# Strong Forms of Weakly $e$-continuous Functions ***This study was supported
by Turkish-German University Scientific Research Projects Commission under the
grant no: 2021BR01.

Burcu Sünbül Ayhan

###### abstract

The main purpose of this study is to introduce and study two new classes of
continuity called $eR$-continuous functions and weakly $eR$-continuous
functions via $e$-regular sets. Both of the forms of continuous functions we
have described are stronger than the weakly $e$-continuity. Furthermore, we
obtain various characterizations of weakly $eR$-continuous functions. In
addition, we examine not only the relations of these functions with some other
forms of existing continuous functions, but also some of their fundamental
properties.

Key Words: $e$-regular set, $e$-connectedness, $eR$-continuity, weakly
$eR$-continuity.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $eR$-continuity and Weakly $eR$-continuity
  4. 4 Some Basic Properties of Weakly $eR$-continuity
  5. 5 Conclusion

2010 Mathematics Subject Classification: 54C08, 54D05, 54D10, 54D30.

##  1 Introduction

There is no doubt that one of the fundamental concepts of general topology is
different forms of the notion of open set. The discussion about $e$-open set
types, one of the generalized open set concepts, is still a rich field to
study in terms of general topology. Some forms of this concept such as
$ge\Lambda$-closed sets [2], generalized $e$-closed sets [3], $\pi ge$-closed
sets [4], $\Lambda_{e}$-sets and $V_{e}$-sets [20] have been investigated in
the recent years. Apart from these, some forms of $e$-continuity and
$e$-openness of functions have been studied in [21, 22] as well.

In recent years, many authors have studied on generalizations of strong
continuity such as strongly $\theta$-continuous functions [16], strongly
$\theta$-precontinuous functions [17], strongly $\theta$-semi continuous
functions [12], strongly $\theta$-$b$-continuous functions [24], strongly
$\theta$-$e$-continuous functions [18]. On the other hand, many researchers
have introduced and investigated some properties of the weakly clopen
functions [25] and weakly $e$-continuous functions [19].

In this paper, we investigate different classes of continuity called
$eR$-continuous functions, weakly $eR$-continuous functions and
strongly-$\theta$-$e$-continuous functions and study some of their fundamental
properties. So, it turns out that weakly $eR$-continuous functions are weaker
than strongly $\theta$-$e$-continuous functions, weakly clopen functions and
$eR$-continuous functions and also stronger than weakly $e$-continuous
functions.

##  2 Preliminaries

Throughout this paper, $X$ and $Y$ represent topological spaces. For a subset
$A$ of a space $X$, $cl(A)$ and $int(A)$ denote the closure of $A$ and the
interior of $A$, respectively. The family of every closed (resp. open, clopen)
sets of $X$ is denoted by $C(X)$ (resp. $O(X),\ CO(X)$). A subset $A$ is
called regular open [26] (resp. regular closed [26]) if $A=$ $int(cl(A))$
$($resp. $A=cl(int(A)))$. A point $x\in X$ is called $\delta$-cluster point
[28] of $A$ if $int(cl(U))\cap A\neq\emptyset$ for every open neighborhood $U$
of $x$. The set of all $\delta$-cluster points of $A$ is called the
$\delta$-closure [28] of $A$ and is denoted by $cl_{\delta}(A)$. If
$A=cl_{\delta}(A)$, then $A$ is called $\delta$-closed [28] and the complement
of a $\delta$-closed set is called $\delta$-open [28]. The set $\\{x|(\exists
U\in O(X,x))(int(cl(U))\subseteq A)\\}$ is called the $\delta$-interior of $A$
and is denoted by $int_{\delta}(A)$.

A subset $A$ of a space $X$ is called semiopen [14] (resp. preopen [15],
$b$-open [1], $e$-open [11], $a$-open [10]) if $A$ $\subseteq cl(int(A))$
(resp. $A$ $\subseteq int(cl(A))$, $A$ $\subseteq cl(int(A))\cup int(cl(A))$,
$A$ $\subseteq cl(int_{\delta}(A))\cup int(cl_{\delta}(A))$, $A\subseteq
int(cl(int_{\delta}(A)))$. The complement of a semiopen (resp. preopen,
$b$-open, $e$-open, $a$-open) set is called semiclosed [14] (resp. preclosed
[15], $b$-closed [1], $e$-closed [11], $a$-closed [10]). The intersection of
all semiclosed (resp. preclosed, $b$-closed, $e$-closed, $a$-closed) sets of
$X$ containing $A$ is called the semi-closure [14] (resp. pre-closure [15],
$b$-closure [1], $e$-closure [11], $a$-closure [10]) of $A$ and is denoted by
$scl(A)$ (resp. $pcl(A)$, $bcl(A)$, $e$-$cl(A)$, $a$-$cl(A)$). The union of
every semiopen (resp. preopen, $b$-open, $e$-open, $a$-open) sets of $X$
contained in $A$ is called the semi-interior [14] (resp. pre-interior [15],
$b$-interior [1], $e$-interior [11], $a$-interior [10]) of $A$ and is denoted
by $sint(A)$ (resp. $pint(A)$, $bint(A)$, $e$-$int(A)$, $a$-$int(A)$).

A point $x$ of $X$ is said to be $\theta$-cluster ($e$-$\theta$-cluster) point
of $A$ if $cl(U)\cap A\neq\emptyset$ ($e$-$cl(U)\cap A\neq\emptyset$) for all
open ($e$-open) set $U$ containing $x$. The set of all $\theta$-cluster
($e$-$\theta$-cluster) points of $A$ is called the $\theta$-closure [28]
($e$-$\theta$-closure [18]) of $A$ and is denoted by $cl_{\theta}(A)$
($e$-$cl_{\theta}(A)$). A subset $A$ is called to be $\theta$-closed
($e$-$\theta$-closed) if $A=cl_{\theta}(A)$ ($A=e$-$cl_{\theta}(A)$). The
complement of a $\theta$-closed ($e$-$\theta$-closed) set is called a
$\theta$-open [28] ($e$-$\theta$-open [18]). A point $x$ of $X$ called to be a
$\theta$-interior [28] ($e$-$\theta$-interior [18]) point of a subset $A$,
denoted by $int_{\theta}(A)$ ($e$-$int_{\theta}(A)$), if there exists an open
($e$-open) set $U$ of $X$ containing $x$ such that $cl(U)\subseteq A$
($e$-$cl(U)\subseteq A$).

A subset $A$ is called $e$-regular [18] if it is $e$-open and $e$-closed.
Also, it is noted in [13] that

$e\mbox{-regular}\Rightarrow e\mbox{-}\theta\mbox{-open}\Rightarrow
e\mbox{-open.}$

The family of every $e$-$\theta$-open $($resp. $e$-$\theta$-closed,
$e$-regular, regular open, regular closed, $\delta$-open, $\delta$-closed,
$\theta$-open, $\theta$-closed, semiopen, semiclosed, preopen, preclosed,
$b$-open, $b$-closed, $e$-open, $e$-closed, $a$-open, $a$-closed$)$ subsets of
$X$ is denoted by $e\theta O(X)$ $($resp. $e\theta C(X)$, $eR(X)$, $RO(X)$,
$RC(X)$, $\delta O(X)$, $\delta C(X)$, $\theta O(X)$, $\theta C(X)$, $SO(X)$,
$SC(X)$, $PO(X)$, $PC(X)$, $BO(X)$, $BC(X)$, $eO(X)$, $eC(X)$, $aO(X)$,
$aC(X))$. The family of every open $($resp. closed, $e$-$\theta$-open,
$e$-$\theta$-closed, $e$-regular, regular open, regular closed, $\delta$-open,
$\delta$-closed, $\theta$-open, $\theta$-closed, semiopen, semiclosed,
preopen, preclosed, $b$-open, $b$-closed, $e$-open, $e$-closed, $a$-open,
$a$-closed$)$ sets of $X$ containing a point $x$ of $X$ is denoted by $O(X,x)$
$($resp. $C(X,x)$, $e\theta O(X,x)$, $e\theta C(X,x)$, $eR(X,x)$, $RO(X,x)$,
$RC(X,x)$, $\delta O(X,x)$, $\delta C(X,x)$, $\theta O(X,x)$, $\theta C(X,x)$,
$SO(X,x)$, $SC(X,x)$, $PO(X,x)$, $PC(X,x)$, $BO(X,x)$, $BC(X,x)$, $eO(X,x)$,
$eC(X,x)$, $aO(X,x)$, $aC(X,x))$.

We shall use the well-known accepted language almost in the whole of the
proofs of the theorems in this article. The following fundamental properties
of $e$-$\theta$-closure is useful in the sequel:

###### Lemma 2.1

[18]; [13] Let $A$ and $B$ be subsets of a space $X$. Then the followings are
hold:
(1) $A\subseteq e$-$cl(A)$ $\subseteq$ $e$-$cl_{\theta}(A)$,
(2) $e$-$cl_{\theta}(A)$ is $e$-$\theta$-closed,
(3) If $A$ is $e$-$\theta$-closed, then $A=e$-$cl_{\theta}(A)$,
(4) If $A\subseteq B$, then $e$-$cl_{\theta}(A)\subseteq e$-$cl_{\theta}(B),$
(5) $e$-$cl_{\theta}(e$-$cl_{\theta}(A))=e$-$cl_{\theta}(A)$,
(6) $e$-$cl_{\theta}(X\setminus A)$ $=X\setminus e$-$int_{\theta}(A)$,
(7) $x\in e$-$cl_{\theta}(A)$ iff $A\cap U\neq\emptyset$ for all $eR(X,x)$,
(8) $e$-$cl_{\theta}(A)=\bigcap\\{V|(A\subseteq V)(V\in
eR(X))\\}=\bigcap\\{V|(A\subseteq V)(V\in e\theta C(X))\\}$,
(9) $A\in e\theta O(X)$ iff for all $x\in A$ there exists $U\in eR(X,x)$ such
that $U\subseteq A$,
(10) Any intersection (union) of $e$-$\theta$-closed ($e$-$\theta$-open) sets
is $e$-$\theta$-closed ($e$-$\theta$-open).

###### Lemma 2.2

[6] Let $A$ be a subset of a space $X$. If $A$ is an open set in $X$, then
$cl(A)=cl_{\theta}(A)$.

###### Definition 2.1

A function $f:X\rightarrow Y$ is called to be:
(a) $e$-continuous (briefly $e$.c.) [11] if the inverse image of all open set
is $e$-open.
(b) strongly $\theta$-continuous (briefly, st.$\theta$.c.) [16] if for all
$x\in X$ and all open set $V$ of $Y$ containing $f(x)$, there exists $U\in
O(X,x)$ such that $f[cl(U)]\subseteq V$.
(c) strongly $\theta$-semicontinuous (briefly, st.$\theta$.s.c.) [12] if for
all $x\in X$ and all open set $V$ of $Y$ containing $f(x)$, there exists $U\in
SO(X,x)$ such that $f[scl(U)]\subseteq V$.
(d) strongly $\theta$-precontinuous (briefly, st.$\theta$.p.c.) [17] if for
all $x\in X$ and all open set $V$ of $Y$ containing $f(x)$, there exists $U\in
PO(X,x)$ such that $f[pcl(U)]\subseteq V$.
(e) strongly $\theta$-$b$-continuous (briefly, st.$\theta$.b.c.) [24] if for
all $x\in X$ and all open set $V$ of $Y$ containing $f(x)$, there exists $U\in
BO(X,x)$ such that $f[bcl(U)]\subseteq V$.
(f) strongly $\theta$-$e$-continuous (briefly, st.$\theta$.e.c.) [18] if for
all $x\in X$ and all open set $V$ of $Y$ containing $f(x)$, there exists $U\in
eO(X,x)$ such that $f[e\mbox{-}cl(U)]\subseteq V$.
(g) weakly clopen (briefly, w.co.) [25] if for all $x\in X$ and all open set
$V$ of $Y$ containing $f(x)$, there exists $U\in CO(X,x)$ such that
$f[U]\subseteq cl(V)$.
(h) weakly $b$-continuous (briefly w.$b$.c.) [27] if for all $x\in X$ and all
open set $V$ of $Y$ containing $f(x)$, there exists $U\in BO(X,x)$ such that
$f[U]\subseteq cl(V)$.
(i) weakly $a$-continuous (briefly w.$a$.c.) [5] if for all $x\in X$ and all
open set $V$ of $Y$ containing $f(x)$, there exists $U\in aO(X,x)$ such that
$f[U]\subseteq cl(V)$.
(j) weakly $e$-continuous (briefly w.$e$.c.) [19] if for all $x\in X$ and all
open set $V$ of $Y$ containing $f(x)$, there exists $U\in eO(X,x)$ such that
$f[U]\subseteq cl(V)$.
(k) weakly $BR$-continuous (briefly, w.$BR$.c.) [8] if for all $x\in X$ and
all open set $V$ of $Y$ containing $f(x)$, there exists $U\in BR(X,x)$ such
that $f[U]\subseteq cl(V)$.
(l) $BR$-continuous (briefly, $BR$.c.) [8] if $f^{-1}[V]$ is $b$-regular in
$X$ for all open set $V$ of $Y$.

##  3 $eR$-continuity and Weakly $eR$-continuity

###### Definition 3.1

A function $f:X\rightarrow Y$ is called weakly $eR$-continuous (briefly
w.$eR$.c.) at $x\in X$ if for each open set $V$ containing $f(x)$, there
exists an $e$-regular set $U$ in $X$ containing $x$ such that
$f\left[U\right]\subseteq cl(V)$. The function $f$ is w.$eR$.c. if and only if
$f$ is w.$eR$.c. for all $x\in X$.

###### Definition 3.2

A function $f:X\rightarrow Y$ is called $eR$-continuous (briefly $eR$.c.) if
$f^{-1}(V)$ is $e$-regular in $X$ for every open set $V$ of $Y$.

###### Theorem 3.1

Let $f:X\rightarrow Y$ be a function. If $f$ is $eR$-continuous, then $f$ is
weakly $eR$-continuous.

Proof: Let $x\in X$ and $V\in O(Y,f(x))$.
$\left.\begin{array}[]{r}(x\in X)(V\in O(Y,f(x)))\\\ f\text{ is
$eR$.c.}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\mbox{}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!f^{-1}[V]\in eR(X,x)\\\
U:=f^{-1}[V]\end{array}\right\\}\Rightarrow(U\in eR(X,x))(f[U]\subseteq
V\subseteq cl(V)).\end{array}$ $\Box$

###### Theorem 3.2

Let $f:X\rightarrow Y$ be a function. If $f$ is $eR$-continuous, then $f$ is
$e$-continuous.

Proof: It is obvious from Definitions 2.1 and 3.2. $\Box$

###### Remark 3.1

From Definitions 2.1, 3.1 and 3.2, we have the following diagram. The
converses of the below implications are not true in general, as shown in the
related articles and following examples.

$\begin{array}[]{ccccccccc}&&\text{st.}\theta\text{.}s\text{.c.}&&&&BR\text{.c.}&\longrightarrow&\text{w.$b$.c.}\\\
&\nearrow&&\searrow&&&\downarrow&\nearrow&\uparrow\\\
\text{st.}\theta\text{.c.}&&&&\text{st.}\theta\text{.}b\text{.c.}&\longrightarrow&\text{w.}BR\text{.c.}&\longleftarrow&\text{w.co.}\\\
&\searrow&&\nearrow&&&&\swarrow&\downarrow\\\
&&\text{st.}\theta\text{.}p\text{.c.}&\longrightarrow&\text{st.}\theta\text{.}e\text{.c.}&\longrightarrow&\text{w.}eR\text{.c.}&\longrightarrow&\text{w.$e$.c.}\\\
&&&&&&\uparrow&\nearrow&\uparrow\\\
&&&&&&eR\text{.c.}&\longrightarrow&\text{$e$.c.}\end{array}$

###### Example 3.1

Let $X=\\{a,b,c,d,e\\}$ and
$\tau=\\{\emptyset,\\{a\\},\\{c\\},\\{a,c\\},\\{c,d\\},\\{a,c,d\\},X\\}$ and
$\sigma=\\{\emptyset,X,\\{b,c,d\\}\\}$. Then the function
$f:(X,\tau)\rightarrow(X,\sigma)$ by $f=\\{(a,e),(b,b),(c,c),(d,d),(e,a)\\}$
is weakly $eR$-continuous but it is not strongly $\theta$-$e$-continuous.

###### Example 3.2

Let $X=\\{a,b,c,d,e\\}$ and
$\tau=\\{\emptyset,\\{a\\},\\{c\\},\\{a,c\\},\\{c,d\\},\\{a,c,d\\},X\\}$ and
$\sigma=\\{\emptyset,X,\\{b,c,d\\}\\}$. Then the function
$f:(X,\tau)\rightarrow(X,\sigma)$ by $f=\\{(a,b),(b,a),(c,c),(d,d),(e,e)\\}$
is weakly $eR$-continuous but it is not $eR$-continuous.

###### Example 3.3

Let $X=\\{a,b,c,d\\}$ and
$\tau=\\{\emptyset,\\{a\\},\\{b\\},\\{a,b\\},\\{a,b,c\\},X\\}$ and
$\sigma=\\{\emptyset,X,\\{a\\},\\{b,c\\},\\{a,b,c\\}\\}$. Then the function
$f:(X,\tau)\rightarrow(X,\sigma)$ by $f=\\{(a,a),(b,c),(c,b),(d,d)\\}$ is both
$e$-continuous and weakly $e$-continuous but it is not $eR$-continuous.

 Question: Is there any weakly $e$-continuous function which is not
weakly $eR$-continuous?

###### Theorem 3.3

For a function $f:X\rightarrow Y$, the followings are equivalent:
(1) $f$ is weakly $eR$-continuous;
(2) for all $x\in X$ and all open set $V$ of $Y$ containing $f(x)$, there
exists an $e$-$\theta$-open set $U$ in $X$ containing $x$ such that
$f[U]\subseteq cl(V);$
(3) $e$-$cl_{\theta}(f^{-1}[U])\subseteq f^{-1}[cl(U)]$ for all preopen set
$U$ of $Y;$
(4) $f^{-1}[U]\subseteq$ $e$-$int_{\theta}(f^{-1}[cl(U)])$ for all preopen set
$U$ of $Y;$
(5) $e$-$cl_{\theta}(f^{-1}[int(cl(B))])\subseteq f^{-1}[cl(B)]$ for all
subset $B$ of $Y;$
(6) $e$-$cl_{\theta}(f^{-1}[int(F)])\subseteq f^{-1}[F]$ for all regular
closed set $F$ of $Y;$
(7) $e$-$cl_{\theta}(f^{-1}[U])\subseteq f^{-1}[cl(U)]$ for all open subset
$U$ of $Y;$
(8) $f^{-1}[U]\subseteq e$-$int_{\theta}(f^{-1}[cl(U)])$ for all open subset
$U$ of $Y;$
(9) $f[e\mbox{-}cl_{\theta}(A)]\subseteq cl_{\theta}(f[A])$ for all subset $A$
of $X;$
(10) $e$-$cl_{\theta}(f^{-1}[B])\subseteq f^{-1}[cl_{\theta}(B)]$ for all
subset $B$ of $Y;$
(11) $e$-$cl_{\theta}(f^{-1}[int(cl_{\theta}(B))])\subseteq
f^{-1}[cl_{\theta}(B)]$ for all subset $B$ of $Y.$

Proof: $(1)\Rightarrow(2):$ Let $x\in X$ and $V\in O(Y,f(x))$.
$\left.\begin{array}[]{r}(x\in X)(V\in O(Y,f(x)))\\\ \mbox{
Hypothesis}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\mbox{}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!(\exists U\in eR(X,x))(f[U]\subseteq
cl(V))\\\ eR(X)\subseteq e\theta
O(X)\end{array}\right\\}\Rightarrow\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in e\theta O(X,x))(f[U]\subseteq
cl(V)).\end{array}$

$(2)\Rightarrow(3):$ Let $x\in X\setminus f^{-1}[cl(G)]$ and $G\in PO(Y)$.
$\left.\begin{array}[]{rr}x\in X\setminus f^{-1}[cl(G)]\Rightarrow f(x)\in
Y\setminus cl(G)\Rightarrow(\exists V\in O(Y,f(x)))(V\cap G=\emptyset)\\\
\mbox{Hypothesis}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists U\in e\theta O(X,x))(f[U]\subseteq
cl(V))(V\cap G=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in e\theta O(X,x))(f[U]\cap
G=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in e\theta O(X,x))(U\cap
f^{-1}[G]=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow x\notin
e\mbox{-}cl_{\theta}(f^{-1}[G])\end{array}$
$\begin{array}[]{l}\Rightarrow x\in X\setminus
e\mbox{-}cl_{\theta}(f^{-1}[G]).\end{array}$

$(3)\Rightarrow(4):$ Let $G\in PO(Y)$.
$\left.\begin{array}[]{rr}G\in PO(Y)\Rightarrow Y\setminus cl(G)\in
O(Y)\subseteq PO(Y)\\\ \mbox{Hypothesis}\end{array}\right\\}\Rightarrow
e\mbox{-}cl_{\theta}(f^{-1}[Y\setminus cl(G)])\subseteq f^{-1}[cl(Y\setminus
cl(G))]$
$\begin{array}[]{l}\Rightarrow X\setminus
e\mbox{-}int_{\theta}(f^{-1}[cl(G)])\subseteq X\setminus
f^{-1}[int(cl(G))]\subseteq X\setminus f^{-1}[G]\end{array}$
$\begin{array}[]{l}\Rightarrow f^{-1}[G]\subseteq
e\mbox{-}int_{\theta}(f^{-1}[cl(G)]).\end{array}$

$(4)\Rightarrow(5):$ Let $B\subseteq Y$.
$\left.\begin{array}[]{rr}B\subseteq Y\Rightarrow Y\setminus cl(B)\in
O(Y)\subseteq PO(Y)\\\ \mbox{Hypothesis}\end{array}\right\\}\Rightarrow
f^{-1}[Y\setminus cl(B)]\subseteq e\mbox{-}int_{\theta}(f^{-1}[cl(Y\setminus
cl(B))])$
$\begin{array}[]{l}\Rightarrow X\setminus f^{-1}[cl(B)]\subseteq X\setminus
e\mbox{-}cl_{\theta}(f^{-1}[int(cl(B))])\end{array}$
$\begin{array}[]{l}\Rightarrow
e\mbox{-}cl_{\theta}(f^{-1}[int(cl(B))])\subseteq f^{-1}[cl(B)].\end{array}$

$(5)\Rightarrow(6):$ Let $F\in RC(Y)$.
$\left.\begin{array}[]{rr}F\in RC(Y)\Rightarrow int(F)\subseteq Y\\\
\mbox{Hypothesis}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow
e\mbox{-}cl_{\theta}(f^{-1}[int(F)])=e\mbox{-}cl_{\theta}(f^{-1}[int(cl(int(F)))])\subseteq
f^{-1}[cl(int(F))]=f^{-1}[F].\end{array}$

$(6)\Rightarrow(7):$ Let $U\in O(Y)$.
$\left.\begin{array}[]{rr}U\in O(Y)\Rightarrow cl(U)\in RC(Y)\\\
\mbox{Hypothesis}\end{array}\right\\}\Rightarrow
e\mbox{-}cl_{\theta}(f^{-1}[U])\subseteq
e\mbox{-}cl_{\theta}(f^{-1}[int(cl(U))])\subseteq f^{-1}[cl(U)].$

$(7)\Rightarrow(8):$ Let $U\in O(Y)$.
$\left.\begin{array}[]{rr}U\in O(Y)\Rightarrow Y\setminus cl(U)\in O(Y)\\\
\mbox{Hypothesis}\end{array}\right\\}\Rightarrow
e\mbox{-}cl_{\theta}(f^{-1}[Y\setminus cl(U)])\subseteq f^{-1}[cl(Y\setminus
cl(U))]$
$\begin{array}[]{l}\Rightarrow X\setminus
e\mbox{-}int_{\theta}(f^{-1}[cl(U)])\subseteq X\setminus
f^{-1}[int(cl(U))]\subseteq X\setminus f^{-1}[U]\end{array}$
$\begin{array}[]{l}\Rightarrow f^{-1}[U]\subseteq f^{-1}[int(cl(U))]\subseteq
e\mbox{-}int_{\theta}(f^{-1}[cl(U)]).\end{array}$

$(8)\Rightarrow(9):$ Let $A\subseteq X$.
$\left.\begin{array}[]{rr}A\subseteq X\Rightarrow int(Y\setminus f[A])\in
O(Y)\\\ \mbox{Hypothesis}\end{array}\right\\}\Rightarrow f^{-1}[int(Y\setminus
f[A])]\subseteq e\mbox{-}int_{\theta}(f^{-1}[cl(int(Y\setminus f[A]))])$
$\begin{array}[]{l}\Rightarrow X\setminus f^{-1}[cl(f[A])]\subseteq X\setminus
e\mbox{-}cl_{\theta}(f^{-1}[int(cl(f[A]))])\end{array}$
$\begin{array}[]{l}\Rightarrow
f[e\mbox{-}cl_{\theta}(f^{-1}[int(cl(f[A]))])]\subseteq
f[f^{-1}[cl(f[A])]]\end{array}$
$\begin{array}[]{l}\Rightarrow f[e\mbox{-}cl_{\theta}(A)]\subseteq
cl_{\theta}(f[A]).\end{array}$

$(9)\Rightarrow(10):$ Let $B\subseteq Y$.
$\left.\begin{array}[]{rr}B\subseteq Y\Rightarrow f^{-1}[B]\subseteq X\\\
\mbox{Hypothesis}\end{array}\right\\}\Rightarrow
f[e\mbox{-}cl_{\theta}(f^{-1}[B])]\subseteq cl_{\theta}(f[f^{-1}[B]])\subseteq
cl_{\theta}(B)$
$\begin{array}[]{l}\Rightarrow e\mbox{-}cl_{\theta}(f^{-1}[B])\subseteq
f^{-1}[f[e\mbox{-}cl_{\theta}(f^{-1}[B])]]\subseteq
f^{-1}[cl_{\theta}(f[f^{-1}[B]])]\subseteq f^{-1}[cl_{\theta}(B)].\end{array}$

$(10)\Rightarrow(11):$ Let $B\subseteq Y$.
$\left.\begin{array}[]{rr}B\subseteq Y\Rightarrow int(cl_{\theta}(B))\subseteq
Y\\\ \mbox{Hypothesis}\end{array}\right\\}\Rightarrow
e\mbox{-}cl_{\theta}(f^{-1}[int(cl_{\theta}(B))])\subseteq
f^{-1}[cl_{\theta}(int(cl_{\theta}(B)))]\subseteq f^{-1}[cl_{\theta}(B)].$

$(11)\Rightarrow(1):$ Let $x\in X$ and $U\in O(Y,f(x))$.
$\left.\begin{array}[]{r}(x\in X)(U\in O(Y,f(x)))\\\ \mbox{
Hypothesis}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\mbox{}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!\\!e\mbox{-}cl_{\theta}(f^{-1}[int(cl_{\theta}(U))])\subseteq
f^{-1}[cl_{\theta}(U)]\\\ U\subseteq
int(cl(U))=int(cl_{\theta}(U))\end{array}\right\\}\Rightarrow\end{array}$
$\begin{array}[]{l}\Rightarrow e\mbox{-}cl_{\theta}(f^{-1}[U])\subseteq
e\mbox{-}cl_{\theta}(f^{-1}[int(cl_{\theta}(U))])\subseteq
f^{-1}[cl_{\theta}(U)]\overset{\text{Lemma
\ref{bsa}}}{=}f^{-1}[cl(U)]\end{array}$

It follows by $(1)\Leftrightarrow(7)$ that $f$ is weakly $eR$ continuous.
$\Box$

###### Definition 3.3

[13] A function $f:X\rightarrow Y$ is called to be contra $e\theta$-continuous
(briefly c.$e\theta$.c.) if $f^{-1}[V]$ is $e$-$\theta$-closed in $X$ for
every open set $V$ of $Y$.

###### Theorem 3.4

If $f:X\rightarrow Y$ is contra $e\theta$-continuous, then $f$ is weakly
$eR$-continuous.

Proof: Let $V\in O(Y)$.
$\left.\begin{array}[]{rr}V\in O(Y)\\\ f\mbox{ is
c.$e\theta$.c.}\end{array}\right\\}\Rightarrow f^{-1}[V]\in e\theta
C(X)\Rightarrow e\mbox{-}cl_{\theta}(f^{-1}[V])=f^{-1}[V]\subseteq
f^{-1}[cl(V)]$
Then by 3.3(7) $f$ is weakly $eR$-continuous. $\Box$

###### Theorem 3.5

For a function $f:X\rightarrow Y$, the followings are equivalent:
(1) $f$ is weakly $eR$-continuous at $x\in X$;
(2) $x\in e\mbox{-}int_{\theta}(f^{-1}[cl(V)])$ for each neighborhood $V$ of
$f(x)$.

Proof: $(1)\Rightarrow(2):$ Let $V\in O(Y,f(x))$.
$\left.\begin{array}[]{rr}V\in O(Y,f(x))\\\
\mbox{Hypothesis}\end{array}\right\\}\Rightarrow(\exists U\in
eR(X,x))(f[U]\subseteq cl(V))\Rightarrow(\exists U\in eR(X,x))(U\subseteq
f^{-1}[cl(V)])$
$\begin{array}[]{l}\Rightarrow(\exists U\in eR(X,x))(U\subseteq
e\mbox{-}int_{\theta}(U)\subseteq
e\mbox{-}int_{\theta}(f^{-1}[cl(V)]))\end{array}$
$\begin{array}[]{l}\Rightarrow x\in
e\mbox{-}int_{\theta}(f^{-1}[cl(V)]).\end{array}$

$(2)\Rightarrow(1):$ Let $V\in O(Y,f(x))$.
$\left.\begin{array}[]{rr}V\in O(Y,f(x))\\\
\mbox{Hypothesis}\end{array}\right\\}\Rightarrow x\in
e\mbox{-}int_{\theta}(f^{-1}[cl(V)])\Rightarrow(\exists U\in
eR(X,x))(f[U]\subseteq cl(V))$

Thus, $f$ is weakly $eR$-continuous at $x\in X$. $\Box$

###### Lemma 3.1

[23] Let $X$ be a topological space and $A,B\subseteq X$. If $A\in aO(X)$ and
$B\in eO(X)$, then $A\cap B\in eO(X)$.

###### Theorem 3.6

If $f:X\rightarrow Y$ is weakly $eR$-continuous at $x\in X$, then there exists
a nonempty $e$-open set $U\subseteq H$ such that $U\subseteq
e\mbox{-}cl_{\theta}(f^{-1}[cl(V)])$ for every neighborhood $V$ of $f(x)$ and
every $a$-open neighborhood $H$ of $x$.

Proof: Let $H\in aO(X,x)$ ve $V\in O(Y,f(x))$.
$\left.\begin{array}[]{rr}(H\in aO(X,x))(V\in O(Y,f(x)))\\\ f\mbox{ is
w.$eR$.c. at $x\in X$}\end{array}\right\\}\overset{\text{Theorem
}\ref{4}}{\Rightarrow}x\in e\mbox{-}int_{\theta}(f^{-1}[cl(V)])$
$\left.\begin{array}[]{rr}\overset{\text{Lemma
}\ref{*}(9)}{\Rightarrow}(\exists G\in eR(X,x))(G\subseteq
e\mbox{-}int_{\theta}(f^{-1}[cl(V)]))\\\ U:=G\cap
H\end{array}\right\\}\overset{\text{Lemma }\ref{5}}{\Rightarrow}$
$\begin{array}[]{l}\Rightarrow(U\in eO(X,x))(U\subseteq H)(U\subseteq
G\subseteq e\mbox{-}int_{\theta}(f^{-1}[cl(V)])\subseteq
e\mbox{-}cl_{\theta}(f^{-1}[cl(V)])).\end{array}$ $\Box$

###### Theorem 3.7

Let $f:X\rightarrow Y$ be a function. If $f^{-1}[cl_{\theta}(U)]$ is
$e$-$\theta$-closed in $X$ for every subset $U$ of $Y$, then $f$ is weakly
$eR$-continuous.

Proof: Let $f^{-1}[cl_{\theta}(U)]$ is $e$-$\theta$-closed in $X$ for every
subset $U$ of $Y$.
$\begin{array}[]{l}(U\subseteq Y)(f^{-1}[cl_{\theta}(U)])\in e\theta
C(X))\Rightarrow e\text{-}cl_{\theta}(f^{-1}[U])\subseteq
e\text{-}cl_{\theta}(f^{-1}[cl_{\theta}(U)])=f^{-1}[cl_{\theta}(U)]\end{array}$
Then by Theorem 3.3(10), $f$ is weakly $eR$-continuous. $\Box$

###### Theorem 3.8

Let $f:X\rightarrow Y$ be a weakly $eR$-continuous function. Then the
followings hold:
(1) $f^{-1}[U]$ is $e$-$\theta$-open in $X$ for each $\theta$-open set $U$ of
$Y$,
(2) $f^{-1}[V]$ is $e$-$\theta$-closed in $X$ for each $\theta$-closed set $V$
of $Y$.

Proof: It follows from Theorem 3.3. $\Box$

###### Theorem 3.9

[18] A function $f:X\rightarrow Y$ strongly $\theta$-$e$-continuous if and
only if for all $x\in X$ and all open set $V$ of $Y$ containing $f(x)$, there
exists $U\in eR(X,x)$ such that $f(U)\subseteq V$.

###### Theorem 3.10

Let $f:X\rightarrow Y$ be a function. If $Y$ is regular, then the followings
are equivalent:
(1) $f$ is weakly $eR$-continuous;
(2) $f$ is strongly $\theta$-$e$-continuous.

Proof: $(1)\Rightarrow(2):$ Let $x\in X$ and $V\in O(Y,f(x))$.
$\left.\begin{array}[]{r}(x\in X)(V\in O(Y,f(x)))\\\ Y\mbox{ is
regular}\end{array}\right\\}\Rightarrow\begin{array}[]{c}\mbox{}\\\
\left.\begin{array}[]{r}\\!\\!\\!\\!(\exists H\in O(Y,f(x)))(cl(H)\subseteq
V)\\\ f\mbox{ is w.$eR$.c.}\end{array}\right\\}\Rightarrow\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in eR(X,x))(f[U]\subseteq
cl(H)\subseteq V)\end{array}$
Then by Theorem 3.9 $f$ is strongly $\theta$-$e$-continuous.

$(2)\Rightarrow(1):$ It is obvious from Definitions 2.1 and 3.1. $\Box$

##  4 Some Basic Properties of Weakly $eR$-continuity

###### Definition 4.1

A space $X$ is called to be $e$-connected [9] if $X$ is not the union of two
disjoint nonempty $e$-open sets.

###### Theorem 4.1

If $f:X\rightarrow Y$ is a weakly $eR$-continuous surjection and $X$ is
$e$-connected, then $Y$ is connected.

Proof: Suppose that $Y$ is not connected.
$\left.\begin{array}[]{c}Y\mbox{ is not connected}\Rightarrow(\exists A,B\in
O(Y)\setminus\\{\emptyset\\})(A\cap B=\emptyset)(A\cup B=Y)\end{array}\right.$
$\left.\begin{array}[]{r}\Rightarrow(A,B\in
CO(Y)\setminus\\{\emptyset\\})(A\cap B=\emptyset)(A\cup B=Y)\\\ f\mbox{ is
w.$eR$.c.}\end{array}\right\\}\overset{\text{Theorem
\ref{3}(8)}}{\Rightarrow}$
$\begin{array}[]{r}\Rightarrow f^{-1}[A]\subseteq
e\mbox{-}int_{\theta}(f^{-1}[cl(A)])=e\mbox{-}int_{\theta}(f^{-1}[A])\end{array}$
$\left.\begin{array}[]{r}\\!\Rightarrow(f^{-1}[B]\subseteq
e\mbox{-}int_{\theta}(f^{-1}[cl(B)])=e\mbox{-}int_{\theta}(f^{-1}[B]))(f^{-1}[A\cap
B]=\emptyset)(f^{-1}[A\cup B]=X)\\\ f\mbox{ is
surjection}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{c}\Rightarrow(f^{-1}[A],f^{-1}[B]\in
eO(X)\setminus\\{\emptyset\\})(f^{-1}[A]\cap
f^{-1}[B]=\emptyset)(f^{-1}[A]\cup f^{-1}[B]=X)\end{array}\right.$

This is a contradiction to the fact that $X$ is $e$-connected. Thus, $Y$ is
connected. $\Box$

###### Definition 4.2

A space $X$ is called:
(1) Urysohn [29] if for all distinct two points $x$ and $y$ in $X$, there
exist open sets $U$ and $V$ such that $x\in U$, $y\in V$ and $cl(U)\cap
cl(V)=\emptyset$.
(2) Clopen $T_{2}$ [7] if for all distinct two points $x$ and $y$ in $X$,
there exist disjoint clopen sets $U$ and $V$ of $X$ such that $x\in U$, $y\in
V$ and $U\cap V=\emptyset$.
(3) $eR$-$T_{1}$ if for all distinct two points $x$ and $y$ in $X$, there
exist $e$-regular sets $U$ and $V$ of $X$ containing $x$ and $y$,
respectively, such that $y\notin U$ and $x\notin V$.
(4) $eR$-$T_{2}$ if for all distinct two points $x$ and $y$ in $X$, there
exist disjoint $e$-regular sets $U$ and $V$ of $X$ such that $x\in U$, $y\in
V$ and $U\cap V=\emptyset$.

###### Remark 4.1

Every clopen $T_{2}$ space is $eR$-$T_{2}$. This implication is not reversible
as shown in the following example.

###### Example 4.1

Let $X=\\{a,b,c,d\\}$ and
$\tau=\\{\emptyset,\\{a\\},\\{b\\},\\{a,b\\},\\{a,b,c\\},\\{a,b,d\\},X\\}$. It
is not difficult to see $CO(X)=\\{\emptyset,X\\}$ and
$eR(X)=2^{X}\setminus\\{\\{c\\},\\{d\\},\\{c,d\\},\\{a,b\\},\\{a,b,c\\},\\{a,b,d\\}\\}$.
Then $(X,\tau)$ is $eR$-$T_{2}$ but it is not clopen $T_{2}$.

###### Theorem 4.2

Let $f:X\rightarrow Y$ is a weakly $eR$-continuous function and
$g:X\rightarrow Y$ is a weakly $a$-continuous function. If $Y$ is Urysohn,
then $A=\\{x|f(x)=g(x)\\}\in eC(X)$.

Proof: Let $x\notin A$.
$\left.\begin{array}[]{r}x\notin A\Rightarrow f(x)\neq g(x)\\\ Y\text{ is
Urysohn}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{r}\Rightarrow(\exists V_{1}\in O(Y,f(x)))(\exists
V_{2}\in O(Y,g(x)))(cl(V_{1})\cap cl(V_{2})=\emptyset)\\\ (f\text{ is
w.$eR$.c.})(g\text{ is w.$a$.c.})\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(\exists U\in eR(X,x))(\exists G\in
aO(X,x))(f[U]\subseteq cl(V_{1}))(g[G]\subseteq cl(V_{2}))\\\ W:=U\cap
G\end{array}\right\\}\overset{\text{Lemma }\ref{5}}{\Rightarrow}$
$\begin{array}[]{l}\Rightarrow(\exists W\in eO(X,x))(f[W]\cap g[W]\subseteq
f[U]\cap g[G]\subseteq cl(V_{1})\cap cl(V_{2})=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists W\in eO(X,x))(W\cap
A=\emptyset)\end{array}$
$\begin{array}[]{l}\Rightarrow x\notin e\text{-}cl(A).\end{array}$ $\Box$

###### Theorem 4.3

Let $f:X\rightarrow Y$ be a weakly $eR$-continuous injection. If $Y$ is
Hausdorff, then $X$ is $eR$-$T_{1}$.

Proof: Let $x,y\in X$ and $x\neq y$.
$\left.\begin{array}[]{r}(x,y\in X)(x\neq y)\overset{f\text{ is
injective}}{\Rightarrow}f(x)\neq f(y)\\\ Y\mbox{ is
Hausdorff}\end{array}\right\\}\Rightarrow(\exists U\in O(Y,f(x))(\exists V\in
O(Y,f(y))(U\cap V=\emptyset)$
$\left.\begin{array}[]{r}\Rightarrow(\exists U\in O(Y,f(x))(\exists V\in
O(Y,f(y))(f(x)\notin cl(V))(f(y)\notin cl(U))\\\ f\mbox{ is
w.$eR$.c.}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists A\in eR(X,x))(\exists B\in
eR(X,y))(f[A]\subseteq cl(U))(f[B]\subseteq cl(V))\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists A\in eR(X,x))(\exists B\in
eR(X,y))(y\notin A)(x\notin B).\end{array}$ $\Box$

###### Theorem 4.4

Let $f:X\rightarrow Y$ be a weakly $eR$-continuous injection. If $Y$ is
Urysohn, then $X$ is $eR$-$T_{2}$.

Proof: Let $x,y\in X$ and $x\neq y$.
$\left.\begin{array}[]{r}(x,y\in X)(x\neq y)\overset{f\text{ is
injective}}{\Rightarrow}f(x)\neq f(y)\\\ Y\mbox{ is
Urysohn}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{r}\Rightarrow(\exists U\in O(Y,f(x))(\exists V\in
O(Y,f(y))(cl(U)\cap cl(V)=\emptyset)\\\ f\mbox{ is
w.$eR$.c.}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists A\in eR(X,x))(\exists B\in
eR(X,y))(f[A]\subseteq cl(U))(f[B]\subseteq cl(V))\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists A\in eR(X,x))(\exists B\in
eR(X,y))(A\subseteq f^{-1}[cl(U)])(B\subseteq f^{-1}[cl(V)])\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists A\in eR(X,x))(\exists B\in
eR(X,y))(A\cap B\subseteq f^{-1}[cl(U)]\cap
f^{-1}[cl(V)]=\emptyset).\end{array}$ $\Box$

###### Corollary 4.4A

Let $f:X\rightarrow Y$ be a weakly clopen injection. If $Y$ is Urysohn, then
$X$ is $eR$\- $T_{2}$.

Let $\\{X_{\alpha}|\alpha\in I\\}$ and $\\{Y_{\alpha}|\alpha\in I\\}$ be any
two families of topological spaces with the same index set $I.$ The product
space of $\\{X_{\alpha}|\alpha\in I\\}$ (resp. $\\{Y_{\alpha}|\alpha\in I\\}$)
is simply denoted by $\Pi X_{\alpha}$ (resp. $\Pi Y_{\alpha}$). Let
$f_{\alpha}:X_{\alpha}\rightarrow Y_{\alpha}$ be a function for all $\alpha\in
I.$ Let $f:\Pi X_{\alpha}\rightarrow\Pi Y_{\alpha}$ be the product function
defined as follows: $f(\\{x_{\alpha}\\})=\\{f_{\alpha}(x_{\alpha})\\}$ for all
$\\{x_{\alpha}\\}\in\Pi X_{\alpha}.$

###### Theorem 4.5

Let $\\{X_{\alpha}|\alpha\in I\\}$ and $\\{Y_{\alpha}|\alpha\in I\\}$ be any
two families of topological spaces. If $f_{\alpha}:X_{\alpha}\rightarrow
Y_{\alpha}$ is weakly $eR$-continuous for each $\alpha\in I$, then the
function $f:\Pi X_{\alpha}\rightarrow\Pi Y_{\alpha}$ is weakly
$eR$-continuous.

Proof: Let $x=\\{x_{\alpha}\\}\in\Pi X_{\alpha}$ and $V\in O(\Pi
Y_{\alpha},f(x))$.
$\begin{array}[]{l}V\in O(\Pi Y_{\alpha},f(x))\Rightarrow(\exists
J=\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\\}\subseteq I)\end{array}$
$\left.\begin{array}[]{r}\left(W_{\alpha}:=\left\\{\begin{array}[]{ccc}W_{\alpha_{j}}\in
O(Y_{\alpha_{j}})&,&\alpha\in J\\\ Y_{\alpha}&,&\alpha\notin
J\end{array}\right.\right)\left(\Pi W_{\alpha}\in O(\Pi
Y_{\alpha},f(x))\right)\left(\newline \Pi W_{\alpha}\subseteq V\right)\\\
(\forall\alpha\in I)(f_{\alpha}\mbox{ is
w.$eR$.c.})\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{r}\Rightarrow(\exists U_{\alpha}\in
eR(X_{\alpha},x_{\alpha}))\left(f_{\alpha}[U_{\alpha}]\subseteq
cl(W_{\alpha})\right)\\\
U:=\prod\limits_{j=1}^{n}U_{\alpha_{j}}\times\prod\limits_{\alpha\notin
J}X_{\alpha}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{c}\Rightarrow(U\in eR(\Pi
X_{\alpha},x))\left(f[U]\subseteq\prod\limits_{j=1}^{n}f_{\alpha}[U_{\alpha_{j}}]\times\prod\limits_{\alpha\notin
J}Y_{\alpha}\subseteq\prod\limits_{j=1}^{n}cl(W_{\alpha_{j}})\times\prod\limits_{\alpha\notin
J}Y_{\alpha}\subseteq cl(V)\right).\end{array}\right.$ $\Box$

Recall that the graph of a function $f:X\rightarrow Y$ is the subset
$\\{(x,f(x))|x\in X\\}$ of the product space $X\times Y$ and denoted by
$G(f)$.

###### Theorem 4.6

Let $f:X\rightarrow Y$ be a function. If the graph function $g$ is weakly
$eR$-continuous, then $f$ is weakly $eR$-continuous.

Proof: Let $x\in X$ and $V\in O(Y,f(x))$.
$\left.\begin{array}[]{rr}(x\in X)(V\in O(Y,f(x)))\Rightarrow X\times V\in
O(X\times Y,g(x))\\\ g\mbox{ is w.$eR$.c.}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists U\in eR(X,x))(g[U]\subseteq cl(X\times
V)=X\times cl(V))\end{array}$
$\begin{array}[]{l}\Rightarrow(\exists U\in eR(X,x))(f[U]\subseteq
cl(V)).\end{array}$ $\Box$

###### Definition 4.3

A function $f:X\rightarrow Y$ has an $er$-graph if for all $(x,y)\notin G(f)$,
there exist $U\in eR(X,x)$ and $V\in O(Y,y)$ such that $(U\times cl(V))\cap
G(f)=\emptyset$.

###### Lemma 4.1

A function $f:X\rightarrow Y$ has an $er$-graph if and only if for all
$(x,y)\notin G(f)$, there exist $U\in eR(X,x)$ and $V\in O(Y,y)$ such that
$f\left[U\right]\cap cl(V)=\emptyset.$

Proof: It is obvious from Definition 4.3. $\Box$

###### Theorem 4.7

If $f:X\rightarrow Y$ is weakly $eR$-continuous and $Y$ is a Urysohn space,
then $G(f)$ is an $er$-graph.

Proof: Let $(x,y)\notin G(f).$
$\left.\begin{array}[]{rr}(x,y)\notin G(f)\Rightarrow y\neq f(x)\\\ Y\mbox{ is
Urysohn}\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{rr}\Rightarrow(\exists A\in O(X,f(x)))(\exists B\in
O(Y,y))(cl(A)\cap cl(B)=\emptyset)\\\ f\mbox{ is
w.$eR$.c.}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{c}\Rightarrow(\exists G\in eR(X,x))(\exists B\in
O(Y,y))(f[G]\subseteq cl(A))\end{array}$
$\begin{array}[]{c}\Rightarrow(\exists G\in eR(X,x))(\exists B\in
O(Y,y))(f[G]\cap cl(B)=\emptyset)\end{array}$
$\begin{array}[]{c}\Rightarrow(\exists G\in eR(X,x))(\exists B\in
O(Y,y))((G\times cl(B))\cap G(f)=\emptyset).\end{array}$ $\Box$

###### Theorem 4.8

If $f:X\rightarrow Y$ has an $er$-graph and a weakly $eR$-continuous
injection, then $X$ is $eR$-$T_{2}$.

Proof: Let $x,y\in X$ and $x\neq y$.
$\left.\begin{array}[]{rr}(x,y\in X)(x\neq y)\overset{f\text{ is
injective}}{\Rightarrow}f(x)\neq f(y)\Rightarrow(x,f(y))\notin G(f)\\\
G(f)\mbox{ is $er$-graph}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{l}\Rightarrow(\exists U\in eR(X,x))(\exists V\in
O(Y,f(y)))((U\times cl(V))\cap G(f)=\emptyset)\end{array}$
$\left.\begin{array}[]{rr}\Rightarrow(\exists U\in eR(X,x))(\exists V\in
O(Y,f(y)))(f[U]\cap cl(V)=\emptyset)\\\ f\mbox{ is
w.$eR$.c.}\end{array}\right\\}\Rightarrow$
$\begin{array}[]{c}\Rightarrow(\exists U\in eR(X,x))(\exists W\in
eR(X,y))(f[W]\subseteq cl(V))\end{array}$
$\begin{array}[]{c}\Rightarrow(\exists U\in eR(X,x))(\exists W\in
eR(X,y))(f[U]\cap f[W]=\emptyset)\end{array}$
$\begin{array}[]{c}\Rightarrow(\exists U\in eR(X,x))(\exists W\in
eR(X,y))(U\cap V=\emptyset).\end{array}$ $\Box$

###### Definition 4.4

A space $X$ is called to be $eR$-compact if every cover of $X$ by $e$-regular
sets has a finite subcover.

###### Theorem 4.9

Let $f:X\rightarrow Y$ be a function having an $er$-graph $G(f)$, then $f[K]$
is $\theta$-closed in $Y$ for all $eR$-compact relative to $X$ subset $K$.

Proof: Let $K$ be $eR$-compact relative to $X$ and $y\notin f[K].$
$\left.\begin{array}[]{r}y\notin f[K]\Rightarrow(\forall x\in
K)(\left(x,y\right)\notin G(f))\\\ G(f)\mbox{ is
$er$-graph}\end{array}\right\\}\overset{\text{Lemma \ref{10}}}{\Rightarrow}$
$\begin{array}[]{c}\Rightarrow(\exists U_{x}\in eR(X,x))(\exists V_{x}\in
O(Y,y))(f[U_{x}]\cap cl(V_{x})=\emptyset)\end{array}$
$\left.\begin{array}[]{r}\Rightarrow(\\{U_{x}|x\in K\\}\subseteq
eR(X))(K\subseteq\bigcup\\{U_{x}|x\in K\\})\\\ K\mbox{ is }eR\mbox{-compact
relative to }X\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{r}\Rightarrow(\exists K^{\ast}\subseteq
K)(|K^{\ast}|<\aleph_{0})\newline (K\subseteq\bigcup\\{U_{x}|x\in
K^{\ast}\\})\\\ V:=\underset{x\in K^{\ast}}{\bigcap}V_{x}\in
O(Y,y)\end{array}\right\\}\Rightarrow$
$\left.\begin{array}[]{c}\Rightarrow\left(V\in O(Y,y)\right)\left(f[K]\cap
V\subseteq\left(\underset{x\in
K^{\ast}}{\bigcup}f\left[U_{x}\right]\right)\cap cl(V)\subseteq\underset{x\in
K^{\ast}}{\bigcup}\left(f\left[U_{x}\right]\cap
cl(V)\right)=\emptyset\right)\newline \end{array}\newline \right.$
$\left.\begin{array}[]{c}\Rightarrow y\notin cl_{\theta}(f[K]).\newline
\end{array}\newline \right.$ $\Box$

##  5 Conclusion

This paper is concerned with the concept of $eR$-continuity and weakly
$eR$-continuity defined by utilizing the notion of $e$-regular sets. It turns
out that both $eR$-continuous and weakly $eR$-continuous functions are
stronger than weakly $e$-continuous functions, as will be seen in Remark 3.1.
We believe that this paper will pave the way for future studies relevant to
continuity and convergence etc. known from functional analysis.

## Acknowledgments

The author would like to thank to anonymous reviewer for his/her valuable
suggestions and comments, which improved this study.

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  * [29] S. Willard, General Topology, London: Addison Wesley Publishing Company, (1970).


\nointerlineskip\hbox{\set@color$\textstyle A^{\prime}CA$}}}{\vbox{
\hbox{\leavevmode\resizebox{17.10373pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle A^{\prime}CA$}}}{\vbox{
\hbox{\leavevmode\resizebox{12.53123pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle A^{\prime}CA$}}}$ and let
$\overline{S_{B}}$ as the circular sector bounded by $\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{24.40251pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle A^{\prime}CA$}}}{\vbox{
\hbox{\leavevmode\resizebox{24.40251pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle A^{\prime}CA$}}}{\vbox{
\hbox{\leavevmode\resizebox{17.10373pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle A^{\prime}CA$}}}{\vbox{
\hbox{\leavevmode\resizebox{12.53123pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle
A^{\prime}CA$}}}\cup\overline{AO}\cup\overline{OA^{\prime}}$. Again, we have

$\underline{S_{B}}\subseteq{\mathcal{G}}(2J_{\alpha{\mathcal{B}}}-1)\subseteq\overline{S_{B}},$

and

$\cos\varphi_{B}=\tfrac{1-\alpha^{2}L^{2}}{1+\alpha^{2}L^{2}}.$

Using the arccosine sum identity [2, p. 80, 4.4.33], we get

$\cos(\varphi_{A}-\varphi_{B})=1-\tfrac{2(\alpha
L-\alpha/\gamma)^{2}}{(1+\alpha^{2}L^{2})(1+\alpha^{2}/\gamma^{2})}.$

When $1/\gamma\leq L$, we have $\varphi_{A}\leq\varphi_{B}$. In this case,

$\varphi_{A}$$\underline{S_{A}}$ | $\times$ |  $\varphi_{B}$$\underline{S_{B}}$ | $=$ |  $\\{z\in\mathbb{C}\,|\,|z|=1\\}$
---|---|---|---|---

Therefore

$1\in(1-\theta)1+\theta\underline{S_{A}}\,\underline{S_{B}}\subseteq{\mathcal{G}}(D_{\alpha,\theta}({\mathcal{A}},{\mathcal{B}})),$

but $1\notin{\mathcal{G}}({\mathcal{L}}_{1-\varepsilon})$ for any
$\varepsilon>0$. Therefore, we conclude
${\mathcal{G}}(D_{\alpha,\theta}({\mathcal{A}},{\mathcal{B}}))\nsubseteq{\mathcal{G}}({\mathcal{L}}_{1-\varepsilon})$.

When $1/\gamma>L$, we have $\varphi_{A}>\varphi_{B}$. In this case,

$\varphi_{A}$$\underline{S_{A}}$ | $\times$ |  $\varphi_{B}$$\underline{S_{B}}$ | $=$ |  $\varphi_{A}-\varphi_{B}$$\underline{S_{A}}\,\underline{S_{B}}$
---|---|---|---|---

$\varphi_{A}$$\overline{S_{A}}$ |   $\times$  |  $\varphi_{B}$$\overline{S_{B}}$ |   $=$  |  $\varphi_{A}-\varphi_{B}$$-\varphi_{A}-\varphi_{B}$$\overline{S_{A}}\,\overline{S_{B}}$ |   $\subseteq$  |  $\varphi_{A}-\varphi_{B}$$\overline{S_{B}}$
---|---|---|---|---|---|---

Using the outer bounds $\overline{S_{A}}$ and $\overline{S_{B}}$ we establish
correctness. Using the inner bounds $\underline{S_{A}}$ and
$\underline{S_{B}}$ we establish tightness.

$\theta$$1-\theta$$\varphi_{A}-\varphi_{B}$$(1-\theta)+\theta\underline{S_{A}}\,\underline{S_{B}}$$\subseteq(1-\theta)+\theta\overline{S_{A}}\,\overline{S_{B}}$${\mathcal{G}}\left({\mathcal{L}}_{R}\right)$$R=\sqrt{1-\frac{4\theta(1-\theta)(1-\gamma
L)^{2}}{(1+\gamma^{2}/\alpha^{2})(1+\alpha^{2}L^{2})}}$

With the Pythagorean theorem, we can verify that the containment holds for
larger $R$ and fails for smaller $R$. Since ${\mathcal{L}}_{R}$ is SRG-full,
the containment of the SRG in $\overline{\mathbb{C}}$ equivalent to the
containment of the class by Theorem 3.7.

The result for the cases $\alpha/\gamma\geq 1$ or $\alpha L\geq 1$ and for the
operator $D_{\alpha,\theta}({\mathcal{B}},{\mathcal{A}})$ follows from similar
reasoning.

######  Corollary D.6.

Let $0<1/\gamma\leq L<\infty$ and $\alpha\in(0,\infty)$. Let
$B\in{\mathcal{M}}\cap{\mathcal{L}}_{L}$ and let $A\in{\mathcal{M}}$ satisfy a
condition weaker than or equal to $\gamma$-inverse Lipschitz continuity, such
as $\gamma$-metric subregularity. It is not possible to establish a strict
contraction of the DRS operators $T(A,B,\alpha,\theta)$ or
$T(B,A,\alpha,\theta)$ for any $\alpha>0$ and $\theta\in\mathbb{R}$ without
further assumptions.


# Predicting Corporate Risk by Jointly Modeling Company Networks and Dialogues
in Earnings Conference Calls

Yunxin Sang
Shanghai Jiao Tong University
<EMAIL_ADDRESS>
&Yang Bao
Shanghai Jiao Tong University
<EMAIL_ADDRESS>

###### Abstract

Earnings conference calls are attracting an increasing number of researchers
due to their free form and rich information. Existing studies, however, do not
take speaker role information into account. Furthermore, current research does
not fully account for the impact of inter-company relationships on company
risk. The only study that integrates company networks and earnings conference
calls constructs an undirected graph for companies holding earnings conference
calls at different dates, failing to meet the requirement of no temporal
information leakage for prediction tasks.

To address the aforementioned issues, we propose a new model called Temporal
Virtual Graph Neural Network (TVGNN), which incorporates earnings conference
calls and company networks to predict company risk. For the first time, our
model incorporates participant role information in dialogue modeling.
Moreover, we develop a new approach to construct company networks that ensures
no temporal information leakage in the graph. In experiments, our proposed
model outperforms all baselines. The supplementary analyses demonstrate the
model’s effectiveness and interpretability.

_K_ eywords Company Risk Prediction $\cdot$ Earnings Conference Call $\cdot$
Company Network $\cdot$ Graph Neural Network

##  1 Introduction

Risk is a core element of the modern financial system [1]. Company risk
prediction is one of the crucial tasks in finance, management, and operations
research [2]. Most early studies in this problem are based on numerical data
such as price and volume [2]. With the advancement of deep learning,
researchers attempted to incorporate unstructured data such as company annual
reports [3] into their models. Among these unstructured data, earnings
conference calls draw many researchers attention due to their unique form.

Earnings conference calls are a way for companies to share financial
information with interested investors or institutions on a quarterly basis.
Participants include executives from the company as well as buy-side analysts.
An earnings conference call consists of two parts: the Presentation and the
Q&A. In the presentation part, company executives present the previous
quarter’s financial performance as well as an outlook for the future. During
the Q&A part, analysts ask executives questions from their perspectives.
Compared with other financial data, the communication between executives and
analysts is more flexible, often reflecting the true picture of the company’s
situation, which aids in the prediction of company risk. For example, AMD
disclosed a higher-than-expected loss during its earnings conference call in
Q1 2017, causing its shares to plunge 16.1% after the bell.

Although earnings conference calls have significant value for company risk
prediction, most existing studies design models based on modality fusion and
long text modeling without taking the dialogue structure in earnings
conference calls into account. [4, 5]. Furthermore, not only does the
information disclosed during the earnings conference call affect company risk,
but so does the person who discloses the information. For example, despite
knowing more about the company’s internal information, executives may avoid
disclosing negative information. Analysts, on the other hand, ask questions
from their own or their institution’s perspective, reflecting the market’s
concerns about the company. Therefore, incorporating participant role
information is beneficial for risk prediction.

Moreover, the relationship between companies is useful for predicting company
risk. The flow of risk in the financial system caused the global financial
crisis that lasted from 2007 to 2009. Although Lehman Brothers and AIG were
not the largest companies in the market, their bankruptcy shocked the entire
system due to their close relationship with other institutions. Thus, some
researchers attempted to integrate company networks into company risk
prediction. However, they construct companies that held earnings conference
calls on different dates in an undirected graph, ignoring the requirement of
no temporal information leakage for prediction tasks. [6].

To address the aforementioned issues, we propose a new model, Temporal Virtual
Graph Neural Network (TVGNN), which jointly models earnings conference calls
and company networks to predict company risk. The contributions of our
research are listed below:

  1. 1.

For the first time, we we incorporate participant role information into the
task of company risk prediction.

  2. 2.

We propose a new method to construct company networks, which guarantees no
temporal information leakage.

  3. 3.

We design two new modules to model company networks and market state for
company risk prediction.

In addition, we create a new dataset to validate TVGNN, and the experimental
results demonstrate the effectiveness and interpretability of our model.

##  2 Related Work

Our study involves two fields: graph neural networks and company risk
prediction.

###  2.1 Graph Neural Network

Graph is everywhere in the real world. For example, the supply chain is a
natural graph. However, it is difficult for traditional neural networks to
deal with graphs. To solve this problem, graph neural networks were proposed.

The concept of graph neural networks was first proposed by Gori et al. [7] and
then developed by Scarselli et al [8]. These early studies were based on
Banach’s fixed point theorem, updating the node embedding iteratively until it
achieves stability, which was inefficient.

Inspired by convolutional neural networks, Bruna et al. proposed graph
convolutional networks (GCN) [9]. Kipf and Welling improved GCN’s
computational efficiency by modeling convolutional kernels with Chebyshev
polynomials and renormalization techniques [10]. Recently, a series of graph
neural networks have been proposed, such as Graph Isomorphism Network [11],
Graph Attention Network [12], and Graphormer [13].

Traditional graph neural networks are focused on static graphs with fixed
nodes and edges, whereas many practical applications involve dynamic graphs
with changing nodes or edges. A simple way to deal with dynamic graphs is to
convert them into static graphs. Liben-Nowell and Kleinberg converted dynamic
graphs to static graphs by adding the adjacency matrices of graphs at
different time steps [14]. Hisano modeled the temporal information using the
formation and dissolution matrices of previous $k$ time steps [15]. Aside from
that, it is more natural to separately model the graphs at different time
steps and then aggregate the results. Yao et al. used graph neural networks to
model the graph snapshots at each time step, then weighted these snapshots
based on their time difference from the current time step [16].

###  2.2 Company Risk Prediction

Company risk prediction is an essential task in the capital market. With the
advancement of deep learning, unstructured data becomes increasingly important
in this task. Kogan et al. predicted the volatility of listed companies’ stock
returns using financial reports and digital indicators [2]. Bao and Datta
developed a variant of the latent Dirichlet allocation topic model for
simultaneously discovering and quantifying risk types from textual risk
disclosures, and they investigated how risk disclosures in 10-K forms affect
investors’ risk perceptions [3].

Compared to annual reports and news, earnings conference calls contain more
information due to their free form. Many researchers attempt to incorporate it
into the task of company risk prediction. Qin and Yang proposed a multimodal
deep regression model to capture CEOs’ textual and audio features in earnings
conference calls for company risk prediction [4].Theil et al., on the other
hand, used a hierarchical recurrent neural network to model the textual
features of the presentation session and the Q&A session separately [17]. Li
et al. collected textual and audio data from the earnings conference calls of
S&P 1500 companies from 2015 to 2018 and aligned the two modalities to
construct a dataset called MAEC [18]. Yang et al. developed a Transformer-
based multi-task architecture to model earnings conference call textual and
audio features, with an auxiliary task of predicting stock price volatility on
a given day [19]. Yang et al. improved his work in 2021 by introducing two new
training tasks: Numeral Category Classification and Magnitude Comparison,
which allowed the model to better capture numerical information [20].

The above studies focus on textual and audio modality modeling and fusion,
ignoring the unique dialogue structure of earnings conference calls. Zhen Ye
et al. proposed a multi-round question-and-answer attention network (MRQA) to
model the dialogue structure in earnings conference calls, employing a
Reinforced Sentence Selector to identify important sentences in the Q&A
session and a Reinforced Bidirectional Attention Network to model the
association between the sentences [21]. These studies, however, do not take
into account the role information of participants in earnings conference
calls.

Furthermore, Ye et al. utilized graph neural networks to incorporate inter-
company relationships for company risk prediction for the first time [6].
However, in this study, companies that held earnings conference calls on
different dates were constructed in an undirected graph, causing temporal
information leakage.

##  3 Methodology

In this section, we formulate our problem of company risk prediction and then
present our proposed TVGNN model.

###  3.1 Problem Definition

Kogan et al. used the stock return volatility to measure the risk of listed
companies [2]. The greater the volatility of a company’s stock return, the
greater the risk for investors holding this company’s stock. Generally, the
stock return volatility $y_{[t,t+\tau]}$ from trading day $t$ to $t+\tau$ is
defined as follows

$y_{[t,t+\tau]}=\sqrt{\sum_{i=0}^{\tau}(r_{t+i}-\overline{r})^{2}/\tau}$ (1)

where $r_{t}$ is the adjusted return of the given stock on the trading day
$t$, $\overline{r}$ is the average of the adjusted returns from $t$ to
$t+\tau$, and $\tau$ is the length of the time window used to calculate stock
return volatility. The adjusted stock return is defined as:

$r_{t}=\frac{p_{t}}{p_{t-1}}-1$ (2)

where $p_{t}$ is the adjusted closing price of the stock on the trading day
$t$ which can reflect the actual change of stock price by removing the stock
splits, dividends, and additions.

According to Earning Momentum, stock prices drift abnormally for a period of
time after the release of earnings [22], so we set the time windows $\tau$ for
calculating stock return volatility as 3, 7, and 15 days.

By quantifying company risk, we define the task of company risk prediction as
a supervised regression task. Specifically, given the earnings conference call
of company $c_{i}$ on day $t$, the earnings conference call can be represented
as a sequence of sentences $S_{i}=[s_{i,1},s_{i,2},\cdots,s_{i,N}]$ in time
order. The goal of company risk prediction is to predict the company’s stock
return volatility $y_{[t+1,t+\tau]}$ by the sequence of sentences.

###  3.2 Proposed Model

We propose a new model, Temporal Virtual Graph Neural Network (TVGNN), to
incorporate company network and earnings conference calls for company risk
prediction, guaranteeing no temporal information leakage. The model has a
hierarchical architecture, including three modules:

1) Sentence Encoder: This module extracts the textual information of the
earnings conference call and incorporates dialogue structural features and
participant role information to output a vector representation for each
sentence.

2) Dialogue Encoder: This module captures the contextual relationship of
sentences in earnings conference calls and encodes the entire dialogue into a
vector.

3) Company Network Encoder: This is the key module of our model, consisting of
company network construction, market encoder, and network encoder. First, we
construct company networks based on company relationships, guaranteeing no
temporal information leakage. The market encoder then models the market state
at each time step. Finally, the network encoder employs a graph neural network
to fuse all information and update the company representations.

At last, the obtained company representations are fed into an output layer for
the downstream task.

####  3.2.1 Sentence Encoder

We use a pre-trained model MPNet 111https://www.sbert.net/ to encode sentences
[23]. The sequence of encoded sentences is denoted as
$S_{i}=[\mathbf{s}_{i,1},\cdots,\mathbf{s}_{i,j},\cdots,\mathbf{s}_{i,N}]$,
where the $\mathbf{s}_{i,j}$ is the encoded vector of the $j$th sentence in
the earnings conference call of company $c_{i}$.

According to the position encoding in BERT [24], we use four structural
embeddings to represent the dialogue structure of an earnings conference call:

1) Position embedding $pos_{i,j}$ encodes the order of sentences in the
dialogue.

2) Utterance embedding $uttr_{i,j}$ encodes to which utterance the sentences
subordinate. All sentences spoken by a speaker at once are called an
utterance.

3) Role embedding $role_{i,j}$ encodes the speaker’s role information of a
given sentence. There are two roles in an earnings conference call: executives
and analysts.

4) Part embedding $part_{i,j}$ encodes in which parts the sentence appears. An
earnings conference call consists of two parts: presentation and Q&A.

Finally, we concatenate the structural embeddings to the sentence vector
$\mathbf{s}_{i,j}$:

$\mathbf{x}_{i,j}=\mathbf{s}_{i,j}\oplus pos_{i,j}\oplus uttr_{i,j}\oplus
role_{i,j}\oplus part_{i,j}$ (3)

In this way, we obtain a sequence of sentence vectors
$X_{i}=[\mathbf{x}_{i,1},\mathbf{x}_{i,2},\cdots,\mathbf{x}_{i,N}]$ that
incorporates dialogue structural information, where each vector
$\mathbf{x}_{i,j}\in\mathbb{R}^{d}$.

####  3.2.2 Dialogue Encoder

The key to text modeling is to model the contextual relationships. To obtain
contextual information, we update sentence vectors with a Transformer encoder
[25]. Following BERT, we add a trainable [CLS] vector $\mathbf{a}$ to the
input sequence $X_{i}$ to obtain the representation of a given earnings
conference call. Then the input sequence becomes
$X_{i}=[\mathbf{a},\mathbf{x}_{i,1},\mathbf{x}_{i,2},\cdots,\mathbf{x}_{i,N}]$.
After $L$-layer Transformer encoders, we take $\mathbf{a}^{(L)}$ in the output
sequence as the representation of the earnings conference call, denoted as
$\mathbf{v}_{i}$.

####  3.2.3 Company Network Encoder

The company network encoder contains two submodules: a market encoder and a
network encoder that model company networks and market state, respectively.

In this section, we first describe how to construct company networks. Then, we
introduce how to apply the market encoder and the network encoder on the
constructed company network to model the market state and the company
relationships.

Company Network Construction Assume there are $M$ companies holding their
earnings conference calls in a given quarter (natural quarter), we treat each
company as a node in the graph, and the node $v_{i}$ corresponds to company
$c_{i}$. If there are some kind of relationship exists between $c_{i}$ and
$c_{j}$ and the date $t_{c_{j}}$ when $c_{i}$ holds its earnings conference
call in the given quarter is not earlier than $t_{c_{j}}$, a weighted directed
edge is connected between nodes $v_{i}$ and $v_{j}$. The weight of the edge is
defined as follows.

$\mathbf{A}_{i,j}=\begin{cases}\frac{1}{(t_{c_{i}}-t_{c_{j}}+1)},\text{If
$v_{i}$ is connected with $v_{j}$ and $t_{c_{i}}-t_{c_{j}}>=0$}\\\
0,\text{Other}\end{cases}$ (4)

The weight defined in Equation (4) ensures that the weight of the edge between
two nodes decreases as the interval between the two corresponding companies’
earnings conference calls increases. Figure reffig:cn shows an example of a
company network constructed in this way.

Figure 1: Example of Constructed Company Network

The constructed company network is a directed static graph.The network’s
directed edges prevent temporal information leakage, i.e., information can
only flow from temporally preceding nodes to temporally following nodes, which
is critical for a temporal prediction task.

Finally, we assign $\mathbf{c}_{i}$ obtained from the dialogue encoder as the
initial embedding of the node $v_{i}$.

Market Encoder The capital asset pricing model points out that the return of
an asset consists of the risk-free return of the market and the return of the
asset [26]. Therefore, when predicting the risk of a given company, we should
consider the impact of both the events (earnings conference calls) of the
company and the market state. Thus, we design the market encoder to model the
market state $\mathbf{m}_{t_{i}}$ at day $t_{i}$. The structure of the market
encoder is shown in Figure 2.

Figure 2: Market Encoder

Suppose there are $T$ different dates $[t_{1},\cdots,t_{i},\cdots,t_{T}]$ for
holding an earnings conference call in a given quarter. The dates are sorted
in chronological order. The set of companies holding earnings conference call
at $t_{i}$ are denoted as $\mathcal{V}_{t_{i}}$.

The market state of $t_{i}$ can be represented as the sum of all events
(earnings conference calls) happening in the market at $t_{i}$. Thus, we can
calculate the market state $\mathbf{m}_{t_{i}}$ by a global attention module:

$\displaystyle\mathbf{m}_{t_{i}}$ $\displaystyle=\sum_{v_{j}\in
V_{t_{i}}}\beta_{j}\mathbf{v}_{j}$ (5) $\displaystyle\mathbf{k}_{j}$
$\displaystyle=\mathbf{v}_{j}\mathbf{w}^{T}_{k}$ (6) $\displaystyle e_{j}$
$\displaystyle=\frac{\mathbf{w}_{q}\mathbf{k}^{T}_{j}}{\sqrt{d}}$ (7)
$\displaystyle\beta_{j}$
$\displaystyle=\frac{e_{j}}{\sum_{v_{u}\in\mathcal{V}_{t_{i}}}e_{u}}$ (8)

where $\beta_{j}$ is the attention sore of node $v_{j}\in\mathcal{V}_{t_{i}}$,
$\mathbf{w}_{q}$ is a trainable parameter.

The market state $\mathbf{m}_{t_{i}}$ is affected not only by the events at
day $t_{i}$, but also by the market state of the past. Thus, we use a Gated
Recurrent Unit Network (GRU) [27] to model the historial market state. Since
the larger the time interval, the weaker the effect of the previous market
state on the current market state, we define a coefficient
$\Delta_{t_{i}}=\sigma(\frac{w_{d}}{t_{i}-t_{i-1}+1})$ which represents the
time interval between $t_{i}$ and $t_{i-1}$, to adjust the value of the reset
gate in the GRU:

$\displaystyle\mathbf{z}_{t_{i}}$
$\displaystyle=\sigma(\mathbf{m}_{t_{i}}\mathbf{w}_{1}^{T}+\mathbf{a}_{t_{i-1}}\mathbf{w}_{2}^{T})$
(9) $\displaystyle\mathbf{r}_{t_{i}}$
$\displaystyle=\sigma(\mathbf{m}_{t_{i}}\mathbf{w}_{1}^{T}+\mathbf{a}_{t_{i-1}}\mathbf{w}_{2}^{T})$
(10) $\displaystyle\tilde{\mathbf{a}}_{t_{i}}$
$\displaystyle=\tanh(\mathbf{m}_{t_{i}}\mathbf{w}_{1}^{T}+(\Delta_{t_{i}}\circ\mathbf{r}_{t_{i}}\circ\mathbf{h}_{t_{i-1}})\mathbf{w}_{2}^{T})$
(11) $\displaystyle\mathbf{a}_{t_{i}}$
$\displaystyle=(1-\mathbf{z}_{t_{i}})\circ\mathbf{a}_{t_{i-1}}+\mathbf{z}_{t_{i}}\circ\tilde{\mathbf{a}}_{t_{i}}$
(12) $\displaystyle\mathbf{m}^{\prime}_{t_{i}}$
$\displaystyle=\mathbf{a}_{t_{i}}\mathbf{w}_{a}^{T}+b_{a}$ (13)

where $\sigma$ is an activation function.

Network Encoder We use a graph attention network with symmetric normalized
adjacency matrix [28] to update node embedding so that each company node can
capture the topology of the company network. Assume using $L$-layer graph
attention networks, the update formula of the company node embedding at the
$l$-layer is as follows.

$\displaystyle\mathbf{g}^{(l-1)}_{i}$
$\displaystyle=\mathbf{v}^{(l-1)}_{i}+\mathbf{m}^{(l)\prime}_{t(c_{i})}$ (14)
$\displaystyle\mathbf{v}^{(l)}_{i}$
$\displaystyle=\sigma(\sum_{v_{j}\in\mathcal{N}(v_{i})}\frac{\gamma_{ij}}{\tilde{D}_{ij}}\mathbf{g}^{(l-1)}_{j}\mathbf{w}^{T}_{0}+\mathbf{g}^{(l-1)}_{i}\mathbf{w}_{1}^{T})$
(15)

where equation (14) adds the company node embedding $\mathbf{v}^{(l-1)}_{i}$
and corresponding market state $\mathbf{m^{(l)}_{t(c_{i})}}$ to obtain the
input vector $\mathbf{g}^{(l-1)}_{i}$ for the $l$-th layer of graph attention
networks. $\gamma_{ij}$ is the attention score between $v_{i}$ and $v_{j}$,
calculated by

$\gamma_{ij}=\frac{\exp\\{\operatorname{LeakyReLU}\\{[(\mathbf{v}_{i}\oplus\mathbf{v}_{j})\mathbf{w}_{1}^{T}](\mathbf{e}_{ij}\mathbf{w}_{2}^{T})\\}\\}}{\sum_{r\in\mathcal{N}(v_{i})}\exp\\{\operatorname{LeakyReLU}\\{[(\mathbf{v}_{i}\oplus\mathbf{v}_{r})\mathbf{w}_{1}^{T}](\mathbf{e}_{ir}\mathbf{w}_{2}^{T})\\}\\}}$
(16)

where $\mathbf{w}_{1}^{T}$ and $\mathbf{w}_{2}^{T}$ are trainable parameters.
Compared to the vanilla graph attention network, we introduce the edge feature
$\mathbf{e}_{ij}$ into the formula of attention score. Because the edge
features commonly measure the strength of connections, we map it to a scalar
coefficient to adjust the attention scores.

Finally, the company network encoder combines the market encoder and network
encoder. The constructed company network and the company embedding are input
into the company network encoder. For the $i$th-layer company network encoder,
it alternately uses a market encoder and a network encoder to update the
company node embedding. In fact, the company encoder can be seen as a model
that adds temporal virtual nodes to the company nodes and then uses extended
GRU and GAT to update node embedding.

####  3.2.4 Output Layer

We use a multilayer perceptron as the output layer to predict stock return
volatility:

$\hat{y}_{i}=\sigma(\mathbf{v}^{(L)}_{i}\mathbf{w}^{T}_{1}+b_{1})\mathbf{w}_{2}+b_{2}$
(17)

where $\mathbf{w}_{1},\mathbf{w}_{2},b_{1},b_{2}$ are trainable parameters,
$\sigma$ is an activation function.

##  4 Experimentation

###  4.1 Dataset

Because the datasets used in previous studies lack the information of
participant role, part segmentation, and company networks, which are required
in our proposed model, we construct a new dataset to verify TVGNN.

For earnings conference calls, we collect the transcripts of companies in S&P
500 from Seeking Alpha 222https://seekingalpha.com/, date ranges from 2008 to
2019. Because the S&P 500 constituent companies are changing, we use all
companies that have entered the index since its creation as target companies
for processing ease
333https://en.wikipedia.org/wiki/List_of_S&P_500_companies.

For the construction of the company network, we use the Text-based Network
Industry Classifications (TNIC) dataset
444https://hobergphillips.tuck.dartmouth.edu/industryclass.htm, which computes
company pairwise similarity scores based on the product descriptions in 10-K
files [29]. We add edges for two companies with a similarity score greater
than 0.15. Since the TNIC dataset is updated annually based on the latest 10-K
files, we use the TNIC dataset from the previous year to extract the company
relationships to avoid temporal information leakage. The statistics of the
constructed dataset are shown in Appendix A.

We split the dataset by time, using samples before 2016 as the training set,
samples from 2016 as the validation set, and samples after 2016 as the test
set. For a natural quarter, all samples are constructed as a company network,
, with each node labeled with the corresponding company’s stock return
volatility.

###  4.2 Baselines

In order to verify the effectiveness of TVGNN, we compare it with baselines
proposed in recent related studies. The baselines are shown below.

1) $v_{past}$[2]. $v_{past}$ is a simple but very effective benchmark model,
which directly uses the return volatility $v_{[t-\tau,t-1]}$ as the prediction
$v_{[t+1,t+\tau]}$, without using any other information.

2) HAN[30]. The model is a widely used long document encoder that employs two
BiGRU to capture contextual relationships at the word and sentence levels, as
well as a simple attention module to obtain encoding at each level. The
obtained document encoding is then used for downstream tasks.

3) ProFET[5]. This model simply considers the dialogue structure of earnings
conference calls and respectively models the presentation and Q&A part by a
BiLSTM and an attention module. We use the text modeling part of ProFET for
experiments.

4) MDRM[4]. The model employs a BiLSTM to extract textual and audio features
from earnings conference calls, which are then fused by another BiLSTM to
predict the company’s risk. We use the text modeling part of this model for
experiments.

5) HTML[19]. The model repectively uses a pre-trained language model to
extract word encoding and Praat to extract audio features. The features are
then fused to obtain a multimodal encoding at the sentence level. The model is
trained in a multi-task framework with an auxiliary task of predicting the
return of the company’s stock on a given day. We use the text modeling and the
multi-task part of the model for experiments.

6) MRQA[21]. The model uses a BiLSTM to encode the textual features of
earnings conference calls. It uses a reinforced sentence selector to select
important sentences in the Q&A, and a reinforced bidirectional attention
network to capture the interaction between questions and answers. The model
directly models the dialogue structure of earnings conference calls.

###  4.3 Evaluation Metrics

We use Mean Square Error (MSE) as the loss function and evaluation metric.

$\text{MSE}=\sum_{i}^{M}(\hat{y}_{i}-y_{i})^{2}/M$ (18)

The lower the MSE of a given model on the test set, the better the model. In
addition, we use the $R^{2}$ metric as another metric to reflect the
improvement of the model over $v_{past}$:

$R^{2}=1-\frac{\text{MSE}}{\text{MSE}_{v_{past}}}$ (19)

###  4.4 Experimental Setting

We use the Adam optimizer to optimize models [31]. The hyperparameters of
TVGNN are tuned on the validation mean square error (MSE) to get: learning
rate is 5e-4, weight decay is 1e-7, hidden state is 64, number of layers of
dialogue encoder is 2, number of attention heads of dialogue encoder is 8,
number of layers of company network is 3, and number of attention head of
company network encoder is 1. For the baselines, we use the hyperparameters
from the proposing paper. Training is stopped if the model’s MSE on the
validation set does not decrease in 10 epochs.

###  4.5 Experimental Results

Table 1: Experimental Results Model | $\overline{\text{MSE}}$ | $\text{MSE}_{3}$ | $\text{MSE}_{7}$ | $\text{MSE}_{15}$ | $R^{2}_{3}$ | $R^{2}_{7}$ | $R^{2}_{15}$
---|---|---|---|---|---|---|---
$v_{past}$ | 0.5843 | 1.1336 | 0.4026 | 0.2167 |  |  |
HAN | 0.6698 | 0.8911 | 0.6199 | 0.4983 | 0.2139 | -0.5397 | -1.2995
MDRM | 0.6135 | 0.7958 | 0.5560 | 0.4888 | 0.2980 | -0.3810 | -1.2557
ProFET | 0.5098 | 0.7527 | 0.4378 | 0.3390 | 0.3360 | -0.0874 | -0.5644
MRQA | 0.4637 | 0.7141 | 0.3796 | 0.2974 | 0.3701 | 0.0571 | -0.3724
HTML | 0.4156 | 0.6969 | 0.3097 | 0.2401 | 0.3852 | 0.2308 | -0.1080
TVGNN | 0.3605 | 0.5980 | 0.2818 | 0.2017 | 0.4725 | 0.3000 | 0.0692

Table 1 shows the result of the comparison experiment, we can see that TVGNN
performs best in the company risk prediction tasks for the three time windows.
Compared to the best baseline models, TVGNN respectively improves 14.19%,
9.01%, and 6.92% for $\tau=3,7,15$. On the overall performance
$\overline{\text{MSE}}$, TVGNN improves 13.26% at the basis of the best
baseline HTML, demonstrating the effectiveness of TVGNN.

Additionally, as the time window gets longer, the difference between $vpast$
and other baselines gets smaller. For $\tau=15$, all baselines are worse than
$v_{past}$, indicating that the additional effect of the earnings conference
calls transcripts gradually diminishes as the time window becomes longer,
which is consistent with the Earning Momentum [22]. The impact of a company’s
earnings information on its stock price fades over time. In our experiment,
the effect lasts no longer than 15 days.

##  5 Supplementary Analysis

###  5.1 Ablation Experiment

There are four modules in TVGNN: sentence encoder, dialogue encoder, market
encoder, and network encoder. To test the effectiveness of each module, we
gradually add a module and build different variants of TVGNN for the ablation
experiment.

Table 2: Ablation Experimental Result Variant |  | Sentence
---
Encoder
| Dialogue
---
Encoder
| Market
---
Encoder
| Network
---
Encoder
$\overline{\text{MSE}}$ | $\text{MSE}_{3}$ | $\text{MSE}_{7}$ | $\text{MSE}_{15}$
1 | + |  |  |  | 0.4162 | 0.6616 | 0.3354 | 0.2517
2 | + | + |  |  | 0.4116 | 0.6451 | 0.3193 | 0.2704
3 | + | + | + |  | 0.3700 | 0.6065 | 0.2773 | 0.2261
TVGNN | + | + | + | + | 0.3605 | 0.5980 | 0.2818 | 0.2017

The results are shown in Table 2, where the "+" in the table indicates that
the variant contains the corresponding module. As we can see:

1) Adding each module can improve the model’s performance in company risk
prediction, demonstrating the effectiveness of the four modules in TVGNN.

2) It is difficult for a model to improve on all three tasks without
introducing new data. When $tau=3$ and $tau=7$, variant 2 outperforms variant
1, but when $tau=15$, it outperforms variant 1. While variant 3 outperforms
variant 2 on all three tasks by incorporating company networks.

3) Modeling the market state can improve the model’s performance on the task
$\tau=15$. TVGNN with the market encoder outperforms variant 3 on $tau=15$.

###  5.2 Transductive Learning

While TVGNN performs well on the company risk prediction task, we have to
divide the samples by quarter for training and prediction, which is extremely
restrictive in the real application. To better use our model in practice, we
propose two transductive learning approaches:

1) TVGNN-T: We divide the nodes in a company’s network into training set,
validation set, and test set in chronological order, in the ratio of 7:1:2.
The constructed dataset is then used for training and prediction.

2) TVGNN-T(fine-tune): We pre-train TVGNN on samples from the previous period
and then fine-tune the pre-trained model on the dataset constructed in 1).

The two transductive learning approaches are tested on the 2017 Q1 company
network, and the results are shown in Table 3. We also compare their
performance with TVGNN (The pre-trained model in 2, without fin-tune).

Table 3: Transductive Learning Model | $\overline{\text{MSE}}$ | $\text{MSE}_{3}$ | $\text{MSE}_{7}$ | $\text{MSE}_{15}$
---|---|---|---|---
TVGNN | 0.3819 | 0.5694 | 0.3103 | 0.2659
TVGNN-T | 0.5847 | 0.7721 | 0.5962 | 0.3857
TVGNN-T (fine-tune) | 0.3708 | 0.5586 | 0.2961 | 0.2576

We can see that TVGNN-T performs poorly because it only trains on a small
dataset, whereas TVGNN and TVGNN-T (fine-tune) benefit from the large pre-
training dataset. Furthermore, TVGNN-T (fine-tune) outperforms TVGNN for it
can obtain helpful information about the company network and market state for
the given quarter.

###  5.3 Case Study

It is a requirement for models applied in finance to be interpretable because
the field’s decisions involve millions of dollars. In this section, we use
GNNExplainer to understand the TVGNN predictions. GNNExplainer can identify
the important subgraph that influences the prediction of a given GNN model
/cite ying2019gnnexplainer. Oracle, whose cloud business soared and increased
its dividend in 2017 Q1, was our study case.

Figure 3: Case Study Table 4: Selected Companies for Case Study Symbol | Security. | GICS Sub-Industry | $v_{\tau=3}$
---|---|---|---
ORCL | Oracle | Application Software | -3.2365
CTXS | Citrix | Application Software | -3.1539
FFIV | F5 | Communications Equipment | -2.9807
FTNT | Fortinet | Systems Software | -2.5418
JNPR | Juniper Networks | Communications Equipment | -3.6766
CSCO | Cisco | Communications Equipment | -4.6155
HPE | Hewlett Packard Enterprise | Technology Hardware, Storage & Peripherals | -3.0791
ANSS | Ansys | Application Software | -4.1327
PTC | PTC | Application Software | -3.7511
MSFT | Microsoft | Systems Software | -5.3507
CRM | Salesforce | Application Software | -3.7586

The important 1-hop subgraph of ORCL (Oracle) computed by GNNExplainer is
shown in Figure 3. The details of the nodes are shown in Table 4. The widths
of the edges in the subgraph represent the edge’s important score. Despite the
fact that all of the companies in the subgraph are in the information
technology sector, not all of them have a significant impact on the risk of
other companies. TVGNN’s prediction $\hat{y}_{\tau=3}$ for ORCL (Cisco) is
mainly influenced by CTXS, HPE, CSCO, FTNT, and FFIV. All five companies are
Oracle partners, and they all sell Oracle products in their cloud services.
Furthermore, each node in the graph has a self-loop, which represents the
impact of the company’s event, i.e., the earnings conference call, on the risk
of the company. We can see that the risk of JNPR (Juniper Networks) and FTNT
(Fortinet) is influenced more by their earnings conference calls, whereas the
rest is influenced more by other companies in the company network. That is
because JNPR and FTNT released some shocking financial information during
their earnings conference calls. JNPR posted a disappointing profit outlook,
causing its stock to plunge 7.5% after the earnings conference call. FTNT’s
stock, on the other hand, jumped 10.8% after its earnings conference call, as
its performance in 2016 gave investors confidence in its 2017 performance.

##  6 Conclusion

In this paper, we propose a model called TVGNN for company risk prediction by
jointly modeling company networks and earnings conference calls. We make a
methodological contribution by designing a new method to construct company
networks and developing a new model based on graph neural networks to model
earnings conference calls, market state, and company networks. Empirical
results on our constructed dataset demonstrate the superiority of our proposed
model over competitive baselines from the extant literature. We also conduct
supplementary analysis to examine the effects of our model’s modules and
interpretability.

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##  Appendix A Dataset Statistics

The quarterly statistics of the constructed dataset are shown in Table
LABEL:tab:dataset.

Table 5: Dataset Statistics Quarterly | Average #Utterance | Average #Sentences | #Nodes (Companies) | #Edges
---|---|---|---|---
2008Q1 | 95.90 | 162.63 | 559 | 4159
2008Q2 | 87.87 | 148.37 | 566 | 4535
2008Q3 | 88.81 | 156.59 | 502 | 3239
2008Q4 | 87.08 | 166.66 | 553 | 4238
2009Q1 | 86.56 | 165.87 | 570 | 4768
2009Q2 | 81.68 | 159.23 | 514 | 4167
2009Q3 | 78.69 | 155.77 | 515 | 4124
2009Q4 | 79.07 | 153.47 | 481 | 3940
2010Q1 | 78.19 | 149.72 | 511 | 3965
2010Q2 | 78.09 | 139.39 | 466 | 3613
2010Q3 | 79.00 | 138.47 | 378 | 2453
2010Q4 | 76.48 | 135.69 | 362 | 2149
2011Q1 | 74.39 | 126.01 | 479 | 3922
2011Q2 | 76.27 | 121.81 | 478 | 3960
2011Q3 | 73.53 | 119.81 | 575 | 5131
2011Q4 | 73.34 | 120.33 | 572 | 5166
2012Q1 | 73.34 | 127.89 | 578 | 5160
2012Q2 | 69.17 | 120.75 | 577 | 5058
2012Q3 | 72.56 | 123.17 | 596 | 5459
2012Q4 | 69.11 | 125.97 | 579 | 4980
2013Q1 | 58.89 | 108.21 | 607 | 5534
2013Q2 | 61.78 | 112.30 | 631 | 5643
2013Q3 | 60.71 | 110.79 | 627 | 5760
2013Q4 | 63.40 | 115.59 | 622 | 5634
2014Q1 | 68.05 | 125.27 | 624 | 5610
2014Q2 | 69.88 | 123.60 | 615 | 5581
2014Q3 | 69.78 | 125.75 | 617 | 5304
2014Q4 | 68.88 | 126.05 | 608 | 5205
2015Q1 | 73.38 | 138.32 | 614 | 5379
2015Q2 | 74.42 | 137.20 | 593 | 5170
2015Q3 | 78.00 | 138.48 | 615 | 5600
2015Q4 | 77.58 | 140.49 | 593 | 4996
2016Q1 | 78.03 | 147.08 | 590 | 5162
2016Q2 | 77.02 | 137.34 | 589 | 5330
2016Q3 | 77.22 | 139.42 | 566 | 4825
2016Q4 | 76.18 | 139.45 | 581 | 5115
2017Q1 | 77.79 | 144.41 | 594 | 4669
2017Q2 | 75.00 | 135.30 | 578 | 4587
2017Q3 | 71.62 | 133.09 | 569 | 4597
2017Q4 | 73.36 | 139.12 | 561 | 4457
2018Q1 | 68.28 | 139.41 | 587 | 4583
2018Q2 | 71.29 | 134.49 | 588 | 4695
2018Q3 | 69.55 | 131.77 | 572 | 4454
2018Q4 | 70.60 | 134.98 | 573 | 4450
2019Q1 | 67.51 | 147.38 | 555 | 4017
2019Q2 | 66.14 | 141.53 | 537 | 4032
2019Q3 | 66.16 | 145.75 | 497 | 3277
2019Q4 | 64.15 | 143.60 | 493 | 3475

The label distribution on the training set, validation set, and test set is
shown in Figure 4. We can see that the distribution of labels in different
splits is consistent.

tp

(a) $\tau=3$

(b) $\tau=7$

(c) $\tau=15$

Figure 4: The distribution of labels.