Munir Abdul Khalik AL-Khafaji, Mohammed Salih Mahdy Hussan
Department of Mathematics, College of Education, AL-Mustinsiryah University, Baghdad, Iraq.
DOI: 10.4236/apm.2018.89047   PDF    HTML   XML   1,531 Downloads   3,612 Views   Citations

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.

Keywords

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

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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., $I=\left[0,1\right]$ .

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 $\widetilde{\stackrel{˜}{A}}$ is characterized by a type-2 memberships function ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)$ , as

${\mu }_{\widetilde{\stackrel{˜}{A}}}:X\times \left[0,1\right]\to {\left[0,1\right]}^{J_x}\left(J_x\subseteq \left[0,1\right]\right)$ , where $x\in X$ and $u\in J_x$ , that is

$\widetilde{\stackrel{˜}{A}}=\left\{\left(\left(x,u\right),{\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)\right):\text{where}x\in X\text{and}u\in J_x\subseteq \left[0,1\right],\text{where}0\le {\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)\le 1\right\}$ (1)

$\widetilde{\stackrel{˜}{A}}$ can also be expressed as

$\begin{array}{c}\widetilde{\stackrel{˜}{A}}={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in Jx}{\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)/\left(x,u\right)}}\\ ={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in Jx}{f}_x\left(u\right)/u/x}},J_x\subseteq \left[0,1\right]\end{array}$ (2)

where $f_x\left(u\right)={\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)$ an ${{\displaystyle \sum }}^{\text{}}{{\displaystyle \sum }}^{\text{}}$ denotes union over all admissible x and u for continuous universes of discourse, ${{\displaystyle \sum }}^{\text{}}$ is replaced by ${{\displaystyle \int }}^{\text{}}$ . The class of all type-2 fuzzy sets of the universe X denoted by ${\widetilde{\stackrel{˜}{\mathbb{F}}}}_{T_2}\left(X\right)$ .

Definition 2 [2] [7] . A vertical slice, denoted ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x^{\prime }\right)$ , of $\widetilde{\stackrel{˜}{A}}$ , is the intersection between the two-dimensional plane whose axes are u and ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x^{\prime },u\right)$ and the three-dimensional type-2membership function $\widetilde{\stackrel{˜}{A}}$ , i.e.,

${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x^{\prime }\right)={\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x=x^{\prime },u\right)={\displaystyle {\sum }_{u\in J_{x^{\prime }}}{f}_{x^{\prime }}\left(u\right)/u},J_{x^{\prime }}\subseteq I$ in which $0\le f_{x^{\prime }}\left(u\right)\le 1$ . $\widetilde{\stackrel{˜}{A}}$ can also be expressed as follows: $\widetilde{\stackrel{˜}{A}}=\left\{\left(x,{\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)\right):\forall x\in X\right\}$ or as following

$\begin{array}{c}\widetilde{\stackrel{˜}{A}}={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in Jx}{\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)/\left(x\right)}}\\ ={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in Jx}{f}_x\left(u\right)/u/x}},J_x\subseteq \left[0,1\right]\end{array}$ (3)

The vertical slice, ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x^{\prime }\right)$ is also called the secondary membership function, and its domain is called the primary membership of x, which is denoted by $J_X$ where $J_X\subseteq I$ for any $x\in X$ . 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 $\widetilde{\stackrel{˜}{X}}$ , such that

$\widetilde{\stackrel{˜}{X}}={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in \left[1,1\right]}1/u/x}}$ (4)

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

A type-2 fuzzy empty set, denoted $\widetilde{\stackrel{˜}{\varnothing }}$ , such that

$\widetilde{\stackrel{˜}{\varnothing }}={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in \left[0,0\right]}1/u/x}}$ (5)

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

When all the secondary grades of types $\widetilde{\stackrel{˜}{A}}$ are equal to 1, that is ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)=1$ for all $x\in X$ and for all $u\in J_x\subseteq \left[0,1\right]$ , $\widetilde{\stackrel{˜}{A}}$ is as an Interval type-2 fuzzy set.

Operation of Types-2 fuzzy sets 6. Consider two type-2 fuzzy sets, $\widetilde{\stackrel{˜}{A}}$ and $\widetilde{\stackrel{˜}{B}}$ , in a universe X. Let ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)$ and ${\mu }_{\widetilde{\stackrel{˜}{B}}}\left(x\right)$ be the membership grades of these two sets, which are represented for each $x\in X$ , ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)={\displaystyle {\sum }_{u\in J_x^u}{f}_x\left(u\right)/u}$ and ${\mu }_{\widetilde{\stackrel{˜}{B}}}\left(x\right)={\displaystyle {\sum }_{w\in J_x^w}{g}_x\left(w\right)/w}$ , respective, where $u\in J_x^u$ , $w\in J_x^w$ indicate the primary memberships of x and $f_x\left(u\right),g_x\left(w\right)\in \left[0,1\right]$ indicate the secondary memberships (grades) of x. The membership grades for the union, intersection and complement of the type-2 fuzzy sets $\widetilde{\stackrel{˜}{A}}$ and $\widetilde{\stackrel{˜}{B}}$ have been defined as follows [5] .

Containment:

$\widetilde{\stackrel{˜}{A}}$ is a subtype-2 fuzzy set of $\widetilde{\stackrel{˜}{B}}$ denoted $\widetilde{\stackrel{˜}{A}}\subseteq \widetilde{\stackrel{˜}{B}}$ if $u\le w$ and $f_x\left(u\right)\le g_x\left(w\right)$ for every $x\in X$ .

Equality:

$\widetilde{\stackrel{˜}{A}}$ and $\widetilde{\stackrel{˜}{B}}$ are type-2 fuzzy sets are equal, denoted $\widetilde{\stackrel{˜}{A}}=\widetilde{\stackrel{˜}{B}}$ if $u=w$ and $f_x\left(u\right)={\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)=g_x\left(w\right)={\mu }_{\widetilde{\stackrel{˜}{B}}}\left(x,w\right)$ for every $x\in X$ .

Union of two type-2 fuzzy sets:

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}\iff {\mu }_{\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}}\left(x\right)={\displaystyle {\sum }_{u\in J_x^u}{\displaystyle {\sum }_{w\in J_x^w}{f}_x\left(u\right)\star g_x\left(w\right)/\left(u\vee w\right)}}\\ \equiv {\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)\bigsqcup {\mu }_{\widetilde{\stackrel{˜}{B}}}\left(x\right),x\in X\end{array}$ (6)

Intersection of two type-2 fuzzy sets:

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\iff {\mu }_{\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}}\left(x\right)={\displaystyle {\sum }_{u\in J_x^u}{\displaystyle {\sum }_{w\in J_x^w}{f}_x\left(u\right)\star g_x\left(w\right)/\left(u\vee w\right)}}\\ \equiv {\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)\sqcap {\mu }_{\widetilde{\stackrel{˜}{B}}}\left(x\right),x\in X\end{array}$ (7)

Complement of a type-2 fuzzy set:

$\sim \widetilde{\stackrel{˜}{A}}={\mu }_{\sim \widetilde{\stackrel{˜}{A}}}\left(x\right)={\displaystyle {\sum }_{u\in J_x^u}{f}_x\left(u\right)/\left(1-u\right)}\equiv \neg {\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right),x\in X$ (8)

Where $\vee$ represent the max t-conorm and $\star$ represent a t-norm. The summation indicate logical unions. We refer to the operations $\bigsqcup ,\sqcap$ and $\neg$ as join, meet and negation respectively and ${\mu }_{\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}}\left(x\right)$ , ${\mu }_{\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}}\left(x\right)$ , ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)$ and ${\mu }_{\widetilde{\stackrel{˜}{B}}}\left(x\right)$ are the secondary membership functions and all are type-1 fuzzy sets. If ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right)$ and ${\mu }_{\widetilde{\stackrel{˜}{B}}}\left(x\right)$ have continuous domains, then the summations in 3, 4 and 5 are replaced by integrals.

Example 7: Let $X=\left\{x_1,x_2,x_3\right\}$ be anon empty set, and let $\widetilde{\stackrel{˜}{A}}$ and $\widetilde{\stackrel{˜}{B}}$ are type-2 fuzzy sets over the same universe X.

$\widetilde{\stackrel{˜}{A}}=\left\{\left(\left(x_1,0.1\right),0.3\right),\left(\left(x_1,0.5\right),1\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.6\right),0.3\right),\left(\left(x_3,0.8\right),1\right)\right\}$

$\widetilde{\stackrel{˜}{B}}=\left\{\left(\left(x_1,0.1\right),0.7\right),\left(\left(x_1,0.2\right),1\right),\left(\left(x_2,0.6\right),1\right),\left(\left(x_3,0.5\right),0.6\right),\left(\left(x_3,0.9\right),1\right)\right\}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}\text{for}x=x_1\text{to}\text{get}\\ {\mu }_{\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}}\left(x_1\right)=\frac{0.3\wedge 0.7}{0.1\vee 0.1}+\frac{0.3\wedge 1}{0.1\vee 0.2}+\frac{1\wedge 0.7}{0.5\vee 0.1}+\frac{1\wedge 1}{0.5\vee 0.2}\\ =\frac{0.3}{0.1}+\frac{0.3}{0.2}+\frac{0.7}{0.5}+\frac{1}{0.5}=\left\{\left(0.1,0.3\right),\left(0.2,0.3\right),\left(0.5,\max \left\{0.7,1\right\}\right)\right\}\\ \widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}\text{for}x=x_1,\left\{\left(\left(x_1,0.1\right),0.3\right),\left(\left(x_1,0.2\right),0.3\right),\left(\left(x_1,0.5\right),1\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}\text{for}x=x_2\text{to}\text{get}\\ {\mu }_{\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}}\left(x_2\right)=\frac{1\wedge 1}{0.5\vee 0.6}+\frac{0.3\wedge 1}{0.6\vee 0.6}=\frac{1}{0.6}+\frac{0.3}{0.6}\Rightarrow \left\{\left(0.6,\max \left\{1,0.3\right\}\right)\right\}\\ \widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}\text{for}x=x_2\Rightarrow \left\{\left(\left(x_2,0.6\right),1\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}\text{for}x=x_3\text{to}\text{get}\\ {\mu }_{\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}}\left(x_3\right)=\frac{1\wedge 0.6}{0.8\vee 0.5}+\frac{1\wedge 1}{0.8\vee 0.9}=\frac{0.6}{0.8}+\frac{1}{0.9}=\left\{\left(0.8,0.6\right),\left(0.9,1\right)\right\}\\ \widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}\text{for}x=x_3,\left\{\left(\left(x_3,0.8\right),0.6\right),\left(\left(x_3,0.9\right),1\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{B}}=\{\left(\left(x_1,0.1\right),0.3\right),\left(\left(x_1,0.2\right),0.3\right),\left(\left(x_1,0.5\right),1\right),\left(\left(x_2,0.6\right),1\right),\\ \left(\left(x_3,0.8\right),0.6\right),\left(\left(x_3,0.9\right),1\right)\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\text{for}x=x_1\text{to}\text{get}\\ {\mu }_{\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}}\left(x_1\right)=\frac{0.3\wedge 0.7}{0.1\wedge 0.1}+\frac{0.3\wedge 1}{0.1\wedge 0.2}+\frac{1\wedge 0.7}{0.5\wedge 0.1}+\frac{1\wedge 1}{0.5\wedge 0.2}\\ =\frac{0.3}{0.1}+\frac{0.3}{0.1}+\frac{0.7}{0.1}+\frac{1}{0.2}=\left\{\left(0.1,\max \left\{0.3,0.3,0.7\right\}\right),\left(0.2,1\right)\right\}\\ \widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\text{for}x=x_1,\left\{\left(\left(x_1,0.1\right),0.7\right),\left(\left(x_1,0.2\right),1\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\text{for}x=x_2\text{to}\text{get}\\ {\mu }_{\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}}\left(x_2\right)=\frac{1\wedge 1}{0.5\wedge 0.6}+\frac{0.3\wedge 1}{0.6\wedge 0.6}=\frac{1}{0.5}+\frac{0.3}{0.6}\Rightarrow \left\{\left(0.5,1\right),\left(0.6,0.3\right)\right\}\\ \widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\text{for}x=x_2,\left\{\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.6\right),0.3\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\text{for}x=x_3\text{to}\text{get}\\ {\mu }_{\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}}\left(x_3\right)=\frac{1\wedge 0.6}{0.8\wedge 0.5}+\frac{1\wedge 1}{0.8\wedge 0.9}=\frac{0.6}{0.5}+\frac{1}{0.8}\Rightarrow \left\{\left(0.5,0.6\right),\left(0.8,1\right)\right\}\\ \widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\text{for}x=x_3,\left\{\left(\left(x_3,0.5\right),0.6\right),\left(\left(x_3,0.8\right),1\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}=\{\left(\left(x_1,0.1\right),0.7\right),\left(\left(x_1,0.2\right),1\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.6\right),0.3\right),\\ \left(\left(x_3,0.5\right),0.6\right),\left(\left(x_3,0.8\right),1\right)\}\end{array}$

The complement of a type-2 fuzzy set $\widetilde{\stackrel{˜}{A}}$ is

$\begin{array}{l}\sim \widetilde{\stackrel{˜}{A}}={\mu }_{\sim \widetilde{\stackrel{˜}{A}}}\left(x\right)={\displaystyle {\sum }_{u\in J_x^u}{f}_x\left(u\right)/\left(1-u\right)}\\ \equiv \neg {\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x\right),x\in X\\ =\left\{\left(\left(x_1,0.9\right),0.3\right),\left(\left(x_1,0.5\right),1\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.4\right),0.3\right),\left(\left(x_3,0.2\right),1\right)\right\}.\end{array}$

Operations under collection of type-2 fuzzy sets 8: Let $\left\{{\widetilde{\stackrel{˜}{A}}}_i:i\in \mathbb{N}\right\}$ be an

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

1) The union ${\cup }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i$ is defined as

$\left[{\cup }_{i\in \mathcal{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]\left(x\right)={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in J_x^u}\frac{{\wedge }_{i\in \mathcal{N}}{\left(f_x\left(u\right)\right)}_i}{{\vee }_{i\in \mathcal{N}}{\left(u\right)}_i}}}$ (9)

2) The intersection ${\cap }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i$ is defined as

$\left[{\cap }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]\left(x\right)={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in J_x^u}\frac{{\wedge }_{i\in \mathcal{N}}{\left(f_x\left(u\right)\right)}_i}{{\wedge }_{i\in \mathcal{N}}{\left(u\right)}_i}}}$ (10)

Proposition 9: Let $\left\{{\widetilde{\stackrel{˜}{A}}}_i:i\in \mathbb{N}\right\}$ be an arbitrary collection of type-2 fuzzy sets

subset of X such that $\mathbb{N}$ is countable set and $\widetilde{\stackrel{˜}{B}}$ be another type-2 fuzzy set of X, then

1) $\widetilde{\stackrel{˜}{B}}\cap \left[{\cup }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]={\cup }_{i\in \mathbb{N}}\left(\widetilde{\stackrel{˜}{B}}\cap {\widetilde{\stackrel{˜}{A}}}_i\right)$ .

2) $\widetilde{\stackrel{˜}{B}}\cup \left[{\cap }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]={\cap }_{i\in \mathbb{N}}\left(\widetilde{\stackrel{˜}{B}}\cup {\widetilde{\stackrel{˜}{A}}}_i\right)$ .

3) $1-\left[{\cup }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]={\cap }_{i\in \mathbb{N}}\left(1-{\widetilde{\stackrel{˜}{A}}}_i\right)$ .

4) $1-\left[{\cap }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]={\cup }_{i\in \mathbb{N}}\left(1-{\widetilde{\stackrel{˜}{A}}}_i\right)$ .

3. General Type-2 Fuzzy Topological Space

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

Definition 1: Let $\widetilde{\stackrel{˜}{\mathfrak{F}}}$ be the collection of type-2 fuzzy set over X; then $\widetilde{\stackrel{˜}{\mathfrak{F}}}$ is said to be general type-2 fuzzy topology on X if

1) $\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{X}}\in \widetilde{\stackrel{˜}{\mathfrak{F}}}$

2) $\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\in \widetilde{\stackrel{˜}{\mathfrak{F}}}$ for any $\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{B}}\in \widetilde{\stackrel{˜}{\mathfrak{F}}}$ .

3) ${\cup }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\in \widetilde{\stackrel{˜}{\mathfrak{F}}}$ for any ${\widetilde{\stackrel{˜}{A}}}_i\in \widetilde{\stackrel{˜}{\mathfrak{F}}}$ , $\mathbb{N}$ countable set.

The pair $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ is called general type-2 fuzzy topological space over X.

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

Proposition 3: Let $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ be general type-2 fuzzy topological space over X then the following conditions hold:

1) $\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{X}}$ 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) $\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{X}}$ are type-2 fuzzy closed sets because they are the complements of the type-2 fuzzy open sets $\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{X}}$ is respectively.

2) Let $\left\{{\widetilde{\stackrel{˜}{A}}}_i:i\in \mathbb{N}\right\}$ be an arbitrary collection of type-2 fuzzy closed sets, then

$\begin{array}{c}\left[{\cap }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]\left(x\right)={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in J_x^u}\frac{{\wedge }_{i\in \mathcal{N}}{\left(f_x\left(u\right)\right)}_i}{{\wedge }_{i\in \mathcal{N}}{\left(u\right)}_i}}}\\ ={\displaystyle {\sum }_{x\in X}{\displaystyle {\sum }_{u\in J_x^u}\frac{{\wedge }_{i\in \mathcal{N}}{\left(f_x\left(u\right)\right)}_i}{1-{\left({\vee }_{i\in \mathcal{N}}\left(1-u\right)\right)}_i}}}\left(\text{proposition}\text{2}\text{.7}\text{part}\text{3}\right)\\ =\left[{\cup }_{i\in \mathbb{N}}~{\widetilde{\stackrel{˜}{A}}}_i\right](x)\end{array}$

since arbitrary union of type-2 fuzzy open sets are open $\left[{\cup }_{i\in \mathbb{N}}~{\widetilde{\stackrel{˜}{A}}}_i\right]\left(x\right)$ is an open and $\left[{\cap }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\right]\left(x\right)$ is a type-2 fuzzy closed sets.

3) If ${\widetilde{\stackrel{˜}{A}}}_i\left(i\in \mathbb{N}\right)$ is type-2 fuzzy closed sets, then ${\cup }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i$ is a type-2 fuzzy closed set, [finite intersection of type-2 fuzzy open sets are open].

Example 4: Let $X=\left\{x_1,x_2\right\}$ and let $\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{\varnothing }}$ and $\widetilde{\stackrel{˜}{X}}$ be three type-2 fuzzy sets in X which are

$\widetilde{\stackrel{˜}{\varnothing }}=\left(\left(x_1,0\right),1\right),\left(\left(x_2,0\right),1\right)$ , $\widetilde{\stackrel{˜}{X}}=\left\{\left(\left(x_1,1\right),1\right),\left(\left(x_2,1\right),1\right)\right\}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}=\{\left(\left(x_1,0.8\right),1\right),\left(\left(x_1,0.6\right),0.7\right),\left(\left(x_1,0.3\right),0.6\right),\\ \left(\left(x_2,0.8\right),0.9\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.4\right),0.5\right)\}.\end{array}$ $\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{X}}\text{for}{x}_1:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{X}}}\left(x_1\right)=\frac{1\wedge 1}{0\vee 1}\Rightarrow =\left(1,1\right)\Rightarrow =\left\{\left(\left(x_1,1\right),1\right)\right\}.$ $\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{X}}\text{for}{x}_2:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{X}}}\left(x_2\right)=\frac{1\wedge 1}{0\vee 1}\Rightarrow =\left(1,1\right)\Rightarrow =\left\{\left(\left(x_2,1\right),1\right)\right\}.$ $\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{X}}=\left\{\left(\left(x_1,1\right),1\right),\left(\left(x_2,1\right),1\right)\right\}=\widetilde{\stackrel{˜}{X}}$

$\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{X}}\text{for}{x}_1:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{X}}}\left(x_1\right)=\frac{1\wedge 1}{0\wedge 1}\Rightarrow =\left(0,1\right)\Rightarrow =\left\{\left(\left(x_1,0\right),1\right)\right\}.$

$\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{X}}\text{for}{x}_2:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{X}}}\left(x_2\right)=\frac{1\wedge 1}{0\wedge 1}\Rightarrow =\left(0,1\right)\Rightarrow =\left\{\left(\left(x_2,0\right),1\right)\right\}.$

$\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{X}}=\left\{\left(\left(x_1,0\right),1\right),\left(\left(x_2,0\right),1\right)\right\}=\widetilde{\stackrel{˜}{\varnothing }}$

$\begin{array}{l}\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{A}}\text{for}{x}_1:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{A}}}\left(x_1\right)=\frac{1\wedge 1}{0\vee 0.8}+\frac{1\wedge 0.7}{0\vee 0.6}+\frac{1\wedge 0.6}{0\vee 0.3}\\ =\left\{\left(\left(x_1,0.\text{8}\right),\text{1}\right),\left(\left(x_1,0.\text{6}\right),0.\text{7}\right),\left(\left(x_1,0.\text{3}\right),0.\text{6}\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{A}}\text{for}{x}_2:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{A}}}\left(x_2\right)=\frac{1\wedge 0.9}{0\vee 0.8}+\frac{1\wedge 1}{0\vee 0.5}+\frac{1\wedge 0.5}{0\vee 0.4}\\ =\left\{\left(\left(x_2,0.\text{8}\right),0.9\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.4\right),0.5\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{\varnothing }}\cup \widetilde{\stackrel{˜}{A}}=\{\left(\left(x_1,0.8\right),1\right),\left(\left(x_1,0.6\right),0.7\right),\left(\left(x_1,0.3\right),0.6\right),\\ \left(\left(x_2,0.8\right),0.9\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.4\right),0.5\right)\}=\widetilde{\stackrel{˜}{A}}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{A}}\text{for}{x}_1:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{A}}}\left(x_1\right)=\frac{1\wedge 1}{0\wedge 0.8}+\frac{1\wedge 0.7}{0\wedge 0.6}+\frac{1\wedge 0.6}{0\wedge 0.3}=\frac{1}{0}+\frac{0.7}{0}+\frac{0.6}{0}\\ =\left(0,\max \left\{1,0.7,0.6\right\}\right)\Rightarrow \left\{\left(\left(x_1,0\right),1\right)\right\},\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{A}}\text{for}{x}_2:{\mu }_{\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{A}}}\left(x_2\right)=\frac{1\wedge 0.9}{0\wedge 0.8}+\frac{1\wedge 1}{0\wedge 0.5}+\frac{1\wedge 0.5}{0\wedge 0.4}=\frac{0.9}{0}+\frac{1}{0}+\frac{0.5}{0}\\ =\left(0,\max \left\{0.9,1,0.5\right\}\right)\Rightarrow \left\{\left(\left(x_2,0\right),1\right)\right\},\end{array}$

$\widetilde{\stackrel{˜}{\varnothing }}\cap \widetilde{\stackrel{˜}{A}}=\left\{\left(\left(x_1,0\right),1\right),\left(\left(x_2,0\right),1\right)\right\}=\widetilde{\stackrel{˜}{\varnothing }}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{X}}\text{for}{x}_1:{\mu }_{\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{X}}}\left(x_1\right)=\frac{1\wedge 1}{1\vee 0.8}+\frac{1\wedge 0.7}{1\vee 0.6}+\frac{1\wedge 0.6}{1\vee 0.3}=\frac{1}{1}+\frac{0.7}{1}+\frac{0.6}{1}\\ =\left(1,\max \left\{1,0.7,0.6\right\}\right)\Rightarrow \left\{\left(\left(x_1,1\right),1\right)\right\},\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{X}}\text{for}{x}_2:{\mu }_{\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{X}}}\left(x_2\right)=\frac{1\wedge 0.9}{1\vee 0.8}+\frac{1\wedge 1}{1\vee 0.5}+\frac{1\wedge 0.5}{1\vee 0.4}=\frac{0.9}{1}+\frac{1}{1}+\frac{0.5}{1}\\ =\left(1,\max \left\{1,0.9,0.5\right\}\right)\Rightarrow \left\{\left(\left(x_2,1\right),1\right)\right\}\end{array}$

$\widetilde{\stackrel{˜}{A}}\cup \widetilde{\stackrel{˜}{X}}=\widetilde{\stackrel{˜}{X}}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{X}}\text{for}{x}_1:{\mu }_{\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{X}}}\left(x_1\right)=\frac{1\wedge 1}{1\wedge 0.8}+\frac{1\wedge 0.7}{1\wedge 0.6}+\frac{1\wedge 0.6}{1\wedge 0.3}=\frac{1}{0.8}+\frac{0.7}{0.6}+\frac{0.6}{0.3}\\ =\left\{\left(\left(x_1,0.8\right),1\right),\left(\left(x_1,0.6\right),0.7\right),\left(\left(x_1,0.3\right),0.6\right)\right\}\end{array}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{X}}=\{\left(\left(x_1,0.8\right),1\right),\left(\left(x_1,0.6\right),0.7\right),\left(\left(x_1,0.3\right),0.6\right),\\ \left(\left(x_2,0.8\right),0.9\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.4\right),0.5\right)\}=\widetilde{\stackrel{˜}{A}}\end{array}$

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

Theorem 5: Let $\left\{{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r:r\in ℝ\right\}$ be a family of all general type-2 fuzzy topologies on X ; then ${\cap }_{r\in ℝ}{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ is general type-2 fuzzy topologies on X.

proof: we must prove three conditions of topologies,

1) $\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{X}}\in \left\{{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r:r\in ℝ\right\}\Rightarrow \widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{X}}\in {\cap }_{r\in ℝ}{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ .

2) Let $\left\{{\widetilde{\stackrel{˜}{A}}}_i:i\in \mathbb{N}\right\}\subseteq {\cap }_{r\in ℝ}{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ , then ${\widetilde{\stackrel{˜}{A}}}_i\in {\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ for all $i\in \mathbb{N}$ so

thus ${\cup }_{i\in \mathbb{N}}{\widetilde{\stackrel{˜}{A}}}_i\in {\cap }_{r\in ℝ}{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ .

3) Let $\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{B}}\in {\cap }_{r\in ℝ}{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ , then $\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{B}}\in {\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ and because ${\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ are all general type-2 fuzzy topologies $\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\in {\stackrel{\approx }{\mathfrak{F}}}_r$ for all $r\in ℝ$ , so $\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\in {\cap }_{r\in ℝ}{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_r$ .

Remark 6: Let $\left(X,{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_1\right)$ and $\left(X,{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_2\right)$ be two general type-2 fuzzy topological spaces over the same universe X then $\left(X,{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_1\cup {\widetilde{\stackrel{˜}{\mathfrak{F}}}}_2\right)$ need not be general type-2 fuzzy topological space over X, we can see that in example 3.7.

Example 7: Let $X=\left\{x_1,x_2\right\}$ and ${\widetilde{\stackrel{˜}{\mathfrak{F}}}}_1=\left\{\widetilde{\stackrel{˜}{X}},\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{A}}\right\}$ , ${\widetilde{\stackrel{˜}{\mathfrak{F}}}}_2=\left\{\widetilde{\stackrel{˜}{X}},\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{B}}\right\}$ be two general type-2fuzzy topologies defined on X where $\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{B}},\widetilde{\stackrel{˜}{\varnothing }}$ and $\widetilde{\stackrel{˜}{X}}$ defined as follows: $\widetilde{\stackrel{˜}{\varnothing }}=\left\{\left(\left(x_1,0\right),1\right),\left(\left(x_2,0\right),1\right)\right\}$ ,

$\widetilde{\stackrel{˜}{X}}=\left\{\left(\left(x_1,1\right),1\right),\left(\left(x_2,1\right),1\right)\right\}$

$\begin{array}{l}\widetilde{\stackrel{˜}{A}}=\{\left(\left(x_1,0.8\right),1\right),\left(\left(x_1,0.6\right),0.7\right),\left(\left(x_1,0.3\right),0.6\right),\\ \left(\left(x_2,0.8\right),0.9\right),\left(\left(x_2,0.5\right),1\right),\left(\left(x_2,0.4\right),0.5\right)\}.\end{array}$

$\widetilde{\stackrel{˜}{B}}=\left\{\left(\left(x_1,0.5\right),1\right),\left(\left(x_1,0.6\right),0.2\right),\left(\left(x_2,0.3\right),0.7\right),\left(\left(x_2,0.9\right),1\right)\right\}.$

Let ${\widetilde{\stackrel{˜}{\mathfrak{F}}}}_1\cup {\widetilde{\stackrel{˜}{\mathfrak{F}}}}_2=\left\{\widetilde{\stackrel{˜}{\varnothing }},\widetilde{\stackrel{˜}{X}},\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{B}}\right\}$ so $\left(X,{\widetilde{\stackrel{˜}{\mathfrak{F}}}}_1\cup {\widetilde{\stackrel{˜}{\mathfrak{F}}}}_2\right)$ is not general type-2 fuzzy topological space over X since $\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\notin {\widetilde{\stackrel{˜}{\mathfrak{F}}}}_1\cup {\widetilde{\stackrel{˜}{\mathfrak{F}}}}_2$ .

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

$\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)=\cup \left\{{\widetilde{\stackrel{˜}{G}}}_i:{\widetilde{\stackrel{˜}{G}}}_i\text{type-2}\text{fuzzy}\text{open}\text{sets}\text{in}X,{\widetilde{\stackrel{˜}{G}}}_i\subseteq \widetilde{\stackrel{˜}{A}},i\in \mathbb{N}\right\}$ , $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$ is the largest type-2 fuzzy open set contained in $\widetilde{\stackrel{˜}{A}}$ .

Theorem 9: Let $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ be general type-2 fuzzy topological space over X, and let $\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{B}}$ be two type-2 fuzzy sets in X. Then

1) $\mathrm{int}\left(\widetilde{\stackrel{˜}{\varnothing }}\right)=\widetilde{\stackrel{˜}{\varnothing }}$ and $\mathrm{int}\left(\widetilde{\stackrel{˜}{X}}\right)=\widetilde{\stackrel{˜}{X}}$ .

2) $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \widetilde{\stackrel{˜}{A}}$ .

3) $\widetilde{\stackrel{˜}{A}}$ is type-2 fuzzy open set if and only if $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)=\widetilde{\stackrel{˜}{A}}$ .

4) $\mathrm{int}\left(\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\right)=\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$ .

5) $\widetilde{\stackrel{˜}{A}}\subseteq \widetilde{\stackrel{˜}{B}}\to \mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)$ .

6) $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)=\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\cap \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)$ .

Proof:

1) $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)=\cup \left\{{\widetilde{\stackrel{˜}{G}}}_i:{\widetilde{\stackrel{˜}{G}}}_i\text{type-2}\text{fuzzy}\text{open}\text{sets}\text{in}X,{\widetilde{\stackrel{˜}{G}}}_i\subseteq \widetilde{\stackrel{˜}{A}},i\in \mathbb{N}\right\}$ , $\widetilde{\stackrel{˜}{\varnothing }}$ is type-2 fuzzy open set in $\widetilde{\stackrel{˜}{\mathfrak{F}}}$ and $\widetilde{\stackrel{˜}{\varnothing }}\subseteq \widetilde{\stackrel{˜}{\varnothing }}\Rightarrow \mathrm{int}\left(\widetilde{\stackrel{˜}{\varnothing }}\right)=\widetilde{\stackrel{˜}{\varnothing }}$ .

Now to prove $\mathrm{int}\left(\widetilde{\stackrel{˜}{X}}\right)=\widetilde{\stackrel{˜}{X}}$ ,

$\mathrm{int}\left(\widetilde{\stackrel{˜}{X}}\right)=\cup \left\{{\widetilde{\stackrel{˜}{G}}}_i:{\widetilde{\stackrel{˜}{G}}}_i\text{type-2}\text{fuzzy}\text{open}\text{sets}\text{in}X,{\widetilde{\stackrel{˜}{G}}}_i\subseteq \widetilde{\stackrel{˜}{X}},i\in \mathbb{N}\right\}$ , $\widetilde{\stackrel{˜}{X}}$ is type-2 fuzzy open set in $\widetilde{\stackrel{˜}{\mathfrak{F}}}$ and $\widetilde{\stackrel{˜}{X}}\subseteq \widetilde{\stackrel{˜}{X}}\Rightarrow \mathrm{int}\left(\widetilde{\stackrel{˜}{X}}\right)=\widetilde{\stackrel{˜}{X}}$ .

2) To prove $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \widetilde{\stackrel{˜}{A}}$, since $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)=\cup \left\{{\widetilde{\stackrel{˜}{G}}}_i:{\widetilde{\stackrel{˜}{G}}}_i\text{type-2}\text{fuzzy}\text{open}\text{sets}\text{in}X,{\widetilde{\stackrel{˜}{G}}}_i\subseteq \widetilde{\stackrel{˜}{A}},i\in \mathbb{N}\right\}$ , such that ${\widetilde{\stackrel{˜}{G}}}_i\subseteq \widetilde{\stackrel{˜}{A}}$ that is $\widetilde{\stackrel{˜}{A}}$ is type-2 membership function ${\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)$ where $x\in X$ and $u\in J_X\subseteq \left[0,1\right]$ less than a type-2 membership function ${\mu }_{{\widetilde{\stackrel{˜}{G}}}_i}\left(x,u\right)$ where $x\in X$ and $w\in J_X\subseteq \left[0,1\right]$ such that $w\le u$ and ${\mu }_{{\widetilde{\stackrel{˜}{G}}}_i}\left(x,u\right)\le {\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right)$ , $\sup \left\{{\mu }_{{\widetilde{\stackrel{˜}{G}}}_i}\left(x,u\right)\le {\mu }_{\widetilde{\stackrel{˜}{A}}}\left(x,u\right),w\le u\right\}$ hence $\cup {\widetilde{\stackrel{˜}{G}}}_i\subseteq \widetilde{\stackrel{˜}{A}}\Rightarrow \cup {\widetilde{\stackrel{˜}{G}}}_i\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$ , therefore $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \widetilde{\stackrel{˜}{A}}$ .

3) If $\widetilde{\stackrel{˜}{A}}$ is type-2 fuzzy open set, then $\widetilde{\stackrel{˜}{A}}\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$ , but $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \widetilde{\stackrel{˜}{A}}$ from part (2), hence $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)=\widetilde{\stackrel{˜}{A}}$ .

4) $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$ is a type-2 fuzzy open set and from part (3) we have $\mathrm{int}\left(\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\right)=\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$

5) If $\widetilde{\stackrel{˜}{A}}\subseteq \widetilde{\stackrel{˜}{B}}$ and from part(2) $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \widetilde{\stackrel{˜}{A}}$ , $\mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)\subseteq \widetilde{\stackrel{˜}{B}}$ , then $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \widetilde{\stackrel{˜}{A}}\subseteq \widetilde{\stackrel{˜}{B}}$ . Therefore $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \widetilde{\stackrel{˜}{B}}$ and $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$ is a type-2 fuzzy open set contained in $\widetilde{\stackrel{˜}{B}}$ , so $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)$ .

6) Because $\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)\subseteq \widetilde{\stackrel{˜}{A}}$ and $\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)\subseteq \widetilde{\stackrel{˜}{B}}$ , from part (5) $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)$ and $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)$ , thus $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\cap \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)$ , since $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)\subseteq \widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}$ , so $\mathrm{int}\left(\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\right)\cap \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)\subseteq \left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)$ from part(5) but $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\cap \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)$ is a type-2 fuzzy open sets then $\mathrm{int}\left(\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\right)\cap \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)\subseteq \mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)$ from part(3).Hence $\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)=\mathrm{int}\left(\widetilde{\stackrel{˜}{A}}\right)\cap \mathrm{int}\left(\widetilde{\stackrel{˜}{B}}\right)$ .

Definition 10: Let $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ be general type-2 fuzzy topological space over $\widetilde{\stackrel{˜}{X}}$ and let $\widetilde{\stackrel{˜}{A}}$ be type-2 fuzzy set over X. Then the type-2 fuzzy closure of $\widetilde{\stackrel{˜}{A}}$ , denoted by $cl\left(\widetilde{\stackrel{˜}{A}}\right)$ , is defined as the intersection of all type-2 fuzzy closed sets containing $\widetilde{\stackrel{˜}{A}}$ . That is

$cl\left(\widetilde{\stackrel{˜}{A}}\right)=\cap \left\{{\widetilde{\stackrel{˜}{M}}}_i:{\widetilde{\stackrel{˜}{M}}}_i\text{type-2}\text{fuzzy}\text{closed}\text{sets}\text{in}X,\widetilde{\stackrel{˜}{A}}\subseteq {\widetilde{\stackrel{˜}{M}}}_i,i\in \mathbb{N}\right\}$ ,

$cl\left(\widetilde{\stackrel{˜}{A}}\right)$ is the smallest type-2 fuzzy closed set containing $\widetilde{\stackrel{˜}{A}}$ .

Theorem 11: Let $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ be general type-2 fuzzy topological space over X, and let $\widetilde{\stackrel{˜}{A}},\widetilde{\stackrel{˜}{B}}$ be two type-2 fuzzy sets in X. Then

1) $cl\left(\widetilde{\stackrel{˜}{\varnothing }}\right)=\widetilde{\stackrel{˜}{\varnothing }}$ and $cl\left(\widetilde{\stackrel{˜}{X}}\right)=\widetilde{\stackrel{˜}{X}}$ .

2) $\widetilde{\stackrel{˜}{A}}\subseteq cl\left(\widetilde{\stackrel{˜}{A}}\right)$ .

3) $\widetilde{\stackrel{˜}{A}}$ is type-2 fuzzy closed set if and only if $cl\left(\widetilde{\stackrel{˜}{A}}\right)=\widetilde{\stackrel{˜}{A}}$ .

4) $cl\left(cl\left(\widetilde{\stackrel{˜}{A}}\right)\right)=cl\left(\widetilde{\stackrel{˜}{A}}\right)$ .

5) $\widetilde{\stackrel{˜}{A}}\subseteq \widetilde{\stackrel{˜}{B}}\to cl\left(\widetilde{\stackrel{˜}{A}}\right)\subseteq cl\left(\widetilde{\stackrel{˜}{B}}\right)$ .

6) $cl\left(\widetilde{\stackrel{˜}{A}}\cap \widetilde{\stackrel{˜}{B}}\right)=cl\left(\widetilde{\stackrel{˜}{A}}\right)\cap cl\left(\widetilde{\stackrel{˜}{B}}\right)$ .

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

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

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

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

Definition 14: Let $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ be a general type-2 fuzzy topological space over X

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

Definition 15: Let $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ and $\left(Y,\widetilde{\stackrel{˜}{\mathfrak{S}}}\right)$ 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 $Y\subseteq X$ and the open subsets of Y are precisely of the form ${\widetilde{\stackrel{˜}{\mathfrak{F}}}}_{\widetilde{\stackrel{˜}{Y}}}=\left\{\widetilde{\stackrel{˜}{\mathcal{Y}}}=\widetilde{\stackrel{˜}{Y}}\cap \widetilde{\stackrel{˜}{\mathcal{X}}}:\widetilde{\stackrel{˜}{\mathcal{X}}}\in \widetilde{\stackrel{˜}{\mathfrak{F}}}\right\}$ . Here we may say that each open subset $\widetilde{\stackrel{˜}{\mathcal{Y}}}$ of Y is the restriction to $\widetilde{\stackrel{˜}{\mathcal{Y}}}$ of an open subset $\widetilde{\stackrel{˜}{\mathcal{X}}}$ of X. That is, $\left(Y,\widetilde{\stackrel{˜}{\mathfrak{S}}}\right)$ is called a subspace of $\left(X,\widetilde{\stackrel{˜}{\mathfrak{F}}}\right)$ if the type-2 fuzzy open sets of Y are the type-2 fuzzy intersection of open sets of X with $\widetilde{\stackrel{˜}{\mathcal{Y}}}$ .

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.