1. Introduction
The fuzzy set theory proposed by Zadeh [1] extended the classical notion of sets and permitted the gradual assessment of membership of elements in a set [2] . After introducing the notion of fuzzy sets and fuzzy set operations, several attempts have been made to develop mathematical structures using fuzzy set theory. In 1968, chang [3] introduced fuzzy topology which provides a natural framework for generalizing many of the concepts of general topology to fuzzy topological spaces and its development can be found in [3] . The concept of a type-2 fuzzy set as extension of the concept of an ordinary fuzzy set (henceforth called a type-1 fuzzy set) in which the membership function falls into a fuzzy set in the interval [0,1], [2] [4] . Many scholars have conducted research on type-2 fuzzy set and their properties, including Mizumoto and Tanaka [5] , Mendel [6] , Karnik and Mendel [4] and Mendel and John [7] . Type-2 fuzzy sets are called “fuzzy”, so, it could be called fuzzy set [6] . In [6] Mendel was introduced the concept of an interval type-2 fuzzy set. Type-2 fuzzy sets have also been widely applied to many fields with two parts general type-2 fuzzy set and interval type-2 fuzzy sets. The interval type-2 fuzzy topological space introduced by [2] . Because the interval type-2 fuzzy set, as a special case of general type-2 fuzzy sets, and general type-2 fuzzy sets may be better that the interval type-2 fuzzy sets to deal with uncertainties and because general type-2 fuzzy sets can obtain more degrees of freedom [8] , we introduce general type-2 fuzzy topological spaces. The paper is organized as follows. Section 2 is the preliminary section which recalls definitions and operations to gather with some properties type-2 fuzzy sets. In Section 3, we introduce the definition of general type-2 fuzzy topology and some of its structural properties such as type-2 fuzzy open sets, type-2 fuzzy closed sets, type-2 fuzzy interior, type-2 fuzzy closure and neighborhood of a type-2 fuzzy set are studied.
2. Preliminaries
In this section, we recall the preliminaries of type-2 fuzzy sets, define type-2 fuzzy and some important associated concepts from [7] [9] and throughout this paper, let X be anon empty set and I be closed unit interval, i.e., .
Definition 1 [7] [9] . Let X be a finite and non empty set, which is referred to as the universe a type-2 fuzzy set, denoted by is characterized by a type-2 memberships function , as
, where and , that is
(1)
can also be expressed as
(2)
where an denotes union over all admissible x and u for continuous universes of discourse, is replaced by . The class of all type-2 fuzzy sets of the universe X denoted by .
Definition 2 [2] [7] . A vertical slice, denoted , of , is the intersection between the two-dimensional plane whose axes are u and and the three-dimensional type-2membership function , i.e.,
in which . can also be expressed as follows: or as following
(3)
The vertical slice, is also called the secondary membership function, and its domain is called the primary membership of x, which is denoted by where for any . The amplitude of a secondary membership function is called the secondary grade.
When configuring any type-2 fuzzy topological structures we must present some special types of type-2 fuzzy sets.
Definition 3 [5] [8] . (Type-2 fuzzy universe set).
A type-2 fuzzy universe set, denoted , such that
(4)
Definition 4 [5] [8] . (Type-2 fuzzy empty set)
A type-2 fuzzy empty set, denoted , such that
(5)
Definition 5 [6] . (Interval type-2 fuzzy set).
When all the secondary grades of types are equal to 1, that is for all and for all , is as an Interval type-2 fuzzy set.
Operation of Types-2 fuzzy sets 6. Consider two type-2 fuzzy sets, and , in a universe X. Let and be the membership grades of these two sets, which are represented for each , and , respective, where , indicate the primary memberships of x and indicate the secondary memberships (grades) of x. The membership grades for the union, intersection and complement of the type-2 fuzzy sets and have been defined as follows [5] .
Containment:
is a subtype-2 fuzzy set of denoted if and for every .
Equality:
and are type-2 fuzzy sets are equal, denoted if and for every .
Union of two type-2 fuzzy sets:
(6)
Intersection of two type-2 fuzzy sets:
(7)
Complement of a type-2 fuzzy set:
(8)
Where represent the max t-conorm and represent a t-norm. The summation indicate logical unions. We refer to the operations and as join, meet and negation respectively and , , and are the secondary membership functions and all are type-1 fuzzy sets. If and have continuous domains, then the summations in 3, 4 and 5 are replaced by integrals.
Example 7: Let be anon empty set, and let and are type-2 fuzzy sets over the same universe X.
The complement of a type-2 fuzzy set is
Operations under collection of type-2 fuzzy sets 8: Let be an
arbitrary collection of type-2 fuzzy sets subset of X such that is countable set, operation are possible under an arbitrary collection of type-2 fuzzy sets.
1) The union is defined as
(9)
2) The intersection is defined as
(10)
Proposition 9: Let be an arbitrary collection of type-2 fuzzy sets
subset of X such that is countable set and be another type-2 fuzzy set of X, then
1) .
2) .
3) .
4) .
3. General Type-2 Fuzzy Topological Space
In this section we introduced the concept general type-2 fuzzy topology.
Definition 1: Let be the collection of type-2 fuzzy set over X; then is said to be general type-2 fuzzy topology on X if
1)
2) for any .
3) for any , countable set.
The pair is called general type-2 fuzzy topological space over X.
Remark 2: Let be general type-2 fuzzy topological space over X; then the members of are said to be type-2 fuzzy open set in X and a type-2 fuzzy set is said to be a type-2 fuzzy closed set in X, if its complement .
Proposition 3: Let be general type-2 fuzzy topological space over X then the following conditions hold:
1) are type-2 fuzzy closed sets.
2) Arbitrary intersection of type-2 fuzzy closed sets is closed sets.
3) Finite union of type-2 fuzzy closed sets is closed sets.
Proof:
1) are type-2 fuzzy closed sets because they are the complements of the type-2 fuzzy open sets is respectively.
2) Let be an arbitrary collection of type-2 fuzzy closed sets, then
since arbitrary union of type-2 fuzzy open sets are open is an open and is a type-2 fuzzy closed sets.
3) If is type-2 fuzzy closed sets, then is a type-2 fuzzy closed set, [finite intersection of type-2 fuzzy open sets are open].
Example 4: Let and let and be three type-2 fuzzy sets in X which are
,
Then is general type-2 fuzzy topologies defined on X and the pair is called general type-2 fuzzy topological space over X, every member of is called type-2 fuzzy open sets.
Theorem 5: Let be a family of all general type-2 fuzzy topologies on X ; then is general type-2 fuzzy topologies on X.
proof: we must prove three conditions of topologies,
1) .
2) Let , then for all so
thus .
3) Let , then and because are all general type-2 fuzzy topologies for all , so .
Remark 6: Let and be two general type-2 fuzzy topological spaces over the same universe X then need not be general type-2 fuzzy topological space over X, we can see that in example 3.7.
Example 7: Let and , be two general type-2fuzzy topologies defined on X where and defined as follows: ,
Let so is not general type-2 fuzzy topological space over X since .
Definition 8: Let be general type-2 fuzzy topological space over X and let be type-2 fuzzy set over X. Then the type-2 fuzzy interior of , denoted by , is defined as the union of all type-2 fuzzy open sets contained in . That is,
, is the largest type-2 fuzzy open set contained in .
Theorem 9: Let be general type-2 fuzzy topological space over X, and let be two type-2 fuzzy sets in X. Then
1) and .
2) .
3) is type-2 fuzzy open set if and only if .
4) .
5) .
6) .
Proof:
1) , is type-2 fuzzy open set in and .
Now to prove ,
, is type-2 fuzzy open set in and .
2) To prove , since , such that that is is type-2 membership function where and less than a type-2 membership function where and such that and , hence , therefore .
3) If is type-2 fuzzy open set, then , but from part (2), hence .
4) is a type-2 fuzzy open set and from part (3) we have
5) If and from part(2) , , then . Therefore and is a type-2 fuzzy open set contained in , so .
6) Because and , from part (5) and , thus , since , so from part(5) but is a type-2 fuzzy open sets then from part(3).Hence .
Definition 10: Let be general type-2 fuzzy topological space over and let be type-2 fuzzy set over X. Then the type-2 fuzzy closure of , denoted by , is defined as the intersection of all type-2 fuzzy closed sets containing . That is
,
is the smallest type-2 fuzzy closed set containing .
Theorem 11: Let be general type-2 fuzzy topological space over X, and let be two type-2 fuzzy sets in X. Then
1) and .
2) .
3) is type-2 fuzzy closed set if and only if .
4) .
5) .
6) .
Proof: The proof this theorem similar to the proof of theorem 3.7.
Definition 12: Let be a general type-2 fuzzy topological space over X and . Then is said to be a neighborhood or nbhd for short, of a type-2 fuzzy set if there exist a type-2 fuzzy open set such that .
Proposition 13: A type-2 fuzzy set is open if and only if for each type-2 fuzzy set contained in , is a neighborhood of .
Proof: If is open and then is a neighborhood of . Conversely, since , there exists a type-2 fuzzy open set such that . Hence and is open.
Definition 14: Let be a general type-2 fuzzy topological space over X
and be a subfamily of . If every member of can be written as the type-2 fuzzy union of some members of , then is called a type-2 fuzzy base for the general type-2 fuzzy topology . We can see that if be type-2 fuzzy base for then equals the collection of type-2 fuzzy unions of elements of .
Definition 15: Let and be two general type-2 fuzzy topological space.The general type-2 fuzzy topological space Y is called a subspace of the general type-2 fuzzy topological space X if and the open subsets of Y are precisely of the form . Here we may say that each open subset of Y is the restriction to of an open subset of X. That is, is called a subspace of if the type-2 fuzzy open sets of Y are the type-2 fuzzy intersection of open sets of X with .
4. Conclusion
The main purpose of this paper is to introduce a new concept in fuzzy set theory, namely that of general type-2 fuzzy topological space. On the other hand, type-2 fuzzy set is a kind of abstract theory of mathematics. First, we present definition and properties of this set before introducing definition of general type-2 fuzzy topological space with the structural properties such as open sets, closed sets, interior, closure and neighborhoods in general type-2 fuzzy set topological spaces and some definitions of a type-2 fuzzy base and subspace of general type-2 fuzzy sets.
Acknowledgements
Great thanks to all those who helped us in accomplishing this research especially Prof. Dr. Kamal El-saady and Prof. Dr. Sherif Abuelenin from Egypt for us as well as all the workers in the magazine.
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