# On $e^{*}$-$\theta$-$D$-sets and Related Topics

D. AKALINa, M. Özkoçb,∗ AMuğla Sıtkı Koçman University, Graduate School of
Natural and Applied Sciences, Mathematics, 48000, Menteşe-Muğla, Turkey.
<EMAIL_ADDRESS>BMuğla Sıtkı Koçman University, Faculty of Science,
Department of Mathematics, 48000, Menteşe-Muğla, Turkey.
<EMAIL_ADDRESS>
(Date: — ∗Corresponding author)

###### Abstract.

This paper aims to study the notion of $e^{*}\text{-}\theta$-open [6] sets and
to investigate new properties of this notion. Also, we define a new type of
set, called $e^{*}\text{-}\theta\text{-}D$-set, via the notion of
$e^{*}\text{-}\theta$-open set. Moreover, we introduce some new separation
axioms by utilizing $e^{*}\text{-}\theta\text{-}D$-sets. We obtain many
results related to these new notions. In addition, the notions of
$e^{*}\text{-}\theta\text{-}$kernel and slightly
$e^{*}\text{-}\theta\text{-}R_{0}$ space are defined. Some characterizations
regarding these new notions have been obtained. Furthermore, we have given
many examples concerning the mentioned notions. Finally, we not only put
forward the definition of $e^{*}\text{-}R_{1}$ space but also obtained some of
its characterizations.

###### Key words and phrases:

$e^{*}$-open set, $e^{*}\text{-}\theta$-open set, $e^{*}\text{-}\theta$-closed
set, $e^{*}\text{-}\theta$-closure.

###### 2010 Mathematics Subject Classification:

Primary 54C08, 54C10; Secondary 54C05

##  1\. Introduction

The concept of open set is very important for general topology. The broad
perspective offered to mathematicians by this fundamental concept in topology
facilitates the organization of elements and the study of topological spaces.
This richness of the open set concept adds depth to mathematical thought and
contributes significantly to the understanding of topological spaces. As we go
deeper into topology, the diversity of open sets increases. This diversity
provides a mathematical basis for separating different points and set
structures with different neighborhoods. At this stage, it is important to
point out that separation axioms play a decisive role. Today, various special
and general forms of separation axioms have been defined and studied in detail
by many mathematicians. Moreover, special and general forms of open sets have
also gained an important place in the mathematical literature. For instance,
in 1966, Velicko introduced the notion of $\theta$-open [12] set which is the
stronger form of open sets in topology. After him, several new forms of
$\theta$-open [12] classes such as pre-$\theta$-open [8], semi-$\theta$-open
[7], $\beta$-$\theta$-open [2, 9], and $e^{*}\text{-}\theta$-open [6] were
defined and studied in the literature. In the first part of this paper, we
present an independent notion of $\beta\text{-}\theta\text{-}$open [2, 9] set,
which was introduced in 2015 by Farhan et. al. The literature on
$e^{*}\text{-}\theta$-open and $e^{*}\text{-}\theta$-closed sets defined by
Yang some new results are obtained that do not hold. In the second section,
the $e^{*}\text{-}\theta\text{-}D$-set and quasi
$e^{*}\text{-}\theta\text{-}D$-set defined in this paper are analyzed. The
concepts of $e^{*}\text{-}\theta\text{-}D_{0}$ space,
$e^{*}\text{-}\theta\text{-}D_{1}$ space, $e^{*}\text{-}\theta\text{-}D_{2}$
space are introduced through the concepts of $\theta$-closed set and the
relations between these concepts are investigated. In the third section, the
concepts of $e^{*}\text{-}\theta\text{-}R_{0}$ space and
$e^{*}\text{-}\theta$-core are partially defined and the relations between
these two concepts are analyzed. In the fourth section, the notions of
$S$-continuous, $\theta\text{-}S\text{-}e^{*}$-continuous and
$S\text{-}e^{*}$-continuous functions are defined and some basic results on
these functions are presented. In the last section, the definition of
$e^{*}\text{-}R_{1}$ space is given and two characterizations of this concept
has been obtained.

##  2\. Preliminaries

Throughout this paper, $X$ and $Y$ refer always to topological spaces on which
no separation axioms are assumed unless otherwise mentioned. For a subset $A$
of $X,$ $cl(A)$ and $int(A)$ denote the closure of $A$ and the interior of $A$
in $X,$ respectively. The family of all open subsets containing $x$ of $X$ is
denoted by $O(X,x).$ A subset $A$ is said to be regular open [11] (resp.
regular closed [11]) if $A=int(cl(A))$ (resp. $A=cl(int(A))$). The
$\delta\text{-}$interior [12] of a subset $A$ of $X$ is the union of all
regular open sets of $X$ contained in $A$ and is denoted by
$\delta\text{-}int(A).$ The subset $A$ of a space $X$ is called
$\delta\text{-}$open [12] if $A=\delta\text{-}int(A),$ i.e., a set is
$\delta\text{-}$open if it is the union of some regular open sets. The
complement of a $\delta\text{-}$open set is called $\delta\text{-}$closed.
Alternatively, a subset $A$ of a space $X$ is called $\delta$-closed [12] if
$A=\delta\text{-}cl(A),$ where $\delta\text{-}cl(A)=\\{x\in X|(\forall U\in
O(X,x))(int(cl(U))\cap A\neq\emptyset)\\}.$ The family of all
$\delta\text{-}$open (resp. $\delta\text{-}$closed) sets in $X$ is denoted by
$\delta O(X)$ (resp. $\delta C(X)$).

$A$ subset $A$ of $X$ is said to be $e^{*}\text{-}$open [5] if $A\subseteq
cl(int(\delta\text{-}cl(A))).$ The complement of an $e^{*}\text{-}$open set is
called $e^{*}\text{-}$closed. A subset $A$ of a space $X$ is said to be
$e^{*}$-regular [6] if it is both $e^{*}$-open and $e^{*}$-closed. The
$e^{*}\text{-}$interior [5] of a subset $A$ of $X$ is the union of all
$e^{*}\text{-}$open sets of $X$ contained in $A$ and is denoted by
$e^{*}\text{-}int(A).$ The $e^{*}$-closure [5] of a subset $A$ of $X$ is the
intersection of all $e^{*}$-closed sets of $X$ containing $A$ and is denoted
by $e^{*}\text{-}cl(A).$ The family of all $e^{*}$-open (resp. $e^{*}$-closed,
$e^{*}$-regular) subsets containing $x$ of $X$ is denoted by $e^{*}O(X,x)$
(resp. $e^{*}C(X,x),$ $e^{*}R(X,x)).$

A point $x$ of $X$ is called an $e^{*}\text{-}\theta$-cluster [6] point of
$A\subseteq X$ if $e^{*}\text{-}cl(U)\cap A=\emptyset$ for every $U\in
e^{*}O(X,x).$ The set of all $e^{*}\text{-}\theta\text{-}$cluster points of
$A$ is called the $e^{*}\text{-}\theta\text{-}$closure of $A$ and is denoted
by $e^{*}\text{-}cl_{\theta}(A).$ A subset $A$ is said to be
$e^{*}\text{-}\theta\text{-}$closed if and only if
$A=e^{*}\text{-}cl_{\theta}(A).$ The complement of an
$e^{*}\text{-}\theta\text{-}$closed set is said to be
$e^{*}\text{-}\theta\text{-}$open. The family of all
$e^{*}\text{-}\theta$-closed (resp. $e^{*}\text{-}\theta$-open) subsets of $X$
is denoted by $e^{*}\theta C(X)$ (resp. $e^{*}\theta O(X)).$ The family of all
$e^{*}\text{-}\theta$-closed (resp. $e^{*}\text{-}\theta$-open) subsets
containing $x$ of $X$ is denoted by $e^{*}\theta C(X,x)$ (resp. $e^{*}\theta
O(X,x)).$ Also, the family of all $e^{*}\text{-}\theta$-open sets containing
the subset $F$ of $X$ will be denoted by $e^{*}\theta O(X,F).$

######  Theorem 2.1.

$($[6]$)$ The following properties hold for a subset $A$ of a topological
space $(X,\tau):$
$(a)$ $A\in e^{*}O(X)$ if and only if $e^{*}\text{-}cl(A)\in e^{*}R(X),$
$(b)$ $A\in e^{*}C(X)$ if and only if $e^{*}\text{-}int(A)\in e^{*}R(X).$

######  Corollary 2.2.

$($[6]$)$ Let $A$ and $A_{\alpha}(\alpha\in\Lambda)$ be any subsets of a space
$(X,\tau).$ Then the following properties hold:
$(a)$ $A$ is $e^{*}\text{-}\theta\text{-}$open in $X$ if and only if for each
$x\in A$ there exists $U\in e^{*}R(X,x)$ such that $x\in U\subseteq A,$
$(b)$ If $A_{\alpha}$ is $e^{*}\text{-}\theta\text{-}$open in $X$ for each
$\alpha\in A,$ then $\cup_{\alpha\in\Lambda}A_{\alpha}$ is
$e^{*}\text{-}\theta\text{-}$open in $X.$

######  Theorem 2.3.

$($[6]$)$ For a subset $A$ of a topological space $X,$ the following
properties hold:
$(a)$ If $A\in e^{*}O(X)$, then
$e^{*}\text{-}cl(A)=e^{*}\text{-}cl_{\theta}(A),$
$(b)$ $A\in e^{*}R(X)$ if and only if $A$ is $e^{*}\text{-}\theta\text{-}open$
and $e^{*}\text{-}\theta\text{-}closed.$

######  Definition 2.4.

$($[6]$)$ $A$ space $X$ is said to be $e^{*}$-regular if for each closed set
$F\subseteq X$ and each point $x\in X\setminus F,$ there exist disjoint
$e^{*}$-open sets $U$ and $V$ such that $x\in U$ and $F\subseteq V.$

######  Theorem 2.5.

$($[6]$)$ For a space $X,$ the following are equivalent:
$(a)$ $X$ is $e^{*}\text{-}regular;$
$(b)$ For each point $x\in X$ and for each open set $U$ of $X$ containing $x,$
there exists $V\in e^{*}O(X)$ such that $x\in V\subseteq
e^{*}\text{-}cl(V)\subseteq U;$
$(c)$ For each subset $U\in e^{*}O(X)$ and each $x\in U,$ there exists $V\in
e^{*}R(X)$ such that $x\in V\subseteq U.$

######  Definition 2.6.

$($[4]$)$ $A$ function $f:X\to Y$ is said to be $e^{*}\text{-}$irresolute if
$f^{-1}[V]\in e^{*}O(X)$ for every $V\in e^{*}O(Y).$

######  Theorem 2.7.

$($[10]$)$ Let $f:X\rightarrow Y$ be a function. The following properties are
equivalent:
$(a)$ $f$ is weakly $e^{*}\text{-}$irresolute;
$(b)$ $f[e^{*}\text{-}cl(A)]\subseteq e^{*}\text{-}cl_{\theta}(f[A])$ for
every subset $A$ of $X,$
$(c)$ $f^{-1}[V]$ is $e^{*}\text{-}\theta\text{-}open$ in $X$ for every
$e^{*}\text{-}\theta\text{-}open$ $V$ of $Y.$

######  Remark 2.8.

$($[6]$)$ It can be easily shown that $e^{*}\text{-}$regular $\Rightarrow$
$e^{*}\text{-}\theta$-open $\Rightarrow$ $e^{*}\text{-}$open.

######  Theorem 2.9.

$($[6]$)$ For each subset $A$ of a topological space $(X,\tau),$ we have:

$\begin{array}[]{rcl}e^{*}\text{-}cl_{\theta}(A)&=&\bigcap\\{V|A\subseteq
V\text{ and }V\text{ is }e^{*}\text{-}\theta\text{-}closed\\}\\\
&=&\bigcap\\{V|A\subseteq V\text{ and }V\in e^{*}R(X)\\}.\end{array}$

######  Remark 2.10.

It is easy to prove that:
$(a)$ the intersection of an arbitrary collection of
$e^{*}\text{-}\theta$-closed sets is $e^{*}\text{-}\theta$-closed.
$(b)$ $X$ and $\emptyset$ are $e^{*}\text{-}\theta$-closed sets.

######  Remark 2.11.

The following example shows that the union of any two
$e^{*}\text{-}\theta$-closed sets of $X$ need not be
$e^{*}\text{-}\theta$-closed in $X.$

######  Example 2.12.

([6]) Let $X=\\{1,2,3\\}$ and
$\tau=\\{\emptyset,X,\\{1\\},\\{2\\},\\{1,2\\}\\}.$ The subsets $\\{1\\}$ and
$\\{2\\}$ are $e^{*}\text{-}\theta$-closed in $(X,\tau)$ but $\\{1,2\\}$ is
not $e^{*}\text{-}\theta$-closed.

######  Theorem 2.13.

$($[6]$)$ For a subset $A$ of a space $X,$ the following are equivalent:
$(a)$ $A\in e^{*}R(X);$
$(b)$ $A=e^{*}\text{-}cl(e^{*}\text{-}int(A));$
$(c)$ $A=e^{*}\text{-}int(e^{*}\text{-}cl(A)).$

######  Theorem 2.14.

$($[6]$)$ Let $X$ be a topological space and $A\subseteq X.$ Then, the
operator $e^{*}\text{-}cl$ is idempotent, i.e.,
$e^{*}\text{-}cl_{\theta}(e^{*}\text{-}cl_{\theta}(A))=e^{*}\text{-}cl_{\theta}(A).$

##  3\. On $e^{*}\text{-}\theta$-open sets

######  Definition 3.1.

$A$ subset $A$ of a topological space $X$ is said to be
$\theta\text{-}$complement $e^{*}\text{-}$open (briefly,
$\theta\text{-}c\text{-}e^{*}\text{-}$open$)$ provided there exists a subset
$U$ of $X$ for which $X\setminus A=e^{*}\text{-}cl_{\theta}(U).$ We call a set
$\theta\text{-}$complement $e^{*}\text{-}$closed if its complement is
$\theta\text{-}c\text{-}e^{*}\text{-}$open.

######  Remark 3.2.

It should be mentioned that by Lemma 2.14, $X\setminus
A=e^{*}\text{-}cl_{\theta}(U)$ is $e^{*}\text{-}\theta\text{-}$closed and $A$
is $e^{*}\text{-}\theta\text{-}$open. Therefore, the equivalence of
$\theta\text{-}c\text{-}e^{*}\text{-}$open and
$e^{*}\text{-}\theta\text{-}$open is obvious from Definition 3.1.

######  Theorem 3.3.

Let $X$ be a topological space and $A\subseteq X.$ If $A$ is
$e^{*}\text{-}$open, then $e^{*}\text{-}int(e^{*}\text{-}cl_{\theta}(A))$ is
$e^{*}\text{-}\theta\text{-}open.$

###### Proof.

Let $A\in e^{*}O(X).$
$\begin{array}[]{l}A\in e^{*}O(X)\Rightarrow e^{*}\text{-}cl(A)\in
e^{*}C(X)\Rightarrow X\setminus e^{*}\text{-}cl(A)\in e^{*}O(X)\\\
\overset{\text{Theorem }\ref{185}}{\Rightarrow}e^{*}\text{-}cl(X\setminus
e^{*}\text{-}cl(A))=e^{*}\text{-}cl_{\theta}(X\setminus
e^{*}\text{-}cl(A))\ldots(1)\end{array}$
$\left.\begin{array}[]{rr}X\setminus
e^{*}\text{-}int(e^{*}\text{-}cl(A))=e^{*}\text{-}cl(X\setminus
e^{*}\text{-}cl(A))\Rightarrow e^{*}\text{-}int(e^{*}\text{-}cl(A))=\setminus
e^{*}\text{-}cl(\setminus e^{*}\text{-}cl(A))\\\ A\in
e^{*}O(X)\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow
e^{*}\text{-}int(e^{*}\text{-}cl_{\theta}(A))=e^{*}\text{-}int(e^{*}\text{-}cl(A))=X\setminus
e^{*}\text{-}cl(X\setminus e^{*}\text{-}cl(A))\ldots(2)\end{array}$
$(1),(2)$ $\Rightarrow e^{*}\text{-}int(e^{*}\text{-}cl_{\theta}(A))=\setminus
e^{*}\text{-}cl_{\theta}(\setminus e^{*}\text{-}cl(A))\Rightarrow
e^{*}\text{-}int(e^{*}\text{-}cl_{\theta}(A))\in e^{*}\theta O(X).$ ∎

######  Theorem 3.4.

Let $X$ be a topological space. Then the notion of
$e^{*}\text{-}\theta\text{-}$open is equivalent to the notion of
$e^{*}\text{-}$regular if and only if $e^{*}\text{-}cl_{\theta}(A)$ is
$e^{*}\text{-}$regular for every set $A\subseteq X.$

###### Proof.

$(\Rightarrow):$ Let $e^{*}\theta O(X)=e^{*}R(X)$ and $A\subseteq X.$
$A\subseteq X\Rightarrow e^{*}\text{-}cl_{\theta}(A)\overset{\text{Theorem
}\ref{8}}{=}e^{*}\text{-}cl_{\theta}(e^{*}\text{-}cl_{\theta}(A))\Rightarrow
e^{*}\text{-}cl_{\theta}(A)\in e^{*}\theta C(X)\ldots(1)$
$\left.\begin{array}[]{rr}e^{*}\text{-}cl_{\theta}(A)\in e^{*}\theta
C(X)\Rightarrow X\setminus e^{*}\text{-}cl_{\theta}(A)\in e^{*}\theta O(X)\\\
e^{*}\theta O(X)=e^{*}R(X)\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow$ $X\setminus e^{*}\text{-}cl_{\theta}(A)\in
e^{*}R(X)\subseteq e^{*}\theta C(X)\end{array}$
$\begin{array}[]{l}\Rightarrow e^{*}\text{-}cl_{\theta}(A)\in e^{*}\theta
O(X)\ldots(2)\end{array}$
$(1),(2)\Rightarrow e^{*}\text{-}cl_{\theta}(A)\in e^{*}R(X).$

$(\Leftarrow):$ Let $U\in e^{*}\theta O(X).$ Our aim is to show that $U\in
e^{*}R(X).$
$\left.\begin{array}[]{rr}U\in e^{*}\theta O(X)\overset{\text{Remark
}\ref{uyarı}}{\Rightarrow}(\exists A\subseteq X)(X\setminus
U=e^{*}\text{-}cl_{\theta}(A))\\\
\text{Hypothesis}\end{array}\right\\}\Rightarrow X\setminus U\in e^{*}R(X)$
$\begin{array}[]{l}\Rightarrow U\in e^{*}R(X).\end{array}$ ∎

######  Theorem 3.5.

Let $X$ be a topological space and $B\subseteq X.$ If $B$ is
$e^{*}\text{-}\theta$-open, then $B$ is an union some of
$e^{*}\text{-}$regular sets.

###### Proof.

Let $B\in e^{*}\theta O(X)$ and $x\in B.$
$\left.\begin{array}[]{rr}B\in e^{*}\theta O(X)\overset{\text{Remark
}\ref{uyarı}}{\Rightarrow}(\exists A\subseteq X)(B=X\setminus
e^{*}\text{-}cl_{\theta}(A))\\\ x\in B\end{array}\right\\}\Rightarrow x\notin
e^{*}\text{-}cl_{\theta}(A)$
$\begin{array}[]{l}\Rightarrow(\exists W_{x}\in
e^{*}O(X,x))(e^{*}\text{-}cl(W_{x})\cap A=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists W_{x}\in
e^{*}O(X,x))(e^{*}\text{-}cl(W_{x})\subseteq\setminus A)\end{array}$
$\left.\begin{array}[]{rr}\Rightarrow(W_{x}\in
e^{*}O(X,x))(e^{*}\text{-}int(e^{*}\text{-}cl(W_{x}))=(e^{*}\text{-}int_{\theta}(e^{*}\text{-}cl(W_{x}))\subseteq
e^{*}\text{-}int_{\theta}(\setminus A)=\setminus
e^{*}\text{-}cl_{\theta}(A)\\\
\mathcal{A}:=\\{e^{*}\text{-}int(e^{*}\text{-}cl(W_{x}))|(\forall x\in
B)(\exists W_{x}\in
e^{*}O(X,x))(e^{*}\text{-}int(e^{*}\text{-}cl(W_{x}))\subseteq\setminus
e^{*}\text{-}cl_{\theta}(A))\\}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\mathcal{A}\subseteq
e^{*}R(X))(B=\bigcup\mathcal{A}).\end{array}$ ∎

######  Corollary 3.6.

Let $X$ be a topological space and $B\subseteq X.$ If $B$ is
$e^{*}\text{-}\theta$-closed, then $B$ is an intersection some of
$e^{*}$-regular sets.

##  4\. On $e^{*}\text{-}\theta\text{-}D_{i}$ and $e^{*}\theta\text{-}T_{i}$
topological spaces

In this chapter, we introduce some classes of sets via the notion of
$e^{*}\text{-}\theta$-open sets. Also, the relationships between these notions
and some other existing notions in the literature are investigated.

######  Definition 4.1.

A subset $A$ of a topological space $X$ is called an
$e^{*}\text{-}\theta\text{-}D$-set if there exist two sets $U,V\in e^{*}\theta
O(X)$ such that $U\neq X$ and $A=U\setminus V.$ The family of all
$e^{*}\text{-}\theta\text{-}D$-set of $X$ and all
$e^{*}\text{-}\theta\text{-}D$-set of $X$ containing $x\in X$ will be denoted
by $e^{*}\theta D(X)$ and $e^{*}\theta D(X,x),$ respectively.

######  Remark 4.2.

It is clear that every $e^{*}\text{-}\theta$-open set $U$ different from $X$
is an $e^{*}\text{-}\theta\text{-}D\text{-}$set. However, the converse of this
implication need not be true as shown by the following example.

######  Example 4.3.

Let $X=\\{a,b,c,d\\}$ and
$\tau=\\{\emptyset,X,\\{a\\},\\{c\\},\\{a,c\\},\\{c,d\\},\\{a,c,d\\}\\}.$
Then, $e^{*}\theta O(X)=2^{X}\setminus\\{\\{b\\}\\}$ and $e^{*}\theta
D(X)=2^{X}\setminus\\{X\\}.$ It is obvious that the set $\\{b\\}$ is an
$e^{*}$-$\theta$-$D$-set but it is not $e^{*}$-$\theta$-open.

######  Definition 4.4.

A topological space $X$ is called $e^{*}\text{-}\theta\text{-}D_{0}$ if for
any distinct pair of points $x$ and $y$ in $X,$ there exists
$e^{*}\text{-}\theta\text{-}D\text{-}$set $U$ of $X$ containing $x$ but not
$y$ or $e^{*}\text{-}\theta\text{-}D\text{-}$set $V$ of $X$ containing $y$ but
not $x.$

######  Definition 4.5.

A topological space $X$ is called $e^{*}\text{-}\theta\text{-}D_{1}$ if for
any distinct pair of points $x$ and $y$ in $X,$ there exists
$e^{*}\text{-}\theta\text{-}D\text{-}$set $U$ in $X$ containing $x$ but not
$y$ and $e^{*}\text{-}\theta\text{-}D\text{-}$set $V$ of $X$ containing $y$
but not $x.$

######  Definition 4.6.

A topological space $X$ is called $e^{*}\text{-}\theta\text{-}D_{2}$ if for
any distinct pair of points $x$ and $y$ in $X,$ there exist two
$e^{*}\text{-}\theta\text{-}D\text{-}$sets $U$ and $V$ of $X$ containing $x$
and $y,$ respectively, such that $U\cap V=\emptyset.$

######  Definition 4.7.

$($[1]$)$ A topological space $X$ is called $e^{*}\theta\text{-}T_{0}$ if for
any distinct pair of points $x$ and $y$ in $X,$ there exists an
$e^{*}\text{-}\theta$-open set $U$ of $X$ containing $x$ but not $y$ and an
$e^{*}\text{-}\theta$-open set $V$ of $X$ containing $y$ but not $x.$

######  Definition 4.8.

$($[1]$)$ A topological space $X$ is called $e^{*}\theta\text{-}T_{1}$ if for
any distinct pair of points $x$ and $y$ in $X,$ there exists an
$e^{*}\text{-}\theta\text{-}$open set $U$ of $X$ containing $x$ but not $y$
and an $e^{*}\text{-}\theta$-open set $V$ of $X$ containing $y$ but not $x.$

######  Definition 4.9.

$($[1]$)$ A topological space $X$ is called $e^{*}\theta\text{-}T_{2}$ if for
any distinct pair of points $x$ and $y$ in $X,$ there exist two
$e^{*}\text{-}\theta$-open sets $U$ and $V$ of $X$ containing $x$ and $y,$
respectively, such that $U\cap V=\emptyset.$

######  Remark 4.10.

From Definitions 4.1 to 4.9, we obtain the following diagram:

$\begin{array}[]{c}\begin{array}[]{ccccccccc}e^{*}\theta\text{-}T_{2}&\Rightarrow&e^{*}\theta\text{-}T_{1}&\Rightarrow&e^{*}\theta\text{-}T_{0}\\\
\Downarrow&&\Downarrow&&\Downarrow&&&&\\\
e^{*}\text{-}\theta\text{-}D_{2}&\Rightarrow&e^{*}\text{-}\theta\text{-}D_{1}&\Rightarrow&e^{*}\text{-}\theta\text{-}D_{0}&&&&\
\mbox{}\end{array}\end{array}$

######  Theorem 4.11.

$($[1]$)$ Let $X$ be a topological space. If $X$ is
$e^{*}\theta\text{-}T_{0},$ then it is $e^{*}\theta\text{-}T_{2}.$

######  Theorem 4.12.

Let $X$ be a topological space. If $X$ is $e^{*}\text{-}\theta\text{-}D_{0},$
then it is $e^{*}\theta\text{-}T_{0}.$

###### Proof.

It suffices to prove that every $e^{*}$-$\theta$-$D_{0}$ space is
$e^{*}\theta$-$T_{0}.$
Let $x,y\in X$ and $x\neq y.$
$\left.\begin{array}[]{rr}(x,y\in X)(x\neq y)\\\ X\text{ is
}e^{*}\text{-}\theta\text{-}D_{0}\end{array}\right\\}$ $\Rightarrow(\exists
A\in e^{*}\theta D(X,x))(y\notin A)\lor(\exists B\in e^{*}\theta
D(X,y))(x\notin B)\\\ \begin{array}[]{l}\Rightarrow(\exists N,M\in e^{*}\theta
O(X))(M\neq X)(A=M\setminus N)(x\in A)(y\notin M\lor y=M\cap N)\end{array}$
First case: Let $y\notin M.$
$\left.\begin{array}[]{rr}U:=M\\\ y\notin
M\end{array}\right\\}\Rightarrow(U\in e^{*}\theta O(X,x))(y\notin U)$
Second case: Let $y\in M\cap N.$
$\left.\begin{array}[]{rr}(y\in M\cap N\subseteq N)(V:=N)\\\ (A=M\setminus
N)(x\in A)\Rightarrow x\notin N\end{array}\right\\}\Rightarrow(V\in
e^{*}\theta O(X,y))(x\notin V.)$ ∎

######  Corollary 4.13.

For a topological space $X$, all notions given in Remark 4.10 are equivalent.

######  Definition 4.14.

Let $X$ be a topological space, $N\subseteq X$ and $x\in X$. The set $N$ is
called an $e^{*}\text{-}\theta\text{-}$neighbourhood of $x$ in $X$ if there
exists an $e^{*}\text{-}\theta\text{-}$open set $U$ of $X$ such that $x\in
U\subseteq N.$ The family of all $e^{*}\text{-}\theta\text{-}$neighbourhood of
a point $x$ is denoted by $\mathcal{N}_{e^{*}\theta}(x).$

######  Definition 4.15.

Let $X$ be a topological space and $x\in X.$ The point $x$ which has only $X$
as the $e^{*}\text{-}\theta\text{-}$neighbourhood is called a point common to
all $e^{*}\text{-}\theta\text{-}$closed sets (briefly,
$e^{*}\text{-}\theta\text{-cc}).$

######  Theorem 4.16.

Let $X$ be a topological space. If $X$ is $e^{*}\text{-}\theta\text{-}D_{1}$,
then $X$ has no $e^{*}\text{-}\theta\text{-cc}\text{-}point.$

###### Proof.

Let $x,y\in X$ and $x\neq y.$
$\left.\begin{array}[]{rr}(x,y\in X)(x\neq y)\\\ X\text{ is
}e^{*}\text{-}\theta\text{-}D_{1}\end{array}\right\\}\Rightarrow(\exists A\in
e^{*}\theta D(X,x))(\exists B\in e^{*}\theta D(X,y))(x\notin B)(y\notin A)$
$\begin{array}[]{l}\Rightarrow(\exists U,V\in e^{*}\theta O(X))(U\neq X)(x\in
A=U\backslash V)\end{array}$
$\begin{array}[]{l}\Rightarrow(U\in e^{*}\theta O(X))(x\in U\subseteq U\neq
X)\end{array}$
$\begin{array}[]{l}\Rightarrow X\neq
U\in\mathcal{N}_{e^{*}\theta}(x)\end{array}$
$\begin{array}[]{l}\Rightarrow\mathcal{N}_{e^{*}\theta}(x)\neq\\{X\\}.\end{array}$
∎

######  Definition 4.17.

A subset $A$ of a topological space $X$ is called a quasi
$e^{*}\text{-}\theta\text{-}$closed set (briefly, $qe^{*}\theta$-closed) if
$e^{*}\text{-}cl_{\theta}(A)\subseteq U$ whenever $A\subseteq U$ and $U$ is
$e^{*}\text{-}\theta$-open in $X.$ The family of all quasi
$e^{*}\text{-}\theta\text{-}$closed set in $X$ will be denoted by
$qe^{*}\theta C(X).$

######  Lemma 4.18.

$($[5]$)$ Let $A$ be any subset of a space $X.$ Then, $x\in
e^{*}\text{-}cl_{\theta}(A)$ if and only if $U\cap A\neq\emptyset$ for each
$U\in e^{*}R(X,x).$

######  Theorem 4.19.

For a topological space $X,$ the following statements hold:
$(a)$ For each pair of points x and y in X, $x\in
e^{*}\text{-}cl_{\theta}(\\{y\\})$ implies $y\in
e^{*}\text{-}cl_{\theta}(\\{x\\});$
$(b)$ For each $x\in X,$ the singleton $\\{x\\}$ is $qe^{*}\theta$-closed in
$X.$

###### Proof.

$(a)$ Let $x,y\in X$ and $y\notin e^{*}\text{-}cl_{\theta}(\\{x\\}).$
$\left.\begin{array}[]{rr}y\notin
e^{*}\text{-}cl_{\theta}(\\{x\\})\Rightarrow(\exists V\in
e^{*}O(X,y))(e^{*}\text{-}cl(V)\cap\\{x\\}=\emptyset)\\\ \text{Theorem
}\ref{1}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(e^{*}\text{-}cl(V)\in
e^{*}R(X,y))(e^{*}\text{-}cl(V)\cap\\{x\\}=\emptyset)\\\
U:=e^{*}\text{-}cl(V)\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(U\in
e^{*}O(X,y))(e^{*}\text{-}cl(U)\cap\\{x\\}=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow x\notin
e^{*}\text{-}cl_{\theta}(\\{y\\}).\end{array}$

$(b)$ Let $x\in X$, $U\in e^{*}\theta O(X)$ and $\\{x\\}\subseteq U.$
$\begin{array}[]{rr}(x\in X)(\\{x\\}\subseteq U)(U\in e^{*}\theta
O(X))\Rightarrow U\in e^{*}\theta O(X,x)\end{array}$
$\left.\begin{array}[]{rr}\overset{\text{Corollary
}\ref{5}}{\Rightarrow}(\exists V\in e^{*}R(X,x))(V\subseteq U)\Rightarrow(V\in
e^{*}R(X,x))(V=e^{*}\text{-}cl(V)\subseteq U)\\\ e^{*}R(X)\subseteq
e^{*}\theta O(X)\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(V\in
e^{*}O(X,x))(e^{*}\text{-}cl_{\theta}(\\{x\\})\subseteq
e^{*}\text{-}cl_{\theta}(V)=e^{*}\text{-}cl(V)\subseteq U).\end{array}$ ∎

######  Definition 4.20.

A space $X$ is said to be $e^{*}\text{-}\theta\text{-}T_{1/2}$ if
$qe^{*}\theta C(X)\subseteq e^{*}\theta C(X).$

######  Theorem 4.21.

For a topological space $X,$ the followings are equivalent:
$(a)$ $X$ is $e^{*}\text{-}\theta\text{-}T_{1/2};$
$(b)$ $X$ is $e^{*}\theta\text{-}T_{1}.$

###### Proof.

$(a)\Rightarrow(b):$ Let $x,y\in X$ and $x\neq y.$
$\left.\begin{array}[]{r}x,y\in X\overset{\text{Theorem
}\ref{4}}{\Rightarrow}\\{x\\},\\{y\\}\in qe^{*}\theta C(X)\\\ X\text{ is
}e^{*}\text{-}\theta\text{-}T_{1/2}\Rightarrow qe^{*}\theta C(X)\subseteq
e^{*}\theta C(X)\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\{x\\},\\{y\\}\in e^{*}\theta C(X)\\\ x\neq
y\end{array}\right\\}\Rightarrow\end{array}$
$\left.\begin{array}[]{rr}\Rightarrow(X\setminus\\{y\\}\in e^{*}\theta
O(X,x))(X\setminus\\{x\\}\in e^{*}\theta O(X,y))\\\
(U:=X\setminus\\{y\\})(V:=X\setminus\\{x\\})\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(U\in e^{*}\theta O(X,x))(V\in e^{*}\theta
O(X,y))(y\notin U)(x\notin V).\\\ \end{array}$

$(b)\Rightarrow(a):$ Let $A\in qe^{*}\theta C(X).$ Suppose that $A\notin
e^{*}\theta C(X).$ We will obtain a contradiction.

$\left.\begin{array}[]{rr}A\notin e^{*}\theta C(X)\Rightarrow A\neq
e^{*}\text{-}cl_{\theta}(A)\Rightarrow(\exists x\in X)(x\in
e^{*}\text{-}cl_{\theta}(A)\setminus A)\\\ X\text{ is
}e^{*}\theta\text{-}T_{1}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(\forall a\in A)(\exists V_{a}\in
e^{*}\theta O(X,a))(x\notin V_{a})\\\ \text{Corollary
}\ref{5}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(A\subseteq\bigcup_{a\in
A}V_{a})(x\notin\bigcup_{a\in A}V_{a}\in e^{*}\theta O(X))\\\ A\in
qe^{*}\theta C(X)\end{array}\right\\}\Rightarrow x\notin
e^{*}\text{-}cl_{\theta}(A)\subseteq\bigcup_{a\in A}V_{a}$

This contradicts with $x\in e^{*}\text{-}cl_{\theta}(A).$ ∎

######  Definition 4.22.

A function $f:X\rightarrow Y$ is said to be weakly $e^{*}$-irresolute [10]
(briefly, $w.e^{*}.i.$) (resp. strongly $e^{*}$-irresolute) if for each $x\in
X$ and each $V\in e^{*}O(Y,f(x)),$ there exists $U\in e^{*}O(X,x)$ such that
$f[U]\subseteq e^{*}\text{-}cl(V)$ $($resp. $f[e^{*}\text{-}cl(V)]\subseteq
V).$

######  Remark 4.23 ([10], Theorem $3.7.(d)(e)$).

A function $f:X\rightarrow Y$ is weakly $e^{*}\text{-}$irresolute if and only
if $f^{-1}[V]$ is $e^{*}\text{-}\theta$-closed (resp.
$e^{*}\text{-}\theta$-open) in $X$ for every $e^{*}\text{-}\theta$-closed
(resp. $e^{*}\text{-}\theta$-open) set $V$ in $Y.$

######  Theorem 4.24.

If $f:X\rightarrow Y$ is a weakly $e^{*}\text{-}$irresolute surjection and $A$
is an $e^{*}\text{-}\theta\text{-}$D-set in $Y,$ then the inverse image of $A$
is an $e^{*}\text{-}\theta\text{-}$D-set in $X$.

###### Proof.

Let $A\in e^{*}\theta D(Y).$
$\left.\begin{array}[]{rr}A\in e^{*}\theta D(Y)\Rightarrow(\exists U,V\in
e^{*}\theta O(Y))(U\neq Y)(A=U\setminus V)\\\ f\text{ is weakly
$e^{*}$-irresolute surjection}\end{array}\right\\}\overset{\text{Theorem
}\ref{3}}{\Rightarrow}$
$\begin{array}[]{l}\Rightarrow(f^{-1}[U],f^{-1}[V]\in e^{*}\theta
O(X))(f^{-1}[U]\neq f^{-1}[Y]=X)(f^{-1}[A]=f^{-1}[U]\setminus
f^{-1}[V])\end{array}$
$\begin{array}[]{l}\Rightarrow f^{-1}[A]\in e^{*}\theta D(X).\end{array}$ ∎

######  Theorem 4.25.

If $Y$ is an $e^{*}\text{-}\theta\text{-}D_{1}$ space and $f:X\rightarrow Y$
is a weakly $e^{*}$-irresolute bijection, then $X$ is
$e^{*}\text{-}\theta\text{-}D_{1}.$

###### Proof.

Let $x,y\in X$ and $x\neq y.$
$\left.\begin{array}[]{r}(x,y\in X)(x\neq y)\\\ f\text{ is
bijective}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!\\!(f(x),f(y)\in Y)(f(x)\neq f(y))\\\
Y\text{ is
}e^{*}\text{-}\theta\text{-}D_{1}\end{array}\right\\}\Rightarrow\end{array}$
$\left.\begin{array}[]{rr}\\!\Rightarrow(\exists U\in e^{*}\theta
D(Y,f(x)))(\exists V\in e^{*}\theta D(Y,f(y)))(f(y)\notin U)(f(x)\notin V)\\\
\text{Theorem }\ref{2}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(y\notin f^{-1}[U]\in e^{*}\theta
D(X,x))(x\notin f^{-1}[V]\in e^{*}\theta D(X,y)).\end{array}$ ∎

######  Theorem 4.26.

For a topological space $X,$ the followings are equivalent:
$(a)$ $X$ is $e^{*}\text{-}\theta\text{-}D_{1};$
$(b)$ For each pair of distinct points $x,y\in X,$ there exists a weakly
$e^{*}\text{-}irresolute$ surjection $f:X\rightarrow Y,$ where $Y$ is an
$e^{*}\text{-}\theta\text{-}D_{1}$ space such that $f(x)\neq f(y).$

###### Proof.

$(a)\Rightarrow(b):$ Let $x,y\in X$ and $x\neq y.$
$\left.\begin{array}[]{rr}(x,y\in X)(x\neq y)\\\ (Y:=X)(f:=\\{(x,x)|x\in
X\\})\end{array}\right\\}\overset{\text{Hypothesis}}{\Rightarrow}$
$\begin{array}[]{l}\Rightarrow(f\text{ is }w.e^{*}.i.\text{
surjection})(Y\text{ is }e^{*}\text{-}\theta\text{-}D_{1})(f(x)\neq
f(y)).\end{array}$

$(b)\Rightarrow(a):$ Let $x,y\in X$ and $x\neq y.$
$\left.\begin{array}[]{rr}(x,y\in X)(x\neq y)\\\
\text{Hypothesis}\end{array}\right\\}\Rightarrow(\exists f\in Y^{X}\
w.e^{*}.i.\text{ surjection})(Y\text{ is
}e^{*}\text{-}\theta\text{-}D_{1})(f(x)\neq f(y))$

$\left.\begin{array}[]{rr}\Rightarrow(f\in Y^{X}\ w.e^{*}.i.\text{
sur.})(\exists U\in e^{*}\theta D(Y,f(x)))(\exists V\in e^{*}\theta
D(Y,f(y)))(f(y)\notin U)(f(x)\notin V)\\\ \text{Theorem
}\ref{2}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(y\notin f^{-1}[U]\in e^{*}\theta
D(X,x))(x\notin f^{-1}[V]\in e^{*}\theta D(X,y)).\end{array}$ ∎

##  5\. Further properties

######  Definition 5.1.

Let $A$ be a subset of a topological space $X.$ The
$e^{*}\text{-}\theta$-kernel of $A,$ denoted by
$e^{*}\text{-}ker_{\theta}(A),$ is defined to be the set $\bigcap\\{U|(U\in
e^{*}\theta O(X))(A\subseteq U)\\}.$

######  Remark 5.2.

For a subset $A$ of a topological space $X$, the sets of
$e^{*}\text{-}ker_{\theta}(A)$ and $\beta\text{-}ker_{\theta}(A)$ need not be
equal to each other as shown by the following example.

######  Example 5.3.

Let $X=\\{a,b,c,d\\}$ and
$\tau=\\{\emptyset,X,\\{a\\},\\{a,b\\},\\{a,b,c\\}\\}.$ Then
$e^{*}R(X)=e^{*}\theta O(X)=e^{*}O(X)=2^{X}$ and $\beta R(X)=\beta\theta
O(X)=\\{\emptyset,X\\},$ $\beta
O(X)=\\{\emptyset,X,\\{a\\},\\{a,b\\},\\{a,c\\},\\{a,d\\},\\{a,b,c\\},\\{a,b,d\\},\\{a,c,d\\}\\}.$
For the subset $A=\\{a,b\\},$ $e^{*}\text{-}ker_{\theta}(A)=A\neq
X=\beta\text{-}ker_{\theta}(A).$

######  Definition 5.4.

A space $X$ is called slightly $e^{*}\text{-}\theta\text{-}R_{0}$ space if
$\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})|x\in X\\}=\emptyset.$

######  Remark 5.5.

A slightly $e^{*}$-$\theta$-$R_{0}$ space need not be a slightly
$\beta$-$\theta$-$R_{0}$ space as shown by the following example.

######  Example 5.6.

Let $X=\\{a,b,c,d\\}$ and
$\tau=\\{\emptyset,X,\\{a\\},\\{a,b\\},\\{a,b,c\\}\\}.$ Since
$\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})|x\in
X\\}=\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{a\\}),e^{*}\text{-}cl_{\theta}(\\{b\\}),e^{*}\text{-}cl_{\theta}(\\{c\\}),e^{*}\text{-}cl_{\theta}(\\{d\\})\\}=\emptyset,$
the space $X$ is a slightly $e^{*}$-$\theta$-$R_{0}$ space. On the other hand,
since $\bigcap\\{\beta\text{-}cl_{\theta}(\\{x\\})|x\in
X\\}=\bigcap\\{\beta\text{-}cl_{\theta}(\\{a\\}),\beta\text{-}cl_{\theta}(\\{b\\}),\beta\text{-}cl_{\theta}(\\{c\\}),\beta\text{-}cl_{\theta}(\\{d\\})\\}=\bigcap\\{X\\}=X\neq\emptyset,$
the space $X$ is not a slightly $\beta$-$\theta$-$R_{0}$ space.

######  Theorem 5.7.

Let $A$ be a subset of a space $X.$ Then,
$e^{*}\text{-}ker_{\theta}(A)=\\{x\in X|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap
A\neq\emptyset\\}.$

###### Proof.

Let $x\notin e^{*}\text{-}ker_{\theta}(A).$
$\begin{array}[]{rcl}x\notin
e^{*}\text{-}ker_{\theta}(A)&\Rightarrow&x\notin\bigcap\\{U|(U\in e^{*}\theta
O(X))(A\subseteq U)\\}\\\ &\Rightarrow&(\exists U\in e^{*}\theta
O(X))(A\subseteq U)(x\notin U)\\\ &\Rightarrow&(\setminus U\in e^{*}\theta
C(X))(\\{x\\}\subseteq\setminus U)(\setminus U\subseteq\setminus A)\\\
&\Rightarrow&(\setminus U\in e^{*}\theta
C(X))(e^{*}\text{-}cl_{\theta}(\\{x\\})\subseteq
e^{*}\text{-}cl_{\theta}(\setminus U)=\setminus U\subseteq\setminus A)\\\
&\Rightarrow&e^{*}\text{-}cl_{\theta}(\\{x\\})\cap A=\emptyset\\\
&\Rightarrow&x\notin\\{x|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap
A=\emptyset\\}\end{array}$

Then, we have

$\\{x|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap A=\emptyset\\}\subseteq
e^{*}\text{-}ker_{\theta}(A)\ldots(1)$

Now, let $x\notin\\{x|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap
A\neq\emptyset\\}.$
$\left.\begin{array}[]{rr}x\notin\\{x|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap
A\neq\emptyset\\}\Rightarrow e^{*}\text{-}cl_{\theta}(\\{x\\})\cap
A=\emptyset\Rightarrow A\subseteq\setminus
e^{*}\text{-}cl_{\theta}(\\{x\\})\\\ U:=\setminus
e^{*}\text{-}cl_{\theta}(\\{x\\})\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(U\in e^{*}\theta O(X))(A\subseteq U)(x\notin
U)\end{array}$
$\begin{array}[]{l}\Rightarrow x\notin\bigcap\\{U|(U\in e^{*}\theta
O(X))(A\subseteq U)\\}=e^{*}\text{-}ker_{\theta}(A)\end{array}$

Then, we have

$e^{*}\text{-}ker_{\theta}(A)\subseteq\\{x|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap
A=\emptyset\\}\ldots(2)$

$(1),(2)\Rightarrow
e^{*}\text{-}ker_{\theta}(A)=\\{x|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap
A=\emptyset\\}.$ ∎

######  Theorem 5.8.

Let $X$ be a topological space. Then, $X$ is slightly
$e^{*}\text{-}\theta\text{-}R_{0}$ if and only if
$e^{*}\text{-}ker_{\theta}(\\{x\\})\neq X$ for any $x\in X.$

###### Proof.

$(\Rightarrow):$ Suppose that there is a point $y$ in $X$ such that
$e^{*}\text{-}ker_{\theta}(\\{y\\})=X.$
$e^{*}\text{-}ker_{\theta}(\\{y\\})=\\{x|e^{*}\text{-}cl_{\theta}(\\{x\\})\cap\\{y\\}\neq\emptyset\\}=X\Rightarrow(\forall
x\in X)(y\in e^{*}\text{-}cl_{\theta}(\\{x\\}))$
$\left.\begin{array}[]{rr}\Rightarrow
y\in\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})|x\in X\\}\\\
\text{Hypothesis}\end{array}\right\\}\overset{\text{Theorem
}\ref{kernel}}{\Rightarrow}y\in\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})|x\in
X\\}=\emptyset$

This is a contradiction.

$(\Leftarrow):$ Suppose that $X$ is not slightly
$e^{*}\text{-}\theta\text{-}R_{0}.$
$\begin{array}[]{rcl}X\text{ is not slightly
}e^{*}\text{-}\theta\text{-}R_{0}&\Rightarrow&\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})|x\in
X\\}\neq\emptyset\\\ &\Rightarrow&(\exists y\in
X)(y\in\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})|x\in X\\})\\\
&\Rightarrow&(\exists y\in X)(\forall x\in X)(y\in
e^{*}\text{-}cl_{\theta}(\\{x\\}))\\\ &\overset{\text{Theorem
}\ref{kapanis}}{\Rightarrow}&(\forall x\in
X)(y\in\bigcap\\{V|(\\{x\\}\subseteq V)(V\in e^{*}R(X))\\})\\\
&\Rightarrow&(\forall x\in X)(\forall V\in e^{*}R(X,y))(\\{x\\}\subseteq V)\\\
&\Rightarrow&(\forall V\in e^{*}R(X,y))(V=X)\\\
&\Rightarrow&e^{*}\text{-}cl_{\theta}(\\{y\\})=X\end{array}$
This is a contradiction. ∎

######  Theorem 5.9.

Let $X$ and $Y$ be two topological spaces. If $X$ is slightly
$e^{*}\text{-}\theta\text{-}R_{0},$ then the product $X\times Y$ is slightly
$e^{*}\text{-}\theta\text{-}R_{0}.$

###### Proof.

Let $X$ be slightly $e^{*}$-$\theta$-$R_{0}.$
$\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{(x,y)\\})|(x,y)\in X\times
Y\\}\subseteq\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})\times
e^{*}\text{-}cl_{\theta}(\\{y\\}|(x,y)\in X\times Y\\}\\\
=\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{x\\})|x\in
X\\}\times\bigcap\\{e^{*}\text{-}cl_{\theta}(\\{y\\})|y\in Y\\}=\emptyset.$ ∎

######  Definition 5.10.

A function $f:X\rightarrow Y$ is $S$-continuous (resp.
$\theta\text{-}S\text{-}e^{*}\text{-continuous},\\\
S\text{-}e^{*}$-continuous) if for each $x\in X$ and each $e^{*}$-open subset
$V$ of $Y$ containing $f(x)$, there exists an open subset $U$ of $X$
containing $x$ such that $cl(f[U])\subseteq V$ (resp.
$e^{*}\text{-}cl_{\theta}(f[U])\subseteq V,\ e^{*}\text{-}cl(f[U])\subseteq
V).$

######  Definition 5.11.

A function $f:X\rightarrow Y$ is said to be $e^{*}\text{-}$open [5] if $f[U]$
is $e^{*}$-open in $Y$ for every open set $U$ of $X.$

######  Remark 5.12.

From Definitions 5.10 and 5.11, we have the following diagram.

$\theta\text{-}S\text{-}e^{*}\text{-}\text{continuous}\longrightarrow
S\text{-}e^{*}\text{-}\text{continuous}\longleftarrow
S\text{-}\text{continuous}$

A function $f$ which is $S$-$e^{*}$-continuous need not be $S$-continuous as
shown by the following example.

######  Example 5.13.

Let $X=\\{a,b,c,d\\}$ and
$\tau=\\{\emptyset,X,\\{a\\},\\{c\\},\\{a,c\\},\\{c,d\\},\\{a,c,d\\}\\}.$
Define the function $f:(X,\tau)\rightarrow(X,\tau)$ by $f(x)=c.$ The function
$f$ is $S$-$e^{*}$-continuous but it is not $S$-continuous.

######  Question 5.14.

Is there any $S$-$e^{*}$-continuous function which is not
$\theta$-$S$-$e^{*}$-continuous?

######  Theorem 5.15.

Let $X$ and $Y$ be two topological spaces. If $f:X\rightarrow Y$ is
$S\text{-}e^{*}\text{-continuous}$ and $e^{*}\text{-open},$ then $f$ is
$\theta$-$S$-$e^{*}$-continuous.

###### Proof.

Let $x\in X$ and $V\in e^{*}O(Y,f(x))$.
$\left.\begin{array}[]{r}(x\in X)(V\in e^{*}O(Y,f(x)))\\\ f\text{ is}\
S\text{-}e^{*}\text{-}\text{continuous}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!(\exists U\in
O(X,x))(e^{*}\text{-}cl(f[U])\subseteq V)\\\ f\text{ is}\
e^{*}\text{-}\text{open}\end{array}\right\\}\Rightarrow\end{array}\\\
\begin{array}[]{l}\overset{\text{Theorem }\ref{1}(a)}{\Rightarrow}(\exists
U\in O(X,x))(e^{*}\text{-}cl_{\theta}(f[U])=e^{*}\text{-}cl(f[U])\subseteq
V).\par\end{array}$ ∎

######  Definition 5.16.

The graph $G(f)$ of a function $f:X\rightarrow Y$ is said to be strongly
$e^{*}\text{-}\theta$-closed if for each point $(x,y)\in(X\times Y)\setminus
G(f),$ there exist subsets $U\in e^{*}O(X,x)$ and $V\in e^{*}\theta O(Y,y)$
such that $(e^{*}\text{-}cl(U)\times V)\cap G(f)=\emptyset.$

######  Lemma 5.17.

The graph G(f) of $f:X\rightarrow Y$ is strongly $e^{*}\text{-}\theta$-closed
in $X\times Y$ if and only if for each point $(x,y)\in(X\times Y)\setminus
G(f),$ there exists $U\in e^{*}O(X,x)$ and $V\in e^{*}\theta O(Y,y)$ such that
$f[e^{*}\text{-}cl(U)]\cap V=\emptyset.$

###### Proof.

Let $(x,y)\notin G(f).$
$\left.\begin{array}[]{rr}(x,y)\notin G(f)\\\ G(f)\text{ is strongly
}e^{*}\text{-}\theta\text{-}\text{closed}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists U\in e^{*}O(X,x))(\exists V\in
e^{*}\theta O(Y,y))((e^{*}\text{-}cl(U)\times V)\cap
G(f)=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in e^{*}O(X,x))(\exists V\in
e^{*}\theta O(Y,y))(\forall x\in X)((x,f(x))\notin e^{*}\text{-}cl(U)\times
V)\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in e^{*}O(X,x))(\exists V\in
e^{*}\theta O(Y,y))(f[e^{*}\text{-}cl(U)]\cap V=\emptyset).\end{array}$ ∎

######  Definition 5.18.

A space $X$ is called to be $e^{*}\text{-}T_{1}$ [3] if for each pair of
distinct points in $X,$ there exist $e^{*}\text{-}$open sets $U$ and $V$
containing $x$ and $y,$ respectively, such that $y\notin U$ and $x\notin V.$

######  Theorem 5.19.

Let $X$ and $Y$ be two topological spaces. If $f:X\rightarrow Y$ is
$\theta\text{-}S\text{-}e^{*}$-continuous weak $e^{*}$-irresolute and $Y$ is
$e^{*}\text{-}T_{1},$ then $G(f)$ is strongly $e^{*}\text{-}\theta$-closed.

###### Proof.

Let $(x,y)\notin G(f).$
$\left.\begin{array}[]{rr}(x,y)\notin G(f)\Rightarrow(y,f(x)\in Y)(y\neq
f(x))\\\ Y\text{ is }e^{*}\text{-}T_{1}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{r}\Rightarrow(\exists V\in e^{*}O(Y,f(x))(y\notin
V))\\\ f\text{ is
}\theta\text{-}S\text{-}e^{*}\text{-}\text{continuous}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!(\exists U\in O(X,x))(y\notin
e^{*}\text{-}cl_{\theta}(f[U]))\\\ f\text{ is weak
}e^{*}\text{-}\text{irresolute}\end{array}\right\\}\Rightarrow\end{array}$
$\begin{array}[]{l}\Rightarrow(U\in e^{*}O(X,x))(\setminus
e^{*}\text{-}cl_{\theta}(f[U])\in e^{*}\theta
O(X,y))(e^{*}\text{-}cl(U)\times(\setminus e^{*}\text{-}cl_{\theta}(f[U]))\cap
G(f)=\emptyset).\end{array}$ ∎

######  Theorem 5.20.

Let $f:X\rightarrow Y$ be a weak $e^{*}\text{-}$irresolute function. Then, $f$
is $\theta\text{-}S\text{-}e^{*}$-continuous if and only if for each $x\in X$
and each $e^{*}\text{-}closed$ subset $F$ of $Y$ with $f(x)\notin F,$ there
exists an open subset $U$ of $X$ containing $x$ and an
$e^{*}\text{-}\theta$-open subset $V$ of $Y$ with $F\subseteq V$ such that
$f[e^{*}\text{-}cl(U)]\cap V=\emptyset.$

###### Proof.

$(\Rightarrow):$ Let $x\in X,$ $F\in e^{*}C(Y)$ and $f(x)\notin F.$
$\left.\begin{array}[]{rr}(x\in X)(F\in e^{*}C(Y))(f(x)\notin F)\Rightarrow
Y\setminus F\in e^{*}O(Y,f(x))\\\ f\text{ is
}\theta\text{-}S\text{-}e^{*}\text{-continuous}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(\exists U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(f[U])\subseteq Y\setminus F)\\\ f\text{ is
weak }e^{*}\text{-irresolute}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(U\in
O(X,x))(f[e^{*}\text{-}cl(U)]\subseteq e^{*}\text{-}cl_{\theta}(f[U])\subseteq
Y\setminus F)\\\ V:=Y\setminus
e^{*}\text{-}cl_{\theta}(f[U])\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(U\in O(X,x))(V\in e^{*}\theta O(Y))(F\subseteq
V\subseteq Y\setminus f[e^{*}\text{-}cl(U)])\end{array}\\\
\begin{array}[]{l}\Rightarrow(U\in O(X,x))(V\in e^{*}\theta
O(Y,F))(f[e^{*}\text{-}cl(U])\cap V=\emptyset.\end{array}$

$(\Leftarrow):$ Let $x\in X$ and $V\in e^{*}O(Y,f(x)).$
$\left.\begin{array}[]{rr}(x\in X)(V\in e^{*}O(Y,f(x)))\Rightarrow f(x)\notin
Y\setminus V\in e^{*}C(Y)\\\ \text{Hypothesis}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists U\in O(X,x))(\exists W\in e^{*}\theta
O(Y,Y\setminus V))(f[e^{*}\text{-}cl(U)]\cap W=\emptyset)\end{array}\\\
\begin{array}[]{l}\Rightarrow(U\in O(X,x))(W\in e^{*}\theta O(Y,Y\setminus
V))(f[U]\subseteq f[e^{*}\text{-}cl(U)]\subseteq Y\setminus W\subseteq
V)\end{array}\\\ \begin{array}[]{l}\Rightarrow(U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(f[U])\subseteq
e^{*}\text{-}cl_{\theta}(Y\setminus W)=Y\setminus W\subseteq V).\end{array}$ ∎

######  Corollary 5.21.

Let $X$ and $Y$ be two topological spaces and $f:X\rightarrow Y$ be a weak
$e^{*}\text{-}$irresolute function. Then, f is
$\theta\text{-}S\text{-}e^{*}\text{-}continuous$ if and only if for each $x\in
X$ and each $e^{*}\text{-}open$ subset $V$ of $Y$ containing $f(x)$, there
exists an open subset $U$ of $X$ containing $x$ such that
$e^{*}\text{-}cl_{\theta}(f[e^{*}\text{-}cl(U)])\subseteq V.$

###### Proof.

$(\Rightarrow):$ Let $x\in X$ and $V\in e^{*}O(Y,f(x)).$
$\left.\begin{array}[]{r}(x\in X)(V\in e^{*}O(Y,f(x)))\\\
\text{Hypothesis}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!\\!(\exists U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(f[U])\subseteq V)\\\ f\text{ is weak
}e^{*}\text{-}\text{irresolute}\end{array}\right\\}\overset{\text{Theorem
}\ref{3}}{\Rightarrow}\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in
O(X,x))(f[e^{*}\text{-}cl(U)]\subseteq e^{*}\text{-}cl_{\theta}(f[U])\subseteq
V)\end{array}$
$\begin{array}[]{l}\Rightarrow(U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(f[e^{*}\text{-}cl(U)])\subseteq
e^{*}\text{-}cl_{\theta}(e^{*}\text{-}cl_{\theta}(f[U]))=e^{*}\text{-}cl_{\theta}(f[U])\subseteq
V).\end{array}$

$(\Leftarrow):$ Let $x\in X$ and $V\in e^{*}O(Y,f(x)).$
$\left.\begin{array}[]{rr}(x\in X)(V\in e^{*}O(Y,f(x)))\\\
\text{Hypothesis}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(f[U])\subseteq
e^{*}\text{-}cl_{\theta}(f[e^{*}\text{-}cl(U)])\subseteq V).\end{array}$ ∎

Now, we discuss some fundamental properties of
$\theta\text{-}S\text{-}e^{*}\text{-}$continuous functions related to
composition and restriction.

######  Theorem 5.22.

Let $f:X\to Y$ and $g:Y\to Z$ be two functions. If $f$ is continuous and $g$
is $\theta\text{-}S\text{-}e^{*}$-continuous, then $g\circ f:X\rightarrow Z$
is $\theta\text{-}S\text{-}e^{*}$-continuous.

###### Proof.

Let $x\in X$ and $W\in e^{*}O(Z,g(f(x))).$
$\left.\begin{array}[]{r}W\in e^{*}O(Z,g(f(x)))\\\ g\text{ is
}\theta\text{-}S\text{-}e^{*}\text{-}\text{continuous}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!\\!(\exists V\in
O(Y,f(x))(e^{*}\text{-}cl_{\theta}(g[V])\subseteq W)\\\ f\text{ is
continuous}\end{array}\right\\}\Rightarrow\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(g(f[U]))\subseteq
e^{*}\text{-}cl_{\theta}(g[V])\subseteq W).\end{array}$ ∎

######  Theorem 5.23.

Let $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ be two functions. If $g\circ f$
is $\theta\text{-}S\text{-}e^{*}\text{-}continuous$ and $f$ is an open
surjection, then $g$ is $\theta\text{-}S\text{-}e^{*}\text{-}continuous.$

###### Proof.

Let $y\in Y$ and $W\in e^{*}O(Z,g(y)).$
$\left.\begin{array}[]{r}(y\in Y)(W\in e^{*}O(Z,g(y)))\\\ f\text{ is
surjective}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!\\!(\exists x\in X)(y=f(x))(W\in
e^{*}O(Z,g(f(x))))\\\ g\circ f\text{ is
}\theta\text{-}S\text{-}e^{*}\text{-continuous}\end{array}\right\\}\Rightarrow\end{array}$
$\left.\begin{array}[]{rr}\Rightarrow(\exists U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(g(f[U]))\subseteq W)\\\ f\text{ is
open}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(f[U]\in
O(Y,y))(e^{*}\text{-}cl_{\theta}(g[f[U]])\subseteq W)\\\
V:=f[U]\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(V\in
O(Y,y))(e^{*}\text{-}cl_{\theta}(g[V])\subseteq W).\end{array}$ ∎

######  Theorem 5.24.

Let $f:X\rightarrow Y$ be a function and $A\subseteq X.$ If $f$ is
$\theta\text{-}S\text{-}e^{*}$-continuous, then $f|_{A}:A\rightarrow Y$ is
$\theta\text{-}S\text{-}e^{*}\text{-}$continuous.

###### Proof.

Let $x\in A$ and $V\in e^{*}O(Y,f(x)).$
$\left.\begin{array}[]{r}(x\in A)(V\in e^{*}O(Y,f(x)))\\\ A\subseteq
X\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!\\!(x\in X)(V\in e^{*}O(Y,f(x)))\\\
f\text{ is
}\theta\text{-}S\text{-}e^{*}\text{-}\text{continuous}\end{array}\right\\}\Rightarrow\end{array}$
$\left.\begin{array}[]{rr}\Rightarrow(\exists W\in
O(X,x))(e^{*}\text{-}cl_{\theta}(f[W])\subseteq V)\\\ U:=W\cap
A\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(U\in
O(A,x))(e^{*}\text{-}cl_{\theta}(f|_{A}[U])=e^{*}\text{-}cl_{\theta}(f[W\cap
A])\subseteq e^{*}\text{-}cl_{\theta}(f[W])\subseteq V).\end{array}$ ∎

##  6\. $e^{*}\text{-}R_{1}$ space

######  Definition 6.1.

A topological space $X$ is said to be $e^{*}\text{-}R_{1}$ if for all $x,y\in
X$ with $e^{*}\text{-}cl(\\{x\\})\neq e^{*}\text{-}cl(\\{y\\}),$ there exist
disjoint $e^{*}$-open sets $U$ and $V$ such that
$e^{*}\text{-}cl(\\{x\\})\subseteq U$ and $e^{*}\text{-}cl(\\{y\\})\subseteq
V.$

######  Remark 6.2.

An $e^{*}$-$R_{1}$ space need not be a $\beta$-$R_{1}$ space as shown by the
following example.

######  Example 6.3.

Let $X=\\{a,b,c,d\\}$ and
$\tau=\\{\emptyset,X,\\{a\\},\\{c\\},\\{a,c\\},\\{c,d\\},\\{a,c,d\\}\\}.$
Then, the space $X$ is an $e^{*}$-$R_{1}$ space but it is not $\beta$-$R_{1}.$

######  Theorem 6.4.

Let $X$ be a topological space. Then, $X$ is $e^{*}\text{-}R_{1}$ if and only
if $e^{*}\text{-}cl_{\theta}(\\{x\\})=e^{*}\text{-}cl(\\{x\\})$ for all $x\in
X.$

###### Proof.

$(\Rightarrow):$ Let $x\in X.$
$x\in X\Rightarrow e^{*}\text{-}cl(\\{x\\})\subseteq
e^{*}\text{-}cl_{\theta}(\\{x\\})\ldots(1)$

Now, let $y\notin e^{*}\text{-}cl(\\{x\\}).$
$\left.\begin{array}[]{rr}y\notin e^{*}\text{-}cl(\\{x\\})\Rightarrow
e^{*}\text{-}cl(\\{x\\})\neq e^{*}\text{-}cl(\\{y\\})\\\ X\text{ is
}e^{*}\text{-}R_{1}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists U,V\in e^{*}O(X))(U\cap
V=\emptyset)(e^{*}\text{-}cl(\\{x\\})\subseteq
U)(e^{*}\text{-}cl(\\{y\\})\subseteq V)\end{array}$
$\begin{array}[]{rcl}\Rightarrow(U\in e^{*}O(X,x))(V\in
e^{*}O(X,y))(e^{*}\text{-}cl(\\{x\\})\cap
e^{*}\text{-}cl(\\{y\\})&\subseteq&e^{*}\text{-}cl(\\{x\\})\cap V\\\
&\subseteq&e^{*}\text{-}cl(\\{x\\})\cap e^{*}\text{-}cl(V)\\\
&\subseteq&e^{*}\text{-}cl(U)\cap e^{*}\text{-}cl(V)=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow(V\in e^{*}O(X,y))(\\{x\\}\cap
e^{*}\text{-}cl(V)\subseteq e^{*}\text{-}cl(\\{x\\})\cap
e^{*}\text{-}cl(V)=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow y\notin
e^{*}\text{-}cl_{\theta}(\\{x\\})\end{array}$

Then, we have $e^{*}\text{-}cl_{\theta}(\\{x\\})\subseteq
e^{*}\text{-}cl(\\{x\\})\ldots(2)$

$(1),(2)\Rightarrow
e^{*}\text{-}cl(\\{x\\})=e^{*}\text{-}cl_{\theta}(\\{x\\}).$

$(\Leftarrow):$ Let $x,y\in X$ and $e^{*}\text{-}cl(\\{x\\})\neq
e^{*}\text{-}cl(\\{y\\}).$
$\begin{array}[]{rr}e^{*}\text{-}cl(\\{x\\})\neq
e^{*}\text{-}cl(\\{y\\})\Rightarrow(\exists z\in X)(z\in
e^{*}\text{-}cl(\\{x\\})\setminus e^{*}\text{-}cl(\\{y\\})\vee z\in
e^{*}\text{-}cl(\\{y\\})\setminus e^{*}\text{-}cl(\\{x\\}))\end{array}$

First Case: Let $z\in e^{*}\text{-}cl(\\{x\\})\setminus
e^{*}\text{-}cl(\\{y\\}).$
$\left.\begin{array}[]{rr}z\in e^{*}\text{-}cl(\\{x\\})\setminus
e^{*}\text{-}cl(\\{y\\})\Rightarrow(z\in e^{*}\text{-}cl(\\{x\\}))(z\notin
e^{*}\text{-}cl(\\{y\\}))\\\ \text{Hypothesis}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(z\in
e^{*}\text{-}cl(\\{x\\})=e^{*}\text{-}cl_{\theta}(\\{x\\}))(z\notin
e^{*}\text{-}cl(\\{y\\})=e^{*}\text{-}cl_{\theta}(\\{y\\}))\end{array}\\\
\begin{array}[]{l}\Rightarrow(\forall W\in
e^{*}R(X,z))(W\cap\\{x\\}\neq\emptyset)(\exists U\in
e^{*}R(X,z))(U\cap\\{y\\}=\emptyset)\end{array}$
$\left.\begin{array}[]{rr}\Rightarrow(U\in e^{*}R(X,z))(\\{x\\}\subseteq
U)(\\{y\\}\subseteq\setminus U)\\\ V:=\setminus
U\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(U,V\in e^{*}O(X,z))(U\cap
V=\emptyset)(e^{*}\text{-}cl(\\{x\\})\subseteq
U)(e^{*}\text{-}cl(\\{y\\})\subseteq V).\end{array}$

Second Case: Similarly proved. ∎

######  Theorem 6.5.

Let $X$ be a topological space. Then, $X$ is $e^{*}\text{-}R_{1}$ if and only
if for each $e^{*}\text{-}open$ set $A$ and each $x\in A$,
$e^{*}\text{-}cl_{\theta}(\\{x\\})\subseteq A.$

###### Proof.

$(\Rightarrow):$ Let $A\in e^{*}O(X,x)$ and $y\notin A.$
$\left.\begin{array}[]{rr}y\notin A\in e^{*}O(X,x)\\\ X\text{ is
}e^{*}\text{-}R_{1}\end{array}\right\\}\Rightarrow x\notin
e^{*}\text{-}cl_{\theta}(\\{y\\})=e^{*}\text{-}cl(\\{y\\})\subseteq X\setminus
A$
$\begin{array}[]{rr}\Rightarrow(\exists V\in
e^{*}O(X,x))(e^{*}\text{-}cl(V)\cap\\{y\\}=\emptyset)\end{array}$
$\left.\begin{array}[]{rr}\Rightarrow(e^{*}\text{-}cl(V)\in
e^{*}R(X,x))(e^{*}\text{-}cl(V)\cap\\{y\\}=\emptyset)\\\ U:=\setminus
e^{*}\text{-}cl(V)\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(U\in e^{*}R(X,y)\subseteq
e^{*}O(X,y))(e^{*}\text{-}cl(U)\cap\\{x\\}=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow y\notin
e^{*}\text{-}cl_{\theta}(\\{x\\}).\end{array}$

$(\Leftarrow):$ Let $x,y\in X$ and $y\in
e^{*}\text{-}cl_{\theta}(\\{x\\})\setminus e^{*}\text{-}cl(\\{x\\}).$
$y\in e^{*}\text{-}cl_{\theta}(\\{x\\})\setminus
e^{*}\text{-}cl(\\{x\\})\Rightarrow(y\in
e^{*}\text{-}cl_{\theta}(\\{x\\}))(y\notin e^{*}\text{-}cl(\\{x\\}))$
$\left.\begin{array}[]{rr}\Rightarrow(y\in
e^{*}\text{-}cl_{\theta}(\\{x\\}))(\exists A\in
e^{*}O(X,y))(A\cap\\{x\\}=\emptyset)\\\
\text{Hypothesis}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(y\in
e^{*}\text{-}cl_{\theta}(\\{x\\}))(e^{*}\text{-}cl_{\theta}(\\{y\\})\cap\\{x\\}=\emptyset)\end{array}\\\
\begin{array}[]{l}\Rightarrow(y\in e^{*}\text{-}cl_{\theta}(\\{x\\})(x\notin
e^{*}\text{-}cl_{\theta}(\\{y\\}))\end{array}\\\
\begin{array}[]{l}\overset{\text{Theorem }\ref{4}}{\Rightarrow}(y\in
e^{*}\text{-}cl_{\theta}(\\{x\\})(y\notin
e^{*}\text{-}cl_{\theta}(\\{x\\}))\end{array}\\\ \begin{array}[]{l}\Rightarrow
y\in e^{*}\text{-}cl_{\theta}(\\{x\\})\setminus
e^{*}\text{-}cl_{\theta}(\\{x\\})=\emptyset.\end{array}$

This is a contradiction. ∎

######  Theorem 6.6.

Let $X$ and $Y$ be two topological spaces. If $f:X\rightarrow Y$ is a
$\theta\text{-}S\text{-}e^{*}\text{-}continuous$ surjection, then $Y$ is an
$e^{*}\text{-}R_{1}$ space.

###### Proof.

Let $V\in e^{*}O(Y,y).$
$\left.\begin{array}[]{r}V\in e^{*}O(Y,y)\\\ f\text{ is
surjective}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!\\!(\exists x\in X)(y=f(x))(V\in
e^{*}O(Y,f(x)))\\\ f\text{ is
}\theta\text{-}S\text{-}e^{*}\text{-}\text{continuous}\end{array}\right\\}\Rightarrow\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in
O(X,x))(e^{*}\text{-}cl_{\theta}(\\{y\\})\subseteq
e^{*}\text{-}cl_{\theta}(f[U])\subseteq V).\end{array}$ ∎

##  7\. Acknowledgements

This study has been supported by the Scientific Research Project Fund of Muğla
Sıtkı Koçman University under the project number 23/153/02/1.

## References

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# Reconstructing the Invisible: Video Frame Restoration through Siamese Masked
Conditional Variational Autoencoder

Yongchen Zhou
LIRA Center, Lancaster University
Lancaster, England
<EMAIL_ADDRESS>Richard Jiang
LIRA Center, Lancaster University
Lancaster, England
<EMAIL_ADDRESS>
###### Abstract

In the domain of computer vision, the restoration of missing information in
video frames is a critical challenge, particularly in applications such as
autonomous driving and surveillance systems. This paper introduces the Siamese
Masked Conditional Variational Autoencoder (SiamMCVAE), leveraging a siamese
architecture with twin encoders based on vision transformers. This innovative
design enhances the model’s ability to comprehend lost content by capturing
intrinsic similarities between paired frames. SiamMCVAE proficiently
reconstructs missing elements in masked frames, effectively addressing issues
arising from camera malfunctions through variational inferences. Experimental
results robustly demonstrate the model’s effectiveness in restoring missing
information, thus enhancing the resilience of computer vision systems. The
incorporation of Siamese Vision Transformer (SiamViT) encoders in SiamMCVAE
exemplifies promising potential for addressing real-world challenges in
computer vision, reinforcing the adaptability of autonomous systems in dynamic
environments.

##  1 Introduction

In the dynamic world of computer vision, where the lens of artificial
intelligence gazes upon the visual landscape, a singular challenge has
continued to captivate the imaginations of researchers and engineers alike.
This challenge lies at the intersection of technology and the human
experience—a quest to restore what has been lost [17], to unveil the unseen,
and to breathe life into the incomplete. In a world fueled by the relentless
pursuit of innovation, the restoration of missing information within video
frames stands as a formidable testament to the artistry of visual intelligence
[26].

In recent years, developments in the field of deep learning have witnessed a
growing movement towards the integration of methodologies to address a wide
array of challenges, encompassing language [24], vision [10, 21], speech [43],
and various other domains. The adaptation of Transformer architectures [32],
initially prevalent in natural language processing, has found successful
integration into the realm of computer vision [14]. The landscape of
predictive learning methods has witnessed an intriguing evolution, driven by
the transformative potential of masked language modeling [13, 5] and its
visual counterpart, masked visual modeling (MVM) [17, 2, 38].

This paper confronts the formidable challenge of restoring large-scale missing
information within video frames, introducing a groundbreaking solution that
harnesses the latest advancements in machine learning and computer vision. Our
model, SiamMCVAE, illustrated in Figure 1, draws inspiration from the
Conditional Variational Autoencoder (CVAE) [29], ushering in a significant
breakthrough in the realm of restoration capability. While siamese networks
[29] have conventionally found applications in classification and comparison
tasks [37, 16, 6, 9], our work extends this architecture to the generative
domain, introducing a novel dimension to its utilization.

The existing Masked Autoencoders (MAE) [17] and their extensions [15, 31]
demonstrate proficiency in restoring large-scale missing information. However,
these models lack comprehensive evaluations specifically focused on image
restoration. To address this void, our meticulous evaluation uniquely
scrutinizes the performance of these models, placing a distinct emphasis on
their efficacy in the context of image restoration. Through this
investigation, we unveil SiamMCVAE’s unparalleled advantages over them.
Notably, its exceptional capability to excel in reconstructing images, even in
scenarios characterized by extensive missing information, establishes it as a
pioneering solution in the field.

frame 1patchify$\vdots$ SiamViT $\vdots$frame 2patchify$\vdots$ SiamViT
$\vdots$ Convert $N+1$$\vdots$$\mathcal{P}$$\mathcal{P}^{\complement}$ Convert
$N+1$$\vdots$ Repeat $\lvert\mathcal{P}\lvert$mask token$\vdots$+$\vdots$
Projection $\mathcal{MN}(0,\mathbf{I},\mathbf{I})$
$\mathbf{M}$$\mathbf{S}$$\mathbf{E}$$\odot$+$\mathbf{Z}$ Projection ViT
$\vdots$ Convert $N$$\mathcal{P}^{\complement}$$\mathcal{P}$$\vdots$ Convert
$N$$\vdots$$\vdots$+$\vdots$target

Figure 1: Our SiamMCVAE architecture. The foundational framework of our
SiamMCVAE is meticulously crafted to address the intricate challenges posed by
missing information in video frames. Embracing a siamese architecture, our
model synergistically integrates twin encoders equipped with vision
transformers. This innovative design augments the model’s ability to discern
and reconstruct missing content by capturing inherent similarities between
paired frames. The siamese encoder configuration, coupled with the
transformative power of vision transformers, empowers SiamMCVAE to
proficiently reconstruct missing elements within masked frames. The
intricacies of our architecture extend further with the incorporation of
variational principles, elevating the model’s capacity to generate diverse and
meaningful representations.

What sets our model apart is its remarkable ability to excel in restoring
information under challenging conditions. SiamMCVAE, with its unique capacity
to learn correspondences and reconstruct lost patches within video frames,
positions itself as a pioneer in the field of computer vision. Our extensive
experiments and results unequivocally demonstrate the superiority of our model
in comparison to existing methods, showcasing its potential to revolutionize
the field.

##  2 Related Work

Autoencoder. Autoencoders, integral to unsupervised learning, aim to distill
intricate data representations and excel in reconstructing the original data
from this condensed form [28]. This architecture encompasses an encoder,
responsible for mapping inputs to a latent representation, and a decoder,
tasked with reconstructing the input. Well-established instances of
autoencoders include Principal Component Analysis (PCA) [22] and k-means [19].
In this domain, Denoising Autoencoders (DAE) [33] represent a specialized
class deliberately introducing corruption to input signals, striving to learn
the reconstruction of the original, uncorrupted signal. Moreover, various
methods can be conceptualized as generalized DAEs employing diverse corruption
techniques, such as masking pixels [34, 27, 8], or removing color channels
[42]. Our work is specifically tailored to restoring frames where information
in a substantial proportion of patches has been lost.

Variational inference. Variational inference [3] is a powerful framework in
probabilistic modeling that enables the approximation of complex posterior
distributions. It is particularly valuable when dealing with intractable
probabilistic models. The primary goal of variational inference is to find an
approximate distribution, usually denoted as $q(\mathbf{z})$, that closely
approximates the true posterior distribution, $p(\mathbf{z}|\mathbf{x})$,
where $\mathbf{z}$ represents latent variables and $\mathbf{x}$ represents
observed data.

The core idea of variational inference is to transform the posterior inference
problem into an optimization problem. By minimizing the Kullback-Leibler (KL)
divergence [25] between the approximate distribution $q(\mathbf{z})$ and the
true posterior $p(\mathbf{z}|\mathbf{x})$, we can find the best approximation:

$q^{*}(\mathbf{z})=\underset{q(\mathbf{z})}{\mathrm{argmin}}\,D_{\mathrm{KL}}(q(\mathbf{z})\|p(\mathbf{z}|\mathbf{x})).$
(1)

Here, the KL divergence measures the information lost when using the
approximate distribution instead of the true posterior. The optimal
approximation, $q^{*}(\mathbf{z})$, provides a trade-off between being
expressive enough to capture the true posterior and being computationally
tractable.

Variational inference has found extensive applications in machine learning,
including in the training of Variational Autoencoders (VAE) [23], Conditional
Variational Autoencoders (CVAE) [29], and other generative models. It enables
the efficient learning of complex probabilistic models and has become an
essential tool in the field of deep learning.

Siamese networks. Siamese networks have emerged as a significant architectural
paradigm in the field of computer vision and machine learning [4]. Their
unique ability to compare entities by means of weight-sharing neural networks
has found broad application across diverse domains, and has been extensively
featured in the contrastive learning approaches [37, 16, 6, 9], showcasing its
versatility and efficacy in capturing complex relationships.

In our work, we transcend the conventional boundaries of siamese networks by
venturing into the generative domain, thereby introducing a novel dimension to
its application. This expansion unlocks new possibilities for leveraging
siamese architectures in tasks related to generative modeling and content
restoration.

Data restoration. Traditional denoising methods [7, 40] demonstrate
proficiency in managing noisy images. However, their efficacy experiences a
considerable decline when faced with scenarios involving substantial missing
regions. In recent years, MAE [17] and its variants [15, 31] have surfaced as
leading methodologies for addressing masked scenarios in video frames. These
models employ sophisticated representations to reconstruct missing
information.

Our work builds upon these foundations, introducing the SiamMCVAE model, which
combines the strengths of Siamese architectures and Vision Transformers for
enhanced data restoration. Unlike some existing approaches that might
prioritize specific aspects of masked scenarios, our model takes a holistic
approach, focusing on comprehensive image restoration, even in situations with
large-scale missing information. This distinctive emphasis positions SiamMCVAE
as a robust and versatile solution in the landscape of data restoration.

##  3 Method

In this section, we undertake an in-depth exploration of the fundamental
components comprising our SiamMCVAE model. Our method amalgamates cutting-edge
technologies in computer vision and machine learning, underpinned by the
principles of SiamViT and variational inference [3]. This synthesis of
innovative concepts culminates in a comprehensive solution designed to tackle
the intricate challenges posed by missing information in video frames, thus
bolstering the efficacy of computer vision systems operating in rapidly
evolving scenarios.

To provide a concrete understanding of the inner workings of SiamMCVAE, we
present the forward propagation function outlined in Algorithm 1. This
algorithm serves as the blueprint for the model’s forward propagation,
elucidating the sequential steps involved in processing input data and
generating meaningful output representations. The subsequent sections delve
into a detailed discussion of the various components of SiamMCVAE, shedding
light on their roles and contributions to the overall framework.

Algorithm 1 Forward Propagation of SiamMCVAE

function Convert($\mathbf{X},\mathcal{P},N$)

$M,D\leftarrow$ rows($\mathbf{X}$), cols($\mathbf{X}$)

for $i\leftarrow 1$ to $N$ do

if $i-1\in\mathcal{P}$ then

$\mathbf{y}_{i}\leftarrow\mathbf{0}$

else

if $N\leq M$ then

$k\leftarrow i+M-N$

else

$k\leftarrow i-\lvert\mathcal{P}\cap\\{1,2,\ldots,i-1\\}\lvert$

end if

$\mathbf{y}_{i}\leftarrow(\mathbf{X}_{k1},\mathbf{X}_{k2},\ldots,\mathbf{X}_{kD})^{\mathsf{T}}$

end if

end for

return $[\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{N}]^{\mathsf{T}}$

end function

function SiamMCVAE($\mathbf{X}_{1},\mathbf{X}_{2},\mathcal{P}$)

$\mathbf{X}_{1}\leftarrow$ Patchify($\mathbf{A}_{1},\\{1,2,\ldots,N\\}$)

$\mathbf{X}_{2}\leftarrow$
Patchify($\mathbf{A}_{2},\mathcal{P}^{\complement}$)

$\mathbf{U}_{1}\leftarrow$ SiamViT($\mathbf{X}_{1}$)

$\mathbf{U}_{2}\leftarrow$ SiamViT($\mathbf{X}_{2}$)

$\mathbf{T}\leftarrow$
Repeat($\mathbf{t}^{\mathsf{T}},\lvert\mathcal{P}^{\complement}\lvert$)

$\mathbf{U}\leftarrow[\mathbf{U}_{1},$
Convert($\mathbf{U}_{2},\mathcal{P},N+1$) $+$ Convert(
$\mathbf{T},\mathcal{P},N+1$) $]$

$\mathbf{Z},\mathbf{M},\mathbf{S}\leftarrow$ Reparametrize($\mathbf{U}$)

$\mathbf{O}\leftarrow$ ViT($[\mathbf{Z},\mathbf{U}_{1}]$)

$\mathbf{G}\leftarrow$
Convert($[\mathbf{0}^{\mathsf{T}};\mathbf{X}_{2}],\mathcal{P},N$) $+$
Convert($\mathbf{O},\newline \mathcal{P}^{\complement},N$)

return $\mathbf{G},\mathbf{M},\mathbf{S}$

end function

Siamese encoder. The encoding process commences with the patchification of
each video frame pair. We perform a transformation on the images
$\mathbf{A}_{1}$ and $\mathbf{A}_{2}\in\mathbb{R}^{H\times W\times C}$ by
converting them into sequences of flattened 2D patches, denoted as
$\mathbf{X}_{1}$ and $\mathbf{X}_{2}\in\mathbb{R}^{N\times(P^{2}\cdot C)}$,
where $H\times W$ represents the resolution of the original images, $C$ is the
number of channels, $P\times P$ denotes the resolution of each image patch,
and $N=\frac{HW}{P^{2}}$ signifies the resulting number of patches. Crafted
explicitly for processing pairs of video frames, the SiamViT adeptly manages
paired data with the utilization of two weight-sharing vanilla Vision
Transformers (ViT) [14]. This independent processing of video frame pairs
involves one intact frame and another subjected to masking.

The SiamViT architecture embodies a sophisticated design, featuring a cascade
of interleaved Multiheaded Self-Attention (MSA) [32] and Multilayer Perceptron
(MLP) [30] blocks. The MSA employs adaptive attention kernel, dynamically
selecting the most optimal implementation based on the characteristics of the
input data. The available implementations include Standard Attention [32],
Flash Attention [12], and Memory-Efficient Attention [20]. The choice among
these implementations is made to maximize efficiency and performance. A
strategic application of Layer Normalization (LN) precedes each block,
augmenting the stability and efficiency of the model. Further bolstering the
network’s expressiveness, residual connections are strategically integrated
after each block, contributing to seamless information flow and facilitating
effective gradient propagation [35, 1]. Mathematically, the SiamViT operations
can be represented as follows:

$\displaystyle\mathbf{Y}_{i,0}=[\mathbf{c},\mathbf{W}_{\mathrm{e}}\mathbf{X}_{i}^{\mathsf{T}}+\mathbf{B}_{\mathrm{e}}]^{\mathsf{T}}+\mathbf{P}_{\mathrm{e}},$
(2)
$\displaystyle\mathbf{Y}^{\prime}_{i,l}=\mathrm{MSA}_{l}(\mathrm{LN}(\mathbf{Y}_{i,l-1}))+\mathbf{Y}_{i,l-1},$
(3)
$\displaystyle\mathbf{Y}_{i,l}=\mathrm{MLP}_{l}(\mathrm{LN}(\mathbf{Y}^{\prime}_{i,l-1}))+\mathbf{Y}^{\prime}_{i,l-1},$
(4)
$\displaystyle\mathbf{U}_{i}=(\mathbf{W}_{\mathrm{u}}\mathrm{LN}(\mathbf{Y}_{i,L})^{\mathsf{T}}+\mathbf{B}_{\mathrm{u}})^{\mathsf{T}},$
(5) $\displaystyle\forall i\in\\{1,2\\},\,l\in\\{1,2,\ldots,L\\},$

where
$\mathbf{c}\in\mathbb{R}^{D},\mathbf{W}_{\mathrm{e}}\in\mathbb{R}^{D\times(P^{2}\cdot
C)}$, $\mathbf{B}_{\mathrm{e}}\in\mathbb{R}^{D\times N}$,
$\mathbf{P}_{\mathrm{e}}\in\mathbb{R}^{(N+1)\times
D},\mathbf{W}_{\mathrm{u}}\in\mathbb{R}^{D^{\prime}\times D}$,
$\mathbf{B}_{\mathrm{u}}\in\mathbb{R}^{D^{\prime}\times(N+1)}$,
$[\,\cdot\,,\,\cdot\,]$ denotes the horizontal concatenation of matrices, and
$L$ represents the number of Transformer blocks in the siamese encoder.

Subsequently, we replicate the trainable mask token $\mathbf{t}$
$\lvert\mathcal{P}\rvert$ times to create a matrix. This matrix is then
incorporated into $\mathbf{U}_{2}$, and the consolidation of $\mathbf{U}_{1}$
and $\mathbf{U}_{2}$ is achieved through the following equations:

$\displaystyle\mathbf{T}=\mathrm{Repeat}(\mathbf{t}^{\mathsf{T}},\lvert\mathcal{P}\lvert),$
(6) $\displaystyle\begin{aligned}
\mathbf{U}=[\mathbf{U}_{1},\quad&\mathrm{Convert}(\mathbf{U}_{2},\mathcal{P},N+1)\\\
+&\mathrm{Convert}(\mathbf{T},\mathcal{P}^{\complement},N+1)],\end{aligned}$
(7)

where $\mathcal{P}$ denotes the set of indices for the masked patches in the
image, and $\lvert\,\cdot\,\rvert$ denotes the cardinality of the set.

Reparameterization. The features extracted by the siamese encoder traverse
through the reparameterization layer, where the latent space is generated
using a Gaussian distribution, enhancing the model’s ability to produce varied
and meaningful representations. From a mathematical standpoint, the
reparameterization layer functions as follows:

$\displaystyle\mathbf{M}=(\mathbf{W}_{\mathrm{m}}\mathbf{U}^{\mathsf{T}}+\mathbf{B}_{\mathrm{m}})^{\mathsf{T}},$
(8)
$\displaystyle\mathbf{S}=(\mathbf{W}_{\mathrm{s}}\mathbf{U}^{\mathsf{T}}+\mathbf{B}_{\mathrm{s}})^{\mathsf{T}},$
(9) $\displaystyle\mathbf{Z}=\mathbf{M}+\mathbf{S}\odot\mathbf{E},$ (10)

where
$\mathbf{W}_{\mathrm{m}},\mathbf{W}_{\mathrm{s}}\in\mathbb{R}^{D^{\prime}\times
2D^{\prime}}$,
$\mathbf{B}_{\mathrm{m}},\mathbf{B}_{\mathrm{s}}\in\mathbb{R}^{D^{\prime}\times(N+1)}$,
$\mathbf{E}\sim\mathcal{MN}_{(N+1)\times
D^{\prime}}(\mathbf{0},\mathbf{I},\mathbf{I})$, $\odot$ denotes the Hadamard
product, and $\mathbf{Z}$ represents the latent matrix.

Decoder. The decoder in our framework is implemented as another vanilla ViT
[14]. The decoder’s core objective is to generate predictions for individual
patches in pixel space, with the ultimate goal of reconstructing the initially
missing content. The reconstruction operation is succinctly expressed through
the following mathematical formulation:

$\displaystyle\mathbf{V}_{0}=(\mathbf{W}_{\mathrm{d}}[\mathbf{Z},\mathbf{U}_{1}]^{\mathsf{T}}+\mathbf{B}_{\mathrm{d}})^{\mathsf{T}}+\mathbf{P}_{\mathrm{d}},$
(11)
$\displaystyle\mathbf{V}^{\prime}_{l}=\mathrm{MSA}^{\prime}_{l}(\mathrm{LN}(\mathbf{V}_{l-1}))+\mathbf{V}_{l-1},$
(12)
$\displaystyle\mathbf{V}_{l}=\mathrm{MLP}^{\prime}_{l}(\mathrm{LN}(\mathbf{V}^{\prime}_{l-1}))+\mathbf{V}^{\prime}_{l-1},$
(13)
$\displaystyle\mathbf{O}=(\mathbf{W}_{\mathrm{o}}\mathrm{LN}(\mathbf{V}_{L^{\prime}})^{\mathsf{T}}+\mathbf{B}_{\mathrm{o}})^{\mathsf{T}},$
(14) $\displaystyle\forall l\in\\{1,2,\ldots,L^{\prime}\\},$

where $\mathbf{W}_{\mathrm{d}}\in\mathbb{R}^{D^{\prime}\times 2D^{\prime}}$,
$\mathbf{B}_{\mathrm{d}}\in\mathbb{R}^{D^{\prime}\times(N+1)}$,
$\mathbf{P}_{\mathrm{d}}\in\mathbb{R}^{(N+1)\times D^{\prime}}$,
$\mathbf{W}_{\mathrm{o}}\in\mathbb{R}^{D^{\prime}\times(P^{2}\cdot C)}$,
$\mathbf{B}_{\mathrm{o}}\in\mathbb{R}^{(P^{2}\cdot C)\times(N+1)}$, and
$L^{\prime}$ represents the number of Transformer blocks in the decoder.

Finally, we integrate the predicted masked patches with the unmasked patches
from the original image using the following operation:

$\mathbf{G}=\mathrm{Convert}([\mathbf{0}^{\mathsf{T}};\mathbf{X}_{2}],\mathcal{P},N)+\mathrm{Convert}(\mathbf{O},\mathcal{P}^{\complement},N),$
(15)

where $[\,\cdot\,;\,\cdot\,]$ denotes the vertical concatenation of matrices.

Loss Function. Inspired by $\beta$-VAE [18], we model the prior as an
isotropic unit Gaussian $\mathcal{MN}(\mathbf{0},\mathbf{I},\mathbf{I})$,
leading to the formulation of the constrained optimization problem:

$\begin{gathered}\mathrm{max}_{\phi,\theta}\mathbb{E}_{\mathbf{X}_{1},\mathbf{X}_{2}\sim\mathcal{D}}[\mathbb{E}_{q_{\phi}(\mathbf{Z}\mid\mathbf{X}_{1},\mathbf{X}_{2})}\log
p_{\theta}(\mathbf{R}\mid\mathbf{Z})],\\\
\mathrm{s.t.}\,D_{\mathrm{KL}}(q_{\phi}(\mathbf{Z}\mid\mathbf{X}_{1},\mathbf{X}_{2})\|p(\mathbf{Z}))\leq\epsilon,\end{gathered}$
(16)

We reformulate it as a Lagrangian under the KKT conditions [6]:

$\begin{gathered}\mathcal{F}(\theta,\phi,\beta;\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{R})=\mathbb{E}_{q_{\phi}(\mathbf{Z}\mid\mathbf{X}_{1},\mathbf{X}_{2})}\log
p_{\theta}(\mathbf{R}\mid\mathbf{Z})\\\
-\beta(D_{\mathrm{KL}}(q_{\phi}(\mathbf{Z}\mid\mathbf{X}_{1},\mathbf{X}_{2})\|p(\mathbf{Z}))-\epsilon),\end{gathered}$
(17)

As $\epsilon$ is a constant, it is disregarded in the optimization. Our
training strategy for SiamMCVAE involves the formulation of a comprehensive
loss function that combines both a reconstruction loss
($\mathcal{L}_{\mathrm{r}}$) and a KL divergence loss
($\mathcal{L}_{\mathrm{KL}}$). The structure of the loss function is
articulated as follows:

$\mathcal{L}=\mathcal{L}_{\mathrm{r}}+\beta\cdot\mathcal{L}_{\mathrm{KL}}$
(18)

where $\beta$ is a hyperparameter that controls the trade-off between the two
components.

The reconstruction loss, integral to our model’s training, quantifies the
disparity between the original and reconstructed data and is formulated as
follows:

$\mathcal{L}_{\mathrm{r}}=\frac{1}{P^{2}C\lvert\mathcal{P}\rvert}\|\mathbf{G}-\mathbf{R}\|^{2}_{\mathrm{F}}$
(19)

where $\mathbf{R}$ represents the patchified target image.

The KL divergence loss, which measures the dissimilarity between the learned
latent distribution and a chosen prior distribution, is given by:

$\mathcal{L}_{\mathrm{KL}}=\frac{\|\mathbf{M}\|_{\mathrm{F}}^{2}+\|\mathbf{S}\|_{\mathrm{F}}^{2}-\sum_{i=1}^{N+1}\sum_{j=1}^{D^{\prime}}\log\mathbf{S}_{ij}}{2(N+1)D^{\prime}}-\frac{1}{2}$
(20)

where $\|\cdot\|_{\mathrm{F}}$ denotes the Frobenius norm.

The overall loss function optimizes the model to minimize the reconstruction
error while encouraging the latent distribution to be close to the chosen
prior. This combination ensures that the SiamMCVAE effectively reconstructs
the lost content in video frames.

##  4 Experiments

In this section, we embark on a comprehensive evaluation of the performance of
our SiamMCVAE model, juxtaposing it against established state-of-the-art
methodologies. This systematic assessment seeks to shed light on the model’s
capabilities and its potential to address real-world challenges.

###  4.1 Experiment Setup

Dataset. Our experiments are conducted on the extensive BDD100K dataset [39],
renowned for its diverse range of driving scenarios. Encompassing a rich
collection of images and videos, BDD100K provides a comprehensive array of
scenarios and environments commonly encountered on roadways [11]. For our
evaluation of the SiamMCVAE model, we meticulously select a curated subset of
video sequences, ensuring a representative sampling across diverse real-world
scenarios and challenges.

Masking. Our masking strategy involves the deliberate occlusion of a segment
within one frame of a paired set of images, while the other frame remains
unaltered. This deliberate masking of a portion of the image serves as a
surrogate for scenarios in which partial data loss or image corruption occurs
in dynamic video sequences.

Evaluation metrics. Our evaluation strategy employs a meticulous selection of
metrics designed to thoroughly assess the quality of the restored frames in
comparison to the ground truth. In addition to the conventional Mean Squared
Error (MSE) and Mean Absolute Error (MAE), we leverage the Peak Signal-to-
Noise Ratio (PSNR), a well-established measure offering valuable insights into
the model’s precision in capturing fine details and minimizing differences in
pixel values.

For a thorough evaluation, we incorporate advanced metrics, notably the
Structural Similarity Index (SSIM) [36] and the Feature-based Similarity Index
(FSIM) [41]. These sophisticated indices augment our assessment by providing a
nuanced perspective on the model’s performance. By scrutinizing the structural
similarity between the restored and ground truth frames, encompassing
considerations such as luminance, contrast, and structure, these metrics go
beyond pixel-level accuracy. They offer valuable insights into the model’s
adeptness in preserving the overall structural coherence and visual fidelity
of the restored frames.

The orchestration of this ensemble of metrics in our evaluation provides a
nuanced and comprehensive view of our model’s prowess in video frame
restoration.

###  4.2 Comparison with Prior Work

We systematically conduct a comprehensive performance analysis, pitting our
SiamMCVAE model against baseline methods, including MAE [17], MAE-ST [15], and
VideoMAE [31], within the domain of video frame restoration. Our meticulous
evaluation focuses on a masking ratio of 75%, representing a scenario
characterized by moderate data degradation. The outcomes specific to this
masking ratio are concisely presented in Table 1, offering valuable insights
into the comparative efficacy of our model and established baselines.

Method | Backbone | MSE | MAE | PSNR | SSIM | FSIM
---|---|---|---|---|---|---
MAE [17] | ViT-B | 197.37 | 6.99 | 25.80 | 0.800 | 0.670
MAE-ST [15] | ViT-B | 258.51 | 8.11 | 24.70 | 0.741 | 0.638
VideoMAE [31] | ViT-B | 198.00 | 6.97 | 25.80 | 0.798 | 0.669
MAE [17] | ViT-L | 146.63 | 5.95 | 27.10 | 0.837 | 0.700
MAE-ST [15] | ViT-L | 221.69 | 7.58 | 25.34 | 0.758 | 0.651
VideoMAE [31] | ViT-L | 133.83 | 5.61 | 27.51 | 0.838 | 0.708
SiamMCVAE (ours) | SiamViT | 123.01 | 5.49 | 27.90 | 0.841 | 0.712
Table 1: Performance comparison with prior work on restoration metrics at a
75% masking ratio. Our proposed method, SiamMCVAE outperforms the existing
approaches across various metrics, showcasing its superior ability in
restoring missing information in video frames.

It is noteworthy that our SiamMCVAE model consistently outperforms the
baseline methods across a spectrum of comprehensive evaluation metrics,
namely, MAE, MSE, PSNR, SSIM, and FSIM. The prominent superiority observed in
these metrics emphasizes the model’s exceptional proficiency in minimizing
both subtle and substantial reconstruction errors. Consequently, SiamMCVAE
stands out as a benchmark in the field of video frame restoration.

These results underscore the efficacy of our SiamMCVAE model, not only in
mitigating the effects of data degradation but also in surpassing established
state-of-the-art methods in the field of video frame restoration. The capacity
to excel in such a challenging scenario further solidifies the model’s
potential for real-world applications where data integrity may be compromised.

###  4.3 Model Robustness

Through extensive experimentation conducted on diverse driving scenarios
extracted from the dataset, we employ a spectrum of masking ratios spanning
from 45% to 90%, encompassing a diverse range of damage severity scenarios.
This intentional variation in mask coverage enables us to perform a nuanced
and thorough assessment of our model’s proficiency in restoring video frames
across a spectrum of degradation conditions. The outcomes depicted in Figure 2
underscore the remarkable superiority of SiamMCVAE over other models in the
face of diverse levels of data degradation. This pronounced ascendancy becomes
particularly conspicuous when the masking ratio attains higher thresholds.

$45$$60$$75$$90$$50$$100$$150$$200$$250$$300$$350$$400$Masking ratio
(%)MSEMAEMAE-STVideoMAESiamMCVAE Figure 2: Performance comparison of different
models across varying masking ratios. In the face of increasing masking
ratios, SiamMCVAE consistently outperforms other models, showcasing its
remarkable resilience and effectiveness in restoring missing information
within video frames.

Furthermore, we evaluate the performance of various models across different
frame gap scenarios, illustrated in Figure 3. What stands out conspicuously is
the persistent dominance of SiamMCVAE, regardless of the frame gap setting.
This sustained advantage serves as a testament to the model’s exceptional
adaptability and robustness.

$24$$36$$48$$60$$150$$200$$250$$300$$350$$400$$450$$500$Frame gapMSEMAE-
STVideoMAESiamMCVAE Figure 3: Performance comparison across different frame
gaps. Notably, the SiamMCVAE consistently outperforms both MAE-ST and
VideoMAE.

###  4.4 Qualitative Analysis

In our pursuit of a comprehensive evaluation, we delve into the qualitative
facets of model performance. To this end, we embark on a visual exploration of
model outputs when faced with masked video frames. The resulting
visualizations, exemplified in Figure 4, offer a nuanced perspective on the
reconstruction capabilities across various models. The visual comparisons
distinctly reveal the superior performance of SiamMCVAE in terms of the
quality of restored images when compared to alternative models.

Figure 4: Comparative visualization of model outputs at a 90% masking ratio.
In the first column, masked video frames are depicted, while the subsequent
columns showcase outputs from various models, including MAE [17], MAE-ST [15],
VideoMAE [31], and our SiamMCVAE, arranged from left to right. The rightmost
column features the unaltered ground truth frames.

###  4.5 Ablation Studies

Attention kernel. In-depth exploration of attention kernels is crucial for
understanding their nuanced impact on the efficacy of our SiamMCVAE model. We
systematically assess the performance by comparing the adaptive attention
kernel with established counterparts such as Standard Attention [32], Flash
Attention [12], and Memory-Efficient Attention [20]. The discerning outcomes
of this comparative analysis are succinctly summarized in Table 2, offering a
comprehensive perspective on how different attention kernels influence the
overall performance of the model.

Kernel | MSE | MAE | PSNR | SSIM | FSIM
---|---|---|---|---|---
SA [32] | 160.29 | 6.32 | 26.75 | 0.806 | 0.687
FA [12] | 143.56 | 5.96 | 27.25 | 0.456 | 0.697
MEA [20] | 156.05 | 6.23 | 26.87 | 0.810 | 0.689
Adaptive | 123.01 | 5.49 | 27.90 | 0.841 | 0.712
Table 2: Comparison of Standard Attention (SA) [32], Flash Attention (FA)
[12], Memory-Efficient Attention (MEA) [20], and the adaptive attention kernel
on SiamMCVAE Performance.

Reparameterization layer. To gain deeper insights into the inner workings of
our SiamMCVAE architecture, we conducted a meticulous comparative analysis
between models with and without the reparameterization layer. The compelling
results, detailed in Table 3, underscore the substantial performance
improvement achieved through the incorporation of the reparameterization
layer. Evident from the reduced MSE and MAE, as well as the elevated PSNR,
SSIM, and FSIM scores, this analysis emphasizes the pivotal role of
reparameterization in enhancing the model’s overall restoration capabilities.

Reparam. | MSE | MAE | PSNR | SSIM | FSIM
---|---|---|---|---|---
× | 174.86 | 6.63 | 26.35 | 0.792 | 0.676
✓ | 123.01 | 5.49 | 27.90 | 0.841 | 0.712
Table 3: Comparison of SiamMCVAE performance: without reparameterization (×)
vs. with reparameterization (✓).

Lagrange multiplier. Within the intricacies of our SiamMCVAE model, we
scrutinize the impact of the Lagrange multiplier, denoted as $\beta$. As
elucidated in Table 4, we conduct a thorough analysis of the model’s
performance across varying $\beta$ values. This examination provides nuanced
insights into the delicate interplay between regularization strength and
restoration efficacy. The results underscore the importance of meticulous
tuning of $\beta$ to strike a balance, ensuring optimal expressiveness while
preserving crucial visual details. Notably, the analysis identifies
$\beta=0.2$ as the optimal value, showcasing superior performance across
multiple evaluation metrics.

$\beta$ | MSE | MAE | PSNR | SSIM | FSIM
---|---|---|---|---|---
0.1 | 151.42 | 6.14 | 26.98 | 0.812 | 0.691
0.2 | 123.01 | 5.49 | 27.90 | 0.841 | 0.712
0.25 | 139.22 | 5.89 | 27.36 | 0.825 | 0.700
0.5 | 172.02 | 6.57 | 26.43 | 0.809 | 0.678
1 | 192.74 | 7.01 | 25.90 | 0.777 | 0.666
Table 4: Impact of Lagrange Multiplier ($\beta$) on SiamMCVAE Performance. The
results demonstrate the model’s sensitivity to the choice of $\beta$. Notably,
the highlighted values indicate the superior performance achieved with a
$\beta$ value of 0.2.

##  5 Discussion

The SiamMCVAE model takes a prominent position in the field of video frame
restoration, showcasing remarkable efficacy in scenarios characterized by
substantial information loss. Through the synergistic integration of the
innovative SiamViT and variational inference, our model excels in the task of
restoration, solidifying its status as a state-of-the-art solution.

Through extensive experimentation conducted on diverse driving scenarios
extracted from the BDD100K dataset [39], SiamMCVAE consistently outshines its
other models across various mask ratios and diverse frame gap settings. This
resounding success underscores its remarkable adaptability, demonstrating
superior performance even in challenging conditions. The robustness of
SiamMCVAE can be attributed to careful design considerations, including the
strategic integration of SiamViT and the judicious application of variational
techniques. These elements collectively contribute to the model’s
adaptability, positioning it as a resilient and superior solution capable of
addressing a spectrum of challenges in video frame restoration.

Our exhaustive ablation study, meticulously scrutinizing the influence of
crucial components, illuminates the efficacy of the SiamMCVAE model’s design.
We explicitly investigate the roles played by attention mechanisms, the
reparameterization layer, and the Lagrange multiplier $\beta$. This in-depth
analysis quantifies the distinct contributions of these elements, offering a
profound insight into the nuanced design choices that form the bedrock of our
model’s success.

##  6 Conclusion

The successful fusion of siamese architectures with advanced vision
transformers, exemplified by SiamMCVAE, presents a significant leap forward in
the domain of video frame restoration under masked scenarios. The
incorporation of variational principles adds another layer of innovation,
enhancing the model’s capacity to generate diverse and meaningful
representations. Beyond the immediate context of video frame restoration, our
work highlights the broader potential of synergizing siamese encoders with
state-of-the-art vision transformers [14] for generative purpose. SiamMCVAE
not only pushes the boundaries of restoration capability but also sets a
precedent for the integration of these advanced architectures, including
variational techniques, in addressing real-world challenges within the
expansive field of computer vision.

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