الفهرس 
Preliminaries
Proximity spacesOnly 14 pages are availabe for public view
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Abstract
The fundamental concept of Efremoviˇc proximity space has been introduced by Efremoviˇc [10]. In addition, Leader [27, 28] and Lodato
[29, 30] have worked with weaker axioms than those of Efremoviˇc proximity space enabling them to introduce an arbitrary topology on the
underlying set. Furthermore, proximity relations are useful in solving
problems based on human perception [40] that arise in areas such as
image analysis [20]. The notion of ideal topological spaces was first
studied by Kuratowski [26] and Vaidyanathaswamy [63, 64]. Compatibility of the topology with an ideal I was first defined by Njastad [36].
In 1990, Jankovic and Hamlett [24] investigated further properties of
ideal topological spaces.
In 1999, Molodtsov [33] proposed the novel concept of soft set theory,
which provides a completely new approach for modeling vagueness and
uncertainty. Soft set theory has a rich potential for applications in
several directions, few of which were shown by Molodtsov in [33]. After
Molodtsov’s work, some different applications of soft sets were studied
in Chen et al. [8]. Further theoretical aspects of soft sets were explored
by Maji et al. [32]. Also the same authors [31] presented the definition
of a fuzzy soft set. The algebraic nature of the soft set has been studied
by several researchers. Aktas and Cagman [2] initiated soft groups, and
Feng [12] defined soft semirings. Babitha and John [3] defined soft set
relations and functions. Yang and Guo [67] introduced kernels and
closures of soft set relations. Hazra et al. [23] introduced the notion of
basic proximity of soft sets. Also, the same authors [22] proposed the
notion of soft proximity.
In classical set theory, a set is a well-defined collection of distinct
objects. If repeated occurrences of any object is allowed in a set, then
a mathematical structure, that is known as multiset (mset [4] or bag [66], for short). Thus, a mset differs from a set in the sense that each
element has a multiplicity a natural number not necessarily one that
indicates how many times it is a member of the mset. One of the most
natural and simplest examples is the mset of prime factors of a positive
integer n. The number 400 has the factorization 400 = 2452 which gives
the mset {2, 2, 2, 2, 5, 5}. Also, the cubic equation x3 −5x2 +3x+9 = 0
has roots 3, 3 and −1 which give the mset {3, 3, −1}.
Classical set theory is a basic concept to represent various situations
in mathematical notation where repeated occurrences of elements are
not allowed. But in various circumstances repetition of elements become mandatory to the system. For example, a graph with loops,
there are many hydrogen atoms, many water molecules, many strands
of identical DNA etc. This leads to effectively three possible relations
between any two physical objects; they are different, they are the same
but separate, or they are coinciding and identical. For example, ammonia NH3, with three hydrogen atoms, say H, H and H, and one
nitrogen atom, say N. Clearly H and N are different. However H, H
and H are the same but separate, while H and H are coinciding and
identical. There are many other examples, for instance, carbon dioxide
CO2, sulfuric acid H2SO4, and water H2O etc.
The overall aim of this thesis is to increase the stock of knowledge
about proximity relations and some related structures. Specifically we
aim to:
• Introduce new approaches of proximity relations based on ideal
notion.
• Investigate new approaches of proximity relations based on soft
set notion.
• Introduce examples of multiset topologies are not tackled before.
• Extending the notions of compact, proximity relations, proximal
neighborhood and proximity mappings to the multiset context.
• Find way to reduce the boundary region of rough sets in the
multiset context.
• Study the notion of mild continuity in relator spaces and its
properties.This Thesis includes six chapters as follow:
Chapter 1 has a collection of all basic definitions and notions for
further study.
In Chapter 2, a new approach of proximity structure based on the
ideal notion has been introduced. For I = {φ}, we have the Efremoviˇc
proximity structure [10] and for the other types of I, we have many
types of proximity structures. Some results on this new approach have
obtained and one of the important results: every ∗−normal T1 space
is I-proximizable space (Theorem 2.2.5). Moreover, δI−neighborhood
in an I−proximity space has been introduced. This provides an alternative description to the study of I−proximity spaces. Furthermore,
the operator ∗ on P (X) with respect to an ideal and uniformity U on
X has been introduced and various properties of it are investigated.
The new generated uniformity via ideal is presented which generated
a topology τ ∗(U) finer than the old one. In addition, τ ∗(U) = τδI is
proved. The notion of generalized proximity has been introduced via
the concept of ideal in the ordinary topology. In addition to that, the
notions of I-Leader, I-Pervin, and I-Lodato proximities have been introduced. It is shown that the relation between the topology generated
via these proximities and the topology τ ∗ which generated via ideal.
It should be noted that some results of this chapter are
published as follow:
• A. Kandil, O. A. Tantawy, S. A. El-Sheikh and A. Zakaria, I−
proximity spaces, J¨okull Journal 63 (2013), 237-245.
• A. Kandil, O. A. Tantawy, S. A. El-Sheikh and A. Zakaria, Generalized I−proximity spaces, Mathematical Sciences Letters 3
(2014), 173-178.
• A. Kandil, O. A. Tantawy, S. A. El-Sheikh, A. Zakaria, On
I−proximity spaces, Journal of Applied Mathematics and Information Science Letters. Accepted.
In Chapter 3, a new approaches of proximity and generalized proximity based on the soft set have been introduced. It is shown that every
soft T4− space is compatible with a proximity relation on P (X)E. In
addition, every soft space is compatible with a Pervin proximity relation on P (X)E. It is also shown that every soft Ro−space is compatible with a Lodato proximity relation on P(X)E. Furthermore, we introduced a new approach of soft proximity structure based on the ideal
notion. For I = {φ}, we have the soft proximity structure [22] and for
the other types of I, we have many types of soft proximity structures.
It should be noted that some results of this chapter are
published as follow:
• A. Kandil, O. A. Tantawy, S. A. El-Sheikh and A. Zakaria, New
structures of proximity spaces, Information Sciences Letters 3
(2014), 85-89.
• A. Kandil, O. A. Tantawy, S. A. El-Sheikh and A. Zakaria, Soft
I−proximity spaces, Ann. Fuzzy Math. Inform. 9 (2015), 675-
682.
Chapter 4, is an attempt to explore the theoretical aspects of
msets by extending the notions of compact, proximity relation, proximal neighborhood and proximity mappings to the mset context. Examples of new mset topologies, open msets cover, compact mset and
many identities involving the concept mset are introduced. A mset
proximity relations and an integral examples of mset proximity are
obtained. In addition, a mset topology induced by a mset proximity
relation on a mset M and study its principal properties. The concept
of mset δ-neighborhood in the mset proximity space which furnishes
an alternative approach to the study of mset proximity spaces has
been introduced. Furthermore, the mset proximity mappings and its
properties are introduced.
It should be noted that some results of this chapter are
submitted as follow:
• A. Zakaria, Note on ”on multiset topologies”. Ann. Fuzzy Math.
Inform. 10 (5) (2015), 13-14.
• A. Kandil, O. A. Tantawy, S. A. El-Sheikh and A. Zakaria, Some
structures in multiset context. Submitted.
Chapter 5 is an attempt to explore a new approach of rough mset
to decrease the boundary region and increase the accuracy measure.
We show that an alleged properties stated in [16] are invalid in general,by giving a counter-examples. A new approach of ideals in the context of msets on the lattice of all submsets with the order relation as
the mset inclusion has been introduced. Properties of mset ideals are
studied. The R∗− upper and R∗− lower mset approximations via mset
ideals have been mentioned. These definitions are different from Girish
et al.’s definitions [16, 18] and more general. If I = {φ}, then our mset
approximations coincide with Girish et al.’s mset approximations. So,
Girish et al.’s mset approximations are special case of our mset approximations. Moreover, Properties of these mset approximations are
studied. Furthermore, an mset topology via this new approach has
been introduced. This mset topology is finer than Girish et al.’s one.
Moreover, this new approach leads to decrease the boundary region
and increase the accuracy measure. In addition, the boundary of a
submset decreases as the mset ideal on a nonempty mset M increases.
Finally, varied examples are introduced to show the significance of this
new approach.
It should be noted that some results of this chapter are
published as follow:
• S. J. John, S. A. El-Sheikh and A. Zakaria, Generalized rough
multiset via multiset ideals, Journal of Intelligent and Fuzzy Systems. Accepted.
• S. A. El-Sheikh and A. Zakaria, Note on ”rough multiset and
its multiset topology”, Ann. Fuzzy Math. Inform. 10 (2015),
235-238.
In Chapter 6, A. Sz´az and A. Zakaria continue the investigations ´
initiated by A. Sz´az [45, 50, 51, 53, 54, 58, 59] on the basic continuity ´
properties of a single relation, and also of a pair of relations, on one relator (generalized uniform) space to another. Furthermore, the notion
of mild continuity in relator spaces has been introduced. Moreover,
several useful consequences of mild continuity have been mentioned.
Finally, the properties of proper, uniform, proximal, topological and
paratopological mild continuity have been presented.
It should be noted that some results of this chapter are
published as follow:
• A. Sz´az and A. Zakaria, ´ Mild continuity properties of relations and relators in relator spaces, Tech. Rep., Inst. Math., Univ.
Debrecen 2015/1, 65 pp.
• A. Sz´az and A. Zakaria, ´ Mild continuity properties of relations
and relators in relator spaces, Essays in Mathematics and its
Applications: In Honor of Vladimir Arnold. To appear 2016