الفهرس 
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Abstract
Multiset theory was introduced in 1986 by Yager [68]. 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 is
allowed in a set, then a mathematical structure, that is known as multiset (mset
[9] or bag [68], for short), is obtained in [11, 25, 59, 60]. 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). Noted that each
multiplicity ki is a positive integer.
In Mathematics, the equation x24x+4 = 0 has a solution x = 2; 2 which gives
the multiset S = f2=2g. Additionally, One of the simplest examples is the multiset
of prime factors of a positive integer n. The number 504 has the factorization
504 = 233271 which gives the mset X = f3=2; 2=3; 1=7g where CX(2) = 3, CX(3) =
2, CX(7) = 1. In Chemistry, a water molecule H2O is represented by the mset
M = f2=H; 1=Og and without one of the two hydrogen atoms, the water molecule
is not created.
In the physical world it is observed that there is enormous repetition. For
instance, there are many hydrogen atoms, many water molecules, many strands
of DNA, etc. This leads to three possible relations between any two physical
objects; they are di erent, they are the same but separate or they coincide and are
identical. Many conclusive results were established by these authors and further
study was carried on by Jena et al. [32] and many others [12, 13, 14, 47]. The
notion of an mset is well established both in mathematics and computer science
[9, 10, 15, 16, 23, 58, 61, 62].A wide application of msets can be found in various branches of mathematics.
Algebraic structures for mset space have been constructed by Ibrahim et al. [29].
In [53], Okamoto et al. used msets in coloring of graphs. Additionally, application
of mset theory in decision making can be seen in [69]. In 2012, Girish and Sunil [26]
introduced multiset topologies induced by mset relations. The same authors in
[27] further studied the notions of open sets, closed sets, basis, sub-basis, closure,
interior, continuity in multiset topological (M-topological, for short) spaces.
The concept of soft sets was rst introduced by Molodtsov [48] in 1999 as a general
mathematical tool for dealing with uncertain objects. In [48, 49], Molodtsov
successfully applied the soft set theory in several directions, such as smoothness
of functions, game theory, operations research, Riemann integration, Perron integration,
probability, theory of measurement, and so on. In 2011, Shabir and Naz
[56] initiated the study of soft topological spaces. They de ned soft topology on
the collection of soft sets over X. Consequently, they de ned basic notions of soft
topological spaces such as open soft sets and closed soft sets, soft subspace, soft
closure, soft nbd of a point, soft separation axioms, soft regular spaces and soft
normal spaces.
In 2013, Babitha et al. [8] and Tokat et al. [64] introduced the concept of soft
mset (F;E) as F : E ! PW(U) where E is a set of parameters and PW(U) is
a power whole mset of an mset U. Moreover, Tokat et al. [65] introduced the
concept of soft mset (F;E) by another way as F : E ! P(U) where E is a set of
parameters and P(U) is a power set of an mset U. The notion of a soft multiset
in this thesis is the same as in [65, 66, 67]. In 2013, Tokat et al. [64] introduced
the concept of soft multi topology and its basic properties. In addition, the notion
of soft multi connectedness was studied in [65]. Additionally, the notion of soft
multi compactness on soft multi topological spaces was presented in [66]. In 2015,
Tokat et al. [67] presented the notion of soft multi continuous functions. The
concept of soft msets which is combining soft sets and msets can be used to solve
some real life problems. Also, this concept can be used in many areas, such as
data storage, computer science, information science, medicine, engineering, etc.
A bitopological space (X; 1; 2) was introduced by Kelly [38] in 1963, as a
method of generalizes topological spaces (X; ). Every bitopological space (X; 1; 2)
can be regarded as a topological space (X; ) if 1 = 2 = . Furthermore, he
extended some of the standard results of separation axioms and mappings in a
topological space to a bitopological space. The notion of connectedness in bitopological
spaces has been studied by Pervin [54], Reily [55] and Swart [63].In 1983 Mashhour et al. [44] introduced the notion of supra topological spaces
by dropping only the intersection condition. Kandil et al. [33] generated a supra
topological space (X; 12) from the bitopological space (X; 1; 2) and studied some
properties of the space (X; 1; 2) via properties of the associated space (X; 12).
Thereafter, a large number of papers have been written to generalize topological
concepts to bitopological setting [17, 18, 19, 20, 34, 57].
Generalized open sets play a very important role in general topology and they
are now the research topies of many topologists worldwide. Andrijevic [4, 7]
introduced a class of generalized open sets in a topological space as b-open sets
and -sets. The class of b-open sets is contained in the class of -open sets and
contains both semi-open sets and pre-open sets. Levine [39], Mashhour et al. [43],
Njastad [52] and Abd El-Monsef et al. [1] introduced semi-open sets, pre-open
sets, -open sets and -open sets respectively.
This thesis is devoted to
1. Initiate supra M-topological spaces.
2. Study
-operation and some types of msets in (supra) M-topological spaces.
3. Introduce generalized closed soft msets in soft multi topological spaces.
4. Study some types of soft msets and some types of soft multi continuous
functions in soft multi topological spaces.
5. Introduce separation axioms, semi-compactness and semi-connectedness on
(soft) multi topological spaces.
6. Given comparisons between the current results and the previous one by using
counter examples.
7. Initiate multiset bitopological spaces and some operators on it.
This thesis contains 5 chapters:-
Chapter 1 is the introductory chapter, so it contains the basic concepts and
properties of set theory and mset theory. It contains also the basic concepts
and properties of topological space such as closure, interior, boundary, functions,
separation axioms. The basic concepts and properties of bitopological spaces are
presented. Further, this chapter contains the basic notions related to soft sets
and soft topological spaces. Additionally, in Subsection 1.6, the concept of soft
msets is introduced by Tokat et al. [65]. Moreover, the soft multi function and
it’s properties are introduced in this subsection.In Chapter 2, the concept of supra M-topological spaces is introduced initially.
Then, the notions of supra
-operation, supra pre-open msets, supra -open msets,
supra semi open msets, supra -open msets and supra b-open msets are presented.
The properties of the present notions are studied and the relationships between
them are given. The importance of this approach is that, the class of supra Mtopological
spaces is wider and more general than the class of M-topological spaces.
For a special case, we introduced the notion of
-operation in M-topological spaces.
Some results of this chapter are:
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat,
-operation in M-topological
space, Gen. Math. Notes 27 (2015) 4054.”
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, Supra M-topological space
and decompositions of some types of supra msets, International Journal of
Mathematics Trends and Technology 20 (2015) 1124.”
The goal of Chapter 3 is to study some (soft) multi topological properties in
(soft) multi topological spaces which are represented by introducing separation
axioms on M-topological spaces and study some of their properties. In addition,
some algebraic structures on soft msets are obtained. Also, we introduced the
notion of soft multi semi-compactness as a generalization of semi-compact in Mtopological
spaces and study its properties. Finally, we see that Theorem 1.6.3
in [65] is not correct and that is explained by a counter example. Moreover, the
concept of semi-connectedness in soft multi topological spaces is introduced.
The results of this chapter are:
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, Separation axioms on
multiset topological space, Journal of New Theory 7 (2015) 1121.”
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, Semi-compact soft multi
spaces, Journal of New Theory 6 (2015) 7687.”
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, A note on ”Connectedness
on soft multi topological space”, Journal of New Results in Science,
submitted.”
In Chapter 4, we introduced the concepts of generalized closed (open) soft msets
and their properties. Also, the relationship between the current work and the previous
one [39] is presented with the help of counter examples. Additionally, we
introduced the concept of separated soft msets in soft multi topological spaces
and study some results about this concept. The main purpose of Subsection 4.2 is to introduce the notions of
-operation, pre-open soft msets, -open msets, semiopen
soft msets, -open soft msets and b-open soft msets in soft multi topological
spaces. The current notions are a generalization of the notions in [35]. In addition,
the relationships among these types are studied. Moreover, the concepts
of pre-continuous (respectively semi-continuous, -continuous, -continuous, b-
continuous) soft multi functions are introduced and their properties are studied
in detail. Also, the concepts of pre-irresolute (respectively semi-irresolute, -
irresolute, -irresolute, b-irresolute) soft multi functions are presented.
Some results of this chapter are:
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, Some types of open soft
multisets and some types of mappings in soft multi topological spaces, Ann.
Fuzzy Math. Inform., to appear.”
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, Generalized closed soft
multiset in soft multi topological spaces, Asian Journal of Mathematics and
Computer Research 9 (2015) 302311.”
The main purpose of Chapter 5 is to introduce the notion of multiset bitopological
spaces and study some M-operators on multiset bitopological spaces. Moreover,
the notions of ij-operators such as ij-pre-open msets, ij- -open msets,
ij-semi-open msets and ij- -open msets are presented on multiset bitopological
spaces. The properties of these operators are studied and the relationships
between them are given. Additionally, some deviations between M-topology and
ordinary topology are given with the help of counter examples. The importance
of this approach is that, the class of multiset bitopological spaces is more general
than the class of bitopological spaces.
Some results of this chapter are:
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, Multiset bitopological
spaces, Asian Journal of Mathematics and Computer Research 8 (2015)
103115.”
\S. A. El-Sheikh, R. A-K. Omar and M. Raafat, Operators on multiset
bitopological spaces, South Asian J. Math. 6 (2016) 19.”