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.