An Introduction to Fuzzy Topological Spaces

Topology has enormous applications on fuzzy set. An attention can be brought to the mathematicians about these topological applications on fuzzy set by this article. In this research, first we have classified the fuzzy sets and topological spaces, and then we have made relation between elements of them. For expediency, with mathematical view few basic definitions about crisp set and fuzzy set have been recalled. Then we have discussed about topological spaces. Finally, in the last section, the fuzzy topological spaces which is our main object we have developed the relation between fuzzy sets and topological spaces. Moreover, this article has been concluded with the examination of some of its properties and certain relationships among the closure of these spaces.


Introduction
Primarily in the area of set theory, fuzzy mathematics differs from conventional mathematics. Fuzzy mathematics had been introduced just few years ago, it is full of topics. It is used widely in many sectors such as, vehicles, traffic system where logic circuit controls anti-skid brakes, transmissions, and other operations. We have discussed in this paper about a set, which is more specified than crisp set. It can take a decision between yes or no, i.e. 1 or 0. Thirty years ago, Black [1], a philosopher in America, predicted some ideas which are obtainable in this article. The author of [1] established a hypothesis whose main things fuzzy sets-the sets having borders which are not specific. An influential paper by Zadeh [2] provided a vital point regarding the development of the up to date ideas of ambiguity.
In this real physical world, it would be better if we are able to study the objects in a classified way. But most often than not, it cannot be done, because the objects do not exist a specifically definite criterion of association. Many illustrations may be written, the group of animals obviously consists of birds, cats, deer, etc. On the other hand, the things like bacteria, virus, starfish, jellyfish, etc., have an uncertain category regarding the animal's class. The similar type of uncertainty arises for a numerical value like 10 regarding to the "class" of the set of all real values which become higher than 1. Obviously, "the group of the set of all real values which become higher than 1" otherwise "the class of beautiful women" otherwise "the class of tall men," can not represent sets otherwise groups in the common mathematical logic of these languages. However, the truth continues that such inaccurately called "classes" take part in a vital position in human being thoughts, mainly for the area of prototype abstraction, informative communication and recognition. Research based on the fuzzy sets hypothesis is increasing gradually from the time at the beginning of the hypothesis in mid -1960 S . Now, the concepts and outcomes containing the hypothesis of fuzzy set are relatively remarkable. In addition, various applications based research has been conducted very vigorously and has found more extraordinary outcomes.
For conciseness, the concepts of the definition of topology on fuzzy sets that may be named as topological fuzzy spaces, Chang [3] confined his attention to the more basic definitions, theorems and proofs. Michálek [4] identified and analyzed another idea of topological fuzzy space in his paper which was quite different of the classic Chang's [3] definition. Lowen [5] provided profound concepts for the construction of topological fuzzy spaces. He introduced two new functions and served the idea of fuzzy compression for the simplification of topological compression which allowed seeing obviously additional relation amid topological fuzzy spaces and topological spaces. Hutton [6] continued his study to find the real meaning of the common theorems of topology, and then took a broad view of their proofs. He also developed "pointless" descriptions for structures and properties that depend merely on the Lattice composition of fuzzy sets compilation, but not on its putrefaction. Cheng-Ming [7] discussed about the product-induced spaces, which is a special fuzzy topological spaces. He showed that each topological fuzzy space is isomorphic topologically by a definite space of topology and also introduced the basic idea of double points and set up a type of fuzzy points neighborhood formation for example the Q-neighborhood, which is very significant conception in topological fuzzy set. Finally he discussed the dilemma of metrization in fuzzy on topological fuzzy spaces in addition to obtain a metrization hypothesis in fuzzy. Zimmermann [8] in his book presented the detail of fuzzy set theory and their applications. Papageorgiou [9] introduced several fuzzy topological concepts, fuzzy multifunction and the conception of neighborhood point in fuzzy which is helpful in the learning of common optimization techniques and games in fuzzy. He remarked in his study that common topological point set may be considered as a particular form of topological fuzzy, Klir and Yuan [10] provided the details of the theoretical advances in fuzzy logic and set theory in his book. The author [10] considered an extensive diversity of usages of both fuzzy logic and sets. Many topological properties for intuitionistic I-topological spaces have been discussed in Lee and Lee [11] and Yang and Wang [12]. The authors [12] proved that the accompaniment of any closed set is an open set and the opposite of this statement is also true. They also demonstrated an illustration for computing the exterior, interior, and boundary of Mikania micrantha pedestaled on aerial snaps of Hong Kong landscape. Wenzhong and Kimfung [13] developed the concept of arithmetical topological fuzzy pedestaled on the closure and interior operators. Next, coherent topological fuzzy is named by these operators. Separation and regularity axioms in fuzzy topology on fuzzy set and their characterizations are defined and studied by Kandil et al. [14]. Moreover, with some necessary examples they investigated some of their basic properties and certain relationship among them. Using a special kind of function Akray [15] introduced three different fuzzy topological spaces. In this study he discussed compactness, connectedness properties and also provides the necessary and sufficient conditions under which properties some of these spaces coincide. Aygunoglu et al. [16] investigated the usages of theory of Soft Set on actual life dilemmas and different disciplines. The authors [16] investigated some of its fundamental properties and concluded that soft To complete the study, first we discussed the concepts of fuzzy sets in Section 2, then we discussed topological space in Section 3, and in Section 4 we mainly classify fuzzy topological spaces, which is our main object. For this we have discussed some important definitions and related theorems. To complete the Section 2 we have used the following references [8] [9], and to complete the Section 3 we have used the following references [18] [19] [20]. Hausdroff fuzzy topological spaces and their theorem. We also have showed some related theorem of these topics.

Fuzzy Sets
Fuzzy set is the more universal concept of classical set which is an impending tool for handling indistinctness and uncertainties. It is typically characterized in the form of membership function, whereas a membership function is characterized by the universal set U  to the set ranging between 0 & 1.
More exclusively, let [ ] 0,1 I = be the unit interval and X be a null set, where x be any particular element in X. Subsequently, a function : X is described as fuzzy set in X. Where, ( ) x µ is defined as the "Grade of membership of x X ∈ in µ " Example 1: In the example three consequences fuzzy sets are explain that stand for the perception of very young, young and middle-age person in a country which also mentioned graphically in Figure 1

Crisp Set
Crisp set is a group of objects that define the precious and definite feature which employs bi-valued (yes/no) logic. That is whether each particular element can either within or not belong to a set S, S X ⊆ It is mainly a classical set which is label by a special type of fuzzy sets. Crisp set can be denoted by The set of teenagers is an example of crisp set.

Classical Set
A collection of individual object identified as the member or elements of the set that can be distinguished from one another and which follows some basic property is known as classical set. It is defined in such a way that, each element of the set is spitted either member or non-member groups. i.e. for a set A either a A ∈ or a A ∉ . There is no chance of existence of partial membership. Example:

Characteristic Function of Crisp Sets
Suppose the set of universal set is denoted by X and S is the subset of X, i.e. φ .
Then for each x X ∈ its characteristic function is denoted by S λ or 1 S is defined as, Figure 2 show the graphical representation of the above example.

Characteristic Function of the Complement
For a set S the complement of the characteristic function is denoted by 1 S or S λ and for each x X Figure 3 express the graphical representation of the above example

Characteristic Functions (Union and Intersection)
Let P and be two sets, the characteristic functions of the union and intersection are denoted by 1 P Q ∪ and 1 P Q ∩ can also be obtain by pertaining the formulas:

Standard Fuzzy Operations
The most three operations that has a special significance in fuzzy set theory are t. 1) union 2) intersection and, 3) complement. These can be generalized to fuzzy sets in many ways.

Standard Fuzzy Union and Intersection
Let A and B be two fuzzy set which is defined on X, where X defined the universal set.
Then, the standard fuzzy union of A and B, denoted by A B ∨ and can be described as where, "max" indicates for maximum value.
and for the set A and B the standard fuzzy intersection is denoted by A B ∧ where, "min" stands for minimum value.

Standard Fuzzy Complement
If A is any fuzzy set which is defined on a universal set X, then its complement denoted by A is another fuzzy set on X. So, the membership function of the complement of the fuzzy set A is symbolized by Example: To demonstrate the significance of this definition, let us consider the fuzzy set B of experienced postgraduate students, whose membership function is given in

Topological Space
Topology which is a basic mathematical discipline and whose name was not coined until 1930s has now its influence on many branches of pure mathematics especially geometry and analysis and some of the applied too. We have just used the word topology in its primary sense, as the name of a branch of mathematics. This word derives from two Greek words, and its literal meaning is "the science of position". In recent times for both graduate and undergraduate students, general topology has become a crucial part in Mathematics. For some time now, topology has been firmly established as one of the basic disciplines of pure mathematics. It has also deeply stimulated the abstract algebra and fuzzy mathematics.

Basic Ideas and Definitions
Suppose X be a non empty set, a class τ which is a subset of X is defined a topology if it satisfies the following conditions: (i) X and ϕ belong to τ .
(ii) The union and intersection of any number of sets in τ belongs to τ . We noticed that 1 τ is a topology on X but 2 τ and 3 τ is not a topology on X.
Then τ is discrete topology on X.

Indiscrete topology
For any non empty set X the collection of set consisting φ and X is a topology on X, is defined as indiscrete topology. Example: φ is a topology on X and is known as indiscrete topology.

Interior, Exterior and Boundary Points
For a open set where B is a subset of topological space.
The interior of C B is known as the exterior of B and is denoted by ext. The point which is interior nor an exterior of B is known as boundary point of B.

Closure of a Set
The intersection of all closed super sets of B is denoted by B or B − is known as closure of B.

Homomorphism Topological Spaces
A continuous one-to-one mapping of a topological space onto another is defined to be homomorphic if there exist a homeomorphism of X onto Y and their points can be set into one-to-one correspondence in such a manner that their open sets also correspond to one another.

Fuzzy Topological Spaces
The perception of Fuzzy set theory bring in [2] provides us a wider structure compared to classical set theory which generalize various concepts of topology. Fuzzy topology merges ordered structure to the topological structure. From the point of pure mathematics, this branch of mathematics was first proposed by enormous Mathematician Ehrenman who encompasses the two most active features of topology on lattice, which affect each other. Chang [3] had introduced first the notation of Fuzzy topology. Later on, numerous of researchers continued the study in this area. We observed that fuzzy topology is consider as special case of general point set topology, where membership functions are presently by characteristic functions.

Basic Definitions
In the past, the leading attempt to build up the fuzzy counterpart of basic topology was commenced by Chang [7] in 1968. From the point of Chang, on a set X a fuzzy topology is a family

{ }
: is fuzzy set in F X = µ µ of fuzzy subsets (i.e. X F I ∈ ) that satisfies the following three axioms: , where j denotes an index set, then Then, ( ) 0, Wherever, X I is the fuzzy set on X.

Discrete fuzzy topology
Discrete fuzzy topology is the set that included all the fuzzy sets.
Example 2: , Then from the properties of fuzzy topological space (i) i.e. finite intersection belongs to 1 F .
where j denotes an index set. so, Hence, we conclude that the pair ( ) , X F is a topological space.

Closure Fuzzy Topological Space
For any µ the closure of fuzzy topological space is denoted by µ and is defined as the smallest closed fuzzy set that containing µ .

Proof (ii):
Since µ is the smallest closed set containing µ and µ itself is closed, From (a) and (b) we have, So from (1) and (2), we have, Proof (iv): Since the whole space 1 F ∈ is open, then its complement i.e. 1 0 C = , is closed Also, 0 is closed. So, we can write 0 0 = .
Definition. Fuzzy topology generated by closure operator is denoted by X F From (4) and (5) we have, From (3) and (6), we have,

Interior of a Fuzzy Topological Space
The smallest superior bound of all interior fuzzy sets of µ is called the interior of µ , and is denoted by  µ .evidently,  Proof (iv): On the other hand, ∧ ≤ ∧ ( ) From (7) and (8) we conclude that

Boundary of Fuzzy Topological Space
For , the fuzzy boundary is denoted by b µ is define to be the minimum of all F-closed sets ρ with the property ( ) ( )  is the support of the fuzzy point β µ .

Neighborhood of Fuzzy Topological Space
Suppose ( )  Here, 1 Also since, Therefore β is a fuzzy n.b.d. of α .

Continuous Function of a Fuzzy Topological Space
Hence the theorem is proved.

Homomorphism of Fuzzy Topological Space
iv) F is called homomorphism if F is bijective (i.e. one-one onto) and biconditional. It means that, both f and Example: , that is if the interior of closure of any fuzzy set µ is empty, then µ is called nowhere dense in X, Definition 3.
Any fuzzy set µ is said to be fuzzy boundary (F-boundary) if and only if

Some Results Obtained from Definitions
(i) If ≥ λ µ and λ is F-dense then µ is F-dense too. Proof: Thus µ is F-dense too.
(iv) If µ is F-nowhere dense, then so is µ .

Base and Subbase of the Fuzzy Topological Spaces
Base Let ( ) , X F be a fuzzy topological space and P is a sub family of F, then P is called a base of F if and only if every member of F can be represent as supremum of member of P.

Hausdroff Fuzzy Topological Spaces
An fuzzy topological space ( ) , X F is said to be Hausdroff iff , x y X ∀ ∈ , x y ≠ , there exist ,

Conclusion
Our main aim of this study is to make simple for the reader to understand the relation between the fuzzy set and topological spaces. For that purpose we have attempted to provide more fundamental definition of these two topics. In the fuzzy topological spaces section we have represented definition and provide proved of some theorem. The presented results in this analysis signify that large numbers of the fundamental ideas from general topology may be expanded enthusiastically to topological fuzzy spaces. Even though the fuzzy sets hypothesis belongs in a developing phase, now a days it is undertaken of having spacious precious applications.