More Functions Associated with Neutrosophic gsα*- Closed Sets in Neutrosophic Topological Spaces

Abstract

The concept of neutrosophic continuous function was very first introduced by A.A. Salama et al. The main aim of this paper is to introduce a new concept of Neutrosophic continuous function namely Strongly Neutrosophic gsα* - continuous functions, Perfectly Neutrosophic gsα* - continuous functions and Totally Neutrosophic gsα* - continuous functions in Neutrosophic topological spaces. These concepts are derived from strongly generalized neutrosophic continuous function and perfectly generalized neutrosophic continuous function. Several interesting properties and characterizations are derived and compared with already existing neutrosophic functions.

1. Introduction

The concept of Neutrosophic set theory was introduced by F. Smarandache [1] and it comes from two concept, one is intuitionistic fuzzy sets introduced by K. Atanassov’s [2] and the other is fuzzy sets introduced by L.A. Zadeh’s [3]. It includes three components, truth, indeterminancy and false membership function. R. Dhavaseelan and S. Jafari [4] has discussed about the concept of strongly generalized neutrosophic continuous function. Further he also introduced the topic of perfectly generalized neutrosophic continuous function. The real life application of neutrosophic topology is applied in Information Systems, Applied Mathematics etc.

In this paper, we introduce some new concepts related to Neutrosophic $\mathit{gs}{\alpha }^{\ast }-$ continuous function namely Strongly Neutrosophic $\mathit{gs}{\alpha }^{\ast }-$ continuous function, Perfectly Neutrosophic $\mathit{gs}{\alpha }^{\ast }-$ continuous function, Totally Neutrosophic $\mathit{gs}{\alpha }^{\ast }-$ continuous function.

2. Preliminaries

Definition 2.1: [5] Let $\mathbb{P}$ be a non-empty fixed set. A Neutrosophic set Ҥ on the universe $\mathbb{P}$ is defined as Ҥ$=\left\{⟨\mathcal{p},\left(t_{\mathrm{Ҥ}}\left(\mathcal{p}\right),i_{\mathrm{Ҥ}}\left(\mathcal{p}\right),f_{\mathrm{Ҥ}}\left(\mathcal{p}\right)\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$ where $t_{\mathrm{Ҥ}}\left(\mathcal{p}\right),i_{\mathrm{Ҥ}}\left(\mathcal{p}\right),f_{\mathrm{Ҥ}}\left(\mathcal{p}\right)$ represent the degree of membership function $t_{\mathrm{Ҥ}}\left(\mathcal{p}\right)$, the degree of indeterminacy $i_{\mathrm{Ҥ}}\left(\mathcal{p}\right)$ and the degree of non-membership function $f_{\mathrm{Ҥ}}\left(\mathcal{p}\right)$ respectively for each element $\mathcal{p}Є\mathbb{P}$ to the set Ҥ. Also, $t_{\mathrm{Ҥ}},i_{\mathrm{Ҥ}},f_{\mathrm{Ҥ}}$ : $\mathbb{P}\to ]{}^{-}0,1^{+}[$ and ⁻ 0 $\le t_{\mathrm{Ҥ}}\left(\mathcal{p}\right)+i_{\mathrm{Ҥ}}\left(\mathcal{p}\right)+f_{\mathrm{Ҥ}}\left(\mathcal{p}\right)\le 3^{+}$. Set of all Neutrosophic set over $\mathbb{P}$ is denoted by N_eu($\mathbb{P}$).

Definition 2.2: [8] Let $\mathbb{P}$ be a non-empty set. $\mathrm{Ⱥ}=\left\{⟨\mathcal{p},\left(t_{\mathit{Ⱥ}}\left(\mathcal{p}\right),i_{\mathit{Ⱥ}}\left(\mathcal{p}\right),f_{\mathit{Ⱥ}}\left(\mathcal{p}\right)\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$ and $\mathit{Ƀ}=\left\{⟨\mathcal{p},\left(t_{\mathit{Ƀ}}\left(\mathcal{p}\right),i_{\mathit{Ƀ}}\left(\mathcal{p}\right),f_{\mathit{Ƀ}}\left(\mathcal{p}\right)\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$ are neutrosophic sets, then

1. Ⱥ ⊆ Ƀ if $t_{\mathrm{Ⱥ}}\left(\mathcal{p}\right)\le t_{\mathrm{Ƀ}}\left(\mathcal{p}\right),i_{\mathrm{Ⱥ}}\left(\mathcal{p}\right)\le i_{\mathrm{Ƀ}}\left(\mathcal{p}\right),f_{\mathrm{Ⱥ}}\left(\mathcal{p}\right)\ge f_{\mathrm{Ƀ}}\left(\mathcal{p}\right)$ for all $\mathcal{p}Є\mathbb{P}$.

2. Intersection of two neutrosophic set $\mathrm{Ⱥ}$ and $\mathrm{Ƀ}$ is defined as $\mathrm{Ⱥ}\cap \mathrm{Ƀ}=$ $\left\{⟨\mathcal{p},\left(min\left(t_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),t_{\mathrm{Ƀ}}\left(\mathcal{p}\right)\right),min\left(i_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),i_{\mathrm{Ƀ}}\left(\mathcal{p}\right)\right),max\left(f_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),f_{\mathrm{Ƀ}}\left(\mathcal{p}\right)\right)\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$.

3. Union of two neutrosophic set $\mathrm{Ⱥ}$ and $\mathrm{Ƀ}$ is defined as $\mathrm{Ⱥ}\cup \mathrm{Ƀ}=\left\{⟨\mathcal{p},\left(max\left(t_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),t_{\mathrm{Ƀ}}\left(\mathcal{p}\right)\right),max\left(i_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),i_{\mathrm{Ƀ}}\left(\mathcal{p}\right)\right),min\left(f_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),f_{\mathrm{Ƀ}}\left(\mathcal{p}\right)\right)\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$.

4. ${\mathrm{Ⱥ}}^c=\left\{⟨\mathcal{p},\left(f_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),1-i_{\mathrm{Ⱥ}}\left(\mathcal{p}\right),t_{\mathrm{Ⱥ}}\left(\mathcal{p}\right)\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$.

5. $0_{N_{\mathit{eu}}}=\left\{⟨\mathcal{p},\left(0,0,1\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$ and $1_{N_{\mathit{eu}}}=\left\{⟨\mathcal{p},\left(1,1,0\right)⟩:\mathcal{p}Є\mathbb{P}\right\}$.

Definition 2.3: [5] A neutrosophic topology (N_euT) on a non-empty set $\mathbb{P}$ is a family ${\tau }_{N_{\mathit{eu}}}$ of neutrosophic sets in $\mathbb{P}$ satisfying the following axioms,

1. $0_{N_{\mathit{eu}}}$, $1_{N_{\mathit{eu}}}$ $\mathrm{Є}{\tau }_{N_{\mathit{eu}}}$.

2. ${\mathrm{Ⱥ}}_1\cap {\mathrm{Ⱥ}}_2\mathrm{Є}{\tau }_{N_{\mathit{eu}}}$ for any ${\mathrm{Ⱥ}}_1,{\mathrm{Ⱥ}}_2\mathrm{Є}{\tau }_{N_{\mathit{eu}}}$.

3. $\bigcup {\mathrm{Ⱥ}}_i\mathrm{Є}{\tau }_{N_{\mathit{eu}}}$ for every family $\left\{{\mathrm{Ⱥ}}_i/i\mathrm{Є}\mathrm{ῼ}\right\}$ ⊆ ${\tau }_{N_{\mathit{eu}}}$.

In this case, the ordered pair $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ or simply $\mathbb{P}$ is called a neutrosophic topological space ($N_{\mathit{eu}}$TS). The elements of ${\tau }_{N_{\mathit{eu}}}$ is neutrosophic open set $\left(N_{\mathit{eu}}-\mathit{OS}\right)$ and ${{\tau }_{N_{\mathit{eu}}}}^c$ is neutrosophic closed set $\left(N_{\mathit{eu}}-\mathit{CS}\right)$.

Definition 2.4: [6] A neutrosophic set Ⱥ in a $N_{\mathit{eu}}$TS $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is called a neutrosophic generalized semi alpha star closed set $\left(N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}\right)$ if $N_{\mathit{eu}}\alpha -\mathit{int}\left(N_{\mathit{eu}}\alpha -\mathit{cl}\left(\mathrm{Ⱥ}\right)\right)\subseteq N_{\mathit{eu}}-\mathit{int}$($\mathcal{G}$), whenever Ⱥ ⊆ $\mathcal{G}$ and $\mathcal{G}$ is $N_{\mathit{eu}}{\alpha }^{\ast }-$ open set.

Definition 2.5: [7] A neutrosophic topological space $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is called a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space if every $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Definition 2.6: A neutrosophic function $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is said to be

1. neutrosophic continuous [8] if the inverse image of each $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

2. $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous [7] if the inverse image of each neutrosophic closed set in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ closed set in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

3. $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute map [7] if the inverse image of each $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ closed set in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$closed set in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

4. strongly neutrosophic continuous [4] if the inverse image of each neutrosophic set in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

5. perfectly neutrosophic continuous [4] if the inverse image of each $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Definition 2.7: [9] Let ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}}\right\}$ is a neutrosophic topological space over $\mathbb{P}$. Then $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is called neutrosophic discrete topological space.

Definition 2.8: A neutrosophic topological space $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is called a neutrosophic clopen set $(N_{\mathit{eu}}-$clopen set) if it is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

3. Strongly neutrosophic $\mathit{gs}{\alpha }^{\ast }$-continuous function

Definition 3.1: A neutrosophic function $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is said to be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous if the inverse image of every $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. (ie) $f^{-1}\left(\mathrm{Ⱥ}\right)$ is a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ for every $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ Ⱥ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$.

Theorem 3.2: Every strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous is neutrosophic continuous, but not conversely.

Proof:

Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be any neutrosophic function. Let Ⱥ be any $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since every $N_{\mathit{eu}}-\mathit{CS}$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$, then Ⱥ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is neutrosophic continuous.

Example 3.3: Let $\mathbb{P}$ =$\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ$=\left\{⟨\mathcal{p},\left(\mathrm{0.6,0.4,0.4}\right)⟩\right\}$ and Ƀ$=$ $\left\{⟨\mathcal{q},\left(\mathrm{0.4,0.6,0.2}\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}+0.2$. Let ${\mathrm{Ƀ}}^c=\left\{⟨\mathcal{q},\left(\mathrm{0.2,0.4,0.4}\right)⟩\right\}$be a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left({\mathrm{Ƀ}}^c\right)=$ $\left\{⟨\mathcal{p},\left(0.4,0.6,0.6\right)⟩\right\}$. Now, $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)={\mathrm{Ⱥ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ⱥ}}^c=f^{-1}\left({\mathrm{Ƀ}}^c\right)$ $\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is neutrosophic continuous, but $f$ is not strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. Let $\mathrm{Ȼ}=\left\{⟨\mathcal{q},\left(0.1,0.2,0.8\right)⟩\right\}$ be a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left(\mathrm{Ȼ}\right)=$ $\left\{⟨\mathcal{p},\left(0.3,0.4,1\right)⟩\right\}$. Now $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left(\mathrm{Ȼ}\right)\right)={\mathrm{Ⱥ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ⱥ}}^c\ne f^{-1}\left(\mathrm{Ȼ}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ȼ}\right)$ is not $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Theorem 3.4: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous iff the inverse image of every $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Proof:

Assume that $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous function. Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then ${\mathrm{Ⱥ}}^c$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left({\mathrm{Ⱥ}}^c\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ ${\left(f^{-1}\left(\mathrm{Ⱥ}\right)\right)}^c$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Conversely, Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then ${\mathrm{Ⱥ}}^c$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. By hypothesis, $f^{-1}\left({\mathrm{Ⱥ}}^c\right)$ is $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ ${\left(f^{-1}\left(\mathrm{Ⱥ}\right)\right)}^c$ is $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 3.5: Every strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, but not conversely.

Proof:

Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be any neutrosophic function. Let Ⱥ be any $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then Ⱥ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Example 3.6: Let $\mathbb{P}=\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ $=\left\{⟨\mathcal{p},\left(\mathrm{0.4,0.5,0.7}\right)⟩\right\}$ and Ƀ $=$ $\left\{⟨\mathcal{q},\left(\mathrm{0.6,0.8,0.4}\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}$. Let ${\mathrm{Ƀ}}^c=\left\{⟨\mathcal{q},\left(\mathrm{0.4,0.2,0.6}\right)⟩\right\}$be a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left({\mathrm{Ƀ}}^c\right)=$ $\left\{⟨\mathcal{p},\left(0.4,0.2,0.6\right)⟩\right\}$. $N_{\mathit{eu}}{\alpha }^{\ast }-\mathit{OS}=N_{\mathit{eu}}\alpha -\mathit{OS}=$ $\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and $N_{\mathit{eu}}\alpha -\mathit{CS}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},{\mathrm{Ⱥ}}^c\right\}$. $N_{\mathit{eu}}\alpha –\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)={\mathrm{Ⱥ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ⱥ}}^c$. Now, $N_{\mathit{eu}}\alpha -\mathit{int}\left(N_{\mathit{eu}}\alpha -\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)\right)=$ $\mathrm{Ⱥ}\subseteq N_{\mathit{eu}}-\mathit{int}\left(1_{N_{\mathit{eu}}}\right)=1_{N_{\mathit{eu}}},$ whenever $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ ⊆ $1_{N_{\mathit{eu}}}\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. But $f$ is not strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$continuous. Let $Ȼ=\left\{⟨\mathcal{q},\left(0.3,0.1,0.7\right)⟩\right\}$ be a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left(Ȼ\right)=$ $\left\{⟨\mathcal{p},\left(0.3,0.1,0.7\right)⟩\right\}$. Now $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left(\mathrm{Ȼ}\right)\right)={\mathrm{Ⱥ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ⱥ}}^c\ne f^{-1}\left(\mathrm{Ȼ}\right)\Rightarrow$ $f^{-1}\left(\mathrm{Ȼ}\right)$ is not $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Theorem 3.7: Every strongly neutrosophic continuous is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, but not conversely.

Proof:

Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be any neutrosophic function. Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly neutrosophic continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Hence, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Example 3.8: Let $\mathbb{P}$ =$\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ},\mathrm{Ȼ}\right\}$ and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ$=\left\{⟨\mathcal{p},\left(\mathrm{0.4,0.6,0.2}\right)⟩\right\},$ $\mathrm{Ȼ}=\left\{⟨\mathcal{p},\left(\left[\mathrm{0.4,1}\right],[0.6,1],\left[\mathrm{0,0.2}\right]\right)⟩\right\}$and $\text{Ƀ}=$ $\left\{⟨\mathcal{q},\left(\mathrm{0.4,0.6,0.2}\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}$. Let $Ⱦ=$ $\left\{⟨\mathcal{q},\left(\left[0,0.2\right],\left[0,0.4\right],\left[0.4,1\right]\right)⟩\right\}$ be a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left(\mathrm{Ⱦ}\right)=$ $\left\{⟨\mathcal{p},\left(\left[0,0.2\right],\left[\mathrm{0,0.4}\right],\left[\mathrm{0.4,1}\right]\right)⟩\right\}$. Now $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left(\mathrm{Ⱦ}\right)\right)={\mathrm{Ⱥ}}^c\cap {\mathrm{Ȼ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ȼ}}^c=f^{-1}\left(\mathrm{Ⱦ}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. But $f$ is not strongly neutrosophic continuous. Let Ɇ$=$ $\left\{⟨\mathcal{q},\left(0.4,0.6,0.2\right)⟩\right\}$ be a neutrosophic set in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left(\mathrm{Ɇ}\right)=$ $\left\{⟨\mathcal{p},\left(0.4,0.6,0.2\right)⟩\right\}$. Now $N_{\mathit{eu}}-\mathit{int}\left(f^{-1}\left(\mathrm{Ɇ}\right)\right)=0_{N_{\mathit{eu}}}\cup \mathrm{Ⱥ}=\mathrm{Ⱥ}=f^{-1}\left(\mathrm{Ɇ}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ɇ}\right)$ is $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Also $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left(\mathrm{Ɇ}\right)\right)=1_{N_{\mathit{eu}}}\ne f^{-1}\left(\mathrm{Ɇ}\right)\Rightarrow$ $f^{-1}\left(\mathrm{Ɇ}\right)$ is not $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f^{-1}\left(\mathrm{Ɇ}\right)$is not both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Remark 3.9: Every strongly neutrosophic continuous is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, but not conversely. (by Theorem 3.5 & 3.7).

Theorem 3.10: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be neutrosophic function and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space. Then the following are equivalent.

1. $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

2. $f$ is neutrosophic continuous.

Proof:

1. $\Rightarrow$ (2), Proof follows from theorem 3.2.

2. $\Rightarrow$ (1), Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then Ⱥ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is neutrosophic continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 3.11: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. Both $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ are $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then Ⱥ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 3.12: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Hence, $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute.

Theorem 3.13: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute and $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 3.14: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $\mathit{gof}$is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 3.15: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$continuous, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is neutrosophic continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $\mathit{gof}$is neutrosophic continuous.

Theorem 3.16: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $\mathit{gof}$is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 3.17: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$continuous, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Hence, $\mathit{gof}$is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute.

Theorem 3.18: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be neutrosophic continuous and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is neutrosophic continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Hence, $\mathit{gof}$is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Inter-relationship 3.19:

4. Perfectly neutrosophic $\mathit{gs}{\mathit{\alpha }}^{\ast }$-continuous function

Definition 4.1: A neutrosophic function $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is said to be perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous if the inverse image of every $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ (ie, $N_{\mathit{eu}}-$ clopen set) in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Theorem 4.2: Every perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, but not conversely.

Proof:

Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$be any neutrosophic function. Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Example 4.3: Let $\mathbb{P}$ =$\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ},\mathrm{Ȼ}\right\}$and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ$=\left\{⟨\mathcal{p},\left(\mathrm{0.7,0.8,0.3}\right)⟩\right\},$ $\mathrm{Ȼ}=\left\{⟨\mathcal{p},\left(\left[\mathrm{0.7,1}\right],\left[\mathrm{0.8,1}\right],\left[\mathrm{0,0.3}\right]\right)⟩\right\}$and $\text{Ƀ}=\left\{⟨\mathcal{q},\left(0.7,0.8,0.3\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}$. Let $\mathrm{Ⱦ}=\left\{⟨\mathcal{q},\left(\left[0,0.3\right],\left[0,0.2\right],\left[0.7,1\right]\right)⟩\right\}$ be a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left(\mathrm{Ⱦ}\right)=\left\{⟨\mathcal{p},\left(\left[0,0.3\right],\left[\mathrm{0,0.2}\right],\left[\mathrm{0.7,1}\right]\right)⟩\right\}$. Now $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left(\mathrm{Ⱦ}\right)\right)={\mathrm{Ⱥ}}^c\cap {\mathrm{Ȼ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ȼ}}^c=f^{-1}\left(\mathrm{Ⱦ}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. But $f$ is not perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, because $f^{-1}\left(\mathrm{Ⱦ}\right)$ is not both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Since, $N_{\mathit{eu}}-\mathit{int}\left(f^{-1}\left(\mathrm{Ⱦ}\right)\right)=0_{N_{\mathit{eu}}}\ne f^{-1}\left(\mathrm{Ⱦ}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱦ}\right)$ is not $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f^{-1}\left(\mathrm{Ⱦ}\right)$is not both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Theorem 4.4: Every perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous is perfectly neutrosophic continuous, but not conversely.

Proof:

Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be any neutrosophic function. Let Ⱥ be any $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then Ⱥ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is perfectly neutrosophic continuous.

Example 4.5: Let $\mathbb{P}=\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ},\mathrm{Ȼ},\mathrm{Ɇ}\right\}$ and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ$=\left\{⟨\mathcal{p},\left(\mathrm{0.4,0.2,0.6}\right)⟩\right\}$, $\mathrm{Ȼ}=\left\{⟨\mathcal{p},\left(\mathrm{0.6,0.8,0.4}\right)⟩\right\},$ $\mathrm{Ɇ}=\left\{⟨\mathcal{p},\left(\left[\mathrm{0,0.4}\right],\left[\mathrm{0,0.2}\right],\left[\mathrm{0.6,1}\right]\right)⟩\right\}$and Ƀ$=$ $\left\{⟨\mathcal{q},\left(0.6,0.8,0.4\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}$. Let ${\mathrm{Ƀ}}^c=\left\{⟨\mathcal{q},\left(0.4,0.2,0.6\right)⟩\right\}$be a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left({\mathrm{Ƀ}}^c\right)=$ $\left\{⟨\mathcal{p},\left(\mathrm{0.4,0.2,0.6}\right)⟩\right\}$. Now $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)={\mathrm{Ⱥ}}^c\cap {\mathrm{Ȼ}}^c\cap {\mathrm{Ɇ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ȼ}}^c=f^{-1}\left({\mathrm{Ƀ}}^c\right)$_. Also, $N_{\mathit{eu}}-\mathit{int}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)=\mathrm{Ⱥ}\cup \mathrm{Ɇ}\cup 0_{N_{\mathit{eu}}}=\mathrm{Ⱥ}=f^{-1}\left({\mathrm{Ƀ}}^c\right)$ $\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is perfectly neutrosophic continuous. But $f$ is not perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. Let Ⱦ $=\left\{⟨\mathcal{q},\left(\left[0,0.4\right],\left[0,0.2\right],[0.6,1]\right)⟩\right\}$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left(\mathrm{Ⱦ}\right)=\left\{⟨\mathcal{p},\left(\left[0,0.4\right],\left[0,0.2\right],[0.6,1]\right)⟩\right\}$. Since, $N_{\mathit{eu}}-\mathit{int}\left(f^{-1}\left(\mathrm{Ⱦ}\right)\right)=\mathrm{Ɇ}\cup 0_{N_{\mathit{eu}}}=\mathrm{Ɇ}=f^{-1}\left(\mathrm{Ⱦ}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱦ}\right)$ is $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Also, $N_{\mathit{eu}}-\mathit{cl}\left(f^{-1}\left(\mathrm{Ⱦ}\right)\right)={\mathrm{Ⱥ}}^c\cap {\mathrm{Ȼ}}^c\cap {\mathrm{Ɇ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ȼ}}^c\ne f^{-1}\left(\mathrm{Ⱦ}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱦ}\right)$ is not $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f^{-1}\left(\mathrm{Ⱦ}\right)$is not both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Theorem 4.6: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous iff the inverse image of every $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Proof:

Assume that $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous function. Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then ${\mathrm{Ⱥ}}^c$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left({\mathrm{Ⱥ}}^c\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ ${\left(f^{-1}\left(\mathrm{Ⱥ}\right)\right)}^c$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Conversely, Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then ${\mathrm{Ⱥ}}^c$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. By hypothesis, $f^{-1}\left({\mathrm{Ⱥ}}^c\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ ${\left(f^{-1}\left(\mathrm{Ⱥ}\right)\right)}^c$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 4.7: Let $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ be a neutrosophic discrete topological space and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be any neutrosophic topological space. Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be a neutrosophic function, then the following statements are true.

1. $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

2. $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

1. $\Rightarrow$ (2), Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is neutrosophic discrete topological space, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

2. $\Rightarrow$ (1), Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 4.8: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be perfectly neutrosophic continuous and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then Ⱥ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right).$Since $f$ is perfectly neutrosophic continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 4.9: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $\mathit{gof}$is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 4.10: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be neutrosophic continuous and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is neutrosophic continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $\mathit{gof}$is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 4.11: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is strongly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $\mathit{gof}$is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

5. Totally neutrosophic $\mathit{gs}{\alpha }^{\ast }-$continuous function

Definition 5.1: A neutrosophic function $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is said to be totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous if the inverse image of every $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is both $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ and $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ (ie, $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ clopen set) in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Definition 5.2: A neutrosophic topological space $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ is called a $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$clopen set $(N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$clopen set) if it is both $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ and $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Example 5.3: Let $\mathbb{P}=\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ$=\left\{⟨\mathcal{p},\left(\mathrm{0.4,0.5,0.7}\right)⟩\right\}$ and Ƀ$=$ $\left\{⟨\mathcal{q},\left(\mathrm{0.2,0.7,0.8}\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}$. Let ${\mathrm{Ƀ}}^c=\left\{⟨\mathcal{q},\left(\mathrm{0.8,0.3,0.2}\right)⟩\right\}$be a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left({\mathrm{Ƀ}}^c\right)=$ $\left\{⟨\mathcal{p},\left(0.8,0.3,0.2\right)⟩\right\}$. $N_{\mathit{eu}}{\alpha }^{\ast }-\mathit{OS}=N_{\mathit{eu}}\alpha -\mathit{OS}=$ $\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and $N_{\mathit{eu}}\alpha -\mathit{CS}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},{\mathrm{Ⱥ}}^c\right\}$. $N_{\mathit{eu}}\alpha –\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)=1_{N_{\mathit{eu}}}.$Now, $N_{\mathit{eu}}\alpha -\mathit{int}\left(N_{\mathit{eu}}\alpha -\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)\right)=$ $1_{N_{\mathit{eu}}}\subseteq N_{\mathit{eu}}-\mathit{int}\left(1_{N_{\mathit{eu}}}\right)=1_{N_{\mathit{eu}}},$ whenever $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ ⊆ $1_{N_{\mathit{eu}}}\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Also, $N_{\mathit{eu}}\alpha –\mathit{int}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)=0_{N_{\mathit{eu}}}.$Now, $N_{\mathit{eu}}\alpha -\mathit{cl}\left(N_{\mathit{eu}}\alpha –\mathit{int}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)\right)=0_{N_{\mathit{eu}}}\supseteq N_{\mathit{eu}}-\mathit{cl}\left(0_{N_{\mathit{eu}}}\right)=0_{N_{\mathit{eu}}},$ whenever $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ ⊇ $0_{N_{\mathit{eu}}}\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Theorem 5.4: Every perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, but not conversely.

Proof:

Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be any neutrosophic function. Let Ⱥ be any $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then Ⱥ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ and $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Example 5.5: Let $\mathbb{P}=\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ $=\left\{⟨\mathcal{p},\left(\mathrm{0.2,0.4,0.6}\right)⟩\right\}$ and Ƀ $=$ $\left\{⟨\mathcal{q},\left(\mathrm{0.6,0.8,0.4}\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}$. Let ${\mathrm{Ƀ}}^c=\left\{⟨\mathcal{q},\left(\mathrm{0.4,0.2,0.6}\right)⟩\right\}$be a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left({\mathrm{Ƀ}}^c\right)=$ $\left\{⟨\mathcal{p},\left(0.4,0.2,0.6\right)⟩\right\}$. $N_{\mathit{eu}}{\alpha }^{\ast }-\mathit{OS}=N_{\mathit{eu}}\alpha -\mathit{OS}=$ $\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and $N_{\mathit{eu}}\alpha -\mathit{CS}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},{\mathrm{Ⱥ}}^c\right\}$. $N_{\mathit{eu}}\alpha –\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)={\mathrm{Ⱥ}}^c\cap 1_{N_{\mathit{eu}}}={\mathrm{Ⱥ}}^c.$Now, $N_{\mathit{eu}}\alpha -\mathit{int}\left(N_{\mathit{eu}}\alpha -\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)\right)=\mathrm{Ⱥ}\cup$ $0_{N_{\mathit{eu}}}=\mathrm{Ⱥ}\subseteq N_{\mathit{eu}}-\mathit{int}\left(1_{N_{\mathit{eu}}}\right)=1_{N_{\mathit{eu}}},$ whenever $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ ⊆ $1_{N_{\mathit{eu}}}\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Also, $N_{\mathit{eu}}\alpha –\mathit{int}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)=0_{N_{\mathit{eu}}}.$ Now, $N_{\mathit{eu}}\alpha -\mathit{cl}\left(N_{\mathit{eu}}\alpha -\mathit{int}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)\right)=0_{N_{\mathit{eu}}}\supseteq N_{\mathit{eu}}-\mathit{cl}\left(0_{N_{\mathit{eu}}}\right)=0_{N_{\mathit{eu}}},$ whenever $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ ⊇ $0_{N_{\mathit{eu}}}\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. But $f$ is not perfectly $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. Let Ⱦ$=\left\{⟨\mathcal{q},\left(\mathrm{0.3,0.1,0.8}\right)⟩\right\}$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left(\mathrm{Ⱦ}\right)=\left\{⟨\mathcal{p},\left(\mathrm{0.3,0.1,0.8}\right)⟩\right\}$. Now, $N_{\mathit{eu}}–\mathit{int}\left(f^{-1}\left(\mathrm{Ⱦ}\right)\right)=0_{N_{\mathit{eu}}}\ne f^{-1}\left(\mathrm{Ⱦ}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱦ}\right)$ is not $N_{\mathit{eu}}-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Also, $N_{\mathit{eu}}–\mathit{cl}\left(f^{-1}\left(\mathrm{Ⱦ}\right)\right)={\mathrm{Ⱥ}}^c\ne f^{-1}\left(\mathrm{Ⱦ}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱦ}\right)$ is not $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f^{-1}\left(\mathrm{Ⱦ}\right)$ is not both $N_{\mathit{eu}}-\mathit{OS}$ and $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Theorem 5.6: Every totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be any neutrosophic function. Let Ⱥ be any $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ and $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Example 5.7: Let $\mathbb{P}=\left\{\mathcal{p}\right\}$ and $\mathbb{Q}$ =$\left\{\mathcal{q}\right\}$. ${\tau }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ}\right\}$ and ${\sigma }_{N_{\mathit{eu}}}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ƀ}\right\}$ are $N_{\mathit{eu}}$TS on $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ respectively. Also Ⱥ$=\left\{⟨\mathcal{p},\left(\mathrm{0.7,0.6,0.5}\right)⟩\right\}$ and $\text{Ƀ}=\left\{⟨\mathcal{q},\left(\mathrm{0.7,0.8,0.3}\right)⟩\right\}$ are $N_{\mathit{eu}}\left(\mathbb{P}\right)$ and $N_{\mathit{eu}}\left(\mathbb{Q}\right)$. Define a map $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ by $f\left(\mathcal{p}\right)=\mathcal{q}$. Let ${\mathrm{Ƀ}}^c=\left\{⟨\mathcal{q},\left(\mathrm{0.3,0.2,0.7}\right)⟩\right\}$ be a $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Then $f^{-1}\left({\mathrm{Ƀ}}^c\right)=$ $\left\{⟨\mathcal{p},\left(\mathrm{0.3,0.2,0.7}\right)⟩\right\}$. $N_{\mathit{eu}}{\alpha }^{\ast }-\mathit{OS}=N_{\mathit{eu}}\alpha -\mathit{OS}=$ $\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},\mathrm{Ⱥ},D\right\}$ and $N_{\mathit{eu}}\alpha -\mathit{CS}=\left\{0_{N_{\mathit{eu}}},1_{N_{\mathit{eu}}},{\mathrm{Ⱥ}}^c,E\right\},$ where $D=\left\{⟨\mathcal{p},\left(\left[0.7,1\right],\left[0.6,1\right],\left[0,0.5\right]\right)⟩\right\},$ $E=\left\{⟨\mathcal{p},\left(\left[0,0.5\right],\left[\mathrm{0,0.4}\right],\left[\mathrm{0.7,1}\right]\right)⟩\right\}$. $N_{\mathit{eu}}\alpha –\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)={\mathrm{Ⱥ}}^c\cap F\cap 1_{N_{\mathit{eu}}}=F,$ where $=\left\{⟨\mathcal{p},\left(\left[\mathrm{0.3,0.5}\right],\left[\mathrm{0.2,0.4}\right],0.7\right)⟩\right\}$. Now, $N_{\mathit{eu}}\alpha -\mathit{int}\left(N_{\mathit{eu}}\alpha -\mathit{cl}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)\right)=0_{N_{\mathit{eu}}}$ $\subseteq N_{\mathit{eu}}-\mathit{int}\left(\mathrm{Ⱥ}\right),N_{\mathit{eu}}-\mathit{int}\left(D\right),N_{\mathit{eu}}-\mathit{int}\left(1_{N_{\mathit{eu}}}\right)=\mathrm{Ⱥ},1_{N_{\mathit{eu}}},$ whenever $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ ⊆$\mathrm{Ⱥ},1_{N_{\mathit{eu}}}$ $\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous. But $f$ is not totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, because $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is not $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Since $N_{\mathit{eu}}\alpha -\mathit{cl}\left(N_{\mathit{eu}}\alpha -\mathit{int}\left(f^{-1}\left({\mathrm{Ƀ}}^c\right)\right)\right)=0_{N_{\mathit{eu}}}⊉N_{\mathit{eu}}-\mathit{cl}\left(J\right)={\mathrm{Ⱥ}}^c,$ whenever $f^{-1}\left({\mathrm{Ƀ}}^c\right)$ ⊇$J,$ where $J=\left\{⟨\mathcal{p},\left(\left[\mathrm{0,0.3}\right],\left[\mathrm{0,0.2}\right],\left[\mathrm{0.7,1}\right]\right)⟩\right\}$ $\Rightarrow f^{-1}\left({\mathrm{Ƀ}}^c\right)$ is not $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$.

Inter-relationship 5.8:

Theorem 5.9: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then Ⱥ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ and $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\Rightarrow$ $f^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $f$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ irresolute.

Theorem 5.10: Let $f:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ and $g:\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ be totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous and $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ be $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $\mathit{gof}:\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$ $\to \left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.

Proof:

Let Ⱥ be any $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{R},{\gamma }_{N_{\mathit{eu}}}\right)$. Since $g$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ and $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$ is $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ space, then $g^{-1}\left(\mathrm{Ⱥ}\right)$ is $N_{\mathit{eu}}-\mathit{CS}$ in $\left(\mathbb{Q},{\sigma }_{N_{\mathit{eu}}}\right)$. Since $f$ is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous, then $f^{-1}\left(g^{-1}\left(\mathrm{Ⱥ}\right)\right)={\left(\mathit{gof}\right)}^{-1}\left(\mathrm{Ⱥ}\right)$ is both $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{OS}$ and $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-\mathit{CS}$ in $\left(\mathbb{P},{\tau }_{N_{\mathit{eu}}}\right)$. Therefore, $\mathit{gof}$is totally $N_{\mathit{eu}}\mathit{gs}{\alpha }^{\ast }-$ continuous.