الفهرس 
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Abstract
Topology is one of the mathematical branches which studies properties of the
spaces that are invariant under any deformation. It is sometimes called \rubbersheet
geometry” because of the objects can be stretched and contracted like rubber,
but cannot be broken. For example, a square can be deformed into a circle
without breaking it. Hence, a square is topologically equivalent to a circle. A
topological space is a set endowed with a structure, called a topology, which allows
de ning continuous deformation of subspaces, and all kinds of continuity.
Euclidean spaces and metric spaces are examples of a topological space. The deformations
that are considered in topology are homeomorphisms. A property that
is invariant under such deformations is a topological property. Basic examples of
topological properties are: the dimension, compactness, connectedness and separation
axioms. Topological concepts have been powerful method and useful tools
to study computer science, information systems, rough multisets, rough topology
(nano topology), soft topology, multi topology, etc.
Rough set theory had been proposed by Pawlak [86] in the early of 1982. The
theory is a new mathematical tool to deal with vagueness and imperfect knowledge
by using the concept of the lower and upper approximations [86]. If the
lower and upper approximations of the set are equal to each other, then it is
called crisp (exact) set, otherwise known as rough (inexact) set. Therefore, the
boundary region is de ned as the di erence between the upper and lower approximations,
and then the accuracy of the set or ambiguous [69] depending on
the boundary region is empty or not respectively. A non-empty boundary region
of a set means that our knowledge about the set is not sucient to de ne
the set precisely. The main aim of rough sets is reducing the boundary region
by increasing the lower approximation and decreasing the upper approximation.
Rough set theory has achieved a large amount of applications for various real-life
elds, like economics, medical diagnosis, biochemistry, environmental science, biology,
chemistry, psychology, con
ict analysis, medicine, pharmacology, banking,
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market research, engineering, speech recognition, material science, information
analysis, data analysis, data mining, linguistics, networking and other elds can
be found in [57, 85, 127]. In the classical rough set model approximations are
based on equivalence relations, but this condition does not always hold in many
practical problems and also this restriction limits the wide applications of this
theory. In recent times, lots of researchers are interested to generalize this theory
in many elds of applications [76, 85, 96, 97, 98, 124, 127].
Various new problems in rough sets have been introduced via ideal. In recent
years, there has been a fast growing interest in applying the notion of ideals in
rough sets theory. Jankovic and Hamlet [54] de ned the concept of ideal. An ideal
is a nonempty collection of sets which is closed under hereditary property and the
nite union [54]. Ideal is a fundamental concept in the topological spaces and
plays an important role in the study of topological problems. The study of ideal
on the topological space is not a new concept today. It was studied from twentieth
century and it is studied till today. Kuratowski [70] and Vaidyanathaswamy [112]
were the rst who studied the notion of the ideal topological spaces. After them,
di erent mathematicians applied the concept of ideals in topological spaces (see:
[10, 28, 46, 54, 64, 81]). The interest in the idealized version of many general
topological properties has grown drastically in the past 20 years. After the advent
of the concept of ideals, several research manuscripts [49, 50, 59, 64, 99] appeared
in the context of ideals and rough set theory.
Multiset theory was established in 1986 by Yager [116]. A multiset is considered
to be the generalization of a classical set. In classical set theory, a set is a wellde
ned collection of distinct objects. It states that a given element can appear
only once in a set without repetition. So, the only possible relation between two
mathematical objects is either they are equal or they are di erent. The situation
in science and in ordinary life is not like this. If the repetitions of any object
are allowed in a set, then a mathematical structure, that is known as multiset
(mset [17] or bag [116], for short), is obtained in [19, 41, 94, 95]. For the sake of
convenience an mset is written as fk1=x1; k2=x2; :::; kn=xng in which the element
xi occurs ki times. The number of occurrences of an object x in an mset A,
which is nite in most of the studies that involve msets, is called its multiplicity
or characteristic value, usually denoted by mA(x) or CA(x) or simply by A(x). It
should be noted that each multiplicity ki is a positive integer.
In 2011 and 2012, Girish and John [[41][43]] introduced rough mset in terms
of lower and upper approximations. Moreover, they used an mset topological
concept to investigate Pawlak’s rough set theory by replacing its universe by
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mset. It is worth to be mentioned that, Zakaria et al. [123] presented the notion
of multiset ideals and the fundamental properties of this notion were also studied.
In addition to, they used this notion and suggested the concept of rough multisets
via multisets ideals as generalization of rough multisets.
In 2002, Pawlak [87] discussed the basic concepts and the applications of rough
set theory with a simple example concerning a churn modeling in telecommunications.
Based on this theory, Thivagar et al. [105] de ned a new topology called
rough topology in terms of approximations and boundary region of a universal set
using equivalence relation on it. Thereafter, Thivagar and Richard [102] named
this topology as nano topology. from this time, the nano topological space became
a new type of modern topology in terms of rough sets. The elements of a
nano topological space are called the nano open sets. But certain nano terms are
satis ed simply to mean \very small”. It originates from the Greek word \Nanos”
which means \dwarf” in its modern scienti c sense, an order to magnitude one
billionth of something. Nano car is an example. The nano topology is named due
to its size, because it has at most ve elements in it. They de ned also nano closed
sets, nano interior and nano closure. Moreover, they introduced the weak forms
of nano open sets namely nano -open sets, nano semi-open sets, nano pre-open
sets and nano regular open sets. After the work of Thivagar and Richard [102] on
nano open and closed sets, various mathematicians turned their attention to the
generalizations of these sets. Generalized nano open and closed sets play a very
important role in nano topology and they are now the research topics of many
topologists worldwide. In this direction, Bhuvaneswari and Mythili Gnanapriya
[15] introduced and investigated the properties of nano generalized closed sets in
nano topological spaces which were the extension of nano closed sets. Since the
advent of these previous notions, several research papers with interesting results
in di erent respects came to existence [48, 90, 103]. The classical nano topological
space is based on an equivalence relation on a set, but in some situation, equivalence
relations are nor suitable for coping with granularity, instead the classical
nano topology is extend to general binary relation based covering nano topological
space. Some researchers [12, 55, 83, 104] recently have shown that the concept of
nano topology can be used as a tool to study some real life problems.
The concept of hesitant fuzzy sets is introduced rstly in 2010 by Torra [111]
which permits the membership to have a set of possible values and presents some
of its basic operations in expressing uncertainty and vagueness. Torra et al. [110]
established the similarities and di erences with the hesitant fuzzy sets and the
previous generalization of fuzzy sets such as intuitionistic fuzzy sets, type 2 fuzzy
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sets and type n fuzzy sets. Therefore, other researchers [14] and [16] introduced
the concept of hesitant fuzzy soft sets and they presented some of the applications
in decision making problems. In 2015, Dey and Pal [26] proposed the concept of
hesitant multi-fuzzy soft topological spaces.
This thesis is devoted to
1. study bi-ideal approximation spaces and its applications,
2. give relationships between the present approximations and the previous approximations,
3. initiate a generalization of rough multisets via multiset ideals,
4. study a generalization of nano topological spaces induced by di erent neighborhoods
based on ideals,
5. de ne the concept of hesitant fuzzy soft multisets,
6. introduce the mappings in hesitant fuzzy soft multisets,
7. study the continuous mappings on hesitant fuzzy soft multi spaces,
8. investigate the connectedness on hesitant fuzzy soft multi topological spaces,
9. present the applications in decision-making problems about the hesitant
fuzzy soft multisets.
This thesis contains 6 chapters:-
Chapter 1 is the introductory chapter and contains the basic concepts and properties
of relations and topological structures. It contains also the basic concepts
and properties of multiset theory, rough set theory, nano topological spaces, soft
set theory, soft multi topological spaces, fuzzy set theory and hesitant fuzzy sets.
In Chapter 2, two kinds of approximation operators via ideals which represent
extensions of Pawlak’s approximation operator have been presented. In both
kinds, the de nitions of upper and lower approximations based on ideals have
been given. Moreover, a new type of approximation spaces via two ideals which is
called bi-ideal approximation spaces was introduced for the rst time. This type
of approximations was analyzed by two di erent methods, their properties are
investigated and the relationship between these methods is proposed. The importance
of these methods was its dependent on ideals as a topological tools and the
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two ideals represent two opinions instead of one opinion. Also, an applied example
had been introduced in the chemistry eld by applying the current methods to
illustrate the de nitions in a friendly way. A covering of a universe is a generalization
of the concept of partition of the universe. Rough sets based on coverings
instead of partitions have been studied in several manuscripts [115, 126, 127, 128].
Finally, we show that [Lemma 3.3, p.538] which was introduced in [1] is incorrect
in general, by giving counter examples. Consequently, [Proposition 3.2, p.539]
is also incorrect. Moreover, the correction form of the incorrect results in [1] is
presented. Some results of this chapter are