On Neutrosophic αψ-Closed Sets

Abstract

The aim of this paper is to introduce the concept of $\alpha \psi$-closed sets in terms of neutrosophic topological spaces. We also study some of the properties of neutrosophic $\alpha \psi$-closed sets. Further, we introduce continuity and contra continuity for the introduced set. The two functions and their relations are studied via a neutrosophic point set.

1. Introduction

Zadeh [1] introduced and studied truth (t), the degree of membership, and defined the fuzzy set theory. The falsehood (f), the degree of nonmembership, was introduced by Atanassov [2,3,4] in an intuitionistic fuzzy set. Coker [5] developed intuitionistic fuzzy topology. Neutrality (i), the degree of indeterminacy, as an independent concept, was introduced by Smarandache [6,7] in 1998. He also defined the neutrosophic set on three components $(t,f,i)=(truth,falsehood,indeterminacy)$. The Neutrosophic crisp set concept was converted to neutrosophic topological spaces by Salama et al. in [8]. This opened up a wide range of investigation in terms of neutosophic topology and its application in decision-making algorithms. Arokiarani et al. [9] introduced and studied $\alpha$-open sets in neutrosophic topoloical spaces. Devi et al. [10,11,12] introduced $\alpha \psi$-closed sets in general topology, fuzzy topology, and intutionistic fuzzy topology. In this article, the neutrosophic $\alpha \psi$-closed sets are introduced in neutrosophic topological space. Moreover, we introduce and investigate neutrosophic $\alpha \psi$-continuous and neutrosophic contra $\alpha \psi$-continuous mappings.

2. Preliminaries

Let neutrosophic topological space (NTS) be $(X,\tau )$. Each neutrosophic set(NS) in $(X,\tau )$ is called a neutrosophic open set (NOS), and its complement is called a neutrosophic open set (NOS).

We provide some of the basic definitions in neutrosophic sets. These are very useful in the sequel.

Definition 1.

[6] A neutrosophic set (NS) A is an object of the following form

$$U=\{⟨x,{\mu }_U\left(x\right),{\nu }_U\left(a\right),{\omega }_U\left(x\right)⟩:x\in X\}$$

where the mappings ${\mu }_U:X\to I$, ${\nu }_U:X\to I$, and ${\omega }_U:X\to I$ denote the degree of membership (namely ${\mu }_U\left(x\right)$), the degree of indeterminacy (namely ${\nu }_U\left(x\right)$), and the degree of nonmembership (namely ${\omega }_U\left(x\right)$) for each element $x\in X$ to the set U, respectively, and $0\le {\mu }_U\left(x\right)+{\nu }_U\left(x\right)+{\omega }_U\left(x\right)\le 3$ for each $a\in X$.

Definition 2.

[6] Let U and V be NSs of the form $U=\{⟨a,{\mu }_U\left(x\right),{\nu }_U\left(x\right),{\omega }_U\left(x\right)⟩:a\in X\}$ and $V=\{⟨x,{\mu }_V\left(x\right),{\nu }_V\left(x\right),{\omega }_V\left(x\right)⟩:x\in X\}$. Then

(i) 

$U\subseteq V$ if and only if ${\mu }_U\left(x\right)\le {\mu }_V\left(x\right)$, ${\nu }_U\left(x\right)\ge {\nu }_V\left(x\right)$ and ${\omega }_U\left(x\right)\ge {\omega }_V\left(x\right)$;

(ii) 

$\overline{U}=\{⟨x,{\nu }_U\left(x\right),{\mu }_U\left(x\right),{\omega }_U\left(x\right)⟩:x\in X\}$;

(iii) 

$U\cap V=\{⟨x,{\mu }_U\left(x\right)\wedge {\mu }_V\left(x\right),{\nu }_U\left(x\right)\vee {\nu }_V\left(x\right),{\omega }_U\left(x\right)\vee {\omega }_V\left(x\right)⟩:x\in X\}$;

(iv) 

$U\cup V=\{⟨x,{\mu }_U\left(x\right)\vee {\mu }_V\left(x\right),{\nu }_U\left(x\right)\wedge {\nu }_V\left(x\right),{\omega }_U\left(x\right)\wedge {\omega }_V\left(x\right)⟩:x\in X\}$.

We will use the notation $U=⟨x,{\mu }_U,{\nu }_U,{\omega }_U⟩$ instead of $U=\{⟨x,{\mu }_U\left(x\right),{\nu }_U\left(x\right),{\omega }_U\left(x\right)⟩:x\in X\}$. The NSs $0_{\sim }$ and $1_{\sim }$ are defined by $0_{\sim }=\{⟨x,\underline{0},\underline{1},\underline{1}⟩:x\in X\}$ and $1_{\sim }=\{⟨x,\underline{1},\underline{0},\underline{0}⟩:x\in X\}$.

Let $r,s,t\in [0,1]$ such that $r+s+t\le 3$. A neutrosophic point (NP) $p_{(r,s,t)}$ is neutrosophic set defined by

$$p_{(r,s,t)}\left(x\right)=\left\{\begin{array}{cc}(r,s,t)\left(x\right)\hfill & \hfill ifx=p\\ (0,1,1)\hfill & \hfill otherwise.\end{array}\right.$$

Let f be a mapping from an ordinary set X into an ordinary set Y. If $V=\{⟨y,{\mu }_V\left(y\right),{\nu }_V\left(y\right),{\omega }_V\left(y\right)⟩:y\in Y\}$ is an NS in Y, then the inverse image of V under f is an NS defined by

$$f^{-1}\left(V\right)=\{⟨x,f^{-1}\left({\mu }_V\right)\left(x\right),f^{-1}\left({\nu }_V\right)\left(x\right),f^{-1}\left({\omega }_V\right)\left(x\right)⟩:x\in X\}.$$

The image of NS $U=\{⟨y,{\mu }_U\left(y\right),{\nu }_U\left(y\right),{\omega }_U\left(y\right)⟩:y\in Y\}$ under f is an NS defined by $f\left(U\right)$ = $\{⟨y,f\left({\mu }_U\right)\left(y\right),f\left({\nu }_U\right)\left(y\right),f\left({\omega }_U\right)\left(y\right)⟩:y\in Y\}$ where

$$f\left({\mu }_U\right)\left(y\right)=\left\{\begin{array}{cc}\underset{x\in f^{-1}\left(y\right)}{sup}{\mu }_U\left(x\right),\hfill & \hfill if{f}^{-1}\left(y\right)\ne 0\\ 0\hfill & \hfill otherwise\end{array}\right.$$

$$f\left({\nu }_U\right)\left(y\right)=\left\{\begin{array}{cc}\underset{x\in f^{-1}\left(y\right)}{inf}{\nu }_U\left(x\right),\hfill & \hfill if{f}^{-1}\left(y\right)\ne 0\\ 1\hfill & \hfill otherwise\end{array}\right.$$

$$f\left({\omega }_U\right)\left(y\right)=\left\{\begin{array}{cc}\underset{x\in f^{-1}\left(y\right)}{inf}{\omega }_U\left(x\right),\hfill & \hfill if{f}^{-1}\left(y\right)\ne 0\\ 1\hfill & \hfill otherwise\end{array}\right.$$

for each $y\in Y$.

Definition 3.

[8] A neutrosophic topology (NT) in a nonempty set X is a family τ of NSs in X satisfying the following axioms:

(NT1) 

$0_{\sim },1_{\sim }\in \tau$;

(NT2) 

$G_1\cap G_2\in \tau$ for any $G_1,G_2\in \tau$;

(NT3) 

$\cup G_i\in \tau$ for any arbitrary family $\{G_i:i\in J\}\subseteq \tau$.

Definition 4.

[8] Let U be an NS in NTS X. Then

Nint$\left(U\right)$ = $\cup \{O:O$ is an NOS in X and O $\subseteq U\}$ is called a neutrosophic interior of U;

Ncl$\left(U\right)$ = $\cap \{O:O$ is an NCS in X and O $\supseteq U\}$ is called a neutrosophic closure of U.

Definition 5.

[8] Let $p_{(r,s,t)}$ be an NP in NTS X. An NS U in X is called a neutrosophic neighborhood (NN) of $p_{(r,s,t)}$ if there exists an NOS V in X such that $p_{(r,s,t)}\in V\subseteq U$.

Definition 6.

[9] A subset U of a neutrosophic space $(X,\tau )$ is called

1. 

a neutrosophic pre-open set if $U\subseteq Nint\left(Ncl\right(U\left)\right)$, and a neutrosophic pre-closed set if $Ncl\left(Nint\right(U\left)\right)\subseteq$ U,

2. 

a neutrosophic semi-open set if $U\subseteq Ncl\left(Nint\right(U\left)\right)$, and a neutrosophic semi-closed set if $Nint\left(Ncl\right(U\left)\right)\subseteq U$,

3. 

a neutrosophic α-open set if $U\subseteq Nint\left(Ncl\right(Nint\left(U\right)\left)\right)$, and a neutrosophic α-closed set if $Ncl\left(Nint\right(Ncl\left(U\right)\left)\right)\subseteq U$.

The pre-closure (respectively, semi-closure and α-closure) of a subset U of a neutrosophic space $(X,\tau )$ is the intersection of all pre-closed (respectively, semi-closed, α-closed) sets that contain U and is denoted by $Npcl\left(U\right)$ (respectively, $Nscl\left(U\right)$ and $N\alpha cl\left(U\right)$).

Definition 7.

A subset A of a neutrosophic topological space $(X,\tau )$ is called

1. 

a neutrosophic semi-generalized closed (briefly, $Nsg$-closed) set if $Nscl\left(U\right)\subseteq G$ whenever $U\subseteq G$ and G is neutrosophic semi-open in $(X,\tau )$;

2. 

a neutrosophic $N\psi$-closed set if $Nscl\left(U\right)\subseteq G$ whenever $U\subseteq G$ and G is $Nsg$-open in $(X,\tau )$.

3. On Neutrosophic $\mathit{\alpha }\mathit{\psi }$-Closed Sets

Definition 8.

A neutrosophic $\alpha \psi$-closed ($N\alpha \psi$-closed) set is defined as if $N\psi cl\left(U\right)\subseteq G$ whenever $U\subseteq G$ and G is an $N\alpha$-open set in $(X,\tau )$. Its complement is called a neutrosophic $\alpha \psi$-open ($N\alpha \psi$-open) set.

Definition 9.

Let U be an NS in NTS X. Then

$N\alpha \psi int\left(U\right)$ = $\cup \{O:O$ is an $N\alpha \psi$OS in X and O $\subseteq U\}$ is said to be a neutrosophic $\alpha \psi$-interior of U;

$N\alpha \psi cl\left(U\right)$ = $\cap \{O:O$ is an $N\alpha \psi$CS in X and O $\supseteq U\}$ is said to be a neutrosophic $\alpha \psi$-closure of U.

Theorem 1.

All $N\alpha$-closed sets and N-closed sets are $N\alpha \psi$-closed sets.

Proof. 

Let U be an $N\alpha$-closed set, then $U=N\alpha cl\left(U\right)$. Let $U\subseteq G$, where G is $N\alpha$-open. Since U is $N\alpha$-closed, $N\psi cl\left(U\right)\subseteq N\alpha cl\left(U\right)\subseteq G$. Thus, U is $N\alpha \psi$-closed. ☐

Theorem 2.

Every Nsemi-closed set in a neutrosophic set is an $N\alpha \psi$-closed set.

Proof. 

Let U be an Nsemi-closed set in $(X,\tau )$, then $U=Nscl\left(U\right)$. Let $U\subseteq G$, where G is $N\alpha$-open in $(X,\tau )$. Since U is Nsemi-closed, $N\psi cl\left(U\right)\subseteq Nscl\left(U\right)\subseteq G$. This shows that U is $N\alpha \psi$-closed set.

The converses of the above theorems are not true, as can be seen by the following counter example. ☐

Example 1.

Let $X=\{u,v,w\}$ and neutrosophic sets $G_1,G_2,G_3,G_4$ be defined by

$$G_1=⟨x,(\frac{u}{0.3},\frac{v}{0.4},\frac{w}{0.2}),(\frac{u}{0.5},\frac{v}{0.1},\frac{w}{0.2}),(\frac{u}{0.2},\frac{v}{0.5},\frac{w}{0.6})⟩$$

$$G_2=⟨x,(\frac{u}{0.6},\frac{v}{0.3},\frac{w}{0.4}),(\frac{u}{0.1},\frac{v}{0.5},\frac{w}{0.1}),(\frac{u}{0.3},\frac{v}{0.2},\frac{w}{0.5})⟩$$

$$G_3=⟨x,(\frac{u}{0.6},\frac{v}{0.4},\frac{w}{0.4}),(\frac{u}{0.1},\frac{v}{0.1},\frac{w}{0.1}),(\frac{u}{0.2},\frac{v}{0.2},\frac{w}{0.5})⟩$$

$$G_4=⟨x,(\frac{u}{0.3},\frac{v}{0.3},\frac{w}{0.2}),(\frac{u}{0.5},\frac{v}{0.5},\frac{w}{0.2}),(\frac{u}{0.3},\frac{v}{0.5},\frac{w}{0.6})⟩$$

$$G_5=⟨x,(\frac{u}{0.3},\frac{v}{0.3},\frac{w}{0.3}),(\frac{u}{0.5},\frac{v}{0.5},\frac{w}{0.4}),(\frac{u}{0.3},\frac{v}{0.5},\frac{w}{0.3})⟩$$

$$G_6=⟨x,(\frac{u}{0.6},\frac{v}{0.4},\frac{w}{0.5}),(\frac{u}{0.1},\frac{v}{0.3},\frac{w}{0.1}),(\frac{u}{0.3},\frac{v}{0.3},\frac{w}{0.4})⟩$$

$$G_7=⟨x,(\frac{u}{0.2},\frac{v}{0.3},\frac{w}{0.3}),(\frac{u}{0.5},\frac{v}{0.5},\frac{w}{0.2}),(\frac{u}{0.3},\frac{v}{0.3},\frac{w}{0.5})⟩.$$

Let $\tau =\{0_{\sim },G_1,G_2,G_3,G_4,1_{\sim }\}$. Here, $G_6$ is an $N\alpha$ open set, and $N\psi cl\left(G_5\right)\subseteq G_6$. Then $G_5$ is $N\alpha \psi$-closed in $(X,\tau )$ but is not $N\alpha$-closed; thus, it is not N-closed and $G_7$ is $N\alpha \psi$-closed in $(X,\tau )$, but not $Nsemi$-closed.

Theorem 3.

$Let(X,\tau )beanNTSandlet$ $U\in NS\left(X\right)$. $IfUis$ an $N\alpha \psi$-$closedsetandU\subseteq V\subseteq N\psi cl\left(U\right)$, $thenVis$ an $N\alpha \psi$-$closedset$.

Proof. 

Let G be an $N\alpha$-open set such that $V\subseteq G$. Since $U\subseteq V$, then $U\subseteq G$. But U is $N\alpha \psi$-closed, so $N\psi cl\left(U\right)\subseteq G$, since $V\subseteq N\psi cl\left(U\right)$ and $N\psi cl\left(V\right)\subseteq N\psi cl\left(U\right)$ and hence $N\psi cl\left(V\right)\subseteq G$. Therefore V is an $N\alpha \psi$-closed set. ☐

Theorem 4.

Let U be an $N\alpha \psi$-open set in X and $N\psi int\left(U\right)\subseteq V\subseteq U$, then V is $N\alpha \psi$-open.

Proof. 

Suppose U is $N\alpha \psi$-open in X and $N\psi int\left(U\right)\subseteq V\subseteq U$. Then $\overline{U}$ is $N\alpha \psi$-closed and $\overline{U}\subseteq \overline{V}\subseteq N\psi cl\left(\overline{U}\right)$. Then $\overline{U}$ is an $N\alpha \psi$-closed set by Theorem 3.5. Hence, V is an $N\alpha \psi$-open set in X. ☐

Theorem 5.

$AnNSU$ $inanNTS(X,\tau )isanN\alpha \psi$-$opensetifandonlyifV\subseteq N\psi int\left(U\right)$ $wheneverVisanN\alpha$-$closedsetandV\subseteq U$.

Proof. 

Let U be an $N\alpha \psi$-open set and let V be an $N\alpha$-closed set such that $V\subseteq U$. Then $\overline{U}\subseteq \overline{V}$ and hence $N\psi cl\left(\overline{U}\right)\subseteq \overline{V}$, since $\overline{U}$ is $N\alpha \psi$-closed. But $N\psi cl\left(\overline{U}\right)=\overline{N\psi int\left(U\right)}$, so $V\subseteq N\psi int\left(U\right)$. Conversely, suppose that the condition is satisfied. Then $\overline{N\psi int\left(U\right)}\subseteq \overline{V}$ whenever $\overline{V}$ is an $N\alpha$-open set and $\overline{U}\subseteq \overline{V}$. This implies that $N\psi cl\left(\overline{U}\right)\subseteq \overline{V}=G$, where G is $N\alpha$-open and $\overline{U}\subseteq G$. Therefore, $\overline{U}$ is $N\alpha \psi$-closed and hence U is $N\alpha \psi$-open. ☐

Theorem 6.

Let U be an $N\alpha \psi$-closed subset of $(X,\tau )$. Then $N\psi cl\left(U\right)-U$ does not contain any non-empty $N\alpha \psi$-closed set.

Proof. 

Assume that U is an $N\alpha \psi$-closed set. Let F be a non-empty $N\alpha \psi$-closed set, such that F ⊆ $N\psi cl\left(U\right)-U=N\psi cl\left(U\right)\cap \overline{U}$. i.e., $F\subseteq N\psi cl\left(U\right)$ and $F\subseteq \overline{U}$. Therefore, $U\subseteq \overline{F}$. Since $\overline{F}$ is an $N\alpha \psi$-open set, $N\psi cl\left(U\right)\subseteq \overline{F}\Rightarrow F\subseteq (N\psi cl\left(U\right)-U)\cap \left(\overline{N\psi cl\left(U\right)}\right)\subseteq N\psi cl\left(U\right)\cap \overline{N\psi cl\left(U\right)}$. i.e., $F\subseteq \varphi$. Therefore, F is empty. ☐

Corollary 1.

Let U be an $N\alpha \psi$-closed set of $(X,\tau )$. Then $N\psi cl\left(U\right)$-U does not contain anynon-empty N-closed set.

Proof. 

The proof follows from the Theorem 3.9. ☐

Theorem 7.

If U is both $N\psi$-open and $N\alpha \psi$-closed, then U is $N\psi$-closed.

Proof. 

Since U is both an $N\psi$-open and $N\alpha \psi$-closed set in X, then $N\psi cl\left(U\right)\subseteq U$. We also have $U\subseteq$ $N\psi cl\left(U\right)$. Thus, $N\psi cl\left(U\right)=U$. Therefore, U is an $N\psi$-closed set in X. ☐

4. On Neutrosophic $\mathbf{\alpha }\mathbf{\psi }$-Continuity and Neutrosophic Contra $\mathbf{\alpha }\mathbf{\psi }$-Continuity

Definition 10.

A function $f:X\to Y$ is said to be a neutrosophic $\alpha \psi$-continuous (briefly, $N\alpha \psi$-continuous) function if the inverse image of every open set in Y is an $N\alpha \psi$-open set in X.

Theorem 8.

Let $g:(X,\tau )\to (Y,\sigma )$ be a function. Then the following conditions are equivalent.

(i) 

g is $N\alpha \psi$-continuous;

(ii) 

The inverse $f^{-1}\left(U\right)$ of each N-open set U in Y is $N\alpha \psi$-open set in X.

Proof. 

The proof is obvious, since $g^{-1}\left(\overline{U}\right)=\overline{g^{-1}\left(U\right)}$ for each N-open set U of Y. ☐

Theorem 9.

If $g:(X,\tau )\to (Y,\sigma )$ is an $N\alpha \psi$-continuous mapping, then the following statements hold:

(i) 

$g\left(N\alpha \psi Ncl\right(U\left)\right)\subseteq Ncl\left(g\right(U\left)\right)$, for all neutrosophic sets U in X;

(ii) 

$N\alpha \psi Ncl\left(g^{-1}\left(V\right)\right)\subseteq g^{-1}\left(Ncl\left(V\right)\right)$, for all neutrosophic sets V in Y.

Proof. 

(i)

Since $Ncl\left(g\right(U\left)\right)$ is a neutrosophic closed set in Y and g is $N\alpha \psi$-continuous, then $g^{-1}\left(Ncl\left(g\left(U\right)\right)\right)$ is $N\alpha \psi$-closed in X. Now, since $U\subseteq g^{-1}\left(Ncl\left(g\left(U\right)\right)\right)$, $N\alpha \psi cl\left(U\right)\subseteq g^{-1}\left(Ncl\left(g\left(U\right)\right)\right)$. Therefore, $g\left(N\alpha \psi Ncl\right(U\left)\right)\subseteq Ncl\left(g\right(U\left)\right)$.

(ii)

By replacing U with V in (i), we obtain $g\left(N\alpha \psi cl\left(g^{-1}\left(V\right)\right)\right)\subseteq Ncl\left(g\left(g^{-1}\left(V\right)\right)\right)\subseteq Ncl\left(V\right)$. Hence, $N\alpha \psi cl\left(g^{-1}\left(V\right)\right)\subseteq g^{-1}\left(Ncl\left(V\right)\right)$.

☐

Theorem 10.

Let g be a function from an NTS $(X,\tau )$ to an NTS $(Y,\sigma )$. Then the following statements are equivalent.

(i) 

g is a neutrosophic $\alpha \psi$-continuous function;

(ii) 

For every NP $p_{(r,s,t)}\in X$ and each NN U of $g\left(p_{(r,s,t)}\right)$, there exists an $N\alpha \psi$-open set V such that $p_{(r,s,t)}\in V\subseteq g^{-1}\left(U\right)$.

(iii) 

For every NP $p_{(r,s,t)}\in X$ and each NN U of $g\left(p_{(r,s,t)}\right)$, there exists an $N\alpha \psi$-open set V such that $p_{(r,s,t)}\in V$ and $g\left(V\right)\subseteq U$.

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$. If $p_{(r,s,t)}$ is an NP in X and if U is an NN of $g\left(p_{(r,s,t)}\right)$, then there exists an NOS W in Y such that $g\left(p_{(r,s,t)}\right)\in W\subset U$. Thus, g is neutrosophic $\alpha \psi$-continuous, $V=g^{-1}\left(W\right)$ is an $N\alpha \psi Oset$, and

$$p_{(r,s,t)}\in g^{-1}\left(g\left(p_{(r,s,t)}\right)\right)\subseteq g^{-1}\left(W\right)=V\subseteq g^{-1}\left(U\right).$$

Thus, (ii) is a valid statement.

$\left(ii\right)\Rightarrow \left(iii\right)$. Let $p_{(r,s,t)}$ be an NP in X and let U be an NN of $g\left(p_{(r,s,t)}\right)$. Then there exists an $N\alpha \psi Oset$U such that $p_{(r,s,t)}\in V\subseteq g^{-1}\left(U\right)$ by (ii). Thus, we have $p_{(r,s,t)}\in V$ and $g\left(V\right)\subseteq g\left(g^{-1}\left(U\right)\right)\subseteq U$. Hence, (iii) is valid.

$\left(iii\right)\Rightarrow \left(i\right)$. Let V be an NO set in Y and let $p_{(r,s,t)}\in g^{-1}\left(V\right)$. Then $g\left(p_{(r,s,t)}\right)\in g\left(g^{-1}\left(V\right)\right)\subset V$. Since V is an NOS, it follows that V is an NN of $g\left(p_{(r,s,t)}\right)$. Therefore, from (iii), there exists an $N\alpha \psi Oset$U such that $p_{(r,s,t)}\in U$ and $g\left(U\right)\subseteq V$. This implies that

$$p_{(r,s,t)}\in U\subseteq g^{-1}\left(g\left(U\right)\right)\subseteq g^{-1}\left(V\right).$$

Therefore, we know that $g^{-1}\left(V\right)$ is an $N\alpha \psi Oset$ in X. Thus, g is neutrosophic $\alpha \psi$-continuous. ☐

Definition 11.

A function is said to be a neutrosophic contra $\alpha \psi$-continuous function if the inverse image of each NOS V in Y is an N$\alpha \psi$C set in X.

Theorem 11.

Let $g:(X,\tau )\to (Y,\sigma )$ be a function. Then the following assertions are equivalent:

(i) 

g is a neutrosophic contra $\alpha \psi$-continuous function;

(ii) 

$g^{-1}\left(V\right)$ is an N$\alpha \psi$ C set in X, for each NOS V in Y.

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$ Let g be any neutrosophic contra $\alpha \psi$-continuous function and let V be any NOS in Y. Then $\overline{V}$ is an NCS in Y. Based on these assumptions, $g^{-1}\left(\overline{V}\right)$ is an $N\alpha \psi Oset$ in X. Hence, $g^{-1}\left(V\right)$ is an $N\alpha \psi Cset$ in X.

The converse of the theorem can be proved in the same way. ☐

Theorem 12.

$Letg:(X,\tau )\to (Y,\sigma )beabijectivemappingfromanNTS(X,T)intoanNTS(Y,T)$. $Themappinggisneutrosophiccontra\alpha \psi$-$continuous,ifNcl\left(g\right(U\left)\right)\subseteq g\left(N\alpha \psi int\right(U\left)\right),foreachNSUinX$.

Proof. 

Let V be any NCS in X. Then $Ncl\left(V\right)=V$, and g is onto, by assumption, which shows that $g\left(N\alpha \psi int\left(g^{-1}\left(V\right)\right)\right)\supseteq Ncl\left(g\left(g^{-1}\left(V\right)\right)\right)=Ncl\left(V\right)=V$. Hence, $g^{-1}\left(g\left(N\alpha \psi int\left(g^{-1}\left(V\right)\right)\right)\right)\supseteq g^{-1}\left(V\right)$. Since g is an into mapping, we have $N\alpha \psi int\left(g^{-1}\left(V\right)\right)=g^{-1}\left(g\left(N\alpha \psi int\left(g^{-1}\left(V\right)\right)\right)\right)\supseteq g^{-1}\left(V\right)$. Therefore, $N\alpha \psi int\left(g^{-1}\left(V\right)\right)=g^{-1}\left(V\right)$, so $g^{-1}\left(V\right)$ is an $N\alpha \psi$O set in X. Hence, g is a neutrosophic contra $\alpha \psi$-continuous mapping. ☐

Theorem 13.

$Letg:(X,\tau )\to (Y,\sigma )beamapping$. $Thenthefollowingstatementsareequivalent$:

(i) 

g $isaneutrosophiccontra\alpha \psi$-$continuousmapping$;

(ii) 

$foreachNP{p}_{(r,s,t)}inXandNCSVcontainingg\left(p_{(r,s,t)}\right)thereexistsanN\alpha \psi OsetUinXcontaining{p}_{(r,s,t)}suchthatA\subseteq f^{-1}\left(B\right)$;

(iii) 

$foreachNP{p}_{(r,s,t)}inXandNCSVcontaining{p}_{(r,s,t)}thereexistsanN\alpha \psi OsetUinXcontaining{p}_{(r,s,t)}suchthatg\left(U\right)\subseteq V$.

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$ Let g be a neutrosophic contra $\alpha \psi$-continuous mapping, let V be any NCS in Y and let $p_{(r,s,t)}$ be an NP in X and such that $g\left(p_{(r,s,t)}\right)\in V$. Then $p_{(r,s,t)}\in g^{-1}\left(V\right)=N\alpha \psi int\left(g^{-1}\left(V\right)\right)$. Let $U=N\alpha \psi int\left(g^{-1}\left(V\right)\right)$. Then U is an $N\alpha \psi Oset$ and $U=N\alpha \psi int\left(g^{-1}\left(V\right)\right)\subseteq g^{-1}\left(V\right)$.

$\left(ii\right)\Rightarrow \left(iii\right)$ The results follow from evident relations $g\left(U\right)\subseteq g\left(g^{-1}\left(V\right)\right)\subseteq V$.

$\left(iii\right)\Rightarrow \left(i\right)$ Let V be any NCS in Y and let $p_{(r,s,t)}$ be an NP in X such that $p_{(r,s,t)}\in g^{-1}\left(V\right)$. Then $g\left(p_{(r,s,t)}\right)\in V$. According to the assumption, there exists an $N\alpha \psi OSU$ in X such that $p_{(r,s,t)}\in U$ and $g\left(U\right)\subseteq V$. Hence, $p_{(r,s,t)}\in U\subseteq g^{-1}\left(g\left(U\right)\right)\subseteq g^{-1}\left(V\right)$. Therefore, $p_{(r,s,t)}\in U=\alpha \psi int\left(U\right)\subseteq N\alpha \psi int\left(g^{-1}\left(V\right)\right)$. Since $p_{(r,s,t)}$ is an arbitrary NP and $g^{-1}\left(V\right)$ is the union of all NPs in $g^{-1}\left(V\right)$, we obtain that $g^{-1}\left(V\right)\subseteq N\alpha \psi int\left(g^{-1}\left(V\right)\right)$. Thus, g is a neutrosophic contra $N\alpha \psi$-continuous mapping. ☐

Corollary 2.

Let $X,X_1$ and $X_2$ be NTS sets, $p_1:X\to X_1\times X_2$ and $p_2:X\to X_1\times X_2$ are the projections of $X_1\times X_2$ onto $X_i$, $(i=1,2)$. If $g:X\to X_1\times X_2$ is a neutrosophic contra $\alpha \psi$-continuous, then $p_{i}garealsoneutrosophiccontra\alpha \psi$-$continuousmapping$.

Proof. 

This proof follows from the fact that the projections are all neutrosophic continuous functions. ☐

Theorem 14.

$Letg:(X_1,\tau )\to (Y_1,\sigma )beafunction$. $Ifthegraphh$: $X_1\to X_1\times$ $Y_{1}ofgisneutrosophiccontra\alpha \psi$-$continuous,thengisneutrosophiccontra\alpha \psi$-$continuous.$

Proof. 

For every NOS, V in $Y_1$ holds $g^{-1}\left(V\right)=1\wedge g^{-1}\left(V\right)=h^{-1}(1\times V)$. Since h is a neutrosophic contra $\alpha \psi$-continuous mapping and $1\times V$ is an NOS in $X_1\times Y_1$, $g^{-1}\left(V\right)$ is an $N\alpha \psi Cset$ in $X_1$, so g is a neutrosophic contra $\alpha \psi$-continuous mapping. ☐