# New bounds on the cardinality of $n$-Hausdorff and $n$-Urysohn spaces

Maddalena Bonanzinga111MIFT Department, University of Messina, Italy,
<EMAIL_ADDRESS>., Nathan Carlson222Department of Mathematics, California
Lutheran University, USA<EMAIL_ADDRESS>., Davide Giacopello333MIFT
Department, University of Messina, Italy<EMAIL_ADDRESS>.

###### Abstract

Two new cardinal functions defined in the class of $n$-Hausdorff and
$n$-Urysohn spaces that extend pseudocharacter and closed pseudocharacter
respectively are introduced. Through these new functions bounds on the
cardinality of $n$-Urysohn spaces that represent variations of known results
are given. Also properties of $n$-Urysohn $n$-H-closed spaces are proved.

Keywords: $n$-Hausdorff spaces, $n$-Urysohn spaces, $n$-H-closed space,
pseudocharacter.

AMS Subject Classification: 54A25, 54D10, 54D20, 03E99.

##  1 Introduction

Throughout the paper $n$ denote an integer greater than or equal to $2$. In
[7] Bonanzinga defined the _Hausdorff number_ $H(X)$ of a topological space
$X$ as the least cardinal number $\kappa$ such that for every subset
$A\subseteq X$ with $|A|\geq\kappa$ there exists an open neighbourhood $U_{a}$
for every $a\in A$ such that $\bigcap_{a\in A}U_{a}=\emptyset$. A space $X$ is
said to be _$n$ -Hausdorff_ if $H(X)\leq n$. Of course, with $|X|\geq 2$, $X$
is Hausdorff iff $H(X)=2$. The _Urysohn number_ $U(X)$ of $X$ is the least
cardinal number $\kappa$ such that for every subset $A\subseteq X$ with
$|A|\geq\kappa$ there exists an open neighbourhood $U_{a}$ for every $a\in A$
such that $\bigcap_{a\in A}\overline{U_{a}}=\emptyset$. A space $X$ is said
_$n$ -Urysohn_ if $U(X)\leq n$. Of course, with $|X|\geq 2$, $X$ is Urysohn
iff $U(X)=2$ (see [9, 10]).

For a subset $A$ of a topological space $X$ we will denote by $[A]^{<\lambda}$
$([A]^{\lambda})$ the family of all subsets of $A$ of cardinality
$<\lambda(=\lambda)$.

We consider cardinal invariants of topological spaces (see [17, 22]) and
except for the Hausdorff number $H(X)$ and the Urysohn number $U(X)$, all the
other cardinal functions are multiplied by $\omega$. In particular, given a
topological space $X$, we will denote with $d(X)$ its density, $\chi(X)$ its
character, $\pi\chi(X)$ its $\pi$-character, $t(X)$ its tightness, $\psi(X)$
its pseudocharacter, $\psi_{c}(X)$ its closed pseudocharacter, $nw(X)$ its
net-weight and $l(X)$ its Lindelöf number. Let $(X,\tau)$ be a topological
space, a family $\mathcal{B}\subseteq\tau$ is called $\psi$-base (or
pseudobase) if $\\{x\\}=\bigcap\\{B\in{\cal B}:x\in B\\}$ for every $x\in X$.
If $X$ is $T_{1}$ we define the cardinal function “pseudo weight” as follows
$\psi w(X)=\min\\{|{\cal B}|\,\,:\,\,{\cal B}\hbox{ is a }\psi\hbox{-base for
}X\\}+\omega$.

Recall that, given a space $X$, the cardinal invariant $d_{\theta}(X)$,
closely related to the density $d(X)$, is defined as follows. A subspace
$D\subseteq X$ is $\theta$-dense in $X$ if $D\cap\overline{U}\neq\emptyset$
for every non-empty open set $U$ of $X$. The $\theta$-density $d_{\theta}(X)$
is the least cardinality of a $\theta$-dense subspace of $X$. Observe that
$d_{\theta}(X)\leq d(X)$ for any space $X$ (see [8] and [13] for more details
about $\theta$-density).

In [8, Example 14] it is shown that in the inequality $|X|\leq d(X)^{\chi(X)}$
which is true for all Hausdorff spaces, one can not replace $d$ with
$d_{\theta}$ (again, for Hausdorff spaces) and the following is proved.

######  Theorem 1.1.

[8] If $X$ is $n$-Urysohn space, then $|X|\leq d_{\theta}(X)^{\chi(X)}$.

For a space $X$, the $n$-Hausdorff pseudocharacter of $X$ (where $n\geq 2$),
denoted by $n$-$H\psi(X)$ is the minimum infinite cardinal $\kappa$ such that
for each point $x$ there is a collection $\\{V(\alpha,x):\alpha<\kappa\\}$ of
open neighbourhoods of $x$ such that if $x_{1},...,x_{n}$ are different points
in $X$ then there exist $\alpha_{1},...,\alpha_{n}<\kappa$ such that
$V(\alpha_{1},x_{1})\cap...\cap V(\alpha_{n},x_{n})=\emptyset$ [7]. This
function is defined for $n$-Hausdorff spaces. Of course, with $|X|\geq 2$,
$2$-Hausdorff pseudocharacter of $X$ is exactly the Hausdorff pseudocharacter
of $X$, denoted by $H\psi(X)$ [21].

It is well known that the following chain holds for every Hausdorff space $X$

$\psi(X)\leq\psi_{c}(X)\leq H\psi(X)\leq\chi(X).$

Of course, $H\psi(X)\leq\kappa\Rightarrow n$-$H\psi(X)\leq\kappa$. Also there
exist spaces having countable $n$-Hausdorff pseudocharacter which are not
first countable [7].

Recall the following proposition

######  Theorem 1.2.

[7, Proposition 30] If $X$ is an $n$-Hausdorff space then $|X|\leq
d(X)^{n\hbox{-}H\psi(X)}$.

For a space $X$ the $n$-Urysohn pseudocharacter of $X$, denoted by
$n$-$U\psi(X)$ is the minimum infinite cardinal $\kappa$ such that for each
point $x$ there is a collection $\\{V(\alpha,x):\alpha<\kappa\\}$ of open
neighbourhoods of $x$ such that if $x_{1},...,x_{n}$ are different points in
$X$ then there exist $\alpha_{1},...,\alpha_{n}<\kappa$ such that
$\overline{V(\alpha_{1},x_{1})}\cap...\cap\overline{V(\alpha_{n},x_{n})}=\emptyset$
[12]. This function is defined for $n$-Urysohn spaces. Of course, $2$-Urysohn
pseudocharacter of $X$ is exactly the Urysohn pseudocharacter of $X$, denoted
by $U\psi(X)$, and introduced in [26]. The following holds for every Urysohn
space:

$\psi(X)\leq\psi_{c}(X)\leq H\psi(X)\leq U\psi(X)\leq\chi(X).$

Of course, $U\psi(X)\leq\kappa\Rightarrow n$-$U\psi(X)\leq\kappa$.

The almost Lindelöf degree with respect to closed sets, $aL_{c}(X)$ is the
supremum of $aL(E,X)$ where $E$ is a closed subset of the space $X$ and
$aL(E,X)$ is the minimum cardinal number $\kappa$ such that for every family
$\cal U$ of open subsets of $X$ that covers $E$ there exist a subfamily $\cal
V$ of $\cal U$ such that $|\cal V|\leq\kappa$ and
$E\subseteq\bigcup\\{\overline{V}:V\in{\cal V}\\}$ [29]. The weak Lindelöf
degree of a space $X$, $wL(X)$, is the minimum cardinal number $\kappa$ such
that for every open cover $\cal U$ of $X$ there exist a subfamily $\cal V$ of
$\cal U$ such that $|\cal V|\leq\kappa$ and $X=\overline{\bigcup{\cal V}}$.

In Section 2 we introduce a new cardinal function, denoted by
$n$-$\psi(\cdot)$ and defined for $n$-Hausdorff spaces, which is an extension
of pseudocharacter $\psi(\cdot)$, defined for $T_{1}$ spaces. Note that
$n$-Hausdorffness and satisfying $T_{1}$ separation axiom are independent
properties. We prove that $|X|\leq nw(X)^{n\hbox{-}\psi(X)}$ for each
$n$-Hausdorff space $X$ which is true for $T_{1}$ spaces with $\psi(X)$ in the
place of $n\hbox{-}\psi(X)$. Also we obtain that for $T_{1}$ $n$-Hausdorff
spaces the inequality $|X|\leq\psi w(X)^{l(X)n\hbox{-}\psi(X)}$ holds.

In Section 3 we introduce a new cardinal function, denoted by
$n$-$\psi_{c}(\cdot)$ and defined for $n$-Urysohn spaces which is an extension
of the cardinal function $\psi_{c}(\cdot)$, defined for Hausdorff spaces.
Using this new cardinal function we prove that $|X|\leq
d(X)^{n\hbox{-}\psi_{c}(X)wt(X)}$ for every $n$-Urysohn space $X$ where
$wt(\cdot)$ is a weaker form of tightness introduced by Carlson in [14], and
the following modification for $n$-Urysohn spaces of the Willard-Dissanayake’s
inequality: $|X|\leq d(X)^{\pi\chi(X)n\hbox{-}\psi_{c}(X)}$. Also we show that
if $X$ is an $n$-Urysohn space, then $|X|\leq\pi\chi(X)^{c(X)\chi(X)}$ and we
give a partial answer to the question whether the inequality
$|X|\leq\pi\chi(X)^{c(X)n\hbox{-}\psi_{c}(X)}$ is true for every $n$-Urysohn
space.

In Section 4 we deal with the class of $n$-H-closed spaces and we give some
properties of $n$-Urysohn $n$-H-closed spaces. Recall that an $n$-Hausdorff
space is called $n$-$H$-closed if it is closed in every $n$-Hausdorff space in
which it is embedded [1]. In particular we notice that if $X$ is an
$n$-Urysohn $n$-H-closed space, then $|X|\leq 2^{n\hbox{-}U\psi(X)}$ hence,
$X\leq 2^{\chi(X)}$). In addition, since $|X|\leq 2^{\psi_{c}(X)}$ for every
H-closed space $X$ [16], we pose the question whether $|X|\leq
2^{n\hbox{-}\psi_{c}(X)}$ is true for $n$-Urysohn $n$-H-closed spaces. Also we
prove that $|X|\leq wL(X)^{\chi(X)}$ holds in the class of $n$-Urysohn locally
$n$-H-closed spaces.

##  2 The $n$-pseudocharacter of an $n$-Hausdorff space

In this section we introduce the following cardinal function which is an
extension of pseudocharacter in the class of $n$-Hausdorff spaces.

######  Definition 2.1.

For a space $X$, the $n$-_pseudocharacter_ of $X$ (where $n\geq 2$), denoted
by $n$-$\psi(X)$, is the minimum infinite cardinal $\kappa$ such that for each
point $x$ there is a collection $\\{V(\alpha,x)\,:\,\alpha<\kappa\\}$ of open
neighbourhoods of $x$ such that if $x_{1},...,x_{n}\in X$ are $n$ different
points, then
$\bigcap_{\alpha<\kappa}V(\alpha,x_{1})\cap....\cap\bigcap_{\alpha<\kappa}V(\alpha,x_{n})=\emptyset$.
In other words, $n$-$\psi(X)$ is the minimum infinite cardinal $\kappa$ such
that for each point $x\in X$ there exists a $G_{\kappa}$-set $A_{x}$
containing $x$ such that if $x_{1},...,x_{n}\in X$ are $n$ different points,
then $A_{x_{1}}\cap...\cap A_{x_{n}}=\emptyset$. Note that the
$2$-pseudocharacter is defined for $T_{1}$ spaces while the
$n$-pseudocharacter, $n>2$, is defined for $n$-Hausdorff spaces.

The $2$-pseudocharacter is exactly the pseudocharacter as the following
proposition shows.

######  Proposition 2.2.

Let $X$ be a $T_{1}$ space, then $2$-$\psi(X)=\psi(X)$.

The proof of $2$-$\psi(X)\leq\psi(X)$ is trivial. For every $x\in X$ let
$\\{V(\alpha,x)\,:\,\alpha<\kappa\\}$ be a family of open neighbourhood of $x$
such that if $x,y\in X$ are two different points, then
$\bigcap_{\alpha<\kappa}V(\alpha,x)\cap\bigcap_{\alpha<\kappa}V(\alpha,y)=\emptyset$.
We want to prove that $\\{x\\}=\bigcap_{\alpha<\kappa}V(\alpha,x)$. By
contradiction let $y\in\bigcap_{\alpha<\kappa}V(\alpha,x)$ be a point
different from $x$. Then if we consider
$\bigcap_{\alpha<\kappa}V(\alpha,x)\cap\bigcap_{\alpha<\kappa}V(\alpha,y)$ it
is certainly nonempty because it contains $y$. So we have a contradiction
because
$\bigcap_{\alpha<\kappa}V(\alpha,x)\cap\bigcap_{\alpha<\kappa}V(\alpha,y)=\emptyset$.

It is straightforward to prove the following proposition.

######  Proposition 2.3.

If $X$ is an $n$-Hausdorff space, $n\geq 2$, then $n$-$\psi(X)\leq
n\hbox{-}H\psi(X)$.

Recall that any Cantor cube $2^{\kappa}$ has a dense subset $X$ with
$|X|=\kappa$ and $\psi(X)=\omega$. For sake of completeness, we give the
contruction of the space $X$. Let $\\{s_{\alpha}:\alpha<\kappa\\}$ be an
enumeration of finite nonempty partial functions from $\kappa$ to 2.
Recursively over $\alpha$, construct a sequence of countable partial functions
$\\{p_{\alpha}:\alpha\in\kappa\\}$ from $\kappa$ to $2$ such that (1)
$p_{\alpha}$ extends $s_{\alpha}$ and (2) $p_{\alpha}^{-1}(1)\setminus
s_{\alpha}^{-1}(1)$ is an infinite subset of
$\kappa\setminus\bigcup_{\beta<\alpha}p_{\beta}^{-1}(1)$. Now let
$f_{\alpha}\in 2^{\kappa}$ be an extension of $p_{\alpha}$ such that
$f_{\alpha}(\beta)=0$ for all $\beta\not\in dom(p_{\alpha})$. It is clear that
$F=\\{f_{\alpha}:\alpha<\kappa\\}$ is a dense subset of $2^{\kappa}$ of
cardinality $\kappa$. Also $F$ has countable pseudocharacter. Indeed, it
follows from (2) that
$\\{f_{\alpha}\\}=F\cap\bigcap\\{U_{\beta}:p_{\alpha}(\beta)=1\\}$, where
$U_{\beta}=\\{x\in 2^{\kappa}:x(\beta)=1\\}$.

Further, recall that in the inequality $|X|\leq 2^{c(X)\chi(X)}$ for any
Hausdorff space $X$ [19, 20], character can be replaced with Hausdorff
pseudocharacter in the way similar to the proof of the same theorem (see [7,
Theorem 51]).

Then, since cellularity is hereditary with respect to dense subspaces, it
follows that $c(X)=\omega$. By the inequality $|X|\leq 2^{c(X)H\psi(X)}$ for
any Hausdorff space, we have that if $\kappa>2^{\omega}$ any dense subset $X$
of $2^{\kappa}$, with $|X|=\kappa$ and $\psi(X)=\omega$, is an example of a
Tychonoff space having countable pseudocharacter and uncountable Hausdorff
pseudocharacter. Note that, $\omega=\psi(X)<H\psi(X)\leq\chi(X)=\kappa$;
recently, in [6] Bella, Carlson and Spadaro constructed a Hausdorff space $X$
such that $\psi(X)<H\psi(X)<\chi(X)$ and asked if there exists a regular space
distinguishing the three cardinal functions. Our example is Tychonoff and at
least distinguishes $\psi(X)$ and $H\psi(X)$, giving a partial answer to this
question.

Recall the following result.

######  Theorem 2.4.

[7] For a 3-Hausdorff space X, $|X|\leq 2^{2^{c(X)3-H\psi(X)}}$.

Theorem 2.4 can be extended to any $n$-Haudorff space in the following way.

######  Theorem 2.5.

For a n-Hausdorff space X, $|X|\leq
2^{2^{\cdot^{\cdot^{2^{c(X)n-H\psi(X)}}}}}$, where the power is made
($n-1$)-many times.

Then we obtain the following example.

######  Example 2.6.

A Tychonoff (hence $n$-Hausdorff) space $X$ such that
$n$-$\psi(X)<n\hbox{-}H\psi(X)$.

Consider $n=3$. Let $\kappa>2^{2^{\omega}}$. Any dense subset $X$ of
$2^{\kappa}$ with $|X|=\kappa$ and $\psi(X)=\omega$, is an example of a
Tychonoff space such that $n$-$\psi(X)=\omega$ for any $n$ and, since
$c(X)=\omega$, by Theorem 2.4 we have that $3\hbox{-}H\psi(X)>\omega$. A
similar proof can be done for every $n$ using Theorem 2.5 and taking
$\kappa>2^{2^{\cdot^{\cdot^{2^{\omega}}}}}$, where the power is made
($n-1$)-many times. $\triangle$

Recall the following result

######  Theorem 2.7.

[23, 2.3(a)] Let $X$ be a $T_{1}$ space, then $|X|\leq nw(X)^{\psi(X)}$.

Now we prove (see Theorem 2.8 below) that in the previous result ${\psi(X)}$
can be replaces by $n$-${\psi(X)}$ when $X$ is $n$-Hausdorff instead of
$T_{1}$. Note that $n$-Hausdorffness and satisfying $T_{1}$ separation axiom
are independent properties (see also [7]). Indeed, consider the set
$X=\omega\cup\\{p\\}\cup\\{*\\}$ topologized as follows: $\omega$ has the
discrete topology, an open basic neighbourhood of $p$ is of the form
$U_{p}=\\{p\\}\cup(\omega\setminus F)$ where $F$ is a finite subset of
$\omega$ and an open neighbourhood of $*$ is of the form
$U_{*}=\\{*\\}\cup\\{p\\}\cup(\omega\setminus F)$, where $F$ is a finite
subset of $\omega$. $X$ is a $3$-Hausdorff not $T_{1}$ space. Moreover,
consider the set $Y=4\cup(4\times\omega)$, where $4=\\{0,1,2,3\\}$,
topologized as follows: $4\times\omega$ has the discrete topology, a basic
neighbourhood of $i=0,1,2,3$ is of the form
$U_{i}=\\{i\\}\cup\\{j\\}\times(\omega\setminus F)$, where $F$ is a finite
subset of $\omega$ and $j\not=i$. $Y$ is a $T_{1}$ not $3$-Hausdorff space.

######  Theorem 2.8.

Let $X$ be an $n$-Hausdorff space, then $|X|\leq nw(X)^{n\hbox{-}\psi(X)}$.

Let $\kappa=n\hbox{-}\psi(X)$ and let $\mathcal{N}$ be a network such that
$|{\cal N}|=nw(X)$.
Since $n\hbox{-}\psi(X)=\kappa$, for each $x\in X$ we can fix a collection
$\\{V(\alpha,x)\,:\,\alpha<\kappa\\}$ of open neighbourhoods of $x$ such that
if $x_{1},...,x_{n}\in X$ are $n$ different points, then
$\bigcap_{i=1}^{n}\bigcap\\{V(\alpha,x_{i})\,:\,\alpha<\kappa\\}=\emptyset$.
Let $x\in X$. Then for every $\alpha<\kappa$ there exists
$K(\alpha,x)\in\mathcal{N}$ such that $x\in K(\alpha,x)\subseteq V(\alpha,x)$.
Denote by ${\cal K}(x)=\\{K(\alpha,x)\,\,:\,\,\alpha<\kappa\\}$. Consider the
mapping $\phi:X\to[{\mathcal{N}}]^{\leq\kappa}$ defined by $x\mapsto{\cal
K}(x)$. We want to prove that the cardinality of each fiber is strictly less
than $n$. This will imply that $|X|\leq nw(X)^{\kappa}$. Let $x_{1},...,x_{n}$
be $n$ different points in $X$. Then
$\bigcap_{i=1}^{n}\bigcap\\{V(\alpha,x_{i}):\alpha<\kappa\\}=\emptyset$,
therefore
$\bigcap_{i=1}^{n}\bigcap\\{K(\alpha,x_{i})\,:\alpha<\kappa\\}=\emptyset$. So
we have that there exists a $j\in\\{1,..,n\\}$ such that ${\cal
K}(x_{j})\not={\cal K}(x_{i})$ for some $i\not=j$.

######  Proposition 2.9.

[23, Proposition 2.3] For a $T_{1}$ space $X$, $nw(X)\leq\psi w(X)^{l(X)}$.

Combining Propositions 2.8 and 2.9 we obtain

######  Corollary 2.10.

For a $T_{1}$ $n$-Hausdorff space $X$, $|X|\leq\psi
w(X)^{l(X)n\hbox{-}\psi(X)}$.

The authors were not able to find the source of the result following result
and we give the proof for sake of completness.

######  Proposition 2.11.

For a Hausdorff space $X$, $\psi w(X)\leq d(X)^{t(X)}$.

Let $D$ be a dense subset of $X$ such that $|D|\leq d(X)$. Let $\kappa=t(X)$
and ${\cal D}=\\{\overline{A}:A\in[D]^{\leq\kappa}\\}$. We have that
$X=\overline{D}=\bigcup{\cal D}$ and $|{\cal D}|\leq d(X)^{\kappa}$. Put
${\cal B}=\\{X\setminus\overline{A}:\overline{A}\in{\cal D}\\}$. We will prove
that it is a $\psi$-base. Let $x,y\in X$, then there exists an open subset $U$
of $X$ such that $x\not\in\overline{U}$ and $y\in U$. Since $D$ is dense and
$t(X)\leq\kappa$, $y\in\overline{U\cap D}$ and there exists a subset
$A\in[U\cap D]^{\leq\kappa}$ such that $y\in\overline{A}$. Therefore $y\not\in
X\setminus\overline{A}$ and $x\in X\setminus\overline{A}$.

######  Question 2.12.

Is it true that $\psi w(X)\leq d(X)^{t(X)}$ (or, at least, $\psi w(X)\leq
d(X)^{t(X)l(X)n\hbox{-}\psi(X)}$), for an n-Hausdorff space $X$?

######  Remark 2.13.

Note that, by Propositions 2.8 and 2.9, a positive answer to the previous
question will allow us to prove that $|X|\leq d(X)^{t(X)l(X)n\hbox{-}\psi(X)}$
for every $T_{1}$ $n$-Hausdorff space $X$ and then to obtain a partial answer
to the following question posed in [7].

######  Question 2.14.

[7] Is it true that if $X$ is a $T_{1}$ $n$-Hausdorff space, then $|X|\leq
2^{l(X)t(X)\psi(X)}$?

##  3 $n$-closed-pseudocharacter of an $n$-Urysohn space

In this section we introduce the following cardinal function which is a
generalization of the closed pseudocharacter in the class of $n$-Urysohn
spaces.

######  Definition 3.1.

For a space $X$, the $n$-_closed pseudocharacter of_ $X$ (where $n\geq 2$)
denoted by $n$-$\psi_{c}(X)$, is the minimum infinite cardinal $\kappa$ such
that for each point $x$ there is a collection
$\\{V(\alpha,x)\,:\,\alpha<\kappa\\}$ of open neighbourhoods of $x$ such that
if $x_{1},...,x_{n}\in X$ are $n$ different points, then
$\bigcap\\{\overline{V(\alpha,x_{1})}\,:\alpha<\kappa\\}\cap....\cap\bigcap\\{\overline{V(\alpha,x_{n})}\,:\alpha<\kappa\\}=\emptyset$.
Note that the $2$-closed pseudocharacter is defined for Hausdorff spaces while
the $n$-closed pseudocharacter is defined for $n$-Urysohn spaces.

It is straightforward to prove, as in Proposition 2.2, that the $2$-closed
pseudocharacter is exactly the closed pseudocharacter.

######  Proposition 3.2.

$2$-$\psi_{c}(X)=\psi_{c}(X)$, for every Hausdorff space $X$.

Also, it follows directly from the definitions that the following inequality
holds.

######  Proposition 3.3.

$n\hbox{-}\psi_{c}(X)\leq n\hbox{-}U\psi(X)$, for every $n$-Uryshon space $X$.

######  Example 3.4.

A Tychonoff (hence $n$-Urysohn) space such that
$n\hbox{-}\psi_{c}(X)<n\hbox{-}U\psi(X)$.

Consider the space $X$ of Example 2.6. Recall that pseudocharacter and closed
pseudocharacter coincide for regular spaces and that $n$-$H\psi(Y)\leq
n$-$U\psi(Y)$, for every space $Y.\hfill\triangle$

Recall the following result.

######  Theorem 3.5.

[3] If $X$ is Hausdorff, then $|X|\leq d(X)^{\psi_{c}(X)t(X)}$.

In [14] Carlson generalized the previous result introducing the following
weaker form of tightness.

######  Definition 3.6.

[14] Let $X$ be a space. The weak tightness $wt(X)$ of $X$ is defined as the
least infinite cardinal $\kappa$ for which there is a cover ${\cal C}$ of $X$
such that $|{\cal C}|\leq 2^{\kappa}$ and for all $C\in\mathcal{C}$,
$t(C)\leq\kappa$ and $X=\bigcup_{B\in[C]^{\leq 2^{\kappa}}}\overline{B}$.

It is clear that $wt(X)\leq t(X)$.

######  Theorem 3.7.

[14] If $X$ is Hausdorff, then $|X|\leq d(X)^{\psi_{c}(X)wt(X)}$.

We prove a variation (Theorem 3.11 below) of the previous theorem in the class
of $n$-Urysohn spaces. First we prove the following general result.

######  Theorem 3.8.

Let $\kappa$ and $\lambda$ be two infinite cardinals. Let $X$ be an n-Urysohn
space such that

  * 1.

for every $p\in X$ there exists a family ${\mathcal{V}}_{p}$ of open
neighbourhoods of $p$ such that $|{\mathcal{V}}_{p}|\leq\kappa$ and for every
$x_{1},...,x_{n}\in X$,
$(\bigcap_{V\in{\mathcal{V}}_{x_{1}}}\overline{V})\cap....\cap(\bigcap_{V\in{\mathcal{V}}_{x_{n}}}\overline{V})=\emptyset$.

  * 2.

there exists a subset $S$ of $X$ such that $|S|\leq\lambda$ and
$X=\bigcup_{A\in[S]^{\leq\kappa}}\overline{A}$.

Then $|X|\leq\lambda^{\kappa}$.

For every $p\in X$ fix $A_{p}\in[S]^{\leq\kappa}$ such that
$p\in\overline{A}_{p}$. For every $V\in{\mathcal{V}}_{p}$, we have that
$p\in\overline{A_{p}\cap V}$ and $A_{p}\cap V\in[S]^{\leq\kappa}$. Define the
function $\phi:X\to[[S]^{\leq\kappa}]^{\leq\kappa}$ such that
$\phi(p)=\\{A_{p}\cap V:V\in{\mathcal{V}}_{p}\\}$ for every $p\in X$. We will
show that $\phi$ is a $n-1$ to $1$ map; this implies that
$|X|\leq|[[S]^{\leq\kappa}]^{\leq\kappa}|\leq(|S|^{\kappa})^{\kappa}\leq\lambda^{\kappa}$.
Assume, by contradiction, there exist $x_{1},...,x_{n}\in X$ such that
$\phi(x_{1})=...=\phi(x_{n})$. Then $\\{A_{x_{1}}\cap
V:V\in{\mathcal{V}}_{x_{1}}\\}=...=\\{A_{x_{n}}\cap
V:V\in{\mathcal{V}}_{x_{n}}\\}$. Therefore
$(\bigcap_{V\in{\mathcal{V}}_{x_{1}}}\overline{V})\cap....\cap(\bigcap_{V\in{\mathcal{V}}_{x_{n}}}\overline{V})\not=\emptyset$;
a contradiction.

In [24], Juhász and van Mill introduced the notion of a
$\mathcal{C}$-saturated subset of a space $X$.

######  Definition 3.9.

Given a cover $\mathcal{C}$ of $X$, a subset $A\subseteq X$ is
$\mathcal{C}$-_saturated_ if $A\cap C$ is dense in $A$ for every
$C\in\mathcal{C}$.

It is clear that the union of $\mathcal{C}$-saturated subsets is
$\mathcal{C}$-saturated. The following is given in [24] in the case
$\kappa=\omega$, and extended to the general case in [14].

######  Lemma 3.10.

[24, 14] Let $X$ be a space, $wt(X)=\kappa$, and let $\mathcal{C}$ be a cover
witnessing that $wt(X)=\kappa$. Then for all $x\in X$ there exists
$S_{x}\in[X]^{\leq 2^{\kappa}}$ such that $x\in S_{x}$ and $S_{x}$ is
$\mathcal{C}$-saturated.

######  Theorem 3.11.

If $X$ is an $n$-Urysohn space, then $|X|\leq
d(X)^{n\hbox{-}\psi_{c}(X)wt(X)}$.

Put $\kappa=n$-$\psi_{c}(X)wt(X)$. Let $\mathcal{C}$ be a cover witnessing
that $wt(X)\leq\kappa$ and $D$ be a dense subset of $X$ such that $|D|=d(X)$.
By Lemma 3.10, for all $x\in X$ there exists a $\mathcal{C}$-saturated set
$S_{x}$ such that $x\in S_{x}$ and $|S_{x}|\leq 2^{\kappa}$. Let
$S=\bigcup_{d\in D}S_{d}$. Then $S$ is $\mathcal{C}$-saturated, as $S$ is the
union of $\mathcal{C}$-saturated sets, and $|S|\leq|D|\cdot
2^{\kappa}\leq|D|^{\kappa}$. Observe that as $D\subseteq S$, we have that $S$
is dense in $X$. Since $n$-$\psi_{c}(X)\leq\kappa$, for every $p\in X$ fix a
collection $\\{V(\alpha,p)\,:\,\alpha\leq\kappa\\}$ of open neighbourhoods of
$p$ such that if $x_{1},...,x_{n}\in X$ are $n$ different points, then
$\bigcap\\{\overline{V(\alpha,x_{1})}\,:\alpha<\kappa\\}\cap....\cap\bigcap\\{\overline{V(\alpha,x_{n})}\,:\alpha<\kappa\\}=\emptyset$.
Fix $x\in X$ and let $C\in\mathcal{C}$ such that $x\in C$. We will show that
for each $\alpha<\kappa$ we have $x\in\overline{V(\alpha,x)\cap S\cap C}^{C}$.
As $S$ is $\mathcal{C}$-saturated (hence $S\cap C$ is dense in $S$) and $S$ is
dense in $X$, we have $S\cap C$ is dense in $X$ and then $x\in
V(\alpha,x)\subseteq\overline{V(\alpha,x)}\subseteq\overline{V(\alpha,x)\cap
C\cap S}.$ Therefore $x\in C\cap\overline{V(\alpha,x)\cap S\cap
C}=\overline{V(\alpha,x)\cap S\cap C}^{C}$. As $t(C)\leq\kappa$, there exists
$A_{\alpha}\subseteq V(\alpha,x)\cap S\cap C$ such that
$x\in\overline{A_{\alpha}}^{C}\subseteq\overline{A_{\alpha}}$ and
$|A_{\alpha}|\leq\kappa$. Now let $A_{x}=\bigcup_{\alpha<\kappa}A_{\alpha}$
and note $x\in\overline{A_{x}}$, $|A_{x}|\leq\kappa\cdot\kappa=\kappa$, and
$A_{x}\subseteq S$.

Unfixing $x$, we see that $X=\bigcup_{x\in
X}\overline{A_{x}}=\bigcup_{A\in[S]^{\leq\kappa}}\overline{A}$. Letting
$\lambda=d(X)^{\kappa}$, we see that the conditions of Theorem 3.8 hold and
thus $|X|\leq\lambda^{\kappa}=(d(X)^{\kappa})^{\kappa}=d(X)^{\kappa}$.

######  Corollary 3.12.

If $X$ is an $n$-Urysohn space, then $|X|\leq
d(X)^{n\hbox{-}\psi_{c}(X)t(X)}$.

Since $n\hbox{-}\psi_{c}(X)t(X)\leq\chi(X)$, we can obtain the following
corollary.

######  Corollary 3.13.

If $X$ is an $n$-Urysohn space, then $|X|\leq d(X)^{\chi(X)}$.

Considering Theorem 1.1, it is natural to pose the following question. Recall
that the $\theta$-closure of a subset $A$ of a space $X$, denoted by
$cl_{\theta}(A)$, is the subset $\\{x\in X:\overline{U}\cap
A\not=\emptyset\hbox{ for every open neighbourhood }U\hbox{ of }x\\}$ and the
$t_{\theta}(X)$ is the minimum cardinal number $\kappa$ such that for every
$x\in X$ and every subset $A$ of $X$ such that $x\in cl_{\theta}(A)$ there
exists a subset $B$ such that $x\in cl_{\theta}(B)$ and $|B|\leq\kappa$.

######  Question 3.14.

Is it true that $|X|\leq d_{\theta}(X)^{n\hbox{-}\psi_{c}(X)t_{\theta}(X)}$
for every $n$-Urysohn space $X$?

Now we give an $n$-Urysohn generalization of the following Willard-Dissanayake
result.

######  Theorem 3.15.

[29] If $X$ is a Hausdorff space then $|X|\leq d(X)^{\pi\chi(X)\psi_{c}(X)}$.

######  Theorem 3.16.

If $X$ is an $n$-Urysohn space then $|X|\leq
d(X)^{\pi\chi(X)n\hbox{-}\psi_{c}(X)}$.

Let $\kappa=\pi\chi(X)n\hbox{-}\psi_{c}(X)$ and let $D$ be a dense set such
that $|D|=d(X)$. For all $x\in X$, let $\\{V(\alpha,x):\alpha<\kappa\\}$ be a
closed $n$-pseudobase at $x$ and let ${\cal U}_{x}$ be a local $\pi$-base at
$x$ such that $|{\cal U}_{x}|\leq\kappa$. Fix $x\in X$. Let ${\cal
U}(\alpha,x)=\\{U\in{\cal U}_{x}:U\subseteq V(\alpha,x)\\}$. For all
$U\in{\cal U}(\alpha,x)$, let $d(U,\alpha,x)\in U\cap D$. Let
$D(\alpha,x)=\\{d(U,\alpha,x):U\in{\cal U}_{x}\\}$. Note
$|D(\alpha,x)|\leq|{\cal U}_{x}|\leq\kappa$. We show that
$x\in\overline{D(\alpha,x)}$ for every $\alpha<\kappa$. Let $x\in W$, where
$W$ is open. Then $x\in W\cap V(\alpha,x)$ for every $\alpha<\kappa$. As
${\cal U}_{x}$ is a local $\pi$-base at $x$, there exists $U_{\alpha}\in{\cal
U}_{x}$ such that $U_{\alpha}\subseteq W\cap V(\alpha,x)$ and thus
$U_{\alpha}\in{\cal U}(\alpha,x)$ for every $\alpha<\kappa$. It follows that
$d(U_{\alpha},\alpha,x)\in U_{\alpha}\subseteq W\cap V(\alpha,x)$ and thus
$W\cap D(\alpha,x)\not=\emptyset$. This shows that
$x\in\overline{D(\alpha,x)}\subseteq\overline{V(\alpha,x)}$ for every
$\alpha<\kappa$. Consider the function
$\phi:X\to[[D]^{\leq\kappa}]^{\leq\kappa}$ such that for each $x\in X$,
$\phi(x)=\\{D(\alpha,x):\alpha<\kappa\\}$. This is a $(n-1)\hbox{-}1$ map, in
fact, suppose, by contradiction, that there exist $x_{1},...,x_{n}$ different
points in $X$ such that
$\\{D(\alpha,x_{1})\\}_{\alpha<\kappa}=...=\\{D(\alpha,x_{n})\\}_{\alpha<\kappa}$.
Since $\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{i})}$ is not empty for
each $i=1,...,n$,
$\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{1})}=...=\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{n})}=E$
and
$\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{i})}\subseteq\bigcap_{\alpha<\kappa}\overline{V(\alpha,x_{i})}$
for every $i=1,...,n$, we have that
$\emptyset\not=E\subseteq\bigcap_{i=1}^{n}\bigcap_{\alpha<\kappa}\overline{V(\alpha,x_{i})}$.
That is a contradiction.

Recall the following result.

######  Theorem 3.17.

[27] If X is a Hausdorff space, then $|X|\leq\pi\chi(X)^{c(X)\psi_{c}(X)}$.

Carlson proved the following.

######  Theorem 3.18.

[13] For any space $X$, $d_{\theta}(X)\leq\pi\chi(X)^{c(X)}$.

Then by Theorems 1.1 and 3.18, we have the following result.

######  Theorem 3.19.

If $X$ is $n$-Urysohn, then $|X|\leq\pi\chi(X)^{c(X)\chi(X)}$.

Note that recently in [11] it is proved that $|X|\leq 2^{c(X)\pi\chi(X)}$, for
every $n$-Urysohn homogeneous space $X$. It is natural to pose the following
question.

######  Question 3.20.

Is $|X|\leq\pi\chi(X)^{c(X)n\hbox{-}\psi_{c}(X)}$ true for every $n$-Urysohn
space?

Theorem 3.23 below gives a partial answer to the previous question in the
class of $n$-Urysohn quasiregular spaces. Recall that a space is said to be
quasiregular if for each open set $U$ there exists a open subset $V$ of $U$
contained with its closure in $U$. The following examples show that
quasiregularity and $n$-Urysohness are independent properties.

######  Example 3.21.

A quasiregular space which is not $3$-Urysohn.

Consider a space similar to the one of Example 2.4. $X=3\cup(\omega\times 3)$,
where $3=\\{0,1,2\\}$. The points from $\omega\times 3$ has the discrete
topology, instead, a basic open neighbourhood of $i=0,1,2$ is
$U(i,F)=\\{i\\}\cup\\{(\omega\times j)\setminus(F\times
j)\,\,:\,\,j\in\\{0,1,2\\}\,\,,\,\,j\not=i\\}$, where $F$ is a finite subset
of $\omega$. $X$ is not $3$-Urysohn since $0,1$ and $2$ belong to the closure
of each open neighbourhood of them. $\triangle$

Carlson and Ridderbos in [15], under $(\mathfrak{c}^{+}=2^{\mathfrak{c}})$,
constructed an example of a Urysohn ccc space which has the $\pi$-character
equal to the continuum and the density equal to the successor of the
continuum. Then such space cannot be quasiregular since the inequality
$d(X)\leq\pi\chi(X)^{c(X)}$, proved by Sapirovski for regular spaces [26],
holds also for quasiregular spaces, that is

######  Theorem 3.22.

[13] For a quasiregular space $X$, $d(X)\leq\pi\chi(X)^{c(X)}$.

The previous theorem follows from Theorem 3.18 and the fact that $d$ and
$d_{\theta}$ coincide in the class of quasiregular spaces.

Then, by Theorems 3.11 and 3.22, we obtain the following partial answer to
Question 3.20.

######  Theorem 3.23.

If $X$ is an $n$-Urysohn quasiregular space, then
$|X|\leq\pi\chi(X)^{c(X)wt(X)n\hbox{-}\psi_{c}(X)}$.

Recall that every Hausdorff space having a compact $\pi$-base is quasiregular.
In [28], Tkachenko introduced the o-tightness of a space, and in [5] Bella,
Carlson and Gotchev used it to give an improvement of Theorem 3.17 in the
class of Hausdorff spaces having a compact $\pi$-base. We recall that the
o-tightness of a space $X$ does not exceed $\kappa$, or $ot(X)\leq\kappa$, if
for every family $\mathcal{U}$ of open subsets of $X$ and for every point
$x\in X$ with $x\in\overline{\bigcup{\mathcal{U}}}$ there exists a subfamily
$\mathcal{V}\subseteq\mathcal{U}$ such that $|{\mathcal{V}}|\leq\kappa$ and
$x\in\overline{\bigcup{\mathcal{V}}}$. We have that $ot(X)\leq t(X)$ and
$ot(X)\leq c(X)$. Also $wL(X)ot(X)\leq c(X)$. Moreover $ot(X)\leq wt(X)$, for
any space $X$ [5].

######  Theorem 3.24.

[5] If X is a Hausdorff space with a compact $\pi$-base, then
$|X|\leq\pi\chi(X)^{wL(X)ot(X)\psi_{c}(X)}$.

It is natural to pose the following question.

######  Question 3.25.

Is $|X|\leq\pi\chi(X)^{wL(X)ot(X)n\hbox{-}\psi_{c}(X)}$ true for every
$n$-Urysohn (quasiregular) space?

##  4 On $n$-Urysohn $n$-H-closed spaces

In this section we present some properties of $n$-Urysohn $n$-H-closed spaces
motivated by similar properties that hold in the class of H-closed spaces.
Recall that a space is said to be H-closed if it is Hausdorff and it is closed
in every Hausdorff space in which it is embedded. An $n$-Hausdorff space is
called $n$-$H$-closed if it is closed in every $n$-Hausdorff space in which it
is embedded [1].

######  Theorem 4.1.

[1] For an $n$-Hausdorff space $X$ the following are equivalent:

  * 1.

$X$ is $n$-H-closed;

  * 2.

for each open ultrafilter $\mathcal{U}$ on $X$, $|a({\cal U})|=n-1$, where
$a(\cal U)$ denotes the adherence of $\cal U$;

  * 3.

for every open filter ${\cal F}$ on $X$, $|a({\cal F})|\geq n-1$;

  * 4.

for every $A\in[X]^{<n-1}$ and for each family of open subsets ${\cal U}$ of
$X$ such that $X\setminus A\subseteq\bigcup{\cal U}$, there exists a finite
subfamily ${\cal V}$ of ${\cal U}$ such that $X=\bigcup_{V\in{\cal
V}}\overline{V}$.

For a topological space $(X,\tau)$ and $U\in\tau$, let $r(U)$ denote
$int_{X}(\overline{U})$ and $X(s)$ denote the space whose set is $X$ and whose
topology is generated by the base $\\{r(U):U\in\tau\\}$. A function $f:X\to Y$
is called $\theta$-continuous at $x_{0}$ if for every open neighbourhood $V$
of $f(x_{0})$ in $Y$, there exists an open neighbourhood $U$ of $x_{0}$ in $X$
such that $f(\overline{U}^{X})\subseteq\overline{V}^{Y}$.

Recall the following result.

######  Proposition 4.2.

[25] Let $f:X\to Y$ be a $\theta$-continuous surjection. If $X$ is a H-closed
space, then $Y$ is H-closed.

Now we prove the following.

######  Proposition 4.3.

Let $f:X\to Y$ be a $\theta$-continuous bijection. If $X$ is an $n$-H-closed
space then $Y$ is an $n$-H-closed space.

Fix $A\in[Y]^{<n-1}$ and a family $\mathcal{U}$ of open subsets of $Y$ with
$Y\setminus A\subseteq\bigcup\mathcal{U}$. For every $x\in X\setminus
f^{-1}(A)$ there exists an open subset $V(x)\in\mathcal{U}$ such that $f(x)\in
V(x)$. Since $f$ is $\theta$-continuous, there is an open neighbourhood $U(x)$
of $x$ in $X$ such that $f(\overline{U(x)}^{X})\subseteq\overline{V(x)}^{Y}$.
Considering that the space $X$ is $n$-H-closed, there exists a finite subset
$F$ of $X$ such that $X=\bigcup_{x\in F}\overline{U(x)}^{X}$. Since
$Y=f(X)=f(\bigcup_{x\in F}\overline{U(x)}^{X})=\bigcup_{x\in
F}f(\overline{U(x)}^{X})\subseteq\bigcup_{x\in F}\overline{V(x)}^{Y}$ holds,
$Y$ is $n$-H-closed. In [25] it is proved that the identity $id:X(s)\to X$ is
a $\theta$-homeomorphism. This fact allows to prove the following proposition.

######  Proposition 4.4.

[25] A space $X$ is H-closed iff $X(s)$ is H-closed.

So, using Proposition 4.3 and the same fact, we can prove the following.

######  Proposition 4.5.

A space $X$ is $n$-H-closed iff $X(s)$ is $n$-H-closed.

######  Proposition 4.6.

[25] Let $X$ be a H-closed space and $U$ be an open subset of $X$. Then
$\overline{U}$ is H-closed.

We can prove that the previous proposition can be extended to $n$-H-closed
spaces.

######  Proposition 4.7.

Let $X$ be an $n$-H-closed space and $U$ be an open subset of $X$. Then
$\overline{U}$ is $n$-H-closed.

Note that the $n$-Hausdorff property is hereditary. Let $A=\overline{U}^{X}$
and $\mathcal{F}$ an open filter on $A$. Then $\\{F\cap U:F\in\mathcal{F}\\}$
is an open filter base on $X$. Let ${\cal G}=\\{W\subseteq X:W\hbox{ is open
in }X\hbox{ and }F\cap U\subseteq W\hbox{ for some }F\in\mathcal{F}\\}$. Then
$\mathcal{G}$ is an open filter on $X$, therefore $|a_{X}({\cal G})|\geq n-1$.
Since $a_{X}({\cal G})\subseteq\bigcap_{F\in{\cal F}}\overline{(F\cap
U)}^{X}=\bigcap_{F\in{\cal F}}\overline{(F\cap
U)}^{A}\subseteq\bigcap_{F\in{\cal F}}\overline{F}^{A}=a_{A}({\cal F})$, we
have that $|a_{A}({\cal F})|\geq|a_{X}({\cal G})|\geq n-1$.

We prove the following surprising result.

######  Proposition 4.8.

Every $n$-Hausdorff $T_{1}$ space with at least one isolated point is not
$n$-H-closed.

Suppose by contradiction that for each $A\in[X]^{n-1}$ and every family of
open subsets $\cal U$ of $X$ with $X\setminus A\subseteq\bigcup{\cal U}$ there
exists a finite subfamily $\cal V$ of $\cal U$ such that
$X=\bigcup\\{\overline{V}:V\in{\cal V}\\}$. Suppose there exists one isolated
point $x_{1}$. Let $A=\\{x_{1}\\}$. Since $X$ is $T_{1}$ there exists a family
$\cal U$ of open subsets such that $X\setminus A=\bigcup{\cal U}$. Since
$x_{1}$ is isolated then it does not belong to $\overline{U}$ for each
$U\in{\cal U}$, therefore one cannot select a subfamily $\cal V$ of $\cal U$
such that $X=\bigcup\\{\overline{V}:V\in{\cal V}\\}$. Recall that
$\psi_{c}(X)=\psi_{c}(X(s))$, for any Hausdorff space $X$. Now we prove the
following result.

######  Proposition 4.9.

If $X$ is an $n$-Urysohn space then
$n\hbox{-}\psi_{c}(X)=n\hbox{-}\psi_{c}(X(s))$.

Since $\overline{U}=\overline{r(U)}=\overline{r(U)}^{X(s)}$ (see [25]), if $X$
is $n$-Urysohn, then $X(s)$ is $n$-Urysohn and
$n\hbox{-}\psi_{c}(X)=n\hbox{-}\psi_{c}(X(s))$.

The following result is assumed without proof in [16]; we give the proof for
sake of completeness.

######  Lemma 4.10.

Let $X$ be a H-closed space. Then $\chi(X(s))\leq\psi_{c}(X)$.

Let $\kappa=\psi_{c}(X)$ and $x\in X$. There is a family $\mathcal{U}$ of open
neighbourhood of $x$ of such that
$x\in\bigcap_{U\in\mathcal{U}}U\subseteq\bigcap_{U\in\mathcal{U}}\overline{U}=\\{x\\}$
and $\kappa=|\mathcal{U}|$. Without loss of generality we can assume that
$\mathcal{U}$ is closed under finite intersections. We want to show that
$\\{int(\overline{U}):U\in\mathcal{U}\\}$ is a neighborhood base of $x$ in
$X(s)$. Let $T$ be an open neighborhood of $x$ in $X(s)$. As $X(s)$ is
semiregular, we can assume that $T=int(\overline{U})$ is regular open. So,
$\\{x\\}=\bigcap_{U\in\mathcal{U}}\overline{U}\subseteq T$ and then
$X\backslash T\subseteq
X\backslash\bigcap_{U\in\mathcal{U}}\overline{U}=\bigcup_{U\in\mathcal{U}}X\backslash\overline{U}$.
Thus, $\\{X\backslash\overline{U}:U\in\mathcal{U}\\}$ is a family of regular
open sets of $X$ that cover $X\backslash T$. Since $X\backslash T$ is a H-set
(i.e. a regular closed subset in a H-closed space), there is a finite subset
$\mathcal{G}\subseteq\mathcal{U}$ such that $X\backslash
T\subseteq\bigcup_{U\in\mathcal{G}}\overline{X\backslash\overline{U}}=\bigcup_{U\in\mathcal{G}}X\setminus
int(\overline{U})=X\setminus\bigcap_{U\in\mathcal{G}}int(\overline{U})$. Then,
$\bigcap_{U\in\mathcal{G}}int(\overline{U})\subseteq T$ implying $x\in
int(\overline{\bigcap_{U\in\mathcal{G}}U})\subseteq
int(\bigcap_{U\in\mathcal{G}}\overline{U})=\bigcap_{U\in\mathcal{G}}int(\overline{U})\subseteq
T$. By the arbitrarity of $x$, we conclude that $\chi(X(s))\leq\kappa$. Since
the inequalities $\psi_{c}(X)\leq\psi_{c}(X(s))\leq\chi(X(s))$ are true for
every space $X$, by the previous lemma, we obtain the following proposition.

######  Proposition 4.11.

[16] Let $X$ be a H-closed space. Then $\chi(X(s))=\psi_{c}(X)$.

Then it is natural to pose the following question.

######  Question 4.12.

Let $X$ be an $n$-Urysohn $n$-H-closed space. Is $\chi(X(s))\leq
n$-$\psi_{c}(X)$ true?

Recall the following amazing Dow and Porter’s result that strongly improved
the Gryzlov’s theorem: if $X$ is a H-closed space with
$\psi_{c}(X)\leq\omega$, then $|X|\leq 2^{\omega}$[18].

######  Theorem 4.13.

[16] Let $X$ be a H-closed space. Then $|X|\leq 2^{\psi_{c}(X)}$.

The previous theorem was proved using Propositions 4.4, 4.11 and the next
result:

######  Theorem 4.14.

[16] Let $X$ be a H-closed space. Then $|X|\leq 2^{\chi(X)}$.

Theorem 4.14 can be extended to the class of $n$-H-closed spaces.

######  Theorem 4.15.

[2] Let $X$ be an $n$-H-closed space. Then $|X|\leq 2^{\chi(X)}$.

Then it is natural to pose also the following question.

######  Question 4.16.

Is it true that $|X|\leq 2^{n\hbox{-}\psi_{c}(X)}$ (or at least $|X|\leq
2^{n\hbox{-}\psi_{c}(X)t(X)}$) for every $n$-Urysohn $n$-H-closed space?

Using the following result, we give a partial answer (Theorem 4.18 below) to
Question 4.16 in the class of $n$-Urysohn $n$-H-closed spaces.

######  Theorem 4.17.

[12] If $X$ is an $n$-Urysohn space. Then $|X|\leq
2^{aL(X)n\hbox{-}U\psi(X)}$.

######  Theorem 4.18.

Let $X$ be an $n$-Urysohn $n$-H-closed space. Then $|X|\leq
2^{n\hbox{-}U\psi(X)}$ (hence $|X|\leq 2^{\chi(X)}$).

######  Remark 4.19.

Note that, by Theorems 4.5 and 4.15, a positive answer to Question 4.12 allows
us to give a positive answer to Question 4.16 too.

Finally, we give a bound for the cardinality of $n$-Urysohn, locally
$n$-H-closed spaces. This result is an analogue of Theorem 4.2 in [4] which
involves locally H-closed spaces. Recall that a space is called locally
$n$-H-closed if for every point $x$ there exists an open neighbourhood which
closure is $n$-H-closed.

######  Theorem 4.20.

Let $X$ be an $n$-Urysohn locally $n$-H-closed space. Then $|X|\leq
wL(X)^{\chi(X)}$.

For every $x\in X$ there exists an open neighbourhood $U_{x}$ of $x$ such that
$\overline{U_{x}}$ is $n$-H-closed. By Theorem 4.15, $|\overline{U_{x}}|\leq
2^{\chi(\overline{U_{x}})}\leq 2^{\chi(X)}$. Clearly, ${\cal U}=\\{U_{x}:x\in
X\\}$ is an open cover of $X$, then there exists ${\cal V}\in[{\cal U}]^{\leq
wL(X)}$ such that $X=\overline{\bigcup{\cal V}}$. The set $\bigcup{\cal V}$ is
dense in $X$ and $|\bigcup{\cal V}|\leq wL(X)2^{\chi(X)}\leq wL(X)^{\chi(X)}$.
By Corollary 3.13 we can conclude the proof.

Acknowledgement: The authors express gratitude to I. Juhasz and L. Zdomskyy
for useful discussions. The research was supported by “National Group for
Algebric and Geometric Structures, and their Applications” (GNSAGA-INdAM).

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# DeepFGS: Fine-Grained Scalable Coding for Learned Image Compression

Yi Ma111These authors contribute equally., Yongqi Zhai111These authors
contribute equally., and Ronggang Wang
Shenzhen Graduate School, Peking University
{mayi@pku,zhaiyongqi<EMAIL_ADDRESS>
###### Abstract

Scalable coding, which can adapt to channel bandwidth variation, performs well
in today’s complex network environment. However, the existing scalable
compression methods face two challenges: reduced compression performance and
insufficient scalability. In this paper, we propose the first learned fine-
grained scalable image compression model (DeepFGS) to overcome the above two
shortcomings. Specifically, we introduce a feature separation backbone to
divide the image information into basic and scalable features, then
redistribute the features channel by channel through an information
rearrangement strategy. In this way, we can generate a continuously scalable
bitstream via one-pass encoding. In addition, we reuse the decoder to reduce
the parameters and computational complexity of DeepFGS. Experiments
demonstrate that our DeepFGS outperforms all learning-based scalable image
compression models and conventional scalable image codecs in PSNR and MS-SSIM
metrics. To the best of our knowledge, our DeepFGS is the first exploration of
learned fine-grained scalable coding, which achieves the finest scalability
compared with learning-based methods.

##  1 Introduction

Image coding technology aims to remove redundant information in images and
compress image data dozens of times. With the coming of the metaverse, massive
amounts of image and video data are generated and transmitted. Therefore,
image and video coding technologies have become more and more important. The
traditional codecs, such as JPEG [36], JPEG2000 [10], BPG [7] and the latest
video standard VVC [32], usually include four hand-crafted modules:
prediction, transform coding, quantization, and entropy coding. These modules
are used to remove spatial redundancy, visual redundancy and statistical
redundancy in the image data, and turn the data into a stream that is easy to
store and transmit.

Figure 1: Illustration of coding performance for each coding technique. ”RQ”
indicates the image quality received by a user. Fine-grained scalable coding
technique is the best solution to deal with channel bitrate changes. Figure 2:
The architecture of our fine-grained scalable coding framework (DeepFGS),
which includes a detailed description of the feature separation backbone.

In recent years, learning-based image compression methods have developed
rapidly. Such methods rely on the image understanding and generation
capabilities of the deep generation model and finally obtain better
compression performance than traditional methods. The most advanced
compression model is based on a variational autoencoder (VAE), which uses an
entropy model to estimate the amount of information in the compressed data,
and uses a parameter to maintain the trade-off between compression efficiency
and the quality of the reconstructed image. Some existing researches [4, 5,
28, 19, 9, 14, 13] have focused on designing a better entropy model to
facilitate the probability estimation and get a rate-distortion performance
boost. Besides, there are also studies [4, 22, 9] focused on improving the VAE
network structure to enhance its feature extraction capabilities. In addition,
[27] introduces perceptual loss and adversarial training strategies to obtain
reconstructed images with excellent visual quality. Through the accumulation
of the above research, learning-based image compression models have completely
surpassed the traditional compression algorithms in terms of MS-SSIM, PSNR and
visual quality. Nevertheless, when we turn our focus from the pursuit of
performance to actual application scenarios, image compression based on
learning still lacks the realization of two important functions: variable rate
and scalable bitstream. When referring to the variable rate model, it refers
to the model which can achieve different compression ratios through a single
model. Traditional codecs use different quantization parameters and can easily
achieve continuous variable-rate adaptation. However, the existing learned
image compression methods require training multiple models to adapt to
different rates. Recently, [11] deployed gain units with exponential
interpolation to achieve continuous rate adaptation. Although [11] can achieve
different compression ratios, it needs to be encoded multiple times to
generate different non-scalable bitstreams. The role of the image compression
is to convert an image into a compact bitstream, which is scalable if a subset
of the bitstream can also generate a useful representation, i.e. the decoder
can selectively decode part of the bitstream depending on the bandwidth. [15]
use a layered coding structure to make the bitstream coarse-grained scalable
(the bitstream has four independently decodable subsets), but due to the use
of multiple codecs, the inferencing process is complicated, and the
scalability of the bitstream is limited. The RNN-based method [34, 35, 16] can
generate a coarse-grained scalable bitstream through multiple iterations.
However, the encoding-decoding process complexity is too high, and the rate-
distortion performance is lower than the CNN-based methods.

In this paper, we propose an advanced fine-grained scalable image coding
framework (DeepFGS). It can provide a highly flexible bitstream with only one
encoding process to cover an entire bitrate range. This means that even if the
bitstream is truncated at any position, it can be reconstructed into a
complete image. Figure 1 shows the performance difference between non-scalable
coding, coarse-grained scalable coding and fine-grained scalable coding in
actual bitstream transmission. It can be observed that fine-grained scalable
coding can flexibly adapt to bandwidth changes and provide decoded images with
the best quality.

The generation of our scalable bitstream requires the following steps: First,
the image information is decomposed into basic features $l_{b}$ and scalable
features $l_{s}$ through the feature separation backbone, and the redundancy
between the two features is eliminated from pixel-level to feature-level.
Next, the information rearrangement strategy redistributes the features in
$l_{s}$ and establishes the forward dependence between channels instead of
two-way dependence to adapt to the scalable decoding process. In this way,
each additional decoded channel will bring continuous quality gain. In
addition, to reduce the error in the entropy estimation process, we improve
the entropy model with the mutual information of the $l_{b}$ and the $l_{s}$.
The experimental results show that our model has excellent scalability, and
each subset of the bitstream can decode images of different quality. Moreover,
the rate-distortion performance of our model is better than all previous
scalable compression methods, whether traditional or learning-based. Our
contributions can be summarized as follows:

  * •

We propose the first fine-grained scalable coding framework for learned image
compression that outperforms all previous scalable methods in PSNR and MS-SSIM
metrics.

  * •

We introduce a feature separation backbone and information rearrangement
strategy to obtain an extremely flexible bitstream, which can provide a
continuously adjustable image quality.

  * •

We design a powerful mutual entropy model, which uses mutual information
between the basic and scalable features to obtain more accurate probability
estimation.

##  2 Related Works

###  2.1 Image Compression

Existing traditional image compression standards, such as JPEG [36], JPEG2000
[10], BPG [7] and the latest VVC-Intra [32], typically consist of prediction,
transformation, quantization, and entropy coding modules. After decades of
development, most standards have been widely used in practice. However,
limited by the hand-crafted design and independent optimization, it is
difficult to improve the performance further.

In recent years, learning-based image compression approaches have attracted
great interest with promising results. The most representative works can be
divided into two branches: recurrent neural network (RNN) based models and
convolutional neural network (CNN) based models. RNN based models [34, 35, 16]
compress images or residual information from the previous step iteratively,
while CNN based models typically transform images into compact latent
representations for further entropy coding. Some early works [33, 4, 1] solve
the problem of non-differential quantization and rate estimation. Afterward,
some works [4, 5, 28, 19, 9, 14, 13] focus on designing powerful entropy
models to improve the accuracy of rate estimation. Some approaches use
generalized divisive normalization (GDN) layers [4], or attention maechanism
[22, 9], or adaptive feature extration models [24] to improve the variational
autoencoder (VAE) architechure. Recently, several studies have been done to
improve the subjective quality. [29, 2, 27] use generative models and
perceptual loss to generate vivid texture. [26, 39, 8, 23] introduce
importance map or region-of-interest (ROI) mask to allocate more bits to the
important/ROI areas.

All the above methods are dedicated to achieving higher compression efficiency
and better visual quality. However, in many real-time applications, bandwidth
resources fluctuate drastically, most codecs cannot adapt the bitrate based on
the network conditions. The best solution is to make the bitstream scalable
and flexible, which means that the decoder is able to decode parts of the
stream to reconstruct the image with different quality.

###  2.2 Traditional Scalable Image Compression

The scalability of the bitstream is supported by many traditional standards,
such as JPEG2000 [10], the H.264/AVC scalable extension (Scalable Video
Coding, SVC) [38, 30] and the HEVC scalable extension (SHVC) [6]. These
standards encode the image once as several dependent bitstreams, including a
base layer and several enhancement layers. During transmission, the encoder
first sends the bitstreams of the base layer to reconstruct the basic quality
of the image, and if the bandwidth is sufficient, it sends the bitstreams of
the enhancement layers in turn. The more bits are received, the better the
quality of image reconstruction is. The scalability of image compression
mainly includes spatial scalability, fidelity scalability, and fine
granularity scalability (FGS) [30, 6, 20]. Spatial scalable compression
generates bitstreams with different resolutions. Fidelity scalability can be
regarded as a special case of spatial scalability, which generates images of
different quality levels. FGS encodes the image into a base layer bitstream
and an enhancement layer bitstream that can be truncated anywhere.
Specifically, the enhancement layer uses bit-plane decomposition to encode the
residuals of the original image DCT [3] coefficients and the quantized DCT
coefficients of the base layer. In bit-plane decomposition, the higher bits
(important parts) of each coefficient are coded first, thus FGS can finely
adjust image quality according to the number of bits received.

###  2.3 Learned Scalable Image Compression

In the early stage, [34] uses RNN structure to compress raw and residual
information iteratively. After each iteration, the bitstreams will increase,
and the quality of the reconstructed image will be enhanced. Following that,
[31] presents an RNNs architecture with quantization and entropy coding to
achieve scalable learned image compression. Inspired by the conventional
scalable coding, [15] designs a scalable auto-encoder to compress the original
images and residual images iteratively. [40] adopts bit-plane decomposition
and bidirectional ConvLSTM to achieve scalable coding. [12] proposes a four
layers scalable compression framework and uses a decorrelation unit to
eliminate redundancy between previous layers and current layers. [25] designs
a spatial/quality scalable compression framework with latent-feature reuse and
prediction.

However, there are still some disadvantages in the above methods. Firstly,
these methods cannot achieve fine granularity scalability. The number of
layers is limited, and the bitrates gap between layers is quite large, which
is too coarse for changes in channel bandwidth. Secondly, they have high
computational complexity. Due to the complex iterative manner, RNN based
models suffer from highly time and memory-consuming. Most methods have
multiple codecs and require multiple complete encoding and decoding processes
to achieve scalability, which is too complicated in practical applications.

##  3 Proposed Method

In this section, we introduce our proposed fine-grained scalable image
compression framework. As shown in Figure 2, the feature separation backbone
divides the image information into basic features $l_{b}$ and scalable
features $l_{s}$. Then through the information rearrangement strategy, the
information in the latent representation is redistributed. We have deployed
feature selection switches and feature fusion modules in the front of the
decoder to implement scalable decoding and decoder reuse. The details of the
network are given in the Appendix.

###  3.1 Feature Separation Backbone

In order to achieve scalable coding, only a certain subset of the bitstream
can be used to reconstruct the image, which will cause the image quality to be
spatially uneven. To this end, we divide the features of the image into two
parts. The first is the basic features $l_{b}$ that contribute the
indispensable basic texture of the image, and the second part is the scalable
features $l_{s}$ that bring optional quality improvement. In this way, $l_{b}$
guarantees the basic quality, and $l_{s}$ makes the bitstream flexible. The
redundancy between $l_{b}$ and $l_{s}$ will be removed from pixel domain to
feature domain in two steps. First, as shown in Figure 2, image $x$ is sent to
the basic encoder $g_{a}$ to extract the basic feature $l_{b}$.

$l_{b}=Q(g_{a}(x))$ (1)

where $Q$ denotes the quantization operation. And the initial scalable
features $x_{s}$ will be obtained by the difference operation between $x$ and
$\hat{x}$ (the reconstruction of $l_{b}$).

$\displaystyle x_{s}=F_{conv}(x||\hat{x})\qquad with\quad\hat{x}=g_{s}(l_{b})$
(2)

where $g_{s}$ and $||$ denotes the decoder and the concat operation.
$F_{conv}$ refers to the convolution operation, which is used to squeeze the
channel. Although the pixel-level redundancy in $x$ and $x_{s}$ has been
removed, there will be feature-level redundancy when they are compressed into
latents respectively. Figure 2 shows our feature-level redundancy removal
module (FRR). We use $l_{b}$ as a cross guidance to filter the redundant
information in the $l_{s}$ in the channel and space domain.

$\displaystyle l_{s}=l_{s^{\prime}}\otimes f_{mlp}(pooling(l_{b}))\otimes
f_{st}(mean(l_{b}))$ (3) $\displaystyle\qquad with\quad
l_{s^{\prime}}=g_{a1}(x_{s})$

where $g_{a1}$ denotes the scalable encoder in Figure 2. $pooling$ and $mean$
refer to the average in channel and space, respectively. $\otimes$ denotes the
channel-wise multiplication, $f_{mlp}$ denotes multi-layer perceptron, and
$f_{st}$ denotes the operation that first unsqueezes the channel and then
squeezes it to 1. In this way, we have obtained compact latent representations
$l_{s}$ and $l_{b}$, and then we will redistribute the features in $l_{s}$
through the information rearrangement strategy.

Figure 3: Visualization of entropy of the latent representation from kodim20.
We divide the 192 channels into 24 groups on average, and the figure shows the
entropy of each group.

###  3.2 Information Rearrangement Strategy

In the learning-based image compression framework, the features of the image
$x$ are extracted into the latent representation $l\in\mathbb{R}^{C\times
H\times W}$ where $c$, $h$, and $w$ represent the number of channels, height
and width of the feature map. As shown in Figure 3 the contribution of $l$ to
the reconstruction quality is not channel-wise uniform. We use the information
rearrangement strategy to redistribute the features in $l_{s}$ and establish
the forward dependence between channels instead of two-way dependence, which
means that each channel only needs to be combined with the previous channel to
complete the reconstruction. As shown in Figure 4, establishing the forward
dependency allows us to obtain continuous quality improvement when
reconstructing through the first $n$ channels (remaining channels are padding
with zeros). Instead of explicitly rearranging features, we designed an
optimization function to guide network learning how to do so. In our
framework, the features to be encoded include $l_{b}\in\mathbb{R}^{C_{1}\times
H\times W}$ and $l_{s}\in\mathbb{R}^{C_{2}\times H\times W}$, both $C_{1}$ and
$C_{2}$ is set to 192. Let $k\in[C_{1},C_{2}]\cap\mathbb{N}$ denote the number
of channels available for decoding, up to 384, which simulates bandwidth
fluctuations. When $k$ is equal to 192, it means that only the basic feature
$l_{b}$ is available. Therefore, the optimization function is:

Figure 4: Reconstruction quality of the latent representation of kodim20. We
divide the 192 channels into 24 groups on average. The abscissa represents the
first $n$ groups, and the ordinate represents the quality of the reconstructed
image through of the first $n$ groups.

$J_{base}=R(l_{b})+\lambda D(x,g_{s}(l_{b}))$ (4)

$D$ denotes distortion metric (such as MSE or MS-SSIM), $R$ refers to
probability estimation(see section 3.4), used to calculate the number of bits,
and $\lambda$ is used to balance them. When $k$ is greater than 192,
additional scalable features can be obtained, and we need to optimize each
forward-dependent channel combination:

$\displaystyle J_{scalable}=R(l_{b})+\lambda D(x,g_{s}(l_{b}))+$ (5)
$\displaystyle\qquad\sum_{i=1}^{k-C_{1}}R(l_{b})+R(l_{s}^{i})+\lambda
w(i)D(x,g_{s}(l_{b}||l_{s}^{i}))$

Where $l_{s}^{i}$ refers to the first $i$ channels in $l_{s}$. $w(\cdot)$
determines the bitrate range covered by our model, which can adjust the
scalability of the bitstream, here we set $w(i)=iMod8$. Note that the same
decoder $g_{s}$ is used in Equation 2 and Equation 5, because the multiplexing
of the decoder can reduce the complexity of the model. However, the
optimization of the second term in Equation 5 requires dozens of forward
propagation and back propagation, resulting in unacceptably high computational
complexity. Therefore, we simplify it by replacing the accumulation operation
with the sampling operation.

Figure 5: Observation of the process of feature fusion. Basic feature, fusion
feature, and difference feature refer to: $FFM(l_{b})$, $FFM(l_{b}||l_{s})$
and $FFM(l_{b}||l_{s})-FFM(l_{b})$. We search for the feature map with the
largest activation value among the features and visualize the energy of each
channel ($maximum-minimum$).

(a)

(b)

(c)

(d)

Figure 6: Comparison of rate-distortion performance of our DeepFGS with
existing compression standards and other learned compression models, including
Toderici [34], Zhang [40], Guo [12], Jia [15], Mei [25]. (a): PSNR performance
evaluation on Kodak dataset: compare with scalable coding models. (b): MS-SSIM
performance evaluation on Kodak dataset: compare with scalable coding models.
(c): PSNR performance evaluation on Kodak dataset: compare with simulcast
coding models. (d): PSNR performance evaluation on Kodak dataset: compare with
non-scalable coding models.

$\displaystyle J_{scalable}=R(l_{b})+\lambda D(x,g_{s}(l_{b}))+$ (6)
$\displaystyle\qquad R(l_{b})+R(l_{s}^{j})+\lambda
w(j)D(x,g_{s}(l_{b}||l_{s}^{j}))$

where $j$ is sampled from $[C_{1},C_{2}]\cap\mathbb{N}$. We design two
experiments to illustrate the role of information rearrangement strategies. In
Figure 3, we show the amount of information in each channel in the latent
representation. It can be observed that the information distribution of the
non-scalable image compression network is messy and irregular, and our model
arranges the channels in descending entropy order. Only in this way can we
achieve continuous and smooth growth of quality. Figure 4 clearly shows the
impact of the information rearrangement strategy on the quality of the
reconstructed image. The non-scalable compression model is volatile when using
the first $n$ channels for decoding, while our model maintains a smooth and
steady growth in an extensive bit rate range.

###  3.3 Decoder Multiplexing and Analysis

Many studies on learned scalable image compression, such as [15] and [25],
generate bitstreams of different quality levels, which use more than two
decoders to reconstruct these bitstreams. Considering that these decoders all
try to reconstruct the same input image and are optimized by similar loss
functions, their reconstruction modes are similar. Therefore, the complexity
can be reduced by multiplexing a part of the decoder parameters. As shown in
Figure 2, our model includes two decoding processes, which are located in the
feature separation backbone and at the end of the entire framework. In order
to reduce the parameters and complexity of the decoding network, the two
locations share the same decoder. Therefore, we design a feature fusion module
(FFM) at the head of the decoder.

$\displaystyle l_{m}=l_{in}\otimes f_{mlp}(pooling(l_{s}))\otimes
f_{st}(mean(l_{s}))$ (7) $\displaystyle\qquad with\quad
l_{in}=Swich(l_{b},l_{b}||l_{s})$

where $l_{m}$ denotes the fusion feature, which is passed to the subsequent
network for reconstruction.

In addition, we design an observation experiment to demonstrate the process of
feature fusion. First, from the first row of Figure 5, we can observe that the
basic feature and the fusion energy distribution are similar. The difference
(differential feature) between the two is reflected in some low-energy
channels. By comparing with the reconstructed image in the second row, we know
that the high-energy channels of the basic features carry low-frequency basic
textures. The differential features include high-frequency textures that can
improve the quality of the reconstruction, which is consistent with our
expectations for the feature separation backbone.

Figure 7: Ablation study on the proposed DeepFGS. The specific ablation
configurations are shown in Table 1.

###  3.4 Quantization and Mutual Entropy Model

Let ${x}$ denotes the original image, ${l}$ is the latents generated by the
encoder, and after quantization $\hat{l}=Q(l)$. In order to make the model
end-to-end trainable, we approximate the quantization operation by adding
uniform noise [4] during training.

Following hyperpriors [5], the rate of quantized latents is estimated by the
mean and scale Gaussian entropy model, which can be formulated as:

$p_{\hat{l}|\hat{z}}(\hat{l}|\hat{z})=\mathcal{N}(\mu,{\sigma}^{2})$ (8)

where ${z}$, $\hat{z}$ represent the hyper latents before and after
quantization, ${\mu}$, ${{\sigma}^{2}}$ are the estimated mean and scale
parameters of latents $\hat{l}$. In our model, $\hat{l}_{b}$, $\hat{z}_{b}$
are the basic latents, and $\hat{l}_{s}$, $\hat{z}_{s}$ are the scalable
latents. The rates of ${l}_{b}$ and ${l}_{s}$ can be rewritten as :

$\displaystyle R_{{b}}$
$\displaystyle=\mathbb{E}[-{\log_{2}}p_{\hat{l}_{b}}({\hat{l}_{b}})]+\mathbb{E}[-{\log_{2}}p_{\hat{z}_{b}}({\hat{z}_{b}})]$
(9) $\displaystyle R_{{s}}$
$\displaystyle=\mathbb{E}[-{\log_{2}}p_{\hat{l}_{s}}({\hat{l}_{s}})]+\mathbb{E}[-{\log_{2}}p_{\hat{z}_{s}}({\hat{z}_{s}})]+R_{{b}}$

In the previous methods [15, 25], they use independent entropy models for the
base and enhancement layers. However, due to the existence of redundancy
between latents, the mutual information ${I(\hat{l}_{s},\hat{l}_{b})}$ is
positive. Let ${H(\hat{l}_{s})}$ denotes the Shannon entropy of $\hat{l}_{e}$,
then the conditional entropy ${H(\hat{l}_{s}|\hat{l}_{b})}$can be obtained as:

$\displaystyle H(\hat{l}_{s}|\hat{l}_{b})$
$\displaystyle=H(\hat{l}_{s})-I(\hat{l}_{s},\hat{l}_{b})<H(\hat{l}_{s})$ (10)

Table 1: The specific configuration of ablation study.

| $FRR$ | $FFM$ | $MEM$
---|---|---|---
$Case1$ | ✓ | ✓ | ✓
$Case2$ | ✓ | ✓ |
$Case3$ | ✓ |  | ✓
$Case4$ |  | ✓ |
$Case5$ |  |  | ✓
$Case6$ |  |  |

Figure 8: Visual quality comparisons with the learning-based layered scalable
method Jia [15] and traditional scalable codec SVC.

Therefore, in our entropy model, $\hat{l}_{s}$ is not only conditioned on the
information of $\hat{z}_{s}$, but also uses $\hat{l}_{b}$ as the prior
information. The equation 9 can be formulated as:

$\displaystyle
R_{{b}}=\mathbb{E}[-{\log_{2}}p_{\hat{l}_{b}|\hat{z}_{b}}({\hat{l}_{b}|\hat{z}_{b}})]+\mathbb{E}[-{\log_{2}}p_{\hat{z}_{b}}({\hat{z}_{b}})]$
(11) $\displaystyle
R_{{s}}=\mathbb{E}[-{\log_{2}}p_{\hat{l}_{s}|\hat{z}_{s},\hat{l}_{b}}({\hat{l}_{s}}|\hat{z}_{s},\hat{l}_{b})]+\mathbb{E}[-{\log_{2}}p_{\hat{z}_{s}}({\hat{z}_{s}})]+R_{{b}}$

With this prior, we can estimate the probability of $\hat{l}_{s}$ more
accurately, thus improving the R-D performance.

##  4 Experiment

###  4.1 Training Details

COCO [21] dataset is used for training, and all images are randomly cropped
into ${256\times 256}$ patches during training. We adopt Adam [17] optimizer
with a batch size of 16 to train our model. Our network is optimized for 50
epochs with an initial learning rate of $1{e^{-4}}$, and the learning rate is
reduced to $1{e^{-5}}$ for the last 15 epochs.

We use $\lambda$ to adjust the minimum bitrate, and use the function
$w(\cdot)$ to adjust the bitrate range covered by the model. During training,
$k$ is randomly selected as an integer in [0,192] to change the size of the
bitstream received by the decoder. Our models are optimized with two quality
metrics, i.e. MSE (mean squared error) and MS-SSIM (multiscale structural
similarity) [37]. For the MSE models, $\lambda$ is set to 0.002. For the MS-
SSIM models, $\lambda$ is set to 7.0.

###  4.2 Rate-distortion Performance

Compare with Scalable Coding Models. We evaluate the rate-distortion
performance on the Kodak image set [18] with 24 uncompressed $768\times 512$
images (more results on the high-definition data set will be given in the
Appendix). The rate is measured by bits per pixel (bpp), and the quality is
measured by PSNR and MS-SSIM. First, we compare performance with other
scalable coding models. For traditional methods SVC and HSVC, coarse-grained
scalable coding mode performs better than fine-grained coding mode, so we
compare with the former. For the learning-based image compression model [34,
40, 12, 15, 25], we also compare coarse-grained methods, because no fine-
grained methods have appeared before. As shown in Figure 6(a) and 6(b),
whether evaluated by PSNR or MS-SSIM, the rate-distortion performance of our
model surpasses the existing traditional methods and learned methods by a big
step. Furthermore, our model is also much more scalable than existing models.
The bitstream of DeepFGS is nearly continuously adjustable, so that DeepFGS
can well decode the bitstream which is truncated at any position. For the
convenience of comparison, in this section, we decode $l_{s}$ with 8 channels
as the interval to draw the RD curve of our model. In particular, for the
enlarged part in Figure 6(a), this interval is 1 channel. Note that due to
text limitations, the larger bitrate range and higher bitrate of DeepFGS are
not shown in Figure 6, but will be discussed in the Appendix.

Compare with Simulcast Coding Models. The goal of simulcast coding is similar
to scalable coding, but simulcast mode encodes each layer independently, and
the rates sent by the server are the sum of these layers. For example, in
order to switch between three quality levels $(Q_{1},Q_{2},Q_{3})$, the
simulcast mode needs to encode three independent streams
$(R_{1},R_{2},R_{3})$. To achieve the same function as the three-layer
scalable coding, the bitrate of the base layer is $R_{1}$, and the enhancement
layers are $R_{1}+R_{2},R_{1}+R_{2}+R_{3}$. We test two well-known learning-
based single-layer models [28, 9] and two traditional single-layer models BPG
[7] and VVC [32] in simulcast mode. As shown in figure 6(c), our model
performs best, and the advantage becomes more obvious as the number of layers
increases. This is because there are lots of redundancies between the
bitstreams of different layers in the simulcast mode.

Compare with Non-Scalable Coding Models. Due to the existence of redundancy
between layers, compared to non-scalable coding, the performance degradation
brought by the scalable method is inevitable. As shown in Figure 6(d), we
compared the performance with the non-scalable model (the same architecture as
ours). It can be seen that the quality loss caused by our scalable method is
very small. It is worth mentioning that our model is the first scalable model
that exceeds BPG in PSNR.

###  4.3 Ablation Study

In order to evaluate the contribution of each module to the performance of our
model, we perform ablation studies shown in Figure 7. We study the following
models: feature-level redundancy removal module (FRR), feature fusion module
(FFM) and mutual entropy model (MEM). Table 1 shows the specific ablation
configuration.The experimental results show that these modules all contribute
to the performance improvement of DeepFGS, and combining them will achieve the
best performance. In addition, by comparing the combinations $(Case2,Case3)$
and $(Case4,Case5)$, it can be found that MEM is more effective at high
bitrates, while FFM is more effective at low bitrates.

###  4.4 Visual Results

As shown in Figure 8, we present some cropped reconstructed images on the
Kodak dataset. Compared with traditional [38] and learning-based [15] scalable
coding models, our model has two main advantages. Firstly, our method is able
to use fewer bitrates to generate more visually pleasing results. For example,
in contour and textural regions (e.g. human face and sweater), our method
offers much more structural details than other codecs. Secondly, our method
can generate reconstructed images of different qualities in fine-grained
manner. As shown in the figure, our DeepFGS can provide dozens of continuous
quality levels in the interval where [38] and [15] can only generate two
decoded images. More subjective quality results are given in the Appendix.

###  4.5 Limitation

The main limitation of our framework is that the information rearrangement
strategy conflicts with the existing pixel-level autoregressive entropy model.
The existing SOTA non-scalable image compression method utilizes this entropy
model and trades decoding complexity for much rate-distortion performance. The
study we are advancing is to design an autoregressive model compatible with
the information rearrangement strategy. It is worth mentioning that our
DeepFGS improves the performance of fine-grained scalable image compression
from far inferior to BPG to beyond BPG. However, the performance will be
upgraded to the VVC level once we complete the compatible autoregressive
models.

##  5 Conclusion

This paper proposes the first learned fine-grained scalable coding model,
DeepFGS, and achieves optimal compression performance and scalability.
Ablation studies demonstrate the effectiveness of DeepFGS modules. In the
following research, we will further explore the scalability of the image
compression model, such as reducing the minimum tunable interval of the
bitstream or increasing the rate range covered by the model.

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##  6 Appendix

###  6.1 Introduction

This Appendix contains many additional experiments and explorations, such as
rate-distortion performance at high bitrates, models for wide bitrate ranges,
and discussions on extreme fine-grained scalability. In addition, the appendix
also includes the specific network structure of our DeepFGS model, as well as
some high-definition image reconstruction results.

###  6.2 Supplementary Note on Experiment

All the experiments are conducted on the $Tesla\,V100$ GPU and
$Intel\,Xeon\,Platinum\,8163$ CPU. Our framework DeepFGS are implemented using
CompressAI ***https://github.com/InterDigitalInc/CompressAI PyTorch library.
In addition, we used Pytorch version 1.9.1 and CompressAI version 1.1.7.

###  6.3 Rate-Distortion Performance at High Bitrates

As shown in Figure 9, we show the rate-distortion performance of our model at
high bit rates. It can be observed that our DeepFGS model also has excellent
performance at high bit rates. In addition, as the bit rate increases, the
growth of PSNR has not slowed down, while the R-D curves of other scalable
compression models tend to be flat.

We also evaluated the performance of DeepFGS on the high-definition dataset
CLIC †††http://www.compression.cc Professional Validation dataset. In order to
simplify the experiment, we directly show the broadcast performance of DeepFGS
and other non-scalable models. As shown in Figure 10, our model also performs
well on the high-definition dataset.

(a)

(b)

Figure 9: Comparison of rate-distortion performance of our DeepFGS with
existing compression standards and other learned compression models at high
bitrates, including Toderici [34], Zhang [40], Guo [12], Jia [15], Mei [25].
(a): PSNR performance evaluation on Kodak dataset. (b): MS-SSIM performance
evaluation on Kodak dataset. Figure 10: Comparison of rate-distortion
performance at CLIC dataset.

###  6.4 Discussion on Scalability

####  6.4.1 Discussion on the Scalable Range of Bitrate

As shown in the experimental chapter, our DeepFGS-low and DeepFGS-high can
provide scalability in the range of 0.5 bpp. The information rearrangement
strategy constrains this scalability range:

$\displaystyle J_{scalable}=R(l_{b})+\lambda D(x,g_{s}(l_{b}))+$ (12)
$\displaystyle\qquad R(l_{b})+R(l_{s}^{j})+\lambda
w(j)D(x,g_{s}(l_{b}||l_{s}^{j}))$

Where $l_{s}^{j}$ refers to the first $j$ channels in $l_{s}$. $w(j)$ is a
function that is positively correlated with $j$, so that when decoding more
features (channels), the constraint on the bit rate can be relaxed. In this
way, the decoder can learn to reconstruct reconstructed images of different
qualities simultaneously. As shown in Figure 11, we take two different
functions of $w(j)=jMod8$, $w(j)=j^{2}Mod64$ as examples, and the scalable
range of bitrate varies with $w(j)$. It can be observed that the wider the
scalable range of the model, the more significant the quality loss (compared
to the non-scalable model). Therefore, we chose $w(j)=jMod8$ with almost no
loss in quality, and a scalable range of 0.5bpp is sufficient to cope with
most bandwidth fluctuations.

Figure 11: The impact of scalable ranges of bitrate on rate-distortion
performance. Including the comparison between $w(j)=jMod8$ (DeepFGS) and
$w(j)=j^{2}Mod64$ (DeepFGS-Wider).

####  6.4.2 The Finest-grained Scalability

In the experimental section, we demonstrated the scalability of DeepFGS
through channel-by-channel decoding. The same bitstream can reconstruct 192
images of different quality levels. Figure 12 shows our further expansion,
half-channel decoding (remaining channels with zero padding), which can
achieve $192\times 2$ quality levels of reconstruction. It can be observed
that the performance of half-channel decoding is still continuous and stable.
However, we can even get the finest scalability by decoding pixel by pixel,
that is, $192\times\frac{w}{16}\times\frac{h}{16}$ quality levels.

In addition, we demonstrated the continuous quality improvement brought by
channel-by-channel decoding in the supplementary material.

Figure 12: Comparison of channel-by-channel decoding and half-channel
decoding. Figure 13: Architecture of network in DeepFGS, where Encoder Network
refers to basic encoder or scalable encoder, and Decoder Network refers to
multiplexed decoder (excluding information fusion module). ”K” represents
kernel size, ”C” denotes number of filters, and ”S” is stride of a
convolutional layer.

###  6.5 Network Details

Figure 13 shows the details of our DeepFGS network structures. We use residual
blocks and subpixel convolution to improve our backbone network. Specificly,
we use 3x3 stacked convolutions with residual connection instead of 5x5
convolution, which can achieve larger receptive field while reducing the
number of parameters. We use subpixel convolution in the upsampling layers.
And after each upsampling or downsampling layer, generalized division
normalization (GDN) layer is used as the activation function. In additional,
The basic encoder and the scalable encoder use the same network structure.

###  6.6 Additional Sample Image

In the following pages, we will show more sample images from the Kadak dataset
and CLIC Professional Validation dataset. In order to show the continuous
improvement in the quality of our DeepFGS reconstructed images, we use
different subsets of the same bitstream for decoding.

Figure 14: Visualization of reconstructed images _kodim18_ from Kodak dataset.
Figure 15: Visualization of reconstructed images _kodim04_ from Kodak dataset.
Figure 16: Visualization of reconstructed images _alejandro-escamilla-6_ from
CLIC dataset. Figure 17: Visualization of reconstructed images _allef-
vinicius-109434_ from CLIC dataset. Figure 18: Visualization of reconstructed
images _sergey-zolkin-1045_ from CLIC dataset.