Characterizations of Nnce-open and Nnce-closed Functions

The purpose of this paper is to introduce and investigate several new classes of functions called, Nnce-open and Nnce-closed functions in Nnc topological spaces by using the concept of Nnce-open sets. Several new characterizations and fundamental properties concerning of these new types of functions are obtained. Furthermore, these kinds of functions have strong application in the area of image processing and have very important applications in quantum particle physics, high energy physics and superstring theory.


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Introduction Smarandache's neutrosophic system have wide range of real time applications for the fields of Computer Science, Information Systems, Applied Matheamatics, Artifical Intelligence, Mechanics, decision making, Medicine, Electrical & Electronic, and Management Science etc [1,2,3,4,31,32]. Topology is a classical subject, as a generalization topological spaces many types of topological spaces introduced over the year. Smarandache [25] defined the Neutrosophic set on three component Neutrosophic sets (T-Truth, F-Falsehood, I-Indeterminacy). Neutrosophic topological spaces (nts's) introduced by Salama and Alblowi [22]. Lellies Thivagar et.al. [12] was given the geometric existence of N topology, which is a non-empty set equipped with N arbitrary topologies. Lellis Thivagar et al. [13] introduced the notion of Nn-open (closed) sets and Nn topological spaces. Al-Hamido [5]  Nnce-open sets. Some characterizations and several interesting properties of these functions are discussed. Additionally, these kinds of functions have strong application in the area of Image Processing and have very important applications in quantum particle physics, high energy physics and superstring theory.

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Preliminaries Salama and Smarandache [24] presented the idea of a neutrosophic crisp set in a set X and defined the inclusion between two neutrosophic crisp sets, the intersection (union) of two neutrosophic crisp sets, the complement of a neutrosophic crisp set, neutrosophic crisp empty (resp., whole) set as more then two types. And they studied some properties related to nutrosophic crisp set operations. However, by selecting only one type, we define the inclusion, the intersection (union), and neutrosophic crisp empty (resp., whole) set again and discover a few properties. Definition 2.1 Let X be a non-empty set. Then H is called a neutrosophic crisp set (in short, ncs) in X if H has the form H = (H1,H2,H3), where H1,H2, and H3 are subsets of X, The neutrosophic crisp empty (resp., whole) set, denoted by ϕ n (resp., Xn) is an ncs in X defined by ϕ n = (ϕ ,ϕ ,X) (resp. Xn = (X,X,ϕ )). We will denote the set of all ncs's in X as ncS(X).
Likewise the following are the quick consequence of Definition 2.2. [23] A neutrosophic crisp topology (briefly, ncts) on a non-empty set X is a family τ of nc subsets of X satisfying the following axioms (iii) ∪Ha ∈ τ, for any {Ha : a ∈ J} ⊆ τ. a Then (X,τ) is a neutrosophic crisp topological space (briefly, ncts ) in X. The τ elements are called neutrosophic crisp open sets (briefly, ncos) in X. A ncs C is closed set (briefly, nccs) iff its complement C c is ncos. Definition 2.4 [5] Let X be a non-empty set. Then ncτ1, ncτ2, ··· , ncτN are N-arbitrary crisp topologies defined on X and the collection Nncτ = {A ⊆ X : A = ( ∪ Hj) ∪ ( ∩ Lj), Hj,Lj ∈ ncτj} is called N neutrosophic crisp (briefly, Nnc)-topology on N N j=1 j=1 X if the axioms are satisfied: Then (X,Nncτ) is called a Nnc-topological space (briefly, Nncts) on X. The Nncτ elements are called Nnc-open sets (Nncos) on X and its complement is called Nnc-closed sets (Nnccs) on X. The elements of X are known as Nnc-sets (Nncs) on X. Definition 2.5 [5] Let (X,Nncτ) be any Nncts. Let H be an Nncs in (X,Nncτ). Then H is said to be a Nnc-regular open [26] set (briefly, Nncros) if H = Nncint (Nnccl(H)). The complement of an Nncros is called an Nnc-regular closed set (briefly, Nncrcs ) in X.
Definition 2.9 A space (X,Nncτ) is said to be: (i) Nnce-T1 [30] if for each pair of distinct points x and y of X, there exist Nnceo sets A and B containing x and y, respectively, such that x /∈ B and y /∈ A. (ii) Nnce-T2 [30] if for each pair of distinct points x and y in X, there exist disjoint Nnceo sets A and B in X such that x ∈ A and y ∈ B. Definition 2.10 A space (X,Nncτ) is said to be: (i) Nnce-compact [30] if every cover of X by Nnceo sets has a Nnc finite sub cover.
(ii) Nnce-Lindelo¨f [30] if every cover of X by Nnceo sets has a countable subcover. Definition 2.11 A space (X,Nncτ) is said to be Nnce-connected [30] if X cannot be written as the union of two nonempty disjoint Nnceo sets. 3 Characterizations of Nnce-open and Nnce-closed functions In this section, we obtain some characterizations and several properties concerning Nnce-open functions and Nnceclosed functions via Nnceo and Nncec sets.
Proof. Suppose that f is NnceO and B ⊆ Y and let x ∈ f −1 (Nncecl(B)). Then, (Nnceint(B)). By Theorem 3.4 we have f is Nnceo. Now we introduce some characterizations concerning Nnce-closed functions.
Conversely ( . Theorem 3.9 If f : (X,Nncτ) → (Y,Nncτ * ) is NnceO bijection. Then the following hold: Proof. (i) Let y1 and y2 be any distinct points in Y. Then there exist x1 and x2 in X such that f(x1) = y1 and f(x2) = y2. Since X is NncT1 then, there exist two Nnco sets U and V in X with x1 ∈ U, x2 ∈/ U and x2 ∈ V, x1 ∈/ V. Now f(U) and (ii) It is similar to (i). Thus is omitted. Theorem 3.10 If f : (X,Nncτ) → (Y,Nncτ * ) is NnceO bijective. Then the following properties are hold: Proof. Suppose that X is not Nnc-connected. Then there exist two non-empty disjoint Nnco sets U and V in X such that X = U∪V. Then f(U) and f(V ) are non-empty disjoint Nnceo sets in Y with Y = f(U) ∪ f(V ) which contradicts the fact that Y is Nnceconnected. 4 Conclusion Generalized open and closed sets play a very a prominent role in general Topology and it applications. And many topologists worldwide are focusing their researches on these topics and this mounted to many important and useful results. Indeed a significant theme in General Topology, Real analysis and many other branches of mathematics concerns the variously modified forms of continuity, separation axioms etc., by utilizing generalized open and closed sets. One of the well-known notions and that expected it will has a wide applying in physics and Topology and their applications is the notion of Nnce-open sets. The importance of general topological spaces rapidly increases in both the pure and applied directions; it plays a significant role in data mining [21]. One can observe the influence of general topological spaces also in computer science and digital topology [8,9,10], computational topology for geometric and molecular design [14], particle physics, high energy physics, quantum physics, and Superstring theory [11,15,16,17,18,19,20]. In this paper we have introduced and investigated the notions of new classes of functions which may have very important applications in quantum particle physics, high energy physics and superstring theory. Furthermore, the fuzzy topological version of the concepts and results introduced in this paper are very important. Since El-Naschie has shown that the notion of fuzzy topology has very important applications in quantum particle physics especially in related to both string theory and ϵ ∞ theory.