On Generalized Closed Sets and Generalized Pre-Closed Sets in Neutrosophic Topological Spaces

In this paper, the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets are introduced. We also study relations and various properties between the other existing neutrosophic open and closed sets. In addition, we discuss some applications of generalized neutrosophic pre-closed sets, namely neutrosophic pT1 2 space and neutrosophic gpT1 2 space. The concepts of generalized neutrosophic connected spaces, generalized neutrosophic compact spaces and generalized neutrosophic extremally disconnected spaces are established. Some interesting properties are investigated in addition to giving some examples.


Introduction
Zadeh [1] introduced the notion of fuzzy sets. After that, there have been a number of generalizations of this fundamental concept. The study of fuzzy topological spaces was first initiated by Chang [2,3] in 1968. Atanassov [4] introduced the notion of intuitionistic fuzzy sets (IFs). This notion was extended to intuitionistic L-fuzzy setting by Atanassov and Stoeva [5], which currently has the name "intuitionistic L-topological spaces". Coker [6] introduced the notion of intuitionistic fuzzy topological space by using the notion of (IFs). The concept of generalized fuzzy closed set was introduced by Balasubramanian and Sundaram [7]. In various recent papers, Smarandache generalizes intuitionistic fuzzy sets and different types of sets to neutrosophic sets (NSs). On the non-standard interval, Smarandache, Peide and Lupianez defined the notion of neutrosophic topology [8][9][10]. In addition, Zhang et al. [11] introduced the notion of an interval neutrosophic set, which is a sample of a neutrosophic set and studied various properties.
Recently, Al-Omeri and Smarandache [12,13] introduced and studied a number of the definitions of neutrosophic closed sets, neutrosophic mapping, and obtained several preservation properties and some characterizations about neutrosophic of connectedness and neutrosophic connectedness continuity.
This paper is arranged as follows. In Section 2, we will recall some notions that will be used throughout this paper. In Section 3, we mention some notions in order to present neutrosophic generalized pre-closed sets and investigate its basic properties. In Sections 4 and 5, we study the neutrosophic generalized pre-open sets and present some of their properties. In addition, we provide an application of neutrosophic generalized pre-open sets. Finally, the concepts of generalized neutrosophic connected space, generalized neutrosophic compact space and generalized neutrosophic extremally disconnected spaces are introduced and established in Section 6 and some of their properties in neutrosophic topological spaces are studied.
This class of sets belongs to the important class of neutrosophic generalized open sets which is very useful not only in the deepening of our understanding of some special features of the already well-known notions of neutrosophic topology but also proves useful in neutrosophic multifunction theory in neutrosophic economy and also in neutrosophic control theory. The applications are vast and the researchers in the field are exploring these realms of research.

Preliminaries
Definition 1. Let Z be a non-empty set. A neutrosophic set (NS for short)S is an object having the form µS(k), and the degree of non-membership (namely γS(k) ), the degree of indeterminacy (namely σS(k)), and the degree of membership function (namely µS(k)), of each element k ∈ Z to the setS, see [14].
Definition 2. LetS = µS(k), σS(k), γS(k) be an NS on Z . [15] The complement of the set S(C(S), for short) may be defined as follows: Neutrosophic sets (NSs) 0 N and 1 N [14] in Z are introduced as follows: 1 − 0 N can be defined as four types: 2-1 N can be defined as four types: Definition 3. Let k be a non-empty set, and generalized neutrosophic sets GNSsS andR be in the form Then, we may consider two possible definitions for subsets (S ⊆R) [14]: (ii) ∪S j can defined as two types: [14].
Definition 11. An NSS in an NT Z is said to be a neutrosophic α generalized closed set (NαgCS [18]) if NαNCl(S) ⊆B whensoeverS ⊆B andB is an NOS in Z . The complement C(S) of an NαgCSS is an NαgOS in Z .
for more details).
is not neutrosophic connected, and there exists a neutrosophic setS such thatS is both NCs and NOs ∈ (Z , Γ).

Since the neutrosophic open set is GN-open and the neutrosophic closed set is
Proof. Suppose that (K , Γ 1 ) is not neutrosophic connected, such that the neutrosophic setS is both neutrosophic open and neutrosophic closed in (K , Γ 1 ). Since g is GN-continuous, g −1 (S) is GN-open and GN-closed in ((K , Γ). Thus, (K , Γ) is not GN connected. Hence, (K , Γ 1 ) is neutrosophic connected.

Proof. Let
Therefore, g(K) is neutrosophic compact. Proposition 3. Let (Z , Γ) be a neutrosophic compact space and suppose that K is a GN-closed set of (Z , Γ). Then, K is a neutrosophic compact set.
Since K is GN-closed, NCl(K) ⊆ i∈J K j . Since (Z , Γ) is a neutrosophic compact space, there exists a finite subcover J 0 ⊆ J. Now, NCl(K) ⊆ i∈J 0 K j . Hence, K ⊆ NCl(K) ⊆ i∈J 0 K j . Therefore, K is a neutrosophic compact set.

Generalized Neutrosophic Pre-Closed Set
Definition 20. An NSS is said to be a neutrosophic generalized pre-closed set (GNPCS in short) in (Z , Γ) if pNCl(S) ⊆B whensoeverS ⊆B andB is an NO in Z . The family of all GNPCSs of an NT (Z , Γ) is defined by GNPC(Z ). Theorem 4. Every NC is a GNPC, but the converse is not true.  The following Figure 1 shows the implication relations between GNPC set and the other existed ones.
Proof. By TheoremS c ⊆R c ⊆ (pN Int(S)) c . LetR c ⊆R andR be NOs. SinceS c ⊆ B c ,S c ⊆R. However,S c is a GNPCs, pNCl(S c ) ⊆R. In addition,R c ⊆ (pN Int(S)) c = pNCl(S c ) (by theorem). Therefore, pNCl(R c ) ⊆ pNCl(S c ) ⊆R. Hence, B c is GNPC. This implies thatR is a GNPO of Z .  Proof. LetS ∈ GNPO(Z ). Then,S c is GnPCS in Z . Therefore, pNCl(S c ) ⊆B whensoeverS c ⊆B andB is an NO in Z . That is, NCl(N Int(S c )) ⊆B. This impliesB c ⊆ N Int(NCl(S)) whensoever B c ⊆S andB c is NCs in Z . ReplacingB c , byR, we getR ⊆ N Int(NCl(S)) whensoeverR ⊆S andR is an NC in Z .

Theorem 10. For NSS,S is an NO and GNPC in Z if and only ifS is an NRO in Z .
Proof. =⇒ LetS be an NO and a GNPCS in Z . Then, pNCl(S) ⊆S. This implies NCl(N Int(S)) ⊆S. SinceS is an NO, it is an NPO. Hence,S ⊆ N Int(NCl(S)). Therefore,S = N Int(NCl(S)). Hence,S is an NRO in Z .
⇐= LetS be an NRO in Z . Therefore,S = N Int(NCl(S)). LetS ⊆B andB be an NO in Z . This implies pNCl(S) ⊆S. Hence,S is GNPC in Z .
Theorem 11. An NSS of an NT (Z , Γ) is a GNPO iff H ⊆ pN Int(S), whensoever H is an NC and H ⊆S.
Proof. =⇒ LetS be GNPO in Z . Let H be an NCs and H ⊆S. Then, H c is an NOS in Z such thatS c ⊆ H c . SinceS c is GNPC, we have pNCl(S c ) ⊆ H c . Hence, (pN Int(S)) c ⊆ H c . Therefore, H ⊆ pN Int(S).
⇐= SupposeS is an NS of Z and let H ⊆ pN Int(S) whensoever H is an NC and H ⊆S. Then,S c ⊆ H c and H c is an NO. By assumption, (pN Int(S)) c ⊆ H c , which implies pNCl(S c ) ⊆ H c . Therefore,S c is GNPCs of Z . Hence,S is a GNPOS of Z . ⇐= SupposeS be an NS of Z and H ⊆ N Int(NCl(S)), whensoever H is an NC and H ⊆S. Then,S c ⊆ H c and H c is an NO. By assumption, (NInt(NCl(S))) c ⊆ H c . Hence, NCl(NInt(S c )) ⊆ H c . This implies pNCl(S c ) ⊆ H c . Hence,S is a GNPOS of Z .

Applications of Generalized Neutrosophic Pre-Closed Sets
Definition 23. An NTS (Z , Γ) is said to be neutrosophic-pT1 Proof. Let Z be an N pT1 2 space andS be GNPC ∈ Z . By assumption,S is NCs in Z . Since every NC is an NPC,S is an NPC in Z . Hence, Z is an NgpT1   Proof. =⇒ LetS be a GNPOs in Z ; then,S c is GNPCs in Z . By assumption,S c is an NPCs in Z . Thus,S is NPOs in Z . Hence, GNPO(Z ) = NPO(Z ).
⇐= LetS be GNPC ∈ Z . Then,S c is GNPO in Z . By assumption,S c is an NPO in Z . Thus,S is an NPC ∈ Z . Therefore, Z is an NgpT1 (ii) =⇒ (i). Let every singleton set of Z be either NPCS or NPOS. LetS be an NPGCS of (Z , Γ). Let x ∈ Z . We show that x ∈ Z in two cases.

Conclusions
We have introduced generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets over neutrosophic topology space. Many results have been established to show how far topological structures are preserved by these neutrosophic pre-closed. We also have provided examples where such properties fail to be preserved. In this paper, we have studied a few ideas only; it will be necessary to carry out more theoretical research to establish a general framework for decision-making and to define patterns for complex network conceiving and practical application.