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 space s w ere 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.

Type-2 Fuzzy Set Interval Type-2 Fuzzy Topological Space General Type-2 Fuzzy Topological Spaces Type-2 Fuzzy Open Sets Type-2 Fuzzy Closed Sets Type-2 Fuzzy Interior Type-2 Fuzzy Closure Neighborhood of a Type-2 Fuzzy Set
1. Introduction

The fuzzy set theory proposed by Zadeh  extended the classical notion of sets and permitted the gradual assessment of membership of elements in a set  . 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  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  . 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],   . Many scholars have conducted research on type-2 fuzzy set and their properties, including Mizumoto and Tanaka  , Mendel  , Karnik and Mendel  and Mendel and John  . Type-2 fuzzy sets are called “fuzzy”, so, it could be called fuzzy set  . In  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  . 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  , 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   and throughout this paper, let X be anon empty set and I be closed unit interval, i.e., I = [ 0 , 1 ] .

Definition 1   . 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   . 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   . (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   . (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  . (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  .

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 ∈ ℕ } be an

arbitrary collection of type-2 fuzzy sets subset of X such that ℕ is countable set, operation are possible under an arbitrary collection of type-2 fuzzy sets.

1) The union ∪ i ∈ ℕ 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 ∈ ℕ 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 (10)

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

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

1) B ˜ ˜ ∩ [ ∪ i ∈ ℕ A ˜ ˜ i ] = ∪ i ∈ ℕ ( B ˜ ˜ ∩ A ˜ ˜ i ) .

2) B ˜ ˜ ∪ [ ∩ i ∈ ℕ A ˜ ˜ i ] = ∩ i ∈ ℕ ( B ˜ ˜ ∪ A ˜ ˜ i ) .

3) 1 − [ ∪ i ∈ ℕ A ˜ ˜ i ] = ∩ i ∈ ℕ ( 1 − A ˜ ˜ i ) .

4) 1 − [ ∩ i ∈ ℕ A ˜ ˜ i ] = ∪ i ∈ ℕ ( 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 ∈ ℕ A ˜ ˜ i ∈ F ˜ ˜ for any A ˜ ˜ i ∈ F ˜ ˜ , ℕ 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 ∈ ℕ } be an arbitrary collection of type-2 fuzzy closed sets, then

[ ∩ i ∈ ℕ 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 ∈ ℕ ~ A ˜ ˜ i ] (x)

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

3) If A ˜ ˜ i ( i ∈ ℕ ) is type-2 fuzzy closed sets, then ∪ i ∈ ℕ 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 ∈ ℝ } be a family of all general type-2 fuzzy topologies on X ; then ∩ r ∈ ℝ F ˜ ˜ r is general type-2 fuzzy topologies on X.

proof: we must prove three conditions of topologies,

1) ∅ ˜ ˜ , X ˜ ˜ ∈ { F ˜ ˜ r : r ∈ ℝ } ⇒ ∅ ˜ ˜ , X ˜ ˜ ∈ ∩ r ∈ ℝ F ˜ ˜ r .

2) Let { A ˜ ˜ i : i ∈ ℕ } ⊆ ∩ r ∈ ℝ F ˜ ˜ r , then A ˜ ˜ i ∈ F ˜ ˜ r for all i ∈ ℕ so

thus ∪ i ∈ ℕ A ˜ ˜ i ∈ ∩ r ∈ ℝ F ˜ ˜ r .

3) Let A ˜ ˜ , B ˜ ˜ ∈ ∩ r ∈ 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 ∈ ℝ , so A ˜ ˜ ∩ B ˜ ˜ ∈ ∩ r ∈ 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 ∈ ℕ } , 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 ∈ ℕ } , ∅ ˜ ˜ 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 ∈ ℕ } , 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 ∈ ℕ } , 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 ∈ ℕ } ,

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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

AL-Khafaji, M.A.K. and Hussan, M.S.M. (2018) General Type-2 Fuzzy Topological Spaces. Advances in Pure Mathematics, 8, 771-781. https://doi.org/10.4236/apm.2018.89047

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