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

Abstract

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 $p{T}_{\frac{1}{2}}$ space and neutrosophic $gp{T}_{\frac{1}{2}}$ 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.

1. 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 Section 4 and Section 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.

2. Preliminaries

Definition 1.

Let $\mathcal{Z}$ be a non-empty set. A neutrosophic set ($NS$ for short) $\tilde{S}$ is an object having the form $\tilde{S}=\{\langle k,{\mu }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\gamma }_{\tilde{S}}(k)\rangle :k\in \mathcal{Z}\}$, where ${\gamma }_{\tilde{S}}(k)$, ${\sigma }_{\tilde{S}}(k),{\mu }_{\tilde{S}}(k)$, and the degree of non-membership (namely ${\gamma }_{\tilde{S}}(k)$ ), the degree of indeterminacy (namely ${\sigma }_{\tilde{S}}(k)$), and the degree of membership function (namely ${\mu }_{\tilde{S}}(k)$), of each element $k\in \mathcal{Z}$ to the set $\tilde{S}$, see [14].

A neutrosophic set $\tilde{S}=\{\langle k,{\mu }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\gamma }_{\tilde{S}}(k)\rangle :k\in \mathcal{Z}\}$ can be identified as $\langle {\mu }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\gamma }_{\tilde{S}}(k)\rangle$ in $\rfloor 0^{-},1^{+}\lfloor$ on $\mathcal{Z}$.

Definition 2.

Let $\tilde{S}=\langle {\mu }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\gamma }_{\tilde{S}}(k)\rangle$ be an $NS$ on $\mathcal{Z}$. [15] The complement of the set $\tilde{S}(C(\tilde{S}),\mathrm{for}\mathrm{short})$ may be defined as follows:

(i) 

$C(\tilde{S})=\{\langle k,1-{\mu }_{\tilde{S}}(k),1-{\gamma }_{\tilde{S}}(k)\rangle :k\in \mathcal{Z}\}$,

(ii) 

$C(\tilde{S})=\{\langle k,{\gamma }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\mu }_{\tilde{S}}(k)\rangle :k\in \mathcal{Z}\}$,

(iii) 

$C(\tilde{S})=\{\langle k,{\gamma }_{\tilde{S}}(k),1-{\sigma }_{\tilde{S}}(k),{\mu }_{\tilde{S}}(k)\rangle :k\in \mathcal{Z}\}$.

Neutrosophic sets ($NSs$) $0_N$ and $1_N$ [14] in $\mathcal{Z}$ are introduced as follows:

$1-0_N$ can be defined as four types:

(i)

$0_N=\{\langle k,0,0,1\rangle :k\in \mathcal{Z}\}$,

(ii)

$0_N=\{\langle k,0,1,1\rangle :k\in \mathcal{Z}\}$,

(iii)

$0_N=\{\langle k,0,1,0\rangle :k\in \mathcal{Z}\}$,

(iv)

$0_N=\{\langle k,0,0,0\rangle :k\in \mathcal{Z}\}$.

2- $1_N$ can be defined as four types:

(i)

$1_N=\{\langle k,1,0,0\rangle :k\in \mathcal{Z}\}$,

(ii)

$1_N=\{\langle k,1,0,1\rangle :k\in \mathcal{Z}\}$,

(iii)

$1_N=\{\langle k,1,1,0\rangle :k\in \mathcal{Z}\}$,

(iv)

$1_N=\{\langle k,1,1,1\rangle :k\in \mathcal{Z}\}$.

Definition 3.

Let k be a non-empty set, and generalized neutrosophic sets $GNSs$ $\tilde{S}$ and $\tilde{R}$ be in the form $\tilde{S}=\{k,{\mu }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\gamma }_{\tilde{S}}(k)\}$, $B=\{k,{\mu }_{\tilde{R}}(k),{\sigma }_{\tilde{R}}(k),{\gamma }_{\tilde{R}}(k)\}$. Then, we may consider two possible definitions for subsets $(\tilde{S}\subseteq \tilde{R})$ [14]:

(i) 

$\tilde{S}\subseteq B\iff {\mu }_{\tilde{S}}(k)\le {\mu }_B(k),{\sigma }_{\tilde{S}}(k)\ge {\sigma }_B(k),\mathrm{and}{\gamma }_{\tilde{S}}(k)\le {\gamma }_B(k)$,

(ii) 

$\tilde{S}\subseteq B\iff {\mu }_{\tilde{S}}(k)\le {\mu }_B(k),{\sigma }_{\tilde{S}}(k)\ge {\sigma }_B(k),\mathrm{and}{\gamma }_{\tilde{S}}(k)\ge {\gamma }_B(k)$.

Definition 4.

Let $\{{\tilde{S}}_j:j\in J\}$ be an arbitrary family of $NSs$ in $\mathcal{Z}$. Then,

(i) 

$\cap {\tilde{S}}_j$ can defined as two types:

$\cap {\tilde{S}}_j=\langle k,\underset{j\in J}{\wedge }{\mu }_{\tilde{S}j}(k),\underset{j\in J}{\wedge }{\sigma }_{\tilde{S}j}(k),\underset{j\in J}{\vee }{\gamma }_{\tilde{S}j}(k)\rangle$,

$\cap {\tilde{S}}_j=\langle k,\underset{j\in J}{\wedge }{\mu }_{\tilde{S}j}(k),\underset{j\in J}{\vee }{\sigma }_{\tilde{S}j}(k),\underset{j\in J}{\vee }{\gamma }_{\tilde{S}j}(k)\rangle$.

(ii) 

$\cup {\tilde{S}}_j$ can defined as two types:

$\cup {\tilde{S}}_j=\langle k,\underset{j\in J}{\vee }{\mu }_{\tilde{S}j}(k),\underset{j\in J}{\vee }{\sigma }_{\tilde{S}j}(k),\underset{j\in J}{\wedge }{\gamma }_{\tilde{S}j}(k)\rangle$,

$\cup {\tilde{S}}_j=\langle k,\underset{j\in J}{\vee }{\mu }_{\tilde{S}j}(k),\underset{j\in J}{\wedge }{\sigma }_{\tilde{S}j}(k),\underset{j\in J}{\wedge }{\gamma }_{\tilde{S}j}(k)\rangle$, see [14].

Definition 5.

A neutrosophic topology ($NT$ for short) [16] and a non empty set $\mathcal{Z}$ is a family $\Gamma$ of neutrosophic subsets of $\mathcal{Z}$ satisfying the following axioms:

(i) 

$0_N,1_N\in \Gamma$,

(ii) 

${\tilde{S}}_1\cap {\tilde{S}}_2\in \Gamma$ for any ${\tilde{S}}_1,{\tilde{S}}_2\in \Gamma$,

(iii) 

$\cup {\tilde{S}}_i\in \Gamma$, $\forall \{{\tilde{S}}_i|j\in J\}\subseteq \Gamma$.

In this case, the pair $(\mathcal{Z},\Gamma )$ is called a neutrosophic topological space ($NTS$ for short) and any neutrosophic set in Γ is known as neutrosophic open set $NOS\in \mathcal{Z}$. The elements of Γ are called neutrosophic open sets. A closed neutrosophic set $\tilde{R}$ if and only if its $C(\tilde{R})$ is neutrosophic open.

Note that, for any $NTS$ $\tilde{S}$ in $(\mathcal{Z},\Gamma )$, we have $NCl({\tilde{S}}^c)={[NInt(\tilde{S})]}^c$ and $NInt({\tilde{S}}^c)={[NCl(\tilde{S})]}^c$.

Definition 6.

Let $\tilde{S}=\{{\mu }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\gamma }_{\tilde{S}}(k)\}$ be a neutrosophic open set and $B=\{{\mu }_B(k),{\sigma }_B(k),{\gamma }_B(k)\}$ a neutrosophic set on a neutrosophic topological space $(\mathcal{Z},\Gamma )$. Then,

(i) 

$\tilde{S}$ is called neutrosophic regular open [14] iff $\tilde{S}=NInt(NCl(\tilde{S}))$.

(ii) 

If $B\in NCS(\mathcal{Z}),$ then B is called neutrosophic regular closed [14] iff $\tilde{S}=NCl(NInt(\tilde{S}))$.

Definition 7.

Let $(k,\Gamma )$ be $NT$ and $\tilde{S}=\{k,{\mu }_{\tilde{S}}(k),{\sigma }_{\tilde{S}}(k),{\gamma }_{\tilde{S}}(k)\}$ an $NS$ in $\mathcal{Z}$. Then,

(i) 

$NCL(\tilde{S})=\cap \{U:U\mathrm{is}\mathrm{an}\mathrm{NCS}\mathrm{in}\mathcal{Z},\tilde{S}\subseteq U\},$

(ii) 

$NInt(\tilde{S})=\cup \{V:V\mathrm{is}\mathrm{an}\mathrm{NOS}\mathrm{in}\mathcal{Z},V\subseteq \tilde{S}\}$, see [14].

It can be also shown that $NCl(\tilde{S})$ is an $NCS$ and $NInt(\tilde{S})$ is an $NOS$ in $\mathcal{Z}$. We have

(i) 

$\tilde{S}$ is in $\mathcal{Z}$ iff $NCl(\tilde{S})$.

(ii) 

$\tilde{S}$ is an $NCS$ in $\mathcal{Z}$ iff $NInt(\tilde{S})=\tilde{S}$.

Definition 8.

Let $\tilde{S}$ be an $NS$ and $(\mathcal{Z},\Gamma )$ an $NT$. Then,

(i) 

Neutrosophic semiopen set $(NSOS)$ [12] if $\tilde{S}\subseteq NCl(NInt(\tilde{S}))$,

(ii) 

Neutrosophic preopen set $(NPOS)$ [12] if $\tilde{S}\subseteq NInt(NNCl(\tilde{S}))$,

(iii) 

Neutrosophic α-open set $(N\alpha OS)$ [12] if $\tilde{S}\subseteq NInt(NNCl(NInt(\tilde{S}))),$

(iv) 

Neutrosophic β-open set $(N\beta OS)$ [12] if $\tilde{S}\subseteq NNCl(NInt(NCl(\tilde{S}))).$

The complement of $\tilde{S}$ is an NSOS, N$\alpha$OS, NPOS, and NROS, which is called NSCS, N$\alpha$CS, NPCS, and NRCS, resp.

Definition 9.

Let $\tilde{S}=\{{\tilde{S}}_1,{\tilde{S}}_2,{\tilde{S}}_3\}$ be an $NS$ and $(\mathcal{Z},\Gamma )$ an $NT$. Then, the *-neutrosophic closure of $\tilde{S}$ ($*-NCl(\tilde{S})$ for short [12]) and *-neutrosophic interior $(*-NInt(\tilde{S})$ for short [12]) of $\tilde{S}$ are defined by

(i) 

$\alpha NCl(\tilde{S})=\cap \{V:VisanNRCin\mathcal{Z},\tilde{S}\subseteq V\}$,

(ii) 

$\alpha NInt(\tilde{S})=\cup \{U:UisanNROin\mathcal{Z},U\subseteq \tilde{S}\}$,

(iii) 

$pNCl(\tilde{S})=\cap \{V:VisanNPCin\mathcal{Z},\tilde{S}\subseteq V\}$,

(iv) 

$pNInt(\tilde{S})=\cup \{U:UisanNPOin\mathcal{Z},U\subseteq \tilde{S}\}$,

(v) 

$sNCl(\tilde{S})=\cap \{V:VisanNSCin\mathcal{Z},\tilde{S}\subseteq V\}$,

(vi) 

$sNInt(\tilde{S})=\cup \{U:UisanNSOin\mathcal{Z},U\subseteq \tilde{S}\}$,

(vii) 

$\beta NCl(\tilde{S})=\cap \{V:VisanNC\beta Cin\mathcal{Z},\tilde{S}\subseteq V\}$,

(viii) 

$\beta NInt(\tilde{S})=\cup \{U:UisaN\beta Oin\mathcal{Z},U\subseteq \tilde{S}\}$,

(ix) 

$rNCl(\tilde{S})=\cap \{V:VisanNRCin\mathcal{Z},\tilde{S}\subseteq V\}$,

(x) 

$rNInt(\tilde{S})=\cup \{U:UisanNROin\mathcal{Z},U\subseteq \tilde{S}\}$.

Definition 10.

An $(NS)$ $\tilde{S}$ of an $NT$ $(\mathcal{Z},\Gamma )$ is called a generalized neutrosophic closed set [17] ($GNC$ in short) if $NCl(\tilde{S})\subseteq \tilde{B}$ wherever $\tilde{S}\subset \tilde{B}$ and $\tilde{B}$ is a neutrosophic closed set in $\mathcal{Z}$.

Definition 11.

An $NS$ $\tilde{S}$ in an $NT$ $\mathcal{Z}$ is said to be a neutrosophic α generalized closed set ($N\alpha gCS$ [18]) if $N\alpha NCl(\tilde{S})\subseteq \tilde{B}$ whensoever $\tilde{S}\subseteq \tilde{B}$ and $\tilde{B}$ is an $NOS$ in $\mathcal{Z}$. The complement $C(\tilde{S})$ of an $N\alpha gCS$ $\tilde{S}$ is an $N\alpha gOS$ in $\mathcal{Z}$.

3. Neutrosophic Generalized Connected Spaces, Neutrosophic Generalized Compact Spaces and Generalized Neutrosophic Extremally Disconnected Spaces

Definition 12.

Let $(\mathcal{Z},\Gamma )$ and $(\mathcal{K},{\Gamma }_1)$ be any two neutrosophic topological spaces.

(i) 

A function $g:(\mathcal{Z},\Gamma )⟶(\mathcal{K},{\Gamma }_1)$ is called generalized neutrosophic continuous( $GN$-continuous) $g^{-1}$ of every closed set in $(\mathcal{Z},{\Gamma }_1)$ is $GN$-closed in $(\mathcal{Z},\Gamma )$.

Equivalently, if the inverse image of every open set in $(\mathcal{Z},{\Gamma }_1)$ is $GN$-open in $(\mathcal{Z},\Gamma )$:

(ii) 

A function $g:(\mathcal{Z},\Gamma )⟶(\mathcal{K},{\Gamma }_1)$ is called generalized neutrosophic irresolute $g^{-1}$ of every $GN$-closed set in $(\mathcal{Z},{\Gamma }_1)$ is $GN$-closed in $(\mathcal{Z},\Gamma )$.

Equivalently $g^{-1}$ of every $GN$-open set in $(\mathcal{Z},{\Gamma }_1)$ is $GN$-open in $(\mathcal{Z},\Gamma )$

(iii) 

A function $g:(\mathcal{Z},\Gamma )⟶(\mathcal{K},{\Gamma }_1)$ is said to be strongly neutrosophic continuous if $g^{-1}(\tilde{S})$ is both neutrosophic open and neutrosophic closed in $(\mathcal{Z},\Gamma )$ for each neutrosophic set $\tilde{S}$ in $(\mathcal{Z},{\Gamma }_1)$.

(iv) 

A function $g:(\mathcal{Z},\Gamma )⟶(\mathcal{K},{\Gamma }_1)$ is said to be strongly $GN$-continuous if the inverse image of every $GN$-open set in $(\mathcal{Z},{\Gamma }_1)$ is neutrosophic open in $(\mathcal{Z},\Gamma )$, see ([17] for more details).

Definition 13.

An $NTS$ $(\mathcal{Z},\Gamma )$ is said to be neutrosophic-$T_{\frac{1}{2}}$ ($N{T}_{\frac{1}{2}}$ in short) space if every $GNC$ in $\mathcal{Z}$ is an $NC$ in $\mathcal{Z}$.

Definition 14.

Let $(\mathcal{Z},\Gamma )$ be any neutrosophic topological space. $(\mathcal{Z},\Gamma )$ is said to be generalized neutrosophic disconnected (in shortly $GN$-disconnected) if there exists a generalized neutrosophic open and generalized neutrosophic closed set $\tilde{R}$ such that $\tilde{R}\ne 0_N$ and $\tilde{R}\ne 1_N$.$(\mathcal{Z},\Gamma )$ is said to be generalized neutrosophic connected if it is not generalized neutrosophic disconnected.

Proposition 1.

Every $GN$-connected space is neutrosophic connected. However, the converse is not true.

Proof. 

For a $GN$-connected $(\mathcal{Z},\Gamma )$ space and let $(\mathcal{Z},\Gamma )$ not be neutrosophic connected. Hence, there exists a proper neutrosophic set, $\tilde{S}=\langle {\mu }_{\tilde{S}}(x),{\sigma }_{\tilde{S}}(x),{\gamma }_{\tilde{S}}(x)\rangle$ $\tilde{S}\ne 0_N,\tilde{S}\ne 1_N$, such that $\tilde{S}$ is both neutrosophic open and neutrosophic closed in $(\mathcal{Z},\Gamma )$. Since every neutrosophic open set is $GN$-open and neutrosophic closed set is $GN$-closed, $\mathcal{Z}$ is not $GN$-connected. Therefore, $(\mathcal{Z},\Gamma )$ is neutrosophic connected. □

Example 1.

Let $\mathcal{Z}=\{u,v,w\}$. Define the neutrosophic sets $\tilde{S},\tilde{R}$ and $\mathcal{Z}$ in $\mathcal{Z}$ as follows: $\tilde{S}=\langle x,(\frac{a}{0.4},\frac{b}{0.5},\frac{c}{0.5}),(\frac{a}{0.4},\frac{b}{0.5},\frac{c}{0.5}),(\frac{a}{0.5},\frac{b}{0.5},\frac{c}{0.5})\rangle$, $\tilde{R}=\langle x,(\frac{a}{0.7},\frac{b}{0.6},\frac{c}{0.5}),(\frac{a}{0.7},\frac{b}{0.6},\frac{c}{0.5}),(\frac{a}{0.3},\frac{b}{0.4},\frac{c}{0.5})\rangle$. Then, the family $\Gamma =\{0_N,1_N,\tilde{S},\tilde{R}\}$ is neutrosophic topology on $\mathcal{Z}$. It is obvious that $(\mathcal{Z},\Gamma )$ is $NTS$. Now, $(\mathcal{Z},\Gamma )$ is neutrosophic connected. However, it is not a $GN$-connected for $\tilde{Z}=\langle x,(\frac{a}{0.5},\frac{b}{0.6},\frac{c}{0.5}),(\frac{a}{0.5},\frac{b}{0.6},\frac{c}{0.5}),(\frac{a}{0.5},\frac{b}{0.6},\frac{c}{0.5})\rangle$ is $GN$ open and $GN$ closed in $(\mathcal{Z},\Gamma )$.

Theorem 1.

Let $(\mathcal{Z},\Gamma )$ be a neutrosophic $T_{\frac{1}{2}}$ space; then, $(\mathcal{Z},\Gamma )$ is neutrosophic connected iff $(\mathcal{Z},\Gamma )$ is $GN$-connected.

Proof. 

Suppose that $(\mathcal{Z},\Gamma )$ is not $GN$-connected, and there exists a neutrosophic set $\tilde{S}$ which is both $GN$-open and $GN$-closed. Since $(\mathcal{Z},\Gamma )$ is neutrosophic $T_{\frac{1}{2}}$, $\tilde{S}$ is both neutrosophic open and neutrosophic closed. Hence, $(\mathcal{Z},\Gamma )$ is $GN$-connected. Conversely, let $(\mathcal{Z},\Gamma )$ is $GN$-connected. Suppose that $(\mathcal{Z},\Gamma )$ is not neutrosophic connected, and there exists a neutrosophic set $\tilde{S}$ such that $\tilde{S}$ is both $NCs$ and $NOs\in (\mathcal{Z},\Gamma )$. Since the neutrosophic open set is $GN$-open and the neutrosophic closed set is $GN$-closed, $(\mathcal{Z},\Gamma )$ is not $GN$-connected. Hence, $(\mathcal{Z},\Gamma )$ is neutrosophic connected. □

Proposition 2.

Suppose $(\mathcal{Z},\Gamma )$ and $(\mathcal{K},{\Gamma }_1)$ are any two $NTSs$. If $g:(\mathcal{Z},\Gamma )⟶(\mathcal{K},{\Gamma }_1)$ is $GN$-continuous surjection and $(\mathcal{Z},\Gamma )$ is $GN$-connected, then $(\mathcal{K},{\Gamma }_1)$ is neutrosophic connected.

Proof. 

Suppose that $(\mathcal{K},{\Gamma }_1)$ is not neutrosophic connected, such that the neutrosophic set $\tilde{S}$ is both neutrosophic open and neutrosophic closed in $(\mathcal{K},{\Gamma }_1)$. Since g is $GN$-continuous, $g^{-1}(\tilde{S})$ is $GN$-open and $GN$-closed in ($(\mathcal{K},\Gamma )$. Thus, $(\mathcal{K},\Gamma )$ is not $GN$ connected. Hence, $(\mathcal{K},{\Gamma }_1)$ is neutrosophic connected. □

Definition 15.

Let $(\mathcal{K},\Gamma )$ be an $NT$. If a family $\{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}$ of $GN$ open sets in $(\mathcal{K},\Gamma )$ satisfies the condition $\bigcup \{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}=1_N$, then it is called a $GN$ open cover of $(\mathcal{K},\Gamma )$. A finite subfamily of a $GN$ open cover $\{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}$ of $(\mathcal{Z},\Gamma )$, which is also a $GN$ open cover of $(\mathcal{K},\Gamma )$ is called a finite subcover of

$$\{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}.$$

Definition 16.

An $NT$ $(\mathcal{K},\Gamma )$ is called $GN$ compact iff every $GN$ open cover of $(\mathcal{K},\Gamma )$ has a finite subcover.

Theorem 2.

Let $(\mathcal{K},\Gamma )$ and $(\mathcal{K},{\Gamma }_1)$ be any two $NTs$, and $g:(\mathcal{Z},\Gamma )⟶(\mathcal{K},{\Gamma }_1)$ be $GN$ continuous surjection. If $(\mathcal{K},\Gamma )$ is $GN$-compact, hence so is $(\mathcal{K},{\Gamma }_1).$

Proof. 

Let $G_i=\{\langle y,{\mu }_{G_i}(x),{\sigma }_{G_i}(x),{\gamma }_{G_i}(x):i\in J\rangle \}$ be a neutrosophic open cover in $(\mathcal{K},{\Gamma }_1)$ with

$$\tilde{\bigcup }\{\langle y,{\mu }_{G_i}(x),{\sigma }_{G_i}(x),{\gamma }_{G_i}(x):i\in J\rangle \}=\underset{i\in J}{\tilde{\bigcup }}{G}_i=1_N.$$

Since g is $GN$ continuous, $g^{-1}(G_i)=G_i=\{\langle y,{\mu }_{g^{-1}(G_i)}(x),{\sigma }_{g^{-1}(G_i)}(x),{\gamma }_{g^{-1}(G_i)}(x):i\in J\rangle \}$ is $GN$ open cover of $(\mathcal{K},\Gamma )$. Now,

$$\underset{i\in J}{\tilde{\bigcup }}{g}^{-1}(G_i)=g^{-1}(\underset{i\in J}{\tilde{\bigcup }}{G}_i)=1_N.$$

Since $(\mathcal{K},\Gamma )$ is $GN$ compact, there exists a finite subcover $J_0\subset J$, such that

$$\underset{i\in J_0}{\tilde{\bigcup }}{g}^{-1}(G_i)=1_N.$$

Hence,

$$g\left(\underset{i\in J_0}{\tilde{\bigcup }}{g}^{-1}(G_i)=1_N\right),g^{-1}\left(\underset{i\in J_0}{\tilde{\bigcup }}(G_i)=1_N\right).$$

That is,

$$\underset{i\in J_0}{\tilde{\bigcup }}(G_i)=1_N.$$

Therefore, $(\mathcal{K},{\Gamma }_1)$ is neutrosophic compact. □

Definition 17.

Let $(\mathcal{K},\Gamma )$ be an $NT$ and K be a neutrosophic set in $(\mathcal{Z},\Gamma )$. If a family $\{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}$ of $GN$ open sets in $(\mathcal{K},\Gamma )$ satisfies the condition $K\subseteq \bigcup \{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}=1_N$, then it is called a $GN$ open cover of K. A finite subfamily of a $GN$ open cover $\{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}$ of K, which is also a $GN$ open cover of K is called a finite subcover of $\{\langle k,{\mu }_{G_i}(k),{\sigma }_{G_i}(k),{\gamma }_{G_i}(k):i\in J\rangle \}$.

Definition 18.

An $NT$ $(\mathcal{K},\Gamma )$ is called $GN$ compact iff every $GN$ open cover of K has a finite subcover.

Theorem 3.

Let $(\mathcal{K},\Gamma )$ and $(\mathcal{K},{\Gamma }_1)$ be any two $NTs$, and $g:(\mathcal{Z},\Gamma )⟶(\mathcal{K},{\Gamma }_1)$ be an $GN$ continuous function. If K is $GN$-compact, then so is $g(K)$ in $(\mathcal{K},{\Gamma }_1)$.

Proof. 

Let $G_i=\{\langle y,{\mu }_{G_i}(x),{\sigma }_{G_i}(x),{\gamma }_{G_i}(x):i\in J\rangle \}$ be a neutrosophicopen cover of $g(K)$ in $(\mathcal{K},{\Gamma }_1)$. That is,

$$g(K)\subseteq \underset{i\in J}{\tilde{\bigcup }}{G}_i.$$

Since g is $GN$ continuous, $g^{-1}(G_i)=\{\langle x,{\mu }_{g^{-1}(G_i)}(x),{\sigma }_{g^{-1}(G_i)}(x),{\gamma }_{g^{-1}(G_i)}(x):i\in J\rangle \}$ is $GN$ open cover of K in $(\mathcal{Z},\Gamma )$. Now,

$$K\subseteq g^{-1}(\underset{i\in J}{\tilde{\bigcup }}{G}_i)\subseteq \underset{i\in J}{\tilde{\bigcup }}{g}^{-1}(G_i).$$

Since K is $(\mathcal{Z},\Gamma )$ is $GN$ compact, there exists a finite subcover $J_0\subset J$, such that

$$K\subseteq \underset{i\in J_0}{\tilde{\bigcup }}{g}^{-1}(G_i)=1_N.$$

Hence,

$$g(K)\subseteq g\left(\underset{i\in J_0}{\tilde{\bigcup }}{g}^{-1}(G_i)\right)\underset{i\in J_0}{\tilde{\bigcup }}(G_i).$$

Therefore, $g(K)$ is neutrosophic compact. □

Proposition 3.

Let $(\mathcal{Z},\Gamma )$ be a neutrosophic compact space and suppose that K is a $GN$-closed set of $(\mathcal{Z},\Gamma )$. Then, K is a neutrosophic compact set.

Proof. 

Let $K_j==\{\langle y,{\mu }_{G_i}(x),{\sigma }_{G_i}(x),{\gamma }_{G_i}(x):i\in J\rangle \}$ be a family of neutrosophic open set in $(\mathcal{Z},\Gamma )$ such that

$$K\subseteq \underset{i\in J}{\tilde{\bigcup }}{K}_j.$$

Since K is $GN$-closed, $NCl(K)\subseteq {\tilde{\mathsf{\bigcup }}}_{i\in J}{K}_j$. Since $(\mathcal{Z},\Gamma )$ is a neutrosophic compact space, there exists a finite subcover $J_0\subseteq J$. Now, $NCl(K)\subseteq {\tilde{\mathsf{\bigcup }}}_{i\in J_0}{K}_j$. Hence, $K\subseteq NCl(K)\subseteq {\tilde{\mathsf{\bigcup }}}_{i\in J_0}{K}_j$. Therefore, K is a neutrosophic compact set. □

Definition 19.

Let $(\mathcal{Z},\Gamma )$ be any neutrosophic topological space. $(\mathcal{Z},\Gamma )$ is said to be $GN$ extremally disconnected if $NCl(K)$ neutrosophic open and K is $GN$ open.

Proposition 4.

For any neutrosophic topological space $(\mathcal{Z},\Gamma )$, the following are equivalent:

(i) 

$(\mathcal{Z},\Gamma )$ is $GN$ extremally disconnected.

(ii) 

For each $GN$ closed set K, $NGNInt(\tilde{S})$ is a $GN$ closed set.

(iii) 

For each $GN$ open set K, we have $NGNCl(K)+NGNCl(1-NGNCl(\tilde{S}))=1$.

(iv) 

For each pair of $GN$ open sets K and M in $(\mathcal{Z},\Gamma )$, $NGNCl(K)+M=1$, we have $NGNCl(K)+NGNCl(B)=1$.

4. Generalized Neutrosophic Pre-Closed Set

Definition 20.

An $NS$ $\tilde{S}$ is said to be a neutrosophic generalized pre-closed set ($GNPCS$ in short) in $(\mathcal{Z},\Gamma )$ if $pNCl(\tilde{S})\subseteq \tilde{B}$ whensoever $\tilde{S}\subseteq \tilde{B}$ and $\tilde{B}$ is an $NO$ in $\mathcal{Z}$. The family of all $GNPCSs$ of an $NT$ $(\mathcal{Z},\Gamma )$ is defined by $GNPC(\mathcal{Z})$.

Example 2.

Let $\mathcal{Z}=\{a,b\}$ and $\Gamma =\{0_N,1_N,T\}$ be a neutrosophic topology on $\mathcal{Z}$, where $T=\langle (0.2,0.3,0.5),(0.8,0.7,0.7)\rangle$. Then, the $NS$ $\tilde{S}=\langle (0.2,0.2,0.2),(0.8,0.7,0.7)\rangle$ is $GNPCs\in \mathcal{Z}$.

Theorem 4.

Every $NC$ is a $GNPC$, but the converse is not true.

Proof. 

Let $\tilde{S}$ be an $NC$ in $\mathcal{Z}$, $\tilde{S}\subseteq \tilde{B}$ and $\tilde{B}$ is $NOS$ in $(\mathcal{Z},\Gamma )$. Since $pNCl(\tilde{S})\subseteq NCl(\tilde{S})$ and $\tilde{S}$ is $NCS$ in $\mathcal{Z}$, $pNCl(\tilde{S})\subseteq NCl(\tilde{S})=\tilde{S}\subseteq \tilde{B}$. Therefore, $\tilde{S}$ is $GNPCs\in \mathcal{Z}$. □

Example 3.

Let $\mathcal{Z}=\{u,v\}$ and $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.2,0.3,0.5),(0.8,0.7,0.7)\rangle$. Then, the $NS$ $\tilde{S}=\langle (0.2,0.2,0.2),(0.8,0.7,0.7)\rangle$ is a $GNPC$ in $\mathcal{Z}$ but not an $NCS\in \mathcal{Z}$.

Theorem 5.

Every $N\alpha CS$ is $GNPC$, but the converse is not true.

Proof. 

Let $\tilde{S}$ be an $N\alpha CS$ in $\mathcal{Z}$ and let $\tilde{S}\subseteq \tilde{B}$ and $\tilde{B}$ is an $NOS$ in $(\mathcal{Z},\Gamma )$. Now, $NCl(NInt(NCl(\tilde{S})))\subseteq \tilde{S}$. Since $\tilde{S}\subseteq NCl(\tilde{S})$, $NCl(NInt(\tilde{S}))\subseteq NCl(NInt(NCl(\tilde{S})))\subseteq \tilde{S}$. Hence, $pNCl(\tilde{S})\subseteq \tilde{S}\subseteq \tilde{B}$. Therefore, $\tilde{S}$ is $GNPCs\in \mathcal{Z}$. □

Example 4.

Let $\mathcal{Z}=\{u,v\}$ and let $\Gamma =\{0_N,1_N,H\}$ is a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.4,0.2,0.5),(0.6,0.7,0.6)\rangle$. Then, the $NS$ $\tilde{S}=\langle (0.3,0.1,0.4),(0.7,0.8,0.7)\rangle$ is a $GNPC$ in $\mathcal{Z}$ but not $N\alpha Cs$ in $\mathcal{Z}$ since $NCl(NInt(NCl(\tilde{S})))=\langle (0.5,0.6,0.5),(0.5,0.3,0.6)\rangle$ $\not\subset \tilde{S}$.

Theorem 6.

Every $GN\alpha C$ is a $GNPC$, but the converse is not true.

Proof. 

Let $\tilde{S}$ be $GN\alpha Cs\in \mathcal{Z}$, $\tilde{S}\subseteq \tilde{B}$, $\tilde{B}$ be an $NOs$ in $(\mathcal{Z},\Gamma )$. By Definition 6, $\tilde{S}\cup NCl(NInt(NCl(\tilde{S})))\subseteq \tilde{B}$. This implies $NCl(NInt(NCl(\tilde{S})))\subseteq \tilde{B}$ and $NCl(NInt(\tilde{S}))\subseteq \tilde{B}$. Therefore, $pNCl(\tilde{S})=\tilde{S}\cup NCl(NInt(\tilde{S}))\subseteq \tilde{B}$. Hence, $\tilde{S}$ is $GNPCs\in \mathcal{Z}$. □

Example 5.

Let $\mathcal{Z}=\{u,v\}$ and $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.5,0.6,0.6),(0.5,0.4,0.4)\rangle$. Then, the $NS$ $\tilde{S}=\langle (0.4,0.5,0.5),(0.6,0.5,0.5)\rangle$ is $GNPC$ in $\mathcal{Z}$ but not $GN\alpha C$ in $\mathcal{Z}$ since $\alpha NCl(\tilde{S})=1_N\not\subset H$.

Definition 21.

An $NS$ $\tilde{S}$ is said to be a neutrosophic generalized pre-closed set ($GNSCS$ ) in $(\mathcal{Z},\Gamma )$ if $SNCl(\tilde{S})\subseteq \tilde{B}$ whensoever $\tilde{S}\subseteq \tilde{B}$ and $\tilde{B}$ is an $NO$ in $\mathcal{Z}$. The family of all $GNSCSs$ of an $NT$ $(\mathcal{Z},\Gamma )$ is defined by $GNSC(\mathcal{Z})$.

Proposition 5.

Let $\tilde{S},B$ be a two $GNPCs$ of an $NT$ $(\mathcal{Z},\Gamma )$. $NGSC$ and $NGPC$ are independent.

Example 6.

Let $\mathcal{Z}=\{u,v\}$, $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.5,0.4,0.4),(0.5,0.6,0.5)\rangle$. Then, the $NS$ $\tilde{S}=H$ is $GNSC$ but not $GNPC$ in $\mathcal{Z}$ since $\tilde{S}\subseteq H$ but $pNCl(\tilde{S})=\langle (0.5,0.6,0.4),(0.5,0.4,0.5)\rangle \not\subset H$

Example 7.

Let $\mathcal{Z}=\{u,v\}$, $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.7,0.9,0.7),(0.3,0.1,0.1)\rangle$. Then, the $NS$ $\tilde{S}=\langle (0.6,0.7,0.6),(0.4,0.3,0.4)\rangle$ is GNPC but not GNsC in $\mathcal{Z}$ since $sNCl(\tilde{S})=1_N\subseteq H$.

Proposition 6.

NSC and GNPC are independent.

Example 8.

Let $\mathcal{Z}=\{a,b\}$, $\Gamma =\{0_N,1_N,T\}$ be a neutrosophic topology on $\mathcal{Z}$, where $T=\langle (0.5,0.2,0.3),(0.5,0.6,0.5)\rangle$. Then, the $NS$ $\tilde{S}=T$ is an $NSC$ but not $GNPC$ in $\mathcal{Z}$ since $\tilde{S}\subseteq T$ but $pNCl(\tilde{S})=1\langle (0.5,0.6,0.5),(0.5,0.2,0.3)\rangle \not\subset T$.

Example 9.

Let $\mathcal{Z}=\{u,v\}$, $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.8,0.8,0.8),(0.2,0.2,0.2)\rangle$. Then, the $NS$ $\tilde{S}=\langle (0.8,0.8,0.8),(0.2,0.2,0.2)\rangle$ is $GNPC$ but not an $NSC$ in $\mathcal{Z}$ since $NInt(NCl(\tilde{S}))\not\subset \tilde{S}$.

The following Figure 1 shows the implication relations between $GNPC$ set and the other existed ones.

Remark 1.

Let $\tilde{S},B$ be a two $GNPCs$ of an $NT$ $(\mathcal{Z},\Gamma )$. Then, the union of any two $GNPCs$ is not a $GNPC$ in general—see the following example.

Example 10.

Let $(\mathcal{Z},\Gamma )$ be a neutrosophic topology set on $\mathcal{Z}$, where $\mathcal{Z}=\{u,v\}$, $T=\langle (0.6,0.8,0.6),(0.4,0.2,0.2)\rangle$. Then, $\Gamma =\{0_N,1_N,T\}$ is neutrosophic topology on $\mathcal{Z}$ and the $NS$ $\tilde{S}=\langle (0.2,0.9,0.3),(0.8,0.2,0.6)\rangle$, $B=\langle (0.6,0.7,0.6),(0.4,0.3,0.4)\rangle$ are $GNPCSs$ but $\tilde{S}\cup B$ is not a GNPC in $\mathcal{Z}$.

5. Generalized Neutrosophic Pre-Open Sets

In this section, we present generalized neutrosophic pre-open sets and investigate some of their properties.

Definition 22.

An $NS$ $\tilde{S}$ is said to be a generalized neutrosophic pre-open set ($GNPOS$ ) in $(\mathcal{Z},\Gamma )$ if the complement ${\tilde{S}}^c$ is a $GNPCS$ in $\mathcal{Z}$. The family of all $GNPOSs$ of $NTS$ $(\mathcal{Z},\Gamma )$ is denoted by $GNPO(\mathcal{Z})$.

Example 11.

Let $\mathcal{Z}=\{u,v\}$ and $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.8,0.7,0.8),(0.3,0.4,0.3)\rangle$. Then, the $NS$ $\tilde{S}=\langle (0.9,0.8,0.8),(0.3,0.3,0.3)\rangle$ is $GNPO\in \mathcal{Z}$.

Theorem 7.

Let $(\mathcal{Z},\Gamma )$ be an $NT$. Then, for every $\tilde{S}\in GNPO(\mathcal{Z})$ and for every $\tilde{R}\in NS(\mathcal{Z})$, $pNInt(\tilde{S})\subseteq \tilde{R}\subseteq \tilde{S}$ implies $\tilde{R}\in GNPO(\mathcal{Z})$.

Proof. 

By Theorem ${\tilde{S}}^c\subseteq {\tilde{R}}^c\subseteq {(pNInt(\tilde{S}))}^c$. Let ${\tilde{R}}^c\subseteq \tilde{R}$ and $\tilde{R}$ be $NOs$. Since ${\tilde{S}}^c\subseteq B^c,{\tilde{S}}^c\subseteq \tilde{R}$. However, ${\tilde{S}}^c$ is a $GNPCs$, $pNCl({\tilde{S}}^c)\subseteq \tilde{R}$. In addition, ${\tilde{R}}^c\subseteq {(pNInt(\tilde{S}))}^c=pNCl({\tilde{S}}^c)$ (by theorem). Therefore, $pNCl({\tilde{R}}^c)\subseteq pNCl({\tilde{S}}^c)\subseteq \tilde{R}$. Hence, $B^c$ is $GNPC$. This implies that $\tilde{R}$ is a $GNPO$ of $\mathcal{Z}$. □

Remark 2.

Let $\tilde{S},\tilde{R}$ be two $GNPOs$ of an $NT$ $(\mathcal{Z},\Gamma )$. The intersection of any two $GNPOSs$ is not a $GNPO$ in general.

Example 12.

Let $\mathcal{Z}=\{u,v\}$ and $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.6,0.8,0.6),(0.4,0.2,0.4)\rangle$. Then, the $NSs$, $\tilde{S}=\langle (0.9,0.2,0.1),(0.1,0.8,0.2)\rangle$ and $\tilde{R}=\langle (0.4,0.3,0.4),(0.6,0.7,0.6)\rangle$ is $GNPO$, but $\tilde{S}\cap \tilde{R}$ is not $GNPO\in \mathcal{Z}$.

Theorem 8.

For any an $NTS$ $(\mathcal{Z},\Gamma )$, the following hold:

(i) 

Every $NO$ is $GNPO$,

(ii) 

Every $NSO$ is $GNPO$,

(iii) 

Every $N\alpha O$ is $GNPO$,

(iv) 

Every $NPO$ is $GNPO$.

Proof. 

The proof is clear, so it has been omitted. □

The converses are not true in general.

Example 13.

Let $\mathcal{Z}=\{u,v\}$ and $H=\langle (0.2,0.3,0.2),(0.8,0.7,0.7)\rangle$. Then, $\Gamma =\{0_N,1_N,H\}$ is a neutrosophic topology on $\mathcal{Z}$, an $NS$ $\tilde{S}=\langle (0.8,0.7,0.7),(0.2,0.2,0.2)\rangle$ is an $NSO$ in $(\mathcal{Z},\Gamma )$ but not an $NO\in \mathcal{Z}$.

Example 14.

Let $\mathcal{Z}=\{u,v\}$ and $\Gamma =\{0_N,1_N,H\}$ be neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.6,0.4,0.7),(0.7,0.4,0.6)\rangle$. Then, an $NS$ $\tilde{S}=\langle (0.2,0.7,0.7),(0.8,0.3,0.8)\rangle$ is $GNPO$ but not an $NSO\in \mathcal{Z}$.

Example 15.

Let $\mathcal{Z}=\{u,v\}$ and $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.4,0.2,0.4),(0.6,0.7,0.6)\rangle$. Then, an $NS$ $\tilde{S}=\langle (0.8,0.9,0.8),(0.4,0.2,0.3)\rangle$ is $GNPO$ but not an $N\alpha O\in \mathcal{Z}$.

Example 16.

Let $\mathcal{Z}=\{u,v\}$ and $\Gamma =\{0_N,1_N,H\}$ be a neutrosophic topology on $\mathcal{Z}$, where $H=\langle (0.6,0.5,0.6),(0.5,0.6,0.5)\rangle$. Then, an $NS$ $\tilde{S}=\langle (0.8,0.7,0.8),(0.4,0.5,0.3)\rangle$ is $GNPO$ but not an $NPO\in \mathcal{Z}$.

Theorem 9.

Let $(\mathcal{Z},\Gamma )$ be an $NT$. If $\tilde{S}\in GNPO(\mathcal{Z})$, then $\tilde{R}\subseteq NInt(NCl(\tilde{S}))$ whensoever $\tilde{R}\subseteq \tilde{S}$ and $\tilde{R}$ is an $NC$ in $\mathcal{Z}$.

Proof. 

Let $\tilde{S}\in GNPO(\mathcal{Z})$. Then, ${\tilde{S}}^c$ is $GnPCS$ in $\mathcal{Z}$. Therefore, $pNCl({\tilde{S}}^c)\subseteq \tilde{B}$ whensoever ${\tilde{S}}^c\subseteq \tilde{B}$ and $\tilde{B}$ is an $NO$ in $\mathcal{Z}$. That is, $NCl(NInt({\tilde{S}}^c))\subseteq \tilde{B}$. This implies ${\tilde{B}}^c\subseteq NInt(NCl(\tilde{S}))$ whensoever ${\tilde{B}}^c\subseteq \tilde{S}$ and ${\tilde{B}}^c$ is $NCs$ in $\mathcal{Z}$. Replacing ${\tilde{B}}^c$, by $\tilde{R}$, we get $\tilde{R}\subseteq NInt(NCl(\tilde{S}))$ whensoever $\tilde{R}\subseteq \tilde{S}$ and $\tilde{R}$ is an $NC$ in $\mathcal{Z}$. □

Theorem 10.

For $NS$ $\tilde{S}$, $\tilde{S}$ is an $NO$ and $GNPC$ in $\mathcal{Z}$ if and only if $\tilde{S}$ is an $NRO$ in $\mathcal{Z}$.

Proof. 

⟹ Let $\tilde{S}$ be an $NO$ and a $GNPCS$ in $\mathcal{Z}$. Then, $pNCl(\tilde{S})\subseteq \tilde{S}$. This implies $NCl(NInt(\tilde{S}))\subseteq \tilde{S}$. Since $\tilde{S}$ is an $NO$, it is an $NPO$. Hence, $\tilde{S}\subseteq NInt(NCl(\tilde{S}))$. Therefore, $\tilde{S}=NInt(NCl(\tilde{S}))$. Hence, $\tilde{S}$ is an $NRO$ in $\mathcal{Z}$.

⟸ Let $\tilde{S}$ be an $NRO$ in $\mathcal{Z}$. Therefore, $\tilde{S}=NInt(NCl(\tilde{S}))$. Let $\tilde{S}\subseteq \tilde{B}$ and $\tilde{B}$ be an $NO$ in $\mathcal{Z}$. This implies $pNCl(\tilde{S})\subseteq \tilde{S}$. Hence, $\tilde{S}$ is $GNPC$ in $\mathcal{Z}$. □

Theorem 11.

An $NS$ $\tilde{S}$ of an $NT$ $(\mathcal{Z},\Gamma )$ is a $GNPO$ iff $H\subseteq pNInt(\tilde{S})$, whensoever H is an $NC$ and $H\subseteq \tilde{S}$.

Proof. 

⟹ Let $\tilde{S}$ be $GNPO$ in $\mathcal{Z}$. Let H be an $NCs$ and $H\subseteq \tilde{S}$. Then, $H^c$ is an $NOS$ in $\mathcal{Z}$ such that ${\tilde{S}}^c\subseteq H^c$. Since ${\tilde{S}}^c$ is $GNPC$, we have $pNCl({\tilde{S}}^c)\subseteq H^c$. Hence, ${(pNInt(\tilde{S}))}^c\subseteq H^c$. Therefore, $H\subseteq pNInt(\tilde{S})$.

⟸ Suppose $\tilde{S}$ is an $NS$ of $\mathcal{Z}$ and let $H\subseteq pNInt(\tilde{S})$ whensoever H is an $NC$ and $H\subseteq \tilde{S}$. Then, ${\tilde{S}}^c\subseteq H^c$ and $H^c$ is an $NO$. By assumption, ${(pNInt(\tilde{S}))}^c\subseteq H^c$, which implies $pNCl({\tilde{S}}^c)\subseteq H^c$. Therefore, ${\tilde{S}}^c$ is $GNPCs$ of $\mathcal{Z}$. Hence, $\tilde{S}$ is a $GNPOS$ of $\mathcal{Z}$. □

Corollary 1.

An $NS$ $\tilde{S}$ of an $NTS$ $(\mathcal{Z},\Gamma )$ is $GNPO$ iff $H\subseteq NInt(NCl(\tilde{S}))$, whensoever H is an $NC$ and $H\subseteq \tilde{S}$.

Proof. 

⟹ Let $\tilde{S}$ is a $GNPOS$ in $\mathcal{Z}$. Let H be an $NCS$ and $H\subseteq \tilde{S}$. Then, $H^c$ is an $NOS$ in $\mathcal{Z}$ such that ${\tilde{S}}^c\subseteq H^c$. Since ${\tilde{S}}^c$ is $GNPC$, we have $pNCl({\tilde{S}}^c)\subseteq H^c$. Therefore, $NCl(NInt({\tilde{S}}^c))\subseteq H^c$. Hence, ${(NInt(NCl(\tilde{S})))}^c\subseteq H^c$. This implies $H\subseteq NInt(NCl(\tilde{S}))$.

⟸ Suppose $\tilde{S}$ be an $NS$ of $\mathcal{Z}$ and $H\subseteq NInt(NCl(\tilde{S}))$, whensoever H is an $NC$ and $H\subseteq \tilde{S}$. Then, ${\tilde{S}}^c\subseteq H^c$ and $H^c$ is an $NO$. By assumption, ${(NInt(NCl(\tilde{S})))}^c\subseteq H^c$. Hence, $NCl(NInt({\tilde{S}}^c))\subseteq H^c$. This implies $pNCl({\tilde{S}}^c)\subseteq H^c$. Hence, $\tilde{S}$ is a $GNPOS$ of $\mathcal{Z}$. □

6. Applications of Generalized Neutrosophic Pre-Closed Sets

Definition 23.

An $NTS$ $(\mathcal{Z},\Gamma )$ is said to be neutrosophic-$p{T}_{\frac{1}{2}}$ ($Np{T}_{\frac{1}{2}}$ in short) space if every $GNPC$ in $\mathcal{Z}$ is an $NCs\in \mathcal{Z}$.

Definition 24.

An $NTS$ $(\mathcal{Z},\Gamma )$ is said to be neutrosophic-$gp{T}_{\frac{1}{2}}$ ($Ngp{T}_{\frac{1}{2}}$ in short) space if every $GNPC$ in $\mathcal{Z}$ is an $NPCs\in \mathcal{Z}$.

Theorem 12.

Every $Np{T}_{\frac{1}{2}}$ space is an $Ngp{T}_{\frac{1}{2}}$ space.

Proof. 

Let $\mathcal{Z}$ be an $Np{T}_{\frac{1}{2}}$ space and $\tilde{S}$ be $GNPC\in \mathcal{Z}$. By assumption, $\tilde{S}$ is $NCs$ in $\mathcal{Z}$. Since every $NC$ is an $NPC$, $\tilde{S}$ is an $NPC$ in $\mathcal{Z}$. Hence, $\mathcal{Z}$ is an $Ngp{T}_{\frac{1}{2}}$ space. □

The converse is not true.

Example 17.

Let $\mathcal{Z}=\{u,v\}$, $H=\langle (0.9,0.9,0.9),(0.1,0.1,0.1)\rangle$ and $\Gamma =\{0_N,1_N,H\}$. Then, $(\mathcal{Z},\Gamma )$ is an $Ngp{T}_{\frac{1}{2}}$ space, but it is not $Np{T}_{\frac{1}{2}}$ since an $NS$ $H=\langle (0.2,0.3,0.3),(0.8,0.7,0.7)\rangle$ is $GNPC$ but not an $NCS\in \mathcal{Z}$.

Theorem 13.

Let $(\mathcal{Z},\Gamma )$ be an $NT$ and $\mathcal{Z}$ is an $Np{T}_{\frac{1}{2}}$ space; then,

(i) 

the union of $GNPCs$ is $GNPC$,

(ii) 

the intersection of $GNPOs$ is $GNPO$.

Proof. 

(i) Let ${\{{\tilde{S}}_i\}}_{i\in J}$ be a collection of $GNPCs$ in an $Np{T}_{\frac{1}{2}}$ space $(\mathcal{Z},\Gamma )$. Thus, every $GNPCs$ is an $NCS$. However, the union of an $NC$ is an $NCS$. Therefore, the Union of $GNPCs$ is $GNPCs$ in $\mathcal{Z}$.

(ii) Proved by taking complement in (i). □

Theorem 14.

An $NT$ $\mathcal{Z}$ is an $Ngp{T}_{\frac{1}{2}}$ space iff $GNPO(\mathcal{Z})=NPO(\mathcal{Z})$.

Proof. 

⟹ Let $\tilde{S}$ be a $GNPOs$ in $\mathcal{Z}$; then, ${\tilde{S}}^c$ is $GNPCs$ in $\mathcal{Z}$. By assumption, ${\tilde{S}}^c$ is an $NPCs$ in $\mathcal{Z}$. Thus, $\tilde{S}$ is $NPOs$ in $\mathcal{Z}$. Hence, $GNPO(\mathcal{Z})=NPO(\mathcal{Z})$.

⟸ Let $\tilde{S}$ be $GNPC\in \mathcal{Z}$. Then, ${\tilde{S}}^c$ is $GNPO$ in $\mathcal{Z}$. By assumption, ${\tilde{S}}^c$ is an $NPO$ in $\mathcal{Z}$. Thus, $\tilde{S}$ is an $NPC\in \mathcal{Z}$. Therefore, $\mathcal{Z}$ is an $Ngp{T}_{\frac{1}{2}}$ space. □

Theorem 15.

For an $NTS$ $(\mathcal{Z},\Gamma ),$ the following are equivalent:

(i) 

$(\mathcal{Z},\Gamma )$ is a neutrosophic pre-$T_{\frac{1}{2}}$ space.

(ii) 

Every non-empty set of $\mathcal{Z}$ is either an $NPCS$ or $NPOS$.

Proof. 

$(i)⟹(ii)$. Suppose that $(\mathcal{Z},\Gamma )$ is a neutrosophic pre-$T_{\frac{1}{2}}$ space. Suppose that $\{x\}$ is not an $NPCS$ for some $x\in \mathcal{Z}$. Then, $\mathcal{Z}-\{x\}$ is not an $NPOS$ and hence $\mathcal{Z}$ is the only an $NPOS$ containing $\mathcal{Z}-\{x\}$. Hence, $\mathcal{Z}-\{x\}$ is an $NPGCS$ in $(\mathcal{Z},\Gamma )$. Since $(\mathcal{Z},\Gamma )$ is a neutrosophic pre-$T_{\frac{1}{2}}$ space, then $\mathcal{Z}-\{x\}$ is an $NPCS$ or equivalently $\{x\}$ is an $NPOS$. $(ii)⟹(i)$. Let every singleton set of $\mathcal{Z}$ be either $NPCS$ or $NPOS$. Let $\tilde{S}$ be an $NPGCS$ of $(\mathcal{Z},\Gamma )$. Let $x\in \mathcal{Z}$. We show that $x\in \mathcal{Z}$ in two cases.

Case (i): Suppose that $\{x\}$ is $NPCS$. If $x\notin \tilde{S}$, then $x\in pNCl(\tilde{S})-\tilde{S}$. Now, $pNCl(\tilde{S})-\tilde{S}$ contains a non—empty $NPCS$. Since $\tilde{S}$ is $NPGCS$, by Theorem 7, we arrived to a contradiction. Hence, $x\in \mathcal{Z}$.

Case (ii): Let $\{x\}$ be $NPOS$. Since $x\in pNCl(\tilde{S})$, then $\{x\}\cap \tilde{S}\ne \varphi$. Thus, $x\in \mathcal{Z}$. Thus, in any case $x\in \mathcal{Z}$. Thus, $PNCl(\tilde{S})\subseteq \tilde{S}$. Hence, $\tilde{S}=pNCl(\tilde{S})$ or equivalently $\tilde{S}$ is an $NPCS$. Thus, every $NPGCS$ is an $NCS$. Therefore, $(\mathcal{Z},\Gamma )$ is neutrosophic pre-$T_{\frac{1}{2}}$ space. □

7. 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.