# deeplenstronomy: A dataset simulation package for strong gravitational
lensing

Robert Morgan111Corresponding author Brian Nord  Simon Birrer  Joshua Yao-Yu
Lin  Jason Poh

††margin:  DOI:
https://doi.org/10.21105/joss.02854https://doi.org10.21105/joss.02854 Software
• https://github.com/openjournals/joss-
reviews/issues/2854https://doi.orgReview •
https://github.com/deepskies/deeplenstronomyhttps://doi.orgRepository •
https://doi.org/10.5281/zenodo.4479712https://doi.orgArchive Submitted: 10
November 2020
Published: 04 February 2021 License
Authors of papers retain copyright and release the work under a Creative
Commons Attribution 4.0 International License
(https://creativecommons.org/licenses/by/4.0/https://doi.orgCC BY 4.0).

## Background

Astronomical observations and statistical modeling permit the high-fidelity
analysis of strong gravitational lensing (SL) systems, which display an
astronomical phenomenon in which light from a distant object is deflected by
the gravitational field of another object along its path to the observer.
These systems are of great scientific interest because they provide
information about multiple astrophysical and cosmological phenomena, including
the nature of dark matter, the expansion rate of the Universe, and
characteristics of galaxy populations. They also serve as standing tests of
the theory of General Relativity and modified theories of gravity.

Traditional searches for SL systems have involved time- and effort-intensive
visual or manual inspection of images by humans to identify characteristic
features — like arcs, particular color combinations, and object orientations.
However, a comprehensive search using the traditional approach is
prohibitively expensive for large numbers of images, like those in
cosmological surveys — e.g., the Sloan Digital Sky Survey (York et al. 2000),
the Dark Energy Survey (Abbott et al. 2018), and the Legacy Survey of Space
and Time (LSST) (Ivezić et al. 2019). To automate the SL detection process,
techniques based on machine learning (ML) are beginning to overtake
traditional approaches for scanning astronomical images. In particular, deep
learning techniques have been the focus, but they require large sets of
labeled images to train these models. Because of the relatively low number of
observed SL systems, simulated datasets of images are often needed. Thus, the
composition and production of these simulated datasets have become integral
parts of the SL detection process.

One of the premier tools for simulating and analyzing SL systems, lenstronomy
(Birrer and Amara 2018), works by the user specifying the properties of the
physical systems, as well as how they are observed (e.g., telescope and
camera) through a python-based application programming interface (API) to
generate a single image. Generating populations of SL systems that are fit for
neural network training requires additional infrastructure.

## Statement of need

Due to the inherent dependence of the performance of ML approaches on their
training data, the deep learning approach to SL detection is in tension with
scientific reproducibility without a clear prescription for the simulation of
the training data. There is a critical need for a tool that simulates full
datasets in an efficient and reproducible manner, while enabling the use of
all the features of the lenstronomy simulation API. Additionally, this tool
should simplify user interaction with lenstronomy and organize the simulations
and associated metadata into convenient data structures for deep learning
problems. Multiple packages have been developed to generate realistic training
data by wrapping around lenstronomy: baobab (Park 2021) generates training
sets for lens modeling and hierarchical inference and the LSST Dark Energy
Science Collaboration’s SLSprinkler (Kalmbach et al. 2020) adds strongly
lensed variable objects into catalogs and images. Nonetheless, the need for a
simple, general tool capable of efficiently simulating any astronomical system
in a reproducible manner while giving the user complete freedom to set the
properties of objects remains.

## Summary

deeplenstronomy generates SL datasets by organizing and expediting user
interaction with lenstronomy. The user creates a single yaml-style
configuration file that describes the aspects of the dataset: number of
images, properties of the telescope and camera, cosmological parameters,
observing conditions, properties of the physical objects, and geometry of the
SL systems. deeplenstronomy parses the configuration file and generates the
dataset, producing both the images and the parameters that led to the
production of each image as outputs. The configuration files can easily be
shared, enabling users to easily reproduce each other’s training datasets.

The premier objective of deeplenstronomy is to help astronomers make their
training datasets as realistic as possible. To that end, deeplenstronomy
contains built-in features for the following functionalities: use any stellar
light profile or mass profile in lenstronomy; simulate a variety of
astronomical systems such as single galaxies, foreground stars, galaxy
clusters, supernovae, and kilonovae, as well as any combination of those
systems; fully control the placements of objects in the simulations; use
observing conditions of real astronomical surveys; draw any parameter from any
probability distribution; introduce any correlation; and incorporate real
images into the simulation. Furthermore, deeplenstronomy facilitates realistic
time-domain studies by providing access to public spectral energy
distributions of observed supernovae and kilonovae and incorporating the
transient objects into time series of simulated images. Finally,
deeplenstronomy provides data visualization functions to enable users to
inspect their simulation outputs. These features and the path from
configuration file to full data set are shown in Figure 1.

deeplenstronomy makes use of multiple open-source software packages:
lenstronomy is used for all gravitational lensing calculations and image
simulation; numpy (Harris et al. 2020) Arrays are used internally to store
image data and perform vectorized calculations; pandas (McKinney et al. 2010)
DataFrames are utilized for storing simulation metadata and file reading and
writing; scipy (Jones et al. 2001) is used for integration and interpolation;
matplotlib (Hunter 2007) functions are used for image visualization; astropy
(Astropy Collaboration et al. 2013) is used for cosmological calculations and
color image production; h5py (Collette 2014) is utilized for saving images;
and PyYAML (Simonov and Net 2006) is used to manage the configuration file.
While not used directly, some python-benedict (Caccamo 2018) functionalities
helped to create deeplenstronomy’s data structures and internal search
algorithms.

deeplenstronomy is packaged and disseminated via
https://pypi.org/project/deeplenstronomy/https://doi.orgPyPI. Documentation
and example notebooks are available on the
https://deepskies.github.io/deeplenstronomy/https://doi.orgdeeplenstronomy
website. Any bugs or feature requests can be opened as issues in the
https://github.com/deepskies/deeplenstronomy/issueshttps://doi.orgGitHub
repository (Morgan 2020).

Figure 1: The deeplenstronomy process. Dataset properties, camera and
telescope properties, observing conditions, object properties (e.g.,
lenstronomy light and mass profiles, point sources, and temporal behavior),
the geometry of the SL systems, and optional supplemental input files (e.g.,
probability distributions, covariance matrices, and image backgrounds) are
specified in the main configuration file. deeplenstronomy then interprets the
configuration file, calls lenstronomy simulation functionalities, and
organizes the resulting images and metadata.

## Acknowledgements

R. Morgan thanks the LSSTC Data Science Fellowship Program, which is funded by
LSSTC, NSF Cybertraining Grant #1829740, the Brinson Foundation, and the Moore
Foundation; his participation in the program has benefited this work. R.
Morgan also thanks the Universities Research Association Fermilab Visiting
Scholar Program for funding his work on this project.

We acknowledge the Deep Skies Lab as a community of multi-domain experts and
collaborators who’ve facilitated an environment of open discussion, idea-
generation, and collaboration. This community was important for the
development of this project. We acknowledge contributions from Joao Caldeira
during the early stages of this project.

Work supported by the Fermi National Accelerator Laboratory, managed and
operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy. The U.S. Government retains and the
publisher, by accepting the article for publication, acknowledges that the
U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide
license to publish or reproduce the published form of this manuscript, or
allow others to do so, for U.S. Government purposes.

## References

Abbott, T. M. C., F. B. Abdalla, A. Alarcon, J. Aleksić, S. Allam, S. Allen,
A. Amara, et al. 2018. “Dark Energy Survey year 1 results: Cosmological
constraints from galaxy clustering and weak lensing” 98 (4): 043526.
doi:https://doi.org/10.1103/PhysRevD.98.043526https://doi.org10.1103/PhysRevD.98.043526.

Astropy Collaboration, T. P. Robitaille, E. J. Tollerud, P. Greenfield, M.
Droettboom, E. Bray, T. Aldcroft, et al. 2013. “Astropy: A community Python
package for astronomy” 558 (October): A33.
doi:https://doi.org/10.1051/0004-6361/201322068https://doi.org10.1051/0004-6361/201322068.

Birrer, Simon, and Adam Amara. 2018. “Lenstronomy: Multi-Purpose Gravitational
Lens Modelling Software Package.” _Physics of the Dark Universe_ 22: 189–201.
doi:https://doi.org/10.1016/j.dark.2018.11.002https://doi.org10.1016/j.dark.2018.11.002.

Caccamo, Fabio. 2018. “Python-Benedict.”
https://github.com/fabiocaccamo/python-benedict.

Collette, Andrew. 2014. _Python and Hdf5_. O’Reilly.

Harris, Charles R., K. Jarrod Millman, Stéfan J van der Walt, Ralf Gommers,
Pauli Virtanen, David Cournapeau, Eric Wieser, et al. 2020. “Array Programming
with NumPy.” _Nature_ 585: 357–62.
doi:https://doi.org/10.1038/s41586-020-2649-2https://doi.org10.1038/s41586-020-2649-2.

Hunter, John D. 2007. “Matplotlib: A 2d Graphics Environment.” _Computing in
Science and Engineering_ 9 (3): 90–95.
doi:https://doi.org/10.1109/MCSE.2007.55https://doi.org10.1109/MCSE.2007.55.

Ivezić, Željko, Steven M. Kahn, J. Anthony Tyson, Bob Abel, Emily Acosta,
Robyn Allsman, David Alonso, et al. 2019. “LSST: From Science Drivers to
Reference Design and Anticipated Data Products” 873 (2): 111\.
doi:https://doi.org/10.3847/1538-4357/ab042chttps://doi.org10.3847/1538-4357/ab042c.

Jones, Eric, Travis Oliphant, Pearu Peterson, and others. 2001. “SciPy: Open
Source Scientific Tools for Python.” http://www.scipy.org/.

Kalmbach, J. Bryce, Ji Won Park, Matthew Wiesner, and James Chiang. 2020\.
_LSSTDESC/SLSprinkler: LSST DESC Strong Lensing Sprinkler for Simulations_
(version v1.1.0). Zenodo.
doi:https://doi.org/10.5281/zenodo.4480392https://doi.org10.5281/zenodo.4480392.

McKinney, Wes, and others. 2010. “Data Structures for Statistical Computing in
Python.” In _Proceedings of the 9th Python in Science Conference_ , 445:51–56.
Austin, TX.
doi:https://doi.org/10.25080/majora-92bf1922-00ahttps://doi.org10.25080/majora-92bf1922-00a.

Morgan, Robert. 2020. “Deeplenstronomy.”
https://github.com/deepskies/deeplenstronomy.

Park, Ji Won. 2021. _Jiwoncpark/Baobab: V0.1.2_ (version v0.1.2). Zenodo.
doi:https://doi.org/10.5281/zenodo.4476822https://doi.org10.5281/zenodo.4476822.

Simonov, K., and I. Net. 2006. “PyYaml.” https://github.com/yaml/pyyaml.

York, Donald G., J. Adelman, Jr. Anderson John E., Scott F. Anderson, James
Annis, Neta A. Bahcall, J. A. Bakken, et al. 2000. “The Sloan Digital Sky
Survey: Technical Summary” 120 (3): 1579–87.
doi:https://doi.org/10.1086/301513https://doi.org10.1086/301513.


# Productivity of cellular-Lindelöf spaces

Alan Dow and R.M. Stephenson, Jr. Department of Mathematics and Statistics,
University of North Carolina at Charlotte, Charlotte, NC 28223<EMAIL_ADDRESS>Department of Mathematics, University of South Carolina, Columbia, SC 29208
<EMAIL_ADDRESS>
###### Abstract.

The main purpose of this note is to prove that the product of a cellular-
Lindelöf space with a space of countable spread need not be cellular-Lindelöf.

###### Key words and phrases:

cellular-Lindelöf space, cellular-compact space, Souslin tree, Moore’s L-space

###### 2010 Mathematics Subject Classification:

54B10, 54G20, 54E65

##  1\. Introduction

All hypothesized spaces are Hausdorff, and by Lindelöf (of countable spread)
we shall mean any space in which every open cover has a countable subcover
(every discrete subspace is countable); otherwise, the terminology and
notation not defined here generally agree with that in [3] or [5].

A. Bella and S. Spadaro in [1] defined a space $X$ to be cellular-Lindelöf
provided that for every family $\mathcal{U}$ of pairwise disjoint nonempty
open subsets of $X$ there is a Lindelöf subspace $L\subset X$ such that $L\cap
U\neq\emptyset$ for every $U\in{\mathcal{U}}$. In that article and also in [2]
they derived some properties of cellular-Lindelöf spaces. Likewise, W. Xuan
and Y. Song studied cellular-Lindelöf spaces in [15] and several other
articles referred to in [15]. A similar family of spaces, the cellular-compact
spaces, were introduced and studied extensively by V.V. Tkachuk and R.G.
Wilson in [13]. A space $X$ is said to be cellular-compact provided that for
every family $\mathcal{U}$ of pairwise disjoint nonempty open subsets of $X$
there is a compact subspace $K\subset X$ such that $K\cap U\neq\emptyset$ for
every $U\in{\mathcal{U}}$.

Among the nice properties of these two cellular concepts, most of which were
noted or shown in several of the articles above, are: each is preserved by
regular closed subsets, extensions, continuous images, and finite unions (and
for the case cellular-Lindelöf, countable unions and open
$F_{\sigma}$-subsets).

In [15] and especially in [13] its authors proved that dense subspaces of
compact product spaces, such as $\Sigma$-product subspaces, provide a useful
source of examples of non-Lindelöf spaces illustrating properties of cellular-
compact or cellular-Lindelöf spaces. In [15] its authors presented several
theorems, examples and questions concerning products of cellular-Lindelöf
spaces. We list some of these results next, but first note one obvious
necessary condition for a product space to be cellular-compact or cellular-
Lindelöf, according to the invariance under continuous maps property.

######  Theorem 1.1.

If a product space is cellular-compact or cellular-Lindelöf, then each of its
factor spaces must have the same property.

In [10] A.D. Rojas-Sánchez and A. Tamariz-Mascarúa provided a very clever
proof that there exist two Lindelöf spaces whose product contains an
uncountable clopen discrete subspace $D$. The authors of [15] referred the
reader to [10], re-described that example, re-derived the properties of $D$,
and noted that it establishes the following:

######  Theorem 1.2.

The product of two Tychonoff Lindelöf spaces need not be cellular-Lindelöf.

They then obtained the following result.

######  Theorem 1.3.

([15], Theorem 3.9) The product of a separable space and a cellular-Lindelöf
space is cellular-Lindelöf.

Proof. We sketch a different proof from the one in [15]: It is immediate that
the product of a countable space and a cellular-Lindelöf space is cellular-
Lindelöf, and hence any extension of such a space must also be cellular-
Lindelöf. $\square$

Noting that this theorem implies the product of a compact metrizable space
with a cellular-Lindelöf space is cellular-Lindelöf, the authors of [15]
asked:

######  Question 1.4.

Is the product of a compact space and a cellular-Lindelöf space cellular-
Lindelöf?

In Section 2 we shall provide a negative answer to this question, and we shall
answer an analogous question not raised in [13] or [15] about whether or not
the property cellular-compact is productive. It will be shown that the product
of a convergent sequence with a cellular-compact space need not be cellular-
compact.

The authors of [15] stated in their Theorem 3.12 that the product of a
cellular-Lindelöf space with a space of countable spread, such as a
hereditarily Lindelöf space, is cellular-Lindelöf. However, as noted in the
Abstract, the main purpose of this note is to provide a proof that this
assertion is not true. In Section 3 the focus will be on properties possible
counterexamples will need to possess. A proof will be given in Section 4
showing it is consistent that the product of a cellular-Lindelöf space with a
hereditarily Lindelöf space need not be cellular-Lindelöf. Then in Section 5 a
construction of Justin Moore will be used to produce very different
counterexamples within ZFC.

##  2\. Products with one factor compact

Let us first recall some needed terminology and a theorem. A family
$\mathcal{V}$ of nonempty open subsets of a space $X$ is called a $\pi$-base
for $X$ (a local $\pi$-base for a point $x\in X$) provided that for every
nonempty open subset $T$ of $X$ (containing $x$) there exists
$V\in{\mathcal{V}}$ such that $V\subset T$. The $\pi$-weight $\pi(X)$ of the
space $X$ ($\pi$-character $\pi\chi(x,X)$ of the space $X$ at the point $x$)
is the minimal cardinality of a $\pi$-base for $X$ (a local $\pi$-base at
$x$). The density ${\rm d}(X)$ of $X$ is the minimal cardinality of a dense
subset of $X$. A point $p\in\beta X\setminus X$ is called a remote point of a
Tychonoff space $X$ if $p\notin{\rm cl}_{\beta X}(Y)$ for any nowhere dense
set $Y\subset X$. Finally, (Theorem 3.13 of [13]) if $X$ is a regular
cellular-compact space and $p$ is a non-isolated point of $X$, then the space
$X\setminus\\{p\\}$ is cellular-compact if and only if there exists no
pairwise disjoint local $\pi$-base at the point $p$ in $X$. We shall use only
a special case of a corollary to this theorem (Corollary 3.14 of [13]), a case
which by itself has an immediate proof, and so we give both here.

######  Corollary 2.1.

([13]) If $K$ is a compact space, $p$ is a non-isolated point of $K$, and
there exists no pairwise disjoint local $\pi$-base at the point $p$ in $K$,
then the space $X=K\setminus\\{p\\}$ is cellular-compact.

Proof. Assume $\mathcal{U}$ is a family of pairwise disjoint nonempty open
subsets of $K$. Choose an open neighborhood $W$ of $p$ which does not contain
any member of $\mathcal{U}$. Then $K\setminus W=X\setminus W$ is a compact set
which meets every member of $\mathcal{U}$. $\square$

######  Theorem 2.2.

Let $X=\beta{\mathbb{Q}}\setminus\\{p\\}$, where $p$ is any remote point of
$\mathbb{Q}$, and let $Y={\mathbb{N}}\cup\\{\infty\\}$ be the one-point
compactification of $\mathbb{N}$. Then the space $X$ is cellular-compact, but
the product space $X\times Y$ is not cellular-compact.

Proof. To prove that $X$ is cellular-compact, it suffices by Corollary 2.1 for
us to prove that there is no pairwise disjoint local $\pi$-base for $p$ in
$\beta{\mathbb{Q}}$. Let $\mathcal{V}$ be an arbitrary family of pairwise
disjoint nonempty open sets in $\beta{\mathbb{Q}}$. Proceeding as in Example
3.21 of [13], we may (and do) choose, for each $V\in\mathcal{V}$, a rational
number $q_{V}\in V$. Then $D=\\{q_{V}:V\in{\mathcal{V}}\\}$ is discrete, and
so the remote point $p\notin{\rm cl}_{\beta{\mathbb{Q}}}(D)$. Hence
$\beta{\mathbb{Q}}\setminus{\rm cl}_{\beta{\mathbb{Q}}}(D)$ is an open
neighborhood of $p$ which contains no member of $\mathcal{V}$. Therefore,
$\mathcal{V}$ cannot be a local $\pi$-base for $p$ in $\beta{\mathbb{Q}}$.

Next, again using the denseness of $\mathbb{Q}$ in $\beta\mathbb{Q}$, let
${\mathcal{W}}=\\{W_{n}:n\in{\mathbb{N}}\\}$ be a $\pi$-base for
$\beta{\mathbb{Q}}$, and for each $n\in{\mathbb{N}}$, let
$U_{n}=(W_{n}\setminus\\{p\\})\times\\{n\\}$. Then
${\mathcal{U}}=\\{U_{n}:n\in{\mathbb{N}}\\}$ is a family of pairwise disjoint
nonempty open subsets of $X\times Y$.

Finally, assume $K$ is any compact subset of $X\times Y$. To complete the
proof, we need to show that $K\cap U_{n}=\emptyset$ for some
$n\in{\mathbb{N}}$. Since the projection ${\rm pr}_{X}(K)$ is a compact subset
of $\beta{\mathbb{Q}}$ not containing $p$, the set
$T=\beta{\mathbb{Q}}\setminus{\rm pr}_{X}(K)$ is an open neighborhood of $p$
in $\beta{\mathbb{Q}}$ which is disjoint from ${\rm pr}_{X}(K)$. As
$\mathcal{W}$ is a $\pi$-base for $\beta{\mathbb{Q}}$, $T$ must contain some
$W_{n}$. Hence, $K\cap U_{n}\subset K\cap((W_{n}\setminus\\{p\\})\times
Y)\subset K\cap(T\times Y)=\emptyset$.

$\square$

We turn next to producing a negative answer to Question 1.4, and we shall do
this by finding a cellular-Lindelöf (actually, another cellular-compact) space
$X$ having the form $X=K\setminus\\{p\\}$, where $K$ is compact, such that the
product space $X\times(\kappa+1)$, where $\kappa=\pi\chi(p,X)$, is not
cellular-Lindelöf.

######  Lemma 2.3.

There exists a compact space $K$ having a $P$-point $p$ such that $K$ has
cellularity $\mathfrak{c}<\pi\chi(p,K)$.

Proof. Consider the $P$-space $Y$ obtained by endowing the product space
$2^{{\mathfrak{c}}^{+}}$ with the $G_{\delta}$-topology. Let $p$ be the
constant 0 function in $Y$ and $K=\beta Y$.

We verify that the $P$-point $p$ of $K$ has $\pi\chi(p,K)={\mathfrak{c}}^{+}$.
Since $K$ is an extension space of $Y$ which is regular, it will be enough to
check that $\pi\chi(p,Y)={\mathfrak{c}}^{+}$.

Let $\mathcal{A}$ be the family of all countable subsets of
${\mathfrak{c}}^{+}$. For each $A\in\mathcal{A}$ let $N_{A}=\\{y\in
Y:y(\alpha)=0\rm{\ for\ all\ }\alpha\in A\\}$. Then
$\\{N_{A}:A\in{\mathcal{A}}\\}$ is a local base for $p$ in $Y$, so
$\pi\chi(p,Y)\leq\pi(p,Y)\leq|\mathcal{A}|={\mathfrak{c}}^{+}$. Assume next
that $\mathcal{V}$ is an arbitrary family of basic clopen $G_{\delta}$-subsets
of $Y$ such that $|\mathcal{V}|\leq{\mathfrak{c}}$. For each
$V\in\mathcal{V}$, let $D_{V}$ be the countable set of restricted coordinates
of $V$, and let $\mathcal{D}=\\{D:V_{D}\in\mathcal{V}\\}$. Then
${\mathcal{V}}$ cannot be a local $\pi$-base for $p$ in $Y$, since for any
$\alpha\in{{\mathfrak{c}}^{+}}\setminus\bigcup{\mathcal{D}}$,
${\pi_{\alpha}}^{-1}(0)$ is a neighborhood of $p$ in $Y$ which contains no
member of $\mathcal{V}$. Thus, $\pi\chi(p,Y)\geq{{\mathfrak{c}}^{+}}$.

Suppose next that $\\{f_{\alpha}:\alpha<{\mathfrak{c}}^{+}\\}$ are countable
partial functions from ${\mathfrak{c}}^{+}$ to 2 coding basic clopen
$G_{\delta}$-sets $\\{[f_{\alpha}]:\alpha<{\mathfrak{c}}^{+}\\}$ of $Y$. Let
this family be an element of $M\prec H({\mathfrak{c}}^{+})$, where $M$ has
cardinality $\mathfrak{c}$ and $M^{\omega}\subset M$. Let
$M\cap\,{\mathfrak{c}}^{+}\subset\delta\in{\mathfrak{c}}^{+}$, and let
$f=f_{\delta}\upharpoonright M$. Since the domain of $f$ is a countable subset
of $M$, $f$ is in $M$ by our choice of $M$. Thus there is an $\alpha\in M$
such that $f\subset f_{\alpha}$. It follows that $f_{\alpha}\cup f_{\delta}$
is a function, and therefore that $[f_{\alpha}]\cap[f_{\delta}]$ is not empty.
This proves that $K$ has cellularity equal to $\mathfrak{c}<\pi\chi(p,K)$.
$\square$

######  Theorem 2.4.

Let $X$ be any space having the form $X=K\setminus\\{p\\}$, where $K$ is a
compact space, $p$ is a $P$-point of $K$, and $K$ has cellularity
$\mathfrak{c}<\pi\chi(p,K)=\kappa$. Topologize the set $\kappa+1$ so that it
is the one-point compactification of the discrete space $\kappa$. Then the
space $X$ is cellular-compact, but the product space $X\times(\kappa+1)$ is
not cellular-Lindelöf.

Proof. It follows from Corollary 2.1 (or Corollary 3.17 of [13]) and the
cellularity inequality satisfied by $K$ that the space $X$ is cellular-
compact.

Now fix a local $\pi$-base $\\{U_{\alpha}:\alpha<\kappa\\}$ for $p$ in $K$. We
may assume that $p\notin U_{\alpha}$ for all $\alpha<\kappa$. For each
$\alpha\in\kappa$, the set $W_{\alpha}={U_{\alpha}}\times\\{\alpha\\}$ is open
in $X\times(\kappa+1)$, and ${\mathcal{W}}=\\{W_{\alpha}:\alpha<\kappa\\}$ is
a pairwise disjoint family of nonempty open sets.

Assume that $Y$ is a subspace of $X\times(\kappa+1)$ such that $Y\cap
W_{\alpha}\neq\emptyset$ for all $\alpha\in\kappa$. We prove that $Y$ cannot
be Lindelöf. For each $y\in Y$, choose a neighborhood $W_{y}$ of $y$ and
subset $V_{y}$ of $X$ such that $W_{y}\subset V_{y}\times(\kappa+1)$ and the
point $p\notin\overline{V_{y}}$. If there is a countable subcover
$\\{W_{y_{n}}:n\in\omega\\}$ of $Y$, then since $p$ is a $P$-point, we may
choose $\alpha<\kappa$ such that $U_{\alpha}\cap V_{y_{n}}=\emptyset$ for all
$n$. This implies that $Y$ is then disjoint from $W_{\alpha}$. $\square$

######  Remark 2.5.

For any space $X$ and $P$-point $p$ of $X$, if $p$ has character $\aleph_{1}$,
then there is a pairwise disjoint local $\pi$-base at $p$, and hence one can
show the space $X\setminus\\{p\\}$ cannot be cellular-Lindelöf.

One corollary to Theorems 2.2 and 2.4 which illustrates a difference between
properties of the spaces studied here and those in [10] is the following.

######  Corollary 2.6.

The property cellular-Lindelöf (cellular-compact) is not an inverse invariant
of open perfect surjections.

##  3\. Seeking a counterexample of countable spread

We consider next some properties spaces $X$ and $Y$ would need to have in
order to be counterexamples to the assertion that if $X$ is cellular-Lindelöf
and $Y$ is of countable spread, then $X\times Y$ is cellular-Lindelöf.

As noted earlier, neither space can be separable. In addition, the following
theorem, a corollary to one due to B. Šapirovskiĭ in [12], shows the space $Y$
must have a dense hereditarily Lindelöf subspace.

######  Theorem 3.1.

([4], Proposition 5.6, or [12]) Every non-separable space of countable spread
has a dense subspace that is hereditarily Lindelöf.

A tool often found useful in producing counterexamples is the Alexandroff
double ${\mathbb{D}}(X)$ of a topological space $X$. We recall that
${\mathbb{D}}(X)=X\times\\{0,1\\}$, topologized so that a subset $T$ of
${\mathbb{D}}(X)$ is defined to be open if and only if for each point
$(x,0)\in T$, there is an open subset $U$ of the space $X$ such that
$(x,0)\in\left((U\times\\{0,1\\})\setminus\\{(x,1)\\}\right)\subset T$. Three
well-known, easily verified properties are: ${\mathbb{D}}(X)$ is Lindelöf if
and only if $X$ is Lindelöf; and since $X\times\\{1\\}$ is an open discrete
subspace, ${\mathbb{D}}(X)$ is separable (hereditarily Lindelöf) if and only
if $X$ is countable.

######  Theorem 3.2.

Suppose $X$ and $Y$ are regular spaces such that ${\mathbb{D}}(X)\times Y$ is
cellular-Lindelöf, $Y$ has $\pi$-weight $\pi(Y)=\kappa>\aleph_{0}$, and for
every nonempty open subset $T$ of $X$, ${\rm d}(T)=\pi(T)=\kappa$. Then
${\mathbb{D}}(X)\times Y$ has a Lindelöf subspace which is dense in
$(X\times\\{0\\})\times Y$.

Proof. A recursive construction shows that there is a family
$\\{D_{\alpha}:\alpha<\kappa\\}$ of pairwise disjoint dense subsets of $X$,
each having cardinality $\kappa$.

Let $\\{U_{\alpha}:\alpha\in\kappa\\}$ enumerate a $\pi$-base for $Y$. Set
$W_{\alpha,d}=\\{(d,1)\\}\times U_{\alpha}$ for all $\alpha<\kappa$ and $d\in
D_{\alpha}$, and $\mathcal{W}=\\{W_{\alpha,d}:\alpha\in\kappa{\rm\ and\ }d\in
D_{\alpha}\\}$. Then $\mathcal{W}$ is a cellular family, so assume that $L$ is
a Lindelöf subspace of ${\mathbb{D}}(X)\times Y$ such that $L\cap
W\neq\emptyset$ for all $W\in\mathcal{W}$. We shall prove that $L$ is dense in
$(X\times\\{0\\})\times Y$.

Fix any nonempty regular closed sets $B\subset X$ and $R\subset Y$. We show
that $L\cap((B\times\\{0\\})\times R)\neq\emptyset$.

Since $\\{U_{\alpha}:\alpha\in\kappa\\}$ is a $\pi$-base for $Y$, there exists
$\beta<\kappa$ such that $U_{\beta}\subset{\rm int}(R)$. As $D_{\beta}$ is a
dense subset of $X$ and $d({\rm int}(B))=\kappa$, it follows that
$L\cap((B\times\\{0,1\\})\times R)$ meets $W_{\beta,d}$ for $\kappa$-many
points $d\in D_{\beta}$. For each such $d$, pick $((d,1),y_{d})\in L\cap
W_{\beta,d}$, and note that $((d,1),y_{d})\in L\cap((B\times\\{0,1\\})\times
R)$, Moreover, $L\cap((B\times\\{0,1\\})\times R)$ is Lindelöf. As this subset
of points $((d,1),y_{d})\in L\cap((B\times\\{0,1\\})\times R)$ has cardinality
$\kappa>\aleph_{0}$, it must have a limit point in
$L\cap((B\times\\{0\\})\times R)$, for every point of
$L\cap((B\times\\{1\\})\times R)$ has a neighborhood meeting that subset in at
most a single point.

$\square$

Based on the preceding results, we seek as counterexamples spaces $X$ and $Y$,
each nowhere separable and of $\pi$-weight $\aleph_{1}$, so that $X$ is
Lindelöf, $Y$ is hereditarily Lindelöf, and ${\mathbb{D}}(X)\times Y$ has no
Lindelöf subspace which is dense in $(X\times\\{0\\})\times Y$.

##  4\. Consistent product space counterexamples

Let $S$ be a full-branching Souslin tree (a subtree of $2^{<\omega_{1}}$). In
this section we show how to get $X$ and $Y$ from $S$. We begin by giving some
notation and terminology that will be used.

Let $s,t\in S$ and $A\subset S$. Then $s\leq t$ is assigned its usual meaning
(as on p. 68 of [5]), namely, $s\leq t$ if and only if $s\subset t$ if and
only if the sequence $t$ extends $s$, and likewise $s\lor t$, $s\land t$, and
$s\perp t$ have their usual meanings with respect to the relation $\leq$. In
addition, $o(s)\in\omega_{1}$ denotes the domain of $s$, $[s]$ denotes
$\\{u\in S:s\leq u\\}$, and $A$ is said to be cofinal above $s$ if for each
$u\in[s]$, $A\cap[u]\neq\emptyset$. The set $A$ is said to be cofinal in $S$
if $A$ is cofinal above $s$ for every $s\in S$. If $o(s)=\xi$ and
$i,j\in\\{0,1\\}$, then $s^{\frown}i$ denotes the sequence
$s\cup\\{(\xi,i)\\}$ whose domain is $\xi+1$, and similarly, $s^{\frown}ij$
denotes $(s^{\frown}i)^{\frown}j$. In case $s\neq t$, we define
$\Delta(s,t)={\rm min}\\{\alpha\in o(s)\cup o(t):s(\alpha)\neq t(\alpha)\\}$
(using the convention that $s(\alpha)\neq t(\alpha)$ if
$\alpha\in(o(s)\setminus o(t))\cup(o(t)\setminus o(s))$), and denote by
$\prec$ the following order: $s\prec t$ if and only if $s(\Delta(s,t))=0$ or
$t(\Delta(s,t))=1$.

The reader familiar with the procedures M.E. Rudin developed on pp. 1116–1118
of [11] may notice that the relation $\prec$ is an example of a general type
of relation presented and studied in [11] for the purpose of demonstrating how
to use a “normalized and pruned” Souslin tree and Dedekind completion to
produce a Souslin line. We include for later reference statements and proof
outlines illustrating the properties of $\prec$ needed for our examples. Some
of these properties were derived or pointed out in [11].

######  Claim 4.1.

The relation $\prec$ is a dense total order on $S$.

Proof. First we prove $\prec$ is transitive. Let $s,t$, and $u\in S$ satisfy
$s\prec t$ and $t\prec u$, and let $\alpha_{0}=\Delta(s,t)$ and
$\beta_{0}=\Delta(t,u)$. We consider three cases.

Suppose $\alpha_{0}=\beta_{0}$. Then
$s\upharpoonright\alpha_{0}=t\upharpoonright\alpha_{0}=u\upharpoonright\alpha_{0}$,
and the only possible values for $s$, $t$, and $u$ at $\alpha_{0}$ in this
case are $s(\alpha_{0})=0$, $t(\alpha_{0})$ is not defined, and
$u(\alpha_{0})=1$, which together imply $s\prec u$.

Suppose $\alpha_{0}<\beta_{0}$. Then
$s\upharpoonright\alpha_{0}=t\upharpoonright\alpha_{0}=u\upharpoonright\alpha_{0}$.
If $s(\alpha_{0})=0$ or is undefined, then $t(\alpha_{0})=1$, and
$t\upharpoonright\beta_{0}=u\upharpoonright\beta_{0}$ implies
$u(\alpha_{0})=1$, and so $s\prec u$.

Assume $\beta_{0}<\alpha_{0}$. Then
$s\upharpoonright\beta_{0}=t\upharpoonright\beta_{0}$, $\beta_{0}\in o(t)$,
and $s(\beta_{0})=t(\beta_{0})$. Since also $t\prec u$, we must have
$t(\beta_{0})=0$. Then $s(\beta_{0})=0$,
$s\upharpoonright\beta_{0}=u\upharpoonright\beta_{0}$, and since $t\prec u$,
either $u(\beta_{0})=1$ or $u(\beta_{0})$ is not defined. Hence
$s(\beta_{0})=0\neq u(\beta_{0})$, and $s\prec u$.

Obviously the partial order $\prec$ is a total order. To prove it is a dense
order, let $s$ and $t$ be as above. We find a sequence $m\in S$ such that
$s\prec m\prec t$, depending on which of the three possibilities occurs.

If $s(\alpha_{0})=0$ and $t(\alpha_{0})=1$, let
$m=s\upharpoonright\alpha_{0}$.

Suppose $s(\alpha_{0})=0$ and $t(\alpha_{0})$ is not defined. Find the first
$\beta\geq\alpha_{0}+1$ where $s(\beta)=0$, and set
$m=(s\upharpoonright\beta)^{\frown}1$; and if no such $\beta$ exists, set
$m=s^{\frown}1$.

Assume $s(\alpha_{0})$ is not defined and $t(\alpha_{0})=1$. Find the first
$\beta\geq\alpha_{0}+1$ where $t(\beta)=1$, and set
$m=(t\upharpoonright\beta)^{\frown}0$; and if no such $\beta$ exists, let
$m=t^{\frown}0$. $\square$

Before stating our next claim, some additional notation needed is, for any
$t\in S$: let $t^{\dagger}=t$ if $t$ is on a limit level of $S$; and let
$t^{\dagger}$ denote the “other” $<$-successor of $t\upharpoonright\alpha$ in
$S$ if $o(t)=\alpha+1$.

######  Claim 4.2.

Let $s,t\in S$ with $o(s)+1<o(t)$. (i) If $s\prec t$ then
$s\prec([t]\cup[t^{\dagger}])$, and (ii) if $t\prec s$ then
$([t]\cup[t^{\dagger}])\prec s$.

Proof. Let $s,t\in S$, and assume the hypothesis of (i) holds. Then whether
$s(\Delta(s,t))$ is not defined or equals 0, since $o(s)+1<o(t)$, it follows
that $t(\Delta(s,t))=1$, $t^{\dagger}(\Delta(s,t))=1$, and
$\Delta(s,t)=\Delta(s,t^{\dagger})$. Thus, for any $u\in S$, if $u\supset t$
or $u\supset t^{\dagger}$ then $\Delta(s,u)=\Delta(s,t)$,
$u(\Delta(s,u))=t(\Delta(s,t))=1$, and hence $s\prec u$. The statement (ii) is
proved similarly. $\square$

For the remainder of this section, $S$ is endowed with the $\prec$-order
topology, and the open interval notation $(s,t)$ will refer to $\prec$, where
$s\prec t$ (except when it is used to denote a point in the product space
$S\times S$). Thus, topological terms referring to $S$ should be understood to
be referring to the topology induced on $S$ by $\prec$ (but not to the one
induced on it by the tree order $<$).

######  Claim 4.3.

For each $s\in S$, $[s]$ is a clopen subset of $S$, and the family
$\\{[s]:s\in S\\}$ is a $\pi$-base for the topology on $S$.

Proof. First, we observe that each $[s]$ is an open set: for any $t\geq s$,
$t^{\frown}0\prec t\prec t^{\frown}1$, and so $I=(t^{\frown}0,t^{\frown}1)$ is
an open interval containing $t$, and one can check that $s\leq t\leq u$ for
any $u\in I$, and hence $I\subset[s]$.

To see that $[s]$ is a closed set, consider any point $t\in S$ such that
$s\nleq t$. Let $\alpha=\Delta(s,t)$. Then $\alpha\in o(s)\cap o(t)$ (if
$s\perp t$) or $\alpha\in o(s)\setminus o(t)$. If $\alpha\in o(t)$, then $[t]$
is an open neighborhood of $t$ disjoint with $[s]$. If $\alpha\notin o(t)$,
then $[t^{\frown}(1-s(\alpha))]$ is an open neighborhood of $t$ disjoint with
$[s]$.

Next, let us prove that the family $\\{(v_{1},v_{2}):v_{1}\prec v_{2}{\rm\
and\ }v_{1}\perp v_{2}\\}$ is a $\pi$-base. Fix any $s\prec t$ with $s\neq t$,
and we find $(v_{1},v_{2})\subset(s,t)$ from this $\pi$-base. If $s\perp t$,
then $v_{1}=s$ and $v_{2}=t$ suffice. So assume $s\not\perp t$, and we
consider the remaining possibilities.

Assume $s\leq t$, i.e., $s\subset t$. Then $s(\Delta(s,t))$ is not defined and
$t(\Delta(s,t))=1$, since $s\prec t$. Thus $s^{\frown}1\subset t$, and we
define $v_{1}=t^{\frown}00$ and $v_{2}=t^{\frown}01$.

Assume $t\leq s$, i.e., $t\subset s$. Then $s(\Delta(s,t))=0$ and
$t(\Delta(s,t))$ is not defined, since $s\prec t$. Hence $t^{\frown}0\subset
s$, and we define $v_{1}=s^{\frown}10$ and $v_{2}=s^{\frown}11$.

In either of these definitions, it follows easily that $v_{1}\perp v_{2}$ and
$s\prec v_{1}\prec v_{2}\prec t$.

To finish the proof, we just prove that for each $v_{1}\prec v_{2}$ with
$v_{1}\perp v_{2}$, there is some $s$ with $v_{1}\prec[s]\prec v_{2}$.
Evidently $\Delta(v_{1},v_{2})\in o(v_{1})\cap o(v_{2})$, so
$v_{1}(\Delta(v_{1},v_{2}))=0<1=v_{2}(\Delta(v_{1},v_{2}))$. Let
$s={v_{1}}^{\frown}1$. Hence $v_{1}\prec[s]$. Furthermore, if $s\subset t$,
then
$t(\Delta(v_{1},v_{2}))=s(\Delta(v_{1},v_{2}))=v_{1}(\Delta(v_{1},v_{2}))=0$,
and
$t\upharpoonright\Delta(v_{1},v_{2})=s\upharpoonright\Delta(v_{1},v_{2})=v_{1}\upharpoonright\Delta(v_{1},v_{2})=v_{2}\upharpoonright\Delta(v_{1},v_{2})$.
Thus $t\prec v_{2}$, which shows $[s]\prec v_{2}$. $\square$

######  Claim 4.4.

The space $S$ is hereditarily Lindelöf.

Proof. Since $S$ is a linearly ordered space (with respect to the
$\prec$-order), to prove that it is hereditarily Lindelöf, it suffices by
Theorem 2.2 in [6] to prove that it satisfies the countable chain condition
($ccc$). It was shown in [11] that the Dedekind completion of any such space
is $ccc$. Using the same method, we apply Claim 4.3 here and observe that if
$\mathcal{A}=\\{[s_{\alpha}]:\alpha\in A\\}$ is a pairwise disjoint listing of
the members of a family of elements of the $\pi$-base for $S$, then
$\\{s_{\alpha}:\alpha\in A\\}$ is an antichain in the Souslin tree $(S,\leq)$,
and so $\mathcal{A}$ must be countable. $\square$

######  Claim 4.5.

Let $\\{(u_{\xi},v_{\xi}):\xi<\omega_{1}\\}$ be a subset of $S\times S$ such
that for all $\xi<\omega_{1}$, there is a $t_{\xi}$ with $o(t_{\xi})\geq\xi,\
t_{\xi}\leq u_{\xi}$, and $t_{\xi}\neq t_{\xi}^{\dagger}\leq v_{\xi}$ (hence
every $o(t_{\xi})$ is a successor). Then this set co-countably converges to
the diagonal in $S\times S$.

Proof. Any open neighborhood of the diagonal $\Delta$ in $S\times S$ will
contain a set of the form W=$\bigcup\\{(s^{-},s^{+})\times(s^{-},s^{+}):s\in
S\\}$, where for all $s\in S$, $s^{-}\prec s\prec s^{+}$. Since $S$ (and hence
$\Delta$) are Lindelöf and $S$ is a Souslin tree, there is a
$\delta<\omega_{1}$ such that the collection $\\{(s^{-},s^{+}):s\in S{\rm\
and\ }o(s)<\delta\\}$ is a countable cover of $S$. Let
$\delta\leq\beta<\omega_{1}$ be large enough so that
max$\\{o(s^{-}),o(s^{+})\\}<\beta$ for all $s\in S$ with $o(s)<\delta$. Choose
any $\xi<\omega_{1}$ such that $\beta+1<\xi$, and note that
$\beta+1<o(t_{\xi})$. Since $t_{\xi}\neq t_{\xi}^{\dagger}$, we can let
${\overline{t}}_{\xi}$ be the member of $S$ so that $t_{\xi}$ and
$t_{\xi}^{\dagger}$ are its immediate successors. Choose $s\in S$ with
$o(s)<\delta$ such that $s^{-}\prec{\overline{t}}_{\xi}\prec s^{+}$. Since
$o({\overline{t}}_{\xi})>\beta\geq{\rm max}\\{o(s^{-}),o(s^{+})\\}+1$, it
follows from Claim 4.2 that $[{\overline{t}}_{\xi}]\subset(s^{-},s^{+})$,
i.e., $s^{-}\prec[{\overline{t}}_{\xi}]$ and $[{\overline{t}}_{\xi}]\prec
s^{+}$. This proves that $(u_{\xi},v_{\xi})\in W$. $\square$

Now we are ready to construct the example. Let $S^{+}$ denote the set of all
$s\in S$ such that $o(s)$ is a successor, $U$ denote the set of all $s\in
S^{+}$ such that for $o(s)=\alpha+1$, $s(\alpha)=0$, and $V=S^{+}\setminus U$.
Then note that $V=\\{u^{\dagger}:u\in U\\}$, each of $U$ and $V$ is cofinal in
$S$ (which means, as defined at the beginning of this section, with respect to
the tree order $\leq$), and thus $U$ and $V$ are disjoint dense subsets of
$S$. We let $X={\mathbb{D}}(U)$, the Alexandroff double of $U$.

######  Theorem 4.6.

The space $X={\mathbb{D}}(U)$ is Lindelöf, and the space $V$ is hereditarily
Lindelöf, but $X\times V$ is not cellular-Lindelöf..

Proof. It suffices to prove that $X\times V$ is not cellular-Lindelöf.

Let $\\{t_{\xi}:\xi\in\omega_{1}\\}$ be an enumeration of a subset of $U$ such
that for each $\xi<\omega_{1}$, $o(t_{\xi})\geq\xi$. For each
$\xi\in\omega_{1}$, choose any $u_{\xi}\in U$ and $w_{\xi}\in V$ such that
$t_{\xi}\leq u_{\xi}$, $u_{\xi}\neq u_{\eta}$ for any $\eta<\xi$, and
$t_{\xi}^{\dagger}<w_{\xi}$. Also choose $w_{\xi}^{-}\prec w_{\xi}\prec
w_{\xi}^{+}$ in $[t_{\xi}^{\dagger}]$, and note that either $t_{\xi}\prec
w_{\xi}^{-}$ or $w_{\xi}^{+}\prec t_{\xi}$. Now, for each $\xi<\omega_{1}$,
the set $U_{\xi}=\\{(u_{\xi},1)\\}\times(V\cap(w_{\xi}^{-},w_{\xi}^{+}))$ is a
nonempty open subset of $X\times V$, and $\\{U_{\xi}:\xi<\omega_{1}\\}$ is
pairwise disjoint.

Let $Y\subset X\times V$, and assume that $Y\cap U_{\xi}$ is nonempty for all
$\xi<\omega_{1}$. We prove that $Y$ is not Lindelöf by proving that it has an
uncountable subset $R$ with no complete accumulation point in $X\times V$. For
each $\xi\in\omega_{1}$ choose $v_{\xi}\in V\cap(w_{\xi}^{-},w_{\xi}^{+})$ so
that $((u_{\xi},1),v_{\xi})\in U_{\xi}\cap Y$, and let
$R=\\{(u_{\xi},1),v_{\xi}):\xi<\omega_{1}\\}$. Note that by construction, each
$v_{\xi}\in[t_{\xi}^{\dagger}]$.

Fix any $(u,v)\in U\times V$. Since $\\{(u,1)\\}\times V$ is a neighborhood of
the point $((u,1),v)$ which intersects $R$ in at most one point, $((u,1),v)$
is not a limit point of $R$. Let us prove that neither is $((u,0),v)$ a
complete accumulation point of $R$. Choose $u^{-}\prec u\prec u^{+}$ and
$v^{-}\prec v\prec v^{+}$ (from $S$) so that $(u^{-},u^{+})\cap(v^{-},v^{+})$
is empty, and so that the closure of $(u^{-},u^{+})\times(v^{-},v^{+})$ is
disjoint from the diagonal in $S\times S$. By Claim 4.5, we may choose
$\eta<\omega_{1}$ so that
$(u_{\xi},v_{\xi})\notin((u^{-},u^{+})\times(v^{-},v^{+}))$ for all
$\eta<\xi<\omega_{1}$. It follows easily that
$((u_{\xi},1),v_{\xi})\notin((u^{-},u^{+})\times\\{0,1\\})\times(v^{-},v^{+})$
for all $\eta<\xi<\omega_{1}$, and that completes the proof. $\square$

######  Remark 4.7.

Another way to prove Theorem 4.6 would be to prove that there is no Lindelöf
subspace of ${\mathbb{D}}(U)\times V$ which is dense in
$(U\times\\{0\\})\times V$, and then appeal to Theorem 3.2.

##  5\. ZFC product space counterexamples

In this section we show that Moore’s L space contains a pair of nowhere
separable hL spaces whose product has no dense Lindelöf subspace.

Moore defines his L space after developing the key properties of a very
special combinatorial object $\mathop{Osc}$. The object $\mathop{Osc}$ is a
function from the ordered pairs $\alpha<\beta<\omega_{1}$ into the finite
subsets of $\alpha$. Similarly, the function $\mathop{osc}(\alpha,\beta)$ is
defined as the cardinality of the set $\mathop{Osc}(\alpha,\beta)$. These
definitions depend on the choice of a family $\vec{\mathcal{C}}=\langle
C_{\alpha}:\alpha\in\omega_{1}\rangle$ where

######  Definition 5.1.

A family $\vec{\mathcal{C}}=\langle C_{\alpha}:\alpha\in\omega_{1}\rangle$ is
a C-sequence on $\omega_{1}$ provided

  1. (1)

$C_{0}=\\{0\\}$,

  2. (2)

for each $0<\alpha<\omega_{1}$, $0\in C_{\alpha}$ is a cofinal subset of
$\alpha$ (in case $\alpha=\beta+1$, this just means $\beta\in C_{\alpha}$),

  3. (3)

for all $\gamma<\alpha$, $C_{\alpha}\cap\gamma$ is finite.

We will be using properties of the function $\mathop{osc}$ and Moore’s L space
from both papers [8] and [9]. In each paper, it is stated that any choice of a
C-sequence will suffice. We mention this here because in order to establish
our main result, we will impose an additional restriction on the C-sequence
that we use.

Here is the definition of Moore’s L space from [8]. First of all, $\mathbb{T}$
denotes the unit circle in the complex plane, and for
$z=e^{i\,r}\in\mathbb{T}$ and $k\in\mathbb{N}$, of course $z^{k}=e^{i(kr)}$ is
also an element of $\mathbb{T}$. The topology on $\mathbb{T}$ is the one
inherited as a subspace of the complex plane. We let ${1}\in\mathbb{T}$ denote
the element $(1,0)$.

######  Definition 5.2.

Let $\\{z_{\alpha}:\alpha\in\omega_{1}\\}$ be a rationally independent set.
For all $\beta\in\omega_{1}$, define $w_{\beta}\in\mathbb{T}^{\omega_{1}}$
with the formula

$w_{\beta}(\xi)=\begin{cases}z_{\xi}^{\mathop{osc}(\xi,\beta)+1}&\xi<\beta\\\
{1}&\beta\leq\xi.\end{cases}$

######  Theorem 5.3 ([8], 7.11).

For all uncountable $X\subset\omega_{1}$,
$\mathcal{L}_{X}=\\{w_{\beta}\restriction X:\beta\in X\\}$ is hereditarily
Lindelöf.

For each $X\subset\omega_{1}$, $\mathcal{L}[X]=\\{w_{\beta}:\beta\in X\\}$. It
is also noted in [8] that the closures of countable subsets of
$\mathcal{L}=\mathcal{L}[{\omega_{1}}]$ are countable, and therefore we have
the following

######  Lemma 5.4.

For all uncountable $X\subset\omega_{1}$, $\mathcal{L}[X]$ is a nowhere
separable hereditarily Lindelöf space.

For convenience we make some assumptions about the family
$\\{z_{\alpha}:\alpha\in\omega_{1}\\}$. We choose a rationally independent set
$\\{r_{\alpha}:\alpha<\omega_{1}\\}$ of reals in the interval
$(0,\frac{1}{2\pi})$, and for each $\alpha<\omega_{1}$ we let $z_{\alpha}$ be
the point in $\mathbb{T}$ obtained by rotating ${1}$ by $2\pi r_{\alpha}$
radians. For convenience we adopt the following metric on $\mathbb{T}$. For
points $z,w$ of $\mathbb{T}$ we let $\rho(z,w)$ be the smallest value $0\leq
r<2\pi$ so that $w\in\\{e^{-2\pi r\,i}z,e^{2\pi r\,i}z\\}$. In other words,
$\rho(z,w)$ is $\frac{1}{2\pi}$ times the arc length within $\mathbb{T}$
between $z$ and $w$. Note that (since $\mathbb{T}$ is a group) for any
$u\in\mathbb{T}$, $\rho(z,w)=\rho(u\cdot z,u\cdot w)$.

For any point $w\in\mathbb{T}^{\omega_{1}}$, let $U[w;F,\epsilon]$ (for finite
$F\subset\omega_{1}$ and real $\epsilon>0$) denote the basic open set
consisting of all points $w^{\prime}\in\mathbb{T}^{\omega_{1}}$ satisfying
that $\rho(w^{\prime}(\xi),w(\xi))<\epsilon$ for all $\xi\in F$.

Let $\Lambda$ denote the set of limit ordinals in $\omega_{1}$, and then let
$\Lambda^{\prime}$ denote those members $\delta$ of $\Lambda$ such that
$\Lambda\cap\delta$ is cofinal in $\delta$ (i.e., $\Lambda^{\prime}$ is the
set of limit points of $\Lambda$).

######  Lemma 5.5.

There is a choice of the C-sequence $\vec{\mathcal{C}}=\langle
C_{\alpha}:\alpha\in\omega_{1}\rangle$ so that there are disjoint uncountable
subsets $X$ and $Y$ of $\omega_{1}$ satisfying that for all
$\delta\in\Lambda^{\prime}$ and $\delta<\alpha\in X$ (respectively $Y$) and
$\alpha<\beta\in Y$ (respectively $X$),
$|\mathop{Osc}(\alpha,\beta)\cap\delta|>1$.

We will prove Lemma 5.5 later. For now we fix the pair of uncountable sets $X$
and $Y$. One of the key properties of $\mathop{osc}$ is proven in Lemma 4.4 of
[8]. This is improved in Lemma 8 of [9] by Peng and Wu. We will just need a
minor variation of a special case of this Lemma 8 that we record here. The set
of two-element subsets of $\omega_{1}$ is denoted by $[\omega_{1}]^{2}$. For
$a\in[\omega_{1}]^{2}$, we let $a(0)=\min(a)$ and $a(1)=\max(a)$. For
$a,b\in[\omega_{1}]^{2}$, we let $a<b$ denote the relation that $a(1)<b(0)$.
We will apply this Lemma to pairs $a\in[\omega_{1}]^{2}$ that have an element
in each of $X$ and $Y$ from Lemma 5.5. It follows that the integer $c$ in the
statement of the next lemma would then be at least $2$.

######  Lemma 5.6.

Suppose that $\mathcal{A}\subset[\omega_{1}]^{2}$ is an uncountable family of
pairwise disjoint sets. Then there are an uncountable collection
$\mathcal{A}^{\prime}\subset\mathcal{A}$ , an integer $c$, and a
$\delta\in\Lambda^{\prime}$ satisfying that for any $a<b$, both in
$\mathcal{A}^{\prime}$, $|\mathop{Osc}(a(0),a(1))\cap\delta|=c$ and

$\mathop{osc}(a(0),b(0))+c-1\leq\mathop{osc}(a(0),b(1))\leq\mathop{osc}(a(0),b(0))+c.$

We defer the proofs of Lemma 5.5 and Lemma 5.6 until after we prove that the
main result is a consequence.

######  Theorem 5.7.

If $X$ and $Y$ are the disjoint uncountable subsets as in Lemma 5.6, then
$\mathcal{L}[X]\times\mathcal{L}[Y]$ does not contain a dense Lindelöf subset.

###### Proof.

Assume that $D\subset X\times Y$ and that
$\tilde{D}=\\{(w_{\beta},w_{\gamma}):(\beta,\gamma)\in D\\}$ is dense in
$\mathcal{L}[X]\times\mathcal{L}[Y]$. We will produce an uncountable subset of
$\tilde{D}$ that has no complete accumulation point in
$\mathcal{L}[X]\times\mathcal{L}[Y]$.

By Lemma 5.4, we have that, for all $\delta\in\omega_{1}$,

$\tilde{D}_{\delta}=\\{(w_{\beta},w_{\gamma}):(\beta,\gamma)\in
D\cap(\delta\times\omega_{1}\cup\omega_{1}\times\delta)\\}$

is not dense. Therefore we may choose an uncountable subset $D_{1}\subset D$
that has the property that if $d_{1},d_{2}\in D_{1}$ are distinct, then there
is a $\delta\in\omega_{1}$ such that (wlog) $d_{1}\in\delta\times\delta$ and
both coordinates of $d_{2}$ are greater than $\delta$. By passing to a subset
of $D_{1}$ we can assume that (wlog) for each $d\in D_{1}$, the first
coordinate of $d$ ($d(0)$ in $X$) is less than the second coordinate of $d$
($d(1)$ which is in $Y$). Let $\mathcal{A}$ be the uncountable set of disjoint
pairs $\\{\\{d(0),d(1)\\}:d\in D_{1}\\}$. By applying Lemma 5.6, we may choose
an integer $c$ and an uncountable $D_{2}\subset D_{1}$ such that for all
$a,b\in D_{2}$ with $a(1)<b(0)$, we have that
$\mathop{osc}(a(0),b(0))+c-1\leq\mathop{osc}(a(0),b(1))\leq\mathop{osc}(a(0),b(0))+c$.
By Lemma 5.5, the value of $c-1$ is positive.

For each real $r$, we let $[r]_{2\pi}$ denote the value $r-2\pi\ell$, where
$2\pi\ell\leq r<2\pi(\ell+1)$, i.e., $e^{i\,r}=e^{i\,[r]_{2\pi}}$. Note that
for $a<d$, both in $D_{2}$ and $k=\mathop{osc}(a(0),d(0))+1$, the value of
$\rho(w_{d(0)}(a(0)),w_{d(1)}(a(0)))$ equals one of

$\rho(z_{a(0)}^{k},z_{a(0)}^{k+c})=\rho({1},z_{a(0)}^{c})=[c\,r_{a(0)}]_{2\pi}\rm{\
and}$
$\rho(z_{a(0)}^{k},z_{a(0)}^{k+c-1})=\rho({1},z_{a(0)}^{c-1})=[(c-1)\,r_{a(0)}]_{2\pi}~{}~{}.$

Now we choose three uncountable subsets of $D_{2}$. First of all, choose any
pair $0<s_{1}<s_{2}$ so that

  1. (1)

$s_{1}$ and $s_{2}$ are complete accumulation points of
$(s_{1},s_{2})\cap\\{r_{d(0)}:d\in D_{2}\\}$,

  2. (2)

and there is an $\ell$ with $\pi\ell<cs_{1}<cs_{2}<\pi(\ell+1),$

and let $s=\frac{s_{1}+s_{2}}{2}$. Note that for $s_{1}<r_{1}<r_{2}<s_{2}$,
$[cr_{2}]_{2\pi}-[cr_{1}]_{2\pi}=c(r_{2}-r_{1})$.

Choose $0<\epsilon<\frac{s_{2}-s_{1}}{5}$ so that $D_{4}=\\{b\in
D_{2}:r_{b(0)}\in(s_{1}+2\epsilon,s_{2}-2\epsilon)\\}$ is uncountable. Also
let $D_{3}=\\{a\in D_{2}:r_{a(0)}\in(s_{1},s_{1}+\epsilon)\\}$, and
$D_{5}=\\{b\in D_{2}:r_{b(0)}\in(s_{2}-\epsilon,s_{2})\\}$. We note that each
of $D_{3},D_{4},D_{5}$ is uncountable.

For any $a\in D_{i}$, $a<b\in D_{j}$ ($3\leq i\neq j\leq 5$), $b<d\in D_{2}$,
and $c^{\prime}\in\\{c-1,c\\}$, we have that

$|\,[c^{\prime}\,r_{a(0)}]_{2\pi}-[c^{\prime}\,r_{b(0)}]_{2\pi}|=c^{\prime}|r_{a(0)}-r_{b(0)}|>\epsilon~{}~{}~{}.$

This implies that if $a_{3}<a_{4}<a_{5}<d$ with $a_{i}\in D_{i}$ and $d\in
D_{2}$, then there is some choice $\\{a,b\\}\subset\\{a_{3},a_{4},a_{5}\\}$ so
that

$|\rho(w_{d(0)}(a(0)),w_{d(1)}(a(0)))-\rho(w_{d(0)}(b(0)),w_{d(1)}(b(0)))|>\epsilon~{}.$

We prove that the uncountable set $\\{(z_{d(0)},z_{d(1)}):d\in D_{2}\\}$ has
no complete accumulation point in $\mathcal{L}[X]\times\mathcal{L}[Y]$. Fix
any pair $(u,v)\in X\times Y$ and choose $a\in D_{3}$ so that
$\max\\{u,v\\}<a(0)$; then choose $b,e$ so that $a<b\in D_{4}$, and $b<e\in
D_{5}$. Use the neighborhood $U(w_{u};\\{a(0),b(0),e(0)\\},\epsilon/4)\times
U(w_{v};\\{a(0),b(0),e(0)\\},\epsilon/4)$ for $(w_{u},w_{v})$ in
$\mathcal{L}[X]\times\mathcal{L}[Y]$. Let $r=\rho(z_{u},z_{v})$ and recall
that $w_{u}(\xi)={1}$ and $w_{v}(\xi)={1}$ for all $a(0)\leq\xi\in\omega_{1}$.
Now suppose that $(w_{d(0)},w_{d(1)})$ is in this neighborhood for some $d\in
D$. It follows that for each $\xi\in\\{a(0),b(0),e(0)\\}$,
$\rho(w_{d(0)}(\xi),{1})<\epsilon/4$ and $\rho(w_{d(1)}(\xi),{1})<\epsilon/4$.
This implies that
$\rho(w_{d(0)}(\xi),w_{d(1)}(\xi))\in(-\epsilon/2,\epsilon/2)$ for each
$\xi\in\\{a(0),b(0),e(0)\\}$. This implies that $d$ is not an element in
$D_{2}$ above $e$ since for such $d\in D_{2}$ there is a pair
$\xi,\zeta\in\\{a(0),b(0),e(0)\\}$ such that
$|\rho(w_{d(0)}(\xi),w_{d(1)}(\xi))-\rho(w_{d(0)}(\zeta),w_{d(1)}(\zeta))|>\epsilon$.
$\square$

Now we prove Lemma 5.5.

###### Proof of Lemma 5.5:.

Let $\Lambda^{\prime\prime}$ be the set of $\lambda\in\Lambda^{\prime}$ such
that $\Lambda^{\prime}\cap\lambda$ is cofinal in $\lambda$. To prove the Lemma
it will suffice to define uncountable subsets $X$ and $Y$ of $\omega_{1}$ and
to then prove that for all $x\in X$ and $y,y^{\prime}\in Y$ with
$y<x<y^{\prime},$

$|\mathop{Osc}(y,x)\cap\omega|>1\ \mbox{and}\
|\mathop{Osc}(x,y^{\prime})\cap\omega|>1~{}~{}.$

In fact, to accomplish this we will be more prescriptive in our definitions
and by the end of this proof we will have proven

######  Fact 5.8.

For all $x\in X$ and $y,y^{\prime}\in Y$ with $y<x<y^{\prime}$,

$\\{30,200\\}\subset\mathop{Osc}(y,x)$ and
$\\{10,90\\}\subset\mathop{Osc}(x,y^{\prime}).$

Of course the specific choice of $\\{10,30,90,200\\}$ is quite arbitrary but
the numbers do have to be some distance apart. Our next steps are to recall
the underlying definitions from [8] related to the Osc function and to then
construct a specific $C$-sequence that will work. Our choice of the
uncountable sets $X$ and $Y$ is also critical and are chosen so as to allow us
to carefully control the values of $\mbox{Osc}(\delta,z)\cap\omega$ for
$\delta\in\Lambda^{\prime}$ and $z\in(X\cup Y)\setminus\delta$.

The first definition we will need is that of the lower trace.

######  Definition 5.9 ([8]).

If $\alpha\leq\beta<\omega_{1}$, then $L(\alpha,\beta)$ is defined recursively
by

$L(\alpha,\alpha)=\emptyset$,

$L(\alpha,\beta)=\left(L(\alpha,\min(C_{\beta}\setminus\alpha))\cup\\{\max(C_{\beta}\cap\alpha)\\}\right)\setminus\max(C_{\beta}\cap\alpha)$.

We will see below that $\mathop{Osc}(x,y)\subset L(x,y)$, and so we will
certainly need $L(x,y)\cap\omega$ to be non-empty (in fact, quite large).
Since $\max(C_{\beta}\cap\alpha)$ is likely to be greater than $\omega$, in
order for us to be ensuring that $\mathop{Osc}(x,y)\cap\omega$ is non-empty
for special pairs $x,y$, we will need to be adding elements to
$\mathop{Osc}(x,y)$ as per the recursive nature of the definition. The
following is Fact 1 from [8].

######  Fact 5.10.

If $\alpha\leq\beta\leq\gamma$ and
$\max(L(\beta,\gamma))<\min(L(\alpha,\beta))$, then

$L(\alpha,\gamma)=L(\alpha,\beta)\cup L(\beta,\gamma)$.

The next definition that we need is the following recursively defined function
$\varrho_{1}$ of Todorčević.

######  Definition 5.11 ([14]).

If $\alpha\leq\beta$, then $\varrho_{1}(\alpha,\beta)$ is defined recursively
by

$\varrho_{1}(\alpha,\alpha)=0,$

$\varrho_{1}(\alpha,\beta)=\max\left(|C_{\beta}\cap\alpha|,\varrho_{1}(\alpha,\min(C_{\beta}\setminus\alpha))\right)$.

Following [8], we let $e_{\beta}:\beta\to\omega$ be the function defined by
$e_{\beta}(\xi)=\varrho_{1}(\xi,\beta)$. Again we loosely note that in order
to ensure that for some integer $k$ and $\omega<x<y\in\omega_{1}$,
$\rho_{1}(k,x)<\rho_{1}(k,y)$, we will somehow need to have different values
for $|C_{\beta}\cap k|$ for appropriate $\beta$ arising in Definition 5.11.
And now we have the definition of $\mathop{Osc}$ and can see why we will need
to have $\rho_{1}(k,x)<\rho_{1}(k,y)$ at critical integers $k$.

######  Definition 5.12.

For $\alpha<\beta\in\omega_{1}$, $\mathop{Osc}(\alpha,\beta)$ is the set of
$\xi\in L(\alpha,\beta)\setminus\min(L(\alpha,\beta))$ such that
$e_{\alpha}(\xi^{-})\leq e_{\beta}(\xi^{-})$ and
$e_{\alpha}(\xi)>e_{\beta}(\xi)$, where $\xi^{-}$ is the maximum of
$L(\alpha,\beta)\cap\xi$.

The value of $\mathop{osc}(\alpha,\beta)$ is equal to the cardinality of
$\mathop{Osc}(\alpha,\beta)$.

We are ready to make our choices of $X$ and $Y$:

$X=\\{\delta+\omega\cdot 8:\delta\in\Lambda^{\prime\prime}\\}\ \ \mbox{and}\ \
\ Y=\\{\delta+\omega\cdot
9:\delta\in\Lambda^{\prime}\setminus\Lambda^{\prime\prime}\\}~{}~{}.$

For each $\min(\Lambda^{\prime})\leq\alpha\in\omega_{1}$, let
$\delta_{\alpha}$ be the supremum of $\Lambda^{\prime}\cap\alpha$ (thus
$\delta_{\alpha}\in\Lambda^{\prime}$), and let
$T=\\{\alpha:\min(\Lambda^{\prime})<\alpha\ \mbox{and}\
\delta_{\alpha}<\alpha\leq\delta_{\alpha}+\omega\cdot 9\\}$.

We now give the definition of our $C$-sequence. We hope that the discussion
above about our goal in ensuring that $L(x,y)\cap\omega$ and
$L(y,x)\cap\omega$ are substantial, as well as occasionally needing large
values of $|C_{\beta}\cap k|$ for integers $k$, partially serves to motivate
the definition. In particular, we make special coherent choices for
$C_{\delta+\omega\cdot k}$ for $\delta\in\Lambda^{\prime}$ and $k\leq 9$. We
also need $\rho(\delta,\delta+\omega\cdot 8)$ and
$\rho(\delta,\delta+\omega\cdot 9)$ to be substantially different and we will
accomplish this by ensuring that the recursive definition of
$L(\delta,\delta+\omega\cdot 8)$ proceeds through $\\{\delta+\omega\cdot
6,\delta+\omega\cdot 4,\delta+\omega\cdot 2\\}$ and that of
$L(\delta,\delta+\omega\cdot 9)$ through $\\{\delta+\omega\cdot
7,\delta+\omega\cdot 5,\delta+\omega\cdot 3,\delta+\omega\\}$. At each stage
an integer from $C_{\delta+\omega\cdot k}$ will be added to
$L(\delta,\delta+\omega\cdot 8)$ but only if it is larger than those added at
earlier steps (see Definition 5.9). We use the integer 1000 in cases when we
do not want small integers added to $L(\alpha,\beta)$. We will use the
following sets $F_{1},F_{2},\ldots,F_{9}$ in presenting our definition of our
$C$-sequence:  $F_{1}=[5,10]\cup[20,40]\cup\\{90\\}\cup[100,200]$
$F_{2}=[2,10]\cup\\{30\\}\cup[40,90]\cup\\{200\\}$
$F_{3}=[5,10]\cup[20,40]\cup\\{90\\}$ $F_{4}=[2,10]\cup\\{30\\}\cup[40,90]$
$F_{5}=[5,10]\cup[20,40]$ $F_{6}=[2,10]\cup\\{30\\}$ $F_{7}=[5,10]$
$F_{8}=\emptyset$ $F_{9}=\emptyset$

Recall that $T=\\{\alpha:\min(\Lambda^{\prime})<\alpha\ \mbox{and}\
\delta_{\alpha}<\alpha\leq\delta_{\alpha}+\omega\cdot 9\\}$.

######  Definition 5.13.

For each ordinal $\alpha<\omega_{1}$, we define $C_{\alpha}$ as follows:

  1. (1)

$C_{0}=\\{0\\}$ and if $\omega<\alpha=\beta+1$, then
$C_{\alpha}=\\{0,1000,\beta\\}$,

  2. (2)

if $0<\alpha<\omega$, then $C_{\alpha}=\\{0,\alpha-1\\}$,

  3. (3)

if $\alpha\in\Lambda\cap\min(\Lambda^{\prime})$ or if
$\alpha\in\Lambda^{\prime}$, then $C_{\alpha}$ is any suitable cofinal subset
of $\alpha$ such that $C_{\alpha}\cap[0,1000]=\\{0,1000\\}$,

  4. (4)

if $\alpha\in T$ and $\alpha=\delta_{\alpha}+\omega\cdot k$, let
$j=\max(0,k-2)$ and
$C_{\alpha}=\\{0\\}\cup F_{k}\cup\\{\delta_{\alpha}+\omega\cdot
j\\}\cup(\delta_{\alpha}+\omega\cdot(k{-}1),\delta_{\alpha}+\omega\cdot k)$.

  5. (5)

if $\delta_{\alpha}<\alpha\in\Lambda\setminus T$, and if
$\beta<\alpha\leq\beta+\omega$ for some $\beta\in\Lambda$, then
$C_{\alpha}=\\{0,1000,\delta_{\alpha}\\}\cup(\beta,\alpha)$.

Let us note that $1000\in C_{\beta}$ for all $\omega\leq\beta\notin T$ and
prove the following:

######  Fact 5.14.

For all $\beta\notin T$ and $0<k<\min(\beta,1000)$, $\varrho_{1}(k,\beta)=1$.

Proof of Fact 5.14: For each $0<n<\omega$, $C_{n}=\\{0,n-1\\}$ and so, by
induction on $n>k$, $\rho_{1}(k,n)=1$. If $\omega\leq\beta\notin T$, then
$C_{\beta}\cap k=\\{0\\}$ and $1000=\min(C_{\beta}\setminus k)$. Therefore,
$\rho_{1}(k,\beta)=1$

It will be helpful to notice that $\max(C_{\beta}\cap\alpha)$ is the minimum
element of $L(\alpha,\beta)$.

######  Fact 5.15.

If $\delta\in\Lambda^{\prime}$ and $\alpha=\delta+\omega\cdot 8\in X$ then
$L(\delta_{\alpha},\alpha)=\\{0,30,90,200\\}.$

Proof of Fact 5.15: For each $0\leq i\leq 4$, let
$\beta_{i}=\delta+\omega\cdot(2i)$. For each $0\leq i<4$,
$\min(C_{\beta_{i+1}}\setminus\delta)=\beta_{i}$ and
$\max(C_{\beta_{i+1}}\cap\delta)=\min(L(\delta,\beta_{i+1}))>\max(C_{\beta_{i}}\cap\delta)$.
This implies that $\max(C_{\beta_{i+1}}\cap\delta)\cup
L(\delta,\beta_{i})\subset L(\delta,\beta_{i+1})$. Now
$L(\delta,\beta_{1})=\\{200\\}$, $L(\delta,\beta_{2})=\\{90,200\\}$,
$L(\delta,\beta_{3})=\\{30,90,200\\}$,
$L(\delta,\beta_{4})=\\{0,30,90,200\\}$.

By a similar argument, which we skip, we also have:

######  Fact 5.16.

If $\delta\in\Lambda^{\prime}$ and $\alpha=\delta+\omega\cdot 9$ then
$L(\delta_{\alpha},\alpha)=\\{0,10,40,90,200\\}.$

Using Definition 5.11 and following the ideas of the proofs of Fact 5.15 and
Fact 5.16 we have the following. We leave the simple checking to the reader.

######  Fact 5.17.

Let $\delta\in\Lambda^{\prime}$, and, for $k=0,\ldots,9$, let
$\beta_{k}=\delta+\omega\cdot k$. Then, for each $j<201$,
$e_{\beta_{8}}(j)=\varrho_{1}(j,\beta_{8})=|C_{\beta_{2}}\cap j|=|F_{2}\cap
j|$ and $e_{\beta_{9}}(j)=\varrho_{1}(j,\beta_{9})=|F_{1}\cap j|$.

Again using that for all $1000<\xi<\delta\in\Lambda^{\prime}$, $1000$ is in
$C_{\delta}$, we have that $1000\leq\min(L(\xi,\delta))$ and we record this
next fact.

######  Fact 5.18.

If $\omega\leq\xi<\delta\in\Lambda^{\prime}$, then

  1. (1)

$L(\delta,\delta+\omega\cdot 9)=L(\xi,\delta+\omega\cdot 9)\cap 1000$, and

  2. (2)

$L(\delta,\delta+\omega\cdot 8)=L(\xi,\delta+\omega\cdot 8)\cap 1000$.

Now we are ready to complete the proof of Lemma 5.5 by proving Fact 5.8. Let
$x\in X$ and $y,y^{\prime}\in Y$ with $y<x<y^{\prime}$. By definition,
$x=\delta_{x}+\omega\cdot 8$, $y=\delta_{y}+\omega\cdot 9$,
$y^{\prime}=\delta_{y^{\prime}}+\omega\cdot 9$ and
$\delta_{y}<\delta_{x}<\delta_{y^{\prime}}$. We must prove that
$\\{30,200\\}\subset\mathop{Osc}(y,x)$ and
$\\{10,90\\}\subset\mathop{Osc}(x,y^{\prime})$.
By Facts 5.15 and 5.16 we have that

$L(\delta_{x},x)=\\{0,30,90,200\\}\ \ \ \mbox{and}\ \ \
L(\delta_{y},y)=L(\delta_{y^{\prime}},y^{\prime})=\\{0,10,40,90,200\\}~{}~{}.$

It then follows from Fact 5.18 that

$L(y,x)=\\{0,30,90,200\\}\ \ \ \mbox{and}\ \ \
L(x,y^{\prime})=\\{0,10,40,90,200\\}~{}~{}.$

By Facts 5.17 and the fact that

$C_{\delta_{x}+\omega\cdot 2}\cap 1000=\\{0,30,200\\}\cup[2,10]\cup[40,90],$

we have the following values:

$e_{x}(0)=0,e_{x}(10)=9,e_{x}(30)=10,e_{x}(40)=11,e_{x}(90)=62,e_{x}(200)=63.$

Similarly

$C_{\delta_{y}+\omega}\cap 1000=C_{\delta_{y^{\prime}}+\omega}\cap
1000=\\{0,90\\}\cup[5,10]\cup[20,40]\cup[100,200],$

and we have the following values for $e_{y}$ (and $e_{y^{\prime}}$):

$e_{y}(0)=0,e_{y}(10)=6,e_{y}(30)=18,e_{y}(40)=28,e_{y}(90)=29,e_{y}(200)=129~{}.$

Now we verify that $\\{30,200\\}\subset\mathop{Osc}(y,x)$. First let
$\xi=30\in L(y,x)=\\{0,30,90,200\\}$ and note that $\xi^{-}=0$. Since
$e_{y}(\xi^{-})=e_{x}(\xi^{-})$ and $e_{y}(30)=18>e_{x}(30)=10$, we have
$30\in\mathop{Osc}(y,x)$. Similarly, with $\xi=200$, we have $\xi^{-}=90$,
$e_{y}(90)=29<e_{x}(90)=62$, and $e_{y}(200)=129>e_{x}(200)=63$.

Next we verify that $\\{10,90\\}\subset\mathop{Osc}(x,y^{\prime})$. With
$\xi=10\in L(x,y^{\prime})=\\{0,10,40,90,200\\}$, we have $\xi^{-}=0$ and
$e_{x}(0)=e_{y^{\prime}}(0)$ while $e_{x}(10)=9>e_{y^{\prime}}(10)=6$. Next
let $\xi=90$ and $\xi^{-}=40$. We again have $e_{x}(40)=11\leq
e_{y^{\prime}}(40)=28$ and $e_{x}(90)=62>e_{y^{\prime}}(90)=29$.

This completes the proof of Lemma 5.5. $\square$

###### Proof of Lemma 5.6:.

Let $\mathcal{A}$ be an uncountable set of pairwise disjoint two element
subsets of $\omega_{1}$. For $a\in\mathcal{A}$ we let $a(0)=\min(a)$ and
$a(1)=\max(a)$. For each limit ordinal $\delta$, choose any
$a_{\delta}\in\mathcal{A}$ so that $\delta\leq\min(a_{\delta})$. Let
$c_{\delta}=|\mathop{Osc}(a_{\delta}(0),a_{\delta}(1))\cap\delta|$.

We will need this next Fact 3 from [8] (proven in [14]).

######  Fact 5.19.

For each $\alpha\leq\beta\in\omega_{1}$, the set of $\xi<\alpha$ such that
$e_{\alpha}(\xi)\neq e_{\beta}(\xi)$ is finite. Also, for each
$\alpha\in\omega_{1}$, the set
$\\{e_{\beta}\restriction\alpha:\alpha\leq\beta\in\omega_{1}\\}$ is countable.

Let $\\{t_{\xi}:\xi\in\omega_{1}\\}$ be an enumeration of the entire family
$\\{e_{\beta}\restriction\alpha:\alpha\leq\beta<\omega_{1}\\}$. By standard
arguments, we may choose a cub $C\subset\omega_{1}$ such that for all
$\delta\in C$,

$\\{t_{\xi}:\xi<\delta\\}=\\{e_{\beta}\restriction\alpha:\alpha<\delta,\
\mbox{and}\ \alpha\leq\beta<\omega_{1}\\}~{}~{}.$

For each $\delta\in C$, there is a $\gamma_{\delta}\in C_{\delta}$ such that

  1. (1)

$L(\delta,a_{\delta}(0))\cup L(\delta,a_{\delta}(1))\subset\gamma_{\delta}$,

  2. (2)

each of $e_{a_{\delta}(0)}\restriction[\gamma_{\delta},\delta)$ and
$e_{a_{\delta}(1)}\restriction[\gamma_{\delta},\delta)$ is equal to
$e_{\delta}\restriction[\gamma_{\delta},\delta)$.

For all $\gamma_{\delta}<\xi<\delta\in C$, $\max(C_{\delta}\cap\xi)$ is the
minimum element of $L(\xi,\delta)$, and so we have that
$\gamma_{\delta}\leq\min(L(\xi,\delta))$. By Fact 5.10, for
$\gamma_{\delta}<\xi<\delta\in C$, and $\delta\leq\beta$, if
$\max(L(\delta,\beta))<\gamma_{\delta}$, then $L(\xi,\beta)=L(\xi,\delta)\cup
L(\delta,\beta)$.

By the pressing down lemma, there is a $\gamma\in\omega_{1}$, functions
$s_{0},s_{1}\in\omega^{\gamma}$, a pair of finite sets
$L_{0},L_{1}\subset\gamma$, an integer $\bar{c}$, and a stationary set
$S\subset C$ satisfying that for all $\delta\in S$,

  1. (1)

$\gamma_{\delta}=\gamma$, $L(\delta,a_{\delta}(0))=L_{0}$,
$L(\delta,a_{\delta}(1))=L_{1}$,

  2. (2)

$e_{a_{\delta}(0)}\restriction\gamma=s_{0}$,
$e_{a_{\delta}(1)}\restriction\gamma=s_{1}$,

  3. (3)

$e_{a_{\delta}(0)}\restriction[\gamma,\delta)=e_{a_{\delta}(0)}\restriction[\gamma,\delta)=e_{\delta}\restriction[\gamma,\delta)$,

  4. (4)

$c_{\delta}=\bar{c}$.

Now let $\eta<\delta$ both be elements of $S$ and we calculate each of
$\mathop{osc}(a_{\eta}(0),a_{\delta}(0))$ and
$\mathop{osc}(a_{\eta}(0),a_{\delta}(1))$. Let
$\zeta=\min(L(a_{\eta}(0),\delta))$ and note that
$\gamma=\gamma_{\delta}\leq\zeta$. To determine the cardinality of
$\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))$, we consider each of

$\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))\cap\gamma$ and
$\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))\setminus\gamma$.

The set $\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))\cap L(\delta,a_{\delta}(0))$
is empty because $L(a_{\eta}(0),\delta)\subset\gamma_{\delta}<a_{\eta}(0)$ and
$e_{a_{\eta}(0)}\restriction\gamma=e_{a_{\delta}(0)}\restriction\gamma$. Also,
$\zeta^{-}<\gamma$ and so
$e_{a_{\eta}(0)}(\zeta^{-})=e_{a_{\delta}(0)}(\zeta^{-})$. Let $k=1$ if
$e_{a_{\eta}(0)}(\zeta)>e_{\delta}(\zeta)$ and otherwise let $k=0$. Since
$L(a_{\eta}(0),a_{\delta}(0))\setminus\gamma$ equals $L(a_{\eta}(0),\delta)$
and
$e_{a_{\delta}}(0)\restriction[\gamma,\delta)=e_{\delta}\restriction[\gamma,\delta)$,
we have that

$\mathop{Osc}(a_{\eta}(0),\delta)\subset\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))\
\setminus\gamma\subset\mathop{Osc}(a_{\eta}(0),\delta)\cup\\{\zeta\\}.$

Similarly, we evaluate each of
$\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\cap\gamma$ and
$\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\setminus\gamma$. We first consider
$\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\setminus\gamma$. Since
$e_{a_{\delta}(1)}\restriction[\gamma,\delta)$ is also equal to
$e_{\delta}\restriction[\gamma,\delta)$, and
$L(a_{\eta}(0),a_{\delta}(1))\setminus\gamma$ is also equal to
$L(a_{\eta}(0),\delta)$, we have that

$\mathop{Osc}(a_{\eta}(0),\delta)\subset\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\
\setminus\gamma\subset\mathop{Osc}(a_{\eta}(0),\delta)\cup\\{\zeta\\}.$

So the only possible difference between

$\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\ \setminus\gamma\ \ \ \mbox{ and}\ \
\ \mathop{Osc}(a_{\eta}(0),a_{\delta}(0))\ \setminus\gamma$

is the singleton $\zeta$. If $k=0$, then
$\zeta\notin\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\setminus\gamma$. However,
if $k=1$, then
$\zeta\in\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))\setminus\gamma$ but $\zeta$
may or may not be in $\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\setminus\gamma$.

Next, we note that $\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\cap\gamma$ equals
$\mathop{Osc}(a_{\eta}(0),a_{\eta}(1))\cap\gamma$, and that this latter set is
the same for all $\eta\in S$. This proves that

$\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))\setminus\\{\zeta\\}\subset\mathop{Osc}(a_{\eta}(0),a_{\delta}(1))\subset\mathop{Osc}(a_{\eta}(0),a_{\delta}(0))~{}.$

Let the value of $c$ stated in Lemma 5.6 be the cardinality of
$\mathop{Osc}(a_{\eta}(0),a_{\eta}(1))\cap\gamma$, and this completes the
proof. $\square$

##  6\. Remarks by the second author and by both authors of this article

While the second author was preparing reviews of [13] and [15], he sent to
several members of the Carolina Seminar some related product questions and
observations, including a counterexample to the method of proof of Theorem
3.12 in [15]. Within a few days of each such communication, the first author
responded with proofs or indications of proofs answering those questions and
disproving Theorem 3.12. At the Seminar’s meeting the following month (on
February 22, 2020), the first author presented a portion of his answers, and
before doing so, he informed the second author that he would like for the two
of them to write this article. This was agreed upon, and then some weeks later
the first author succeeded in proving (and writing up) the results in Section
5.

We acknowledge with appreciation the careful, thorough and detailed review of
this article and the numerous corrections and important suggestions provided
by the referee. These have led to more than 29 significant improvements in the
original version of this article.

## References

  * [1] A. Bella and S. Spadaro, On the cardinality of almost discretely Lindelöf spaces, Monatsh. Math. 186 (2018), no. 2, 345–353; MR3808659
  * [2] XXXXX, Cardinal invariants of cellular-Lindelöf spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2805-2811; MR3956284
  * [3] R. Engelking, General Topology, Sigma Series in Pure Mathematics, 6 Heldermann Verlag, Berlin, 1989; MR1039321
  * [4] R. Hodel, Cardinal functions I, 1–61, in Handbook of Set-Theoretic Topology, by K. Kunen and J.E. Vaughan, North Holland, Amsterdam, 1984; MR0776620
  * [5] K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980;MR0756630
  * [6] D.J. Lutzer and H.R. Bennett, Separability, the countable chain condition and the Lindelöf property in linearly orderable spaces, Proc. Amer. Math. Soc. 23 (1969), 664–667; MR248762
  * [7] J. T. Moore, An $L$-space with a d-separable square, Topology Appl. 155 (2008), no. 4, 304–307; MR2380267
  * [8] XXXXX, A solution to the $L$-space problem, J. Amer. Math. Soc. 19 (2006), no. 3, 717–736; MR2220104
  * [9] Y. Peng and L. Wu, A Lindelöf group with non-Lindelöf square, Adv. Math. 325 (2018), 215–242; MR3742590
  * [10] A.D. Rojas-Sánchez and A. Tamariz-Mascarúa, Spaces with star countable extent, Comment. Math Univ. Caroin. 57 (2016), no. 3, 381–395; MR3554518
  * [11] M.E. Rudin, Souslin’s conjecture, Amer. Math. Monthly 76 (1969), 1113–1119; MR270322
  * [12] B. Šapirovskiĭ, On discrete subspaces of topological spaces: weight, tightness and Suslin number, Soviet Math. Dokl. 13 (1972), 215–219; MR0292012
  * [13] V.V. Tkachuk and R.G. Wilson, Cellular-compact spaces and their applications, Acta Math. Hungar. 159 (2) (2019), 674–688; MR4022157
  * [14] S. Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3–4, 261–294; MR908147
  * [15] W.-F. Xuan and Y.-K. Song, More on cellular-Lindelöf spaces, Topology and its Appl. 266 (2019), 12 pp; MR3997193