11institutetext: Department of Biomedical Engineering, School of Biomedical
Engineering & Imaging Sciences, King’s College London, UK 22institutetext:
Data Science and Sharing Team, Functional Magnetic Resonance Imaging Facility,
National Institute of Mental Health, Bethesda, MD, 20892, USA 33institutetext:
Machine Learning Team, Functional Magnetic Resonance Imaging Facility,
National Institute of Mental Health, Bethesda, MD, 20892, USA 44institutetext:
Institute of Psychiatry, Psychology & Neuroscience, King’s College London, UK
55institutetext: Centre for Brain and Cognitive Development, Birkbeck College,
UK 66institutetext: Institute of Neurology, University College London, UK

# Brain Imaging Generation with Latent Diffusion Models

Walter H. L. Pinaya Equal contribution11 Petru-Daniel Tudosiu 0 11 Jessica
Dafflon 2233 Pedro F Da Costa 4455 Virginia Fernandez 11 Parashkev Nachev 66
Sebastien Ourselin 11 M. Jorge Cardoso 11

###### Abstract

Deep neural networks have brought remarkable breakthroughs in medical image
analysis. However, due to their data-hungry nature, the modest dataset sizes
in medical imaging projects might be hindering their full potential.
Generating synthetic data provides a promising alternative, allowing to
complement training datasets and conducting medical image research at a larger
scale. Diffusion models recently have caught the attention of the computer
vision community by producing photorealistic synthetic images. In this study,
we explore using Latent Diffusion Models to generate synthetic images from
high-resolution 3D brain images. We used T1w MRI images from the UK Biobank
dataset (N=31,740) to train our models to learn about the probabilistic
distribution of brain images, conditioned on covariables, such as age, sex,
and brain structure volumes. We found that our models created realistic data,
and we could use the conditioning variables to control the data generation
effectively. Besides that, we created a synthetic dataset with 100,000 brain
images and made it openly available to the scientific community.

###### Keywords:

Synthetic data Diffusion models Generative models Brain Imaging.

##  1 Introduction

Deep neural networks fuelled several ground-breaking advancements in areas
such as natural language processing and computer vision, where part of these
improvements was attributed to the large amount of rich data used to train
these networks, with some public datasets reaching millions of images and text
sentences [8, 26]. During the same period, medical image analysis also made
remarkable breakthroughs by applying deep neural networks to solve tasks such
as segmentation, structure detection, and computer-aided diagnosis (detailed
review available at [20, 27]). However, one current limitation of medical
imaging projects is the lack of availability of large datasets. Medical data
are costly and laborious to collect, and privacy concerns create challenges to
data sharing by restricting publicly available medical datasets to up to a few
thousand examples. This limitation creates a bottleneck on models’
generalizability and hampers the rate at which cutting-edge methods are
deployed in the clinical routine.

Generating synthetic data with privacy guarantees provides a promising
alternative, allowing meaningful research to be carried out at scale [15, 14,
33]. Together with traditional data augmentation techniques (e.g., geometric
transformations), these synthetic data could complement real data to
dramatically increase the training set of machine learning models. Generative
models learn the probability density function underlying the data they are
trained on and can create realistic representations of examples which are
different from the ones present in the training data by sampling from the
learned distribution. However, generating meaningful synthetic data is not
easy, especially when considering complex organs like the brain.

Nowadays, Generative Adversarial Networks (GANs) have been applied in various
fields to create synthetic images, producing realistic and clear images and
achieving impressive performance [7, 32]. In the medical field, for example,
[17] combine variational autoencoders with GANs to generate various modalities
of whole brain volumes from a small training set and achieved a better
performance compared to several baselines. However, since their study resized
the images to a small volume before training, with a size of 64 × 64 × 64
voxels, their synthetic medical images did not replicate many essential finer
details. In addition, due to the prevalence of 3D high-resolution data in the
field, researchers tend to have their models restrained by the amount of GPU
memory available. To mitigate this problem, [31] proposed a 3D GAN with a
hierarchical structure which is able to generate a low-resolution version of
the image and anchor the generation of high-resolution sub-volumes on it. With
this approach, the authors were able to generate impressive realistic 3D
thorax CT and brain MRI with resolutions up to 256 × 256 × 256 voxels. Despite
generating great interest, GANs still come with inherent challenges, such as
being notoriously unstable during training and failing to converge or to
capture the variability of the generated data due to mode collapse issues
[16].

Recently, diffusion models caught the attention of the machine learning
community by showing promising results when synthesizing natural images. They
have rivalled GANs in sample quality [9] while building upon a solid
theoretical foundation. Not only have they reported impressive photorealism
unconditioned images, but they have also been used to create images
conditioned in classes and text sentences (using techniques like classifier-
free guidance [13]), with exceptional results on models like Latent Diffusion
Models [23], DALLE 2 [22], and Imagen [25].

In this study, we used diffusion models to create synthetic MRI images of the
adult human brain. For that, we used 31,740 training images from the UK
Biobank [30]to train our models. In order to efficiently scale the application
of diffusion models to these high-resolution 3D data, we combined our
diffusion models with compression models following the architecture of Latent
Diffusion Models (LDM) [23]. Furthermore, we conditioned the image generations
on age, gender, ventricular volume, and brain volume relative to the
intracranial volume in order to generate realistic examples of brain scans
with specific covariate values. We compared our synthetic images to state-of-
the-art methods based on GANs, and we made our synthetic dataset comprising
100,000 brain images publicly available to the scientific community.

Figure 1: Real and synthetic samples of head MRI generated using VAE-GAN,
LSGAN, LDM and LDM+DDIM.

##  2 Methods

###  2.1 Datasets and Image Preprocessing

In this study, we used images from the UK Biobank (UKB) [30] to train our
generative models. The UKB is a study that aims to follow the health and well-
being of volunteer participants across the United Kingdom. Here, we used an
early release of the project’s data comprising 31,740 participants with T1w
images. The dataset consists of healthy individuals aged between 44 and 82
years with average age of 63.6 $\pm$ 7.5 years (average $\pm$ SD) and 14,942
male subjects (47%). In our experiments, we also conditioned for the volume of
ventricular cerebrospinal fluid (min-max: 6995.68 - 171375.0 $mm^{3}$; UKB
Data-Field 25004) and brain volume normalised for head size (min-max: 1144240
- 1793910 $mm^{3}$; UKB Data-Field 25009). All variables used for model
conditioning were normalised using min-max normalisation before feeding them
to our models.

For the image pre-processing, we used
UniRes111https://github.com/brudfors/UniRes [2, 3] to perform a rigid body
registration to a common MNI space. The final images had 1 $mm^{3}$ as voxel
size, and we cropped the image to obtain a volume of the head measuring 160 ×
224 × 160 voxels.

###  2.2 Generative models

In our experiments, we used LDMs, which combine the use of autoencoders to
compress the input data into a lower-dimensional latent representation with
the generative modelling properties of diffusion models. The compression model
was an essential step to allow us to scale to high-resolution medical images.
We trained the autoencoder with a combination of L1 loss, perceptual loss
[34], a patch-based adversarial objective [10], and a KL regularization of the
latent space. The encoder maps the brain image to a latent representation with
a size of 20 × 28 × 20. After training the compression model, the latent
representations of the training set are used as input to the diffusion model.
Diffusion models [12, 28] are generative models that convert Gaussian noise
into samples from a learned data distribution via an iterative denoising
process. Given a latent representation of an example from our training set,
the diffusion process gradually destroys the structure of the data via a fixed
Markov chain over 1000 steps by adding Gaussian noise using a fixed linear
variance schedule. The reverse process is also modelled as a Markov chain
which learns to recover the original input from the noisy one. We conditioned
our models according to age, gender, ventricular volume, and brain volume
relative to the intracranial volume. To perform this conditioning, we used a
hybrid approach combining the concatenation of the conditioning with the input
data and the use of cross-attention mechanisms, as proposed in [23]. Training
and model details are available in the supplementary material.

##  3 Experiments

###  3.1 Sampling Quality

Fig. 1 shows images generated using LDMs compared to real images and the
baselines (i.e., VAE-GAN [18] and LSGAN [21]). Unlike the baselines, we
observe that the LDMs were able to sample high-quality images with sharp
details and realistic textures. Besides that, training the diffusion models at
such a high resolution was much more stable and easier to achieve convergence
when compared to the GAN-based baselines. The baselines required a meticulous
design of the interaction between discriminator and generator, and they
presented problems of mode collapse, showcasing the problems of GAN-based
applied in such high-resolution 3D images. Therefore, we will refine and
expand our comparisons with other baselines in future works.

We also obtained quantitative metrics about the performance of our models. We
used the Fréchet Inception Distance (FID) [11]to measure how realistic the
synthetic images are. A small FID indicates that the distribution of the
generated images is similar to the distribution of the real images. The FID
was calculated using an approach similar to [31], where features were
extracted using a pre-trained Med3D [5]. We also measured the generation
diversity with the Multi-Scale Structural Similarity Metric (MS-SSIM) and
4-G-R-SSIM [24, 4, 19], where a value close to 0 suggests high diversity.
Here, we presented the MS-SSIM for comparison with previous studies, but we
also added the 4-G-R-SSIM as it has been shown to have better image quality
assessment. We compute the average values from 1000 sample pairs. Table 1
shows the quantitative results for different models used for the image
synthesis.

Table 1: Quantitative evaluation of the synthetic images on the UK Biobank. We used the Fréchet Inception Distance (FID) to verify how realistic are the images and the multi-scale structural similarity metric (MS-SSIM) and 4-G-R-SSIM to evaluate generation diversity. We used 50 timesteps when sampling our models with DDIM sampler. | FID $\downarrow$ | MS-SSIM $\downarrow$ | 4-G-R-SSIM $\downarrow$
---|---|---|---
LSGAN | 0.0231 | 0.9997 | 0.9969
VAE-GAN | 0.1576 | 0.9671 | 0.8719
LDM | 0.0076 | 0.6555 | 0.3883
LDM + DDIM | 0.0080 | 0.6704 | 0.3957
Real images | 0.0005 | 0.6536 | 0.3909

Recently, different methods have been proposed to speed up the reverse process
(e.g., Denoising Diffusion Implicit Models - DDIM), reducing by 10× 50× the
number of necessary reverse steps [29]. Using the DDIM sampler, we reduced the
number of timesteps from 1000 steps to only 50. This improved our sampling
time from an average of 142.3$\pm$1.6s per sample to 7.6$\pm$0.2 s per sample
@ NVIDIA TITAN RTX with minimum loss in performance (Table 1). Because of this
boost in processing time and a minimal performance loss, we are using the LDM
with the DDIM sampler for all the remaining analyses.

Figure 2: Conditioned sampling varying the ventricular volume and the brain
volume normalized by the intracranial volume. In both rows, we kept the other
variables constant.

###  3.2 Conditioning Evaluation

Using the hybrid conditioning approach [23], we were able to condition our
models and generate brain images where we can specify the age, sex,
ventricular volume, and brain volume. As we can observe in Fig. 2, our model
was able to learn representations conditioned on regional (i.e., ventricular
volume) and global (i.e., brain volume) volumes.

In order to quantitatively evaluate the conditioning, we used SynthSeg
222https://github.com/BBillot/SynthSeg [1] to measure the volumes of the
ventricles of 1000 synthetic brains. In this analysis, we measured the
combination of the left and right lateral ventricles and the left and right
inferior lateral ventricles. We then computed the Pearson correlation between
the obtained volumes and the inputted conditioning values. Using this
approach, we observed a high correlation coefficient of 0.972, which
demonstrates the effectiveness of conditioning on our model (Fig. 3).

Figure 3: Conditioning analysis. Left) Correlation between inputted
ventricular volume vs ventricular measured with SynthSeg. Right) Correlation
between inputted age and predicted brain age.

Additionally, we verified how well the effectiveness of conditioning brain
generation by age. To this end, we used discriminative models to perform the
task of brain age prediction, where we predict chronological age based on the
brain image. In our study, we used the 3D convolution neural network proposed
in [6], trained on the same training set used in the LDM training. After
training the model, we verify how well the predicted age approximated the
inputted age of the synthetic dataset. As shown in Fig. 3, our model presented
a high correlation between the inputted conditioning and the predicted age
(r=0.692).

Finally, we verified how our model extrapolates the conditioning variables for
values never shown during training. Fig. 4 presents samples where we used a
normalised ventricular value higher than 1; in this case, we can see
abnormally huge ventricles when using values of 1.5 and 1.9. If we use a
negative value (e.g., -0.5), an image without ventricles is generated.
Similarly, if we use negative values for the brain normalised for head size,
the brain exhibits signs of neurodegeneration, showing smaller volumes of
white and grey matter. These findings suggest that our models learned the
concepts behind these conditioning variables during training.

Figure 4: Extrapolating values of conditioning variables. During the training
of the models, the inputted values of the conditioning variables were scaled
between 0 and 1. In this experiment, we tried values outside of this range,
and we observed that our model could extrapolate the representation of brain
and ventricular volumes, showing that it learned the concept of these
variables.

###  3.3 Synthetic Dataset

We made a synthetic dataset of 100,000 human brain images generated by our
model openly available to the community. This dataset is available at
Academics
Torrents333https://academictorrents.com/details/63aeb864bbe2115ded0aa0d7d36334c026f0660b,
FigShare444https://figshare.com/, and HDRUK
Gateway555https://www.healthdatagateway.org/, together with the conditioning
information.

##  4 Conclusions

In our study, we were able to train diffusion models to effectively generate
synthetic brain images that replicate properties from the training images. As
it is the case with natural image generation, our diffusion models outperform
alternative GANs-based methods in an unconditioned scenario. Additionally, we
demonstrated how our methods could be conditioned on covariates such as age,
sex, and brain structure volumes to produce the expected representation. In
future works, we will develop models that use other scanning modalities as
conditioning, such as images and radiological reports. By making the synthetic
dataset openly available, this work also addresses one of the biggest
limitations in medical machine learning – the challenge of obtaining large
imaging datasets – while not posing threats to privacy infringements. In sum,
our results show that LDMs are promising models to be explored in medical
image generation.

####  4.0.1 Acknowledgements

WHLP and MJC are supported by Wellcome Innovations [WT213038/Z/18/Z]. PTD is
supported by the EPSRC Research Council, part of the EPSRC DTP, grant Ref:
[EP/R513064/1]. JD is supported by the Intramural Research Program of the NIMH
(ZIC-MH002960 and ZIC-MH002968). PFDC is supported by the European Union’s
HORIZON 2020 Research and Innovation Programme under the Marie Sklodowska-
Curie Grant Agreement No 814302. PN is supported by Wellcome Innovations
[WT213038/Z/18/Z] and the UCLH NIHR Biomedical Research Centre. This research
has been conducted using the UK Biobank Resource (Project number: 58292).

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# On some topological games involving networks

Leandro F. Aurichi, Maddalena Bonanzinga
and Davide Giacopello

###### Abstract

In these notes we introduce and investigate two new games called R-nw-
selective game and the M-nw-selective game. These games naturally arise from
the corresponding selection principles involving networks introduced in [7].

Keywords: Countable network weight, Menger space, Rothberger space, Menger
game, Rothberger game, M-separable space, R-separable space, $G_{fin}({\cal
D},{\cal D})$, $G_{1}({\cal D},{\cal D})$, M-nw-selective space, R-nw-
selective space.

AMS Subject Classification: 54D65, 54A25, 54A20.

##  1 Introduction

Throughout the paper we mean by “space”a topological Hausdorff space.
A family $\mathcal{N}$ of sets is called a network for $X$ if for every $x\in
X$ and for every open neighbourhood $U$ of $x$ there exists an element $N$ of
$\mathcal{N}$ such that $x\in N\subseteq U$;
$nw(X)=min\\{|{\mathcal{N}}|:{\mathcal{N}}$ is a network for $X\\}$ is the
network weight of $X$. A family $\mathcal{P}$ of open sets is called a
$\pi$-base for $X$ if every nonempty open set in $X$ contains a nonempty
element of $\mathcal{P}$; $\pi w(X)=min\\{|{\mathcal{P}}|:{\mathcal{P}}$ is a
$\pi$-base for $X\\}$ is the $\pi$-weight of $X$. It is known that $nw(X)\leq
w(X)$, where $w(X)$ denotes the weight of the space $X$, and in the class of
compact Hausdorff spaces $nw(X)=w(X)$ (see [2]).
$\delta(X)=\sup\\{d(Y):Y\hbox{ is a dense subset of }X\\}$, where $d(X)$
denotes the density of the space $X$.
Recall that for $f,g\in\omega^{\omega}$, $f\leq^{*}g$ means that $f(n)\leq
g(n)$ for all but finitely many $n$ (and $f\leq g$ means that $f(n)\leq g(n)$
for all $n\in\omega$). A subset $D\subseteq\omega^{\omega}$ is dominating if
for each $g\in\omega^{\omega}$ there is $f\in D$ such that $g\leq^{*}f$. The
minimal cardinality of a dominating subset of $\omega^{\omega}$ is denoted by
${\mathfrak{d}}$. The value of ${\mathfrak{d}}$ does not change if one
considers the relation $\leq$ instead of $\leq^{*}$ [9, Theorem 3.6]. $\cal M$
denotes the family of all meager subsets of $\mathbb{R}$. $cov(\cal M)$ is the
minimum of the cardinalities of subfamilies ${\cal U}\subseteq{\cal M}$ such
that $\bigcup{\cal U}=\mathbb{R}$. However, another description of the
cardinal $cov(\cal M)$ is the following. $cov(\cal M)$ is the minimum
cardinality of a family $F\subset\omega^{\omega}$ such that for every
$g\in\omega^{\omega}$ there is $f\in F$ such that $f(n)\not=g(n)$ for all but
finitely many $n\in\omega$ (see [3] and also [4, Theorem 2.4.1]). Thus if
$F\subset\omega^{\omega}$ and $|F|<cov(\cal M)$, then there is
$g\in\omega^{\omega}$ such that for every $f\in F$, $f(n)=g(n)$ for infinitely
many $n\in\omega$; it is often said that $g$ guesses $F$. Also if $\mathbb{P}$
is a countable poset and $\cal D$ is a family of dense sets of cardinality
strictly less than $cov({\cal M})$ then there exists a generic that meets all
the dense sets of the family [4, Section 3].

In [13, 15] a systematic approach was considered to describe selection
principles. Given two collections $\cal A$ and $\cal B$ of some particular
topological objects on a space $X$, Scheepers introduced the following
notation:

  * $S_{1}({\cal A},{\cal B})$

: For every sequence $({\cal U}_{n}:n\in\omega)$ of elements of $\cal A$ there
exists $U_{n}\in{\cal U}_{n}$, $n\in\omega$, such that
$\\{U_{n}:n\in\omega\\}$ belongs to $\cal B$.

  * $S_{fin}({\cal A},{\cal B})$

: For every sequence $({\cal U}_{n}:n\in\omega)$ of elements of $\cal A$ there
exists a finite subset ${\cal F}_{n}\in{\cal U}_{n}$, $n\in\omega$, such that
$\bigcup_{n\in\omega}{\cal F}_{n}$ belongs to $\cal B$.

If one denotes by ${\cal O}$ the family of all open covers of a space $X$ and
by ${\cal D}$ the family of all dense subsets of a space $X$, a space is said
to be Rothberger if it satisfies $S_{1}({\cal O},{\cal O})$, Menger if it
satisfies $S_{fin}({\cal O},{\cal O})$, R-separable if it has the property
$S_{1}({\cal D},{\cal D})$ and M-separable if has the property $S_{fin}({\cal
D},{\cal D})$.
$\delta(X)=\omega$ for every M-separable space $X$ [6] and if
$\delta(X)=\omega$ and $\pi w(X)<{\mathfrak{d}}$, then $X$ is M-separable (a
stronger version of this fact is estabilished in [15, Theorem 40]); moreover,
if $\delta(X)=\omega$ and $\pi w(X)<$ cov$({\cal M})$, then $X$ is R-separable
(a stronger version of this fact is estabilished in [15, Theorem 29]).
Every space having a countable base is R-separable, therefore M-separable.
However, not every space with countable network weight is M-separable. Hence
in [7] the authors asked under which conditions a space with countable network
weight must be M-separable and introduced and studied the following classes of
spaces.

######  Definition 1.1.

Let $X$ be a space with $nw(X)=\omega$.

  * •

$X$ is M-nw-selective if for every sequence $({\cal N}_{n}:n\in\omega)$ of
countable networks for $X$ one can select finite ${\cal F}_{n}\subset{\cal
N}_{n}$, $n\in\omega$, such that $\bigcup_{n\in\omega}{\cal F}_{n}$ is a
network for $X$.

  * •

$X$ is R-nw-selective if for every sequence $({\cal N}_{n}:n\in\omega)$ of
countable networks for $X$ one can pick $F_{n}\in{\cal N}_{n}$, $n\in\omega$,
such that $\\{F_{n}:n\in\omega\\}$ is a network for $X$.

In [7] it was proved that any R-nw-selective (M-nw-selective) space is both
Rothberger and R-separable (Menger and M-separable). See also [8] for more
details about these two properties.
Recall that topological games, introduced with a systematical structure in
[13, 15], are infinite games played by two different players, Alice and Bob,
on a topological space $X$ (see also [1]). We assume that the lenght (number
of innings) of the games is $\omega$ and the two players pick in each inning
some topological objects of a fixed space. The strategies of the two players
are a priori defined; they are some functions that take care of the game
history. At the end there is only one winner, so a draw is not allowed.
Playing a game $G$ on a space $X$ gives rise to two properties: “Alice has a
winning strategy in the game $G$ on $X$”; “Bob has a winning strategy in the
game $G$ on $X$”. Of course, since there is not draw, it is impossible for a
space to have both these properties, but it can happen that the negation of
both of them holds. In this case we say that the game $G$ is undeterminate on
the space $X$.
Given two families of topological objects $\cal A$ and $\cal B$, the
followings are two games associated to selection principles.

  * $G_{1}({\cal A},{\cal B})$

: is played according to the following rules.

    * –

for every $n\in\omega$ Alice chooses $A_{n}\in{\cal A}$;

    * –

Bob answers picking $b_{n}\in A_{n}$ for each $n\in\omega$;

    * –

the winner is Bob if $\\{b_{n}:n\in\omega\\}\in{\cal B}$, otherwise Alice
wins.

  * $G_{fin}({\cal A},{\cal B})$

: is played according to the following rules.

    * –

for every $n\in\omega$ Alice chooses $A_{n}\in{\cal A}$;

    * –

Bob answers picking a finite subset $B_{n}\subseteq A_{n}$ for each
$n\in\omega$;

    * –

the winner is Bob if $\bigcup\\{B_{n}:n\in\omega\\}\in{\cal B}$, otherwise
Alice wins.

The game $G_{1}({\cal O},{\cal O})$, called Rothberger game, is strictly
related to the Rothberger property. In the following we denote this game by
Rothberger$(X)$. The game $G_{fin}({\cal O},{\cal O})$, called Menger game, is
strictly related to the Menger property. In the following we denote this game
by Menger$(X)$. The game $G_{1}({\cal D},{\cal D})$ is strictly related to the
R-separarability, in the following we denote this game by R-separable$(X)$.
Similarly, the game $G_{fin}({\cal D},{\cal D})$ is strictly related to the
M-separability and in the following we denote this game M-separable$(X)$.
This games were largely studied and some important characterizations of “Alice
does not have a winning strategy”and “Bob has a winning strategy”have been
given [1, 15, 17, 18]. Despite this some questions are still open. We denote
by ${\sc Bob}\uparrow G$ on $X$, the fact that “Bob has a winning strategy in
the game $G$ on $X$”and by ${\sc Alice}\not\uparrow G$ on $X$, the fact that
“Alice does not have a winning strategy in the game $G$ on $X$”.

######  Remark 1.2.

In general the following implications hold.

  1. 1.

${\sc Bob}\uparrow G_{1}({\cal A},{\cal B})\implies{\sc Bob}\uparrow
G_{fin}({\cal A},{\cal B})$;

  2. 2.

${\sc Alice}\not\uparrow G_{1}({\cal A},{\cal B})\implies{\sc
Alice}\not\uparrow G_{fin}({\cal A},{\cal B})$;

  3. 3.

${\sc Bob}\uparrow G_{1}({\cal A},{\cal B})\implies{\sc Alice}\not\uparrow
G_{1}({\cal A},{\cal B})\implies S_{1}({\cal A},{\cal B})$;

  4. 4.

${\sc Bob}\uparrow G_{fin}({\cal A},{\cal B})\implies{\sc Alice}\not\uparrow
G_{fin}({\cal A},{\cal B})\implies S_{fin}({\cal A},{\cal B})$.

For some properties the last two implications of points 3 and 4 are, in fact,
characterizations, that is ${\sc Alice}\not\uparrow G({\cal A},{\cal B})\iff
S({\cal A},{\cal B})$.

In [12] it is proved that a space $X$ is Rothberger if, and only if, ${\sc
Alice}\not\uparrow$ Rothberger$(X)$. In [17, 11] it is proved that if $X$ is a
space in which each point is a $G_{\delta}$, Bob $\uparrow$ Rothberger$(X)$
if, and only if, $X$ is countable. Similar arguments are valid for the Menger
case: in [14, 16] it is proved that a space $X$ is Menger if, and only if,
Alice$\not\uparrow$ Menger$(X)$ and in [18] that if $X$ is a metrizable space,
Bob $\uparrow$ Menger$(X)$ if, and only if, $X$ is $\sigma$-compact. In [15]
it is proved that Bob$\uparrow$ R-separable$(X)$ if, and only if, $\pi
w(X)=\omega$ and, under CH, it is given an example of a R-separable space $X$
such that Alice $\uparrow$ R-separable$(X)$.

Two topological games $G$ and $G^{\prime}$ are called dual if both “${\sc
Alice}\uparrow G\iff{\sc Bob}\uparrow G^{\prime}$”and “${\sc Alice}\uparrow
G^{\prime}\iff{\sc Bob}\uparrow G$”hold. Sometimes this dual vision could be
useful to apply different techniques in demonstrations. For instance, the
Point-open game is the dual of the Rothberger game (see [11]), the Point-
picking game is the dual of $G_{1}({\cal D},{\cal D})$ (see [15]), the
Compact-open game is a possible dual of the Menger game (see [18]), but the
question about the hypotesis to add to let them be dual is still open.

In Section 2 we study the R-nw-selective game. We present a characterization
of the “Bob having a winning strategy”property and a sufficient condition for
“Alice not having a winning”strategy property. We also give a consistent
characterization, in terms of games, of the R-nw-selective property in the
class of spaces without isolated points, with countable netweight and weight
strictly less than $cov({\cal M})$. Moreover, we introduce the (Point,
Open)-Set game and we prove that it is a promising candidate to be the dual of
the R-nw-selective game.

In Section 3 we study the M-nw-selective game. We present, under some
consistent hypotesis, a sufficient condition for “Alice not having a winning
strategy”property and some necessary conditions for the “Bob having a winning
strategy”property. We also give a consistent characterization, in terms of
games, of the M-nw-selective property in the class of spaces with countable
netweight and weight strictly less than $\mathfrak{d}$.

##  2 The R-nw-selective game

######  Definition 2.1.

Let $X$ be a space with $nw(X)=\omega$. The R-nw-selective game, denoted by
R-nw-selective$(X)$, is played according to the following rules. Alice chooses
a countable network ${\cal N}_{0}$ and Bob answers picking an element
$N_{0}\in{\cal N}_{0}$. Then Alice chooses another countable network ${\cal
N}_{1}$ and Bob answers in the same way and so on for countably many innings.
At the end Bob wins if the set $\\{N_{n}:n\in\omega\\}$ of his selections is a
network.

Simultaneously we consider the possible dual version of the R-nw-selective
game.

######  Definition 2.2.

The (Point, Open)-Set game on a space $X$, denoted by PO-set$(X)$, is played
according to the following rules. Alice chooses a point $x_{0}$ and an open
set $U_{0}$ containing $x_{0}$. Then Bob picks $N_{0}$ a subset of $X$ such
that $x_{0}\in N_{0}\subseteq U_{0}$. The game go ahead in this way for every
$n\in\omega$ and Alice wins if the set $\\{N_{n}:n\in\omega\\}$ of Bob’s
choices is a network.

######  Proposition 2.3.

Let $X$ be a space. Bob $\uparrow$ R-nw-selective$(X)$ if, and only if, the
space $X$ is countable and second countable.

Let ${\mathbb{M}}$ be the collection of all countable networks of $X$. Let
$\sigma$ a winning strategy for Bob.
First we prove that the space is countable.
Claim 1.
$|\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}|\leq
1$.
Indeed, suppose to have two points, say $x$ and $y$, that are in all the
clousure of the possible answers to $\mathcal{N}$, for any
${\mathcal{N}}\in{\mathbb{M}}$. Since the space is $T_{1}$, there exist a
closed set $C_{x}$ containing $x$ and not $y$, and a closed set $C_{y}$
containing $y$ and not $x$. Pick a countable network $\mathcal{N}$. If
$N\in\mathcal{N}$ is such that $x\in N$, then $C_{x}\cap N\subset C_{x}$ and
if $N\in\mathcal{N}$ is such that $y\in N$, then $C_{y}\cap N\subset C_{y}$;
then it is possible to construct a countable network such that any elements of
it contains at most one of the points.
Claim 2. There exists a countable ${\mathbb{M}}^{\prime}\subset{\mathbb{M}}$
such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\sigma({\mathcal{N}})}=\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\cal
N})}$.
Indeed, if $\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\cal
N})}=\\{x\\}$ (it is the same if
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\cal
N})}=\emptyset$), the complements of all the closures form an open cover of
$X\setminus\\{x\\}$ (or $X$) and then, since having countable network implies
hereditary Lindelöfness, we can obtain a countable subcover of
$X\setminus\\{x\\}$ (or of the all space $X$).
Note that it is straightforward to prove the claims 1. and 2. for any inning.
Consider the following tree of possible evolution of the R-nw-selective game
on $X$. By claim 2. there exists
$({\mathcal{N}}_{\emptyset}^{n})_{n\in\omega}$, that is countably many
possible choices of Alice in the first inning
${\mathcal{N}}_{\emptyset}^{0},{\mathcal{N}}_{\emptyset}^{1},{\mathcal{N}}_{\emptyset}^{2},...$,
such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}=\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}_{\emptyset}^{n})}$.
Fix, for example, the branch with ${\mathcal{N}}^{0}_{\emptyset}$ then there
exists a sequence $({\mathcal{N}}_{<0>}^{n})_{n\in\omega}$ such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}})}=\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{n})}$.
Again consider, for example, ${\mathcal{N}}^{1}_{<0>}$, then there exists a
sequence $({\mathcal{N}}_{<0,1>}^{n})_{n\in\omega}$ such that
$\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{1},{\mathcal{N}}_{<0,1>}^{n})}=\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{1},{\mathcal{N}})}$.
By Claim 1, each intersection is empty or contains only one element. If the
intersection
$\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{n}_{\emptyset})}$ is non-
empty we call this element $x^{\emptyset}$, otherwise we go on; if
$\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{n})}$
is not empty we call this element $x^{<0>}$; if the intersection
$\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{1},{\mathcal{N}}_{<0,1>}^{n})}$
is not empty we call this element $x^{<0,1>}$, and so on. We obtain a subset
$X_{0}=\\{x^{s}:s\in\omega^{<\omega}\\}$ and now we want to prove that
$X_{0}=X$. By contradiction, assume there exists $y\in X\setminus X_{0}$. Then
$y\notin\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{n}_{\emptyset})}$;
hence there exists an element of the sequence
$\\{\sigma({\mathcal{N}}_{\emptyset}^{n}):n\in\omega\\}$, say
$\sigma({\mathcal{N}}_{\emptyset}^{k_{0}})$, such that $y$ does not belong to
it. By hypotesis,
$y\notin\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{k_{0}}_{\emptyset},{\mathcal{N}}_{<k_{0}>}^{n})}$;
hence there exists an element of
$\\{\sigma({\mathcal{N}}_{<k_{0}>}^{n}):n\in\omega\\}$, say
$\sigma({\mathcal{N}}_{<k_{0}>}^{k_{1}})$, such that $y$ does not belong to
it. Again,
$y\not\in\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{k_{0}}_{\emptyset},{\mathcal{N}}_{<k_{0}>}^{k_{1}},{\mathcal{N}}_{<k_{0},k_{1}>}^{n})}$,
there exists an element of
$\\{\sigma({\mathcal{N}}_{<k_{0},k_{1}>}^{n}):n\in\omega\\}$, say
$\sigma({\mathcal{N}}_{<k_{0},k_{1}>}^{k_{2}})$, such that $y$ does not belong
to it. Proceeding in this way we obtain a branch consisting of elements that
do not contain $y$; a contradiction, because such a branch is a network
because $\sigma$ is a winning strategy for Bob. Then $X$ is countable.
Now we prove that $X$ is secound countable.
Claim 3. If
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}=\\{x\\}$,
there exists an open set $V$ such that $x\in
V\subset\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}$.
Indeed, assume by contradiction that for every open set $V$ such that $x\in V$
there exists $y_{V}\in V\setminus\overline{\sigma({\mathcal{N}})}$, for every
${\mathcal{N}}\in{\mathbb{M}}$. Let ${\mathcal{N}}$ be a countable network and
consider the family
${\mathcal{N}}^{\prime}=({\mathcal{N}}\setminus{\mathcal{N}}_{x})\cup\\{(x,y_{V}):V\in\tau_{x}\\}$,
where $\tau_{x}$ denotes the family of all open sets containing $x$ and
${\mathcal{N}}_{x}=\\{N\in{\mathcal{N}}:x\in\overline{N}\\}$. Since $X$ is
countable, ${\mathcal{N}}^{\prime}$ is countable. Now we prove that
${\mathcal{N}}^{\prime}$ is a network. Let $y\in X$, $y\neq x$, and $A$ be an
open set such that $y\in A$. Since $X$ is $T_{2}$, there exists an open set
$B$ such that $y\in B$ and $x\notin\overline{B}$. Then there exists
$N\in{\mathcal{N}}$ such that $y\in N\subset A\cap B$. Therefore
$N\in{\mathcal{N}}\setminus{\mathcal{N}}_{x}$.
Claim 4. If
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}=\\{x\\}$,
there exists ${\mathbb{M}}^{\prime}\subset{\mathbb{M}}$ countable such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\sigma({\mathcal{N}})}=\\{x\\}$
and also such that
$\bigcup_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\sigma({\mathcal{N}})}=\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}$.
Recall that, by Claim 2 there exists a countable subset
${\mathbb{M}}^{*}\subset{\mathbb{M}}$ such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}=\bigcap_{{\mathcal{N}}\in{\mathbb{M}}^{*}}\overline{\sigma({\mathcal{N}})}$;
further, since $X$ is countable,
$\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}$ is
countable and then there exists a countable subset
${\mathbb{M}}^{**}\subset{\mathbb{M}}$ such that
$\bigcup_{{\mathcal{N}}\in{\mathbb{M}}^{**}}\overline{\sigma({\mathcal{N}})}=\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\sigma({\mathcal{N}})}$.
Then ${\mathbb{M}}^{\prime}={\mathbb{M}}^{*}\cup{\mathbb{M}}^{**}$.
Even Claims 3 and 4 are clearly true for each inning. Consider the
construction of the tree in the previous part of the proof. We know that
$|\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{n}_{\emptyset})}|\leq
1$. If
$\bigcap_{n\in\omega}\overline{\sigma({\mathcal{N}}^{n}_{\emptyset})}\neq\emptyset$,
fix $V_{\emptyset}$ as in Claim 3. If
$\bigcap_{k\in\omega}\overline{\sigma({\mathcal{N}}^{n}_{\emptyset},{\mathcal{N}}^{k}_{<n>})}\neq\emptyset$,
fix $V_{<n>}$ as in Claim 3 and so on. Now we prove that
$\\{V_{s}:s\in\omega^{<\omega}\\}$ is a base. Assuming the contrary, there
exist $x\in X$ and an open set $A$ with $x\in A$ such that for every
$s\in\omega^{<\omega}$ such that $x\in V_{s}$, $V_{s}$ is not contained in
$A$. In the first inning, we have a family ${\mathbb{M}}^{\prime}$ of
countably many networks obtained as in Claim 4. Consider the intersection
$\bigcap_{N\in{\mathbb{M}}^{\prime}}\overline{\sigma({\mathcal{N}})}$. If
$\bigcap_{N\in{\mathbb{M}}^{\prime}}\overline{\sigma({\mathcal{N}})}=\emptyset$,
we can pick a ${\mathcal{N}}\in{\mathbb{M}}^{\prime}$, such that
$x\notin\sigma({\mathcal{N}})$. If
$\bigcap_{N\in{\mathbb{M}}^{\prime}}\overline{\sigma({\mathcal{N}})}=\\{y\\}$
we have two cases: if $y\neq x$, we can pick a
${\mathcal{N}}\in{\mathbb{M}}^{\prime}$, such that
$x\notin\sigma({\mathcal{N}})$; if $y=x$, then we can pick, if there exists a
${\mathcal{N}}\in{\mathbb{M}}^{\prime}$, such that
$x\not\in\sigma({\mathcal{N}})$, otherwise by hypothesis and Claim 3 we can
pick a ${\mathcal{N}}\in{\mathbb{M}}^{\prime}$, such that
$\overline{\sigma({\mathcal{N}})}$ is not contained in $A$. Then, proceeding
in this way for each inning, we find a branch of the tree, i.e., a R-nw-
selective$(X)$ in which Alice has a winning strategy, a contradiction.

The following proposition shows that the (Point, Open)-set game is a good
candidate to be the dual of the R-nw-selective game.

######  Proposition 2.4.

Let $X$ be a space. The following implications hold.

  1. 1.

Alice $\uparrow$ PO-set$(X)\implies$ Bob $\uparrow$ R-nw-selective$(X)$.

  2. 2.

Alice $\uparrow$ R-nw-selective$(X)\implies$ Bob $\uparrow$ PO-set$(X)$.

  3. 3.

Bob $\uparrow$ R-nw-selective$(X)\implies$ Alice $\uparrow$ PO-set$(X)$.

The proof of points 1. and 2. is trivial. By Proposition 2.3 it is
straightforward to prove the point 3.

######  Question 2.5.

Does Bob $\uparrow$ PO-set$(X)$ imply Alice $\uparrow$ R-nw-selective$(X)$?

Now we study the determinacy of the R-nw-selective game.

######  Proposition 2.6.

Let $X$ be a space with $nw(X)=\omega$. If $|X|<cov({\cal M})$ and
$w(X)<cov({\cal M})$, then Alice $\not\uparrow$ R-nw-selective$(X)$.

Suppose, by contradiction, that $\sigma$ is a winning strategy for Alice in
the R-nw-selective$(X)$ and fix a base $\cal B$ of cardinality $w(X)$.
Construct a countable tree using the strategy $\sigma$ in such a way that
$\sigma(\langle\rangle)={\cal N}_{0}$; for each $N_{0}\in{\cal N}_{0}$ apply
the strategy and so on. Look at this tree as the poset of all finite branches
ordered with the inverse natural extention. The nodes in this tree are the
countable networks that are images through the function $\sigma$. Fix $x\in X$
and $B\in{\cal B}$ containing $x$, the set $D_{(x,B)}$ of all the finite
sequences of the tree such that there exists an element of the sequence that
is a $\sigma(\langle...,N\rangle)$ with $x\in N\subset B$, is dense in the
poset. Since the cardinality of the family $\\{D_{(x,B)}:x\in X\hbox{ and
}B\in{\cal B}\\}$ is less than $cov({\cal M})$ there exists a generic filter
whose union is a branch of the tree and that intersects all the dense sets of
the family. Therefore we obtain the contradiction, in particular this branch
is a R-nw-selective$(X)$ in which Bob wins.

######  Example 2.7.

($\omega_{1}<cov({\cal M})$) Consider a subspace $X\subset{\mathbb{R}}$ with
cardinality $\omega_{1}$. By Proposition 2.3 and Proposition 2.6 the R-nw-
selective game on $X$ is indeterminate.

######  Question 2.8.

Is there any ZFC example of a space in which the R-nw-selective game turns out
to be indeterminate?

The following diagram shows all the relations found above.

Alice$\not\uparrow$ R-nw-selective$(X)$$X$is countable + secound
countableBob$\uparrow$ R-nw-selective$(X)$$nw(X)\leq\omega+\,\,|X|<cov({\cal
M})$+$w(X)<cov({\cal M})$R-nw-selective

######  Question 2.9.

Does R-nw-selective property on a space $X$ imply Alice $\not\uparrow$ R-nw-
selective$(X)$?

Recall the following result.

######  Proposition 2.10.

[8] Let $X$ be a space without isolated points, with $nw(X)=\omega$ and
$w(X)<cov({\cal M})$. Then the following are equivalent.

  1. 1.

$|X|<cov({\cal M})$;

  2. 2.

$X$ is R-nw-selective.

Then it is possible to give a partial answer to Question 2.9.

######  Proposition 2.11.

Let $X$ be a space without isolated points, with $nw(X)=\omega$ and
$w(X)<cov({\cal M})$. Then the following are equivalent.

  1. 1.

$|X|<cov({\cal M})$;

  2. 2.

Alice $\not\uparrow$ R-nw-selective$(X)$;

  3. 3.

$X$ is R-nw-selective.

Now we show that if Bob is forced to select a fixed number of element from
each network the obtained game is equivalent to the R-nw-selective game. Let
${\sc Nw}$ denote the class of all countable networks of a fixed space $X$.
Let $k\in\omega$ and $G_{k}({\sc Nw},{\sc Nw})$ on $X$ be the game played in
the following way: Alice chooses a countable network ${\cal N}_{0}$ and Bob
answers picking a subset ${\cal F}_{0}\subset{\cal N}_{0}$ such that $|{\cal
F}_{0}|=k$. Then Alice chooses another countable network ${\cal N}_{1}$ and
Bob answers picking a subset ${\cal F}_{1}\subset{\cal N}_{1}$ such that
$|{\cal F}_{1}|=k$ and so on for countably many innings. At the end Bob wins
if the set $\bigcup\\{{\cal F}_{n}:n\in\omega\\}$ of his selections is a
network.

######  Proposition 2.12.

Alice $\uparrow$ R-nw-selective($X$) if, and only if, Alice $\uparrow
G_{k}({\sc Nw},{\sc Nw})$ on $X$.

We only prove that if Alice $\uparrow$ R-nw-selective($X$) then Alice
$\uparrow G_{k}({\sc Nw},{\sc Nw})$ on $X$. Suppose that Alice selects a
network ${\cal N}_{0}$ in the $G_{k}({\sc Nw},{\sc Nw})$. Then Bob answers by
picking ${\cal F}_{0}=\\{F_{1}^{0},...,F^{0}_{k}\\}$. Let Alice play the
network ${\cal N}_{0}$ in the R-nw-selective$(X)$ for $k$-many innings and
assume that Bob chooses $F_{i}^{0}$, $i=1,...,k$, in the $i$-th inning. Now
suppose that Alice selects a network ${\cal N}_{1}$ in the $G_{k}({\sc
Nw},{\sc Nw})$. Then Bob answers by picking ${\cal
F}_{1}=\\{F_{1}^{1},...,F^{1}_{k}\\}$. Let Alice play the network ${\cal
N}_{1}$ in the R-nw-selective$(X)$ for $k$-many innings and assume that Bob
chooses $F_{i}^{1}$, $i=1,...,k$, in the $(k+i)$-th inning and so on. Then by
hypotesis $\bigcup_{n\in\omega}{\cal F}_{n}$ is not a network.

######  Proposition 2.13.

Bob $\uparrow$ R-nw-selective($X$) if, and only if, Bob $\uparrow G_{k}({\sc
Nw},{\sc Nw})$ on $X$.

We prove that Bob $\uparrow G_{k}({\sc Nw},{\sc Nw})$ on $X$ implies that the
space $X$ is countable and secound countable. In fact the proof is similar to
the one of Proposition 2.3. Let $\sigma$ be a winning strategy for Bob in the
$G_{k}({\sc Nw},{\sc Nw})$ on $X$ and ${\mathbb{M}}$ be the collection of all
countable networks of the space $X$. We just need to prove the following
claims.
Claim 1. $|\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\cal
N})}|\leq k$.
We prove the claim for $k=2$. Suppose, by contradiction, that there are three
different points
$x_{1},x_{2},x_{3}\in\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\cal
N})}$. Since the space is $T_{1}$, there exist a closed set $C_{1}$ containing
$x_{1}$ but not $x_{2},x_{3}$, a closed set $C_{2}$ containing $x_{2}$ but not
$x_{1},x_{3}$ and a closed set $C_{3}$ containing $x_{3}$ but not
$x_{1},x_{2}$. Pick a countable network $\mathcal{N}$. For every
$N\in\mathcal{N}$ such that $x_{1},x_{2},x_{3}\in N$, replace $N$ with $N\cap
C_{i}$, $i=1,2,3$. In this way we construct a countable network such that any
elements of it contains at most one of the points $x_{1},x_{2},x_{3}$.
Claim 2. There exists ${\mathbb{M}}^{\prime}\subset{\mathbb{M}}$ countable
such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\bigcup\sigma({\mathcal{N}})}=\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\cal
N})}$.
Claim 3. If
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}=\\{x\\}$,
there exists an open set $V$ such that $x\in
V\subset\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}$.
Claim 4. If
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}=\\{x\\}$,
there exists ${\mathbb{M}}^{\prime}\subset{\mathbb{M}}$ countable such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\bigcup\sigma({\mathcal{N}})}=\\{x\\}$
and also such that
$\bigcup_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\bigcup\sigma({\mathcal{N}})}=\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}$.
The proof of Claims 2, 3 and 4 are similar to the ones in Proposition 2.3.

##  3 M-nw-selective game

######  Definition 3.1.

Let $X$ be a space with $nw(X)=\omega$. The M-nw-selective game, denoted by
M-nw-selective$(X)$, is played according to then following rules. Alice
chooses a countable network ${\cal N}_{0}$ and Bob answers picking a finite
subset ${\cal F}_{0}\subset{\cal N}_{0}$. Then Alice chooses another countable
network ${\cal N}_{1}$ and Bob answers in the same way and so on for countably
many innings. At the end Bob wins if the set $\bigcup\\{{\cal
F}_{n}:n\in\omega\\}$ of his selections is a network.

######  Proposition 3.2.

$(MA[{\mathfrak{d}}])$ Let $X$ be a space with $nw(X)=\omega$. If
$|X|<\mathfrak{d}$ and $w(X)<\mathfrak{d}$, then Alice $\not\uparrow$ M-nw-
selective$(X)$.

Similar to the proof of Proposition 2.6. Recall the following result.

######  Proposition 3.3.

[8] Let $X$ be a space with $nw(X)=\omega$, $w(X)<{\mathfrak{d}}$. Then the
following conditions are equivalent:

  1. 1.

$|X|<{\mathfrak{d}}$;

  2. 2.

$X$ is M-nw-selective.

Then it is possible to give the following equivalences.

######  Proposition 3.4.

$(MA[{\mathfrak{d}}])$ Let $X$ be a space with $nw(X)=\omega$ and
$w(X)<{\mathfrak{d}}$. The following are equivalent:

  1. 1.

$|X|<{\mathfrak{d}}$;

  2. 2.

Alice $\not\uparrow$ M-nw-selective$(X)$;

  3. 3.

$X$ is M-nw-selective.

However, it is worthwhile to pose the following question.

######  Question 3.5.

Does the M-nw-selective property imply that Alice $\not\uparrow$ M-nw-
selective$(X)$?

######  Proposition 3.6.

Let $X$ be a regular space such that Bob $\uparrow$ M-nw-selective(X). Then
$X$ is $\sigma$-compact.

Let $\mathbb{M}$ be the collection of all countable networks of $X$ and
$\sigma$ a winning strategy for Bob in M-nw-selective(X).
Claim 1: $\bigcap_{{\cal N}\in{\mathbb{M}}}\overline{\bigcup\sigma({\cal N})}$
is compact.
Indeed, put $K=\bigcap_{{\cal N}\in{\mathbb{M}}}\overline{\bigcup\sigma({\cal
N})}$, let $\cal U$ be a cover made by open sets of $X$ and ${\cal
N}\in{\mathbb{M}}$. Consider the network ${\cal N}^{\prime}=\\{N\in{\cal
N}:\overline{N}\subset U\hbox{ for some }U\in{\cal U}\\}\cup\\{N\in{\cal
N}:\overline{N}\cap K=\emptyset\\}$. Then $K\subset\sigma(\langle{\cal
N}^{\prime}\rangle)$ and considering the corresponding open sets we have the
compactness of $K$.
Claim 2: There exists a countable subset
${\mathbb{M}}^{\prime}\subset{\mathbb{M}}$ such that $\bigcap_{{\cal
N}\in{\mathbb{M}}^{\prime}}\overline{\bigcup\sigma({\cal N})}=\bigcap_{{\cal
N}\in{\mathbb{M}}}\overline{\bigcup\sigma({\cal N})}$.
The proof is similar to the one of Claim 2. in Proposition 2.3 and, as in
there, these claims are true also for all the other innings.
There exists $({\mathcal{N}}_{\emptyset}^{n})_{n\in\omega}$, that is countably
many possible first inning
${\mathcal{N}}_{\emptyset}^{0},{\mathcal{N}}_{\emptyset}^{1},{\mathcal{N}}_{\emptyset}^{2},...$,
such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}=\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}_{\emptyset}^{n})}$.
If Alice chooses ${\mathcal{N}}^{0}_{\emptyset}$, we can find
$({\mathcal{N}}_{<0>}^{n})_{n\in\omega}$ such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}})}=\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{n})}$.
If then Alice chooses ${\mathcal{N}}^{1}_{<0>}$, we can find
$({\mathcal{N}}_{<0,1>}^{n})_{n\in\omega}$ such that
$\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{1},{\mathcal{N}}_{<0,1>}^{n})}=\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{1},{\mathcal{N}})}$.
By claims 1 and 2, each intersection, if it is not empty, is a compact subset.
If the intersection is empty, we do not do anything and if
$\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}^{n}_{\emptyset})}$
is a compact, we call this subset $K^{\emptyset}$. If
$\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{n})}$
is a compact subset, we call it $K^{<0>}$. If
$\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}^{0}_{\emptyset},{\mathcal{N}}_{<0>}^{1},{\mathcal{N}}_{<0,1>}^{n})}$
is a compact subset, we call this element $K^{<0,1>}$, and so on. Consider the
set $X_{0}=\bigcup\\{K^{s}:s\in\omega^{<\omega}\\}$. Now we prove that
$X_{0}=X$. By contradiction, assume there exists $y\in X\setminus X_{0}$. Then
$y\notin\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}^{n}_{\emptyset})}$;
hence there exists $n_{0}\in\omega$ such that
$y\not\in\overline{\bigcup\sigma({\mathcal{N}}^{n_{0}}_{\emptyset})}$. Again,
$y\notin\bigcap_{n\in\omega}\overline{\bigcup\sigma({\mathcal{N}}^{n_{0}}_{\emptyset},{\mathcal{N}}^{n}_{\langle
n_{0}\rangle})}$; hence there exists $n_{1}\in\omega$ such that
$y\not\in\overline{\bigcup\sigma({\mathcal{N}}^{n_{0}}_{\emptyset},{\mathcal{N}}^{n_{1}}_{\langle
n_{0}\rangle})}$. Proceeding in this way we obtain a branch (or an evolution
of the M-nw-selective($X$)) in which Bob does not win, a contradiction,
because $\sigma$ is a winning strategy. Then $X$ is $\sigma$-compact.

######  Corollary 3.7.

Let $X$ be a regular space in which Bob $\uparrow$ M-nw-selective$(X)$. Then
$X$ is secound countable.

Recall that every regular secound countable space is metrizable and a space is
called $\sigma$-(metrizable compact) if it is union of countably many
metrizable compact spaces. Then it is possible to obtain the following
corollary.

######  Corollary 3.8.

Let $X$ be a regular space in which Bob $\uparrow$ M-nw-selective$(X)$. Then
$X$ is metrizable.

######  Corollary 3.9.

Let $X$ be a regular space in which Bob $\uparrow$ M-nw-selective$(X)$. Then
$X$ is $\sigma$-(metrizable compact).

The following is a consistent example showing that the M-nw-selective game can
be indeterminate. It also shows that the vice versa of the Corollaries 3.7 and
3.8 does not hold.

######  Example 3.10.

$(MA[{\mathfrak{d}}]+\omega_{1}<{\mathfrak{d}})$ Consider a subset $X$ of the
irrational numbers having cardinality $\omega_{1}$. By Proposition 3.2, Alice
$\not\uparrow$ M-nw-selective$(X)$. Since $X$ is not $\sigma$-compact, by
Proposition 3.6 we have that Bob $\not\uparrow$ M-nw-selective$(X)$.

The hypotesis of regularity in Corollary 3.7 can be replaced by countability
of the space as the following shows.

######  Proposition 3.11.

If $X$ is a countable space in which Bob $\uparrow$ M-nw-selective$(X)$. Then
$X$ is secound countable.

Similar to the the proof of Proposition 2.3 replacing Claims 3 and 4 with
following.
Claim $3^{\prime}$. If
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}=\\{x\\}$,
there exists an open set $V$ such that $x\in
V\subset\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}$.
Claim $4^{\prime}$. If
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}=\\{x\\}$,
there exists ${\mathbb{M}}^{\prime}\subset{\mathbb{M}}$ countable such that
$\bigcap_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\bigcup\sigma({\mathcal{N}})}=\\{x\\}$
and also such that
$\bigcup_{{\mathcal{N}}\in{\mathbb{M}}^{\prime}}\overline{\bigcup\sigma({\mathcal{N}})}=\bigcup_{{\mathcal{N}}\in{\mathbb{M}}}\overline{\bigcup\sigma({\mathcal{N}})}$.

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