A New Study on Tri-Lindelöf Spaces
HAMZA QOQAZEH1, ALI A. ATOOM2, ALI JARADAT1, EMAN ALMUHUR3,
NABEELA ABU-ALKISHIK4
1Department of Mathematics, Faculty of Arts and Science,
Amman Arab University,
P.O.Box 24 Amman, 11953,
JORDAN
2Department of Mathematics, Faculty of Science,
Ajloun National University,
P.O.Box 43, Ajloun, 26810,
JORDAN
3Department of Basic Science and Humanities,
Faculty of Arts and Science,
Applied Science Private University,
P.O.Box 541350, Amman, 11937,
JORDAN
4Department of Mathematics,
Faculty of Science,
Jerash University,
P.O.Box 2600, Jerash, 21220,
JORDAN
Abstract: - This paper defines new covering properties in tri-topological spaces called tri-Lindelöf space and
the properties of this topological property and its relationship with some other types of tri-topological spaces
will be studied. The effect of some types of functions on tri-Lindelöf spaces will be studied. This paper also
investigates the necessary conditions through which the tri-topological space is reduced into a single
topological space. Many and varied illustrative examples will be discussed and many well-known facts and
theorems are generalized concerning Lindelöf spaces.
Key-Words: - Topological space, Tri-topological space, Tri-Lindelöf space, S- compact space, T- compact
space, C- compact space, S- Lindelöf space, T- Lindelöf space, C- Lindelöf space.
Received: March 1, 2023. Revised: October 27, 2023. Accepted: November 9, 2023. Published: November 30, 2023.
1 Introduction
The study of Tri-topological spaces is considered a
generalization of the same study in bi-topological
spaces, which in turn was a generalization of single
topological spaces. This study is based on choosing
any non-empty set with three topologies
defined on which is called the tri-
topological space denoted by
. To
clarify what we started in our introduction, we
present the following: The first person to discuss the
idea of bi-topological spaces was, [1], who built this
idea on a non-empty set defined by two
topologies
and
, and this was denoted by
.He then proceeded to generalize
the definition of a number of topological spaces to
bi-topological spaces, including Hausdörff, regular
and normal spaces. This idea gained the attention of
researchers of that era, and their efforts led to the
definition of pairwise compact spaces with a
comprehensive study of their features by scientists,
[2], [3], and, [4]. In the 1980s, specifically in 1983,
a study was conducted that focused on Lindelöff’s
theories in bi-topological spaces, during which the
two scientists, [5], reached distinctive results and
generalizations that led to the development and
confirmation of many topological concepts in this
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field, while supporting those concepts with
illustrative examples that illustrate the relationship
between those concepts, [6], and, [7]. This is a
simple summary of the concepts, research, and
studies in bi-topological spaces that formed a
research base for researchers later to generalize
them in tritopological spaces and elsewhere. It
will also be presented in our introduction to the
topic of the study. The study and discussion of the
basic principles of the tri-topological spaces began
in the year 2000 by, [8], where many important
separation axioms were defined in tri-topological
spaces, and those concepts were clarified by setting
various and distinctive examples. These studies
encourage many researchers to conduct numerous
studies in that field. In 2011 new types of separation
axioms were defined in tri-topological spaces based
on the open sets in those spaces called
and spaces, [9]. These
concepts paved the way for their definitions of some
types of separation axioms in trispaces that do not
depend on open sets, in those spaces, but rather on
other types of open sets in tri-topological spaces,
where three new types of separation axioms define
bopen sets and called them
and spaces, to benefit more, [9]. Also, In
2011, the doctoral dissertation submitted by the
researcher, [10], includes in its main topic a detailed
study of the tri-topological spaces and discussed
them. Many results were reached, the most
important of which is the development of definitions
of the tri continuous functions and
tri continuous functions. Also In the same year
2011, [11], introduced the concept of continuous
functions with a detailed study of their properties in
tritopological spaces. Subsequent to the above and
in 2016, [12], a new study was conducted aimed at
defining two types of open sets in tritopological
spaces: semi-open sets and pre-open sets among
their basic properties. Two important concepts of
continuous functions with many associated
properties were also discussed namely:
trisemicontinuous and tripricontinuous
functions. In 2017, [13], studied the main
topological characteristics of tri open sets and
tribclosed sets and used them to define
tri continuous functions. They studied the
properties of these functions and their nature and
connected them to other types of functions of
tritopological spaces. Distinguished scientific
efforts In 2017, we summarize as follows: a new
type of open sets has been discussed in
tritopological spaces by the researchers, [14],
dubbed it open sets where properties that they
studied. It was used in defining one of the types of
continuous functions associated with it called
functions and obtained some of their
properties. The efforts of the researchers continued
in the field of research in the tritopological spaces
so in 2017, a scientific paper was presented entitled
"Soft tritopological spaces" through the
researcher, [15], in which good results were reached
and generalizations were issued. For more studies
on this subject note the following researcher's
references, [16], and, [17]. In 2018, [18], give the
concept of fuzzy connectedness as a new study in
tritopological spaces. They also discussed several
separation properties in this topic. In 2019, [19],
presented an important study that discussed the
main basics of the subject of (fuzzy soft
tritopological space) and drew a number of
conclusions and recommendations. In 2021, [20], a
new topic has been addressed in tri-topological
spaces based on tri closed sets and closed
set which is tri continuous functions. Also,
during this research, the following topics were
discussed which are: quasi-tri-, perfectly tri and
tri continuous functions. It is worth noting that
qualitative conclusions were reached in this work.
The topic of neutrosophic Tritopological space
was studied in 2021 by, [21]. This study aimed to
generalize the topic of neutrosophic topological
spaces and several results and characteristics related
to this topic. Further, different types of open and
closed sets were defined in the neutrosophic
trispaces and their different properties were
compared and discussed. Also, during the year
2021, an important study was conducted entitled
"neutrosophic soft tritopological spaces" that
highlighted the discussion of a variety of topological
characteristics in "neutrosophic soft topological
spaces" and its generalization in tritopological
spaces. The following sets have been defined and
their basic characteristics studied through this study:
neutrosophic soft triopen and tri-closed sets, [22].
In 2022, [23], studied the concepts of supra
tritopological spaces and gave the definitions of
some open sets such as topen sets and supra
trisemipre open sets in supra tritopological
spaces. In 2023, fuzzy soft tripreopen sets were
investigated and studied by, [24]. This study
focused mostly on studying the basic properties of
those sets. In this paper, First, a new definition of
open covers in tri-topological spaces and their types
was presented, where these types were used to
define several types of tritopological spaces such
as Scompact spaces, Tcompact spaces,
Ccompact spaces, S Lindelöf spaces,
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TLindelöf spaces, C Lindelöf spaces,
Scountably compact spaces, T countably
compact spaces and Ccountably compact spaces.
second, The relationship between these concepts
has been studied and many illustrative examples of
these concepts have been developed. these examples
also show the relationship between these concepts.
Third, many theories were proved and introduced
to these new concepts, the most important of which
was the study of the effect of some types of
functions on them, and a number of results and
characteristics were reached on this subject.
Fourth and finally, the necessary and sufficient
conditions that are required to reduce tri-topological
spaces into a single topological space have been
studied.
In this paper, we use the following letters: the
letters closure, interior of a set will be
denoted by respectively. The
product of
and
will be denoted by
.
Let denote the set of all ( real,
integer, natural, and rational ) numbers,
respectively. Let
denote the ( discrete, indiscrete, usual, Sorgenfrey
line, cocountable, cofinite, left-ray, and right-ray )
topologies, respectively.
2 TriLindlöf Spaces
In this section, we will discuss the concept of
triLindlöf topological spaces and extract some of
their characteristics and their relationship to several
other types of tritopological spaces with
illustrative examples for each case. Also, many
theories on this subject will be discussed and
proven.
Definition 2.1. A cover of a tritopological
space
is called
open if
. If in addition, contains at
least one non-empty member of each
,
then is called
open cover.
Definition 2.2. (1) A tritopological space
is called Scompact if every
open the cover of the space
has a finite subcover.
(2) A tritopological space
is
called Tompact if every
open the cover of
the space
has a finite subcover.
(3) A tritopological space
is said
to be C-compact if every
open cover of
, has a
finite subcover,
and
finite subcover , where
.
Example 2.1 The tri-topological space
is Tcompact, since every
open cover for must contain . So
is a finite subcover of for . But
is not Scompact since
is a open cover of , so it is
a open cover which has no finite
subcover.
Definition 2.3 A tritopological space
is called SLindelöf if every
open the cover of the space
has a countable subcover.
(2) A tritopological space
is
called TLindelöf if every
open the cover of
the space
has a countable subcover.
(3) A tritopological space
is said
to be CLindelöf if every
open cover of
has a
countable subcover,
and
countable subcover
, where .
Example 2.2 The tritopological space
is T Lindelöf.
Example 2.3 The tritopological space
is SLindelöf and TLindelöf
space.
Example 2.4 is not C
Lindelöf.
Example 2.5 is C
Lindelöf.
Theorem 2.1 A tritopological space
is SLindelöf iff is
Lindelöf space, where is the least
upper bound topology of τ¹,τ²and τ³.
Proof: Assume that is Lindelöf space, let
be a open cover of
, then
. But
is lindelöf space, so has a countable subcover.
Remark 2.1 It is clear that if
is
C Lindelöf then each of
and
is Lindelöf.
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Remark 2.2 A tritopological space
has a particular topological
property if , and have this property. For
instance,
is said to be space if
all
and
are spaces.
Example 2.6 The tritopological space
is space.
Theorem 2.2 A tritopological space
is Slindelöf iff it is lindelöf and
Tlindelöf.
Proof: Assume that
is S lindelöf.
Since any Topen or
open or
open or
open cover of
is a
open cover, then the result is as
follows.
Conversely, if a
open cover of
is not T-open, then it is
open
or
open or
open cover of
. Since
is Lindelöf,
then the result is as follows.
Example 2.7 The tritopological space
is Lindelöf. So it is
Lindelöf and Lindelöf. But not Lindelöf, since
the open cover of has no
open countable subcover.
Example 2.8 Let
let and be the
topologies defined on , which are generated by
the bases and , respectively. Then
is CLindelöf. Since any open
cover of or any open cover of or any
open cover of must contain , so
is a countable subcover of any open cover
of , . It is clear that is not
Lindelöf, for the open cover
has no countable subcover, also,
is Lindelöf, but not SLindelöf since it is not
Lindelöf.
Example 2.9 The tritopological space
is TLindelöf but not
SLindelöf.
Example 2.10 The tritopological space
is TLindelöf but not
SLindelöf. because is not Lindelöf.
Example 2.11 Consider the three typologies
and on , generated by the basis
and ,
respectively. Then is T Lindelöf but
not Lindelöf. It is clear that is not
Slindelöf, since the open cover
of which exactly a open cover but it has
no countable subcover.
Definition 2.4 (1) A tritopological space
is called Tcountably compact if
every countably
open cover of the space
has a finite subcover.
(2) A tritopological space
is
called Scountably compact if every countably
open cover of the space
has a finite subcover.
(3) A tri-topological space
is called
Ccountably compact if every countably
open cover of the space
, ,
has a finite
open subcover , and a
finite
open subcover .
Example 2.12 The tritopological space
is TLindelöf space which is
neither Tcountably compact nor Tcompact.
Example 2.13 Let ,
and let and be the topologies on
which are generated by and
respectively, then is C Lindelöf;
since for any open cover of or open
cover of or open cover of must contains
as a member. So is a countable subcover of
each open cover, . However,
is not TLindelöf, for the
open cover of has no countable
subcover. It is clear that is neither
Tcompact nor Tcountably compact. Also,
is Ccompact and C-countably
compact. We can see also, it is not S-
Lindelöf so not S-compact.
It is easy to prove the following theorem
Theorem 2.3 (1) Every T(respectively S, C)
compact space is T(respectively S, C) countably
compact and T(respectively S, C) Lindelöf space.
(2) Every T(respectively S, C) countably compact
T(respectively S, C) Lindelöf is T(respectively
S, C) compact.
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Proof (1) Assume that
is a
Tcompact space. Let be a
open cover
of then has a finite subcover , so is a
countable subcover for .
The remaining parts in (1) are intelligibly evident in
a similar fashion.
(2) Assume that
is a Tcountably
compact, TLindelöf space. Let be a
open
cover of . Since
is TLindelöf,
has a countable subcover . Since
is a Tcountably compact then
has a finite subcover for .
The remaining parts in (2) are intelligibly evident in
a similar fashion.
Theorem 2.4 If a tritopological space
is a hereditary Lindelöf space, then
it is SLindelöf.
Proof Let
be a
an open cover of a
non-empty set , where
and
. Since
is a hereditary Lindelöf, then
is
Lindelöf, then there exists a
countable set
such that
Similarly,
is
Lindelöf, then there exists a countable set
such that
. Also,
is
Lindelöf, then there exists a
countable set
such that
. Now,
is a countable subcover of for . Hence
is S- Lindelöf.
Corollary 2.1 Every second countable tri-
topological space
is S- Lindelöf.
Proof Since every second countable space is
hereditary Lindelöf. Hence the result.
Theorem 2.5 A
closed proper subset of a S
Lindelöf space
is
Lindelöf
and
Lindelöf, where
.
Proof Let be a nonempty
closed proper
subset of and let be
open
cover t of such that
is a
countable subcover of for W. Since
, then for all then
and then is a
countable subcover for , hence is a
Lindelöf. Similarly, we can prove that every
closed proper subset of is
Lindelöf.of
. Since
open, then for each
there exists a
open set such that
. Since and ,
then is a
an open cover of hence it is a
an open cover of an SLindelöf space . Thus
there exists a countable set of and a countable
set.
Corollary 2.2 A
closed proper subset of an S-
compact space
is
compact
and
compact, where
Definition 2.7 A family
of nonvoid
subsets of a tritopological space
is said to be
closed if every member
of is
closed or
closed or
closed
for all . If contains members
such that is
closed and is
closed
and is
closed, then it is called Tclosed.
Theorem 2.6 A tri topological space
is Lindelöf if and only if every
closed family with countable intersection
property has a nonempty intersection.
Proof Let
be a family of Tclosed proper subset of
with countable intersection property. Suppose that
. Now,
is a
open
cover of because
. But is
TLindelöf so has a countable subcover
Hence i.e
, which is
a contradiction. Hence .
Conversely; assume that is not TLindelöf. Let
be a
an open cover of which has no countable
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subcover. Now
is a Tclosed
family which has the countable intersection
property; by assumption i.e.
.
Which is a contradiction since .
Theorem 2.7 A tri-topological space
is SLindelöf space if and only if
every
closed family with countable
intersection property has a nonempty intersection.
Proof The proof is similar to the last theorem.
Theorem 2.8 Let
be a
Ccompact and
be a CLindelöf.
Then
is
CLindelöf.
Definition 2.6 A tritopological space
is said to be THausdörff if for
any two distinct points and in there exists a
open set of w and
open set of
such that , where .
Remark 2.3 If a tritopological space
is THausdörff, then
and
are spaces.
Theorem 2.9 If a tritopological space
is THausdörff, then for each
we have is a
neighbourhood of
Proof Assume that
is
THausdörff space. Let such that
and
. Since
is THausdörff, then there is a
open set of
and a
open set of such that
. Hence , then ,
but , then so
which is a contradiction. Hence
Recall that: A topological space
is
said to be P-space if the intersection of any
countable number of open sets is open.
Theorem 2.10 If
is a THousdörf
and Pspace, then any
Lindelöf proper subset
of is
closed .
Proof Let be a
Lindelöf proper subset of
and , then by theorem [2.9] we have
is a
. Since
, then
, therefore, is
a
open cover of so there exists a
countable subset subset of such that
is a
open cover of . Let
, then is a
open set such that
and . Hence ,
this implies that is a
closed
.
Corollary 2.3 If
is THausdörff ,
then every
compact subset of is
closed .
Proof Using the same technique as the above
theorem.
Definition 2.7 A function
is said to be Tcontinuous (Topen,
Tclosed, Thomomorphism, respectively) if
,
and
are continuous (open, closed,
homomorphism, respectively).
Theorem 2.11 Let
be a Tcontinuous onto function,
then
(1) If
is TLindelöf ( SLindelöf,
CLindelöf ,respectively), then
is
TLindelöf ( SLindelöf, CLindelöf
,respectively).
(2) If
is T-compact( Scompact,
Ccompact,respectively), then
is T-
compact( Scompact, Ccompact,respectively).
Proof
(1) Let
be a
open cover of where
and
. Since is
Tcontinuous and onto, then
open cover of . Since
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is TLindelöf, then there exists a
countable subset of such that
is a
countable sub-cover of for , So
is a countable
subcover of for . Hence is TLindelöf. the
remaining part of (1) Is easy and similarly proved.
(2) The proof is similar to that in the statement (1).
Theorem 2.12 Let
a be T-continuous onto the map, then
(1) If is one-to-one,
is THousdörf,
Pspace and
is SLindelöf, then
is Thomomorphism.
(2) If is one-to-one,
is THousdörf
and
is Scompact, then is
Thomomorphism.
Proof
(1) Since is continuous, onto and one-to-one
map, it is sufficient to show that is closed. Let
be a
closed proper subset of a Lindelöf
space
.Then, by theorem [2.5] is
Lindelöf, where , Hence
is
Lindelöf because
is continuous. By theorem [2.10] is
closed, where . Hence
is closed for each ,
so is Tclosed.
(2) The proof is similar to that in the statement (1).
3 Reduce a tri-Topological Space to a
Single Topology
In this section, we will introduce and discuss the
necessary conditions so that the tri-topological space
is reduced to a single topology.
Dear reader, here are some facts that will be used
in this section.
(1) Every Lindelöf subset of a Hausdörff P-space
is closed.
(2) Every compact subset of a Hausdörff space is
closed.
Theorem 3.1 Let
be a Hausdörff,
SLindelöf, Pspace, then
.
Proof First: we show that
. Let
. Then is
closed proper subset of a
SLindelöf space . By theorem [2.5] is
Lindelöf. The fact " every
Lindelöf
subset of a
Hausdörff Pspace is
closed"
Thus is
closed. So,
. Hence
. Similarly, we can show that
.
Using the same technique we can show that
and
.
Theorem 3.2 Let
be a Hausdörff,
Scompact space, then
.
Proof First: we show that
. Let
. Then is
closed proper subset of a
Scompact space . By corollary [2.2] is
compact. The fact " every
compact
subset of a
Housdörf space is
closed"
Thus is
closed. So,
. Hence
. Similarly, we can show that
.
Using the same technique we can show that
and
.
Theorem 3.3 Let
be a Lindelöf
THousdörf, Pspace, then
.
Proof Let
. Then is
closed
proper subset of a Lindelöf space
so
is
Lindelöf. By theorem
is
closed ; .Hence
u is
open. This implies that
. Similarly,
we can show that
where
. Thus;
.
Corollary 3.1 Let
be a compact
THousdörf, space, then
.
Proof Let
. Then is
closed
proper subset of a compact space
so
is
compact. By corollary [2.3] is
closed; where . Hence u is
open. This implies that
. Similarly, we
can show that
where .
Thus;
.
Lemma 3.1 Let
be a Scompact
space and be a
closed set such that
, then is
compact
Proof Let , then there exists a
open set such that . Let
be a
open cover for . For each
there exists a
open set such
that . Since
is a
open
cover of a Scompact space , then there exists a
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finite subsets and
such that
is a cover of .
Since for all and , then
. Thus is a
finite subcover of for . Hence is
compact.
Theorem 3.4 Let
be a
THausdörff S-compact space. If for
all
, then
where
.
Proof Let
, then is
closed with . By the above
lemma, W- is
compact. By corollary [2.3]
is
closed. So
. Thus
.
Corollary 3.2 Let
be a
Hausdörff S-compact space. If for
all
and for all
, then
where
.
4 Results
Among the most important main and general results
of this research are the following :
1. A definition of open covers in tri-topological
spaces has been developed.
2. The definition of various concepts in singular or
bi-topological spaces was generalized and studied in
tri-topological spaces, with various examples. The
topological relationships between those concepts
were also clarified.
3. The main subject of the study is a generalization
of scientific results, theories, and facts in Lindelöf’s
topological spaces, whether singular or pairwise.
4. A generalization of known results and theories in
Lindelöf spaces and pairwise Lindelöf spaces in tri-
topological spaces.
5. The necessary conditions for transforming the tri-
topological space into a single topological space
have been studied and developed.
6. The effect and characteristics of some types of
functions on tri-Lindlöf spaces were studied.
5 Conclusions
Dear reader, as we have noticed in our study
through the introduction and through the content of
this research, this study is an extension of what the
previous studies have reached in the field of tri-
topological spaces in their various subjects and this
is evident in the introduction to this research. Also,
all the results that we obtained are generalizations of
theories and known results on the main subject of
the study, which is Lindelöf topological spaces.
Here are the most important points of the results of
this research work:
1. Formulating the following definitions in the tri-
topological spaces: tri-Lindlöf spaces, tri-compact
spaces, and tri-countably compact spaces. Also, we
clarify the relationship between them.
2. Illustrative examples were provided for all the
concepts and theories contained in the research,
which clarify the rationale of the theories or show
the relationship between these concepts.
3. Special types of functions were defined in tri-
spaces and their impact on the concepts of the study,
especially its main subject was studied.
4. The necessary and sufficient conditions were
studied in the tri-topological spaces
through which it is reduced into a
single space so that
.
6 Recommendations and Possible
Future Studies
1. Researchers recommended the need to study the
covering properties of the various tri-topological
spaces in order to reach further relationships
between them, especially Tri-compact and Tri-
countably compact spaces.
2. This study can be applied in quadrilateral,
pentagonal, and other topological spaces in a
manner similar to what was achieved in this study or
other studies in the same field.
Acknowledgment:
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.100
Hamza Qoqazeh, Ali A. Atoom,
Ali Jaradat, Eman Almuhur, Nabeela Abu-Alkishik
E-ISSN: 2224-2880
922
Volume 22, 2023
The researchers express their gratitude to faculty
members who participated in conducting this
research and enhanced its scientific quality. Many
thanks and respect are due to all esteemed scholars
whose scientific contributions have been cited.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.100
Hamza Qoqazeh, Ali A. Atoom,
Ali Jaradat, Eman Almuhur, Nabeela Abu-Alkishik
E-ISSN: 2224-2880
923
Volume 22, 2023
Contribution of individual authors to the
creation of a scientific article (ghostwriting
policy)
- Hamza Qoqazeh: presented the main idea of the
research, provided the basic definition of the
research, and enriched it with several theories and
illustrative examples.
- Ali A.Atoom: wrote the introduction of the
research and added some theories and scientific
facts to the research.
- Ali Jaradat: wrote the introduction of the research
and added some theories and scientific facts to the
research.
- Eman Almuhur: conducted scientific and
linguistic scrutiny of the research, as well as
reviewed and verified the academic references.
- Nabeela Abu-Alkishik: conducted scientific and
linguistic scrutiny of the research, as well as
reviewed and verified the academic references.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interests
The researchers state that no personal objectives are
gained through publishing this paper and confirm
the originality of the work. The main objective of
the current paper is to contribute to scientific
research in the field of general topology. Also, all
researchers acknowledge and attest that this research
is not taken from any other source and is not
published, sent for publication, or accepted for
publication in any journal.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.100
Hamza Qoqazeh, Ali A. Atoom,
Ali Jaradat, Eman Almuhur, Nabeela Abu-Alkishik
E-ISSN: 2224-2880
924
Volume 22, 2023
