الفهرس 
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Abstract
SUMMARY
We are not able to use classical methods to solve some kinds of problems given
in sociology, economics, environment, engineering etc., since, these kinds of problems
have their own uncertainties. Fuzzy set theory, which was …rstly proposed by Zadeh
[121] in 1965, has become a very important tool to solve these kinds of problems
and provides an appropriate framework for representing vague concepts by allowing
partial membership. Fuzzy set theory has been studied by both mathematicians and
computer scientists and many applications of fuzzy set theory have arisen over the
years, such as fuzzy control systems, fuzzy automata, fuzzy logic, fuzzy topology etc.
Beside this theory, there are also theory of probability, rough set theory which deal
with to solve these problems. The notion of fuzzy topology was introduced by Chang
[28] . Lowen [81] introduced an other de…nition of fuzzy topology.
The concept of soft sets was …rstly introduced by Molodtsov [91] in 1999 as a
general mathematical tool for dealing with uncertain objects. Molodtsov [91], suc-
cessfully applied the soft theory in several directions, such as smoothness of functions,
game theory, operations research, Riemann integration, Perron integration, probabil-
ity and theory of measurement. After presentation of the operations on soft sets [91],
the properties and applications of soft set theory have been studied increasingly [13,
101].
Recently, in 2011, Shabir et al. [108] 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, closed soft
sets, soft subspace, soft closure, soft nbd of a point, soft regular spaces, soft normal
spaces and established their several properties. Hussain et al. [55] investigated the
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properties of open (closed) soft sets, soft nbd and soft closure. They also de…ned and
discussed the properties of soft interior, soft exterior and soft boundary which are
fundamental for further research on soft topology.
Maji et al. [85] initiated the study involving both fuzzy sets and soft sets. In
this paper, the notion of fuzzy soft sets was introduced as a fuzzy generalizations of
soft sets and some basic properties of fuzzy soft sets are discussed in detail. Maji et
al. combined fuzzy sets and soft sets and introduced the concept of fuzzy soft sets.
Tanay et al. [110] and Simsekler [109] gave the topological structure of fuzzy soft sets
and generalized by Chakraborty et al. [27] and Goswami et al. [46]. The fuzzy soft
sets have many application such as: making dicision [19, 24, 30, 32, 45, 61, 79, 111]
and mobile network [115].
The local properties of a space which may also be in certain cases the properties
of the whole space, are important …eld for study in general topology, fuzzy topology,
and soft topology. The notion of ideal in general topology was introduced by Kura-
towski [81], Vaidyanathaswamy [116, 117] and several other authors carried out such
analyses. Recently, there has been an extensive study on the importance of ideal in
general topology in the paper of Jankovi´ and Hamlett [58], in fuzzy topology: byc
Nasef et al. [95], Mahmoud [86] and D. Sarker [107], in soft set theory: by Kandil et.
al. [69] in 2014.
The main aims of this thesis can be summarized, as follows:
1- Introducing fuzzy soft ideal theory, fuzzy soft local function and generating a
new fuzzy soft topological space by two di¤erent methods.
2- Genaralized fuzzy soft sets and decompositions of some forms of fuzzy soft
contiuities via fuzzy soft ideals.
3- Introducing some fuzzy soft topological properties such as: fuzzy soft separation
and regularity axioms, fuzzy soft some classes of compactness in fuzzy soft topological
spaces.
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4- Introducing some types of fuzzy soft separated sets, some types fuzzy soft
connected sets and study the relation between them.
5- Introducing an adiitional types of connectedness in fuzzy soft topological spaces
such as: fuzzy soft extremally disconnected spaces, fuzzy soft hyperconnected, fuzzy
soft connectedness based on fuzzy soft -open sets and ?-connectedness in fuzzy soft
ideal topological sapces.
This thesis contains six chapters, as follows:
Chapter (I): Initiate generalization, providing the reader with results concern-
ing, fuzzy topological spaces, soft topological spaces, fuzzy soft topological spaces,
fuzzy soft point and its neigbourhood structure, fuzzy soft closure, fuzzy soft interior,
fuzzy soft accumulation point, fuzzy soft boundary. Also, the represent the notions
of fuzzy soft continuity, fuzzy soft separation axioms and fuzzy soft comaptness.
Chapter (II): Our aim of this chapter is to extend those ideal of general topology,
fuzzy topology, and soft topology to fuzzy soft setting. In Section 4.1, we de…ne
fuzzy soft ideal and introduce the notion of fuzzy soft local function corresponding
to a fuzzy soft topological space. We have deduce some characterization theorems
for such concepts exactly analogous to general topology, fuzzy topology, and soft
topology and succeeded in …nding out the generated new fuzzy soft topologies for any
fuzzy topological space. In Section 4.2, we discuss the basic structure of new fuzzy
soft topology and it is established that the new fuzzy soft topology cannot be further
generated with the same fuzzy soft ideal. Finally, in Section 4.3, we de…ne the local
function by using the quasi-coincident relation and study its properties. Also, we
introduce the concept of quasi-cover of a fuzzy soft set and introduce the notion of
compatibility of fuzzy soft ideal with a fuzzy soft topological space and obtain some
results concerning this concept.
Some results of this chapter are:
1- A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Sawsan S. S. El-Sayed,
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Fuzzy soft ideal topological spaces, South Asian Journal of Mathematics, 6 ( 4 )
(2016), 186-198. [65]
2- A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Sawsan S. S. El-Sayed, Fuzzy
Soft Ideal Theory: Fuzzy Soft Local Function and Generated Fuzzy Soft Topological
Spaces, The Journal of Fuzzy Mathematics, 25 (2), 2017. [66]
Chapter (III): The purpose of this chapter is to introduce the notions of fuzzy
eee
soft semi-I-open sets (respectively, fuzzy soft I-open sets, fuzzy soft pre-I-open sets,
eee
fuzzy soft -I-open sets, fuzzy soft -I-open sets, fuzzy soft almost I-open, fuzzy soft
ee
functions (respectively, fuzzy soft I-continuous functions, fuzzy soft pre-I-continuous
functions, fuzzy soft
ee
fuzzy soft almost I-continuous functions, fuzzy soft -I-continuous functions) More-
Chapter (IV):The object of this chapter is to inroduce a set of new regularity and
e
-I-continuous functions, fuzzy soft
e
-I-continuous functions,
e
-dense-in-itself). Furthermore, we present the notions of fuzzy soft semi-I-continuous
over, the decomposition of such forms of fuzzy soft continuity is studied.
separation axioms which are called (F SRi ; i = 0; 1; 2; 3) and (F STi ; i = 0; 1; 2; 3; 4)
by using fuzzy soft quasi-coincident and neighborhood system. The notion of fuzzy
soft hereditary property is examined. Furthermore, we introduced the ideas fuzzy soft
semi-compactness, fuzzy soft -compactness, fuzzy soft -compactness, and fuzzy soft
strongly compactness. Also, the notions of fuzzy soft S-closed, fuzzy soft s-closed,
fuzzy soft P -closed, fuzzy soft
-closed and fuzzy soft -closed are studied. Finite
intersection property is used to characterize these concepts. A comparison between
these types of compactness in fuzzy soft topological spaces is established.
In Chapter (V): The notions of fuzzy soft connected sets and fuzzy soft con-
nected components are very important in fuzzy soft topological spaces which in turn
re‡ the intrinsic nature of it that is in fact its peculiarity. In this Chapter, weect
introduce some types of fuzzy soft separated sets, some types of connectedness in
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fuzzy soft topological spaces and study the relationship between them. Also, we in-
troduce an equivalence relation on fuzzy soft points and de…ne a fuzzy soft connected
components as an equivalence class induced by this equivalence relation.
Some results of this chapter are:
1- A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Sawsan S. S. El-Sayed,
Fuzzy Soft Connected Sets in Fuzzy Soft Topological Spaces I, Journal of Advances
in Mathematics, 12 (8) (2016), 6473-6488. [64]
2- A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Sawsan S. S. El-Sayed,
Fuzzy Soft Connected Sets in Fuzzy Soft Topological Spaces II, Journal of Egyptian
Mathematical Society, (accepted).
Chapter (VI): In this chapter, we introduce the concept of fuzzy soft extremally
disconnected spaces, fuzzy soft D-space and fuzzy soft hyperconnected space. The
relation between these concepts is investigated. Furthermore, we introduce the no-
tions of fuzzy soft -separated sets and use it to introduce the notions of fuzzy -s-
connectness in fuzzy soft topological spaces and study its basic properties. Moreover,
we extend the notion of fuzzy soft connectedness via fuzzy soft ideal.
Some results of this chapter are:
1- A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Sawsan S. S. El-Sayed,
Fuzzy Soft Hyperconnected spaces, Annals of fuzzy mathematics and informatics,
(accepted).
2- A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-Latif, S. El-Sayed,
Fuzzy soft connectedness based on fuzzy
-open soft sets, Journal of Mathematics
and Computer Applications Research (JMCAR), 5 (2) (2015), 37-48. [63]
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