N-Methods

Abstract

In this paper, we present the concept of $G_N$-topological space by introducing a topology defined on a subfamily of the family consisting of all convergent nets in any general topological space. Inspired by the method concept, which is a type of function whose domain is a set of convergent sequences and whose range is a set of real numbers, we define a different type of function, which we call the N-method, whose range is a general topological space and whose domain is the $G_N$-topological space defined on this topological space. After stating the elements that led us to conduct this research and the sources of inspiration for us, we examine the properties of some types of continuities on the concept of the N-method, whose basis is the concept of an h-open set, and reveal the relations between them. In the following section, we similarly introduce the nano $G_N$-topological space and nano N-method concepts and examine the behaviors of the continuity types, which are based on the concept of the nano h-open set and nano N-method. In the conclusion, we outline the impact we expect our studies to have on the scientific world.

1. Introduction

The concept of functions has always been one of the indispensable characters of the world of mathematics. This concept has been used in the majority of studies in the various subbranches of mathematics. In addition, it has made it possible to reinterpret other concepts that are among the indispensable characters of mathematics. For example, studies examining the images of a compact or connected space under functions with different properties and revealing their properties have attracted attention in the scientific world and have been a source of inspiration for different studies conducted later. However, as time goes by, the classical concept of a function has became inadequate for conducting new research, and so it has been divided into different types. For example, the concept of a sequence, which is a function whose domain is the set of natural numbers and whose range is the set of real numbers, has been introduced. With this sequence concept, many new concepts have been introduced to the scientific world and have provided scientists with opportunities to conduct many research studies, especially in the field of analysis, which is a subbranch of mathematics. We can also add concepts such as polynomial, exponential function, isomorphism, monomorphism, homomorphism, and homeomorphism to the types of functions that provide important opportunities for researchers to present new products and produce different ideas in the scientific world. One of the newest types of functions is the method concept introduced by Cakalli in 2008 in [1]. The most important feature that distinguishes the method concept from other classical function concepts is that each element of the domain is a convergent sequence. Another distinguishing feature of the method concept is that the image of every convergent sequence in the domain under a method is a real number. This new type of function has paved the way for the reinterpretation of indispensable characters of topology, especially compactness and connectedness, which were previously introduced not only in general topology but also in some other non-standard types of topological space, and for the introduction of different types of these characters. G-compactness [2], G-sequential continuity [3], G-continuity [4], G-connectedness [5], G-topological group [6], G-space [7], G-neighborhood [8], G-derived [9], G-mapping [10], G-sequentially compactness [11], $GO$-compactness [12], G-convergence [13], and G-closure operator [14] are among the most striking examples of the reinterpretation of the indispensable characters of the general topology with the method concept. Also, G-connectedness and the G-sequential method in product spaces studied in [15,16] and G-continuity in neutrosophic spaces studied in [17] may be admitted as examples of the reinterpretation of these terms in some other non-standard types of topological space. Later, different types of method concepts, such as the sequential method and product method, have formed the cornerstones of giving new identities to the concepts of connectedness and G-connectedness previously introduced in general topology in [18,19].

In this study, we introduce the N-method concept inspired by the method concept. The difference between the N-method concept and the method concept is that each element of the domain is not a convergent sequence but rather a convergent net. In addition, another feature that distinguishes the N-method concept from the method concept is that its range is not the set of real numbers but rather a topological space. In addition, we present the concept of $G_N$-topological space defined over a topological space, which is necessary for the introduction of the N-method concept, and examine the properties of various types of continuity, whose basis is the concept of the h-open set, which is among the newest types of open sets, under the N-method concept.

2. Preliminaries

In this section, we present the basic definitions related to N-method concepts and the $G_N$-h-open set.

Definition 1 

([20]). Let $(W,\tau )$ be a topological space. A subset $C\subseteq W$ is defined to be h-open if for every non-empty open set $T\in \tau$ with $T\ne W$, the inclusion $C\subseteq Int(C\cup T)$ holds. The complement of an h-open set is referred to as an h-closed set. The family of all h-open subsets of W is denoted by ${\tau }^h$.

Theorem 1 

([20]). In any topological space $(W,\tau )$, every open set is also an h-open set.

Definition 2 

([20]). A function $j:(W,\tau )\to (Z,\vartheta )$ is referred to as h-continuous if given any open set T in Z, the set $j^{-1}\left(T\right)$ is h-open in W.

Definition 3 

([20]). A function $j:(W,\tau )\to (Z,\vartheta )$ is regarded as h-open when for each open subset T of W, its image $j\left(T\right)$ is h-open in Z.

Definition 4 

([20]). A function $j:(W,\tau )\to (Z,\vartheta )$ is called h-irresolute if for every h-open subset T of Z, the preimage $j^{-1}\left(T\right)$ is an h-open set in W.

Definition 5 

([20]). A function $j:(W,\tau )\to (Z,\vartheta )$ is called h-totally continuous if for every h-open subset T of Z, the preimage $j^{-1}\left(T\right)$ is a clopen set in W.

Definition 6 

([21]). Let S be a non-empty finite set, referred to as the universe, and let R be an equivalence relation on S. The structure $(S,R)$ is termed an approximation space. For any subset $W\subseteq S$, we define the following:

1. 

The lower approximation of W with respect to R, denoted $L_R\left(W\right)$, as $L_R\left(W\right)={\bigcup }_{x\in S}\{R\left(W\right):R\left(W\right)\subseteq W\}$, where $R\left(W\right)$ is the equivalence class containing W.

2. 

The upper approximation of W with respect to R, denoted $S_R\left(W\right)$, by $S_R\left(W\right)={\bigcup }_{x\in S}\{R\left(W\right):R\left(W\right)\cap W\ne \varnothing \}$.

3. 

The boundary region of W with respect to R, denoted $B_R\left(W\right)$, as $B_R\left(W\right)=S_R\left(W\right)\setminus L_R\left(W\right)$.

Definition 7 

([21]). Consider the universe S and an equivalence relation R on S. Then, for $W\subseteq S$, the collection ${\tau }_R\left(W\right)=\{\varnothing ,S,L_R\left(W\right),S_R\left(W\right),B_R\left(W\right)\}$ is defined as the nano-topology on S. The pair $(S,{\tau }_R\left(W\right))$ is referred to as a nano-topological space. The members of ${\tau }_R\left(W\right)$ are known as nano open sets, and the complement of a nano open set is called a nano closed set.

Definition 8 

([21]). In the nano-topological space $\left(S,{\tau }_R\left(W\right)\right)$ associated with $W\subseteq S$, the nano interior of a subset $D\subseteq S$, written $nInt\left(D\right)$, is the largest nano open set contained in D, while the nano closure of D, denoted $nCl\left(D\right)$, is the smallest nano closed set containing D.

Definition 9 

([22]). Let $\left(S,{\tau }_R\left(W\right)\right)$ be a nano-topological space. A subset D of S is called nano h-open if $D\subseteq nInt(D\cup E)$ for every non-empty nano open set E distinct from S. Its complement is known as a nano h-closed set.

Theorem 2 

([22]). In a nano-topological space $\left(S,{\tau }_R\left(W\right)\right)$, every nano open set B is also nano h-open within $\left(S,{\tau }_R\left(W\right)\right)$.

3. ${\mathit{G}}_{\mathit{N}}$-$\mathit{h}$-Continuous $\mathit{N}$-Methods and ${\mathit{G}}_{\mathit{N}}$-$\mathit{h}$-$\mathit{N}$-Homeomethod

In this section, we introduce new classes of methods called $G_N$-h-continuous N-methods, $G_N$-h-open N-methods, $G_N$-h-irresolute N-methods, $G_N$-h-totally continuous N-methods, $G_N$-h-contra-continuous N-methods, and the $G_N$-h-N-homeomethod and study some properties of these N-methods.

Definition 10. 

Let $(W,\tau )$ be a topological space and $\mathsf{\Gamma }$ be a subset of the set of all convergent nets in $(W,\tau )$. Let η be the collection of subsets of $\mathsf{\Gamma }$; then, η is said to be a $G_N$-topology on $\mathsf{\Gamma }$ if

(1) 

$\varnothing ,\mathsf{\Gamma }$ belong to η;

(2) 

The union of any number of sets in η belongs to η;

(3) 

The intersection of any two sets in η belongs to η.

Then, $(\mathsf{\Gamma },\eta )$ is said to be a $G_N$-topological space over $(W,\tau )$. And, every element of η is a $G_N$-open set in $(\mathsf{\Gamma },\eta )$. The complement of a $G_N$-open set is called a $G_N$-closed set in $(\mathsf{\Gamma },\eta )$.

Definition 11. 

Consider a $G_N$-topological space $(\mathsf{\Gamma },\eta )$ over $(W,\tau )$. Let $\lambda \subseteq \mathsf{\Gamma }$, then

(i) 

The $G_N$-interior of the set λ is defined as the union of all $G_N$-open subsets contained in λ and is denoted by ${\lambda }_{G_N}^{\circ }$;

(ii) 

The $G_N$-closure of the set λ is defined as the intersection of all $G_N$-closed

subsets containing λ and is denoted by ${\overline{\lambda }}_{G_N}$.

Definition 12. 

A subset λ of the $G_N$-topological space $(\mathsf{\Gamma },\eta )$ is called a $G_N$-h-open set if for every non-empty set δ in $(\mathsf{\Gamma },\eta )$, $\delta \ne \mathsf{\Gamma }$ and $\delta \in \eta$, $\lambda \subseteq {(\lambda \cup \delta )}_{G_N}^{\circ }$. The complement of the $G_N$-h-open set is called $G_N$-h-closed.

Theorem 3. 

Every $G_N$-open set in any $G_N$-topological space $(\mathsf{\Gamma },\eta )$ is $G_N$-h-open.

Proof. 

Omitted. □

Remark 1. 

The converse situation may not be true as shown in the following example.

Example 1. 

Let $(W,\tau )$ be a topological space and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I},{\left\{{\omega }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I}\}$, $\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\},\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I},{\left\{{\omega }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Then, $\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\}$ is $G_N$-h-open but not $G_N$-open in $(\mathsf{\Gamma },\eta )$.

Definition 13. 

An N-method is a function $G_N$ defined from a $G_N$-topological space $(\mathsf{\Gamma },\eta )$ over $(W,\tau )$ to $(W,\tau )$.

Definition 14. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is considered $G_N$-continuous if for each open set S in $(W,\tau )$, $G_N^{-1}\left(S\right)$ is $G_N$-open in $(\mathsf{\Gamma },\eta )$.

Definition 15. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is called $G_N$-h-continuous if for every open set S in $(W,\tau )$, the preimage $G_N^{-1}\left(S\right)$ is $G_N$-h-open in $(\mathsf{\Gamma },\eta )$.

Definition 16. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be $G_N$-open if $G_N\left(S\right)$ is open in $(W,\tau )$ for every $G_N$-open set S in $(\mathsf{\Gamma },\eta )$.

Example 2. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,W,\{a,c\left\}\right\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\},\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Since the family of $G_N$-h-open sets in $(\mathsf{\Gamma },\eta )$ is $\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, $G_N$ is $G_N$-h-continuous.

Theorem 4. 

Every $G_N$-continuous N-method is $G_N$-h-continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-continuous N-method and S be any open subset in $(W,\tau )$. Since $G_N$ is $G_N$-continuous, then $G_N^{-1}\left(S\right)$ is a $G_N$-open set in $(\mathsf{\Gamma },\eta )$. Since every $G_N$-open set is a $G_N$-h-open set by Theorem 3, then $G_N^{-1}\left(S\right)$ is a $G_N$-h-open set in $(\mathsf{\Gamma },\eta )$. Therefore, $G_N$ is $G_N$-h-continuous. □

Remark 2. 

The converse of Theorem 4 need not be true as shown in the following example.

Example 3. 

Let $(W,\tau )$ be a topological space, where $W=\{a,b,c\}$, $\tau =\{\varnothing ,$W$,\left\{a\right\},\{b,c\}\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is $G_N$-h-continuous but not $G_N$-continuous.

Theorem 5. 

If $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is $G_N$-h-continuous and $j:(W,\tau )\to (Z,\vartheta )$ is continuous, then $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be $G_N$-h-continuous and $j:(W,\tau )\to (Z,\vartheta )$ be continuous. Let S be an open set in $(Z,\vartheta )$. Since j is continuous, then $j^{-1}\left(S\right)$ is an open set in $(W,\tau )$. Since $G_N$ is $G_N$-h-continuous, then $G_N^{-1}\left(j^{-1}\left(S\right)\right)={\left(jo{G}_N\right)}^{-1}\left(S\right)$ is a $G_N$-h-open set in $(\mathsf{\Gamma },\eta )$. Therefore, $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-continuous. □

Definition 17. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be $G_N$-h-open if $G_N\left(S\right)$ is an h-open set in $(W,\tau )$ for every $G_N$-open set S in $(\mathsf{\Gamma },\eta )$.

Example 4. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,$W$,\left\{a\right\}\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is $G_N$-h-open.

Theorem 6. 

Every $G_N$-open N-method is $G_N$-h-open.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-open N-method and S be any $G_N$-open set in $(\mathsf{\Gamma },\eta )$. Since $G_N$ is $G_N$-open, then $G_N\left(S\right)$ is an open set in $(W,\tau )$. Since every open set is an h-open set by Theorem 1, then $G_N\left(S\right)$ is an h-open set in $(W,\tau )$. Therefore, $G_N$ is $G_N$-h-open. □

Remark 3. 

The converse of Theorem 6 need not be true as shown in the following example.

Example 5. 

In Example 4, $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is $G_N$-h-open but not $G_N$-open.

Theorem 7. 

If $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is $G_N$-open and $j:(W,\tau )\to (Z,\vartheta )$ is h-open, then $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-open.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be $G_N$-open and $j:(W,\tau )\to (Z,\vartheta )$ be h-open. Let S be an $G_N$-open set in $(\mathsf{\Gamma },\eta )$. Since $G_N$ is $G_N$-open, then $G_N\left(S\right)$ is an open set in $(W,\tau )$. Since j is h-open, then $\left(jo{G}_N\right)\left(S\right)=j\left(G\left(S\right)\right)$ is an h-open set in $(Z,\vartheta )$. Therefore, $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-open. □

Definition 18. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be $G_N$-h-irresolute if $G_N^{-1}\left(S\right)$ is a $G_N$-h-open set in $(\mathsf{\Gamma },\eta )$ for every h-open set S in $(W,\tau )$.

Example 6. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,W,\{b\left\}\right\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\},\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is $G_N$-h-irresolute.

Theorem 8. 

If a bijective N-method is $G_N$-h-open, then it is $G_N-h$-irresolute.

Proof. 

Consider that $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is a bijective and $G_N$-h-open N-method. Let S be any h-open set in $(W,\tau )$. Since $G_N$ is bijective and $G_N$-h-open, then $G_N^{-1}\left(S\right)$ is a $G_N$-open set in $(\mathsf{\Gamma },\eta )$ and hence a $G_N-h$-open set in $(W,\tau )$ by Theorem 3. Therefore, $G_N$ is $G_N$-h-irresolute. □

Remark 4. 

The converse of Theorem 8 need not be true, as shown in the following example.

Example 7. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,W,\{a\left\}\right\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is bijective. And $G_N$ is $G_N$-h-irresolute but not $G_N-h$-open.

Theorem 9. 

Every $G_N$-h-irresolute N-method is $G_N$-h-continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-h-irresolute N-method and U be any open set in $(W,\tau )$. Since every open set is an h-open set by Theorem 1 and $G_N$ is $G_N$-h-irresolute, then $G_N^{-1}\left(S\right)$ is an $G_N$-h-open set in $(\mathsf{\Gamma },\eta )$. Therefore $G_N$ is $G_N$-h-continuous. □

Remark 5. 

The converse of Theorem 9 need not be true, as shown in the following example.

Example 8. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,W,\{b,c\left\}\right\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is $G_N$-h-continuous but not $G_N-h$-irresolute.

Theorem 10. 

If $G:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is a $G_N$-h-irresolute N-method and $j:(W,\tau )\to (Z,\vartheta )$ is an h-irresolute function, then $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-irresolute.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-h-irresolute N-method and $j:(W,\tau )\to (Z,\vartheta )$ be an h-irresolute function. Let S be any h-open set in $(Z,\vartheta )$. Since f is h-irresolute, then $j^{-1}\left(S\right)$ is h-open set in $(W,\tau )$. Since $G_N$ is $G_N-h$-irresolute, then $G^{-1}\left(j^{-1}\left(S\right)\right)={\left(jo{G}_N\right)}^{-1}\left(S\right)$ is $G_N$-h-open in $(\mathsf{\Gamma },\eta )$. Therefore, $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-irresolute. □

Theorem 11. 

If $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is $G_N$-h-irresolute and $j:(W,\tau )\to (Z,\vartheta )$ is h-continuous, then $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-h-irresolute N-method and $j:(W,\tau )\to (Z,\vartheta )$ be an h-continuous function. Let S be an open set in $(Z,\vartheta )$. Since j is h-continuous and $j^{-1}\left(S\right)$ is h-open in $(W,\tau )$, and since $G_N$ is $G_N$-h-irresolute, $G_N^{-1}\left(j^{-1}\left(S\right)\right)$ is $G_N$-h-open in $(\mathsf{\Gamma },\eta )$. This implies that ${\left(jo{G}_N\right)}^{-1}\left(S\right)$ is $G_N$-h-open in $(\mathsf{\Gamma },\eta )$. Therefore, $jo{G}_N:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-continuous. □

Definition 19. 

A bijective N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be an N-$homeomethod$ if $G_N$ is a $G_N$-continuous and $G_N$-open N-method.

Definition 20. 

A bijective N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be an h-N-homeomethod if $G_N$ is a $G_N$-h-continuous and $G_N$-h-open N-method.

Theorem 12. 

If $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is an N-homeomethod, then $G_N$ is an h-N-homeomethod.

Proof. 

Since every $G_N$-continuous N-method is $G_N$-h-continuous by Theorem 4, every $G_N$-open N-method is $G_N$-h-open by Theorem 6, and $G_N$ is bijective, therefore, $G_N$ is an h-homeomethod. □

Remark 6. 

The converse of Theorem 12 need not be true, as shown in the following example.

Example 9. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,$W$,\{b,c\}\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is an h-N-homeomethod, but it is not an N-homeomethod.

Definition 21. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be $G_N$-totally continuous if $G_N^{-1}\left(S\right)$ is a $G_N$-clopen set in $(\mathsf{\Gamma },\eta )$ for every open set S in $(W,\tau )$.

Definition 22. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be $G_N$-h-totally continuous if $G_N^{-1}\left(S\right)$ is a $G_N$-clopen set in $(\mathsf{\Gamma },\eta )$ for every h-open set S in $(W,\tau )$.

Example 10. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,W,\{a\left\}\right\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is $G_N$-h-totally continuous.

Theorem 13. 

Every $G_N$-h-totally continuous N-method is $G_N$-totally continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be $G_N$-h-totally continuous and S be any open set in $(W,\tau )$. Since every open set is an h-open set by Theorem 1, then S is an h-open set in $(W,\tau )$. Since $G_N$ is a $G_N$-h-totally continuous N-method, then $G_N^{-1}\left(S\right)$ is a $G_N$-clopen set in $(\mathsf{\Gamma },\eta )$. Therefore, $G_N$ is $G_N$-totally continuous. □

Remark 7. 

The converse of Theorem 13 need not be true, as shown in the following example.

Example 11. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,W,\{b,c\left\}\right\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is $G_N$-totally continuous, but it is not $G_N$-h-totally continuous.

Theorem 14. 

Every $G_N$-h-totally continuous N-method is $G_N$-h-irresolute.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be $G_N$-h-totally continuous and S be any h-open set in $(W,\tau )$. Since $G_N$ is a $G_N$-h-totally continuous N-method, then $G_N^{-1}\left(S\right)$ is a $G_N$-clopen set in $(\mathsf{\Gamma },\eta )$, which implies $G_N^{-1}\left(S\right)$$G_N$-open, and it follows that $G_N^{-1}\left(S\right)$ is a $G_N$-h-open set in $(\mathsf{\Gamma },\eta )$. Therefore, $G_N$ is $G_N$-h-irresolute. □

Remark 8. 

The converse of Theorem 14 need not be true, as shown in the following example.

Example 12. 

In Example 6, the N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is $G_N$-h-irresolute but not $G_N$-h-totally continuous.

Theorem 15. 

If $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is a $G_N$-h-totally continuous N-method and $j:(W,\tau )\to (Z,\vartheta )$ is an h-totally continuous function, then $joG:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-totally continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-h-totally continuous N-method and $j:(W,\tau )\to (Z,\vartheta )$ be an h-totally continuous function. Let S be any h-open in $(Z,\vartheta )$. Since j is h-totally continuous, then $j^{-1}\left(S\right)$ is a clopen set in $(W,\tau )$, which implies that $j^{-1}\left(S\right)$ is an open set, and it follows that $j^{-1}\left(S\right)$ is an h-open set. Since $G_N$ is $G_N$-h-totally continuous, then $G_N^{-1}\left(j^{-1}\left(S\right)\right)={\left(joG\right)}^{-1}\left(S\right)$ is $G_N$-clopen in $(\mathsf{\Gamma },\eta )$. Therefore, $joG:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-totally continuous. □

Theorem 16. 

If $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is a $G_N$-h-totally continuous N-method and $j:(W,\tau )\to (Z,\vartheta )$ is an h-irresolute function, then $joG:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is a $G_N$-h-totally continuous N-method.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-h-totally continuous N-method and $j:(W,\tau )\to (Z,\vartheta )$ be an h-irresolute function. Let S be an h-open set in $(Z,\vartheta )$. Since j is h-irresolute, then $j^{-1}\left(S\right)$ is an h-open set in $(W,\tau )$. Since $G_N$ is $G_N$-h-totally continuous, then $G_N^{-1}\left(j^{-1}\left(S\right)\right)={\left(joG\right)}^{-1}\left(S\right)$ is a $G_N$-clopen set in $(\mathsf{\Gamma },\eta )$. Therefore, $JoG:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-h-totally continuous. □

Theorem 17. 

If $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is a $G_N$-h-totally continuous N-method and $j:(W,\tau )\to (Z,\vartheta )$ is an h-continuous function, then $joG:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-totally continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-h-totally continuous N-method and $j:(W,\tau )\to (Z,\vartheta )$ be an h-continuous function. Let S be an open set in $(Z,\vartheta )$. Since j is h-continuous, then $j^{-1}\left(S\right)$ is an h-open set in $(W,\tau )$. Since $G_N$ is $G_N$-h-totally continuous, then $G_N^{-1}\left(j^{-1}\left(S\right)\right)={\left(joG\right)}^{-1}\left(S\right)$ is a $G_N$-clopen set in $(\mathsf{\Gamma },\eta )$. Therefore, $joG:(\mathsf{\Gamma },\eta )\to (Z,\vartheta )$ is $G_N$-totally continuous. □

Definition 23. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be $G_N$-$contra$-$continuous$ if $G_N^{-1}\left(U\right)$ is a $G_N$-closed set in $(\mathsf{\Gamma },\eta )$ for every open set S in $(W,\tau )$.

Definition 24. 

An N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is said to be $G_N$-h-contra-continuous if $G_N^{-1}\left(U\right)$ is a $G_N$-h-closed set in $(\mathsf{\Gamma },\eta )$ for every open set S in $(W,\tau )$.

Example 13. 

Let $(W,\tau )$ be a topological space where $W=\{a,b,c\}$, $\tau =\{\varnothing ,W,\{a\left\}\right\}$ and $(\mathsf{\Gamma },\eta )$ be a $G_N$-topological space over $(W,\tau )$, defined as $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ and $\eta =\{\varnothing ,\mathsf{\Gamma },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I}\}\}$, where ${\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}$ are convergent nets in $(W,\tau )$. Consider an N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Then, $G_N$ is $G_N$-h-contra-continuous.

Theorem 18. 

Every $G_N$-$contra$-continuous N-method is $G_N$-h-contra-continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be a $G_N$-$contra$-continuous N-method and S be any open set in $(W,\tau )$. Since $G_N$ is $G_N$-$contra$-continuous, then $G_N^{-1}\left(S\right)$ is a $G_N$-closed set in $(\mathsf{\Gamma },\eta )$. Since every $G_N$-closed set is a $G_N$-h-closed set, then $G_N^{-1}\left(S\right)$ is a $G_N$-h-closed set in $(\mathsf{\Gamma },\eta )$. Therefore, $G_N$ is $G_N$-h-contra-continuous. □

Remark 9. 

The converse of Theorem 18 need not be true, as shown in the following example.

Example 14. 

In Example 13, the N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is $G_N$-h-contra-continuous but not $G_N$-$contra$-continuous.

Theorem 19. 

Every $G_N$-totally continuous N-method is $G_N$-h-contra-continuous.

Proof. 

Let $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ be $G_N$-totally continuous and S be any open set in $(W,\tau )$. Since $G_N$ is a $G_N$-totally continuous function, then $G_N^{-1}\left(S\right)$ is a $G_N$-clopen set in $(\mathsf{\Gamma },\eta )$, and hence $G_N$-closed, and it follows that it is a $G_N$-h-closed set. Therefore, $G_N$ is $G_N$-h-contra-continuous. □

Remark 10. 

The converse of Theorem 19 need not be true, as shown in the following example.

Example 15. 

In Example 13, the N-method $G_N:(\mathsf{\Gamma },\eta )\to (W,\tau )$ is $G_N$-h-contra-continuous but not $G_N$-totally continuous.

4. Nano ${\mathit{G}}_{\mathit{N}}$-$\mathit{h}$-Continuity

In this section, we introduce the nano N-method concept and some classes of this new concept, such as nano $G_N$-h-continuous nano N-methods, nano $G_N$-h-open nano N-methods, nano $G_N$-h-irresolute nano N-methods, nano $G_N$-h-totally continuous nano N-methods, nano $G_N$-h-contra-continuous nano N-methods, and nano $G_N$-h-N-homeomethods, and investigate some properties of them.

Definition 25. 

Let $\left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano-topological space and Φ be a subset of all convergent nets in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Then, a relation $R_2$ is a nano $G_N$-equivalence relation on Φ if it satisfies the following three properties:

(a) 

For any ${\left\{{\alpha }_i\right\}}_{i\in I}\in \mathsf{\Phi }$, $({\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\alpha }_i\right\}}_{i\in I})\in R_2$;

(b) 

If $({\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I})\in R_2$ then,$({\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\alpha }_i\right\}}_{i\in I})\in R_2$;

(c) 

If $({\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I})\in R_2$ and $({\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I})\in R_2$ then, $({\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I})\in R_2$.

Definition 26. 

Let $\left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano-topological space, Φ be a subset of all convergent nets in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ and $R_2$ be a nano $G_N$-equivalence relation on Φ. For any ${\left\{{\alpha }_i\right\}}_{i\in I}\in \mathsf{\Phi }$, $\{{\left\{{\beta }_i\right\}}_{i\in I}\in \mathsf{\Phi }|({\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I})\in R_2\}$ is a nano $G_N$-equivalence class of ${\left\{{\alpha }_i\right\}}_{i\in I}$ and denoted as $\left[{\left\{{\alpha }_i\right\}}_{i\in I}\right]$.

Definition 27. 

Let $\left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano-topological space, Ψ be a non-empty finite set of convergent nets in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ and $R_2$ be an nano $G_N$-equivalence relation on Ψ. The pair $(\mathsf{\Psi },R_2)$ is called a nano $G_N$-approximation space, and Ψ is called the universe of $(\mathsf{\Psi },R_2)$. Let $\mathsf{\Gamma }\subseteq \mathsf{\Psi }$:

1. 

The lower nano $G_N$-approximation of $\mathsf{\Gamma }$ with respect to $R_2$ is denoted by $L{R}_2\left(\mathsf{\Gamma }\right)$ and $L{R}_2\left(\mathsf{\Gamma }\right)={\bigcup }_{{\left\{{\alpha }_i\right\}}_{i\in I}\in \mathsf{\Psi }}\left\{\left[{\left\{{\alpha }_i\right\}}_{i\in I}\right]\right|\left[{\left\{{\alpha }_i\right\}}_{i\in I}\right]\subseteq \mathsf{\Gamma }\}$,

2. 

Upper nano $G_N$-approximation of $\mathsf{\Gamma }$ with respect to $R_2$ is denoted by $U_{R_2}\left(\mathsf{\Gamma }\right)$ and $U_{R_2}\left(\mathsf{\Gamma }\right)={\bigcup }_{{\left\{{\alpha }_i\right\}}_{i\in I}\in \mathsf{\Psi }}\left\{\left[{\left\{{\alpha }_i\right\}}_{i\in I}\right]\right|\left[{\left\{{\alpha }_i\right\}}_{i\in I}\right]\bigcap \mathsf{\Gamma }\ne \varnothing \}$;

3. 

The boundary region of $\mathsf{\Gamma }$ with respect to $R_2$ is denoted by $B_{R_2}\left(\mathsf{\Gamma }\right)$ and $B_{R_2}\left(\mathsf{\Gamma }\right)=U_{R_2}\left(\mathsf{\Gamma }\right)\setminus L_{R_2}\left(\mathsf{\Gamma }\right)$.

Definition 28. 

Let $\left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano-topological space, Ψ be a non-empty finite set of convergent nets in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ and $R_2$ be an nano $G_N$-equivalence relation on Ψ. Then, for $\mathsf{\Gamma }\subseteq \mathsf{\Psi }$, ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)=\{\varnothing ,\mathsf{\Gamma },L_{R_2}\left(\mathsf{\Gamma }\right),U_{R_2}\left(\mathsf{\Gamma }\right),B_{R_2}\left(\mathsf{\Gamma }\right)\}$ is called the nano $G_N$-topology on Ψ and $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ is called a nano $G_N$-topological space over $\left(U,{\tau }_{R_1}\left(W\right)\right)$. The elements of ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)$ are called nano $G_N$-open sets, and the complement of a nano $G_N$-open set is called a nano $G_N$-closed set.

Definition 29. 

Consider a nano $G_N$-topological space $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ with respect to $\mathsf{\Gamma }$, where $\mathsf{\Gamma }\subseteq \mathsf{\Psi }$. Let $\lambda \subseteq \mathsf{\Psi }$, then the following apply:

(i) 

The nano $G_N$-interior of the set λ is defined as the union of all nano $G_N$-open subsets contained in λ, and is denoted by $n{\left(\lambda \right)}_{G_N}^{\circ }$;

(ii) 

The nano $G_N$-closure of the set λ is defined as the intersection of all nano $G_N$-closed subsets containing λ, and it is denoted by $n{\overline{\left(\lambda \right)}}_{G_N}$.

Definition 30. 

A subset λ of the nano $G_N$-topological space $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ is called a nano $G_N$-h-open set if for every non-empty set δ in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$, $\delta \ne \mathsf{\Psi }$ and $\delta \in {\eta }_{R_2}\left(\mathsf{\Gamma }\right)$, $\lambda \subseteq n{(\lambda \cup \delta )}_{G_N}^{\circ }$. The complement of the nano $G_N$-h-open set is called nano $G_N$-h-closed.

Theorem 20. 

Every nano $G_N$-open set in any nano $G_N$-topological space $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ is nano $G_N$-h-open.

Definition 31. 

Let $\left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano-topological space and $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ be a nano $G_N$-topological space over $\left(U,{\tau }_{R_1}\left(W\right)\right)$. A nano N-method is a function $G_N$ defined from $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ to $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Definition 32. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$-continuous if $G_N^{-1}\left(\delta \right)$ is a nano $G_N$-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ for every nano open set δ in $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Definition 33. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$-h-continuous if $G_N^{-1}\left(\delta \right)$ is a nano $G_N$-h-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ for every nano open set δ in $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Example 16. 

Let $U=\{a,b,c\}$ with $R_1=\left\{\{a,b\},\left\{c\right\}\right\}$ and $\mathsf{\Psi }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ with $R_2=\left\{\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\}$. Take $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\subset \mathsf{\Psi }$ and $W=\{a,c\}\subset U$. Then, ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)=\{\varnothing ,\mathsf{\Psi },\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$ and ${\tau }_{R_1}\left(W\right)=\{\varnothing ,U,\{a,b\},\left\{c\right\}\}$. Consider a nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Since the family of nano $G_N$-h-open sets in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ is $P\left(\mathsf{\Gamma }\right)$, $G_N$ is nano $G_N$-h-continuous.

Theorem 21. 

Every nano $G_N$-continuous nano N-method is nano $G_N$-h-continuous.

Proof. 

Let $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano $G_N$-continuous N-method and V be any nano open subset in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since $G_N$ is nano $G_N$-continuous, then $G_N^{-1}\left(V\right)$ is a nano $G_N$-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Since every nano $G_N$-open set is a nano $G_N$-h-open set by Theorem 20, then $G_N^{-1}\left(U\right)$ is a nano $G_N$-h-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Therefore, $G_N$ is nano $G_N$-h-continuous. □

Remark 11. 

The converse of Theorem 21 need not be true; as we can see in Example 16, $\left\{c\right\}$ is nano open in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ but $G_N^{-1}\left(\left\{c\right\}\right)=\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}$ is not nano $G_N$-open in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$.

Definition 34. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$-h-open if $G_N\left(V\right)$ is a nano h-open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ for every nano $G_N$-open set V in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$.

Example 17. 

$U=\{a,b,c\}$ with $R_1=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\}$ and $\mathsf{\Psi }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ with $R_2=\left\{\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\}$. Take $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}\subset \mathsf{\Psi }$ and $W=\{a,c\}\subset U$. Then, ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)=\{\varnothing ,\mathsf{\Psi },\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I}\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\}$ and ${\tau }_{R_1}\left(W\right)=\{\varnothing ,U,\{a,c\}\}$. Consider a nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Since the family of nano $G_N$-h-open sets in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ is $P\left(\mathsf{\Gamma }\right)$, $G_N$ is nano $G_N$-h-open.

Theorem 22. 

Every nano $G_N$-open nano N-method is nano $G_N$-h-open.

Proof. 

Let $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano $G_N$-open nano N-method and V be any nano $G_N$-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Since $G_N$ is nano $G_N$-open, then $G_N\left(V\right)$ is a nano open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since every nano open set is a nano h-open set by Theorem 2, then $G_N\left(V\right)$ is a nano h-open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Therefore, $G_N$ is nano $G_N$-h-open. □

Remark 12. 

The converse of Theorem 22 need not be true, as shown in the following example.

Example 18. 

In Example 17, $\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}$ is nano $G_N$-open in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. But $G_N\left(\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\})=\left\{c\right\}$ is not open in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. So, $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is nano $G_N$-h-open but not nano $G_N$-open.

Definition 35. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$-h-irresolute if $G_N^{-1}\left(V\right)$ is a nano $G_N$-h-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ for every nano h-open set V in $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Example 19. 

Let $U=\{a,b,c\}$ with $R_1=\left\{\{a,b\},\left\{c\right\}\right\}$ and $\mathsf{\Psi }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ with $R_2=\left\{\left\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\}$. Take $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I}\}\subset \mathsf{\Psi }$ and $W=\{a,b\}\subset U$. Consider a nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Since the families of nano $G_N$-h-open sets in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ and $\left(U,{\tau }_{R_1}\left(W\right)\right)$ are $P\left(\mathsf{\Psi }\right)$ and $P\left(U\right)$, $G_N$ is nano $G_N$-h-irresolute.

Theorem 23. 

Every nano $G_N$-h-irresolute nano N-method is nano $G_N$-h-continuous.

Proof. 

Let $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano $G_N$-h-irresolute nano N-method and V be any nano open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since every nano open set is nano h-open and $G_N$ is $G_N$-h-irresolute, then V is a nano h-open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ and $G_N^{-1}\left(V\right)$ is a nano $G_N$-h-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Therefore, $G_N$ is $G_N$-h-continuous. □

Remark 13. 

The converse of Theorem 23 need not be true, as shown in the following example.

Example 20. 

Let $U=\{a,b,c\}$ with $R_1=\left\{\left\{a\right\},\{b,c\}\right\}$ and $\mathsf{\Psi }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ with $R_2=\left\{\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\}$. Take $\mathsf{\Gamma }=\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\}\subset \mathsf{\Psi }$ and $W=\{b,c\}\subset U$. Then, ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)=\{\varnothing ,\mathsf{\Psi },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\}\}$ and ${\tau }_{R_1}\left(W\right)=\{\varnothing ,U,\{b,c\}\}$. Consider a nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Clearly, $G_N$ is nano $G_N$-h-continuous. Since $\left\{b\right\}$ is nano h-open $\left(U,{\tau }_{R_1}\left(W\right)\right)$ and $G_N^{-1}\left(\left\{b\right\}\right)=\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\}$ is not a nano $G_N$-h-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$, $G_N$ is not nano $G_N$-h-irresolute.

Definition 36. 

A bijective nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be a nano N-homeomethod if $G_N$ is a nano $G_N$-continuous and nano $G_N$-open N-method.

Definition 37. 

A bijective nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be an h-N-homeomethod if $G_N$ is nano $G_N$-h-continuous and nano $G_N$-h-open N-method.

Example 21. 

In Example 20, the nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is an h-N-homeomethod.

Theorem 24. 

If $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is a nano N-homeomethod, then $G_N$ is a nano h-N-homeomethod.

Proof. 

Since every $G_N$-continuous nano N-method is nano $G_N$-h-continuous and every $G_N$-open N-method is $G_N$-h-open, $G_N$ is $G_N$-h-continuous and $G_N$-h-open. Further, $G_N$ is bijective. Therefore, $G_N$ is a nano h-N-homeomethod. □

Remark 14. 

In Example 20, $\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\}$ is nano $G_N$-open in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ but $G_N\left(\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\}\right)=\left\{a\right\}$ is not open in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. This implies that $G_N$ is not $G_N$-open, and the converse of Theorem 22 need not be true.

Definition 38. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$- totally continuous if $G_N^{-1}\left(\delta \right)$ is a nano $G_N$-clopen set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ for every nano open set δ in $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Definition 39. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$-h-totally continuous if $G_N^{-1}\left(V\right)$ is a nano $G_N$-clopen set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ for every nano h-open set V in $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Example 22. 

Let $U=\{a,b,c\}$ with $R_1=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\}$ and $\mathsf{\Psi }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ with $R_2=\left\{\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\}$. Take $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\subset \mathsf{\Psi }$ and $W=\left\{a\right\}\subset U$. Then, ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)=\{\varnothing ,\mathsf{\Psi },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$ and ${\tau }_{R_1}\left(W\right)=\{\varnothing ,U,\left\{a\right\}\}$. Consider a nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Clearly, $G_N$ is nano $G_N$-h-totally continuous.

Theorem 25. 

Every nano $G_N$-h-totally continuous nano N-method is nano $G_N$-totally continuous.

Proof. 

Let $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ be nano $G_N$-h-totally continuous and V be any nano open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since every nano open set is a nano h-open set by Theorem 2, then V is a nano h-open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since $G_N$ is a nano $G_N$-h-totally continuous nano N-method, then $G_N^{-1}\left(V\right)$ is a nano $G_N$-clopen set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Therefore, $G_N$ is nano $G_N$-totally continuous. □

Remark 15. 

The converse of Theorem 25 need not be true, as shown in the following example.

Example 23. 

Let $U=\{a,b,c\}$ with $R_1=\left\{\left\{a\right\},\{b,c\}\right\}$ and $\mathsf{\Psi }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ with $R_2=\left\{\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\}$. Take $\mathsf{\Gamma }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\subset \mathsf{\Psi }$ and $W=\{b,c\}\subset U$. Then, ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)=\{\varnothing ,\mathsf{\Psi },\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$ and ${\tau }_{R_1}\left(W\right)=\{\varnothing ,U,\{b,c\}\}$. Consider a nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Clearly, $G_N$ is nano $G_N$-totally continuous but not nano $G_N$-h-totally continuous.

Theorem 26. 

Every nano $G_N$-h-totally continuous nano N-method is nano $G_N$-h-irresolute.

Proof. 

Let $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ be nano $G_N$-h-totally continuous and V be a nano h-open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since $G_N$ is a nano $G_N$-h-totally continuous nano N-method, then $G_N^{-1}\left(V\right)$ is a nano $G_N$-clopen set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$, which implies $G_N^{-1}\left(V\right)$ nano $G_N$-open, and it follows that $G_N^{-1}\left(V\right)$ is a nano $G_N$-h-open set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Therefore, $G_N$ is nano $G_N$-h-irresolute. □

Remark 16. 

In Example 19, $\{a,c\}$ is nano $G_N$-h-open in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ but $G_N^{-1}\left(\{a,c\}\right)=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ is not nano $G_N$-clopen in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. This implies that $G_N$ is not nano $G_N$-h-totally continuous, and the converse of Theorem 24 need not be true.

Definition 40. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$-contra continuous if $G_N^{-1}\left(\delta \right)$ is a nano $G_N$-closed set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ for every nano open set δ in $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Definition 41. 

A nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ is said to be nano $G_N$-h-contra-continuous if $G_N^{-1}\left(V\right)$ is a nano $G_N$-h-closed set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$ for every nano open set V in $\left(U,{\tau }_{R_1}\left(W\right)\right)$.

Example 24. 

Let $U=\{a,b,c\}$ with $R_1=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\}$ and $\mathsf{\Psi }=\{{\left\{{\alpha }_i\right\}}_{i\in I},{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}$ with $R_2=\left\{\left\{{\left\{{\alpha }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\},\left\{{\left\{{\gamma }_i\right\}}_{i\in I}\right\}\right\}$. Take $\mathsf{\Gamma }=\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\subset \mathsf{\Psi }$ and $W=\left\{b\right\}\subset U$. Then, ${\eta }_{R_2}\left(\mathsf{\Gamma }\right)=\{\varnothing ,\mathsf{\Psi },\{{\left\{{\beta }_i\right\}}_{i\in I},{\left\{{\gamma }_i\right\}}_{i\in I}\}\}$ and ${\tau }_{R_1}\left(W\right)=\{\varnothing ,U,\left\{b\right\}\}$. Consider a nano N-method $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ defined as $G_N\left({\left\{{\alpha }_i\right\}}_{i\in I}\right)=a$, $G_N\left({\left\{{\beta }_i\right\}}_{i\in I}\right)=b$, $G_N\left({\left\{{\gamma }_i\right\}}_{i\in I}\right)=c$. Clearly, $G_N$ is nano $G_N$-h-contra-continuous.

Theorem 27. 

Every nano $G_N$-$contra$-continuous N-method is nano $G_N$-h-contra-continuous.

Proof. 

Let $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ be a nano $G_N$-$contra$-continuous nano N-method and V be any nano open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since $G_N$ is nano $G_N$-contra-continuous, then $G_N^{-1}\left(V\right)$ is nano $G_N$-closed in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Since every nano $G_N$-closed set is a nano $G_N$-h-closed set, then $G_N^{-1}\left(V\right)$ is a nano $G_N$-h-closed set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. Therefore, $G_N$ is nano $G_N$-h-contra-continuous. □

Remark 17. 

In Example 24, $\left\{b\right\}$ is nano $G_N$-open in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ but $G_N^{-1}\left(\left\{b\right\}\right)=\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\}$ is not nano $G_N$-closed in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. This implies that $G_N$ is not nano $G_N$-$contra$-continuous, and the converse of Theorem 27 need not be true.

Theorem 28. 

Every nano $G_N$-totally continuous nano N-method is nano $G_N$-h-contra-continuous.

Proof. 

Let $G_N:\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)\to \left(U,{\tau }_{R_1}\left(W\right)\right)$ be nano $G_N$-totally continuous and V be any nano open set in $\left(U,{\tau }_{R_1}\left(W\right)\right)$. Since $G_N$ is a nano $G_N$-totally continuous function, then $G_N^{-1}\left(V\right)$ is a nano $G_N$-clopen set in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$, and hence nano $G_N$-closed, and it follows that it is a nano $G_N$-h-closed set. Therefore, $G_N$ is nano $G_N$-h-contra-continuous. □

Remark 18. 

In Example 24, $\left\{b\right\}$ is nano $G_N$-open in $\left(U,{\tau }_{R_1}\left(W\right)\right)$ but $G_N^{-1}\left(\left\{b\right\}\right)=\left\{{\left\{{\beta }_i\right\}}_{i\in I}\right\}$ is not nano $G_N$-clopen in $\left(\mathsf{\Psi },{\eta }_{R_2}\left(\mathsf{\Gamma }\right)\right)$. This implies that $G_N$ is not nano $G_N$-totally continuous, and the converse of Theorem 28 need not be true.

5. Conclusions

In this study, we have introduced the N-method concept by taking inspiration from the method concept, which is a previously introduced function type. As we have stated before, the most obvious features that distinguish the method concept from other types of functions are that its domain consists of convergent sequences and its range is the set of real numbers.

One of the features that distinguishes the N-method concept, which we have just introduced to the mathematical world, from the method concept is that its domain consists of convergent nets. It is known that the concept of a sequence is a type of function whose domain is the set of natural numbers and whose range is the set of real numbers. In the net concept, the set of natural numbers, which is the domain of the sequence concept, is replaced by the directed set concept, and the set of real numbers, which is the range of the sequence concept, is replaced by the topological space concept. So, we are not be wrong if we say that the sequence concept is a special case of the net concept. Therefore, we can say that the N-method concept is a slightly modified version of the method concept. As is known, the image of a convergent sequence under a method is a real number. That is, the image of a convergent sequence under the method is an element of the range of that sequence.

Similarly, the image of a convergent net under an N-method is a point in the topological space where the convergent net is defined. Therefore, we have introduced the concept of $G_N$-topological space, where each element is a convergent net defined in the topological space that itself is the range of the N-method. After clearly defining the N-method concept, we examined the relationships between the N-method concept and various types of continuity that are relatively new in the world of mathematics.

Our expectation while conducting this study is that the N-method concept, like the method concept, enables the reinterpretation of many characters that we can describe as the leading actors of mathematical science and gives these characters new identities.