Y})$. Let us put
$g^{\prime}_{X}=\left.g_{X}\right|_{X^{\prime}}$ and
$g^{\prime}_{Y}=\left.g_{Y}\right|_{Y^{\prime}}$. Then
$f^{\prime}=h\circ(g^{\prime}_{X}\times g^{\prime}_{Y})$. Hence $f^{\prime}$
is a (separately) $\mathsf{T}$-factorizable function. ∎

######  Corollary 3.

Let $X$ and $Y$ be topological spaces, $X^{\prime}$ retract $X$ and
$Y^{\prime}$ retract $Y$. If the product $X\times Y$ is separately
$\mathsf{TS}_{cnw}$-factorizable, then the product $X^{\prime}\times
Y^{\prime}$ is separately $\mathsf{TS}_{cnw}$-factorizable.

Define groups

$\displaystyle\mathsf{{G}_{1}}$ $\displaystyle=B(\mathsf{L}_{[A]}),$
$\displaystyle\mathsf{{H}_{1}}$ $\displaystyle=B(\mathsf{R}_{[B]}),$
$\displaystyle\mathsf{{G}_{2}}$ $\displaystyle=B(\mathsf{L}_{(A]}),$
$\displaystyle\mathsf{{H}_{2}}$ $\displaystyle=B(\mathsf{R}_{[B)}).$

######  Theorem 5.

The topological groups $\mathsf{{G}_{1}}$, $\mathsf{{H}_{1}}$,
$\mathsf{{G}_{2}}$, $\mathsf{{H}_{2}}$ are Lindelöf to countable degree.

  1. (1)

The product $\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ is not a
$\mathsf{SG}_{cnw}$-factorizable (and then not $\mathbb{R}$-factorizable)
group and has a discrete Suslin number $2^{\omega}$.

  2. (2)

The product $\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ is not a
$\mathsf{SG}_{cnw}$-factorizable (and then not $\mathbb{R}$-factorizable) non-
normal separable group and has extend $2^{\omega}$.

###### Proof.

It follows from Proposition 1 and Theorem 1 that the groups
$\mathsf{{G}_{1}}$, $\mathsf{{H}_{1}}$, $\mathsf{{G}_{2}}$, $\mathsf{{H}_{2}}$
are Lindelöf to countable degree and $\mathsf{L}_{[A]}$, $\mathsf{R}_{[B]}$,
$\mathsf{L}_{(A]}$, $\mathsf{R}_{[B)}$ is a retract of $\mathsf{{G}_{1}}$,
$\mathsf{{H}_{1}}$, $\mathsf{{G}_{2}}$, $\mathsf{{H}_{2}}$, respectively. From
Proposition 2, Corollaries 1 and 3 it follows that the groups
$\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ and
$\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ are not
$\mathsf{SG}_{cnw}$-factorizable.

(1) Since $\mathsf{L}_{[A]}\times\mathsf{R}_{[B]}$ is a retract of
$\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$, then from Proposition 2(1) it
follows that $\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ has a discrete Suslin
number $2^{\omega}$.

(2) Since $\mathsf{L}_{(A]}\times\mathsf{R}_{[B)}$ embeds in
$\mathsf{{G}_{2}}\times\mathsf{{H}_{2}}$ and generates this group, from
Proposition 2(2) implies that $\mathsf{{G}_{1}}\times\mathsf{{H}_{1}}$ is
separable, has an extend $2^{\omega}$, and is not normal. ∎

##  4 $\mathbb{R}$-factorizability and dimension

######  Proposition 5.

If a product of spaces $X\times Y$ is $\mathbb{R}$-factorizable, then $X\times
Y$ is rectangular.

###### Proof.

Let ${\mathcal{U}}=\\{U_{1},...,U_{n}\\}$ is a ﬁnite cozero cover of $X\times
Y$. For $i=1,...,n$, we fix a continuous function $f_{i}$ on $X\times Y$ such
that $X\times Y\setminus U_{i}=f_{i}^{-1}(0)$. Since $X\times Y$ is
$\mathbb{R}$-factorizable, for $i=1,...,n$, there exists separable metrizable
spaces $X_{i}$ and $Y_{i}$, continuous mappings $p_{i}:X\to X_{i}$, and
$q_{i}:Y\to Y_{i}$ and a continuous function $h_{i}:X_{i}\times
X_{i}\to\mathbb{R}$, such that $f_{i}=h_{i}\circ(p_{i}\times q_{i})$. Let us
put

$\displaystyle X_{*}$ $\displaystyle=\prod_{i=1}^{n}X_{i},$ $\displaystyle
Y_{*}$ $\displaystyle=\prod_{i=1}^{n}Y_{i},$ $\displaystyle p$
$\displaystyle=\operatornamewithlimits{\mathchoice{\vbox{\hbox{\huge$\Delta$}}}{\vbox{\hbox{\Large$\Delta$}}}{\mathrm{\Delta}}{\mathrm{\Delta}}}_{i=1}^{n}p_{i},$
$\displaystyle q$
$\displaystyle=\operatornamewithlimits{\mathchoice{\vbox{\hbox{\huge$\Delta$}}}{\vbox{\hbox{\Large$\Delta$}}}{\mathrm{\Delta}}{\mathrm{\Delta}}}_{i=1}^{n}q_{i}.$

Then $X_{*}$ and $Y_{*}$ are separable metrizable spaces and $U_{i}=(p\times
q)^{-1}(V_{i})$ for $i=1,...,n$ for some open $V_{i}\subset X_{*}\times
Y_{*}$. There is a countable family $\gamma_{i}$ consisting of rectangular
open sets of the form $U\times V$, such that $V_{i}=\bigcup\gamma_{i}$. Then
the family

$\mu=\bigcup_{i=1}^{n}\\{(p\times q)^{-1}(U\times V)\,:\,U\times
V\in\gamma_{i}\\}$

is countable (and even more so $\sigma$-locally ﬁnite), consists of
rectangular cozero sets, and is reﬁnement of ${\mathcal{U}}$. Therefore
$X\times Y$ is rectangular. ∎

Corollary 2, Proposition 5, and Theorem 4 imply the following assertions.

######  Proposition 6.

Let $G$ and $H$ be groups with topology and let $G\times H$ be
$\mathsf{PG}_{sm}$-factorizable. Then

$\operatorname{dim}_{0}G\times
H\leq\operatorname{dim}_{0}G+\operatorname{dim}_{0}H.$

######  Theorem 6.

Let $G$ and $H$ be (para)topological groups and the (para)topological group
$G\times H$ be $\mathbb{R}$-factorizable. Then

$\operatorname{dim}_{0}G\times
H\leq\operatorname{dim}_{0}G+\operatorname{dim}_{0}H.$

In [7] the strongly zero-dimensional topological groups $\mathsf{{G}_{3}}$ and
$\mathsf{{H}_{3}}$ are constructed, such that the group $\mathsf{{G}_{3}}$ is
Lindelöf, the group $\mathsf{{H}_{3}}$ is separable metrizable111Group
$\mathsf{{H}_{3}}$ in the original paper with a countable network, but with a
slight modification of the construction it is possible to make
$\mathsf{{H}_{3}}$ become metrizable separable, and
$\operatorname{dim}_{0}\mathsf{{G}_{3}}\times\mathsf{{H}_{3}}>0$. There are
two arguments why the topological group
$\mathsf{{G}_{3}}\times\mathsf{{H}_{3}}$ is not $\mathbb{R}$-factorizable.
First, because

$\operatorname{dim}_{0}\mathsf{{G}_{3}}\times\mathsf{{H}_{3}}>0=\operatorname{dim}_{0}\mathsf{{G}_{3}}+\operatorname{dim}_{0}\mathsf{{H}_{3}},$

then the theorem 6 implies that $\mathsf{{G}_{3}}\times\mathsf{{H}_{3}}$ is
not $\mathbb{R}$-factorizable. Second, since the group
$\mathsf{{G}_{3}}\times\mathsf{{H}_{3}}$ is zero-dimensional and is not
strongly zero-dimensional, Theorem 3(2) implies that
$\mathsf{{G}_{3}}\times\mathsf{{H}_{3}}$ is not $\mathbb{R}$-factorizable.

In [5], topological groups $\mathsf{{G}_{4}}$ and $\mathsf{{H}_{4}}$ are
constructed with the same properties as the groups $\mathsf{{G}_{3}}$ and
$\mathsf{{H}_{3}}$ and, additionally, the following condition is satisfied:
the group $\mathsf{{G}_{4}}\times\mathsf{{H}_{4}}$ embeds closed in some
strongly zero-dimensional group. The first and second arguments also work for
the group $\mathsf{{G}_{4}}\times\mathsf{{H}_{4}}$, hence
$\mathsf{{G}_{4}}\times\mathsf{{H}_{4}}$ is not $\mathbb{R}$-factorizable. In
[5] a third argument was found why the group
$\mathsf{{G}_{4}}\times\mathsf{{H}_{4}}$ is not $\mathbb{R}$-factorizable.
Thirdly, $\mathsf{{G}_{4}}\times\mathsf{{H}_{4}}$ is not strongly zero-
dimensional and is embedded in some strongly zero-dimensional group, hence, by
Theorem 3(1), $\mathsf{{G}_{4}}\times\mathsf{{H}_{4}}$ is not
$\mathbb{R}$-factorizable.

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# Alljoined - A dataset for EEG-to-Image decoding

Jonathan Xu * 1 2, Bruno Aristimunha * 3 4, Max Emanuel Feucht * 1 5,
Emma Qian †1, Charles Liu †1 2, Tazik Shahjahan †1 2, Martyna Spyra †1, Steven
Zifan Zhang1,6,
Nicholas Short1,2, Jioh Kim1,6, Paula Perdomo1,6, Ricky Renfeng Mao1,2,
Yashvir Sabharwal1,
Michael Ahedor1, Moaz Shoura6, Adrian Nestor6

###### Abstract

We present Alljoined, a dataset built specifically for EEG-to-Image decoding.
Recognizing that an extensive and unbiased sampling of neural responses to
visual stimuli is crucial for image reconstruction efforts, we collected data
from 8 participants looking at 10,000 natural images each. We have currently
gathered 46,080 epochs of brain responses recorded with a 64-channel EEG
headset. The dataset combines response-based stimulus timing, repetition
between blocks and sessions, and diverse image classes with the goal of
improving signal quality. For transparency, we also provide data quality
scores. We publicly release the dataset and all code at
https://linktr.ee/alljoined1.

††* equal contribution. † core contribution. 1AllJoined 2The University of
Waterloo 3Université Paris-Saclay, Inria TAU, CNRS, LISN 4Federal University
of ABC 5Vrije Universiteit Amsterdam 6University of Toronto

##  1 Introduction

In the fields of cognitive neuroscience and medical imaging, advancements in
deep learning have demonstrated unparalleled precision in decoding brain
activity [36, 20, 38, 37, 6]. Researchers have translated the intricate
patterns of brain activity during various cognitive processes by utilizing
neuroimaging modalities, such as functional Magnetic Resonance Imaging (fMRI)
and electroencephalography (EEG).

In this context, one particular area of interest is image reconstruction,
which involves the decoding of neural responses to visual stimuli, offering
insights into how the brain encodes and processes visual information [35, 37,
6, 24, 9, 10].

While fMRI has traditionally been the primary tool for image reconstruction
due to its excellent spatial resolution, the slow temporal components severely
delimit actual clinical usage. On the other hand, EEG is a medical modality
available in everyday clinical contexts with an excellent time resolution [33,
40, 39]. As neurons fire at millisecond scales, the high temporal resolution
provided by EEG is crucial for real-time monitoring of neural dynamics [45,
12, 17]. Additionally, EEG is portable, more accessible to set up, and much
more cost-effective than fMRI, making it suitable for real-world applications,
including brain-computer interfaces and clinical diagnostics.

The development of very large fMRI-to-image datasets has proven foundational
for recent breakthroughs in deep-learning image reconstruction projects.
Inspired by the need for such datasets in the EEG domain, we present
Alljoined, a novel, large-scale dataset covering a wide range of naturalistic
stimuli that allows for robust, generalizable image reconstruction efforts.
Our contributions are as follows:

  * •

We propose a stimulus presentation approach that tailors trial duration and
session and block repetitions to maximize the signal-to-noise (SNR) ratio.

  * •

We introduce a diverse dataset of EEG responses to 9k unique naturalistic
images for each of the eight participants, with 1k additional images shared
between participants.

  * •

We perform qualitative comparisons against current EEG-to-image datasets.

##  2 Related Work

###  2.1 EEG-to-Image Datasets

EEG-to-image datasets consist of EEG waveforms recorded while participants
watch visual stimuli, enabling the study of neural representations in the
brain. However, previous research on EEG-based image reconstruction has often
relied on datasets exhibiting severe limitations regarding acquisition design
or generalizability to naturalistic stimuli [48, 27, 39].

A popular EEG-image dataset is _Brain2Image_ [22], which consists of evoked
responses to a visual stimulus from distinct image classes. Each block
consists of stimuli corresponding only to a single image class. There are 40
classes, with 50 unique images in each class. This dataset has been criticized
for having no train-test separation during recording, block-specific stimuli
patterns, and lack of consistency across different frequency bands. These
factors can incorrectly boost model performance by giving extraneous proxy
information about the block rather than the actual image-specific brain
responses [28, 3]. Moreover, the limited number of classes may encourage
models to perform 40-way classification rather than image reconstruction.

Recent studies achieving impressive reconstruction results have relied on this
dataset [26, 5, 23], which may affect the validity of their results. As
recommended by [28, 3], the stimuli within each block in our dataset were
chosen randomly across a variety of natural images, effectively minimizing the
risk for block-class correlations.

The diversity of decoding stimuli further limits current EEG-based image
reconstruction datasets. While Brain2Image consists of 40 classes, several
studies utilize a dataset of visual imagery of characters and objects
belonging to only 10 different classes, ThoughtViz [46]. Both Brain2Image and
ThoughtViz fail to represent the continuous, diverse quality of naturalistic
stimuli. The same limitation applies to studies utilizing a severely limited
quantity of naturalistic stimuli. Approaches to EEG-based image reconstruction
derived from the ThoughtViz [32, 39], Brain2Image [26, 5, 23], or other
equally selective datasets [2, 48], may thus suffer from generalizing well to
diverse, real-world stimuli.

To account for the diverse and continuous nature of naturalistic images,
Alljoined consists of 1) 10,000 images per participant and 2) that belong to
at least one of 80 MS-COCO [30] object categories. Importantly, each MS-COCO
category is broader than a single object class (e.g. the _things_ category
includes car, skateboard, hat, etc.), and each image can belong to up to 5
classes [29].

There are also existing datasets that include naturalistic stimuli, but
compromise in other domains. The MindBigData initiative [47] captures a wide
sample of images from the ImageNet dataset, but is derived only from a single
individual, limiting the potential of training image reconstruction models
that generalize to other individuals. The THINGS-EEG1 [16] and THINGS-EEG2
[13] datasets were acquired using short image presentation times of 50 and 100
ms, and a stimulus onset asynchrony of 100 and 200 ms.

Although the rapid serial visual presentation (RSVP) [15] paradigm proposes
disentangling the temporal dynamics of visual processing and categorical
abstraction of non-target stimuli, it is not ideal for capturing cortical
image processing beyond early visual activity with low noise. We see that [41]
obtained the highest accuracy with their EEG-image classifier when focusing on
320-480 ms after stimulus onset, and [34] is able to extract relevant decoding
features even around 550 ms after stimulus onset. This suggests that while it
takes 50-120 ms for object recognition of a stimulus to register in the visual
cortex, a longer stimulus period is beneficial for accuracy on downstream
tasks. Alljoined consists of extensive data from n=8 participants, measured
with an inter-stimulus interval of 300 ms, which captures important hallmarks
in visual processing while maintaining a high presentation frequency [44, 16].

###  2.2 fMRI-Image Datasets

The recent development of large functional magnetic resonance imaging (fMRI)
datasets has enabled researchers to decode and reconstruct images observed by
humans with unprecedented accuracy.

The Brain, Object, Landscape Dataset (BOLD5000) [8] contains brain responses
from 4 human participants who viewed 5,254 images depicting natural scenes
from the Scene UNderstanding (SUN) [49], MS-COCO [30], and ImageNet datasets
[11]. Similarly, in the Generic Object Decoding Dataset (GOD) [21], 1,200
images from the ImageNet database were cropped and shown to 5 participants,
resulting in one of the first datasets to establish methods for decoding
generic object categories from brain activity.

The Natural Scenes Dataset (NSD) [4] consists of the brain responses of 8
human participants passively viewing 9,000–10,000 color natural scenes from
MS-COCO. This magnitudes-larger dataset has fueled leaps in reconstruction
accuracy seen in recent work like MindEye2 [38]. However, the adaptation of
such impressive achievements to real-life contexts is quite limited, as MRI
scanners are notoriously expensive and difficult to access.

##  3 Methods and Materials

###  3.1 Participants

We collected data from eight participants (six male, two female), with an
average age of 22 $\pm$ 0.64 years, all with normal or corrected-to-normal
vision, right-handed. All participants were healthy, with no neurocognitive
impairments, except 2 participants who reported a history of mental health
disorders (e.g. GAD, ADHD). Each participant provided informed consent. The
Research Ethics Board approved the procedures as suppressed for double-blind
review.

###  3.2 Stimuli

We use the same visual stimuli as what was shown in the fMRI Natural Scenes
Dataset (NSD) [4], consisting of $70,566$ images portraying everyday objects
and situations in their natural context. All NSD images are drawn from the MS-
COCO dataset [29], including annotations about objects and their corresponding
category contained in the image. Each image can contain more than one object
and more than one object category. These fine-grained object categories are
further grouped into _supercategories_ , each of which comprehensively
includes all related categories as defined subsets.

The current study uses a subset of the first $960$ images in the $1000$ images
shown across all participants in the NSD study. These images are drawn from
the _shared1000_ subset of the NSD dataset, which comprises 1000 specially
curated images that all participants in the original NSD study were presented
with [4]. Within this subset of the NSD dataset, the supercategory _person_
was most represented, occurring in 50.94% of all images, followed by animal
(23.54%) and vehicle (23.33%). The distribution of the supercategories is
shown in Figure 1.

Figure 1: Top 12 most frequently occurring _supercategories_ in our dataset.

###  3.3 Procedure

Images were displayed to participants over the course of multiple one-hour-
long sessions. Each session consisted of 16 blocks, wherein images in the
first 8 unique blocks were in the second 8 blocks.

The repeated blocks (e.g., blocks 1 & 8, 2 & 9, etc.) contained the same
stimuli but in a shuffled order to avoid sequence effects. Within each block,
120 images from the NSD dataset were presented twice, as well as 24 oddball
stimuli, amounting to 264 images per block.

Given the within-block and the between-block repetitions of NSD images, each
NSD image was presented 4 times to obtain a higher signal-to-noise ratio of
the corresponding brain activity. Within each trial, an image (NSD or oddball)
was presented for $300$ ms, followed by $300$ ms of a black screen; a white
fixation cross was visible on the screen throughout the entire trial.

At the end of each trial, an extra jitter time between 0-50 ms was added for
randomness. To ensure focus, participants were prompted to press the space bar
when two consecutive trials contained the same image. These oddball trials
occurred 24 times within each block; brain activity in response to oddballs
has been discarded from the dataset due to motion artifacts, EEG repetition
suppression, and other issues.

Figure 2: Schematic overview of the structure of trials, blocks, and sessions.
Each of the 120 block-specific NSD images is presented twice within each
block, and each of the 8 session-specific blocks is presented twice within
each session. Each participant performed two sessions on different days. Each
of the 10 sessions thus consists of 960 NSD images repeated four times within
and across blocks, totaling 9600 unique NSD images per participant.

###  3.4 Hardware Setup

We recorded data using a 64-electrode BioSemi ActiveTwo system, digitized at a
rate of $512$ Hz with 24-bit A/D conversion. The montage was arranged in the
International 10-20 System, and the electrode offset was kept below 40 mV. We
used a $22$ inch Dell monitor at a resolution of 1080p/60Hz to display the
visual stimulus. As depicted in Figure 3, the monitor was positioned centrally
and placed at a distance of $80$ cm to maintain a $3.5$° visual angle of
stimuli. We avoided larger angles to minimize the occurrence of gaze drift.

Figure 3: Experimental setup with monitor 80 cm from participant.

###  3.5 Pre-processing

Regarding the dataset pre-processing, we follow recent work on the importance
of separating the biomarkers from the central nervous and peripheral systems,
as described in [7], and applied the minimum necessary steps. This dataset was
pre-processed entirely using the MNE-Python library [14].

#### Filtering

Initially, we applied the band-pass filtering with a low frequency of $0.5$ Hz
and a high frequency of $125$ Hz with overlap-add finite impulse response
filtering, with range based on [43]. We then apply a notch filter at 60Hz to
eliminate power line noise.

#### Independent Component Analysis (ICA)

Next, we performed an ICA decomposition using a FastICA model [18, 1] to
separate non-gaussian biological artifacts noise from the signal source. We
used a decomposition that retained 95% of the variance and excluded ICs
corresponding to eye blinks on the raw data.

#### Epoching

We segment our data into intervals starting at $-50$ ms onset stimulus and
ending at the end of each trial at $600$ ms, as the stimulus section describes
in Section 3.2. We discard the jitter time between trials.

#### Artifact correction

We use a peak-to-peak threshold for each sensor to identify low-quality trials
with the Autoreject algorithm [19] to automatically determine if a trial
should be (i) repaired by interpolation of neighboring sensors or (ii)
excluded from further analysis. Autoreject performs grid search to determine
appropriate values for $\rho$, the number of channels to interpolate, and
$\kappa$, the percentage of channels that must agree as a fraction of total
channels for consensus. By looking at the number of erroneous sensors per
trial, this approach allows correction on a per-trial basis instead of
applying a single global threshold to all trials. A mean of 130.75 epochs was
dropped per trial, with a standard deviation of 260.44.

#### Baseline correction

Finally, we re-reference our channels using an average reference scheme,
before applying a baseline correction window from $-50$ ms to $0$ ms relative
to stimulus onset, following [42] recommendation for ERP baseline. The epoch
data subtracts the average activation during the baseline interval to remove
noise from the signal.

##  4 Analysis

(a) First session of the fifth participant

(b) All sessions across all participants

Figure 4: EEG topographic maps and corresponding signals at all 64 electrodes
averaged over a) 3823 events for the fifth participant (left) and b) across
all sessions for all participants (43070 events) in the Alljoined1 dataset
(right), highlighting individual and common brain activity patterns associated
with image presentation.

###  4.1 ERP Analysis

The distribution of event-related potentials (ERPs) across all 64 channels is
displayed for a single session of one participant and averaged over all
participants and sessions in Figure 5. We observe a strong consistent rise in
activity beginning after 150 ms, with a peak between 250 and 300 ms. Note that
this latency aligns with the timing parameters of our experimental design,
which involves a 300ms presentation followed by a 300ms rest period, with an
additional 0-50ms jitter. Both participant- and cohort-level activity exhibits
a sustained high level of activity up until 500 ms after stimulus onset, where
a consistent dip in activity is observed for both the single participant as
well as the whole cohort.

Topographies of activation additionally reveal a strong concentration of
positive activation at occipital, parietal, and partly temporal electrodes and
a consistently negative activation at central and frontal areas. This
topographical distribution was stable across the duration of the ERP and
corresponded well between the single participant and the cohort. Given the
strong peak in activity at the occipital and parietal areas, we further
investigated the distribution of ERPs across individual participants and
sessions at the occipital and parietal electrodes, as displayed in figure 5.
While the magnitude of activation differs between participants, we conclude a
by-and-large consistent activation pattern across participants and sessions.

Figure 5: ERPs averaged over occipital and parietal electrodes for all
participants and sessions. Shaded areas around the grand average ERP indicate
standard deviations at all timepoints.

###  4.2 SNR Analysis

Figure 6: Signal to Noise Rate (SNR) averaged across each session, across each
block, and within each block for participant 5. Left: SNR for only the first
session 1, Right: SNR for all sessions.

The Signal-to-Noise Ratio (SNR) serves as a pivotal metric in evaluating the
efficacy of our dataset. To ascertain the SNR, we employ the Standard
Measurement Error (SME) as a gauge for noise assessment [31]. The SME is
determined by calculating the standard deviation of the aggregated waveform
average for each event type across all trials and then dividing this by the
square root of the event type’s occurrence count. The SNR is subsequently
derived by dividing the mean signal values by their corresponding SME.

Figure 6 compares the average SNR across all events in a single session for
participant 5 with the average SNR across all events for both sessions
concatenated. We see that the SNR is noticeably lower in the multi session
graph. This is due to the increased number of repetitions for a given event at
different timepoints. This attributes to a disproportionately higher standard
deviation value and consequently a higher SME and lower SNR. However, that is
not to say that the quality of the data is worse. It actually reflects more
accurate SNR values as there are more data points, distributed across
different sessions. It is also observed that the single sessions graph is more
volatile across time, demonstrating a greater variance in SNR values which are
captured by have a less accurate metric for noise with less trials to average
between. This fluctuation underscores the limited accuracy of noise metrics
derived from fewer trials, thus highlighting the critical importance of
incorporating repeated measures across sessions or blocks for robust SNR
evaluation.

Furthermore, we observe a strong SNR increase 150 ms after stimulus onset.
Note that this increase in SNR exhibits the same spiking timing we see in our
earlier topographic maps and averaged ERP graphs, suggesting that meaningful
activity starts to surface with a considerable delay with respect to stimulus
onset.

###  4.3 Discussion

The ERP and topography analyses, as well as our analysis of the SNR reveal and
reinforce several benefits of the acquired dataset, with regard to the
stimulus design and timing.

  1. 1.

Stimulus Duration and Stimulus Onset Asynchrony: A 300 ms presentation window
as well as a subsequent 300 ms rest period allows the capture of both early
and late cognitive processes, as evidenced by the single subject peaks at
around 262 ms up to 479 ms in Figure 6 a), and the averaged peaks at 293 ms
and 521 ms in Figure 6 b), respectively. The duration of 300 ms for image
presentation is sufficient for the brain to engage in both perceptual encoding
and initial stages of memory processing, which may not be as effectively
captured with shorter presentation times. The subsequent 300ms rest period
provides a window to measure the brain’s higher-level visual and semantic
response to the stimuli. The whole ERP thus not only reflects the initial
feed-forward transfer of sensory information to visual cortical areas but also
the subsequent recurrent interactions involved in attention and semantic
analysis, that unfold over hundreds of milliseconds after stimulus onset. The
relevance of longer presentation times and longer stimulus-onset asynchrony is
additionally supported by the sustained ERP activation presented in Figure 5,
as well as the latency of SNR increase and peak in Figure 6.

  2. 2.

Comparison with Prior Studies: Presentation times of only 100 ms, or stimulus
onset asynchronies of only 200 ms fail to capture the rich neural dynamics
associated with image processing, involving both lower and higher level
processing. In THINGS EEG2 [13], with a shorter 100ms presentation time
followed by 100ms of rest, the stimulus exposure may have been insufficient to
elicit the full range of cognitive processes to occur. The limited time window
could explain the lesser degree of neural activity in the corresponding time
window. Similarly, THINGS EEG1 [16] employed a shorter 50ms presentation
window followed by 50ms rest, which, while suitable for examining the earliest
stages of sensory processing in the visual cortex, likely precluded the phases
of the cognitive processes that unfold over a longer period. This includes
higher-order mechanisms such as selective attention, working memory updating,
and retrieval of semantic associations from long-term memory stores [25].

  3. 3.

Phase Locking Mitigation: The inclusion of a jitter ranging from 0-50ms helps
mitigate phase locking, a phenomenon where the participant’s alpha-wave
activity becomes synchronously aligned with the pattern of the stimuli after
repeated presentations.

  4. 4.

Anticipatory Bias Minimization: Additionally, the jitter prevents the
participants from predicting the exact onset of the next stimulus, thus
reducing the potential for anticipatory neural activity that could confound
the data.

In conclusion, the design parameters of our experiment not only accommodate
the full breadth of neural responses captured in the critical window of 300ms
but also provide a comparative advantage over the shorter intervals used in
THINGS EEG2 and THINGS EEG1. The ideal timing of our experiment ensures the
acquisition of a comprehensive ERP waveform, contributing to a more nuanced
understanding of cognitive processes and neural dynamics.

##  5 Conclusion

We introduce Alljoined1, an EEG-image dataset that uses well-timed stimuli,
repetitions between blocks and sessions, and a wide distribution of natural
images to create an improved dataset for image decoding tasks. We believe that
its size, diversity, and quality will help promote work to better understand
the mechanisms of visual processing, and in decoding visual responses in
clinical and consumer BCI contexts.

Future Directions: We are eager to explore high-density EEG recording of
exclusively the occipital and parietal regions to better target regions of the
brain most responsive to visual stimuli. We are also interested in conducting
ablation studies on the generalizability of responses to imagined mental
imagery. We also believe there is great potential in exploring continuous data
collection in natural environments with a wireless headset.

Data availability: The preprocessed EEG dataset is available on OSF. Labels to
the corresponding NSD image IDs are included in the object files.

Code availability: The stimulus and preprocessing code to reproduce all the
results is available on anonymized GitHub here and here.

##  6 Acknowledgements

This work was sponsored by ZFellows. BA work was supported by DATAIA
Convergence Institute as part of the “Programme d’Investissement d’Avenir”,
(ANR-17-CONV-0003) operated by LISN-CNRS. We would like to thank Dr Sylvain
Chevallier for his valuable feedback on this manuscript.

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