Fuzzy Topological Approaches via r-Fuzzy γ-Open Sets in the Sense of Sostak

1. Introduction

The concept of a fuzzy set ($\mathcal{F}$-set) of a nonempty set M is a mapping $\mathcal{D}:M\to I$ (where $I=[0,1]$), this concept was first defined in 1965 by Zadeh [1]. The theory of $\mathcal{F}$-sets provides a framework for mathematical modeling of those real world situations, which involve an element of uncertainty, imprecision, or vagueness in their description. After the introduction of the concept of $\mathcal{F}$-sets, several research studies were conducted on the generalizations of $\mathcal{F}$-sets. The integration between $\mathcal{F}$-sets and some uncertainty approaches such as soft sets ($\mathcal{S}$-sets) and rough sets ($\mathcal{R}$-sets) has been discussed in [2-4]. The concept of a fuzzy topology ($\mathcal{F}$-topology) was presented in 1968 by Chang [5]. Several authors have successfully generalized the theory of general topology to the fuzzy setting with crisp methods. According to Šostak [6], these notions, an $\mathcal{F}$-topology is a crisp subclass of the class of $\mathcal{F}$-sets and fuzziness in the notion of openness of an $\mathcal{F}$-set has not been considered, which seems to be a drawback in the process of fuzzification of the notion of a topological space. Therefore, Šostak [6] defined a novel definition of the notion of an $\mathcal{F}$-topology as the concept of openness of $\mathcal{F}$-sets. It is an extension of an $\mathcal{F}$-topology introduced by Chang [5]. Thereafter, many researchers (Ramadan [7], Chattopadhyay et. al. [8], El Gayyar et. al. [9], Höhle and Šostak [10], Ramadan et. al. [11], Kim et. al. [12], Abbas [13,14], Kim and Abbas [15], Aygun and Abbas [16,17], Li and Shi [18,19], Shi and Li [20], Fang and Guo [21], El-Dardery et. al. [22], Kalaivani and Roopkumar [23], Solovyov [24], Minana and Šostak [25]) have redefined the same notion and studied $\mathcal{FTS}s$ being unaware of Šostak work.

The generalizations of fuzzy open sets ($\mathcal{F}$-open sets) play an effective role in an $\mathcal{F}$-topology through their use to improve on many results, or to open the door to explore and discuss several of fuzzy topological notions such as $\mathcal{F}$-continuity [7,8], $\mathcal{F}$-connectedness [8], $\mathcal{F}$-compactness [8,9], $\mathcal{F}$-separation axioms [18], etc. Overall, the notions of r-$\mathcal{F}$-semi-open, r-$\mathcal{F}$-pre-open, r-$\mathcal{F}$-$\alpha$-open, and r-$\mathcal{F}$-$\beta$-open sets were defined and studied by the authors of [12,14] on $\mathcal{FTS}s$ in the sense of Šostak [6]. Also, Kim et al. [12] defined and discussed some weaker forms of $\mathcal{F}$-continuity, called $\mathcal{F}$-semi-continuity (resp. $\mathcal{F}$-pre-continuity and $\mathcal{F}$-$\alpha$-continuity) between $\mathcal{FTS}s$ in the sense of Šostak [6]. Furthermore, Abbas [14] explored and characterized the notions of $\mathcal{F}$-$\beta$-continuous (resp. $\mathcal{F}$-$\beta$-irresolute) functions between $\mathcal{FTS}s$ in the sense of Šostak [6]. Additionally, Kim and Abbas [15] introduced several types of r-$\mathcal{F}$-compactness on $\mathcal{FTS}s$ in the sense of Šostak [6].

The notion of fuzzy soft sets ($\mathcal{FS}$-sets) was first presented in 2001 by the author of [26], which combines $\mathcal{S}$-set [27] and $\mathcal{F}$-set [1]. Thereafter, the notion of an $\mathcal{FS}$-topology was defined and many of its properties such as $\mathcal{FS}$-continuity, $\mathcal{FS}$-closure operators, $\mathcal{FS}$-interior operators, and $\mathcal{FS}$-subspaces were introduced in [28,29]. Also, a novel approach to discussing $\mathcal{FS}$-separation and $\mathcal{FS}$-regularity axioms using $\mathcal{FS}$-sets was introduced by Taha [30,31] based on the approach developed by Aygünoǧlu et al. [28]. Moreover, the notions of r-$\mathcal{FS}$-regularly-open, r-$\mathcal{FS}$-pre-open, r-$\mathcal{FS}$-semi-open, r-$\mathcal{FS}$-$\alpha$-open, and r-$\mathcal{FS}$-$\beta$-open sets were introduced by the authors of [32-35] based on the approach developed by Aygünoǧlu et al. [28]. Additionally, Alshammari et al. [36] defined and investigated the notions of r-$\mathcal{FS}$-$\delta$-open sets and $\mathcal{FS}$-$\delta$-continuous functions. Overall, Taha [37] introduced and discussed the notions of $\mathcal{FS}$-almost (resp. $\mathcal{FS}$-weakly) r-minimal continuity, which are weaker forms of $\mathcal{FS}$-r-minimal continuity [33] based on the approach developed by Aygünoǧlu et al. [28].

We lay out the remainder of this article as follows. Section 2 contains some basic definitions and results that help in understanding the obtained results. In Section 3, we display a novel class of $\mathcal{F}$-open sets, called r-$\mathcal{F}$-$\gamma$-open sets on $\mathcal{FTS}s$ in the sense of Šostak [6]. The class of r-$\mathcal{F}$-$\gamma$-open sets is contained in the class of r-$\mathcal{F}$-$\beta$-open sets and contains all r-$\mathcal{F}$-$\alpha$-open, r-$\mathcal{F}$-pre-open, and r-$\mathcal{F}$-semi-open sets. Some properties of r-$\mathcal{F}$-$\gamma$-open sets along with their mutual relationships have been specified with the help of some illustrative examples. After that, we define the concepts of $\mathcal{F}$-$\gamma$-closure and $\mathcal{F}$-$\gamma$-interior operators, and study some of their properties. In Section 4, we explore and investigate the concepts of $\mathcal{F}$-$\gamma$-continuous (resp. $\mathcal{F}$-$\gamma$-irresolute) functions between $\mathcal{FTS}s$$(M,\Im )$ and $(N,\mathsf{Ϝ})$. Moreover, we define and study the concepts of $\mathcal{F}$-almost (resp. $\mathcal{F}$-weakly) $\gamma$-continuous functions, which are weaker forms of $\mathcal{F}$-$\gamma$-continuous functions. We also showed that $\mathcal{F}$-$\gamma$-continuity ⟹$\mathcal{F}$-almost $\gamma$-continuity ⟹$\mathcal{F}$-weakly $\gamma$-continuity, but the converse may not be true. In Section 5, we introduce and discuss some novel $\mathcal{F}$-functions using r-$\mathcal{F}$-$\gamma$-open and r-$\mathcal{F}$-$\gamma$-closed sets, called $\mathcal{F}$-$\gamma$-open (resp. $\mathcal{F}$-$\gamma$-irresolute open, $\mathcal{F}$-$\gamma$-closed, $\mathcal{F}$-$\gamma$-irresolute closed, and $\mathcal{F}$-$\gamma$-irresolute homeomorphism) functions. Furthermore, we define some new types of $\mathcal{F}$-separation axioms, called r-$\mathcal{F}$-$\gamma$-regular (resp. r-$\mathcal{F}$-$\gamma$-normal) spaces, and study some properties of them. Also, we explore and discuss some new types of $\mathcal{F}$-compactness, called r-$\mathcal{F}$-almost (resp. r-$\mathcal{F}$-nearly) $\gamma$-compact sets using r-$\mathcal{F}$-$\gamma$-open sets. In the last section, we close this article with conclusions and proposed future papers.

2. Preliminaries

In this manuscript, nonempty sets will be denoted by M, N, W, etc. On M, $I^M$ is the class of all $\mathcal{F}$-sets. For $\mathcal{D}\in I^M$, ${\mathcal{D}}^c\left(m\right)=1-\mathcal{D}\left(m\right)$, for each $m\in M$. Also, for $\sigma \in I,$$\underline{\sigma }\left(m\right)=\sigma$, for each $m\in M$.

An $\mathcal{F}$-point $m_{\sigma }$ on M is an $\mathcal{F}$-set, defined as follows: $m_{\sigma }\left(u\right)=\sigma$ if $u=m$, and $m_{\sigma }\left(u\right)=0$ for any $u\in M-\left\{m\right\}$. Moreover, we say that $m_{\sigma }$ belong to $\mathcal{D}\in I^M$ ($m_{\sigma }\in \mathcal{D}$), if $\sigma \le \mathcal{D}\left(m\right)$. On M, $P_{\sigma }\left(M\right)$ is the class of all $\mathcal{F}$-points.

On M, an $\mathcal{F}$-set $\mathcal{D}\in I^M$ is a quasi-coincident with $\mathcal{P}\in I^M$ ($\mathcal{D}q\mathcal{P}$), if there is $m\in M$, with $\mathcal{D}\left(m\right)+\mathcal{P}\left(m\right)>1$. Otherwise, $\mathcal{D}$ is not quasi-coincident with $\mathcal{P}$ ($\mathcal{D}\overline{q}\mathcal{P}$).

Lemma 2.1.

$\left[38\right]$ Let $\mathcal{D},\mathcal{P}\in I^M$. Thus,

(1) $\mathcal{D}q\mathcal{P}$ iff there is $m_{\sigma }\in \mathcal{D}$ such that $m_{\sigma }q\mathcal{P}$,

(2) if $\mathcal{D}q\mathcal{P}$, then $\mathcal{D}\wedge \mathcal{P}\ne \underline{0}$,

(3) $\mathcal{D}\overline{q}\mathcal{P}$ iff $\mathcal{D}\le {\mathcal{P}}^c$,

(4) $\mathcal{D}\le \mathcal{P}$ iff $m_{\sigma }\in \mathcal{D}$ implies $m_{\sigma }\in \mathcal{P}$ iff $m_{\sigma }q\mathcal{D}$ implies $m_{\sigma }q\mathcal{P}$ iff $m_{\sigma }\overline{q}\mathcal{P}$ implies $m_{\sigma }\overline{q}\mathcal{D}$,

(5) $m_{\sigma }\overline{q}{\bigvee }_{j\in \Omega }{\mathcal{P}}_j$ iff there is $j_{\circ }\in \Omega$ such that $m_{\sigma }\overline{q}{\mathcal{P}}_{j_{\circ }}$.

Definition 2.1.

$[6,7]$ A mapping $\Im :I^M⟶I$ is said to be a fuzzy topology on M if it satisfies the following conditions:

(1) $\Im \left(\underline{1}\right)=\Im \left(\underline{0}\right)=1.$

(2) $\Im (\mathcal{D}\wedge \mathcal{P})\ge \Im \left(\mathcal{D}\right)\wedge \Im \left(\mathcal{P}\right),$ for each $\mathcal{D},\mathcal{P}\in I^M.$

(3) $\Im \left({\bigvee }_{j\in \Omega }{\mathcal{D}}_j\right)\ge {\bigwedge }_{j\in \Omega }\Im \left({\mathcal{D}}_j\right),$ for each ${\mathcal{D}}_j\in I^M.$

Thus, $(M,\Im )$ is said to be a fuzzy topological space ($\mathcal{FTS}$) in the sense of Šostak.

Definition 2.2.

$[8,11]$ In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $r\in I_{\circ }$ (where $I_{\circ }=(0,1]$), we define $\mathcal{F}$-operators $C_{\Im }$ and $I_{\Im }$$:I^M\times I_{\circ }\to I^M$ as follows:

$$C_{\Im }(\mathcal{D},r)=\bigwedge \{\mathcal{P}\in I^M:\mathcal{D}\le \mathcal{P},\Im \left({\mathcal{P}}^c\right)\ge r\}.$$

$$I_{\Im }(\mathcal{D},r)=\bigvee \{\mathcal{P}\in I^M:\mathcal{P}\le \mathcal{D},\Im \left(\mathcal{P}\right)\ge r\}.$$

Definition 2.3.

$[12,14]$ Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is said to be r-$\mathcal{F}$-regularly-open (resp. r-$\mathcal{F}$-pre-open, r-$\mathcal{F}$-$\beta$-open, r-$\mathcal{F}$-semi-open, r-$\mathcal{F}$-$\alpha$-open, and r-$\mathcal{F}$-open) if $\mathcal{D}=I_{\Im }(C_{\Im }(\mathcal{D},r),r)$ (resp. $\mathcal{D}\le I_{\Im }(C_{\Im }(\mathcal{D},r),r)$, $\mathcal{D}\le C_{\Im }(I_{\Im }(C_{\Im }(\mathcal{D},r),r),r)$, $\mathcal{D}\le C_{\Im }(I_{\Im }(\mathcal{D},r),r)$, $\mathcal{D}\le I_{\Im }(C_{\Im }(I_{\Im }(\mathcal{D},r),r),r)$, and $\mathcal{D}\le I_{\Im }(\mathcal{D},r)$).

Definition 2.4.

$[9,15]$ Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is said to be r-$\mathcal{F}$-compact (resp. r-$\mathcal{F}$-nearly compact and r-$\mathcal{F}$-almost compact) iff for every family ${\{{\mathcal{P}}_j\in I^M|\Im \left({\mathcal{P}}_j\right)\ge r\}}_{j\in \Omega }$, with $\mathcal{D}\le {\bigvee }_{j\in \Omega }{\mathcal{P}}_j$, there is a finite sub-set ${\Omega }_{\circ }$ of $\Omega$, with $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{\mathcal{P}}_j$ (resp. $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{I}_{\Im }(C_{\Im }({\mathcal{P}}_j,r),r)$ and $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{C}_{\Im }({\mathcal{P}}_j,r)$).

Definition 2.5.

$[7,12]$ Let $(M,\Im )$ and $(N,\mathsf{Ϝ})$ be $\mathcal{FTS}s$. An $\mathcal{F}$-function $h:I^M⟶I^N$ is said to be

(1) $\mathcal{F}$-continuous if $\Im \left(h^{-1}\left(\mathcal{P}\right)\right)\ge \mathsf{Ϝ}\left(\mathcal{P}\right)$, for every $\mathcal{P}\in I^N$;

(2) $\mathcal{F}$-open if $\mathsf{Ϝ}\left(h\right(\mathcal{D}\left)\right)\ge \Im \left(\mathcal{D}\right)$, for every $\mathcal{D}\in I^M$;

(3) $\mathcal{F}$-closed if $\mathsf{Ϝ}\left({\left(h\left(\mathcal{D}\right)\right)}^c\right)\ge \Im \left({\mathcal{D}}^c\right)$, for every $\mathcal{D}\in I^M$.

Definition 2.6.

$[12,14]$ Let $(M,\Im )$ and $(N,\mathsf{Ϝ})$ be $\mathcal{FTS}s$ and $r\in I_{\circ }$. An $\mathcal{F}$-function $h:I^M⟶I^N$ is said to be $\mathcal{F}$-$\alpha$-continuous (resp. $\mathcal{F}$-pre-continuous, $\mathcal{F}$-semi-continuous, and $\mathcal{F}$-$\beta$-continuous) if $h^{-1}\left(\mathcal{P}\right)$ is an r-$\mathcal{F}$-$\alpha$-open (resp. r-$\mathcal{F}$-pre-open, r-$\mathcal{F}$-semi-open, and r-$\mathcal{F}$-$\beta$-open) set, for every $\mathcal{P}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{P}\right)\ge r$.

Some basic notations and results that we need in the sequel are found in [7-15].

3. On r-Fuzzy $\gamma$-Open Sets

Here, we define and study a new class of $\mathcal{F}$-open sets, called r-$\mathcal{F}$-$\gamma$-open sets on $\mathcal{FTS}s$ in the sense of Šostak [6]. The class of r-$\mathcal{F}$-$\gamma$-open sets is contained in the class of r-$\mathcal{F}$-$\beta$-open sets and contains all r-$\mathcal{F}$-$\alpha$-open, r-$\mathcal{F}$-pre-open, and r-$\mathcal{F}$-semi-open sets. Also, we explore the concepts of $\mathcal{F}$-$\gamma$-closure and $\mathcal{F}$-$\gamma$-interior operators, and investigate some of their properties.

Definition 3.1.

Let $(M,\Im )$ be an $\mathcal{FTS}$ and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}\in I^M$ is said to be

(1) r-$\mathcal{F}$-$\gamma$-open set if $\mathcal{D}\le C_{\Im }(I_{\Im }(\mathcal{D},r),r)\vee I_{\Im }(C_{\Im }(\mathcal{D},r),r)$;

(2) r-$\mathcal{F}$-$\gamma$-closed set if $\mathcal{D}\ge C_{\Im }(I_{\Im }(\mathcal{D},r),r)\wedge I_{\Im }(C_{\Im }(\mathcal{D},r),r)$.

Remark 3.1.

The complement of r-$\mathcal{F}$-$\gamma$-open set (resp., r-$\mathcal{F}$-$\gamma$-closed set) is r-$\mathcal{F}$-$\gamma$-closed set (resp., r-$\mathcal{F}$-$\gamma$-open set).

Proposition 3.1.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $r\in I_{\circ }$,

(1) every r-$\mathcal{F}$-pre-open set is r-$\mathcal{F}$-$\gamma$-open;

(2) every r-$\mathcal{F}$-$\gamma$-open set is r-$\mathcal{F}$-$\beta$-open;

(3) every r-$\mathcal{F}$-semi-open set is r-$\mathcal{F}$-$\gamma$-open.

Proof. 

(1) If $\mathcal{D}$ is an r-$\mathcal{F}$-pre-open set,

$$\mathcal{D}\le I_{\Im }(C_{\Im }(\mathcal{D},r),r)\le I_{\Im }(C_{\Im }(\mathcal{D},r),r)\vee I_{\Im }(\mathcal{D},r)\le I_{\Im }(C_{\Im }(\mathcal{D},r),r)\vee C_{\Im }(I_{\Im }(\mathcal{D},r),r).$$

Thus, $\mathcal{D}$ is r-$\mathcal{F}$-$\gamma$-open set.

(2) If $\mathcal{D}$ is an r-$\mathcal{F}$-$\gamma$-open set,

$$\mathcal{D}\le C_{\Im }(I_{\Im }(\mathcal{D},r),r)\vee I_{\Im }(C_{\Im }(\mathcal{D},r),r)\le C_{\Im }(I_{\Im }(C_{\Im }(\mathcal{D},r),r),r)\vee I_{\Im }(C_{\Im }(\mathcal{D},r),r)$$

$$\le C_{\Im }(I_{\Im }(C_{\Im }(\mathcal{D},r),r),r).$$

Thus, $\mathcal{D}$ is r-$\mathcal{F}$-$\beta$-open set.

(3) If $\mathcal{D}$ is an r-$\mathcal{F}$-semi-open set,

$$\mathcal{D}\le C_{\Im }(I_{\Im }(\mathcal{D},r),r)\le C_{\Im }(I_{\Im }(\mathcal{D},r),r)\vee I_{\Im }(\mathcal{D},r)\le C_{\Im }(I_{\Im }(\mathcal{D},r),r)\vee I_{\Im }(C_{\Im }(\mathcal{D},r),r).$$

Thus, $\mathcal{D}$ is r-$\mathcal{F}$-$\gamma$-open set.

□

Remark 3.2.

From the previous discussions and definitions, we have the following diagram.

$$r\text{-}\mathcal{F}\text{-}\mathbf{pre}\text{-}\mathbf{open}\mathbf{set}$$

$$\nearrow \searrow$$

$$r\text{-}\mathcal{F}\text{-}\mathbf{open}\mathbf{set}⟶r\text{-}\mathcal{F}\text{-}\mathbf{ff}\text{-}\mathbf{open}\mathbf{set}⟶r\text{-}\mathcal{F}\text{-}\mathbf{fl}\text{-}\mathbf{open}\mathbf{set}⟶r\text{-}\mathcal{F}\text{-}\mathbf{fi}\text{-}\mathbf{open}\mathbf{set}$$

$$\searrow \nearrow$$

$$r\text{-}\mathcal{F}\text{-}\mathbf{semi}\text{-}\mathbf{open}\mathbf{set}$$

Remark 3.3.

The converse of the above diagram fails as Examples 3.1, 3.2, and 3.3.

Example 3.1.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{P},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.4}},{\displaystyle \frac{m_2}{0.3}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.2}},{\displaystyle \frac{m_2}{0.6}}\}$, $\mathcal{V}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.7}}\}$. Define $\Im :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\mathcal{C}=\mathcal{P},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ {\displaystyle \frac{2}{3}},\mathrm{if}\mathcal{C}=\mathcal{P}\wedge \mathcal{D},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{P}\vee \mathcal{D},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, $\mathcal{V}$ is ${\displaystyle \frac{1}{3}}$-$\mathcal{F}$-$\gamma$-open set, but it is neither ${\displaystyle \frac{1}{3}}$-$\mathcal{F}$-pre-open nor ${\displaystyle \frac{1}{3}}$-$\mathcal{F}$-$\alpha$-open.

Example 3.2.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{P},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.3}},{\displaystyle \frac{m_2}{0.2}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.7}},{\displaystyle \frac{m_2}{0.8}}\},$$\mathcal{V}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.4}}\}$. Define $\Im :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{2}{3}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{P},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, $\mathcal{V}$ is ${\displaystyle \frac{1}{2}}$-$\mathcal{F}$-$\gamma$-open set, but it is not ${\displaystyle \frac{1}{2}}$-$\mathcal{F}$-semi-open.

Example 3.3.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.4}}\}$, $\mathcal{V}=\{{\displaystyle \frac{m_1}{0.4}},{\displaystyle \frac{m_2}{0.5}}\}$. Define $\Im :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, $\mathcal{V}$ is ${\displaystyle \frac{1}{4}}$-$\mathcal{F}$-$\beta$-open set, but it is not ${\displaystyle \frac{1}{4}}$-$\mathcal{F}$-$\gamma$-open.

Corollary 3.1.

In an $\mathcal{FTS}$$(M,\Im )$ and $r\in I_{\circ }$,

(1) the union of r-$\mathcal{F}$-$\gamma$-open sets is r-$\mathcal{F}$-$\gamma$-open;

(2) the intersection of r-$\mathcal{F}$-$\gamma$-closed sets is r-$\mathcal{F}$-$\gamma$-closed.

Proof. 

Easily proved by Definition 3.1. □

Corollary 3.2.

In an $\mathcal{FTS}$$(M,\Im )$, for each r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{D}\in I^M$:

(1) If $\mathcal{D}$ is an r-$\mathcal{F}$-regularly-open set, then $\mathcal{D}$ is r-$\mathcal{F}$-pre-closed.

(2) If $\mathcal{D}$ is an r-$\mathcal{F}$-regularly-closed set, then $\mathcal{D}$ is r-$\mathcal{F}$-semi-closed.

(3) If $I_{\Im }(\mathcal{D},r)=\underline{0}$, then $\mathcal{D}$ is r-$\mathcal{F}$-semi-closed.

(4) If $C_{\Im }(\mathcal{D},r)=\underline{0}$, then $\mathcal{D}$ is r-$\mathcal{F}$-pre-closed.

Proof. 

The proof follows by Definitions 2.3 and 3.1. □

Corollary 3.3.

In an $\mathcal{FTS}$$(M,\Im )$, for each r-$\mathcal{F}$-$\gamma$-open set $\mathcal{P}\in I^M$:

(1) If $\mathcal{P}$ is an r-$\mathcal{F}$-regularly-open set, then $\mathcal{P}$ is r-$\mathcal{F}$-semi-open.

(2) If $\mathcal{P}$ is an r-$\mathcal{F}$-regularly-closed set, then $\mathcal{P}$ is r-$\mathcal{F}$-pre-open.

(3) If $I_{\Im }(\mathcal{P},r)=\underline{0}$, then $\mathcal{P}$ is r-$\mathcal{F}$-pre-open.

(4) If $C_{\Im }(\mathcal{P},r)=\underline{0}$, then $\mathcal{P}$ is r-$\mathcal{F}$-semi-open.

Proof. 

The proof follows by Definitions 2.3 and 3.1. □

Definition 3.2.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $r\in I_{\circ }$, we define an $\mathcal{F}$-$\gamma$-closure operator $\gamma C_{\Im }$$:I^M\times I_{\circ }⟶I^M$ as follows: $\gamma C_{\Im }(\mathcal{D},r)=\bigwedge \{\mathcal{P}\in I^M:\mathcal{D}\le \mathcal{P},\mathcal{P}\mathrm{is}r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{closed}\}.$

Proposition 3.2.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is r-$\mathcal{F}$-$\gamma$-closed iff $\gamma C_{\Im }(\mathcal{D},r)=\mathcal{D}$.

Proof. 

Easily proved from Definition 3.2. □

Theorem 3.1.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D},\mathcal{P}\in I^M$ and $r\in I_{\circ }$. An $\mathcal{F}$-operator $\gamma C_{\Im }$$:I^M\times I_{\circ }⟶I^M$ satisfies the following properties.

(1) $\gamma C_{\Im }(\underline{0},r)=\underline{0}$.

(2) $\mathcal{D}\le \gamma C_{\Im }(\mathcal{D},r)\le C_{\Im }(\mathcal{D},r)$.

(3) $\gamma C_{\Im }(\mathcal{D},r)\le \gamma C_{\Im }(\mathcal{P},r)$ if $\mathcal{D}\le \mathcal{P}$.

(4) $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)=\gamma C_{\Im }(\mathcal{D},r)$.

(5) $\gamma C_{\Im }(\mathcal{D}\vee \mathcal{P},r)\ge \gamma C_{\Im }(\mathcal{D},r)\vee \gamma C_{\Im }(\mathcal{P},r)$.

(6) $\gamma C_{\Im }(C_{\Im }(\mathcal{D},r),r)=C_{\Im }(\mathcal{D},r)$.

Proof. 

(1), (2), and (3) are easily proved by Definition 3.2.

(4) From (2) and (3), $\gamma C_{\Im }(\mathcal{D},r)\le \gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)$. Now, we show $\gamma C_{\Im }(\mathcal{D},r)\ge \gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)$. If $\gamma C_{\Im }(\mathcal{D},r)$ does not contain $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)$, there is $m\in M$ and $\sigma \in (0,1)$ with $\gamma C_{\Im }(\mathcal{D},r)\left(m\right)<\sigma <\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)\left(m\right).\left(Z\right)$

Since $\gamma C_{\Im }(\mathcal{D},r)\left(m\right)<\sigma$, by Definition 3.2, there is $\mathcal{V}\in I^M$ as an r-$\mathcal{F}$-$\gamma$-closed set and $\mathcal{D}\le \mathcal{V}$ with $\gamma C_{\Im }(\mathcal{D},r)\left(m\right)\le \mathcal{V}\left(m\right)<\sigma$. Since $\mathcal{D}\le \mathcal{V}$, then $\gamma C_{\Im }(\mathcal{D},r)\le \mathcal{V}$. Again, by the definition of $\gamma C_{\Im }$, $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)\le \mathcal{V}$.

Hence, $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)\left(m\right)\le \mathcal{V}\left(m\right)<\sigma$, which is a contradiction for $\left(Z\right)$. Thus, $\gamma C_{\Im }(\mathcal{D},r)\ge \gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)$. Therefore, $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},r),r)=\gamma C_{\Im }(\mathcal{D},r)$.

(5) Since $\mathcal{D}\le \mathcal{D}\vee \mathcal{P}$ and $\mathcal{P}\le \mathcal{D}\vee \mathcal{P}$, hence by (3), $\gamma C_{\Im }(\mathcal{D},r)\le \gamma C_{\Im }(\mathcal{D}\vee \mathcal{P},r)$ and $\gamma C_{\Im }(\mathcal{P},r)\le \gamma C_{\Im }(\mathcal{D}\vee \mathcal{P},r)$. Thus, $\gamma C_{\Im }(\mathcal{D}\vee \mathcal{P},r)\ge \gamma C_{\Im }(\mathcal{D},r)\vee \gamma C_{\Im }(\mathcal{P},r)$.

(6) From Proposition 3.2 and $C_{\Im }(\mathcal{D},r)$ is an r-$\mathcal{F}$-$\gamma$-closed set, then $\gamma C_{\Im }(C_{\Im }(\mathcal{D},r),r)=C_{\Im }(\mathcal{D},r)$.

□

Definition 3.3.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $r\in I_{\circ }$, we define an $\mathcal{F}$-$\gamma$-interior operator $\gamma I_{\Im }$$:I^M\times I_{\circ }⟶I^M$ as follows: $\gamma I_{\Im }(\mathcal{D},r)=\bigvee \{\mathcal{P}\in I^M:\mathcal{P}\le \mathcal{D},\mathcal{P}\mathrm{is}r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}\}.$

Proposition 3.3.

Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $r\in I_{\circ }$. Then

(1) $\gamma C_{\Im }({\mathcal{D}}^c,r)={\left(\gamma I_{\Im }(\mathcal{D},r)\right)}^c$;

(2) $\gamma I_{\Im }({\mathcal{D}}^c,r)={\left(\gamma C_{\Im }(\mathcal{D},r)\right)}^c$.

Proof. 

(1) For each $\mathcal{D}\in I^M$ and $r\in I_{\circ }$, we have $\gamma C_{\Im }({\mathcal{D}}^c,r)=\bigwedge \{\mathcal{P}\in I^M:{\mathcal{D}}^c\le \mathcal{P},\mathcal{P}\mathrm{is}r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{closed}\}={[\bigvee \{{\mathcal{P}}^c\in I^M:{\mathcal{P}}^c\le \mathcal{D},{\mathcal{P}}^c\mathrm{is}r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}\}]}^c$ = ${\left(\gamma I_{\Im }(\mathcal{D},r)\right)}^c$.

(2) Similar to that of (1).

□

Proposition 3.4.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is r-$\mathcal{F}$-$\gamma$-open iff $\gamma I_{\Im }(\mathcal{D},r)=\mathcal{D}$.

Proof. 

Easily proved from Definition 3.3. □

Theorem 3.2.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D},\mathcal{P}\in I^M$ and $r\in I_{\circ }$. An $\mathcal{F}$-operator $\gamma I_{\Im }$$:I^M\times I_{\circ }⟶I^M$ satisfies the following properties.

(1) $\gamma I_{\Im }(\underline{1},r)=\underline{1}$.

(2) $I_{\Im }(\mathcal{D},r)\le \gamma I_{\Im }(\mathcal{D},r)\le \mathcal{D}$.

(3) $\gamma I_{\Im }(\mathcal{D},r)\le \gamma I_{\Im }(\mathcal{P},r)$ if $\mathcal{D}\le \mathcal{P}$.

(4) $\gamma I_{\Im }(\gamma I_{\Im }(\mathcal{D},r),r)=\gamma I_{\Im }(\mathcal{D},r)$.

(5) $\gamma I_{\Im }(\mathcal{D},r)\wedge \gamma I_{\Im }(\mathcal{P},r)\ge \gamma I_{\Im }(\mathcal{D}\wedge \mathcal{P},r)$.

Proof. 

The proof is similar to that of Theorem 3.1.

□

4. On Fuzzy $\gamma$-Continuity

Here, we define and discuss the concepts of $\mathcal{F}$-$\gamma$-continuous and $\mathcal{F}$-$\gamma$-irresolute functions between $\mathcal{FTS}s$$(M,\Im )$ and $(N,\mathsf{Ϝ})$. We also define and study the concepts of $\mathcal{F}$-almost and $\mathcal{F}$-weakly $\gamma$-continuous functions, which are weaker forms of $\mathcal{F}$-$\gamma$-continuous functions. We showed that $\mathcal{F}$-$\gamma$-continuity ⟹$\mathcal{F}$-almost $\gamma$-continuity ⟹$\mathcal{F}$-weakly $\gamma$-continuity, but the converse may not be true.

Definition 4.1.

An $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is called

(1) $\mathcal{F}$-$\gamma$-continuous if $h^{-1}\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-$\gamma$-open set, for every $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$;

(2) $\mathcal{F}$-$\gamma$-irresolute if $h^{-1}\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-$\gamma$-open set, for every r-$\mathcal{F}$-$\gamma$-open set $\mathcal{D}\in I^N$.

Remark 4.1.

From the previous definitions, we have the following diagram.

$$\mathcal{F}\text{-}\mathbf{pre}\text{-}\mathbf{continuity}$$

$$\nearrow \searrow$$

$$\mathcal{F}\text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\mathbf{ff}\text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\mathbf{fl}\text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\mathbf{fi}\text{-}\mathbf{continuity}$$

$$\searrow \nearrow$$

$$\mathcal{F}\text{-}\mathbf{semi}\text{-}\mathbf{continuity}$$

Remark 4.2.

The converse of the above diagram fails as Examples 4.1, 4.2, and 4.3.

Example 4.1.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{P},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.4}},{\displaystyle \frac{m_2}{0.3}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.2}},{\displaystyle \frac{m_2}{0.6}}\}$, $\mathcal{V}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.7}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\mathcal{C}=\mathcal{P},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{P}\wedge \mathcal{D},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{P}\vee \mathcal{D},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\mathcal{C}=\mathcal{V},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-$\gamma$-continuous, but it is neither $\mathcal{F}$-pre-continuous nor $\mathcal{F}$-$\alpha$-continuous.

Example 4.2.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{P},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.3}},{\displaystyle \frac{m_2}{0.2}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.7}},{\displaystyle \frac{m_2}{0.8}}\}$, $\mathcal{V}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.4}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{P},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{6}},\mathrm{if}\mathcal{C}=\mathcal{V},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-$\gamma$-continuous, but it is not $\mathcal{F}$-semi-continuous.

Example 4.3.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.4}}\}$, $\mathcal{V}=\{{\displaystyle \frac{m_1}{0.4}},{\displaystyle \frac{m_2}{0.5}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{2}{3}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{V},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-$\beta$-continuous, but it is not $\mathcal{F}$-$\gamma$-continuous.

Theorem 4.1.

An $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is $\mathcal{F}$-$\gamma$-continuous iff for any $m_{\sigma }\in P_{\sigma }\left(M\right)$ and any $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ containing $h\left(m_{\sigma }\right)$, there is $\mathcal{A}\in I^M$ that is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le \mathcal{D}$.

Proof. 

(⇒) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ containing $h\left(m_{\sigma }\right)$, then $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)$. Since $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$, then we obtain $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)=\mathcal{A}$ (say). Hence, $\mathcal{A}\in I^M$ is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le \mathcal{D}$.

(⇐) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ and $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$. According to the assumption there is $\mathcal{A}\in I^M$ that is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le \mathcal{D}$. Hence, $m_{\sigma }\in \mathcal{A}\le h^{-1}\left(\mathcal{D}\right)$ and $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)$. Thus, $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)$, so $h^{-1}\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-$\gamma$-open set. Then, h is $\mathcal{F}$-$\gamma$-continuous.

□

Theorem 4.2.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function and $r\in I_{\circ }$, the following statements are equivalent for every $\mathcal{P}\in I^M$ and $\mathcal{D}\in I^N$:

(1) h is $\mathcal{F}$-$\gamma$-continuous.

(2) $h^{-1}\left(\mathcal{D}\right)$ is r-$\mathcal{F}$-$\gamma$-closed, for every $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left({\mathcal{D}}^c\right)\ge r$.

(3) $h\left(\gamma C_{\Im }(\mathcal{P},r)\right)\le C_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),r)$.

(4) $\gamma C_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right)$.

(5) $h^{-1}\left(I_{\mathsf{Ϝ}}(\mathcal{D},r)\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)$.

Proof. 

(1) ⇔ (2) The proof follows by $h^{-1}\left({\mathcal{D}}^c\right)={\left(h^{-1}\left(\mathcal{D}\right)\right)}^c$ and Definition 4.1.

(2) ⇒ (3) Let $\mathcal{P}\in I^M$. By (2), we have $h^{-1}\left(C_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),r)\right)$ is r-$\mathcal{F}$-$\gamma$-closed. Thus,

$$\gamma C_{\Im }(\mathcal{P},r)\le \gamma C_{\Im }(h^{-1}\left(h\left(\mathcal{P}\right)\right),r)\le \gamma C_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),r)\right),r)=h^{-1}\left(C_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),r)\right).$$

Therefore, $h\left(\gamma C_{\Im }(\mathcal{P},r)\right)\le C_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),r)$.

(3) ⇒ (4) Let $\mathcal{D}\in I^N$. By (3), $h\left(\gamma C_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\right)\le C_{\mathsf{Ϝ}}(h\left(h^{-1}\left(\mathcal{D}\right)\right),r)$$\le C_{\mathsf{Ϝ}}(\mathcal{D},r)$. Thus, $\gamma C_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\le h^{-1}\left(h\left(\gamma C_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\right)\right)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right)$.

(4) ⇔ (5) The proof follows by $h^{-1}\left({\mathcal{D}}^c\right)={\left(h^{-1}\left(\mathcal{D}\right)\right)}^c$ and Proposition 3.3.

(5) ⇒ (1) Let $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$. By (5), we obtain $h^{-1}\left(\mathcal{D}\right)=h^{-1}\left(I_{\mathsf{Ϝ}}(\mathcal{D},r)\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\le h^{-1}\left(\mathcal{D}\right)$. Then, $\gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)=h^{-1}\left(\mathcal{D}\right)$. Thus, $h^{-1}\left(\mathcal{D}\right)$ is r-$\mathcal{F}$-$\gamma$-open, so h is $\mathcal{F}$-$\gamma$-continuous. □

Lemma 4.1.

Every $\mathcal{F}$-$\gamma$-irresolute function is $\mathcal{F}$-$\gamma$-continuous.

Proof. 

The proof follows by Definition 4.1. □

Remark 4.3.

The converse of Lemma 4.1 fails as Example 4.4.

Example 4.4.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{P}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.5}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.4}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{C}=\mathcal{P},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-$\gamma$-continuous, but it is not $\mathcal{F}$-$\gamma$-irresolute.

Theorem 4.3.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function and $r\in I_{\circ }$, the following statements are equivalent for every $\mathcal{D}\in I^M$ and $\mathcal{P}\in I^N$:

(1) h is $\mathcal{F}$-$\gamma$-irresolute.

(2) $h^{-1}\left(\mathcal{P}\right)$ is r-$\mathcal{F}$-$\gamma$-closed, for every r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{P}$.

(3) $h\left(\gamma C_{\Im }(\mathcal{D},r)\right)\le \gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{D}\right),r)$.

(4) $\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\le h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$.

(5) $h^{-1}\left(\gamma I_{\mathsf{Ϝ}}(\mathcal{P},r)\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{P}\right),r)$.

Proof. 

(1) ⇔ (2) The proof follows by $h^{-1}\left({\mathcal{P}}^c\right)={\left(h^{-1}\left(\mathcal{P}\right)\right)}^c$ and Definition 4.1.

(2) ⇒ (3) Let $\mathcal{D}\in I^M$. By (2), we have $h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{D}\right),r)\right)$ is r-$\mathcal{F}$-$\gamma$-closed. Thus,

$$\gamma C_{\Im }(\mathcal{D},r)\le \gamma C_{\Im }(h^{-1}\left(h\left(\mathcal{D}\right)\right),r)\le \gamma C_{\Im }(h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{D}\right),r)\right),r)=h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{D}\right),r)\right).$$

Therefore, $h\left(\gamma C_{\Im }(\mathcal{D},r)\right)\le \gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{D}\right),r)$.

(3) ⇒ (4) Let $\mathcal{P}\in I^N$. By (3), $h\left(\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\right)\le \gamma C_{\mathsf{Ϝ}}(h\left(h^{-1}\left(\mathcal{P}\right)\right),r)$$\le \gamma C_{\mathsf{Ϝ}}(\mathcal{P},r)$. Thus, $\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\le h^{-1}\left(h\left(\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\right)\right)\le h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$.

(4) ⇔ (5) The proof follows by $h^{-1}\left({\mathcal{P}}^c\right)={\left(h^{-1}\left(\mathcal{P}\right)\right)}^c$ and Proposition 3.3.

(5) ⇒ (1) Let $\mathcal{P}\in I^N$ be an r-$\mathcal{F}$-$\gamma$-open set. By (5),

$$h^{-1}\left(\mathcal{P}\right)=h^{-1}\left(\gamma I_{\mathsf{Ϝ}}(\mathcal{P},r)\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\le h^{-1}\left(\mathcal{P}\right).$$

Thus, $\gamma I_{\Im }(h^{-1}\left(\mathcal{P}\right),r)=h^{-1}\left(\mathcal{P}\right)$. Therefore, $h^{-1}\left(\mathcal{P}\right)$ is r-$\mathcal{F}$-$\gamma$-open, so h is $\mathcal{F}$-$\gamma$-irresolute. □

Proposition 4.1.

Let $(M,\Im )$, $(W,\eta )$ and $(N,\mathsf{Ϝ})$ be $\mathcal{FTS}s$, and $h:(M,\Im )⟶(W,\eta )$, $f:(W,\eta )⟶(N,\mathsf{Ϝ})$ be two $\mathcal{F}$-functions. The composition $f\circ h$ is $\mathcal{F}$-$\gamma$-irresolute (resp., $\mathcal{F}$-$\gamma$-continuous) if, h is $\mathcal{F}$-$\gamma$-irresolute and f is $\mathcal{F}$-$\gamma$-irresolute (resp., $\mathcal{F}$-$\gamma$-continuous).

Proof. 

The proof follows from Definition 4.1. □

Definition 4.2.

An $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is called $\mathcal{F}$-almost $\gamma$-continuous if $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)\right),r)$, for every $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$.

Lemma 4.2.

Every $\mathcal{F}$-$\gamma$-continuous function is $\mathcal{F}$-almost $\gamma$-continuous.

Proof. 

The proof follows by Definitions 4.1 and 4.2. □

Remark 4.4.

The converse of Lemma 4.2 fails as Example 4.5.

Example 4.5.

Let $M=\{m_1,m_2,m_3\}$ and define $\mathcal{D},\mathcal{P},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.4}},{\displaystyle \frac{m_2}{0.2}},{\displaystyle \frac{m_3}{0.4}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.5}},{\displaystyle \frac{m_3}{0.4}}\},$$\mathcal{V}=\{{\displaystyle \frac{m_1}{0.3}},{\displaystyle \frac{m_2}{0.2}},{\displaystyle \frac{m_3}{0.6}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{U}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{U}\in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{2}{3}},\mathrm{if}\mathcal{U}=\mathcal{D},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{U}=\mathcal{P},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{U}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{U}\in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{U}=\mathcal{V},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-almost $\gamma$-continuous, but it is not $\mathcal{F}$-$\gamma$-continuous.

Theorem 4.4.

An $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is $\mathcal{F}$-almost $\gamma$-continuous iff for any $m_{\sigma }\in P_{\sigma }\left(M\right)$ and any $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ containing $h\left(m_{\sigma }\right)$, there is $\mathcal{A}\in I^M$ that is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)$.

Proof. 

(⇒) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ containing $h\left(m_{\sigma }\right)$, then $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)\right),r)$. Since $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$, then $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)\right),r)=\mathcal{A}$ (say). Therefore, $\mathcal{A}\in I^M$ is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)$.

(⇐) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ such that $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$. According to the assumption there is $\mathcal{A}\in I^M$ that is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)$. Hence, $m_{\sigma }\in \mathcal{A}\le h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)\right)$ and $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)\right),r)$. Thus, $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{D},r),r)\right),r)$. Therefore, h is $\mathcal{F}$-almost $\gamma$-continuous.

□

Theorem 4.5.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function, $\mathcal{P}\in I^N$, and $r\in I_{\circ }$, the following statements are equivalent:

(1) h is $\mathcal{F}$-almost $\gamma$-continuous.

(2) $h^{-1}\left(\mathcal{P}\right)$ is r-$\mathcal{F}$-$\gamma$-open, for every r-$\mathcal{F}$-regularly open set $\mathcal{P}$.

(3) $h^{-1}\left(\mathcal{P}\right)$ is r-$\mathcal{F}$-$\gamma$-closed, for every r-$\mathcal{F}$-regularly closed set $\mathcal{P}$.

(4) $\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$, for every r-$\mathcal{F}$-$\gamma$-open set $\mathcal{P}$.

(5) $\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$, for every r-$\mathcal{F}$-semi-open set $\mathcal{P}$.

Proof. 

(1) ⇒ (2) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{P}$ be an r-$\mathcal{F}$-regularly open set with $m_{\sigma }\in h^{-1}\left(\mathcal{P}\right)$. Hence, by (1), there is $\mathcal{A}\in I^M$ that is an r-$\mathcal{F}$-$\gamma$-open with $m_{\sigma }\in \mathcal{A}$ and $h\left(\mathcal{A}\right)\le I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},r),r)$. Thus, $\mathcal{A}\le h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},r),r)\right)=h^{-1}\left(\mathcal{P}\right)$ and $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(\mathcal{P}\right),r)$. Therefore, $h^{-1}\left(\mathcal{P}\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{P}\right),r)$, so $h^{-1}\left(\mathcal{P}\right)$ is r-$\mathcal{F}$-$\gamma$-open.

(2) ⇒ (3) If $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-regularly closed, then by (2), $h^{-1}\left({\mathcal{P}}^c\right)={\left(h^{-1}\left(\mathcal{P}\right)\right)}^c$ is r-$\mathcal{F}$-$\gamma$-open. Thus, $h^{-1}\left(\mathcal{P}\right)$ is r-$\mathcal{F}$-$\gamma$-closed.

(3) ⇒ (4) If $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-b-open and since $C_{\mathsf{Ϝ}}(\mathcal{P},r)$ is r-$\mathcal{F}$-regularly closed, then by (3), $h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$ is r-$\mathcal{F}$-$\gamma$-closed. Since $h^{-1}\left(\mathcal{P}\right)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$, hence

$$\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right).$$

(4) ⇒ (5) The proof follows from the fact that any r-$\mathcal{F}$-semi-open set is r-$\mathcal{F}$-$\gamma$-open set.

(5) ⇒ (3) If $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-regularly closed, then $\mathcal{P}$ is r-$\mathcal{F}$-semi-open. By (5), $\gamma C_{\Im }(h^{-1}\left(\mathcal{P}\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)=h^{-1}\left(\mathcal{P}\right)$. Hence, $h^{-1}\left(\mathcal{P}\right)$ is r-$\mathcal{F}$-b-closed.

(3) ⇒ (1) If $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{P}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{P}\right)\ge r$ such that $m_{\sigma }\in h^{-1}\left(\mathcal{P}\right)$, then $m_{\sigma }\in h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},r),r)\right)$. Since ${\left[I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},r),r)\right]}^c$ is r-$\mathcal{F}$-regularly closed, then by (3), we have $h^{-1}\left({\left[I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},r),r)\right]}^c\right)$ is r-$\mathcal{F}$-$\gamma$-closed. Hence, $h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}\left(\mathcal{P}\right),r)\right)$ is r-$\mathcal{F}$-$\gamma$-open and

$$m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},r),r)\right),r).$$

Thus, $h^{-1}\left(\mathcal{P}\right)\le \gamma I_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},r),r)\right),r)$. Therefore, h is $\mathcal{F}$-almost $\gamma$-continuous.

□

Definition 4.3.

An $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is called $\mathcal{F}$-weakly $\gamma$-continuous if $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right),r)$, for every $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$.

Lemma 4.3.

Every $\mathcal{F}$-$\gamma$-continuous function is $\mathcal{F}$-weakly $\gamma$-continuous.

Proof. 

The proof follows by Definitions 4.1 and 4.3. □

Remark 4.5.

The converse of Lemma 4.3 fails as Example 4.6.

Example 4.6.

Let $M=\{m_1,m_2,m_3\}$ and define $\mathcal{D},\mathcal{P},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.4}},{\displaystyle \frac{m_2}{0.2}},{\displaystyle \frac{m_3}{0.4}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.5}},{\displaystyle \frac{m_3}{0.4}}\},$ $\mathcal{V}=\{{\displaystyle \frac{m_1}{0.3}},{\displaystyle \frac{m_2}{0.2}},{\displaystyle \frac{m_3}{0.6}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{U}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\mathcal{U}=\mathcal{D},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{U}=\mathcal{P},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{U}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\mathcal{U}=\mathcal{V},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-weakly $\gamma$-continuous, but it is not $\mathcal{F}$-$\gamma$-continuous.

Theorem 4.6.

An $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is $\mathcal{F}$-weakly $\gamma$-continuous iff for any $m_{\sigma }\in P_{\sigma }\left(M\right)$ and any $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ containing $h\left(m_{\sigma }\right)$, there is $\mathcal{A}\in I^M$ that is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le C_{\mathsf{Ϝ}}(\mathcal{D},r)$.

Proof. 

(⇒) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ containing $h\left(m_{\sigma }\right)$, then $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right),r)$. Since $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$, then $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right),r)=\mathcal{A}$ (say). Hence, $\mathcal{A}\in I^M$ is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le C_{\mathsf{Ϝ}}(\mathcal{D},r)$.

(⇐) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{D}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge r$ such that $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$. According to the assumption there is $\mathcal{A}\in I^M$ that is r-$\mathcal{F}$-$\gamma$-open containing $m_{\sigma }$ with $h\left(\mathcal{A}\right)\le C_{\mathsf{Ϝ}}(\mathcal{D},r)$. Hence, $m_{\sigma }\in \mathcal{A}\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right)$ and $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right),r)$. Thus, $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{D},r)\right),r)$. Therefore, h is $\mathcal{F}$-weakly $\gamma$-continuous.

□

Theorem 4.7.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function, the following statements are equivalent:

(1) h is $\mathcal{F}$-weakly $\gamma$-continuous.

(2) $h^{-1}\left(\mathcal{P}\right)\ge \gamma C_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(\mathcal{P},r)\right),r)$, if $\mathcal{P}\in I^N$ with $\mathsf{Ϝ}\left({\mathcal{P}}^c\right)\ge r$.

(3) $\gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right),r)\ge h^{-1}\left(I_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$.

(4) $\gamma C_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(\mathcal{P},r)\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right)$.

Proof. 

(1) ⇔ (2) The proof follows by Proposition 3.3 and Definition 4.3.

(2) ⇒ (3) Let $\mathcal{P}\in I^N$. Hence by (2),

$$\gamma C_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}({\mathcal{P}}^c,r),r)\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}({\mathcal{P}}^c,r)\right).$$

Thus, $h^{-1}\left(I_{\mathsf{Ϝ}}(\mathcal{P},r)\right)\le \gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right),r)$.

(3) ⇔ (4) The proof follows from Proposition 3.3.

(4) ⇒ (1) Let $\mathcal{P}\in I^N$ with $\mathsf{Ϝ}\left(\mathcal{P}\right)\ge r$. Hence by (4), $\gamma C_{\Im }(h^{-1}\left(I_{\mathsf{Ϝ}}({\mathcal{P}}^c,r)\right),r)\le h^{-1}\left(C_{\mathsf{Ϝ}}({\mathcal{P}}^c,r)\right)=h^{-1}\left({\mathcal{P}}^c\right)$. Thus, $h^{-1}\left(\mathcal{P}\right)\le \gamma I_{\Im }(h^{-1}\left(C_{\mathsf{Ϝ}}(\mathcal{P},r)\right),r)$, so h is $\mathcal{F}$-weakly $\gamma$-continuous. □

Lemma 4.4.

Every $\mathcal{F}$-almost $\gamma$-continuous function is $\mathcal{F}$-weakly $\gamma$-continuous.

Proof. 

The proof follows by Definitions 4.2 and 4.3. □

Remark 4.6.

The converse of Lemma 4.4 fails as Example 4.7.

Example 4.7.

Let $M=\{m_1,m_2,m_3\}$ and define $\mathcal{D},\mathcal{P},\mathcal{V}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.6}},{\displaystyle \frac{m_2}{0.2}},{\displaystyle \frac{m_3}{0.4}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.3}},{\displaystyle \frac{m_2}{0.2}},{\displaystyle \frac{m_3}{0.5}}\},$$\mathcal{V}=\{{\displaystyle \frac{m_1}{0.3}},{\displaystyle \frac{m_2}{0.2}},{\displaystyle \frac{m_3}{0.4}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{U}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\mathcal{U}=\mathcal{D},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mathcal{U}=\mathcal{V},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{U}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{U}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\mathrm{if}\mathcal{U}=\mathcal{P},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-weakly $\gamma$-continuous, but it is not $\mathcal{F}$-almost $\gamma$-continuous.

Remark 4.7.

From the previous discussions and definitions, we have the following diagram.

$$\mathcal{F}\text{-}\mathbf{fl}\text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\mathbf{almost}\mathbf{fl}\text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\mathbf{weakly}\mathbf{fl}\text{-}\mathbf{continuity}$$

Proposition 4.2.

Let $(M,\Im )$, $(W,\eta )$ and $(N,\mathsf{Ϝ})$ be $\mathcal{FTS}s$, and $h:(M,\Im )⟶(W,\eta )$, $g:(W,\eta )⟶(N,\mathsf{Ϝ})$ be two $\mathcal{F}$-functions. The composition $g\circ h$ is $\mathcal{F}$-almost $\gamma$-continuous if, h is $\mathcal{F}$-$\gamma$-irresolute (resp., $\mathcal{F}$-$\gamma$-continuous) and g is $\mathcal{F}$-almost $\gamma$-continuous (resp., $\mathcal{F}$-continuous).

Proof. 

The proof follows by the previous definitions. □

5. Further Selected Topics

Here, we introduce and establish some new $\mathcal{F}$-functions using r-$\mathcal{F}$-$\gamma$-open and r-$\mathcal{F}$-$\gamma$-closed sets, called $\mathcal{F}$-$\gamma$-open (resp. $\mathcal{F}$-$\gamma$-irresolute open, $\mathcal{F}$-$\gamma$-closed, $\mathcal{F}$-$\gamma$-irresolute closed, and $\mathcal{F}$-$\gamma$-irresolute homeomorphism) functions. Furthermore, we define some new types of $\mathcal{F}$-separation axioms, called r-$\mathcal{F}$-$\gamma$-regular and r-$\mathcal{F}$-$\gamma$-normal spaces, and study some properties of them. Also, we explore and discuss some new types of $\mathcal{F}$-compactness, called r-$\mathcal{F}$-almost and r-$\mathcal{F}$-nearly $\gamma$-compact sets using r-$\mathcal{F}$-$\gamma$-open sets.

• Some new fuzzy functions:

Definition 5.1.

An $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is called

(1) $\mathcal{F}$-$\gamma$-open if $h\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-$\gamma$-open set, for every $\mathcal{D}\in I^M$ with $\Im \left(\mathcal{D}\right)\ge r$;

(2) $\mathcal{F}$-$\gamma$-closed if $h\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-$\gamma$-closed set, for every $\mathcal{D}\in I^M$ with $\Im \left({\mathcal{D}}^c\right)\ge r$;

(3) $\mathcal{F}$-$\gamma$-irresolute open if $h\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-$\gamma$-open set, for every r-$\mathcal{F}$-$\gamma$-open set $\mathcal{D}\in I^M$;

(4) $\mathcal{F}$-$\gamma$-irresolute closed if $h\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-$\gamma$-closed set, for every r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{D}\in I^M$.

Lemma 5.1.

(1) Each $\mathcal{F}$-$\gamma$-irresolute open function is $\mathcal{F}$-$\gamma$-open.

(2) Each $\mathcal{F}$-$\gamma$-irresolute closed function is $\mathcal{F}$-$\gamma$-closed.

Proof. 

The proof follows from Definition 5.1. □

Remark 5.1.

The converse of Lemma 5.1 fails as Example 5.1.

Example 5.1.

Let $M=\{m_1,m_2\}$ and define $\mathcal{D},\mathcal{P}\in I^M$ as follows: $\mathcal{D}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.5}}\}$, $\mathcal{P}=\{{\displaystyle \frac{m_1}{0.5}},{\displaystyle \frac{m_2}{0.4}}\}$. Define $\mathcal{F}$-topologies $\Im ,\eta :I^M⟶I$ as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\mathrm{if}\mathcal{C}=\mathcal{D},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\eta \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\mathrm{if}\mathcal{C}=\mathcal{P},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity $\mathcal{F}$-function $f:(M,\Im )⟶(M,\eta )$ is $\mathcal{F}$-$\gamma$-open, but it is not $\mathcal{F}$-$\gamma$-irresolute open.

Theorem 5.1.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function, the following statements are equivalent for every $\mathcal{A}\in I^M$ and $\mathcal{D}\in I^N$:

(1) h is $\mathcal{F}$-$\gamma$-open.

(2) $h\left(I_{\Im }(\mathcal{A},r)\right)\le \gamma I_{\mathsf{Ϝ}}(h\left(\mathcal{A}\right),r)$.

(3) $I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\le h^{-1}\left(\gamma I_{\mathsf{Ϝ}}(\mathcal{D},r)\right)$.

(4) For every $\mathcal{D}$ and every $\mathcal{A}$ with $\Im \left({\mathcal{A}}^c\right)\ge r$ and $h^{-1}\left(\mathcal{D}\right)\le \mathcal{A}$, there is $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-$\gamma$-closed with $\mathcal{D}\le \mathcal{P}$ and $h^{-1}\left(\mathcal{P}\right)\le \mathcal{A}$.

Proof. 

(1) ⇒ (2) Since $h\left(I_{\Im }(\mathcal{A},r)\right)\le h\left(\mathcal{A}\right)$, hence by (1), $h\left(I_{\Im }(\mathcal{A},r)\right)$ is r-$\mathcal{F}$-$\gamma$-open. Thus,

$$h\left(I_{\Im }(\mathcal{A},r)\right)\le \gamma I_{\mathsf{Ϝ}}(h\left(\mathcal{A}\right),r).$$

(2) ⇒ (3) Put $\mathcal{A}=h^{-1}\left(\mathcal{D}\right)$, hence by (2), $h\left(I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\right)\le \gamma I_{\mathsf{Ϝ}}(h\left(h^{-1}\left(\mathcal{D}\right)\right),r)\le \gamma I_{\mathsf{Ϝ}}(\mathcal{D},r)$. Thus, $I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\le h^{-1}\left(\gamma I_{\mathsf{Ϝ}}(\mathcal{D},r)\right).$

(3) ⇒ (4) Let $\mathcal{D}\in I^N$ and $\mathcal{A}\in I^M$ with $\Im \left({\mathcal{A}}^c\right)\ge r$ such that $h^{-1}\left(\mathcal{D}\right)\le \mathcal{A}$. Since ${\mathcal{A}}^c\le h^{-1}\left({\mathcal{D}}^c\right)$, ${\mathcal{A}}^c=I_{\Im }({\mathcal{A}}^c,r)\le I_{\Im }(h^{-1}\left({\mathcal{D}}^c\right),r)$. Hence by (3), ${\mathcal{A}}^c\le I_{\Im }(h^{-1}\left({\mathcal{D}}^c\right),r)\le h^{-1}\left(\gamma I_{\mathsf{Ϝ}}({\mathcal{D}}^c,r)\right)$. Then, we have $\mathcal{A}\ge {\left(h^{-1}\left(\gamma I_{\mathsf{Ϝ}}({\mathcal{D}}^c,r)\right)\right)}^c=h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{D},r)\right).$ Thus, there is $\gamma C_{\mathsf{Ϝ}}(\mathcal{D},r)\in I^N$ is r-$\mathcal{F}$-$\gamma$-closed with $\mathcal{D}\le \gamma C_{\mathsf{Ϝ}}(\mathcal{D},r)$ and $h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{D},r)\right)\le \mathcal{A}.$

(4) ⇒ (1) Let $\mathcal{B}\in I^M$ with $\Im \left(\mathcal{B}\right)\ge r$. Put $\mathcal{D}={\left(h\left(\mathcal{B}\right)\right)}^c$ and $\mathcal{A}={\mathcal{B}}^c$, then $h^{-1}\left(\mathcal{D}\right)=h^{-1}\left({\left(h\left(\mathcal{B}\right)\right)}^c\right)\le \mathcal{A}$. Hence by (4), there is $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-$\gamma$-closed with $\mathcal{D}\le \mathcal{P}$ and $h^{-1}\left(\mathcal{P}\right)\le \mathcal{A}={\mathcal{B}}^c$. Thus, $h\left(\mathcal{B}\right)\le h\left(h^{-1}\left({\mathcal{P}}^c\right)\right)\le {\mathcal{P}}^c$. On the other hand, since $\mathcal{D}\le \mathcal{P}$, $h\left(\mathcal{B}\right)={\mathcal{D}}^c\ge {\mathcal{P}}^c$. Hence, $h\left(\mathcal{B}\right)={\mathcal{P}}^c$, so $h\left(\mathcal{B}\right)$ is an r-$\mathcal{F}$-$\gamma$-open set. Therefore, h is $\mathcal{F}$-$\gamma$-open.

□

Theorem 5.2.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function, the following statements are equivalent for every $\mathcal{B}\in I^M$ and $\mathcal{D}\in I^N$:

(1) h is $\mathcal{F}$-$\gamma$-closed.

(2) $\gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{B}\right),r)\le h\left(C_{\Im }(\mathcal{B},r)\right)$.

(3) $h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{D},r)\right)\le C_{\Im }(h^{-1}\left(\mathcal{D}\right),r)$.

(4) For every $\mathcal{D}$ and every $\mathcal{B}$ with $\Im \left(\mathcal{B}\right)\ge r$ and $h^{-1}\left(\mathcal{D}\right)\le \mathcal{B}$, there is $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-$\gamma$-open with $\mathcal{D}\le \mathcal{P}$ and $h^{-1}\left(\mathcal{P}\right)\le \mathcal{B}$.

Proof. 

The proof is similar to that of Theorem 5.1.

□

Theorem 5.3.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function, the following statements are equivalent for every $\mathcal{B}\in I^M$ and $\mathcal{D}\in I^N$:

(1) h is $\mathcal{F}$-$\gamma$-irresolute open.

(2) $h\left(\gamma I_{\Im }(\mathcal{B},r)\right)\le \gamma I_{\mathsf{Ϝ}}(h\left(\mathcal{B}\right),r)$.

(3) $\gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),r)\le h^{-1}\left(\gamma I_{\mathsf{Ϝ}}(\mathcal{D},r)\right)$.

(4) For every $\mathcal{D}$ and every $\mathcal{B}$ is an r-$\mathcal{F}$-$\gamma$-closed set with $h^{-1}\left(\mathcal{D}\right)\le \mathcal{B}$, there is $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-$\gamma$-closed with $\mathcal{D}\le \mathcal{P}$ and $h^{-1}\left(\mathcal{P}\right)\le \mathcal{B}$.

Proof. 

The proof is similar to that of Theorem 5.1.

□

Theorem 5.4.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function, the following statements are equivalent for every $\mathcal{B}\in I^M$ and $\mathcal{D}\in I^N$:

(1) h is $\mathcal{F}$-$\gamma$-irresolute closed.

(2) $\gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{B}\right),r)\le h\left(\gamma C_{\Im }(\mathcal{B},r)\right)$.

(3) $h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{D},r)\right)\le \gamma C_{\Im }(h^{-1}\left(\mathcal{D}\right),r)$.

(4) For every $\mathcal{D}$ and every $\mathcal{B}$ is an r-$\mathcal{F}$-$\gamma$-open set with $h^{-1}\left(\mathcal{D}\right)\le \mathcal{B}$, there is $\mathcal{P}\in I^N$ is r-$\mathcal{F}$-$\gamma$-open with $\mathcal{D}\le \mathcal{P}$ and $h^{-1}\left(\mathcal{P}\right)\le \mathcal{B}$.

Proof. 

The proof is similar to that of Theorem 5.1.

□

Proposition 5.1.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be a bijective $\mathcal{F}$-function, then h is $\mathcal{F}$-$\gamma$-irresolute open iff h is $\mathcal{F}$-$\gamma$-irresolute closed.

Proof. 

The proof follows from;

$$h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{V},r)\right)\le \gamma C_{\Im }(h^{-1}\left(\mathcal{V}\right),r)⟺h^{-1}\left(\gamma I_{\mathsf{Ϝ}}({\mathcal{V}}^c,r)\right)\le \gamma I_{\Im }(h^{-1}\left({\mathcal{V}}^c\right),r).$$

□

Definition 5.2.

A bijective $\mathcal{F}$-function $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ is called $\mathcal{F}$-$\gamma$-irresolute homeomorphism if $h^{-1}$ and h are $\mathcal{F}$-$\gamma$-irresolute.

The proof of the following corollary is easy.

Corollary 5.1.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be a bijective $\mathcal{F}$-function, the following statements are equivalent for every $\mathcal{P}\in I^M$ and $\mathcal{V}\in I^N$:

(1) h is $\mathcal{F}$-$\gamma$-irresolute homeomorphism.

(2) h is $\mathcal{F}$-$\gamma$-irresolute closed and $\mathcal{F}$-$\gamma$-irresolute.

(3) h is $\mathcal{F}$-$\gamma$-irresolute open and $\mathcal{F}$-$\gamma$-irresolute.

(4) $h\left(\gamma I_{\Im }(\mathcal{P},r)\right)=\gamma I_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),r)$.

(5) $h\left(\gamma C_{\Im }(\mathcal{P},r)\right)=\gamma C_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),r)$.

(6) $\gamma I_{\Im }(h^{-1}\left(\mathcal{V}\right),r)=h^{-1}\left(\gamma I_{\mathsf{Ϝ}}(\mathcal{V},r)\right)$.

(7) $\gamma C_{\Im }(h^{-1}\left(\mathcal{V}\right),r)=h^{-1}\left(\gamma C_{\mathsf{Ϝ}}(\mathcal{V},r)\right)$.

• r-fuzzy $\gamma$-regular and $\gamma$-normal spaces:

Definition 5.3.

Let $m_{\sigma }\in P_{\sigma }\left(M\right)$, $\mathcal{A},\mathcal{B}\in I^M$, and $r\in I_{\circ }$. An $\mathcal{FTS}$$(M,\Im )$ is called

(1) r-$\mathcal{F}$-$\gamma$-regular space if $m_{\sigma }\overline{q}\mathcal{A}$ for each r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{A}$, there is ${\mathcal{C}}_j\in I^M$ with $\Im \left({\mathcal{C}}_j\right)\ge r$ for $j\in \{1,2\}$, such that $m_{\sigma }\in {\mathcal{C}}_1$, $\mathcal{A}\le {\mathcal{C}}_2$, and ${\mathcal{C}}_1\overline{q}{\mathcal{C}}_2$.

(2) r-$\mathcal{F}$-$\gamma$-normal space if $\mathcal{A}\overline{q}\mathcal{B}$ for each r-$\mathcal{F}$-$\gamma$-closed sets $\mathcal{A}$ and $\mathcal{B}$, there is ${\mathcal{C}}_j\in I^M$ with $\Im \left({\mathcal{C}}_j\right)\ge r$ for $j\in \{1,2\}$, such that $\mathcal{A}\le {\mathcal{C}}_1$, $\mathcal{B}\le {\mathcal{C}}_2$, and ${\mathcal{C}}_1\overline{q}{\mathcal{C}}_2$.

Theorem 5.5.

Let $(M,\Im )$ be an $\mathcal{FTS}$, $m_{\sigma }\in P_{\sigma }\left(M\right)$, $\mathcal{A},\mathcal{P}\in I^M$, the following statements are equivalent:

(1) $(M,\Im )$ is r-$\mathcal{F}$-$\gamma$-regular space.

(2) If $m_{\sigma }\in \mathcal{A}$ for each r-$\mathcal{F}$-$\gamma$-open set $\mathcal{A}$, there is $\mathcal{P}$ with $\Im \left(\mathcal{P}\right)\ge r$ and $m_{\sigma }\in \mathcal{P}\le C_{\Im }(\mathcal{P},r)\le \mathcal{A}.$

(3) If $m_{\sigma }\overline{q}\mathcal{A}$ for each r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{A}$, there is ${\mathcal{D}}_j\in I^M$ with $\Im \left({\mathcal{D}}_j\right)\ge r$ for $j\in \{1,2\}$, such that $m_{\sigma }\in {\mathcal{D}}_1$, $\mathcal{A}\le {\mathcal{D}}_2$, and $C_{\Im }({\mathcal{D}}_1,r)\overline{q}{C}_{\Im }({\mathcal{D}}_2,r)$.

Proof. 

(1) ⇒ (2) Let $m_{\sigma }\in \mathcal{A}$ for each r-$\mathcal{F}$-$\gamma$-open set $\mathcal{A}$, then $m_{\sigma }\overline{q}{\mathcal{A}}^c$. Since $(M,\Im )$ is r-$\mathcal{F}$-$\gamma$-regular, then there is $\mathcal{P},\mathcal{D}\in I^M$ with $\Im \left(\mathcal{P}\right)\ge r$ and $\Im \left(\mathcal{D}\right)\ge r$, such that $m_{\sigma }\in \mathcal{P}$, ${\mathcal{A}}^c\le \mathcal{D}$, and $\mathcal{P}\overline{q}\mathcal{D}$. Thus, $m_{\sigma }\in \mathcal{P}\le {\mathcal{D}}^c\le \mathcal{A}$, so $m_{\sigma }\in \mathcal{P}\le C_{\Im }(\mathcal{P},r)\le \mathcal{A}$.

(2) ⇒ (3) Let $m_{\sigma }\overline{q}\mathcal{A}$ for each r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{A}$, then $m_{\sigma }\in {\mathcal{A}}^c$. By (2), there is $\mathcal{D}$ with $\Im \left(\mathcal{D}\right)\ge r$ and $m_{\sigma }\in \mathcal{D}\le C_{\Im }(\mathcal{D},r)\le {\mathcal{A}}^c$. Since $\Im \left(\mathcal{D}\right)\ge r$, then $\mathcal{D}$ is an r-$\mathcal{F}$-$\gamma$-open set and $m_{\sigma }\in \mathcal{D}$. Again, by (2), there is $\mathcal{U}$ with $\Im \left(\mathcal{U}\right)\ge r$ and $m_{\sigma }\in \mathcal{U}\le C_{\Im }(\mathcal{U},r)\le \mathcal{D}\le C_{\Im }(\mathcal{D},r)\le {\mathcal{A}}^c$. Hence, $\mathcal{A}\le {\left(C_{\Im }(\mathcal{D},r)\right)}^c=I_{\Im }({\mathcal{D}}^c,r)\le {\mathcal{D}}^c$. Put $\mathcal{V}=I_{\Im }({\mathcal{D}}^c,r)$, thus $\Im \left(\mathcal{V}\right)\ge r$. Then, $C_{\Im }(\mathcal{V},r)\le {\mathcal{D}}^c\le {\left(C_{\Im }(\mathcal{U},r)\right)}^c$. Therefore, $C_{\Im }(\mathcal{V},r)\overline{q}{C}_{\Im }(\mathcal{U},r)$.

(3) ⇒ (1) Easily proved by Definition 5.3. □

Theorem 5.6.

Let $(M,\Im )$ be an $\mathcal{FTS}$, $m_{\sigma }\in P_{\sigma }\left(M\right)$, $\mathcal{A},\mathcal{B}\in I^M$, the following statements are equivalent:

(1) $(M,\Im )$ is r-$\mathcal{F}$-$\gamma$-normal space.

(2) If $\mathcal{B}\le \mathcal{A}$ for each r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{B}$ and r-$\mathcal{F}$-$\gamma$-open set $\mathcal{A}$, there is $\mathcal{D}$ with $\Im \left(\mathcal{D}\right)\ge r$ and $\mathcal{B}\le \mathcal{D}\le C_{\Im }(\mathcal{D},r)\le \mathcal{A}$.

(3) If $\mathcal{A}\overline{q}\mathcal{B}$ for each r-$\mathcal{F}$-$\gamma$-closed sets $\mathcal{A}$ and $\mathcal{B}$, there is ${\mathcal{D}}_j\in I^M$ with $\Im \left({\mathcal{D}}_j\right)\ge r$ for $j\in \{1,2\}$, such that $\mathcal{A}\le {\mathcal{D}}_1$, $\mathcal{B}\le {\mathcal{D}}_2$, and $C_{\Im }({\mathcal{D}}_1,r)\overline{q}{C}_{\Im }({\mathcal{D}}_2,r)$.

Proof. 

The proof is similar to that of Theorem 5.5.

□

Theorem 5.7.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be a bijective $\mathcal{F}$-$\gamma$-irresolute and $\mathcal{F}$-open function. If $(M,\Im )$ is an r-$\mathcal{F}$-$\gamma$-regular space (resp., r-$\mathcal{F}$-$\gamma$-normal space), then $(N,\mathsf{Ϝ})$ is an r-$\mathcal{F}$-$\gamma$-regular space (resp., r-$\mathcal{F}$-$\gamma$-normal space).

Proof. 

If $n_{\sigma }\overline{q}\mathcal{B}$ for each r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{B}\in I^N$ and $\mathcal{F}$-$\gamma$-irresolute function h, then $h^{-1}\left(\mathcal{B}\right)$ is an r-$\mathcal{F}$-$\gamma$-closed set. Put $n_{\sigma }=h\left(m_{\sigma }\right)$, then $m_{\sigma }\overline{q}{h}^{-1}\left(\mathcal{B}\right)$. Since $(M,\Im )$ is r-$\mathcal{F}$-$\gamma$-regular, there is ${\mathcal{D}}_1,D_2\in I^M$ with $\Im \left({\mathcal{D}}_1\right)\ge r$ and $\Im \left({\mathcal{D}}_2\right)\ge r$ such that $m_{\sigma }\in {\mathcal{D}}_1$, $h^{-1}\left(\mathcal{B}\right)\le {\mathcal{D}}_2$, and ${\mathcal{D}}_1\overline{q}{\mathcal{D}}_2$. Since h is a bijective $\mathcal{F}$-open, hence $n_{\sigma }\in h\left({\mathcal{D}}_1\right),$$\mathcal{B}=h\left(h^{-1}\left(\mathcal{B}\right)\right)\le h\left({\mathcal{D}}_2\right),$ and $h\left({\mathcal{D}}_1\right)\overline{q}h\left({\mathcal{D}}_2\right).$ Therefore, $(N,\mathsf{Ϝ})$ is an r-$\mathcal{F}$-$\gamma$-regular space. The other case also follows similar lines. □

Theorem 5.8.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an injective $\mathcal{F}$-continuous and $\mathcal{F}$-$\gamma$-irresolute closed function. If $(N,\mathsf{Ϝ})$ is an r-$\mathcal{F}$-$\gamma$-regular space (resp., r-$\mathcal{F}$-$\gamma$-normal space), then $(M,\Im )$ is an r-$\mathcal{F}$-$\gamma$-regular space (resp., r-$\mathcal{F}$-$\gamma$-normal space).

Proof. 

If $m_{\sigma }\overline{q}\mathcal{B}$ for each r-$\mathcal{F}$-$\gamma$-closed set $\mathcal{B}\in I^M$ and injective $\mathcal{F}$-$\gamma$-irresolute closed function h, hence $h\left(\mathcal{B}\right)$ is an r-$\mathcal{F}$-$\gamma$-closed set and $h\left(m_{\sigma }\right)\overline{q}h\left(\mathcal{B}\right)$. Since $(N,\mathsf{Ϝ})$ is r-$\mathcal{F}$-$\gamma$-regular, there is ${\mathcal{D}}_1,{\mathcal{D}}_2\in I^N$ with $\mathsf{Ϝ}\left({\mathcal{D}}_1\right)\ge r$ and $\mathsf{Ϝ}\left({\mathcal{D}}_2\right)\ge r$ such that $h\left(m_{\sigma }\right)\in {\mathcal{D}}_1$, $h\left(\mathcal{B}\right)\le {\mathcal{D}}_2$, and ${\mathcal{D}}_1\overline{q}{\mathcal{D}}_2$. Since h is $\mathcal{F}$-continuous, then $m_{\sigma }\in h^{-1}\left({\mathcal{D}}_1\right)$, $\mathcal{B}\le h^{-1}\left({\mathcal{D}}_2\right)$ with $\Im \left(h^{-1}\left({\mathcal{D}}_1\right)\right)\ge r$, $\Im \left(h^{-1}\left({\mathcal{D}}_2\right)\right)\ge r$, and $h^{-1}\left({\mathcal{D}}_1\right)\overline{q}{h}^{-1}\left({\mathcal{D}}_2\right)$. Hence, $(M,\Im )$ is an r-$\mathcal{F}$-$\gamma$-regular space. The other case also follows similar lines.

□

Theorem 5.9.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be a surjective $\mathcal{F}$-$\gamma$-irresolute, $\mathcal{F}$-open, and $\mathcal{F}$-closed function. If $(M,\Im )$ is an r-$\mathcal{F}$-$\gamma$-regular space (resp., r-$\mathcal{F}$-$\gamma$-normal space), then $(N,\mathsf{Ϝ})$ is an r-$\mathcal{F}$-$\gamma$-regular space (resp., r-$\mathcal{F}$-$\gamma$-normal space).

Proof. 

The proof is similar to that of Theorem 5.7.

□

• Several new types of fuzzy compactness:

Definition 5.4.

Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is called r-$\mathcal{F}$-$\gamma$-compact if for each family ${\{{\mathcal{B}}_j\in I^M|{\mathcal{B}}_j\mathrm{is}r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}\}}_{j\in \Omega }$ with $\mathcal{D}\le {\bigvee }_{j\in \Omega }{\mathcal{B}}_j$, there is a finite sub-set ${\Omega }_{\circ }$ of $\Omega$ with $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{\mathcal{B}}_j$.

Lemma 5.2.

In an $\mathcal{FTS}$$(M,\Im )$, every r-$\mathcal{F}$-$\gamma$-compact set is r-$\mathcal{F}$-compact.

Proof. 

The proof follows from Definitions 2.4 and 5.4.

□

Theorem 5.10.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-$\gamma$-continuous function, then $h\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-compact set if $\mathcal{D}\in I^M$ is an r-$\mathcal{F}$-$\gamma$-compact set.

Proof. 

Let ${\{{\mathcal{B}}_j\in I^N|\mathsf{Ϝ}\left({\mathcal{B}}_j\right)\ge r\}}_{j\in \Omega }$ with $h\left(\mathcal{D}\right)\le {\bigvee }_{j\in \Omega }{\mathcal{B}}_j$, then $\{h^{-1}\left({\mathcal{B}}_j\right)\in I^M|h^{-1}\left({\mathcal{B}}_j\right)$ is $r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}$} (by h is $\mathcal{F}$-$\gamma$-continuous) with $\mathcal{D}\le {\bigvee }_{j\in \Omega }{h}^{-1}\left({\mathcal{B}}_j\right)$. Since $\mathcal{D}$ is r-$\mathcal{F}$-$\gamma$-compact, there is a finite sub-set ${\Omega }_{\circ }$ of $\Omega$ with $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{h}^{-1}\left({\mathcal{B}}_j\right)$. Hence, $h\left(\mathcal{D}\right)\le {\bigvee }_{j\in {\Omega }_{\circ }}{\mathcal{B}}_j$. Therefore, $h\left(\mathcal{D}\right)$ is r-$\mathcal{F}$-compact. □

Definition 5.5.

Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is called r-$\mathcal{F}$-almost $\gamma$-compact if for each family ${\{{\mathcal{B}}_j\in I^M|{\mathcal{B}}_j\mathrm{is}r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}\}}_{j\in \Omega }$ with $\mathcal{D}\le {\bigvee }_{j\in \Omega }{\mathcal{B}}_j$, there is a finite sub-set ${\Omega }_{\circ }$ of $\Omega$ with $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{C}_{\Im }({\mathcal{B}}_j,r)$.

Lemma 5.3.

In an $\mathcal{FTS}$$(M,\Im )$, every r-$\mathcal{F}$-almost $\gamma$-compact set is r-$\mathcal{F}$-almost compact.

Proof. 

The proof follows from Definitions 2.4 and 5.5.

□

Lemma 5.4.

In an $\mathcal{FTS}$$(M,\Im )$, every r-$\mathcal{F}$-$\gamma$-compact set is r-$\mathcal{F}$-almost $\gamma$-compact.

Proof. 

The proof follows from Definitions 5.4 and 5.5.

□

Remark 5.2.

The converse of Lemma 5.4 fails as Example 5.2.

Example 5.2.

Let $W=[0,1]$, $t\in \mathbb{N}-\left\{1\right\}$, and $\mathcal{A},{\mathcal{B}}_t\in I^W$ defined as follows:

$$\mathcal{A}\left(w\right)=\left\{\begin{array}{c}1,\mathrm{if}w=0,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{otherwise},\hfill \end{array}\right.{\mathcal{B}}_t\left(w\right)=\left\{\begin{array}{c}0.8,\mathrm{if}w=0,\hfill \\ tw,\mathrm{if}0<w\le {\displaystyle \frac{1}{t}},\hfill \\ 1,\mathrm{if}{\displaystyle \frac{1}{t}}<w\le 1.\hfill \end{array}\right.$$

Also, ℑ defined on W as follows:

$$\Im \left(\mathcal{C}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{C}\in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{2}{3}},\mathrm{if}\mathcal{C}\le \mathcal{A},\hfill \\ {\displaystyle \frac{t}{t+1}},\mathrm{if}\mathcal{C}\le {\mathcal{B}}_t,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, W is ${\displaystyle \frac{1}{2}}$-$\mathcal{F}$-almost $\gamma$-compact, but it is not ${\displaystyle \frac{1}{2}}$-$\mathcal{F}$-$\gamma$-compact.

Theorem 5.11.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-continuous function, then $h\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-almost compact set if $\mathcal{D}\in I^M$ is an r-$\mathcal{F}$-almost $\gamma$-compact set.

Proof. 

Let ${\{{\mathcal{B}}_j\in I^N|\mathsf{Ϝ}\left({\mathcal{B}}_j\right)\ge r\}}_{j\in \Omega }$ with $h\left(\mathcal{D}\right)\le {\bigvee }_{j\in \Omega }{\mathcal{B}}_j$, then $\{h^{-1}\left({\mathcal{B}}_j\right)\in I^M|h^{-1}\left({\mathcal{B}}_j\right)$ is $r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}$} (by h is $\mathcal{F}$-$\gamma$-continuous), such that $\mathcal{D}\le {\bigvee }_{j\in \Omega }{h}^{-1}\left({\mathcal{B}}_j\right)$. Since $\mathcal{D}$ is r-$\mathcal{F}$-almost $\gamma$-compact, there is a finite sub-set ${\Omega }_{\circ }$ of $\Omega$ with $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{C}_{\Im }(h^{-1}\left({\mathcal{B}}_j\right),r)$. Since h is $\mathcal{F}$-continuous function,

$$\mathcal{D}\le \underset{j\in {\Omega }_{\circ }}{\bigvee }{C}_{\Im }(h^{-1}\left({\mathcal{B}}_j\right),r)$$

$$\le \underset{j\in {\Omega }_{\circ }}{\bigvee }{h}^{-1}\left(C_{\mathsf{Ϝ}}({\mathcal{B}}_j,r)\right)$$

$$=h^{-1}\left(\underset{j\in {\Omega }_{\circ }}{\bigvee }{C}_{\mathsf{Ϝ}}({\mathcal{B}}_j,r)\right).$$

Hence, $h\left(\mathcal{D}\right)\le {\bigvee }_{j\in {\Omega }_{\circ }}{C}_{\mathsf{Ϝ}}({\mathcal{B}}_j,r)$. Therefore, $h\left(\mathcal{D}\right)$ is r-$\mathcal{F}$-almost compact. □

Definition 5.6.

Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $r\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is called r-$\mathcal{F}$-nearly $\gamma$-compact if for each family ${\{{\mathcal{B}}_j\in I^M|{\mathcal{B}}_j\mathrm{is}r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}\}}_{j\in \Omega }$ with $\mathcal{D}\le {\bigvee }_{j\in \Omega }{\mathcal{B}}_j$, there is a finite sub-set ${\Omega }_{\circ }$ of $\Omega$ with $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{I}_{\Im }(C_{\Im }({\mathcal{B}}_j,r),r)$.

Lemma 5.5.

In an $\mathcal{FTS}$$(M,\Im )$, every r-$\mathcal{F}$-nearly $\gamma$-compact set is r-$\mathcal{F}$-nearly compact.

Proof. 

The proof follows from Definitions 2.4 and 5.6.

□

Lemma 5.6.

In an $\mathcal{FTS}$$(M,\Im )$, every r-$\mathcal{F}$-$\gamma$-compact set is r-$\mathcal{F}$-nearly $\gamma$-compact.

Proof. 

The proof follows from Definitions 5.4 and 5.6.

□

Remark 5.3.

The converse of Lemma 5.6 fails as Example 5.3.

Example 5.3.

Let $W=[0,1]$, $0<t<1$, and $\mathcal{A},\mathcal{B},{\mathcal{D}}_t\in I^W$ defined as follows:

$$\mathcal{A}\left(w\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}},\mathrm{if}0\le w<1,\hfill \\ 1,\mathrm{if}w=1,\hfill \end{array}\right.\mathcal{B}\left(w\right)=\left\{\begin{array}{c}1,\mathrm{if}w=0,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}0<w\le 1,\hfill \end{array}\right.$$

$${\mathcal{D}}_t\left(w\right)=\left\{\begin{array}{c}{\displaystyle \frac{w}{t}},\mathrm{if}0\le w<t,\hfill \\ {\displaystyle \frac{1-w}{1-t}},\mathrm{if}t<w\le 1.\hfill \end{array}\right.$$

Also, ℑ defined on W as follows:

$$\Im \left(\mathcal{P}\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathcal{P}\in \{\mathcal{A},\mathcal{B},\underline{1},\underline{0}\},\hfill \\ \max \left(\{1-t,t\}\right),\mathrm{if}\mathcal{P}={\mathcal{D}}_t,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, W is ${\displaystyle \frac{1}{2}}$-$\mathcal{F}$-nearly $\gamma$-compact, but it is not ${\displaystyle \frac{1}{2}}$-$\mathcal{F}$-$\gamma$-compact.

Theorem 5.12.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-continuous and $\mathcal{F}$-open, then $h\left(\mathcal{D}\right)$ is an r-$\mathcal{F}$-nearly compact set if $\mathcal{D}\in I^M$ is an r-$\mathcal{F}$-nearly $\gamma$-compact set.

Proof. 

Let ${\{{\mathcal{B}}_j\in I^N|\mathsf{Ϝ}\left({\mathcal{B}}_j\right)\ge r\}}_{j\in \Omega }$ with $h\left(\mathcal{D}\right)\le {\bigvee }_{j\in \Omega }{\mathcal{B}}_j$, then $\{h^{-1}\left({\mathcal{B}}_j\right)\in I^M|h^{-1}\left({\mathcal{B}}_j\right)$ is $r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}$} (by h is $\mathcal{F}$-$\gamma$-continuous), such that $\mathcal{D}\le {\bigvee }_{j\in \Omega }{h}^{-1}\left({\mathcal{B}}_j\right)$. Since $\mathcal{D}$ is r-$\mathcal{F}$-nearly $\gamma$-compact, there is a finite sub-set ${\Omega }_{\circ }$ of $\Omega$, such that $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{I}_{\Im }(C_{\Im }(h^{-1}\left({\mathcal{B}}_j\right),r),r)$. Since h is $\mathcal{F}$-continuous and $\mathcal{F}$-open,

$$h\left(\mathcal{D}\right)\le \underset{j\in {\Omega }_{\circ }}{\bigvee }h\left(I_{\Im }(C_{\Im }(h^{-1}\left({\mathcal{B}}_j\right),r),r)\right)$$

$$\le \underset{j\in {\Omega }_{\circ }}{\bigvee }{I}_{\mathsf{Ϝ}}(h\left(C_{\Im }(h^{-1}\left({\mathcal{B}}_j\right),r)\right),r)$$

$$\le \underset{j\in {\Omega }_{\circ }}{\bigvee }{I}_{\mathsf{Ϝ}}(h\left(h^{-1}\left(C_{\mathsf{Ϝ}}({\mathcal{B}}_j,r)\right)\right),r)$$

$$\le \underset{j\in {\Omega }_{\circ }}{\bigvee }{I}_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}({\mathcal{B}}_j,r),r).$$

Therefore, $h\left(\mathcal{D}\right)$ is r-$\mathcal{F}$-nearly compact. □

Lemma 5.7.

In an $\mathcal{FTS}$$(M,\Im )$, every r-$\mathcal{F}$-nearly $\gamma$-compact set is r-$\mathcal{F}$-almost $\gamma$-compact.

Proof. 

The proof follows from Definitions 5.5 and 5.6.

□

Remark 5.4.

From the previous discussions and definitions, we have the following diagram.

$$r\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathbf{compact}\mathbf{set}\to r\text{-}\mathcal{F}\text{-}\mathbf{compact}\mathbf{set}$$

$$\downarrow \downarrow$$

$$r\text{-}\mathcal{F}\text{-}\mathbf{nearly}\gamma \text{-}\mathbf{compact}\mathbf{set}\to r\text{-}\mathcal{F}\text{-}\mathbf{nearly}\mathbf{compact}\mathbf{set}$$

$$\downarrow \downarrow$$

$$r\text{-}\mathcal{F}\text{-}\mathbf{almost}\gamma \text{-}\mathbf{compact}\mathbf{set}\to r\text{-}\mathcal{F}\text{-}\mathbf{almost}\mathbf{compact}\mathbf{set}$$

6. Conclusions and Future Works

In the present manuscript, a novel class of $\mathcal{F}$-open sets, called r-$\mathcal{F}$-$\gamma$-open sets has been introduced on $\mathcal{FTS}s$ in Šostak’s sense [6]. Some characterizations of r-$\mathcal{F}$-$\gamma$-open sets along with their mutual relationships have been discussed with the help of some illustrative examples. Furthermore, the notions of $\mathcal{F}$-$\gamma$-interior and $\mathcal{F}$-$\gamma$-closure operators have been defined and investigated. After that, the notions of $\mathcal{F}$-$\gamma$-continuous (resp. $\mathcal{F}$-$\gamma$-irresolute) functions between $\mathcal{FTS}s$$(M,\Im )$ and $(N,\mathsf{Ϝ})$ has been explored and discussed. Moreover, the notions of $\mathcal{F}$-almost (resp. $\mathcal{F}$-weakly) $\gamma$-continuous functions, which are weaker forms of $\mathcal{F}$-$\gamma$-continuous functions have been defined and characterized. We also showed that $\mathcal{F}$-$\gamma$-continuity ⟹$\mathcal{F}$-almost $\gamma$-continuity ⟹$\mathcal{F}$-weakly $\gamma$-continuity, but the converse may not be true. Thereafter, we defined and studied some new $\mathcal{F}$-functions using r-$\mathcal{F}$-$\gamma$-open and r-$\mathcal{F}$-$\gamma$-closed sets, called $\mathcal{F}$-$\gamma$-open (resp. $\mathcal{F}$-$\gamma$-irresolute open, $\mathcal{F}$-$\gamma$-closed, $\mathcal{F}$-$\gamma$-irresolute closed, and $\mathcal{F}$-$\gamma$-irresolute homeomorphism) functions. Also, we introduced and studied some new types of $\mathcal{F}$-separation axioms, called r-$\mathcal{F}$-$\gamma$-regular (resp. r-$\mathcal{F}$-$\gamma$-normal) spaces using r-$\mathcal{F}$-$\gamma$-closed sets. Finally, some new types of $\mathcal{F}$-compactness, called r-$\mathcal{F}$-almost (resp. r-$\mathcal{F}$-nearly) $\gamma$-compact sets have been defined and discussed via r-$\mathcal{F}$-$\gamma$-open sets. In the next works, we intend to the following topics:

• Defining upper (lower) $\gamma$-continuous $\mathcal{F}$-multifunctions and r-$\mathcal{F}$-$\gamma$-connected sets.

• Extending these notions given here to include fuzzy soft topological (r-minimal) spaces [28,33,37].

• Finding a use for these notions given here in the frame of fuzzy ideals as defined in [39-41].

• Introducing the notions as defined in [42-44] by using r-$\mathcal{F}$-$\gamma$-open sets.