(𝔫, 𝔪)-Fuzzy e-Open Set in Šostak’s Sense with Applications via Double Fuzzy Topological Spaces

Abstract

In this paper, we present and describe a new type of fuzzy open sets, called fuzzy $(\mathfrak{n},\mathfrak{m})$-e-open sets in double fuzzy topological spaces (DFT Ss,) based on Šostak’s approach. This type belongs to the group of fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-$\beta$-open sets and includes all fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-$\alpha$-open sets, fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-pre-open sets, and fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-semi-open sets. We then explore the idea of $DF$-e-continuity between DFT Ss $(Q,\Gamma ,{\Gamma }^{*})$ and $(K,\mathcal{T},{\mathcal{T}}^{*})$. We also introduce and examine the concepts of $DF$-almost e-continuity and $DF$-weakly e-continuity, which are less strict than $DF$-e-continuity. Subsequently, we introduce and analyze new $DF$ mappings through Fuzzy $(\mathfrak{n},\mathfrak{m})$-e-open and Fuzzy $(\mathfrak{n},\mathfrak{m})$-e-closed sets. Finally, we present and introduce some novel new types of $DF$-separation axioms, named Fuzzy $(\mathfrak{n},\mathfrak{m})$-e-regular and Fuzzy $(\mathfrak{n},\mathfrak{m})$-e-normal spaces, and we examine some of their properties.

1. Historical Background

The idea of a fuzzy set (denoted as an F-set) within a non-empty set Q can be understood as a mapping $f:Q\to I$, where $I=[0,1]$. This concept was introduced by Zadeh in 1965 [1]. In 1968, the concept of F-topology was developed by the same author mentioned in [2]. Numerous researchers have successfully expanded general topology into the fuzzy realm using precise techniques. Notably, Šostak [3] pointed out that the concept of an F-topology as a crisp subclass of F-sets, along with the notion of fuzziness regarding the openness of an F-set, has not been thoroughly explored. This gap in understanding could hinder the process of fuzzifying topological spaces. To address this, Šostak proposed a new definition of F-topology, framing it through the lens of openness for F-sets, which builds upon the prior work by Chang [2]. Several other scholars (including Al-omeri [4,5], El Gayyar et al. [6], Höhle and Šostak [7], Ramadan et al. [8], Priyalatha et al. [9], and Al-omeri [10]) have redefined and examined the same concepts in fuzzy topology, often overlooking Šostak’s contributions.

The notion of an intuitionistic fuzzy set was defined by Atanassov [11,12], which is a generalization of a fuzzy set [1]. In conjunction with this, Coker [13,14] described intuitionistic fuzzy topology inspired by Chang’s perspective [2]. Subsequently, the authors of [15,16] launched an intuitionistic fuzzy topology based on Šostak’s interpretation [3]. Additionally, in 2005, Garcia and Rodabaugh substituted the term ‘intuitionistic’ with ‘double’ [17]. In 2025, Al-omeri presented robust new definitions including double fuzzy $\delta$-cluster points, $(\mathfrak{n},\mathfrak{m})$-fuzzy $\delta$-closure of $\rho$ denoted as $\delta C_{{\Gamma }^{*}}$, as well as $(\mathfrak{n},\mathfrak{m})$-fuzzy $\delta$-closed and $\delta$-open sets and the concept of a double fuzzy $\delta$-continuous function, all grounded in Šostak’s interpretations [18] as published in the Eur. J. Pure Appl. Math. We will clarify how the double fuzzy structure generalizes the standard fuzzy topology in cases where the second structure becomes degenerate or when the parameters $(\mathfrak{n},\mathfrak{m})$ are held at their extreme values.

Consider $(Q,\Gamma ,{\Gamma }^{*})$ to be a Double Fuzzy Topological Space (DFTS). For each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and $\rho \in I^Q$, if we have a double fuzzy point $x_{\left(\alpha ,\beta \right)}\in FP\left(X\right)$ representing the collection of all fuzzy points in Q then we can identify $x_{\left(\alpha ,\beta \right)}$ as a double fuzzy $\delta$-cluster point of a fuzzy set [18] $\rho$ within a Double Fuzzy Topological Space Q. This is true if every (r, s)-fuzzy regular open set that includes the double fuzzy point $x_{\left(\alpha ,\beta \right)}$ (which shares the same support as $x_{\left(\alpha ,\beta \right)}$) intersects non-trivially with $\rho$. Alternatively, for every $(\mathfrak{n},\mathfrak{m})$-fuzzy regular open Q-neighborhood $\phi$ around $x_{\left(\alpha ,\beta \right)}$, if it is q-coincident with $\rho$ then we can state that ${\mathcal{I}}_{{\Gamma }^{*}}(C_{{\Gamma }^{*}}(\phi ,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})q\rho$.

The following aspects are addressed in this discussion:

1

In Section 2, we delve into a new class of fuzzy open sets within DFT Ss, inspired by Šostak’s perspective [3]. We refer to these sets as double fuzzy $(\mathfrak{n},\mathfrak{m})$-e-open sets. Additionally, we define and explore the concepts of double fuzzy $(\mathfrak{n},\mathfrak{m})$-e-closure operators and double fuzzy $(\mathfrak{n},\mathfrak{m})$-e-interior operators. Moreover, we introduce the terms double fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-$\beta$-open sets, double fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-$\alpha$-open sets, double fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-pre-open sets, and double fuzzy $(\mathfrak{n},\mathfrak{m})$-$\delta$-semi-open sets.

2

In Section 3, we present the notion of double fuzzy $(\mathfrak{n},\mathfrak{m})$-e-continuity between DFTSs $(Q,\Gamma ,{\Gamma }^{*})$ and $(K,\Gamma ,{\Gamma }^{*})$. Furthermore, we examine and define the concepts of $DF$-weakly e-continuity and $DF$-almost e-continuity, which serve as weaker variants of $DF$-e-continuity.

3

Section 4 focuses on investigating and defining innovative DF-mappings that utilize double fuzzy $(\mathfrak{n},\mathfrak{m})$-e-open sets. We also put forth new classifications of double fuzzy separation axioms, identified as double fuzzy $(\mathfrak{n},\mathfrak{m})$-e-regular and double fuzzy $(\mathfrak{n},\mathfrak{m})$-e-normal spaces, and we will discuss their properties in detail.

4

Finally, in Section 6 we wrap up the paper by presenting our conclusions and suggesting topics for future research.

2. Preliminaries

Definition 1

([19,20,21]). A mapping is said to be double fuzzy topology on Q for the pair of mapping $\Gamma ,{\Gamma }^{*}:I^Q\to I$ if it satisfies the following conditions:

1. 

$\Gamma \left(\mu \right)+{\Gamma }^{*}\left(\mu \right)\le 1$,

2. 

${\Gamma }^{*}({\mu }_1\bigwedge {\mu }_2)\ge {\Gamma }^{*}\left({\mu }_1\right)\bigwedge {\Gamma }^{*}\left({\mu }_2\right)$ and $\Gamma ({\mu }_1\bigwedge {\mu }_2)\le \Gamma \left({\mu }_1\right)\bigvee \Gamma \left({\mu }_2\right)$,

3. 

$\Gamma \left({\bigvee }_{i\in I}{\mu }_i\right)\ge {\bigwedge }_{i\in I}\Gamma \left({\mu }_i\right)$ and ${\Gamma }^{*}\left({\bigvee }_{i\in I}{\mu }_i\right)\le {\bigvee }_{i\in I}{\Gamma }^{*}\left({\mu }_i\right)$ for each ${\mu }_I\in I^X$, $i\in I$.

Definition 2

([22]). Let $(Q,\Gamma ,{\Gamma }^{*})$ be a DFTS. $\mu \in I^Q$, $x_i\in FP\left(Q\right)$, $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, a fuzzy set μ is called the $(\mathfrak{n},\mathfrak{m})$-Q-neighborhood of $x_i$, if $\mathcal{T}\left(\mu \right)\ge \mathfrak{n},{\mathcal{T}}^{*}\left(\mu \right)\le s$, and $x_{i}q\mu$.

Definition 3

([23]). Let $(X,\Gamma ,{\Gamma }^{*})$ be a DFTS. Then, for each $\mathfrak{r}\in I_0$, $\mathfrak{s}\in I_1$ and $\mu \in I^Q$ is called $(\mathfrak{r},\mathfrak{s})$-fuzzy regular open (or $(\mathfrak{r},\mathfrak{s})$-FRO, for short) if $\mu ={\mathcal{I}}_{\Gamma ,{\Gamma }^{*}}(C_{\Gamma ,{\Gamma }^{*}}(\mu ,\mathfrak{r},\mathfrak{s}),\mathfrak{r},\mathfrak{s})$. A fuzzy set μ is called $(\mathfrak{r},\mathfrak{s})$-fuzzy regular closed (or $(\mathfrak{r},\mathfrak{s})$-FRC, for short) if $\underline{1}-\mu$ is a $(\mathfrak{r},\mathfrak{s})$-FRO set.

Definition 4

([23]). $(Q,\Gamma ,{\Gamma }^{*})$ is an example of a DFTS. For every $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and $\mu \in I^Q$, a fuzzy set μ is said to be $(\mathfrak{n},\mathfrak{m})$-fuzzy regular open (or $(\mathfrak{n},\mathfrak{m})$-FRO, for short). Assume $\mu ={\mathcal{I}}_{{\Gamma }^{*}}(C_{\mathcal{T},{\mathcal{T}}^{*}}(\mu ,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$. A fuzzy set μ is called $(\mathfrak{n},\mathfrak{m})$-fuzzy regular closed (or $(\mathfrak{n},\mathfrak{m})$-FRC, for short) if $\underline{1}-\mu$ is an $(\mathfrak{n},\mathfrak{m})$-FRO set.

Definition 5

([18]). Consider the DFTS $(Q,\Gamma ,{\Gamma }^{*})$. For every $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and $\mu \in I^Q$, where μ is a fuzzy subset of DFTS Q. Assume that φ is a fuzzy subset of Q that meets the requirements listed below:

(a) 

For any double fuzzy points $x_{\left(\alpha ,\beta \right)}$ in φ are $DF$-δ-cluster points of μ;

(b) 

For any $DF$-set ν, which satisfies this condition $\phi \le \nu$, there is a $DF$-point $x_{\left(\alpha ,\beta \right)}\in \nu$ that is not a $DF$-δ-cluster point of μ.

Definition 6

([18]). We define $(\mathfrak{n},\mathfrak{m})$-fuzzy δ-closure of μ, the set of all $DF$-δ-cluster points of μ, denoted by $\delta C_{{\Gamma }^{*}}$. Consequently, a characteristic associated with the notation $\delta C_{{\Gamma }^{*}}$, will always suggest that the property belongs to every $(\mathfrak{n},\mathfrak{m})$-fuzzy δ-closure of μ. A double fuzzy set μ is called an $(\mathfrak{n},\mathfrak{m})$-fuzzy δ-closed set if $\mu =\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})$. The complement of an $(\mathfrak{n},\mathfrak{m})$-F-δ-closed set is called an $(\mathfrak{n},\mathfrak{m})$-F-δ-open set.

Definition 7

([18]). Let $(Q,\Gamma ,{\Gamma }^{*})$ be a DFTS. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$ and $\mu \in I^Q$. The $\delta C_{{\Gamma }^{*}}$ and ${\mathcal{I}}_{{\Gamma }^{*}}$ operators are defined as follows:

1. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})=\bigvee \{\phi \in I^X|\mu \ge \phi ,\phi is(r,s)-FRO\}$;

2. 

$\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})=\bigwedge \{\phi \in I^X|\mu \le \phi ,\phi is(r,s)-FRC\}$.

Definition 8

([18]).

1. 

$\underline{1}-\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})=\delta {\mathcal{I}}_{{\Gamma }^{*}}(\underline{1}-\mu ,\mathfrak{n},\mathfrak{m})$;

2. 

$\delta C_{{\Gamma }^{*}}(\underline{1}-\mu ,\mathfrak{n},\mathfrak{m})=\underline{1}-\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})$.

Theorem 1

([18]). Let $(Q,\Gamma ,{\Gamma }^{*})$ be a DFTS. If for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$ and $\mu \in I^Q$ then the operator $\delta {\mathcal{I}}_{{\Gamma }^{*}}$ satisfies the following statements:

1. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})\le \delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,{\mathfrak{n}}_1,{\mathfrak{m}}_1)$, if $\mathfrak{n}\ge {\mathfrak{n}}_1$ and $\mathfrak{m}\le {\mathfrak{m}}_1$;

2. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu \bigwedge \phi ,\mathfrak{n},\mathfrak{m})=\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})\bigwedge \delta {\mathcal{I}}_{{\Gamma }^{*}}(\phi ,\mathfrak{n},\mathfrak{m})$;

3. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}\left(\delta {\mathcal{I}}_{{\Gamma }^{*}}((\mu ,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)=\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})$;

4. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}(\underline{0},\mathfrak{n},\mathfrak{m})=\underline{0}$, $\delta {\mathcal{I}}_{{\Gamma }^{*}}(\underline{1},\mathfrak{n},\mathfrak{m})=\underline{1}$;

5. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})\le \mu$.

Theorem 2

([18]). Let $(Q,\Gamma ,{\Gamma }^{*})$ be a DFTS. If for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$ and $\mu \in I^Q$ then the operator $\delta C_{{\Gamma }^{*}}$ satisfies the following statements:

1. 

$\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})\le \delta C_{{\Gamma }^{*}}(\mu ,{\mathfrak{n}}_1,{\mathfrak{m}}_1)$, if $\mathfrak{n}\le {\mathfrak{n}}_1$ and $\mathfrak{m}\ge {\mathfrak{m}}_1$;

2. 

$\delta C_{{\Gamma }^{*}}(\mu \vee \phi ,\mathfrak{n},\mathfrak{m})=\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})\vee \delta C_{{\Gamma }^{*}}(\phi ,\mathfrak{n},\mathfrak{m})$;

3. 

$\delta C_{{\Gamma }^{*}}\left(\delta C_{{\Gamma }^{*}}((\mu ,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)=\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})$;

4. 

$\delta C_{{\Gamma }^{*}}(\underline{0},\mathfrak{n},\mathfrak{m})=\underline{0}$, $\delta C_{{\Gamma }^{*}}(\underline{1},\mathfrak{n},\mathfrak{m})=\underline{1}$;

5. 

$\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})\ge \mu$.

Definition 9

([18]). A function f from a Double Fuzzy Topological Space (DFTS) $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ to another DFTS $(K,{\Gamma }_2,{\Gamma }_2^{*})$ is considered double fuzzy δ-continuous at a double fuzzy point $x_{\alpha ,\beta }$ in Q if certain conditions are met. In this context, $f^{-1}\left(\upsilon \right)$ is an $(\mathfrak{n},\mathfrak{m})$-fuzzy δ-open set, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and υ is an $(\mathfrak{n},\mathfrak{m})$-Fuzzy Regular open Open (FRO) set in $I^Y$ with certain properties. These properties include the condition that the measure of the FRO set under the second DFTS, denoted as ${\mathcal{T}}_2\left(\upsilon \right)$, should be greater than or equal to a certain value $\mathfrak{n}$, and the dual measure, denoted as ${\mathcal{T}}_2^{*}\left(\upsilon \right)$, should be less than or equal to a certain value $\mathfrak{m}$. In simpler terms, for every double fuzzy point $x_{\alpha }$ in Q and for every $(\mathfrak{n},\mathfrak{m})$-FRO set neighborhood γ of $f\left(x_{\alpha }\right)$ in $I^K$ there must exist an $(\mathfrak{n},\mathfrak{m})$-FRO set neighborhood β of $x_{\alpha }$ in Q such that $f\left(\beta \right)$ is less than or equal to γ.

3. On (𝔫, 𝔪)-Fuzzy e-Open and e-Closed Sets

Here, we present and study a new class of F-$(\mathfrak{n},\mathfrak{m})$-open sets, called F-$(\mathfrak{n},\mathfrak{m})$-e-open sets and other open sets in DFTS $(Q,\Gamma ,{\Gamma }^{*})$ based on Šostak’s sense [3]. In addition, we explore and investigate the concepts of $DF$-e-interior operators and $DF$-e-closure operators.

Definition 10.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and F-set $A\in I^Q$ is said to be:

1. 

F-$(\mathfrak{n},\mathfrak{m})$-δpre-open set if $A\le {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

2. 

F-$(\mathfrak{n},\mathfrak{m})$-δsemi-open set if $A\le C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

3. 

F-$(\mathfrak{n},\mathfrak{m})$-$\delta \beta$-open set if $A\le {\mathrm{Cl}}_{{\Gamma }^{*}}({\mathcal{I}}_{{\Gamma }^{*}}(\delta {\mathrm{Cl}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\mathfrak{n},\mathfrak{m})$.

4. 

F-$(\mathfrak{n},\mathfrak{m})$-a-open set if $A\le {\mathcal{I}}_{{\Gamma }^{*}}(C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\mathfrak{n},\mathfrak{m})$.

5. 

F-$(\mathfrak{n},\mathfrak{m})$-e-open set if $A\le C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\vee {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

Definition 11.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and F-set $A\in I^Q$ is said to be:

1. 

F-$(\mathfrak{n},\mathfrak{m})$-δpre-closed set if $A\ge C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

2. 

F-$(\mathfrak{n},\mathfrak{m})$-δsemi-closed set if $A\ge {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

3. 

F-$(\mathfrak{n},\mathfrak{m})$-$\delta \beta$-closed set if $A\ge {\mathcal{I}}_{{\Gamma }^{*}}(C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\mathfrak{n},\mathfrak{m})$.

4. 

F-$(\mathfrak{n},\mathfrak{m})$-a-closed set if $A\ge C_{{\Gamma }^{*}}({\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\mathfrak{n},\mathfrak{m})$.

5. 

F-$(\mathfrak{n},\mathfrak{m})$-e-closed set if $A\ge C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\wedge {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

The complements of F-$(\mathfrak{n},\mathfrak{m})$-e-open sets (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-closed sets) are F-$(\mathfrak{n},\mathfrak{m})$-e-closed sets (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-open sets).

Theorem 3.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. IF for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$ then

1. 

Every F-$(\mathfrak{n},\mathfrak{m})$-δpre-open set is F-$(\mathfrak{n},\mathfrak{m})$-e-open.

2. 

Every F-$(\mathfrak{n},\mathfrak{m})$-e-open set is F-$(\mathfrak{n},\mathfrak{m})$-$\delta \beta$-open.

3. 

Every F-$(\mathfrak{n},\mathfrak{m})$-δsemi-open set is F-$(\mathfrak{n},\mathfrak{m})$-e-open.

Proof. 

(1) If A is an F-$(\mathfrak{n},\mathfrak{m})$-$\delta$pre-open set then

$$\begin{array}{cc}\hfill & A\le {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\hfill \\ \hfill & \le {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\vee \delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m})\hfill \\ \hfill & \le C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\vee {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m}).\hfill \end{array}$$

Thus, A is F-$(\mathfrak{n},\mathfrak{m})$-e-open.

(2) If A is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set then

$$\begin{array}{cc}\hfill & A\le C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\vee {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\hfill \\ \hfill & \le C_{{\Gamma }^{*}}({\mathcal{I}}_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\vee {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\hfill \\ \hfill & \le C_{{\Gamma }^{*}}(I_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\mathfrak{n},\mathfrak{m}).\hfill \end{array}$$

Thus, A is an F-$(\mathfrak{n},\mathfrak{m})$-$\delta \beta$-open.

(3) If A is an F-$(\mathfrak{n},\mathfrak{m})$-$\delta$semi-open set then

$$\begin{array}{cc}\hfill & A\le C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\hfill \\ \hfill & \le C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\vee \delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m})\hfill \\ \hfill & \le C_{{\Gamma }^{*}}(\delta {\mathcal{I}}_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\vee {\mathcal{I}}_{{\Gamma }^{*}}(\delta C_{{\Gamma }^{*}}(A,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m}).\hfill \end{array}$$

Thus, A is F-$(\mathfrak{n},\mathfrak{m})$-e-open. □

Remark 1.

Based on our earlier discussions and definitions, here is the diagram:

Remark 2.

The opposite of the diagram presented above does not hold true, as demonstrated by Examples 1–3.

Example 1.

Let $Q=\{q_1,q_2\}$ and suppose $\mathfrak{X},\mathfrak{Y},\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.4}},{\displaystyle \frac{q_2}{0.3}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_2}{0.6}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.7}}\}$. Define $\Gamma ,{\Gamma }^{*}:I^Q-\to I$ as follows:

$$\Gamma \left(\mathcal{C}\right)=\left\{\begin{array}{cc}1,\hfill & if\rho \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathcal{C}=\mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathcal{C}=\mathfrak{X}\vee \mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{X}\wedge \mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }^{*}\left(\mathcal{C}\right)=\left\{\begin{array}{cc}0,\hfill & if\rho \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathcal{C}=\mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathcal{C}=\mathfrak{X}\vee \mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{X}\wedge \mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Therefore, $\mathfrak{Z}$ is an F-$({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}})$-e-open set; however, it does not meet the criteria for being an F-$({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}})$-$\delta$pre-open nor F-$({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}})$-a-open.

Example 2.

Let $Q=\{q_1,q_2\}$ and suppose $\mathfrak{X},\mathfrak{Y},\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.3}},{\displaystyle \frac{q_2}{0.2}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.7}},{\displaystyle \frac{q_2}{0.8}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.4}}\}$. Define $\Gamma ,{\Gamma }^{*}:I^Q-\to I$ as follows:

$$\Gamma \left(\mathcal{C}\right)=\left\{\begin{array}{cc}1,\hfill & if\rho \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{C}=\mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{X},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }^{*}\left(\mathcal{C}\right)=\left\{\begin{array}{cc}0,\hfill & if\rho \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{C}=\mathfrak{X},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Consequently, $extZ$ qualifies as an F-$({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{2}})$-e-open set; however, it does not meet the criteria for being an F-$({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{2}})$-$\delta$semi-open set.

Example 3.

Let $Q=\{q_1,q_2\}$ and suppose $\mathfrak{X},\mathfrak{Y},\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.4}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.4}},{\displaystyle \frac{q_2}{0.5}}\}$. Define $\Gamma ,{\Gamma }^{*}:I^Q-\to I$ as follows:

$$\Gamma \left(\mathcal{C}\right)=\left\{\begin{array}{cc}1,\hfill & if\rho \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{X},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }^{*}\left(\mathcal{C}\right)=\left\{\begin{array}{cc}0,\hfill & if\rho \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{C}=\mathfrak{X},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Therefore, $\mathfrak{Z}$ is an F-$({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{2}})$-$e\delta \beta$-open set; however, it does not meet the criteria for being an F-$({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{2}})$-e open.

Proposition 1.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$ we have the following properties:

1. 

The union of F-$(\mathfrak{n},\mathfrak{m})$-e-open sets is F-$(\mathfrak{n},\mathfrak{m})$-e-open.

2. 

The intersection of F-$(\mathfrak{n},\mathfrak{m})$-e-closed sets is F-$(\mathfrak{n},\mathfrak{m})$-e-closed.

Proof. 

The proof follows by Definitions 10 and 11. □

Proposition 2.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and every F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}\in I^Q$, we have:

1. 

If $\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})=\underline{0}$ then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δsemi-closed.

2. 

If $\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})=\underline{0}$ then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δpre-closed.

3. 

If $\mathfrak{X}$ is $(\mathfrak{n},\mathfrak{m})$-FRO set then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δpre-closed.

4. 

If $\mathfrak{X}$ is $(\mathfrak{n},\mathfrak{m})$-FRC set then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δsemi-closed.

Proof. 

This is easily proved by Definitions 10 and 11. □

Proposition 3.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. If for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and every F-$(\mathfrak{n},\mathfrak{m})$-e-open set $\mathfrak{Y}\in I^Q$ then we have:

1. 

If $\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})=\underline{0}$ then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δpre-closed.

2. 

If $\delta C_{{\Gamma }^{*}}(\mu ,\mathfrak{n},\mathfrak{m})=\underline{0}$ then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δsemi-closed.

3. 

If $\mathfrak{Y}$ is $(\mathfrak{n},\mathfrak{m})$-FRO set then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δsemi-closed.

4. 

If $\mathfrak{Y}$ is $(\mathfrak{n},\mathfrak{m})$-FRC set then $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-δpre-closed.

Proof. 

This is easily proved by Definitions 10 and 11. □

Definition 12.

In a $DFTSs$ $(Q,\Gamma ,{\Gamma }^{*})$, for every $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, $\mathfrak{X}\in I^Q$, and we define a $DF$-e-closure operator ${\delta }_e{C}_{{\Gamma }^{*}}:I^Q\times I_0\times I_1\to I^Q$ as follows: ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})=\bigwedge \{\mathfrak{Y}\in I^Q|\mathfrak{X}\le \mathfrak{Y},\mathfrak{Y}is$ F-$(\mathfrak{n},\mathfrak{m})$-e-closed set}.

Corollary 1.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, $\mathfrak{X}\in I^Q$. Then, $\mathfrak{X}$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-closed set if and only if ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})=\mathfrak{X}$.

Proof. 

This is followed by Definition 12. □

Theorem 4.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, $\mathfrak{X},\mathfrak{Y}\in I^Q$. Then, for any $DF$-operator ${\delta }_e{C}_{{\Gamma }^{*}}:I^Q\times I_0\times I_1\to I^Q$ the following properties are satisfying:

1. 

${\delta }_e{C}_{{\Gamma }^{*}}(\underline{0},\mathfrak{n},\mathfrak{m})=\underline{0}$, ${\delta }_e{C}_{{\Gamma }^{*}}(\underline{1},\mathfrak{n},\mathfrak{m})=\underline{1}$;

2. 

$C_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\ge {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\ge \mathfrak{X}$;

3. 

${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{Y},{\mathfrak{n}}_1,{\mathfrak{m}}_1)$, if $\mathfrak{n}\le {\mathfrak{n}}_1$ and $\mathfrak{X}\le \mathfrak{Y}$;

4. 

${\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)={\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})$;

5. 

${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X}\vee \mathfrak{Y},\mathfrak{n},\mathfrak{m})\ge {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\vee {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$;

6. 

${\delta }_e{C}_{{\Gamma }^{*}}\left(C_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)=C_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})$.

Proof. 

Statements $\left(1\right)$, $\left(2\right)$, and $\left(3\right)$ are easily proved by referring to Definition 12.

(4): From ($\left(1\right)$ and $\left(2\right)$, ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le {\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$. Now, we show that ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\ge {\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$. If ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})$ does not contain ${\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$ then there is $q\in Q$ and $\omega \in (0,1)$ with ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\left(q\right)\le \omega \le {\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$. (I)

From ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\left(q\right)\le \omega$, and by Definition 12, there is $\mathfrak{Z}\in I^Q$ as an F-$(\mathfrak{n},\mathfrak{m})$-e-closed set and $\mathfrak{X}\le \mathfrak{Z}$ with ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\left(q\right)\le \mathfrak{Z}\left(q\right)\le \omega$. Since $\mathfrak{X}\le \mathfrak{Z}$, this implies ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le \mathfrak{Z}$. Now, by the definition of ${\delta }_e{C}_{{\Gamma }^{*}}$, it follows that ${\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)\le \mathfrak{Z}$.

Hence, ${\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)\left(q\right)\le \mathfrak{Z}\left(q\right)\le \omega$, which is a contradiction for (I). Therefore, ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\ge {\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$. Then, ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})={\delta }_e{C}_{{\Gamma }^{*}}\left({\delta }_e{C}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$.

$\left(5\right):$ Since $\mathfrak{X}\le \mathfrak{X}\vee \mathfrak{Y}$ and $\mathfrak{Y}\le \mathfrak{X}\vee \mathfrak{Y}$, it follows that by (3), ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X}\vee \mathfrak{Y},\mathfrak{n},\mathfrak{m})$ and ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\le {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X}\vee \mathfrak{Y},\mathfrak{n},\mathfrak{m})$. Therefore, ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X}\vee \mathfrak{Y},\mathfrak{n},\mathfrak{m})\ge {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\vee {\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$.

$\left(6\right):$ By the Corollary 1 and the fact that $C_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-closed set, then ${\delta }_e{C}_{{\Gamma }^{*}}\left(C_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)=$ $C_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})$. □

Definition 13.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for every $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, $\mathfrak{X},\mathfrak{Y}\in I^Q$ we define a $DF$-e-interior operator ${\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}:I^Q\times I_0\times I_1\to I^Q$ as follows: $\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})=\bigvee \{\mathfrak{Y}\in I^Q|\mathfrak{Y}\le \mathfrak{X},\mathfrak{Y}\mathit{is}$ F-$(\mathfrak{n},\mathfrak{m})$-e-open set}.

Corollary 2.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, $\mathfrak{X}\in I^Q$. Then:

1. 

${\delta }_e{C}_{{\Gamma }^{*}}(\underline{1}-\mathfrak{X},\mathfrak{n},\mathfrak{m})=\underline{1}-{\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})$.

2. 

$\underline{1}-{\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})={\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\underline{1}-\mathfrak{X},\mathfrak{n},\mathfrak{m})$.

Proof. 

(1) For every $\mathfrak{X}\in I^Q$, we have ${\delta }_e{C}_{{\Gamma }^{*}}(\underline{1}-\mathfrak{X},\mathfrak{n},\mathfrak{m})=\bigwedge \{\mathfrak{Y}\in I^Q:\underline{1}-\mathfrak{X}\le \mathfrak{Y},\mathfrak{Y}\mathrm{is}$ F-$(\mathfrak{n},\mathfrak{m})$-e-closed} = $\bigvee \underline{1}-[\{\underline{1}-\mathfrak{Y}\in I^Q:\underline{1}-\mathfrak{Y}\le \mathfrak{X},\underline{1}-\mathfrak{Y}\mathrm{is}$ F-$(\mathfrak{n},\mathfrak{m})$-e-open set}] $=\underline{1}-\left(\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\right)$.

(2) This is comparable to (1). □

Proposition 4.

In a $DFTS$$(Q,\Gamma ,{\Gamma }^{*})$, for every $\mathfrak{X}\in I^Q,\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$. An F-set $\mathfrak{X}$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open if and only if ${\delta }_e{C}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})=\mathfrak{X}$.

Proof. 

This is followed by Definition 13. □

Theorem 5.

Let $(Q,\Gamma ,{\Gamma }^{*})$ be a $DFTSs$. Then, for each $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, $\mathfrak{X},\mathfrak{Y}\in I^Q$. Then, for any $DF$-operator ${\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}:I^Q\times I_0\times I_1\to I^Q$, the following properties are satisfying:

1. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}(\underline{1},\mathfrak{n},\mathfrak{m})=\underline{1}$;

2. 

$\delta {\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le {\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le \mathfrak{X}$;

3. 

${\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le {\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{Y},{\mathfrak{n}}_1,{\mathfrak{m}}_1)$, if $\mathfrak{X}\le \mathfrak{Y}$;

4. 

${\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X}\bigwedge \phi ,\mathfrak{n},\mathfrak{m})\le {\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\bigwedge {\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\phi ,\mathfrak{n},\mathfrak{m})$;

5. 

${\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}\left({\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}((\mathfrak{X},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)={\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})$.

Proof. 

This is comparable to the proof of Theorem 4. □

4. On Double Fuzzy e-Irresoluteness and e-Continuity

In this section, we present and examine the idea of $DF$-$\delta$-e-continuity among DFT Ss, as defined by Šostak’s sense [3]. Additionally, we introduce and analyze the concepts of $DF$-$\delta$-almost e-continuity and $DF$-$\delta$-weakly e-continuity. Within this section, we consider $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$ (where $I_0=(0,1]$ and $I_1=[0,1)$) for each $\mathfrak{n},\mathfrak{m}$.

Definition 14.

A function $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is said to be double fuzzy δ-e-continuous ($DF$-δ-e-continuity, for short), if $f^{-1}\left(\mathfrak{Y}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set, for every $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$, and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$.

Remark 3.

Based on our earlier discussions and definitions, here is the diagram:

Example 4.

Let $Q=\{q_1,q_2\}$, and suppose $\mathfrak{X},\mathfrak{Y},\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.4}},{\displaystyle \frac{q_2}{0.3}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_2}{0.6}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.7}}\}$. Define $\Gamma ,{\Gamma }^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q\to I$ as follows:

$${\Gamma }_1\left(\mathfrak{Z}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathfrak{Z}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathfrak{Z}=\mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathfrak{Z}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathfrak{Z}=\mathfrak{X}\vee \mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathfrak{Z}=\mathfrak{X}\wedge \mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathfrak{Z}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathfrak{Z}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathfrak{Z}=\mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathfrak{Z}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathfrak{Z}=\mathfrak{X}\vee \mathfrak{Y},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathfrak{Z}=\mathfrak{X}\wedge \mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathfrak{Z}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathfrak{Z}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathfrak{Z}=\mathfrak{X},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathfrak{Z}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathfrak{Z}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathfrak{Z}=\mathfrak{X},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Therefore, the identity F-function $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is $DF$-$\delta$-e-continuous, but it is neither $DF$-$\delta$pre-continuous nor $DF$-$\delta \alpha$-continuous.

Example 5.

Let $Q=\{q_1,q_2\}$, and suppose $\mathfrak{X},\mathfrak{Y},\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.3}},{\displaystyle \frac{q_2}{0.2}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.7}},{\displaystyle \frac{q_2}{0.8}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.4}}\}$. Define $\Gamma ,{\Gamma }^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q\to I$ as follows:

$${\Gamma }_1\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Therefore, the identity F-function $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is $DF$-$\delta$-e-continuous, but it is not $DF$-$\delta$semi-continuous.

Example 6.

Let $Q=\{q_1,q_2\}$, and suppose $\mathfrak{X},\mathfrak{Y},\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.4}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.4}},{\displaystyle \frac{q_2}{0.5}}\}$. Define $\Gamma ,{\Gamma }^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q-\to I$ as follows:

$${\Gamma }_1\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Therefore, the identity F-function $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is $DF$-$\delta \beta$-continuous, but it is not $DF$-$\delta$-e-continuous.

Recall that $f_{\omega }\left(Q\right)$ denotes the fuzzy point (or induced fuzzy image) corresponding to the point Q under the mapping f.

Theorem 6.

An F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is $DF$-δ-e-continuous if and only if for any $q_{\omega }\in f_{\omega }\left(Q\right)$ and any $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f\left(q_{\omega }\right)$ there is $\mathfrak{X}\in I^Q$, such that we have F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \mathfrak{Y}$.

Proof. 

$(\Rightarrow )$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f\left(q_{\omega }\right)$, and therefore $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$. Since $q_{\omega }\in f^{-1}\left(\mathfrak{Y}\right)$, this implies $q_{\omega }\in {\delta }_e{C}_{{\Gamma }^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})=\mathfrak{X}$. Then, $\mathfrak{X}\in I^Q$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \mathfrak{Y}$.

$(\Leftarrow )$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f\left(q_{\omega }\right)$. According to this assumption, $\mathfrak{X}\in I^Q$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \mathfrak{Y}$. Then, $q_{\omega }\in \mathfrak{X}\le f^{-1}\left(\mathfrak{Y}\right)$ and $q_{\omega }\in {\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$. Thus, $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$, so $f^{-1}\left(\mathfrak{Y}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set. Therefore, f is $DF$-$\delta$-e-continuous. □

Theorem 7.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function. Then, $DF$-δ-e-continuous is equivalent to the following statements for any $\mathfrak{n}\in I_0$, and $\mathfrak{m}\in I_1$, $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-e-continuous;

2. 

$f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed, for each $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left({\mathfrak{Y}}^c\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left({\mathfrak{Y}}^c\right)\le \mathfrak{m}$;

3. 

$f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\right)\le \delta C_{{\Gamma }_2^{*}}(f\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})$;

4. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

5. 

$f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$.

Proof. 

$\left(1\right)⟹\left(2\right)$ The proof follows by $f^{-1}\left({\mathfrak{Y}}^c\right)={\left(f^{-1}\left(\mathfrak{Y}\right)\right)}^c$ and Definition 14.

$\left(2\right)⟹\left(3\right)$ Let $\mathfrak{X}\in I^Q$. By (2), we got that $f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(f\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed. Therefore, ${\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le {\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(f\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m})$ $\le {\delta }_e{C}_{{\Gamma }_1^{*}}\left(f^{-1}({\delta }_e{C}_{{\Gamma }_2^{*}}\left(f\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$ $=f^{-1}({\delta }_e{C}_{{\Gamma }_2^{*}}\left(f\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m}))$. This implies that $f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\right)\le \delta C_{{\Gamma }_2^{*}}(f\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})$.

$\left(3\right)⟹\left(4\right)$ Let $\mathfrak{Y}\in I^K$. By (3), $f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})\right)\le \delta C_{{\Gamma }_2^{*}}(f\left(f^{-1}\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m})$ $\le \delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$. Then, ${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\right)\right)\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$.

$\left(4\right)⟹\left(5\right)$ The proof follows by $f^{-1}{\left(\mathfrak{Y}\right)}^c=(f^{-1}{\left(\mathfrak{Y}\right))}^c$ and Corollary 2.

$\left(5\right)⟹\left(1\right)$ Let $\mathfrak{Y}\in I^K$ be an F-$(\mathfrak{n},\mathfrak{m})$-e-open set. By (5), $f^{-1}\left(\mathfrak{Y}\right)=f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\mathfrak{Y}\right)$. Hence, ${\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})=f^{-1}\left(\mathfrak{Y}\right)$. Therefore, $f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open, so f is $DF$-$\delta$-e-continuous. □

Definition 15.

For an F-function $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is called $DF$-δ-e-irresolute if $f^{-1}\left(\mathfrak{Y}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set for each F-$(\mathfrak{n},\mathfrak{m})$-e-open set $\mathfrak{Y}\in I^K$.

Lemma 1.

Every $DF$-δ-e-irresolute function is $DF$-δ-e-continuous.

Proof. 

The proof is derived from Definitions 14 and 15. □

Remark 4.

As demonstrated by Example 7, the opposite of Lemma 1 fails.

Example 7.

Let $Q=\{q_1,q_2\}$ and suppose $\mathfrak{X},\mathfrak{Y}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.5}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.4}}\}$. Define ${\Gamma }_1,{\Gamma }_1^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q-\to I$ as follows:

$${\Gamma }_1\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Consequently, the F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$, which is the identity, is not $DF$-δ-e-irresolute but is $DF$-δ-e-continuous.

Theorem 8.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function. Then, $DF$-δ-e-continuous is equivalent to the following statements for any $\mathfrak{n}\in I_0$, and $\mathfrak{m}\in I_1$, $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-e-irresolute;

2. 

$f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed, for each $\mathfrak{Y}\in I^K$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed;

3. 

$f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{C}_{{\Gamma }_2^{*}}(f\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})$;

4. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

5. 

$f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$.

Proof. 

$\left(1\right)⟹\left(2\right)$ The proof follows by $f^{-1}\left({\mathfrak{Y}}^c\right)={\left(f^{-1}\left(\mathfrak{Y}\right)\right)}^c$ and Definition 14.

$\left(2\right)⟹\left(3\right)$ Let $\mathfrak{X}\in I^Q$. By (2), we got that $f^{-1}\left({\delta }_e{C}_{{\Gamma }_2^{*}}(f\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed. Therefore, ${\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\le {\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(f\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m})$ $\le {\delta }_e{C}_{{\Gamma }_2^{*}}\left(f^{-1}({\delta }_e{C}_{{\Gamma }_1^{*}}\left(f\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$ $=f^{-1}({\delta }_e{C}_{{\Gamma }_2^{*}}\left(f\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m}))$. This implies that $f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{X},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{C}_{{\Gamma }_2^{*}}(f\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})$.

$\left(3\right)⟹\left(4\right)$ Let $\mathfrak{Y}\in I^K$. By (3), $f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{X}\right),\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{C}_{{\Gamma }_2^{*}}(f\left(f^{-1}\left(\mathfrak{X}\right)\right),\mathfrak{n},\mathfrak{m})$ $\le {\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$. Then, ${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\right)\right)\le f^{-1}\left({\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$.

$\left(4\right)⟹\left(5\right)$ The proof follows by $f^{-1}{\left(\mathfrak{Y}\right)}^c=(f^{-1}{\left(\mathfrak{Y}\right))}^c$ and Corollary 2.

$\left(5\right)⟹\left(1\right)$ Let $\mathfrak{Y}\in I^K$ be an F-$(\mathfrak{n},\mathfrak{m})$-e-open set. By (5), $f^{-1}\left(\mathfrak{Y}\right)=f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\mathfrak{Y}\right)$. Hence, ${\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})=f^{-1}\left(\mathfrak{Y}\right)$. Therefore, $f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open, so f is $DF$-$\delta$-e-irresolute. □

Corollary 3.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ and $g:(K,{\mathcal{T}}_1,{\mathcal{T}}_1^{*})\to (Z,{\mathcal{T}}_2,{\mathcal{T}}_2^{*})$ be two F-functions. Hence, the the composition $f\circ g$ is $DF$-δ-e-irresolute (resp. $DF$-δ-e-continuous) if f is $DF$-δ-e-irresolute and K is $DF$-δ-e-irresolute (resp. $DF$-δ-e-continuous).

Proof. 

The proof is derived from Definitions 14 and 15. □

Definition 16.

An F-function $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is said to be $DF$-δ-almost-e-continuous if $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}({\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$ for each $\mathfrak{Y}\in I^K$ with ${\Gamma }_1\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$.

Lemma 2.

Every $DF$-δ-e-continuous function is $DF$-δ-almost-e-continuous.

Proof. 

The proof is derived from Definitions 14 and 16. □

Remark 5.

As demonstrated by Example 8, the opposite of Lemma 2 fails.

Example 8.

Let $Q=\{q_1,q_2,q_3\}$ and suppose $\mathfrak{X},\mathfrak{Y},\mathit{and}\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.4}},{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_3}{0.4}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.5}},{\displaystyle \frac{q_3}{0.4}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.3}},{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_3}{0.6}}\}$. Define ${\Gamma }_1,{\Gamma }_1^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q-\to I$ as follows:

$${\Gamma }_1\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{2}{3}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Consequently, the F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ identifies with $DF$-δ-almost-e-irresolute, but it is not $DF$-δ-e-continuous.

Theorem 9.

An F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is $DF$-δ-almost-e-continuous if and only if for any $q_{\omega }\in f_{\omega }\left(Q\right)$ and any $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f_{\omega }\left(Q\right)$ there is $\mathfrak{X}\in I^Q$ that is F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

Proof. 

$(\Rightarrow )$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f_{\omega }\left(Q\right)$, and therefore $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$. Since $q_{\omega }\in f^{-1}\left(\mathfrak{Y}\right)$, this implies that $q_{\omega }\in {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})=\mathfrak{X}$. Then, $\mathfrak{X}\in I^Q$ is F-$\left(\mathfrak{n}\mathfrak{m}\right)$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$.

$(\Leftarrow )$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f_{\omega }\left(Q\right)$. According to this assumption, $\mathfrak{X}\in I^Q$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$, since $q_{\omega }\in \mathfrak{X}\le f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$ and $q_{\omega }\le \delta {\mathcal{I}}_{{\Gamma }_2^{*}}(f^{-1}(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$. Hence, $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f^{-1}(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$, Therefore, f is $DF$-$\delta$-almost-e-continuous. □

Theorem 10.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function, $\mathfrak{n}\in I_0$, and $\mathfrak{m}\in I_1$. Then, the following statements are comparable for every $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-almost-e-continuous;

2. 

$f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open for each $(\mathfrak{n},\mathfrak{m})$-FRO $\mathfrak{Y}\in I^K$;

3. 

$f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed for each $(\mathfrak{n},\mathfrak{m})$-FRC $\mathfrak{Y}\in I^K$;

4. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$ for each F-$(\mathfrak{n},\mathfrak{m})$-e-open set $\mathfrak{Y}\in I^K$;

5. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$ for each F-$(\mathfrak{n},\mathfrak{m})$-δsemi-open set $\mathfrak{Y}\in I^K$.

Proof. 

$\left(1\right)⟹\left(2\right)$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ be an $(\mathfrak{n},\mathfrak{m})$-FRO set with $q_{\omega }\in f^{-1}\left(\mathfrak{Y}\right)$. Hence, by (1) there is $\mathfrak{X}\in I^Q$ that is F-$(\mathfrak{n},\mathfrak{m})$-e-open with $q_{\omega }\in \mathfrak{X}$ and $f\left(\mathfrak{X}\right)\le \delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})$. Hence, $\mathfrak{X}\le f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)=f^{-1}\left(\mathfrak{Y}\right)$ and $q_{\omega }\in {\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$. Then, $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$, so $f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open.

$\left(2\right)⟹\left(3\right)$ If we consider that $\mathfrak{Y}\in I^K$ is $(\mathfrak{n},\mathfrak{m})$-FRC set then by (2) $f^{-1}\left({\mathfrak{Y}}^c\right)={\left(f^{-1}\left(\mathfrak{Y}\right)\right)}^c$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open. This implies that $f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed.

$\left(3\right)⟹\left(4\right)$ If $\mathfrak{Y}\in I^K$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open, and since $\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$ is $(\mathfrak{n},\mathfrak{m})$-FRC set, then by (3) $f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed. Since $f^{-1}\left(\mathfrak{Y}\right)\le f^{-1}\left(\delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$, it follows that ${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$.

$\left(4\right)⟹\left(5\right)$ The proof is derived from the principle that any F-$(\mathfrak{n},\mathfrak{m})$-$\delta$semi-open set is F-$(\mathfrak{n},\mathfrak{m})$-e-open.

$\left(5\right)⟹\left(3\right)$ If $\mathfrak{Y}\in I^K$ is $(\mathfrak{n},\mathfrak{m})$-FRC set then $\mathfrak{Y}$ is F-$(\mathfrak{n},\mathfrak{m})$-$\delta$semi-open. By (5), ${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)=f^{-1}\left(\mathfrak{Y}\right)$. Thus, $f^{-1}\left(\mathfrak{Y}\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed.

$\left(3\right)⟹\left(1\right)$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$, such that $q_{\omega }\in f^{-1}\left(\mathfrak{Y}\right)$, and then $q_{\omega }\in f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$. Since $\left[\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right){]}^c$ is $(\mathfrak{n},\mathfrak{m})$-FRC set, then by (3) $f^{-1}{\left(\left[\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)\right]}^c)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed. Hence, $f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open and $q_{\omega }\in {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$. Hence, $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$. Then, f is $DF$-$\delta$-almost-e-continuous. □

Definition 17.

If $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is an F-function then f is said to be $DF$-δ-weakly-e-continuous if $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}(\delta C_{{\Gamma }_1^{*}}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$ for each $\mathfrak{Y}\in I^K$ with ${\Gamma }_1\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$.

Lemma 3.

Every $DF$-δ-e-continuous function is $DF$-δ-weakly-e-continuous.

Proof. 

The proof is derived from Definitions 14 and 17. □

Remark 6.

As demonstrated by Example 9, the opposite of Lemma 3 fails.

Example 9.

Let $Q=\{q_1,q_2,q_3\}$ and suppose $\mathfrak{X},\mathfrak{Y},\mathit{and}\mathfrak{Z}\in I^Q$ define as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.4}},{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_3}{0.4}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.5}},{\displaystyle \frac{q_3}{0.4}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.3}},{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_3}{0.6}}\}$. Define ${\Gamma }_1,{\Gamma }_1^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q-\to I$ as follows:

$${\Gamma }_1\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Consequently, the F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$, which is the identity, is not $DF$-δ-e-continuous but is $DF$-δ-weakly-e-continuous.

Theorem 11.

An F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is $DF$-δ-weakly-e-continuous if and only if for any $q_{\omega }\in f_{\omega }\left(Q\right)$ and any $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f_{\omega }\left(Q\right)$ there is $\mathfrak{X}\in I^Q$ that is F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$.

Proof. 

$(\Rightarrow )$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f_{\omega }\left(Q\right)$, and therefore $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$. Since $q_{\omega }\in f^{-1}\left(\mathfrak{Y}\right)$, this implies that $q_{\omega }\in {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})=\mathfrak{X}$. Then, $\mathfrak{X}\in I^Q$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$.

$(\Leftarrow )$ Suppose $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$ and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$ containing $f_{\omega }\left(Q\right)$. According to this assumption, $\mathfrak{X}\in I^Q$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open containing $q_{\omega }$ with $f\left(\mathfrak{X}\right)\le \delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$, since $q_{\omega }\in \mathfrak{X}\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$ and $q_{\omega }\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$. Hence, $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$. Therefore, f is $DF$-$\delta$-weakly-e-continuous. □

Theorem 12.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function. Then the following statements are equivalent:

1. 

f is $DF$-δ-weakly-e-continuous;

2. 

$f^{-1}\left(\mathfrak{Y}\right)\ge {\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$ if $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left({\mathfrak{Y}}^c\right)\ge \mathfrak{n}$, and ${\Gamma }_2^{*}\left({\mathfrak{Y}}^c\right)\le \mathfrak{m}$;

3. 

${\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})\ge f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

4. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$.

Proof. 

$\left(1\right)⟹\left(2\right)$ The proof follows by Corollary 2 and Definition 17.

$\left(2\right)⟹\left(3\right)$ Let $\mathfrak{Y}\in I^K$. Hence, by (2) ${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}({\mathfrak{Y}}^c,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}({\mathfrak{Y}}^c,\mathfrak{n},\mathfrak{m})\right)$. Then, $f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$.

$\left(3\right)⟹\left(4\right)$ The proof follows from Corollary 2.

$\left(4\right)⟹\left(1\right)$ Let $\mathfrak{Y}\in I^K$ with ${\Gamma }_2\left(\mathfrak{Y}\right)\ge \mathfrak{n}$, and ${\Gamma }_2^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$. Then, by (4) ${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\delta C_{{\Gamma }_2^{*}}({\mathfrak{Y}}^c,\mathfrak{n},\mathfrak{m}),\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}({\mathfrak{Y}}^c,\mathfrak{n},\mathfrak{m})\right)=f^{-1}\left({\mathfrak{Y}}^c\right)$. This implies that $f^{-1}\left(\mathfrak{Y}\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right),\mathfrak{n},\mathfrak{m})$, so f is $DF$-$\delta$-weakly-e-continuous. □

Lemma 4.

Every $DF$-δ-almost-e-continuous function is $DF$-δ-weakly-e-continuous.

Proof. 

The proof is derived from Definitions 16 and 17. □

Remark 7.

As demonstrated by Example 10, the opposite of Lemma 4 fails.

Example 10.

Let $Q=\{q_1,q_2,q_3\}$ and suppose $\mathfrak{X},\mathfrak{Y},\mathit{and}\mathfrak{Z}\in I^Q$ defined as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.6}},{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_3}{0.4}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.3}},{\displaystyle \frac{q_2}{0.2}},{\displaystyle \frac{q_3}{0.5}}\}$, $\mathfrak{Z}=\{{\displaystyle \frac{q_1}{0.3}},{\displaystyle \frac{q_1}{0.2}},{\displaystyle \frac{q_3}{0.4}}\}$. Define ${\Gamma }_1,{\Gamma }_1^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q-\to I$ as follows:

$${\Gamma }_1\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Z},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Consequently, the F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$, which is the identity, is $DF$-δ-weakly-e-continuous, but it is not $DF$-δ-almost-e-continuous.

Remark 8.

Based on our earlier discussions and definitions, here is the diagram:

Corollary 4.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ and $g:(K,{\mathcal{T}}_1,{\mathcal{T}}_1^{*})\to (Z,{\mathcal{T}}_2,{\mathcal{T}}_2^{*})$ be two F-functions. Hence, the the composition $f\circ g$ is $DF$-δ-almost-e-continuous if f is $DF$-δ-e-irresolute (resp. $DF$-δ-e-continuous) and $\mathfrak{Y}$ $DF$-δ-almost-e-continuous (resp. $DF$-δ-continuous).

Proof. 

The proof is derived from previous Definitions 16 and 17. □

5. Some Applications via F-(𝔫, 𝔪)-e-Open

Based on Šostak’s Sense [7], we present and investigate some new $DF$-$\delta$-mappings between DFTSs $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ and $(K,{\Gamma }_2,{\Gamma }_2^{*})$. Then, using F-$(\mathfrak{n},\mathfrak{m})$-e-closed sets, we present and discuss new kinds of $DF$-$\delta$-separation axioms. These are called F-$(\mathfrak{n},\mathfrak{m})$-e-regular and F-$(\mathfrak{n},\mathfrak{m})$-e-normal spaces. Within this section, we consider $\mathfrak{n}\in I_0$, $\mathfrak{m}\in I_1$ (where $I_0=(0,1]$ and $I_1=[0,1)$) for each $\mathfrak{n},\mathfrak{m}$.

Definition 18.

An F-δ-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is said to be $DF$-δ-e-open if $f\left(\mathfrak{X}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set for every $\mathfrak{X}\in I^Q$ with ${\Gamma }_1\left(\mathfrak{X}\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left(\mathfrak{X}\right)\le \mathfrak{m}$.

Definition 19.

An F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is said to be $DF$-δ-e-irresolute open if $f\left(\mathfrak{X}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set for every F-$(\mathfrak{n},\mathfrak{m})$-e-open set $\mathfrak{X}\in I^Q$.

Lemma 5.

Every $DF$-δ-e-irresolute open function is $DF$-δ-e-open.

Proof. 

The proof is derived from Definitions 18 and 19. □

Remark 9.

As demonstrated by Example 11, the opposite of Lemma 5 fails.

Example 11.

Let $Q=\{q_1,q_2,q_3\}$ and suppose $\mathfrak{X},\mathfrak{Y},\mathit{and}\mathfrak{Z}\in I^Q$ define as $\mathfrak{X}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.5}}\}$, $\mathfrak{Y}=\{{\displaystyle \frac{q_1}{0.5}},{\displaystyle \frac{q_2}{0.5}}\}$. Define ${\Gamma }_1,{\Gamma }_1^{*},{\Gamma }_2,{\Gamma }_2^{*}:I^Q-\to I$ as follows:

$${\Gamma }_1\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_1^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\hfill & if\mathcal{U}=\mathfrak{X},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

$${\Gamma }_2\left(\mathcal{U}\right)=\left\{\begin{array}{cc}1,\hfill & if\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.{\Gamma }_2^{*}\left(\mathcal{U}\right)=\left\{\begin{array}{cc}0,\hfill & ifnot\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\hfill & if\mathcal{U}=\mathfrak{Y},\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Consequently, the F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$, which is the identity, is $DF$-δ-e-open, but it is not $DF$-δ-e-irresolute open.

Theorem 13.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function, $\mathfrak{n}\in I_0$, and $\mathfrak{m}\in I_1$. Then, the following statements are equivalent for every $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-e-open;

2. 

$f\left(\delta {\mathcal{I}}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$;

3. 

${\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left({\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

4. 

For each $\mathfrak{Y}$ and each $\mathfrak{Y}$, $\mathfrak{X}$ with ${\Gamma }_1\left({\mathfrak{Y}}^c\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathfrak{Y}}^c\right)\le \mathfrak{m}$ and $f^{-1}\left(\mathfrak{Y}\right)\le \mathfrak{X}$, there is $\mathcal{V}\in I^K$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed with $\mathfrak{Y}\le \mathcal{V}$ and $f^{-1}\left(\mathcal{V}\right)\le \mathfrak{X}$.

Proof. 

$\left(1\right)⟹\left(2\right)$ Since $f\left(\delta {\mathcal{I}}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le f\left(\mathfrak{X}\right)$, then by (1) $f\left(\delta {\mathcal{I}}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open. This implies that $f\left(\delta {\mathcal{I}}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$.

$\left(2\right)⟹\left(3\right)$ Set $\mathfrak{X}=f^{-1}\left(\mathfrak{Y}\right)$; hence, by (2) $f\left(\delta {\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f\left(f^{-1}\left(\mathfrak{Y}\right)\right),\mathfrak{n},\mathfrak{m})$ $\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$. Therefore, ${\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left({\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$.

$\left(3\right)⟹\left(4\right)$ Let $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$ with ${\Gamma }_1\left({\mathfrak{X}}^c\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathfrak{X}}^c\right)\le \mathfrak{m}$, such that $f^{-1}\left(\mathfrak{Y}\right)\le \mathfrak{X}$. Hence, ${\mathfrak{X}}^c\le f^{-1}\left({\mathfrak{Y}}^c\right)$, ${\mathfrak{X}}^c=\delta {\mathcal{I}}_{{\Gamma }_1^{*}}({\mathfrak{X}}^c,\mathfrak{n},\mathfrak{m})\le \delta {\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left({\mathfrak{Y}}^c\right),\mathfrak{n},\mathfrak{m})$. Then, by (3) ${\mathfrak{X}}^c\le \delta {\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left({\mathfrak{Y}}^c\right),\mathfrak{n},\mathfrak{m})$ $\le f^{-1}\left({\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}({\mathfrak{Y}}^c,\mathfrak{n},\mathfrak{m})\right)$. Therefore, we have $\mathfrak{X}\ge {\left(f^{-1}\left({\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}({\mathfrak{Y}}^c,\mathfrak{n},\mathfrak{m})\right)\right)}^c=f^{-1}\left({\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$. Then, ${\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\in I^K$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed with $\mathfrak{Y}\le {\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})$ and $f^{-1}\left({\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le \mathfrak{X}$.

$\left(4\right)⟹\left(1\right)$ Let $\mathcal{U}\in I^Q$ with ${\Gamma }_1\left(\mathcal{U}\right)\ge mathfrakr$ and ${\Gamma }_1^{*}\left(\mathcal{U}\right)\le mathfraks$. Set $\mathfrak{Y}={\left(f\left(\mathcal{U}\right)\right)}^c$ and $\mathfrak{X}={\mathcal{U}}^c$, then $f^{-1}\left(\mathfrak{Y}\right)=f^{-1}\left({\left(f\left(\mathcal{U}\right)\right)}^c\right)\le \mathfrak{X}$. By (4), $\mathcal{V}\in I^K$ is F-$(\mathfrak{n},\mathfrak{m})$-e-closed with $\mathfrak{Y}\le \mathcal{V}$ and $f^{-1}\left(\mathcal{V}\right)\le \mathfrak{X}={\mathcal{U}}^c$. Thus, $P\left(\mathcal{U}\right)\le f\left(f^{-1}\left({\mathcal{V}}^c\right)\right)$ $\le {\mathcal{V}}^c$. Hence, $\mathfrak{Y}\le \mathcal{V}$, $f\left(\mathcal{U}\right)={\mathfrak{Y}}^c\ge {\mathcal{V}}^c$. Therefore, $f\left(\mathcal{U}\right)={\mathcal{V}}^c$, so $f\left(\mathcal{U}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set. Then, f is $DF$-$\delta$-e-open. □

Theorem 14.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function, $\mathfrak{n}\in I_0$, and $\mathfrak{m}\in I_1$. Then, the following statements are equivalent for every $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-e-irresolute open;

2. 

$f\left({\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(f\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$;

3. 

${\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left({\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

4. 

For each $\mathfrak{Y}$ and each $\mathfrak{Y}$, $\mathfrak{X}$ with ${\Gamma }_1\left({\mathfrak{Y}}^c\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathfrak{Y}}^c\right)\le \mathfrak{m}$ and $f^{-1}\left(\mathfrak{Y}\right)\le \mathfrak{X}$, there is $\mathcal{V}\in I^K$ that is F-$(\mathfrak{n},\mathfrak{m})$-e-closed with $\mathfrak{Y}\le \mathcal{V}$ and $f^{-1}\left(\mathcal{V}\right)\le \mathfrak{X}$.

Proof. 

The proof is akin to that of Theorem 13. □

Definition 20.

An F-δ-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is said to be $DF$-δ-e-closed if $f\left(\mathfrak{X}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-closed set for every $\mathfrak{X}\in I^Q$ with ${\Gamma }_1\left({\mathfrak{Y}}^c\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}{\left(\mathfrak{Y}\right)}^c\le \mathfrak{m}$.

Definition 21.

An F-mapping $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is said to be $DF$-δ-e-irresolute closed if $f\left(\mathfrak{X}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-closed set for every F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}\in I^Q$.

Lemma 6.

Every $DF$-δ-e-irresolute closed function is $DF$-δ-e-closed.

Proof. 

The proof is derived from Definitions 20 and 21. □

Theorem 15.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function, $\mathfrak{n}\in I_0$, and $\mathfrak{m}\in I_1$. Then, the following statements are equivalent for every $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-e-closed;

2. 

$f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le \delta C_{{\Gamma }_2^{*}}(f\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$;

3. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

4. 

For each $\mathfrak{Y}$ and each $\mathfrak{Y}$, $\mathfrak{X}$ with ${\Gamma }_1\left({\mathfrak{Y}}^c\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathfrak{Y}}^c\right)\le \mathfrak{m}$ and $f^{-1}\left(\mathfrak{Y}\right)\le \mathfrak{X}$, there is $\mathcal{V}\in I^K$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open with $\mathfrak{Y}\le \mathcal{V}$ and $f^{-1}\left(\mathcal{V}\right)\le \mathfrak{X}$.

Proof. 

The proof is comparable to Theorem 13. □

Theorem 16.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be an F-function, $\mathfrak{n}\in I_0$, and $\mathfrak{m}\in I_1$. Then, the following statements are equivalent for every $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-e-irresolute closed;

2. 

$f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{C}_{{\Gamma }_2^{*}}(f\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})$;

3. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\le f^{-1}\left({\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

4. 

For each $\mathfrak{Y}$ and each $\mathfrak{Y}$, $\mathfrak{X}$ with ${\Gamma }_1\left({\mathfrak{Y}}^c\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathfrak{Y}}^c\right)\le \mathfrak{m}$ and $f^{-1}\left(\mathfrak{Y}\right)\le \mathfrak{X}$, there is $\mathcal{V}\in I^K$ is F-$(\mathfrak{n},\mathfrak{m})$-e-open with $\mathfrak{Y}\le \mathcal{V}$ and $f^{-1}\left(\mathcal{V}\right)\le \mathfrak{X}$.

Proof. 

The proof is comparable to Theorem 13. □

Lemma 7.

Let $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ be a bijective F-function; then, f is $DF$-δ-e-irresolute open if and only if f is $DF$-δ-e-irresolute closed.

Proof. 

The proof is derived from: $f^{-1}\left({\delta }_e{C}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})\iff$ $f^{-1}\left({\delta }_e{\mathcal{I}}_{{\Gamma }_2^{*}}({\mathfrak{Y}}^c,\mathfrak{n},\mathfrak{m})\right)\le {\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left({\mathfrak{Y}}^c\right),\mathfrak{n},\mathfrak{m})$. □

Definition 22.

Suppose that $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is bijective F-mapping; then, it is called $DF$-δ-e-irresolute closed homeomorphism if f and $f^{-1}$ are $DF$-δ-e-irresolute.

Theorem 17.

For a F-bijective function $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$, the following statements are equivalent for every $\mathfrak{Y}\in I^K$ and $\mathfrak{X}\in I^Q$:

1. 

f is $DF$-δ-e-irresolute homeomorphism;

2. 

f is $DF$-δ-e-irresolute closed and $DF$-δ-e-irresolute;

3. 

f is $DF$-δ-e-irresolute open and $DF$-δ-e-irresolute;

4. 

$f\left({\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)=\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(f\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m}))$;

5. 

$f\left({\delta }_e{C}_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)=\delta C_{{\Gamma }_2^{*}}(f\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m}))$;

6. 

${\delta }_e{\mathcal{I}}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})=f^{-1}\left(\delta {\mathcal{I}}_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$;

7. 

${\delta }_e{C}_{{\Gamma }_1^{*}}(f^{-1}\left(\mathfrak{Y}\right),\mathfrak{n},\mathfrak{m})=f^{-1}\left(\delta C_{{\Gamma }_2^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\right)$.

Proof. 

The proof of the following Theorem is easy and so is omitted. □

Definition 23.

Let $q_{\omega }\in f_{\omega }\left(Q\right)$, $\mathfrak{X}\in I^Q$, $\mathfrak{n}\in I^0$, and $\mathfrak{m}\in I^1$. A DFTS $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is said to be an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space if $q_{\omega }\overline{q}\mathfrak{X}$ for every F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}$ there is ${\mathcal{V}}_i\in I^Q$ with ${\Gamma }_1\left({\mathcal{V}}_i\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathcal{V}}_i\right)\le \mathfrak{m}$ for $i=1,2,\dots$, such that $q_{\omega }\in {\mathcal{V}}_1$, $\mathfrak{X}\le {\mathcal{V}}_2$, and ${\mathcal{V}}_1\overline{q}{\mathcal{V}}_2$.

Definition 24.

Let $\mathfrak{X},\mathfrak{Y}\in I^Q$, $\mathfrak{n}\in I^0$, and $\mathfrak{m}\in I^1$. A DFTS $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is said to be an F-$(\mathfrak{n},\mathfrak{m})$-e-normal space if $\mathfrak{X}\overline{q}\mathfrak{Y}$ for each F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}$ and $\mathfrak{Y}$ there is ${\mathcal{V}}_i\in I^Q$ with ${\Gamma }_1\left({\mathcal{V}}_i\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathcal{V}}_i\right)\ge \mathfrak{m}$ for $i=1,2,\dots$, such that $\mathfrak{X}\le {\mathcal{V}}_1$, $\mathfrak{Y}\le {\mathcal{V}}_2$, and ${\mathcal{V}}_1\overline{q}{\mathcal{V}}_2$.

Theorem 18.

For every DFTS $(Q,{\Gamma }_1,{\Gamma }_1^{*})$, such that $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{X}\in I^Q$, the following statements are equivalent:

1. 

$(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is F-$(\mathfrak{n},\mathfrak{m})$-e-regular space.

2. 

If $q_{\omega }\in \mathfrak{X}$ for every F-$(\mathfrak{n},\mathfrak{m})$-e-open set $\mathfrak{X}$ then there is $\mathfrak{Y}\in I^Q$ with ${\Gamma }_1\left(\mathfrak{Y}\right)\ge \mathfrak{n}$, ${\Gamma }_1^{*}\left(\mathfrak{Y}\right)\le \mathfrak{n}$, and $q_{\omega }\in \mathfrak{Y}\le \delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\le \mathfrak{X}$.

3. 

If $q_{\omega }\overline{q}\mathfrak{X}$ for every F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}$ then there is ${\mathcal{V}}_i\in I^Q$ with ${\Gamma }_1\left({\mathcal{V}}_i\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left({\mathcal{V}}_i\right)\le \mathfrak{m}$ for $i=1,2,\dots$, such that $q_{\omega }\in {\mathcal{V}}_1$, $\mathfrak{X}\le {\mathcal{V}}_2$, and $\delta C_{{\Gamma }_1^{*}}({\mathcal{V}}_1,\mathfrak{n},\mathfrak{m})\overline{q}\delta C_{{\Gamma }_1^{*}}({\mathcal{V}}_2,\mathfrak{n},\mathfrak{m})$.

Proof. 

$\left(1\right)⟹\left(2\right)$ Let $q_{\omega }\in \mathfrak{X}$ for each F-$(\mathfrak{n},\mathfrak{m})$-e-open set $\mathfrak{X}$; then, $q_{\omega }\overline{q}{\mathfrak{X}}^c$. Since $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is F-$(\mathfrak{n},\mathfrak{m})$-e-regular, then there is $\mathfrak{Y},\mathcal{V}\in I^Q$ with ${\Gamma }_1\left(\mathfrak{Y}\right)\ge \mathfrak{n}$, ${\Gamma }_1^{*}\left(\mathfrak{Y}\right)\le \mathfrak{m}$, ${\Gamma }_1\left(\mathcal{V}\right)\ge \mathfrak{n}$, and ${\Gamma }_1^{*}\left(\mathcal{V}\right)\le \mathfrak{m}$, such that $q_{\omega }\in \mathfrak{Y}$, ${\mathfrak{X}}^c\le \mathcal{V}$, and $\mathfrak{Y}\overline{q}\mathcal{V}$. Hence, $q_{\omega }\in \mathfrak{Y}\le$ ${\mathcal{V}}^c\le \mathfrak{X}$; then, $q_{\omega }\in \mathfrak{Y}\le \delta C_{{\Gamma }_1^{*}}(\mathfrak{Y},\mathfrak{n},\mathfrak{m})\le \mathfrak{X}$.

$\left(2\right)⟹\left(3\right)$ Let $q_{\omega }\overline{q}\mathfrak{X}$ for every F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}$; then, $q_{\omega }\in {\mathfrak{X}}^c$. By (2), there is $\mathcal{W}\in I^Q$ with ${\Gamma }_1\left(\mathcal{W}\right)\ge \mathfrak{n}$, ${\Gamma }_1^{*}\left(\mathcal{W}\right)\le \mathfrak{m}$ and $q_{\omega }\in \mathcal{W}\le \delta C_{{\Gamma }_2^{*}}(\mathcal{W},\mathfrak{n},\mathfrak{m})\le {\mathfrak{X}}^c$. Since ${\Gamma }_1\left(\mathcal{W}\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left(\mathcal{W}\right)\le \mathfrak{m},\mathrm{then} \mathcal{W}$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-open set and $q_{\omega }\in \mathcal{W}$. In addition, by (2) there is $\mathcal{U}\in I^Q$ with ${\Gamma }_1\left(\mathcal{U}\right)\ge \mathfrak{n},{\Gamma }_1^{*}\left(\mathcal{U}\right)\le \mathfrak{m}$, and $q_{\omega }\in \mathcal{U}\le \delta C_{{\Gamma }_1^{*}}(\mathcal{U},\mathfrak{n},\mathfrak{m})\le \mathcal{W}\le \delta C_{{\Gamma }_1^{*}}(\mathcal{W},\mathfrak{n},\mathfrak{m})\le {\mathfrak{X}}^c$. Therefore, $\mathfrak{X}\le {\left(\delta C_{{\Gamma }_1^{*}}(\mathcal{W},\mathfrak{n},\mathfrak{m})\right)}^c=\delta {\mathcal{I}}_{{\Gamma }_1^{*}}({\mathcal{W}}^c,\mathfrak{n},\mathfrak{m})\le {\mathcal{W}}^c$. Set $\mathcal{V}=\delta {\mathcal{I}}_{{\Gamma }_1^{*}}({\mathcal{W}}^c,\mathfrak{n},\mathfrak{m})$; thus, ${\Gamma }_1\left(\mathcal{V}\right)\ge \mathfrak{n}$ and ${\Gamma }_1^{*}\left(\mathcal{V}\right)\le \mathfrak{m}$. Hence, $\delta C_{{\Gamma }_1^{*}}(\mathcal{V},\mathfrak{n},\mathfrak{m})\le {\mathcal{W}}^c\le {\left(\delta C_{{\Gamma }_1^{*}}(\mathcal{U},\mathfrak{n},\mathfrak{m})\right)}^c$. Therefore, $\delta C_{{\Gamma }_1^{*}}(\mathcal{V},\mathfrak{n},\mathfrak{m})\overline{q}\delta {\mathcal{I}}_{{\Gamma }_1^{*}}(\mathcal{U},\mathfrak{n},\mathfrak{m})$.

$\left(3\right)⟹\left(1\right)$ This is readily demonstrated by Definition 23. □

Theorem 19.

For every DFTS $(Q,{\Gamma }_1,{\Gamma }_1^{*})$, such that $q_{\omega }\in f_{\omega }\left(Q\right)$ and $\mathfrak{X}\in I^Q$, the following statements are equivalent:

1. 

$(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is F-$(\mathfrak{n},\mathfrak{m})$-e-normal space.

2. 

If $\mathfrak{Y}\le \mathfrak{X}$ for each F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{Y}$, and F-$(\mathfrak{n},\mathfrak{m})$-e-open set $\mathfrak{X}$ then there is $\mathcal{W}\in I^Q$ with ${\Gamma }_1\left(\mathcal{W}\right)\ge \mathfrak{n}$, ${\Gamma }_1^{*}\left(\mathcal{W}\right)\le \mathfrak{m}$, and $\mathfrak{Y}\le \mathcal{W}\le \delta C_{{\Gamma }_1^{*}}(\mathcal{W},\mathfrak{n},\mathfrak{m})\le \mathfrak{X}$.

3. 

If $\mathfrak{Y}\overline{q}\mathfrak{X}$ for each F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{Y},\mathfrak{X}$ then there is ${\mathcal{W}}_i\in I^Q$ with ${\Gamma }_1\left({\mathcal{W}}_i\right)\ge \mathfrak{n}$, ${\Gamma }_1^{*}\left({\mathcal{W}}_i\right)\le \mathfrak{m}$ for $i=1,2,\dots$, such that $\mathfrak{X}\le {\mathcal{W}}_1$, $\mathfrak{Y}\le {\mathcal{W}}_2$, and $\delta C_{{\Gamma }_1^{*}}({\mathcal{W}}_1,\mathfrak{n},\mathfrak{m})\overline{q}\delta C_{{\Gamma }_1^{*}}({\mathcal{W}}_2,\mathfrak{n},\mathfrak{m})$.

Proof. 

The proof is comparable to that of Theorem 18. □

Theorem 20.

Suppose $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is a DF-open function and a bijective $DF$-δ-e-irresolute. If $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-normal space) then $(K,{\Gamma }_2,{\Gamma }_2^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-normal space).

Proof. 

If $k_{\omega }\overline{q}\mathfrak{Y}$ for each F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}\in I^K$ and f is $DF$-$\delta$-e-irresolute then $f^{-1}\left(\mathfrak{Y}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-closed set. Set $k_{\omega }=f\left(q_{\omega }\right)$, and then $q_{\omega }\overline{q}{f}^{-1}\left(\mathfrak{Y}\right)$. Hence, $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is F-$(\mathfrak{n},\mathfrak{m})$-e-regular, there is ${\mathcal{W}}_1,{\mathcal{W}}_2\in I^Q$ with ${\Gamma }_1\left({\mathcal{W}}_1\right)\ge \mathfrak{n}$, ${\Gamma }_1^{*}\left({\mathcal{W}}_1\right)\le \mathfrak{m}$, $\Gamma \left({\mathcal{W}}_2\right)\ge \mathfrak{n}$, and ${\Gamma }_1^{*}\left({\mathcal{W}}_2\right)\le \mathfrak{m}$, such that $q_{\omega }\in {\mathcal{W}}_1,f^{-1}\left(\mathfrak{Y}\right)\le {\mathcal{W}}_2$, and ${\mathcal{W}}_1\overline{q}{\mathcal{W}}_2$. Hence, f is a bijective $DF$-$\delta$-open mapping; this implies $k_{\omega }\in f\left({\mathcal{W}}_1\right)$, $\mathfrak{Y}=f\left(f^{-1}\left(\mathfrak{Y}\right)\right)\le f\left({\mathcal{W}}_2\right)$, and $f\left({\mathcal{W}}_1\right)\overline{q}f\left({\mathcal{W}}_2\right)$. Therefore, $(K,{\Gamma }_2,{\Gamma }_2^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space. The other case also follows similar lines. □

Theorem 21.

Suppose that $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is a $DF$-δ-e-irresolute and an injective $DF$-δ-continuous mapping. If $(K,{\Gamma }_2,{\Gamma }_2^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-normal space) then $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-normal space).

Proof. 

If $q_{\omega }\overline{q}\mathfrak{X}$ for every F-$(\mathfrak{n},\mathfrak{m})$-e-closed set $\mathfrak{X}\in I^Q$ and f is injective DF-$\delta$-e-irresolute closed then $f\left(\mathfrak{X}\right)$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-closed set and $f\left(q_{\omega }\right)\overline{q}f\left(\mathfrak{X}\right)$. Thus, $(K,{\Gamma }_2,{\Gamma }_2^{*})$ is F-$(\mathfrak{n},\mathfrak{m})$-e-regular, there is ${\mathcal{W}}_1,{\mathcal{W}}_2\in I^K$ with ${\Gamma }_2\left({\mathcal{W}}_1\right)\ge \mathfrak{n}$, ${\Gamma }_2^{*}\left({\mathcal{W}}_1\right)\le \mathfrak{m}$, ${\Gamma }_2\left({\mathcal{W}}_2\right)\ge \mathfrak{n}$, and ${\Gamma }_2^{*}\left({\mathcal{W}}_2\right)\le s$, such that $f\left(q_{\omega }\right)\in {\mathcal{W}}_1$, $f\left(\mathfrak{X}\right)\le {\mathcal{W}}_2$, and ${\mathcal{W}}_1\overline{q}{\mathcal{W}}_2$. Since f is a $DF$-$\delta$-continuous mapping, then $q_{\omega }\in f^{-1}\left({\mathcal{W}}_1\right)$ and $\mathfrak{X}\le f^{-1}\left({\mathcal{W}}_2\right)$ with ${\Gamma }_2\left(f^{-1}\left({\mathcal{W}}_1\right)\right)\ge r$, ${\Gamma }_2^{*}\left(f^{-1}\left({\mathcal{W}}_1\right)\right)\le \mathfrak{m}$, ${\Gamma }_2\left(f^{-1}\left({\mathcal{W}}_2\right)\right)\ge \mathfrak{n}$, ${\Gamma }_2^{*}\left(f^{-1}\left({\mathcal{W}}_2\right)\right)\ge s$, and $f^{-1}\left({\mathcal{W}}_1\right)\overline{q}{f}^{-1}\left({\mathcal{W}}_2\right)$. Hence, $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space. The other case also follows similar lines. □

Theorem 22.

Suppose that $f:(Q,{\Gamma }_1,{\Gamma }_1^{*})\to (K,{\Gamma }_2,{\Gamma }_2^{*})$ is a surjective $DF$-δ-e-irresolute, $DF$-δ-open, and $DF$-δ-closed mapping. If $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-normal space) then $(K,{\Gamma }_2,{\Gamma }_2^{*})$ is an F-$(\mathfrak{n},\mathfrak{m})$-e-regular space (resp. F-$(\mathfrak{n},\mathfrak{m})$-e-normal space).

Proof. 

The proof is similar to that of Theorem 20. □

6. Conclusions

In this paper, we introduced a new class of generalized F-open sets, referred to as F-$(\mathfrak{n},\mathfrak{m})$-e-open sets, within the context of DFT S, inspired by Šostak’s framework. We explored various characterizations of these sets and examined their interrelations. Additionally, we presented and analyzed the concepts of $DF$-$\delta$-e-closure and $DF$-$\delta$-e-interior operators.

We then defined and discussed the idea of $DF$-$\delta$-e-continuity between DFTS pairs $(Q,{\Gamma }_1,{\Gamma }_1^{*})$ and $(K,{\Gamma }_2,{\Gamma }_2^{*})$. Furthermore, we investigated and characterized the weaker forms of this continuity, namely $DF$-$\delta$-almost-e-continuity and $DF$-$\delta$-weakly-e-continuity. Following this, we defined and studied new DF-mappings that utilize F-$(\mathfrak{n},\mathfrak{m})$-e-closed and F-$(\mathfrak{n},\mathfrak{m})$-e-open sets.

It is important to emphasize that the present theory properly generalizes the classical fuzzy topological framework of Zadeh (and Chang). Indeed, when the double fuzzy structure collapses to a single fuzzy topology (i.e., $\Gamma ={\Gamma }^{*}$ and the parameters are chosen at extremal values $(\mathfrak{n},\mathfrak{m})=(1,0)$, the operators $C_{\Gamma }^{*},{\mathcal{I}}_{\Gamma }^{*}$, and the $\delta$-operator reduce to the usual fuzzy interior and closure operators. Consequently, the definition of a fuzzy $(\mathfrak{n},\mathfrak{m})$-e-open set reduces to the classical notion of fuzzy open sets. Thus, the classical theory appears as a special case of our generalized framework.

The introduction of the parameters $(\mathfrak{n},\mathfrak{m})$ provides additional structural flexibility, allowing a finer gradation between different levels of openness and regularity. This demonstrates that fuzzy $(\mathfrak{n},\mathfrak{m})$-e-open sets form a proper and nontrivial extension of Zadeh-type fuzzy topology, enriching the theoretical foundation of double fuzzy topological spaces and opening new directions for further research.

Finally, we introduced new types of DF-separation axioms based on F-$(\mathfrak{n},\mathfrak{m})$-e-closed sets, and we outlined their properties. Future research may explore several areas, including: (1) the definition of upper (lower) e-continuous DF-multifunctions and F-$(\mathfrak{n},\mathfrak{m})$-e-connected sets; (2) extending the novel concepts introduced here within the framework of fuzzy ideals; and (3) applying these notions to fuzzy soft topological (r-minimal) spaces, as outlined in references [24,25].