APM  Vol.8 No.9 , September 2018
General Type-2 Fuzzy Topological Spaces
ABSTRACT
In this paper, a presented definition of type-2 fuzzy sets and type-2 fuzzy set operation on it was given. The aim of this work was to introduce the concept of general topological spaces were extended in type-2 fuzzy sets with the structural properties such as open sets, closed sets, interior, closure and neighborhoods in topological spaces were extended to general type-2 fuzzy topological spaces and many related theorems are proved.

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., I = [ 0 , 1 ] .

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 A ˜ ˜ is characterized by a type-2 memberships function μ A ˜ ˜ ( x , u ) , as

μ A ˜ ˜ : X × [ 0 , 1 ] [ 0 , 1 ] J x ( J x [ 0 , 1 ] ) , where x X and u J x , that is

A ˜ ˜ = { ( ( x , u ) , μ A ˜ ˜ ( x , u ) ) : where x X and u J x [ 0 , 1 ] , where 0 μ A ˜ ˜ ( x , u ) 1 } (1)

A ˜ ˜ can also be expressed as

A ˜ ˜ = x X u J x μ A ˜ ˜ ( x , u ) / ( x , u ) = x X u J x f x ( u ) / u / x , J x [ 0 , 1 ] (2)

where f x ( u ) = μ A ˜ ˜ ( x , u ) 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 F ˜ ˜ T 2 ( X ) .

Definition 2 [2] [7] . A vertical slice, denoted μ A ˜ ˜ ( x ) , of A ˜ ˜ , is the intersection between the two-dimensional plane whose axes are u and μ A ˜ ˜ ( x , u ) and the three-dimensional type-2membership function A ˜ ˜ , i.e.,

μ A ˜ ˜ ( x ) = μ A ˜ ˜ ( x = x , u ) = u J x f x ( u ) / u , J x I in which 0 f x ( u ) 1 . A ˜ ˜ can also be expressed as follows: A ˜ ˜ = { ( x , μ A ˜ ˜ ( x ) ) : x X } or as following

A ˜ ˜ = x X u J x μ A ˜ ˜ ( x ) / ( x ) = x X u J x f x ( u ) / u / x , J x [ 0 , 1 ] (3)

The vertical slice, μ A ˜ ˜ ( x ) is also called the secondary membership function, and its domain is called the primary membership of x, which is denoted by J X where J X I for any x X . 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 X ˜ ˜ , such that

X ˜ ˜ = x X u [ 1 , 1 ] 1 / u / x (4)

Definition 4 [5] [8] . (Type-2 fuzzy empty set)

A type-2 fuzzy empty set, denoted ˜ ˜ , such that

˜ ˜ = x X u [ 0 , 0 ] 1 / u / x (5)

Definition 5 [6] . (Interval type-2 fuzzy set).

When all the secondary grades of types A ˜ ˜ are equal to 1, that is μ A ˜ ˜ ( x , u ) = 1 for all x X and for all u J x [ 0 , 1 ] , A ˜ ˜ is as an Interval type-2 fuzzy set.

Operation of Types-2 fuzzy sets 6. Consider two type-2 fuzzy sets, A ˜ ˜ and B ˜ ˜ , in a universe X. Let μ A ˜ ˜ ( x ) and μ B ˜ ˜ ( x ) be the membership grades of these two sets, which are represented for each x X , μ A ˜ ˜ ( x ) = u J x u f x ( u ) / u and μ B ˜ ˜ ( x ) = w J x w g x ( w ) / w , respective, where u J x u , w J x w indicate the primary memberships of x and f x ( u ) , g x ( w ) [ 0 , 1 ] indicate the secondary memberships (grades) of x. The membership grades for the union, intersection and complement of the type-2 fuzzy sets A ˜ ˜ and B ˜ ˜ have been defined as follows [5] .

Containment:

A ˜ ˜ is a subtype-2 fuzzy set of B ˜ ˜ denoted A ˜ ˜ B ˜ ˜ if u w and f x ( u ) g x ( w ) for every x X .

Equality:

A ˜ ˜ and B ˜ ˜ are type-2 fuzzy sets are equal, denoted A ˜ ˜ = B ˜ ˜ if u = w and f x ( u ) = μ A ˜ ˜ ( x , u ) = g x ( w ) = μ B ˜ ˜ ( x , w ) for every x X .

Union of two type-2 fuzzy sets:

A ˜ ˜ B ˜ ˜ μ A ˜ ˜ B ˜ ˜ ( x ) = u J x u w J x w f x ( u ) g x ( w ) / ( u w ) μ A ˜ ˜ ( x ) μ B ˜ ˜ ( x ) , x X (6)

Intersection of two type-2 fuzzy sets:

A ˜ ˜ B ˜ ˜ μ A ˜ ˜ B ˜ ˜ ( x ) = u J x u w J x w f x ( u ) g x ( w ) / ( u w ) μ A ˜ ˜ ( x ) μ B ˜ ˜ ( x ) , x X (7)

Complement of a type-2 fuzzy set:

A ˜ ˜ = μ A ˜ ˜ ( x ) = u J x u f x ( u ) / ( 1 u ) ¬ μ A ˜ ˜ ( x ) , x X (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 μ A ˜ ˜ B ˜ ˜ ( x ) , μ A ˜ ˜ B ˜ ˜ ( x ) , μ A ˜ ˜ ( x ) and μ B ˜ ˜ ( x ) are the secondary membership functions and all are type-1 fuzzy sets. If μ A ˜ ˜ ( x ) and μ B ˜ ˜ ( x ) have continuous domains, then the summations in 3, 4 and 5 are replaced by integrals.

Example 7: Let X = { x 1 , x 2 , x 3 } be anon empty set, and let A ˜ ˜ and B ˜ ˜ are type-2 fuzzy sets over the same universe X.

A ˜ ˜ = { ( ( x 1 , 0.1 ) , 0.3 ) , ( ( x 1 , 0.5 ) , 1 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.6 ) , 0.3 ) , ( ( x 3 , 0.8 ) , 1 ) }

B ˜ ˜ = { ( ( x 1 , 0.1 ) , 0.7 ) , ( ( x 1 , 0.2 ) , 1 ) , ( ( x 2 , 0.6 ) , 1 ) , ( ( x 3 , 0.5 ) , 0.6 ) , ( ( x 3 , 0.9 ) , 1 ) }

A ˜ ˜ B ˜ ˜ for x = x 1 to get μ A ˜ ˜ B ˜ ˜ ( x 1 ) = 0.3 0.7 0.1 0.1 + 0.3 1 0.1 0.2 + 1 0.7 0.5 0.1 + 1 1 0.5 0.2 = 0.3 0.1 + 0.3 0.2 + 0.7 0.5 + 1 0.5 = { ( 0.1 , 0.3 ) , ( 0.2 , 0.3 ) , ( 0.5 , max { 0.7 , 1 } ) } A ˜ ˜ B ˜ ˜ for x = x 1 , { ( ( x 1 , 0.1 ) , 0.3 ) , ( ( x 1 , 0.2 ) , 0.3 ) , ( ( x 1 , 0.5 ) , 1 ) }

A ˜ ˜ B ˜ ˜ for x = x 2 to get μ A ˜ ˜ B ˜ ˜ ( x 2 ) = 1 1 0.5 0.6 + 0.3 1 0.6 0.6 = 1 0.6 + 0.3 0.6 { ( 0.6 , max { 1 , 0.3 } ) } A ˜ ˜ B ˜ ˜ for x = x 2 { ( ( x 2 , 0.6 ) , 1 ) }

A ˜ ˜ B ˜ ˜ for x = x 3 to get μ A ˜ ˜ B ˜ ˜ ( x 3 ) = 1 0.6 0.8 0.5 + 1 1 0.8 0.9 = 0.6 0.8 + 1 0.9 = { ( 0.8 , 0.6 ) , ( 0.9 , 1 ) } A ˜ ˜ B ˜ ˜ for x = x 3 , { ( ( x 3 , 0.8 ) , 0.6 ) , ( ( x 3 , 0.9 ) , 1 ) }

A ˜ ˜ B ˜ ˜ = { ( ( x 1 , 0.1 ) , 0.3 ) , ( ( x 1 , 0.2 ) , 0.3 ) , ( ( x 1 , 0.5 ) , 1 ) , ( ( x 2 , 0.6 ) , 1 ) , ( ( x 3 , 0.8 ) , 0.6 ) , ( ( x 3 , 0.9 ) , 1 ) }

A ˜ ˜ B ˜ ˜ for x = x 1 to get μ A ˜ ˜ B ˜ ˜ ( x 1 ) = 0.3 0.7 0.1 0.1 + 0.3 1 0.1 0.2 + 1 0.7 0.5 0.1 + 1 1 0.5 0.2 = 0.3 0.1 + 0.3 0.1 + 0.7 0.1 + 1 0.2 = { ( 0.1 , max { 0.3 , 0.3 , 0.7 } ) , ( 0.2 , 1 ) } A ˜ ˜ B ˜ ˜ for x = x 1 , { ( ( x 1 , 0.1 ) , 0.7 ) , ( ( x 1 , 0.2 ) , 1 ) }

A ˜ ˜ B ˜ ˜ for x = x 2 to get μ A ˜ ˜ B ˜ ˜ ( x 2 ) = 1 1 0.5 0.6 + 0.3 1 0.6 0.6 = 1 0.5 + 0.3 0.6 { ( 0.5 , 1 ) , ( 0.6 , 0.3 ) } A ˜ ˜ B ˜ ˜ for x = x 2 , { ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.6 ) , 0.3 ) }

A ˜ ˜ B ˜ ˜ for x = x 3 to get μ A ˜ ˜ B ˜ ˜ ( x 3 ) = 1 0.6 0.8 0.5 + 1 1 0.8 0.9 = 0.6 0.5 + 1 0.8 { ( 0.5 , 0.6 ) , ( 0.8 , 1 ) } A ˜ ˜ B ˜ ˜ for x = x 3 , { ( ( x 3 , 0.5 ) , 0.6 ) , ( ( x 3 , 0.8 ) , 1 ) }

A ˜ ˜ B ˜ ˜ = { ( ( x 1 , 0.1 ) , 0.7 ) , ( ( x 1 , 0.2 ) , 1 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.6 ) , 0.3 ) , ( ( x 3 , 0.5 ) , 0.6 ) , ( ( x 3 , 0.8 ) , 1 ) }

The complement of a type-2 fuzzy set A ˜ ˜ is

A ˜ ˜ = μ A ˜ ˜ ( x ) = u J x u f x ( u ) / ( 1 u ) ¬ μ A ˜ ˜ ( x ) , x X = { ( ( x 1 , 0.9 ) , 0.3 ) , ( ( x 1 , 0.5 ) , 1 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.4 ) , 0.3 ) , ( ( x 3 , 0.2 ) , 1 ) } .

Operations under collection of type-2 fuzzy sets 8: Let { A ˜ ˜ i : i } 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 i A ˜ ˜ i is defined as

[ i N A ˜ ˜ i ] ( x ) = x X u J x u i N ( f x ( u ) ) i i N ( u ) i (9)

2) The intersection i A ˜ ˜ i is defined as

[ i A ˜ ˜ i ] ( x ) = x X u J x u i N ( f x ( u ) ) i i N ( u ) i (10)

Proposition 9: Let { A ˜ ˜ i : i } be an arbitrary collection of type-2 fuzzy sets

subset of X such that is countable set and B ˜ ˜ be another type-2 fuzzy set of X, then

1) B ˜ ˜ [ i A ˜ ˜ i ] = i ( B ˜ ˜ A ˜ ˜ i ) .

2) B ˜ ˜ [ i A ˜ ˜ i ] = i ( B ˜ ˜ A ˜ ˜ i ) .

3) 1 [ i A ˜ ˜ i ] = i ( 1 A ˜ ˜ i ) .

4) 1 [ i A ˜ ˜ i ] = i ( 1 A ˜ ˜ i ) .

3. General Type-2 Fuzzy Topological Space

In this section we introduced the concept general type-2 fuzzy topology.

Definition 1: Let F ˜ ˜ be the collection of type-2 fuzzy set over X; then F ˜ ˜ is said to be general type-2 fuzzy topology on X if

1) ˜ ˜ , X ˜ ˜ F ˜ ˜

2) A ˜ ˜ B ˜ ˜ F ˜ ˜ for any A ˜ ˜ , B ˜ ˜ F ˜ ˜ .

3) i A ˜ ˜ i F ˜ ˜ for any A ˜ ˜ i F ˜ ˜ , countable set.

The pair ( X , F ˜ ˜ ) is called general type-2 fuzzy topological space over X.

Remark 2: Let ( X , F ˜ ˜ ) be general type-2 fuzzy topological space over X; then the members of F ˜ ˜ are said to be type-2 fuzzy open set in X and a type-2 fuzzy set A ˜ ˜ is said to be a type-2 fuzzy closed set in X, if its complement ~ A ˜ ˜ F ˜ ˜ .

Proposition 3: Let ( X , F ˜ ˜ ) be general type-2 fuzzy topological space over X then the following conditions hold:

1) ˜ ˜ , X ˜ ˜ 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) ˜ ˜ , X ˜ ˜ are type-2 fuzzy closed sets because they are the complements of the type-2 fuzzy open sets ˜ ˜ , X ˜ ˜ is respectively.

2) Let { A ˜ ˜ i : i } be an arbitrary collection of type-2 fuzzy closed sets, then

[ i A ˜ ˜ i ] ( x ) = x X u J x u i N ( f x ( u ) ) i i N ( u ) i = x X u J x u i N ( f x ( u ) ) i 1 ( i N ( 1 u ) ) i ( proposition 2 .7 part 3 ) = [ i ~ A ˜ ˜ i ] (x)

since arbitrary union of type-2 fuzzy open sets are open [ i ~ A ˜ ˜ i ] ( x ) is an open and [ i A ˜ ˜ i ] ( x ) is a type-2 fuzzy closed sets.

3) If A ˜ ˜ i ( i ) is type-2 fuzzy closed sets, then i A ˜ ˜ i is a type-2 fuzzy closed set, [finite intersection of type-2 fuzzy open sets are open].

Example 4: Let X = { x 1 , x 2 } and let A ˜ ˜ , ˜ ˜ and X ˜ ˜ be three type-2 fuzzy sets in X which are

˜ ˜ = ( ( x 1 , 0 ) , 1 ) , ( ( x 2 , 0 ) , 1 ) , X ˜ ˜ = { ( ( x 1 , 1 ) , 1 ) , ( ( x 2 , 1 ) , 1 ) }

A ˜ ˜ = { ( ( x 1 , 0.8 ) , 1 ) , ( ( x 1 , 0.6 ) , 0.7 ) , ( ( x 1 , 0.3 ) , 0.6 ) , ( ( x 2 , 0.8 ) , 0.9 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.4 ) , 0.5 ) } . ˜ ˜ X ˜ ˜ for x 1 : μ ˜ ˜ X ˜ ˜ ( x 1 ) = 1 1 0 1 = ( 1 , 1 ) = { ( ( x 1 , 1 ) , 1 ) } . ˜ ˜ X ˜ ˜ for x 2 : μ ˜ ˜ X ˜ ˜ ( x 2 ) = 1 1 0 1 = ( 1 , 1 ) = { ( ( x 2 , 1 ) , 1 ) } . ˜ ˜ X ˜ ˜ = { ( ( x 1 , 1 ) , 1 ) , ( ( x 2 , 1 ) , 1 ) } = X ˜ ˜

˜ ˜ X ˜ ˜ for x 1 : μ ˜ ˜ X ˜ ˜ ( x 1 ) = 1 1 0 1 = ( 0 , 1 ) = { ( ( x 1 , 0 ) , 1 ) } .

˜ ˜ X ˜ ˜ for x 2 : μ ˜ ˜ X ˜ ˜ ( x 2 ) = 1 1 0 1 = ( 0 , 1 ) = { ( ( x 2 , 0 ) , 1 ) } .

˜ ˜ X ˜ ˜ = { ( ( x 1 , 0 ) , 1 ) , ( ( x 2 , 0 ) , 1 ) } = ˜ ˜

˜ ˜ A ˜ ˜ for x 1 : μ ˜ ˜ A ˜ ˜ ( x 1 ) = 1 1 0 0.8 + 1 0.7 0 0.6 + 1 0.6 0 0.3 = { ( ( x 1 , 0. 8 ) , 1 ) , ( ( x 1 , 0. 6 ) , 0. 7 ) , ( ( x 1 , 0. 3 ) , 0. 6 ) }

˜ ˜ A ˜ ˜ for x 2 : μ ˜ ˜ A ˜ ˜ ( x 2 ) = 1 0.9 0 0.8 + 1 1 0 0.5 + 1 0.5 0 0.4 = { ( ( x 2 , 0. 8 ) , 0.9 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.4 ) , 0.5 ) }

˜ ˜ A ˜ ˜ = { ( ( x 1 , 0.8 ) , 1 ) , ( ( x 1 , 0.6 ) , 0.7 ) , ( ( x 1 , 0.3 ) , 0.6 ) , ( ( x 2 , 0.8 ) , 0.9 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.4 ) , 0.5 ) } = A ˜ ˜

˜ ˜ A ˜ ˜ for x 1 : μ ˜ ˜ A ˜ ˜ ( x 1 ) = 1 1 0 0.8 + 1 0.7 0 0.6 + 1 0.6 0 0.3 = 1 0 + 0.7 0 + 0.6 0 = ( 0 , max { 1 , 0.7 , 0.6 } ) { ( ( x 1 , 0 ) , 1 ) } ,

˜ ˜ A ˜ ˜ for x 2 : μ ˜ ˜ A ˜ ˜ ( x 2 ) = 1 0.9 0 0.8 + 1 1 0 0.5 + 1 0.5 0 0.4 = 0.9 0 + 1 0 + 0.5 0 = ( 0 , max { 0.9 , 1 , 0.5 } ) { ( ( x 2 , 0 ) , 1 ) } ,

˜ ˜ A ˜ ˜ = { ( ( x 1 , 0 ) , 1 ) , ( ( x 2 , 0 ) , 1 ) } = ˜ ˜

A ˜ ˜ X ˜ ˜ for x 1 : μ A ˜ ˜ X ˜ ˜ ( x 1 ) = 1 1 1 0.8 + 1 0.7 1 0.6 + 1 0.6 1 0.3 = 1 1 + 0.7 1 + 0.6 1 = ( 1 , max { 1 , 0.7 , 0.6 } ) { ( ( x 1 , 1 ) , 1 ) } ,

A ˜ ˜ X ˜ ˜ for x 2 : μ A ˜ ˜ X ˜ ˜ ( x 2 ) = 1 0.9 1 0.8 + 1 1 1 0.5 + 1 0.5 1 0.4 = 0.9 1 + 1 1 + 0.5 1 = ( 1 , max { 1 , 0.9 , 0.5 } ) { ( ( x 2 , 1 ) , 1 ) }

A ˜ ˜ X ˜ ˜ = X ˜ ˜

A ˜ ˜ X ˜ ˜ for x 1 : μ A ˜ ˜ X ˜ ˜ ( x 1 ) = 1 1 1 0.8 + 1 0.7 1 0.6 + 1 0.6 1 0.3 = 1 0.8 + 0.7 0.6 + 0.6 0.3 = { ( ( x 1 , 0.8 ) , 1 ) , ( ( x 1 , 0.6 ) , 0.7 ) , ( ( x 1 , 0.3 ) , 0.6 ) }

A ˜ ˜ X ˜ ˜ = { ( ( x 1 , 0.8 ) , 1 ) , ( ( x 1 , 0.6 ) , 0.7 ) , ( ( x 1 , 0.3 ) , 0.6 ) , ( ( x 2 , 0.8 ) , 0.9 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.4 ) , 0.5 ) } = A ˜ ˜

Then F ˜ ˜ = { X ˜ ˜ , ˜ ˜ , A ˜ ˜ } is general type-2 fuzzy topologies defined on X and the pair ( X , F ˜ ˜ ) is called general type-2 fuzzy topological space over X, every member of F ˜ ˜ is called type-2 fuzzy open sets.

Theorem 5: Let { F ˜ ˜ r : r } be a family of all general type-2 fuzzy topologies on X ; then r F ˜ ˜ r is general type-2 fuzzy topologies on X.

proof: we must prove three conditions of topologies,

1) ˜ ˜ , X ˜ ˜ { F ˜ ˜ r : r } ˜ ˜ , X ˜ ˜ r F ˜ ˜ r .

2) Let { A ˜ ˜ i : i } r F ˜ ˜ r , then A ˜ ˜ i F ˜ ˜ r for all i so

thus i A ˜ ˜ i r F ˜ ˜ r .

3) Let A ˜ ˜ , B ˜ ˜ r F ˜ ˜ r , then A ˜ ˜ , B ˜ ˜ F ˜ ˜ r and because F ˜ ˜ r are all general type-2 fuzzy topologies A ˜ ˜ B ˜ ˜ F r for all r , so A ˜ ˜ B ˜ ˜ r F ˜ ˜ r .

Remark 6: Let ( X , F ˜ ˜ 1 ) and ( X , F ˜ ˜ 2 ) be two general type-2 fuzzy topological spaces over the same universe X then ( X , F ˜ ˜ 1 F ˜ ˜ 2 ) need not be general type-2 fuzzy topological space over X, we can see that in example 3.7.

Example 7: Let X = { x 1 , x 2 } and F ˜ ˜ 1 = { X ˜ ˜ , ˜ ˜ , A ˜ ˜ } , F ˜ ˜ 2 = { X ˜ ˜ , ˜ ˜ , B ˜ ˜ } be two general type-2fuzzy topologies defined on X where A ˜ ˜ , B ˜ ˜ , ˜ ˜ and X ˜ ˜ defined as follows: ˜ ˜ = { ( ( x 1 , 0 ) , 1 ) , ( ( x 2 , 0 ) , 1 ) } ,

X ˜ ˜ = { ( ( x 1 , 1 ) , 1 ) , ( ( x 2 , 1 ) , 1 ) }

A ˜ ˜ = { ( ( x 1 , 0.8 ) , 1 ) , ( ( x 1 , 0.6 ) , 0.7 ) , ( ( x 1 , 0.3 ) , 0.6 ) , ( ( x 2 , 0.8 ) , 0.9 ) , ( ( x 2 , 0.5 ) , 1 ) , ( ( x 2 , 0.4 ) , 0.5 ) } .

B ˜ ˜ = { ( ( x 1 , 0.5 ) , 1 ) , ( ( x 1 , 0.6 ) , 0.2 ) , ( ( x 2 , 0.3 ) , 0.7 ) , ( ( x 2 , 0.9 ) , 1 ) } .

Let F ˜ ˜ 1 F ˜ ˜ 2 = { ˜ ˜ , X ˜ ˜ , A ˜ ˜ , B ˜ ˜ } so ( X , F ˜ ˜ 1 F ˜ ˜ 2 ) is not general type-2 fuzzy topological space over X since A ˜ ˜ B ˜ ˜ F ˜ ˜ 1 F ˜ ˜ 2 .

Definition 8: Let ( X , F ˜ ˜ ) be general type-2 fuzzy topological space over X and let A ˜ ˜ be type-2 fuzzy set over X. Then the type-2 fuzzy interior of A ˜ ˜ , denoted by int ( A ˜ ˜ ) , is defined as the union of all type-2 fuzzy open sets contained in A ˜ ˜ . That is,

int ( A ˜ ˜ ) = { G ˜ ˜ i : G ˜ ˜ i type-2 fuzzy open sets in X , G ˜ ˜ i A ˜ ˜ , i } , int ( A ˜ ˜ ) is the largest type-2 fuzzy open set contained in A ˜ ˜ .

Theorem 9: Let ( X , F ˜ ˜ ) be general type-2 fuzzy topological space over X, and let A ˜ ˜ , B ˜ ˜ be two type-2 fuzzy sets in X. Then

1) int ( ˜ ˜ ) = ˜ ˜ and int ( X ˜ ˜ ) = X ˜ ˜ .

2) int ( A ˜ ˜ ) A ˜ ˜ .

3) A ˜ ˜ is type-2 fuzzy open set if and only if int ( A ˜ ˜ ) = A ˜ ˜ .

4) int ( int ( A ˜ ˜ ) ) = int ( A ˜ ˜ ) .

5) A ˜ ˜ B ˜ ˜ int ( A ˜ ˜ ) int ( B ˜ ˜ ) .

6) int ( A ˜ ˜ B ˜ ˜ ) = int ( A ˜ ˜ ) int ( B ˜ ˜ ) .

Proof:

1) int ( A ˜ ˜ ) = { G ˜ ˜ i : G ˜ ˜ i type-2 fuzzy open sets in X , G ˜ ˜ i A ˜ ˜ , i } , ˜ ˜ is type-2 fuzzy open set in F ˜ ˜ and ˜ ˜ ˜ ˜ int ( ˜ ˜ ) = ˜ ˜ .

Now to prove int ( X ˜ ˜ ) = X ˜ ˜ ,

int ( X ˜ ˜ ) = { G ˜ ˜ i : G ˜ ˜ i type-2 fuzzy open sets in X , G ˜ ˜ i X ˜ ˜ , i } , X ˜ ˜ is type-2 fuzzy open set in F ˜ ˜ and X ˜ ˜ X ˜ ˜ int ( X ˜ ˜ ) = X ˜ ˜ .

2) To prove int ( A ˜ ˜ ) A ˜ ˜ , since int ( A ˜ ˜ ) = { G ˜ ˜ i : G ˜ ˜ i type-2 fuzzy open sets in X , G ˜ ˜ i A ˜ ˜ , i } , such that G ˜ ˜ i A ˜ ˜ that is A ˜ ˜ is type-2 membership function μ A ˜ ˜ ( x , u ) where x X and u J X [ 0 , 1 ] less than a type-2 membership function μ G ˜ ˜ i ( x , u ) where x X and w J X [ 0 , 1 ] such that w u and μ G ˜ ˜ i ( x , u ) μ A ˜ ˜ ( x , u ) , sup { μ G ˜ ˜ i ( x , u ) μ A ˜ ˜ ( x , u ) , w u } hence G ˜ ˜ i A ˜ ˜ G ˜ ˜ i int ( A ˜ ˜ ) , therefore int ( A ˜ ˜ ) A ˜ ˜ .

3) If A ˜ ˜ is type-2 fuzzy open set, then A ˜ ˜ int ( A ˜ ˜ ) , but int ( A ˜ ˜ ) A ˜ ˜ from part (2), hence int ( A ˜ ˜ ) = A ˜ ˜ .

4) int ( A ˜ ˜ ) is a type-2 fuzzy open set and from part (3) we have int ( int ( A ˜ ˜ ) ) = int ( A ˜ ˜ )

5) If A ˜ ˜ B ˜ ˜ and from part(2) int ( A ˜ ˜ ) A ˜ ˜ , int ( B ˜ ˜ ) B ˜ ˜ , then int ( A ˜ ˜ ) A ˜ ˜ B ˜ ˜ . Therefore int ( A ˜ ˜ ) B ˜ ˜ and int ( A ˜ ˜ ) is a type-2 fuzzy open set contained in B ˜ ˜ , so int ( A ˜ ˜ ) int ( B ˜ ˜ ) .

6) Because ( A ˜ ˜ B ˜ ˜ ) A ˜ ˜ and ( A ˜ ˜ B ˜ ˜ ) B ˜ ˜ , from part (5) int ( A ˜ ˜ B ˜ ˜ ) int ( A ˜ ˜ ) and int ( A ˜ ˜ B ˜ ˜ ) int ( B ˜ ˜ ) , thus int ( A ˜ ˜ B ˜ ˜ ) int ( A ˜ ˜ ) int ( B ˜ ˜ ) , since int ( A ˜ ˜ B ˜ ˜ ) A ˜ ˜ B ˜ ˜ , so int ( int ( A ˜ ˜ ) ) int ( B ˜ ˜ ) ( A ˜ ˜ B ˜ ˜ ) from part(5) but int ( A ˜ ˜ ) int ( B ˜ ˜ ) is a type-2 fuzzy open sets then int ( int ( A ˜ ˜ ) ) int ( B ˜ ˜ ) int ( A ˜ ˜ B ˜ ˜ ) from part(3).Hence int ( A ˜ ˜ B ˜ ˜ ) = int ( A ˜ ˜ ) int ( B ˜ ˜ ) .

Definition 10: Let ( X , F ˜ ˜ ) be general type-2 fuzzy topological space over X ˜ ˜ and let A ˜ ˜ be type-2 fuzzy set over X. Then the type-2 fuzzy closure of A ˜ ˜ , denoted by c l ( A ˜ ˜ ) , is defined as the intersection of all type-2 fuzzy closed sets containing A ˜ ˜ . That is

c l ( A ˜ ˜ ) = { M ˜ ˜ i : M ˜ ˜ i type-2 fuzzy closed sets in X , A ˜ ˜ M ˜ ˜ i , i } ,

c l ( A ˜ ˜ ) is the smallest type-2 fuzzy closed set containing A ˜ ˜ .

Theorem 11: Let ( X , F ˜ ˜ ) be general type-2 fuzzy topological space over X, and let A ˜ ˜ , B ˜ ˜ be two type-2 fuzzy sets in X. Then

1) c l ( ˜ ˜ ) = ˜ ˜ and c l ( X ˜ ˜ ) = X ˜ ˜ .

2) A ˜ ˜ c l ( A ˜ ˜ ) .

3) A ˜ ˜ is type-2 fuzzy closed set if and only if c l ( A ˜ ˜ ) = A ˜ ˜ .

4) c l ( c l ( A ˜ ˜ ) ) = c l ( A ˜ ˜ ) .

5) A ˜ ˜ B ˜ ˜ c l ( A ˜ ˜ ) c l ( B ˜ ˜ ) .

6) c l ( A ˜ ˜ B ˜ ˜ ) = c l ( A ˜ ˜ ) c l ( B ˜ ˜ ) .

Proof: The proof this theorem similar to the proof of theorem 3.7.

Definition 12: Let ( X , F ˜ ˜ ) be a general type-2 fuzzy topological space over X and N ˜ ˜ F ˜ ˜ . Then is said to be a neighborhood or nbhd for short, of a type-2 fuzzy set A ˜ ˜ if there exist a type-2 fuzzy open set W ˜ ˜ such that A ˜ ˜ W ˜ ˜ N ˜ ˜ .

Proposition 13: A type-2 fuzzy set A ˜ ˜ is open if and only if for each type-2 fuzzy set B ˜ ˜ contained in A ˜ ˜ , A ˜ ˜ is a neighborhood of B ˜ ˜ .

Proof: If A ˜ ˜ is open and B ˜ ˜ A ˜ ˜ then A ˜ ˜ is a neighborhood of B ˜ ˜ . Conversely, since A ˜ ˜ A ˜ ˜ , there exists a type-2 fuzzy open set W ˜ ˜ such that A ˜ ˜ W ˜ ˜ A ˜ ˜ . Hence A ˜ ˜ = W ˜ ˜ and A ˜ ˜ is open.

Definition 14: Let ( X , F ˜ ˜ ) be a general type-2 fuzzy topological space over X

and B ˜ ˜ be a subfamily of F ˜ ˜ . If every member of F ˜ ˜ can be written as the type-2 fuzzy union of some members of B ˜ ˜ , then B ˜ ˜ is called a type-2 fuzzy base for the general type-2 fuzzy topology F ˜ ˜ . We can see that if B ˜ ˜ be type-2 fuzzy base for F ˜ ˜ then F ˜ ˜ equals the collection of type-2 fuzzy unions of elements of B ˜ ˜ .

Definition 15: Let ( X , F ˜ ˜ ) and ( Y , S ˜ ˜ ) 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 Y X and the open subsets of Y are precisely of the form F ˜ ˜ Y ˜ ˜ = { Y ˜ ˜ = Y ˜ ˜ X ˜ ˜ : X ˜ ˜ F ˜ ˜ } . Here we may say that each open subset Y ˜ ˜ of Y is the restriction to Y ˜ ˜ of an open subset X ˜ ˜ of X. That is, ( Y , S ˜ ˜ ) is called a subspace of ( X , F ˜ ˜ ) if the type-2 fuzzy open sets of Y are the type-2 fuzzy intersection of open sets of X with Y ˜ ˜ .

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.

Cite this paper
AL-Khafaji, M. and Hussan, M. (2018) General Type-2 Fuzzy Topological Spaces. Advances in Pure Mathematics, 8, 771-781. doi: 10.4236/apm.2018.89047.
References
[1]   Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.
https://doi.org/10.1016/S0019-9958(65)90241-X

[2]   Zhang, Z. (2013) On Characterization of Generalized Interval Type-2 Fuzzy Rough Sets. Information Sciences, 219, 124-150.
https://doi.org/10.1016/j.ins.2012.07.013

[3]   Chang, C.L. (1968) Fuzzy Topological Space. The Journal of Mathematical Analysis and Applications, 24, 182-190.
https://doi.org/10.1016/0022-247X(68)90057-7

[4]   Karnik, N.N. and Mendel, J.M. (2001) Operations on Type-2 Fuzzy Sets. Fuzzy Sets and Systems, 122, 327-348.
https://doi.org/10.1016/S0165-0114(00)00079-8

[5]   Mizumoto, M. and Tanaka, K. (1976) Some Properties of Fuzzy Sets of Type-2. Information and Control, 31, 312-340.
https://doi.org/10.1016/S0019-9958(76)80011-3

[6]   Mendel, J.M. (2001) Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper Saddle River, NJ.

[7]   Mendel, J.M. and John, R.I. (2002) Type-2 Fuzzy Sets Made Simple. IEEE Transactions on Fuzzy Systems, 10, 117-127.
https://doi.org/10.1109/91.995115

[8]   Zhao, T. and Wei, Z. (2015) On Characterization of Rough Type-2 Fuzzy Sets. Mathematical Problems in Engineering, 2016, 13.

[9]   Zadeh, L.A. (1975) The Concept of a Linguistic Variable and Its Application to Approximate Reasoning-1. Information Sciences, 8, 199-249.
https://doi.org/10.1016/0020-0255(75)90036-5

 
 
Top