Some Characterizations of k-Fuzzy $\gamma$-Open Sets and Fuzzy $\gamma$-Continuity with Further Selected Topics

Abstract

In the present paper, we first introduced the notion of k-fuzzy $\gamma$-open (k-$\mathcal{F}$-$\gamma$-open) sets as a generalized novel class of fuzzy open ($\mathcal{F}$-open) sets on fuzzy topological spaces ($\mathcal{FTS}s$) in the sense of Šostak. The class of k-$\mathcal{F}$-$\gamma$-open sets is contained in the class of k-$\mathcal{F}$-$\beta$-open sets and contains all k-$\mathcal{F}$-semi-open and k-$\mathcal{F}$-pre-open sets. Also, we introduced the closure and interior operators with respect to the classes of k-$\mathcal{F}$-$\gamma$-closed and k-$\mathcal{F}$-$\gamma$-open sets and discussed some of their properties. After that, we defined and studied the notions of $\mathcal{F}$-$\gamma$-continuous (resp. $\mathcal{F}$-$\gamma$-irresolute) functions between $\mathcal{FTS}s$$(M,\Im )$ and $(N,\mathsf{Ϝ})$. However, we displayed and investigated the notions of $\mathcal{F}$-almost (resp. $\mathcal{F}$-weakly) $\gamma$-continuous functions, which are weaker forms of $\mathcal{F}$-$\gamma$-continuous functions. Next, we presented and characterized some new $\mathcal{F}$-functions via k-$\mathcal{F}$-$\gamma$-open and k-$\mathcal{F}$-$\gamma$-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. The relationships between these classes of functions were investigated with the help of some examples. We also introduced some new types of $\mathcal{F}$-separation axioms called k-$\mathcal{F}$-$\gamma$-regular (resp. k-$\mathcal{F}$-$\gamma$-normal) spaces via k-$\mathcal{F}$-$\gamma$-closed sets and discussed some properties of them. Lastly, we explored and studied some new types of $\mathcal{F}$-compactness called k-$\mathcal{F}$-almost (resp. k-$\mathcal{F}$-nearly) $\gamma$-compact sets.

1. Introduction

The need for theories that cope with uncertainty emerges from daily experiences with complicated challenges requiring ambiguous facts. In 1965, the theory of fuzzy sets ($\mathcal{F}$-sets ) was first defined by Zadeh [1] as a suitable approach to address uncertainty cases that cannot be efficiently managed using classical techniques. The concept of an $\mathcal{F}$-set of a nonempty set M is a mapping $\mathcal{D}:M\to I$ (where $I=[0,1]$). Over the last decades, the research on fuzzy sets has had a vital role in mathematics and applied sciences and garnered significant attention due to its ability to handle uncertain and vague information in various real-life applications such as artificial intelligence [2,3], control systems [4,5], decision making [6,7], image processing [8,9], etc. 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 [10,11,12]. The concept of a fuzzy topology ($\mathcal{F}$-topology) was presented in 1968 by Chang [13]. Several authors have successfully generalized the theory of general topology to the fuzzy setting with crisp methods. According to Šostak [14], the notion of an $\mathcal{F}$-topology being a crisp subclass of the class of $\mathcal{F}$-sets and fuzziness in the notion of openness of an $\mathcal{F}$-set have not been considered, which seems to be a drawback in the process of fuzzification of a topological space. Therefore, Šostak [14] defined a novel definition 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 [13]. Thereafter, many researchers (see [15,16,17,18,19,20,21,22,23]) have redefined the same notion and studied $\mathcal{FTS}s$, being unaware of Šostak’s work.

The generalizations of $\mathcal{F}$-open sets play an effective role in an $\mathcal{F}$-topology through their ability to improve on many results and to open the door to display and investigate several fuzzy topological notions such as $\mathcal{F}$-continuity [15,16], $\mathcal{F}$-connectedness [16], $\mathcal{F}$-compactness [16,17], etc. Overall, the notions of k-$\mathcal{F}$-semi-open, k-$\mathcal{F}$-pre-open, k-$\mathcal{F}$-$\alpha$-open, and k-$\mathcal{F}$-$\beta$-open sets were defined and studied by the authors of [20,22] on $\mathcal{FTS}s$ in the sense of Šostak [14]. Also, Kim et al. [20] 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. Furthermore, Abbas [22] 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 [14]. Additionally, Kim and Abbas [23] introduced some types of k-$\mathcal{F}$-compactness on $\mathcal{FTS}s$ in the sense of Šostak.

The notion of fuzzy soft sets ($\mathcal{FS}$-sets) was first presented in 2001 by the author of [24], which combines the $\mathcal{S}$-set [25] 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 [26,27]. Moreover, the notions of k-$\mathcal{FS}$-regularly open, k-$\mathcal{FS}$-pre-open, and k-$\mathcal{FS}$-$\beta$-open sets were introduced by the authors of [28,29] based on the approach developed by Aygünoǧlu et al. [26]. Overall, Taha [30] introduced and discussed the notions of $\mathcal{FS}$-almost (resp. $\mathcal{FS}$-weakly) k-minimal continuity, which are weaker forms of $\mathcal{FS}$-k-minimal continuity [29] based on the approach developed by Aygünoǧlu et al. [26].

We lay out the remainder of this manuscript as follows:

• Section 2 contains some basic definitions that help in understanding the obtained results.
• In Section 3, we define a novel class of $\mathcal{F}$-open sets called k-$\mathcal{F}$-$\gamma$-open sets on $\mathcal{FTS}s$ in the sense of Šostak [14]. This class is contained in the class of k-$\mathcal{F}$-$\beta$-open sets and contains all k-$\mathcal{F}$-$\alpha$-open, k-$\mathcal{F}$-pre-open, and k-$\mathcal{F}$-semi-open sets. Some properties of k-$\mathcal{F}$-$\gamma$-open sets, along with their mutual relationships, are discussed 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.
• In Section 5, we introduce and discuss some novel $\mathcal{F}$-functions using k-$\mathcal{F}$-$\gamma$-open and k-$\mathcal{F}$-$\gamma$-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 present and investigate some new types of $\mathcal{F}$-separation axioms called k-$\mathcal{F}$-$\gamma$-regular (resp. k-$\mathcal{F}$-$\gamma$-normal) spaces. However, we display and discuss some new types of $\mathcal{F}$-compactness called k-$\mathcal{F}$-almost (resp. k-$\mathcal{F}$-nearly) $\gamma$-compact sets.
• In Section 6, we close this paper with conclusions and propose 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 and is 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 $m_{\sigma }$ belongs 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 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 $\mathcal{P}$ ($\mathcal{D}\overline{q}\mathcal{P}$). The following results and notations will be used in the sequel.

Lemma 1

([31]). Let $\mathcal{D},\mathcal{P}\in I^M$. Thus, we have the following:

(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 1

([14,15]). 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 $\mathcal{D},$ and $\mathcal{P}\in I^M.$

(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

([16,19]). In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $k\in I_{\circ }$ (where $I_{\circ }=(0,1]$), we define $\mathcal{F}$-closure and $\mathcal{F}$-interior operators $C_{\Im }$ and $I_{\Im }$$:I^M\times I_{\circ }\to I^M$ as follows:

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

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

Definition 3

([20,22]). Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $k\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is said to be k-$\mathcal{F}$-regularly open (resp. k-$\mathcal{F}$-pre-open, k-$\mathcal{F}$-β-open, k-$\mathcal{F}$-semi-open, k-$\mathcal{F}$-α-open, and k-$\mathcal{F}$-open) if $\mathcal{D}=I_{\Im }(C_{\Im }(\mathcal{D},k),k)$ (resp. $\mathcal{D}\le I_{\Im }(C_{\Im }(\mathcal{D},k),k)$, $\mathcal{D}\le C_{\Im }(I_{\Im }(C_{\Im }(\mathcal{D},k),k),k)$, $\mathcal{D}\le C_{\Im }(I_{\Im }(\mathcal{D},k),k)$, $\mathcal{D}\le I_{\Im }(C_{\Im }(I_{\Im }(\mathcal{D},k),k),k)$, and $\mathcal{D}\le I_{\Im }(\mathcal{D},k)$).

Definition 4

([17,23]). Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and $k\in I_{\circ }$. An $\mathcal{F}$-set $\mathcal{D}$ is said to be k-$\mathcal{F}$-compact (resp. k-$\mathcal{F}$-nearly compact and k-$\mathcal{F}$-almost compact) iff for every family ${\{{\mathcal{P}}_j\in I^M|\Im \left({\mathcal{P}}_j\right)\ge k\}}_{j\in \Omega }$, with $\mathcal{D}\le {\bigvee }_{j\in \Omega }{\mathcal{P}}_j$, there is a finite subset ${\Omega }_{\circ }$ of Ω, with $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{\mathcal{P}}_j$ (resp. $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{I}_{\Im }(C_{\Im }({\mathcal{P}}_j,k),k)$ and $\mathcal{D}\le {\bigvee }_{j\in {\Omega }_{\circ }}{C}_{\Im }({\mathcal{P}}_j,k)$).

Definition 5

([15,20]). Let $(M,\Im )$ and $(N,\mathsf{Ϝ})$ be $\mathcal{FTS}s$. An $\mathcal{F}$-function $h:I^M⟶I^N$ is said to be defined as follows:

(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 6

([20,22]). Let $(M,\Im )$ and $(N,\mathsf{Ϝ})$ be $\mathcal{FTS}s$ and $k\in I_{\circ }$. An $\mathcal{F}$-function $h:I^M⟶I^N$ is said to be $\mathcal{F}$-α-continuous (resp. $\mathcal{F}$-pre-continuous, $\mathcal{F}$-semi-continuous, and $\mathcal{F}$-β-continuous) if $h^{-1}\left(\mathcal{P}\right)$ is a k-$\mathcal{F}$-α-open (resp. k-$\mathcal{F}$-pre-open, k-$\mathcal{F}$-semi-open, and k-$\mathcal{F}$-β-open) set for every $\mathcal{P}\in I^N$, with $\mathsf{Ϝ}\left(\mathcal{P}\right)\ge k$.

Some basic notations and results that we need in the sequel are found in [15,16,17,18,19,20,21,22,23].

3. Some Characterizations of $\mathit{k}$-Fuzzy $\mathit{\gamma }$-Open Sets

Here, we define and study a new class of $\mathcal{F}$-open sets called k-$\mathcal{F}$-$\gamma$-open sets on $\mathcal{FTS}s$ in the sense of Šostak [14]. Also, we explore and investigate the concepts of $\mathcal{F}$-$\gamma$-closure and $\mathcal{F}$-$\gamma$-interior operators.

Definition 7.

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

(1) k-$\mathcal{F}$-γ-open set if $\mathcal{D}\le C_{\Im }(I_{\Im }(\mathcal{D},k),k)\vee I_{\Im }(C_{\Im }(\mathcal{D},k),k)$.

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

Remark 1.

The complement of k-$\mathcal{F}$-γ-open sets (resp. k-$\mathcal{F}$-γ-closed sets) are k-$\mathcal{F}$-γ-closed sets (resp. k-$\mathcal{F}$-γ-open sets).

Proposition 1.

In an $\mathcal{FTS}$$(M,\Im )$, for each $\mathcal{D}\in I^M$ and $k\in I_{\circ }$, we have the following:

(1) Every k-$\mathcal{F}$-pre-open set is k-$\mathcal{F}$-γ-open.

(2) Every k-$\mathcal{F}$-γ-open set is k-$\mathcal{F}$-β-open.

(3) Every k-$\mathcal{F}$-semi-open set is k-$\mathcal{F}$-γ-open.

Proof. 

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

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

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

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

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

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

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

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

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

Thus, $\mathcal{D}$ is a k-$\mathcal{F}$-$\gamma$-open set. □

Remark 2.

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

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

$$\nearrow \searrow$$

$$k\text{-}\mathcal{F}\text{-}\mathbf{open}\mathbf{set}⟶k\text{-}\mathcal{F}\text{-}\alpha \text{-}\mathbf{open}\mathbf{set}⟶k\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathbf{open}\mathbf{set}⟶k\text{-}\mathcal{F}\text{-}\beta \text{-}\mathbf{open}\mathbf{set}$$

$$\searrow \nearrow$$

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

Remark 3.

The converse of the above diagram fails, as Examples 1, 2, and 3 will show.

Example 1.

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

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

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

Example 2.

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

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

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

Example 3.

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

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

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

Definition 8.

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

Proposition 2.

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

Proof. 

This is easily proved from Definition 8. □

Theorem 1.

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

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

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

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

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

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

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

Proof. 

Examples (1), (2), and (3) are easily proved by Definition 8:

(4) From (2) and (3), $\gamma C_{\Im }(\mathcal{D},k)\le \gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)$. Now, we show that $\gamma C_{\Im }(\mathcal{D},k)\ge \gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)$. If $\gamma C_{\Im }(\mathcal{D},k)$ does not contain $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)$, there are $m\in M$ and $\sigma \in (0,1)$ with

$$\gamma C_{\Im }(\mathcal{D},k)\left(m\right)<\sigma <\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)\left(m\right).\left(1\right)$$

Since $\gamma C_{\Im }(\mathcal{D},k)\left(m\right)<\sigma$, by Definition 8, there are $\mathcal{V}\in I^M$ as a k-$\mathcal{F}$-$\gamma$-closed set and $\mathcal{D}\le \mathcal{V}$ with $\gamma C_{\Im }(\mathcal{D},k)\left(m\right)\le \mathcal{V}\left(m\right)<\sigma$. Since $\mathcal{D}\le \mathcal{V}$, then $\gamma C_{\Im }(\mathcal{D},k)\le \mathcal{V}$. Again, this is by the definition of $\gamma C_{\Im }$, $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)\le \mathcal{V}$. Hence, $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)\left(m\right)\le \mathcal{V}\left(m\right)<\sigma$, which is a contradiction for $\left(Z\right)$. Thus, $\gamma C_{\Im }(\mathcal{D},k)\ge \gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)$. Therefore, $\gamma C_{\Im }(\gamma C_{\Im }(\mathcal{D},k),k)=\gamma C_{\Im }(\mathcal{D},k)$.

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

(6) From Proposition 2 and the fact that $C_{\Im }(\mathcal{D},k)$ is a k-$\mathcal{F}$-$\gamma$-closed set, then

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

□

Definition 9.

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

Proposition 3.

Let $(M,\Im )$ be an $\mathcal{FTS}$, $\mathcal{D}\in I^M$, and let $k\in I_{\circ }$. Then, we have the following:

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

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

Proof. 

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

(2) This is similar to that of (1). □

Proposition 4.

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

Proof. 

This is easily proved from Definition 9. □

Theorem 2.

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

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

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

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

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

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

Proof. 

The proof is similar to that of Theorem 1. □

4. On Fuzzy $\mathit{\gamma }$-Continuity and $\mathit{\gamma }$-Irresoluteness

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.

Definition 10.

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

(1) $\mathcal{F}$-γ-continuous if $h^{-1}\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-γ-open set for every $\mathcal{D}\in I^N$, with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge k$ and $k\in I_{\circ }$;

(2) $\mathcal{F}$-γ-irresolute if $h^{-1}\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-γ-open set for every k-$\mathcal{F}$-γ-open set $\mathcal{D}\in I^N$ and $k\in I_{\circ }$.

Remark 4.

From the previous definitions, we have the following diagram:

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

$$\nearrow \searrow$$

$$\mathcal{F}\text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\alpha \text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\gamma \text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\beta \text{-}\mathbf{continuity}$$

$$\searrow \nearrow$$

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

Remark 5.

The converse of the above diagram fails, as Examples 4–6 will show.

Example 4.

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:

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

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

Example 5.

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:

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

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

Example 6.

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:

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

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

Theorem 3.

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

Proof. 

(⇒) Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{D}\in I^N$, with $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge k$ containing $h\left(m_{\sigma }\right)$, and then $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),k)$. Since $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$, then we obtain $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),k)=\mathcal{A}$ (say). Hence, $\mathcal{A}\in I^M$ is k-$\mathcal{F}$-$\gamma$-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 $\mathsf{Ϝ}\left(\mathcal{D}\right)\ge k$ and $m_{\sigma }\in h^{-1}\left(\mathcal{D}\right)$. According to the assumption, there is $\mathcal{A}\in I^M$ that is k-$\mathcal{F}$-$\gamma$-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 $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),k)$. Thus, $h^{-1}\left(\mathcal{D}\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{D}\right),k)$, so $h^{-1}\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-$\gamma$-open set. Then, h is $\mathcal{F}$-$\gamma$-continuous. □

Theorem 4.

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

(1) h is $\mathcal{F}$-γ-continuous.

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

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

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

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

Proof. 

(1) ⇔ (2): The proof follows $h^{-1}\left({\mathcal{D}}^c\right)={\left(h^{-1}\left(\mathcal{D}\right)\right)}^c$ and Definition 10.

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

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

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

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

(4) ⇔ (5): The proof follows $h^{-1}\left({\mathcal{D}}^c\right)={\left(h^{-1}\left(\mathcal{D}\right)\right)}^c$ and Proposition 3.

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

Lemma 2.

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

Proof. 

The proof follows Definition 10. □

Remark 6.

The converse of Lemma 2 fails, as Example 7 will show.

Example 7.

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:

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

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

Theorem 5.

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

(1) h is $\mathcal{F}$-γ-irresolute.

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

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

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

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

Proof. 

(1) ⇔ (2): The proof follows $h^{-1}\left({\mathcal{P}}^c\right)={\left(h^{-1}\left(\mathcal{P}\right)\right)}^c$ and Definition 10.

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

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

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

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

(4) ⇔ (5): The proof follows $h^{-1}\left({\mathcal{P}}^c\right)={\left(h^{-1}\left(\mathcal{P}\right)\right)}^c$ and Proposition 3.

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

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

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

Proposition 5.

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

Proof. 

The proof follows Definition 10. □

Definition 11.

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

Lemma 3.

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

Proof. 

The proof follows Definitions 10 and 11. □

Remark 7.

The converse of Lemma 3 fails, as Example 8 will show.

Example 8.

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:

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

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

Theorem 6.

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

Proof. 

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

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

Theorem 7.

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

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

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

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

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

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

Proof. 

(1) ⇒ (2): Let $m_{\sigma }\in P_{\sigma }\left(M\right)$ and $\mathcal{P}$ be a k-$\mathcal{F}$-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 a k-$\mathcal{F}$-$\gamma$-open set, with $m_{\sigma }\in \mathcal{A}$ and $h\left(\mathcal{A}\right)\le I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},k),k)$. Thus, $\mathcal{A}\le h^{-1}\left(I_{\mathsf{Ϝ}}(C_{\mathsf{Ϝ}}(\mathcal{P},k),k)\right)=h^{-1}\left(\mathcal{P}\right)$, and $m_{\sigma }\in \gamma I_{\Im }(h^{-1}\left(\mathcal{P}\right),k)$. Therefore, $h^{-1}\left(\mathcal{P}\right)\le \gamma I_{\Im }(h^{-1}\left(\mathcal{P}\right),k)$, so $h^{-1}\left(\mathcal{P}\right)$ is k-$\mathcal{F}$-$\gamma$-open.

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

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

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

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

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

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

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

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

Definition 12.

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

Lemma 4.

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

Proof. 

The proof follows Definitions 10 and 12. □

Remark 8.

The converse of Lemma 4 fails, as Example 9 will show.

Example 9.

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:

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

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

Theorem 8.

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

Proof. 

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

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

Theorem 9.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-function, and let $k\in I_{\circ }$. Then, the following statements are equivalent:

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

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

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

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

Proof. 

(1) ⇔ (2): The proof follows Proposition 3 and Definition 12.

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

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

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

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

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

Lemma 5.

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

Proof. 

The proof follows Definitions 11 and 12. □

Remark 9.

The converse of Lemma 5 fails, as Example 10 will show.

Example 10.

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:

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

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

Remark 10.

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

$$\mathcal{F}\text{-}\gamma \text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\mathbf{almost}\gamma \text{-}\mathbf{continuity}⟶\mathcal{F}\text{-}\mathbf{weak}\gamma \text{-}\mathbf{continuity}$$

Proposition 6.

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

Proof. 

The proof follows the previous definitions. □

5. Further Selected Topics

Here, we introduce and establish some new $\mathcal{F}$-functions using k-$\mathcal{F}$-$\gamma$-open and k-$\mathcal{F}$-$\gamma$-closed sets, which are 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 explore and study some new types of $\mathcal{F}$-compactness called k-$\mathcal{F}$-almost and k-$\mathcal{F}$-nearly $\gamma$-compact sets using k-$\mathcal{F}$-$\gamma$-open sets.

Definition 13.

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

(1) $\mathcal{F}$-γ-open if $h\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-γ-open set for every $\mathcal{D}\in I^M$ with $\Im \left(\mathcal{D}\right)\ge k$.

(2) $\mathcal{F}$-γ-closed if $h\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-γ-closed set for every $\mathcal{D}\in I^M$ with $\Im \left({\mathcal{D}}^c\right)\ge k$.

(3) $\mathcal{F}$-γ-irresolute open if $h\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-γ-open set for every k-$\mathcal{F}$-γ-open set $\mathcal{D}\in I^M$.

(4) $\mathcal{F}$-γ-irresolute closed if $h\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-γ-closed set for every k-$\mathcal{F}$-γ-closed set $\mathcal{D}\in I^M$.

Lemma 6.

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

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

Proof. 

The proof follows Definition 13. □

Remark 11.

The converse of Lemma 6 fails, as Example 11 will show.

Example 11.

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:

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

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

Theorem 10.

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

(1) h is $\mathcal{F}$-γ-open.

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

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

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

Proof. 

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

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

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

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

(4) ⇒ (1): Let $\mathcal{B}\in I^M$, with $\Im \left(\mathcal{B}\right)\ge k$. Set $\mathcal{D}={\left(h\left(\mathcal{B}\right)\right)}^c$ 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 $\mathcal{P}\in I^N$ that is k-$\mathcal{F}$-$\gamma$-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 a k-$\mathcal{F}$-$\gamma$-open set. Therefore, h is $\mathcal{F}$-$\gamma$-open. □

Theorem 11.

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

(1) h is $\mathcal{F}$-γ-closed.

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

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

(4) For every $\mathcal{D}$ and every $\mathcal{B}$ with $\Im \left(\mathcal{B}\right)\ge k$ and $h^{-1}\left(\mathcal{D}\right)\le \mathcal{B}$, there is $\mathcal{P}\in I^N$ that is k-$\mathcal{F}$-γ-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 10. □

Theorem 12.

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

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

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

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

(4) For every $\mathcal{D}$ and every $\mathcal{B}$ that define a k-$\mathcal{F}$-γ-closed set with $h^{-1}\left(\mathcal{D}\right)\le \mathcal{B}$, there is $\mathcal{P}\in I^N$ that is k-$\mathcal{F}$-γ-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 10. □

Theorem 13.

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

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

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

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

(4) For every $\mathcal{D}$ and every $\mathcal{B}$ that define a k-$\mathcal{F}$-γ-open set with $h^{-1}\left(\mathcal{D}\right)\le \mathcal{B}$, there is $\mathcal{P}\in I^N$ that is k-$\mathcal{F}$-γ-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 10. □

Proposition 7.

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

Proof. 

The proof follows from the following:

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

□

Definition 14.

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

The proof of the following corollary is easy and so is omitted.

Corollary 1.

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

(1) h is an $\mathcal{F}$-$\gamma$-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) $h\left(\gamma I_{\Im }(\mathcal{P},k)\right)=\gamma I_{\mathsf{Ϝ}}(h\left(\mathcal{P}\right),k)$.

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

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

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

Definition 15.

Let $m_{\sigma }\in P_{\sigma }\left(M\right)$, $\mathcal{A},\mathcal{B}\in I^M$, and let $k\in I_{\circ }$. An $\mathcal{FTS}$ $(M,\Im )$ is defined as follows:

(1) k-$\mathcal{F}$-γ-regular space if $m_{\sigma }\overline{q}\mathcal{A}$ for each k-$\mathcal{F}$-γ-closed set $\mathcal{A}$, there is ${\mathcal{C}}_j\in I^M$ with $\Im \left({\mathcal{C}}_j\right)\ge k$ 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) k-$\mathcal{F}$-γ-normal space if $\mathcal{A}\overline{q}\mathcal{B}$ for each k-$\mathcal{F}$-γ-closed set $\mathcal{A}$ and $\mathcal{B}$, there is ${\mathcal{C}}_j\in I^M$ with $\Im \left({\mathcal{C}}_j\right)\ge k$ 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 14.

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

(1) $(M,\Im )$ is a k-$\mathcal{F}$-γ-regular space.

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

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

Proof. 

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

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

(3) ⇒ (1): This is easily proved by Definition 15. □

Theorem 15.

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

(1) $(M,\Im )$ is a k-$\mathcal{F}$-γ-normal space.

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

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

Proof. 

The proof is similar to that of Theorem 14. □

Theorem 16.

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

Proof. 

If $n_{\sigma }\overline{q}\mathcal{B}$ for each k-$\mathcal{F}$-$\gamma$-closed set $\mathcal{B}\in I^N$ and $\mathcal{F}$-$\gamma$-irresolute function h, then $h^{-1}\left(\mathcal{B}\right)$ is a k-$\mathcal{F}$-$\gamma$-closed set. Set $n_{\sigma }=h\left(m_{\sigma }\right)$, and then $m_{\sigma }\overline{q}{h}^{-1}\left(\mathcal{B}\right)$. Since $(M,\Im )$ is k-$\mathcal{F}$-$\gamma$-regular, there is ${\mathcal{D}}_1,D_2\in I^M$ with $\Im \left({\mathcal{D}}_1\right)\ge k$ and $\Im \left({\mathcal{D}}_2\right)\ge k$ 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 bijective $\mathcal{F}$-open, then $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 a k-$\mathcal{F}$-$\gamma$-regular space. □

Theorem 17.

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

Proof. 

If $m_{\sigma }\overline{q}\mathcal{B}$ for each k-$\mathcal{F}$-$\gamma$-closed set $\mathcal{B}\in I^M$ and injective $\mathcal{F}$-$\gamma$-irresolute closed function h, then $h\left(\mathcal{B}\right)$ is a k-$\mathcal{F}$-$\gamma$-closed set, and $h\left(m_{\sigma }\right)\overline{q}h\left(\mathcal{B}\right)$. Since $(N,\mathsf{Ϝ})$ is k-$\mathcal{F}$-$\gamma$-regular, there is ${\mathcal{D}}_1,{\mathcal{D}}_2\in I^N$ with $\mathsf{Ϝ}\left({\mathcal{D}}_1\right)\ge k$ and $\mathsf{Ϝ}\left({\mathcal{D}}_2\right)\ge k$ 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 $m_{\sigma }\in h^{-1}\left({\mathcal{D}}_1\right)$, and $\mathcal{B}\le h^{-1}\left({\mathcal{D}}_2\right)$, with $\Im \left(h^{-1}\left({\mathcal{D}}_1\right)\right)\ge k$, $\Im \left(h^{-1}\left({\mathcal{D}}_2\right)\right)\ge k$ and $h^{-1}\left({\mathcal{D}}_1\right)\overline{q}{h}^{-1}\left({\mathcal{D}}_2\right)$. Hence, $(M,\Im )$ is a k-$\mathcal{F}$-$\gamma$-regular space. □

Theorem 18.

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

Proof. 

The proof is similar to that of Theorem 16. □

Definition 16.

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

Lemma 7.

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

Proof. 

The proof follows Definitions 4 and 16. □

Theorem 19.

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

Proof. 

Let ${\{{\mathcal{B}}_j\in I^N|\mathsf{Ϝ}\left({\mathcal{B}}_j\right)\ge k\}}_{j\in \Omega }$ with $h\left(\mathcal{D}\right)\le {\bigvee }_{j\in \Omega }{\mathcal{B}}_j$, and then $\{h^{-1}\left({\mathcal{B}}_j\right)\in I^M|h^{-1}\left({\mathcal{B}}_j\right)$ is $k\text{-}\mathcal{F}\text{-}\gamma \text{-}\mathrm{open}$} (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 k-$\mathcal{F}$-$\gamma$-compact, there is a finite subset ${\Omega }_{\circ }$ 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 k-$\mathcal{F}$-compact. □

Definition 17.

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

Lemma 8.

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

Proof. 

The proof follows Definitions 4 and 17. □

Lemma 9.

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

Proof. 

The proof follows Definitions 16 and 17. □

Remark 12.

The converse of Lemma 9 fails, as Example 12 will show.

Example 12.

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

$$\mathcal{A}\left(w\right)=\left\{\begin{array}{c}1,ifw=0,\hfill \\ {\displaystyle \frac{1}{2}},otherwise,\hfill \end{array}\right.{\mathcal{B}}_t\left(w\right)=\left\{\begin{array}{c}0.8,ifw=0,\hfill \\ tw,if0<w\le {\displaystyle \frac{1}{t}},\hfill \\ 1,if{\displaystyle \frac{1}{t}}<w\le 1.\hfill \end{array}\right.$$

Also, ℑ is defined on W as follows:

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

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

Theorem 20.

Let $h:(M,\Im )⟶(N,\mathsf{Ϝ})$ be an $\mathcal{F}$-continuous function, and let $k\in I_{\circ }$. Then, $h\left(\mathcal{D}\right)$ is a k-$\mathcal{F}$-almost compact set if $\mathcal{D}\in I^M$ is a k-$\mathcal{F}$-almost γ-compact set.

Proof. 

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

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

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

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

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

Definition 18.

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

Lemma 10.

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

Proof. 

The proof follows Definitions 4 and 18. □

Lemma 11.

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

Proof. 

The proof follows Definitions 16 and 18. □

Remark 13.

The converse of Lemma 11 fails, as Example 12 will show.

Example 13.

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

$$\mathcal{A}\left(w\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}},if0\le w<1,\hfill \\ 1,ifw=1,\hfill \end{array}\right.\mathcal{B}\left(w\right)=\left\{\begin{array}{c}1,ifw=0,\hfill \\ {\displaystyle \frac{1}{2}},if0<w\le 1,\hfill \end{array}\right.$$

$${\mathcal{D}}_t\left(w\right)=\left\{\begin{array}{c}{\displaystyle \frac{w}{t}},if0\le w<t,\hfill \\ {\displaystyle \frac{1-w}{1-t}},ift<w\le 1.\hfill \end{array}\right.$$

Also, ℑ is defined on W as follows:

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

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

Theorem 21.

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

Proof. 

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

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

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

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

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

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

Lemma 12.

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

Proof. 

The proof follows Definitions 17 and 18. □

Remark 14.

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

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

$$\downarrow \downarrow$$

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

$$\downarrow \downarrow$$

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

6. Conclusions and Future Work

In the present manuscript, a novel class of $\mathcal{F}$-open sets, called k-$\mathcal{F}$-$\gamma$-open sets, has been introduced on $\mathcal{FTS}s$ in Šostak’s sense [14]. Some characterizations of k-$\mathcal{F}$-$\gamma$-open sets, along with their mutual relationships, have been studied 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 $(N,\mathsf{Ϝ})$ have 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. Thereafter, we defined and studied some new $\mathcal{F}$-functions using k-$\mathcal{F}$-$\gamma$-open and k-$\mathcal{F}$-$\gamma$-closed sets, which are 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 k-$\mathcal{F}$-$\gamma$-regular (resp. k-$\mathcal{F}$-$\gamma$-normal) spaces using k-$\mathcal{F}$-$\gamma$-closed sets. Finally, some new types of $\mathcal{F}$-compactness, called k-$\mathcal{F}$-almost (resp. k-$\mathcal{F}$-nearly) $\gamma$-compact sets, have been defined and discussed.

In the next works, we intend to explore the following topics: (1) defining upper (lower) $\gamma$-continuous $\mathcal{F}$-multifunctions and k-$\mathcal{F}$-$\gamma$-connected sets; (2) extending these notions given here to include fuzzy soft topological (k-minimal) spaces [26,30]; (3) finding a use for these notions given here in the frame of fuzzy ideals, as defined in [32,33,34]; (4) defining new types of $\mathcal{F}$-compactness via the other definitions of crisp compactness; and (5) introducing these notions given here based on lattice-valued fuzzy sets.