Novel Approach to Study Soft α-Open Sets and Its Applications via Fuzzy Soft Topologies

1. Introduction

Theory of soft sets was first introduced by Molodtsov [1], which is a completely new approach for modeling uncertainty and vagueness. He demonstrated many applications of these theory in solving several practical problems in mathematics, engineering, economics, and social science etc. Akdag and Ozkan [2] defined the concept of soft $\alpha$-open sets on soft topological spaces, and some properties are specified. The concept of soft $\beta$-open sets was defined and studied by the authors of [3,4]. Also, the concepts of soft semi-open, somewhere dense and Q-sets were studied by the authors of [5,6]. Al-shami et al. [7] introduced the concept of weakly soft semi-open sets and obtained its main properties. Moreover, Al-shami et al. [8] initiated the concept of weakly soft $\beta$-open sets and examined weakly soft $\beta$-continuity. Kaur et al. [9] introduced a new approach to studying soft continuous mappings using an induced mapping based on soft sets. Al Ghour and Al-Mufarrij [10] defined two new concepts of mappings over soft topological spaces: soft somewhat-r-continuity and soft somewhat-r-openness. In 2024, Ameen et al. [11] explored more properties of soft somewhere dense continuous mappings.

The concept of fuzzy soft sets was introduced by Maji et al. [12], which combines soft sets [1] and fuzzy sets [13]. The concept of fuzzy soft topology is defined and some characterized such as fuzzy soft interior (closure) set, fuzzy soft continuity and fuzzy soft subspace is studied in [14,15] based on fuzzy topologies in the sense of Šostak [16]. A new approach to studying separation and regularity axioms via fuzzy soft sets was introduced by the author of [17,18] based on the paper Aygünoǧlu et al. [14]. The concept of r-fuzzy soft regularly open sets was introduced by Çetkin and Aygün [19]. Also, the concepts of r-fuzzy soft pre-open (resp. $\beta$-open) sets were defined by Taha [20].

In our study, the layout is designed as follows.

• Firstly, we introduce the concepts of fuzzy soft $\alpha$-closure ($\alpha$-interior) operators in fuzzy soft topological space $(W,{\tau }_N)$ based on the paper Aygünoǧlu et al. [14], and examine some of its properties. Also, the concept of r-fuzzy soft $\alpha$-connected sets is introduced and studied.

• Secondly, we are going to investigate some properties of fuzzy soft $\alpha$-continuous mappings between two fuzzy soft topological spaces $(W,{\tau }_N)$ and $(V,{\eta }_F)$. Moreover, we define and study the concepts of fuzzy soft weakly (almost) $\alpha$-continuous mappings, which are weaker forms of fuzzy soft $\alpha$-continuous mappings. Also, the relationships between these classes of mappings are investigated with the help of some examples.

• Finally, several types of fuzzy soft compactness via r-fuzzy soft $\alpha$-open sets are defined, and the relationships between them are specified.

• In the end, we close this manuscript with some conclusions and proposed some future works in Section 5.

In this work, nonempty sets will be denoted by W, V etc. N is the set of all parameters for W and $C\subseteq N$. The family of all fuzzy sets on W is denoted by $I^W$ (where $I_{\circ }=(0,1],I=[0,1]$), and for $s\in I,$$\underline{s}\left(w\right)=s$, for all $w\in W.$

The following concepts and results will be used in the next sections.

Definition 1.1.

[14,21,22] A fuzzy soft set $h_C$ on W is a mapping from N to $I^W$ such that $h_C\left(n\right)$ is a fuzzy set on W, for each $n\in C$ and $h_C\left(n\right)=\underline{0}$, if $n\notin C$. The family of all fuzzy soft sets on W is denoted by $\widetilde{(W,N)}$.

Definition 1.2.

[23] The difference between two fuzzy soft sets $h_C$ and $g_B$ is a fuzzy soft set defined as follows, for each $n\in N$: $\left(h_C\overline{\sqcap }{g}_B\right)\left(n\right)=\left\{\begin{array}{cc}\hfill \underline{0},& \mathrm{i}\mathrm{f}{h}_C\left(n\right)\le g_B\left(n\right),\\ \hfill h_C\left(n\right)\wedge {\left(g_B\left(n\right)\right)}^c,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}\right.$

Definition 1.3.

[24] A fuzzy soft point $n_{w_s}$ on W is a fuzzy soft set defined as follows:

$$n_{w_s}\left(k\right)=\left\{\begin{array}{ccc}\hfill w_s,& \mathrm{i}\mathrm{f}& k=n,\hfill \\ \hfill \underline{0},& \mathrm{i}\mathrm{f}& k\in N-\left\{n\right\},\hfill \end{array}\right.$$

where $w_s$ is a fuzzy point on W. $n_{w_s}$ is called belong to a fuzzy soft set $f_A$, denoted by $n_{w_s}\tilde{\in }{f}_A$, if $s\le f_A\left(n\right)\left(w\right)$. The family of all fuzzy soft points on W is denoted by $\widetilde{P_s\left(W\right)}$.

Definition 1.4.

[25] A fuzzy soft point $n_{w_s}\in \widetilde{P_s\left(W\right)}$ is called a soft quasi-coincident with $h_C\in \widetilde{(W,N)}$ and denoted by $n_{w_s}\tilde{q}{h}_C$, if $s+h_C\left(n\right)\left(w\right)>1$. A fuzzy soft set $h_C\in \widetilde{(W,N)}$ is called a soft quasi-coincident with $g_B\in \widetilde{(W,N)}$ and denoted by $h_C\tilde{q}{g}_B$, if there is $n\in N$ and $w\in W$ such that $h_C\left(n\right)\left(w\right)+g_B\left(n\right)\left(w\right)>1$. If $h_C$ is not soft quasi-coincident with $g_B$, $h_C\neg \tilde{q}{g}_B$.

Definition 1.5.

[14] A mapping $\tau :N⟶{[0,1]}^{\widetilde{(W,N)}}$ is called a fuzzy soft topology on W if it satisfies the following, for each $n\in N$:

(1) ${\tau }_n\left(\Phi \right)={\tau }_n\left(\tilde{N}\right)=1,$

(2) ${\tau }_n(h_C\sqcap g_B)\ge {\tau }_n\left(h_C\right)\wedge {\tau }_n\left(g_B\right),$ for each $h_C,g_B\in \widetilde{(W,N)},$

(3) ${\tau }_n\left({\bigsqcup }_{\delta \in \Delta }{\left(h_C\right)}_{\delta }\right)\ge {\wedge }_{\delta \in \Delta }{\tau }_n\left({\left(h_C\right)}_{\delta }\right),$ for each ${\left(h_C\right)}_{\delta }\in \widetilde{(W,N)},\delta \in \Delta .$

Then, $(W,{\tau }_N)$ is called a fuzzy soft topological space (briefly, FSTS) in the sense of Šostak.

Definition 1.6.

[14] Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs. A fuzzy soft mapping ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ is called a fuzzy soft continuous if, ${\tau }_n\left({\phi }_{\psi }^{-1}\left(h_C\right)\right)\ge {\eta }_k\left(h_C\right)$ for each $h_C\in \widetilde{(V,F)}$, $n\in N$ and $(k=\psi (n\left)\right)\in F$.

Definition 1.7.

[15,19] In a FSTS $(W,{\tau }_N)$, for each $h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$, we define the fuzzy soft operators $C_{\tau }$ and $I_{\tau }$$:N\times \widetilde{(W,N)}\times I_{\circ }\to \widetilde{(W,N)}$ as follows:

$C_{\tau }(n,h_C,r)=\sqcap \{g_B\in \widetilde{(W,N)}:h_C⊑g_B,{\tau }_n\left(g_B^c\right)\ge r\}$,

$I_{\tau }(n,h_C,r)=\bigsqcup \{g_B\in \widetilde{(W,N)}:g_B⊑h_C,{\tau }_n\left(g_B\right)\ge r\}.$

Definition 1.8.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_0$. A fuzzy soft set $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft regularly open [19] (resp. $\beta$-open [20], pre-open [20], $\alpha$-open [26] and semi-open [26]) if, $h_C=I_{\tau }(n,C_{\tau }(n,h_C,r),r)$ (resp. $h_C⊑C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r)$, $h_C⊑I_{\tau }(n,C_{\tau }(n,h_C,r),r)$, $h_C⊑I_{\tau }(n,C_{\tau }(n,I_{\tau }(n,h_C,r),r),r)$ and $h_C⊑C_{\tau }(n,I_{\tau }(n,h_C,r),r)$) for each $n\in N$.

Definition 1.9.

[19] Let $(W,{\tau }_N)$ be a FSTS and $r\in I_0$. A fuzzy soft set $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft regularly closed if, $h_C=C_{\tau }(n,I_{\tau }(n,h_C,r),r)$ for each $n\in N$.

Definition 1.10.

[26] Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs and $r\in I_0$. A fuzzy soft mapping ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ is called a fuzzy soft almost (resp. weakly) continuous if, for any $n_{w_s}\in \widetilde{P_s\left(W\right)}$ and any $f_A\in \widetilde{(V,F)}$ with ${\eta }_k\left(f_A\right)\ge r$ containing ${\phi }_{\psi }\left(n_{w_s}\right)$, there is $h_C\in \widetilde{(W,N)}$ with ${\tau }_n\left(h_C\right)\ge r$ containing $n_{w_s}$ such that ${\phi }_{\psi }\left(h_C\right)⊑I_{\eta }(k,C_{\eta }(k,f_A,r),r)$ (resp. ${\phi }_{\psi }\left(h_C\right)⊑C_{\eta }(k,f_A,r)$).

Remark 1.1.

[26] From Definitions 1.6 and 1.10, we have: Fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft weakly continuity, but the converse may not be true.

Lemma 1.1.

Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs and $r\in I_0$. A fuzzy soft mapping ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ is fuzzy soft almost continuous if, ${\tau }_n\left({\phi }_{\psi }^{-1}\left(h_C\right)\right)\ge r$ for each $h_C\in \widetilde{(V,F)}$ is r-fuzzy soft regularly open, $n\in N$ and $(k=\psi (n\left)\right)\in F$.

Proof. 

Easily proved from Definition 1.10.

□

Definition 1.11.

Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs. A fuzzy soft mapping ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ is called a fuzzy soft open if, ${\eta }_k\left({\phi }_{\psi }\left(h_C\right)\right)\ge {\tau }_n\left(h_C\right)$ for each $h_C\in \widetilde{(W,N)}$, $n\in N$ and $(k=\psi (n\left)\right)\in F$.

The basic concepts and results are found in [14,15], which we need in the next sections.

2. On r-Fuzzy Soft $\alpha$-Open Sets

Here, we introduce and discuss the notions of fuzzy soft $\alpha$-closure ($\alpha$-interior) operators in fuzzy soft topological spaces based on the paper Aygünoǧlu et al. [14]. Also, the notion of r-fuzzy soft $\alpha$-connected sets is defined and studied with help of fuzzy soft $\alpha$-closure operators.

Definition 2.1.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_0$. A fuzzy soft set $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft $\alpha$-closed (resp. semi-closed, $\beta$-closed and pre-closed) if, $C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r)⊑h_C$ (resp. $I_{\tau }(n,C_{\tau }(n,h_C,r),r)⊑h_C$, $I_{\tau }(n,C_{\tau }(n,I_{\tau }(n,h_C,r),r),r)⊑h_C$ and $C_{\tau }(n,I_{\tau }(n,h_C,r),r)⊑h_C$) for each $n\in N$.

Remark 2.1.

The complement of an r-fuzzy soft $\alpha$-open [26] (resp. semi-open [26], $\beta$-open [20] and pre-open [20]) set is an r-fuzzy soft $\alpha$-closed (resp. semi-closed, $\beta$-closed and pre-closed) set.

Lemma 2.1.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_0$. Then, any intersection (resp. union) of r-fuzzy soft $\alpha$-closed (resp. $\alpha$-open) sets is an r-fuzzy soft $\alpha$-closed (resp. $\alpha$-open) set.

Proof. 

Easily proved from Definitions 1.8 and 2.1. □

Proposition 2.1.

Let $(W,{\tau }_N)$ be a FSTS, $h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$. Then, the following statements are equivalent.

(1) $h_C$ is r-fuzzy soft $\alpha$-closed.

(2) $h_C$ is r-fuzzy soft semi-closed and r-fuzzy soft pre-closed.

Proof. (1) ⇒ (2) Let $h_C$ be an r-fuzzy soft $\alpha$-closed, $h_C⊒C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r)⊒I_{\tau }(n,C_{\tau }(n,h_C,r),r)$. This shows that $h_C$ is r-fuzzy soft semi-closed.

Since $h_C⊒C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r)$ and $C_{\tau }(n,h_C,r)⊒h_C$, then $h_C⊒C_{\tau }(n,I_{\tau }(n,h_C,r),r)$. Therefore, $h_C$ is r-fuzzy soft pre-closed

(2) ⇒ (1) Let $h_C$ be an r-fuzzy soft semi-closed and r-fuzzy soft pre-closed, then $h_C⊒C_{\tau }(n,I_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r),r)=C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r)$. This shows that $h_C$ is r-fuzzy soft $\alpha$-closed.

□

Proposition 2.2.

Let $(W,{\tau }_N)$ be a FSTS, $g_B,h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$. If $g_B$ is r-fuzzy soft semi-closed set such that $g_B⊒h_C⊒C_{\tau }(n,I_{\tau }(n,g_B,r),r)$, $h_C$ is r-fuzzy soft $\alpha$-closed.

Proof. 

Let $g_B$ be an r-fuzzy soft semi-closed and $g_B⊒h_C$, then $g_B⊒I_{\tau }(n,C_{\tau }(n,g_B,r),r)⊒I_{\tau }(n,C_{\tau }(n,h_C,r),r).$ Let $h_C⊒C_{\tau }(n,I_{\tau }(n,g_B,r),r)$, then $h_C⊒C_{\tau }(n,I_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r),r)=C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,h_C,r),r),r)$. Therefore, $h_C$ is r-fuzzy soft $\alpha$-closed.

□

Lemma 2.2.

Let $(W,{\tau }_N)$ be a FSTS, $g_B,h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$. If $g_B$ is r-fuzzy soft $\alpha$-closed set such that $g_B⊒h_C⊒C_{\tau }(n,I_{\tau }(n,g_B,r),r)$, $h_C$ is r-fuzzy soft $\alpha$-closed.

Proof. 

It is easily proved from every r-fuzzy soft $\alpha$-closed set is r-fuzzy soft semi-closed set.

□

Remark 2.2.

From the previous definition, we can summarize the relationships among different types of fuzzy soft sets as in the next diagram.

$$\alpha -closedset$$

$$\Downarrow \Downarrow$$

$$semi-closedsetpre-closedset$$

$$\Downarrow \Downarrow$$

$$\beta -closedset$$

Remark 2.3.

The converses of the above relationships may not be true, as shown by Examples 2.1 and 2.2.

Example 2.1.

Let $W=\{w_1,w_2\}$, $N=\{n_1,n_2\}$ and define $f_N,g_N,h_N\in \widetilde{(W,N)}$ as follows: $f_N=\{(n_1,\{\frac{w_1}{0.3},\frac{w_2}{0.4}\}),(n_2,\{\frac{w_1}{0.3},\frac{w_2}{0.4}\})\}$, $g_N=\{(n_1,\{\frac{w_1}{0.6},\frac{w_2}{0.2}\}),(n_2,\{\frac{w_1}{0.6},\frac{w_2}{0.2}\})\}$, $h_N=\{(n_1,\{\frac{w_1}{0.3},\frac{w_2}{0.5}\}),(n_2,\{\frac{w_1}{0.3},\frac{w_2}{0.5}\})\}$. Define fuzzy soft topology ${\tau }_N:N⟶{[0,1]}^{\widetilde{(W,N)}}$ as follows:

${\tau }_{n_1}\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=f_N,\hfill \\ \frac{2}{3},\mathrm{if}{t}_N=g_N,\hfill \\ \frac{2}{3},\mathrm{if}{t}_N=f_N\sqcap g_N,\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=f_N\bigsqcup g_N,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$

${\tau }_{n_2}\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{4},\mathrm{if}{t}_N=f_N,\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=g_N,\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=f_N\sqcap g_N,\hfill \\ \frac{1}{4},\mathrm{if}{t}_N=f_N\bigsqcup g_N,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$

Then, $h_N$ is $\frac{1}{4}$-fuzzy soft semi-closed and $\frac{1}{4}$-fuzzy soft $\beta$-closed, but it is neither $\frac{1}{4}$-fuzzy soft $\alpha$-closed nor $\frac{1}{4}$-fuzzy soft pre-closed.

Example 2.2.

Let $W=\{w_1,w_2\}$, $N=\{n_1,n_2\}$ and define $f_N,h_N\in \widetilde{(W,N)}$ as follows: $f_N=\{(n_1,\{\frac{w_1}{0.4},\frac{w_2}{0.5}\}),(n_2,\{\frac{w_1}{0.4},\frac{w_2}{0.5}\})\}$, $h_N=\{(n_1,\{\frac{w_1}{0.7},\frac{w_2}{0.6}\}),(n_2,\{\frac{w_1}{0.7},\frac{w_2}{0.6}\})\}$. Define fuzzy soft topology ${\tau }_N:N⟶{[0,1]}^{\widetilde{(W,N)}}$ as follows:

${\tau }_{n_1}\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{3},\mathrm{if}{t}_N=f_N,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$

${\tau }_{n_2}\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=f_N,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$

Then, $h_N$ is $\frac{1}{3}$-fuzzy soft pre-closed and $\frac{1}{3}$-fuzzy soft $\beta$-closed, but it is neither $\frac{1}{3}$-fuzzy soft semi-closed nor $\frac{1}{3}$-fuzzy soft $\alpha$-closed.

Definition 2.2.

In a FSTS $(W,{\tau }_N)$, for each $h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$, we define a fuzzy soft $\alpha$-closure operator $\alpha C_{\tau }$$:N\times \widetilde{(W,N)}\times I_{\circ }\to \widetilde{(W,N)}$ as follows: $\alpha C_{\tau }(n,h_C,r)=\sqcap \{f_A\in \widetilde{(W,N)}:h_C⊑f_A,f_A\mathrm{is}r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{closed}\}.$

Theorem 2.1.

In a FSTS $(W,{\tau }_N)$, for each $g_B,h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$, the operator $\alpha C_{\tau }$$:N\times \widetilde{(W,N)}\times I_{\circ }\to \widetilde{(W,N)}$ satisfies the following properties.

(1) $\alpha C_{\tau }(n,\Phi ,r)=\Phi$.

(2) $h_C⊑\alpha C_{\tau }(n,h_C,r)⊑C_{\tau }(n,h_C,r)$.

(3) $\alpha C_{\tau }(n,h_C,r)⊑\alpha C_{\tau }(n,g_B,r)$ if, $h_C⊑g_B$.

(4) $\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)=\alpha C_{\tau }(n,h_C,r)$.

(5) $\alpha C_{\tau }(n,h_C\bigsqcup g_B,r)⊒\alpha C_{\tau }(n,h_C,r)\bigsqcup \alpha C_{\tau }(n,g_B,r)$.

(6) $\alpha C_{\tau }(n,h_C,r)=h_C$ iff $h_C$ is r-fuzzy soft $\alpha$-closed.

(7) $\alpha C_{\tau }(n,C_{\tau }(n,h_C,r),r)=C_{\tau }(n,h_C,r)$.

Proof. (1), (2), (3) and (6) are easily proved from Definition 2.2.

(4) From (2) and (3), $\alpha C_{\tau }(n,h_C,r)⊑\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)$. Now we show that $\alpha C_{\tau }(n,h_C,r)⊒\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)$. Suppose that $\alpha C_{\tau }(n,h_C,r)$ is not contain $\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)$. Then, there is $w\in W$ and $s\in (0,1)$ such that $\alpha C_{\tau }(n,h_C,r)\left(n\right)\left(w\right)<s<\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)\left(n\right)\left(w\right).\left(A\right)$

Since $\alpha C_{\tau }(n,h_C,r)\left(n\right)\left(w\right)<s$, by the definition of $\alpha C_{\tau }$, there is $g_B$ is r-fuzzy soft $\alpha$-closed with $h_C⊑g_B$ such that $\alpha C_{\tau }(n,h_C,r)\left(n\right)\left(w\right)\le g_B\left(n\right)\left(w\right)<s$. Since $h_C⊑g_B$, we have $\alpha C_{\tau }(n,h_C,r)⊑g_B$. Again, by the definition of $\alpha C_{\tau }$, we have $\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)⊑g_B$. Hence $\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)\left(n\right)\left(w\right)\le g_B\left(n\right)\left(w\right)<s$, it is a contradiction for $\left(A\right)$. Thus, $\alpha C_{\tau }(n,h_C,r)⊒\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)$. Then, $\alpha C_{\tau }(n,\alpha C_{\tau }(n,h_C,r),r)=\alpha C_{\tau }(n,h_C,r)$.

(5) Since $h_C$ and $g_B$$⊑h_C\bigsqcup g_B$, hence by (3), $\alpha C_{\tau }(n,h_C,r)⊑\alpha C_{\tau }(n,h_C\bigsqcup g_B,r)$ and $\alpha C_{\tau }(n,g_B,r)⊑\alpha C_{\tau }(n,h_C\bigsqcup g_B,r)$. Thus, $\alpha C_{\tau }(n,h_C\bigsqcup g_B,r)⊒\alpha C_{\tau }(n,h_C,r)\bigsqcup \alpha C_{\tau }(n,g_B,r)$.

(7) From (6) and $C_{\tau }(n,h_C,r)$ is r-fuzzy soft $\alpha$-closed set, hence $\alpha C_{\tau }(n,C_{\tau }(n,h_C,r),r)=C_{\tau }(n,h_C,r)$.

□

Theorem 2.2.

In a FSTS $(W,{\tau }_N)$, for each $h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$, we define a fuzzy soft $\alpha$-interior operator $\alpha I_{\tau }$$:N\times \widetilde{(W,N)}\times I_{\circ }\to \widetilde{(W,N)}$ as follows: $\alpha I_{\tau }(n,h_C,r)=\bigsqcup \{f_A\in \widetilde{(W,N)}:f_A⊑h_C,f_A\mathrm{is}r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}.$ Then, for each $g_B$ and $h_C\in \widetilde{(W,N)}$, the operator $\alpha I_{\tau }$ satisfies the following properties.

(1) $\alpha I_{\tau }(n,\tilde{N},r)=\tilde{N}$.

(2) $I_{\tau }(n,h_C,r)⊑\alpha I_{\tau }(n,h_C,r)⊑h_C$.

(3) $\alpha I_{\tau }(n,h_C,r)⊑\alpha I_{\tau }(n,g_B,r)$ if, $h_C⊑g_B$.

(4) $\alpha I_{\tau }(n,\alpha I_{\tau }(n,h_C,r),r)=\alpha I_{\tau }(n,h_C,r)$.

(5) $\alpha I_{\tau }(n,h_C,r)\sqcap \alpha I_{\tau }(n,g_B,r)⊒\alpha I_{\tau }(n,h_C\sqcap g_B,r)$.

(6) $\alpha I_{\tau }(n,h_C,r)=h_C$ iff $h_C$ is r-fuzzy soft $\alpha$-open.

(7) $\alpha I_{\tau }(n,h_C^c,r)={\left(\alpha C_{\tau }(n,h_C,r)\right)}^c$.

Proof. (1), (2), (3) and (6) are easily proved from the definition of $\alpha I_{\tau }$.

(4) and (5) are easily proved by a similar way in Theorem 2.1.

(7) For each $h_C\in \widetilde{(W,N)}$, $n\in N$ and $r\in I_0$, we have $\alpha I_{\tau }(n,h_C^c,r)=\bigsqcup \{f_A\in \widetilde{(W,N)}:f_A⊑h_C^c,f_A\mathrm{is}r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}$ = [$\sqcap \{f_A^c\in \widetilde{(W,N)}:h_C⊑f_A^c,f_A^c\mathrm{is}r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{closed}\}{]}^c$ = ${\left(\alpha C_{\tau }(n,h_C,r)\right)}^c$.

□

Definition 2.3.

Let $(W,{\tau }_N)$ be a FSTS, $r\in I_0$ and $g_B,h_C\in \widetilde{(W,N)}$. Then, we have:

(1) Two fuzzy soft sets $g_B$ and $h_C$ are called r-fuzzy soft $\alpha$-separated iff $g_B\neg \tilde{q}\alpha C_{\tau }(n,h_C,r)$ and $h_C\neg \tilde{q}\alpha C_{\tau }(n,g_B,r)$ for each $n\in N$.

(2) Any fuzzy soft set which cannot be expressed as the union of two r-fuzzy soft $\alpha$-separated sets is called an r-fuzzy soft $\alpha$-connected.

Theorem 2.3.

In a FSTS $(W,{\tau }_N)$, we have:

(1) If $f_A$ and $g_B$$\in \widetilde{(W,N)}$ are r-fuzzy soft $\alpha$-separated and $h_C$, $t_D$$\in \widetilde{(W,N)}$ such that $h_C⊑f_A$ and $t_D⊑g_B$, then $h_C$ and $t_D$ are r-fuzzy soft $\alpha$-separated.

(2) If $f_A\neg \tilde{q}{g}_B$ and either both are r-fuzzy soft $\alpha$-open or both r-fuzzy soft $\alpha$-closed, then $f_A$ and $g_B$ are r-fuzzy soft $\alpha$-separated.

(3) If $f_A$ and $g_B$ are either both r-fuzzy soft $\alpha$-open or both r-fuzzy soft $\alpha$-closed, then $f_A\sqcap g_B^c$ and $g_B\sqcap f_A^c$ are r-fuzzy soft $\alpha$-separated.

Proof. (1) and (2) are obvious.

(3) Let $f_A$ and $g_B$ be an r-fuzzy soft $\alpha$-open. Since $f_A\sqcap g_B^c⊑g_B^c$, $\alpha C_{\tau }(n,f_A\sqcap g_B^c,r)⊑g_B^c$ and hence $\alpha C_{\tau }(n,f_A\sqcap g_B^c,r)\neg \tilde{q}{g}_B$. Then, $\alpha C_{\tau }(n,f_A\sqcap g_B^c,r)\neg \tilde{q}(g_B\sqcap f_A^c).$

Again, since $g_B\sqcap f_A^c⊑f_A^c$, $\alpha C_{\tau }(n,g_B\sqcap f_A^c,r)⊑f_A^c$ and hence $\alpha C_{\tau }(n,g_B$$\sqcap f_A^c,r)\neg \tilde{q}{f}_A$. Then, $\alpha C_{\tau }(n,g_B\sqcap f_A^c,r)\neg \tilde{q}(f_A\sqcap g_B^c).$ Thus, $f_A\sqcap g_B^c$ and $g_B\sqcap f_A^c$ are r-fuzzy soft $\alpha$-separated. The other case follows similar lines.

□

Theorem 2.4.

In a FSTS $(W,{\tau }_N)$, then $f_A$, $g_B$$\in \widetilde{(W,N)}$ are r-fuzzy soft $\alpha$-separated iff there exist two r-fuzzy soft $\alpha$-open sets $h_C$ and $t_D$ such that $f_A⊑h_C$, $g_B⊑t_D$, $f_A\neg \tilde{q}{t}_D$ and $g_B\neg \tilde{q}{h}_C$.

Proof. (⇒) Let $f_A$ and $g_B$$\in \widetilde{(W,N)}$ be an r-fuzzy soft $\alpha$-separated, $f_A⊑{\left(\alpha C_{\tau }(n,g_B,r)\right)}^c=h_C$ and $g_B⊑{\left(\alpha C_{\tau }(n,f_A,r)\right)}^c=t_D$, where $t_D$ and $h_C$ are r-fuzzy soft $\alpha$-open, then $t_D\neg \tilde{q}\alpha C_{\tau }(n,f_A,r)$ and $h_C\neg \tilde{q}\alpha C_{\tau }(n,g_B,r)$. Thus, $g_B\neg \tilde{q}{h}_C$ and $f_A\neg \tilde{q}{t}_D$. Hence, we obtain the required result.

(⇐) Let $h_C$ and $t_D$ be an r-fuzzy soft $\alpha$-open such that $g_B⊑t_D$, $f_A⊑h_C$, $g_B\neg \tilde{q}{h}_C$ and $f_A\neg \tilde{q}{t}_D$. Then, $g_B⊑h_C^c$ and $f_A⊑t_D^c$. Hence, $\alpha C_{\tau }(n,g_B,r)⊑h_C^c$ and $\alpha C_{\tau }(n,f_A,r)⊑t_D^c$. Then, $\alpha C_{\tau }(n,g_B,r)\neg \tilde{q}{f}_A$ and $\alpha C_{\tau }(n,f_A,r)\neg \tilde{q}{g}_B$. Thus, $g_B$ and $f_A$ are r- fuzzy soft $\alpha$-separated. Hence, we obtain the required result.

□

Theorem 2.5.

In a FSTS $(W,{\tau }_N)$, if $g_B\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-connected such that $g_B⊑f_A⊑\alpha C_{\tau }(n,g_B,r)$, then $f_A$ is r-fuzzy soft $\alpha$-connected.

Proof. 

Suppose that $f_A$ is not r-fuzzy soft $\alpha$-connected, then there is r-fuzzy soft $\alpha$-separated sets $h_C^{*}$ and $t_D^{*}$$\in \widetilde{(W,N)}$ such that $f_A=h_C^{*}\bigsqcup t_D^{*}$. Let $h_C=g_B\sqcap h_C^{*}$ and $t_D=g_B\sqcap t_D^{*}$, then $g_B=t_D\bigsqcup h_C$. Since $h_C⊑h_C^{*}$ and $t_D⊑t_D^{*}$, hence by Theorem 2.3(1), $h_C$ and $t_D$ are r-fuzzy soft $\alpha$-separated, it is a contradiction. Thus, $f_A$ is r-fuzzy soft $\alpha$-connected, as required.

□

3. Continuity via r-Fuzzy Soft $\alpha$-Open Sets

Here, we investigate some properties of fuzzy soft $\alpha$-continuous mappings. Additionally, we introduce and study the notions of fuzzy soft almost (weakly) $\alpha$-continuous mappings, which are weaker forms of fuzzy soft $\alpha$-continuous mappings. Also, we show that fuzzy soft $\alpha$-continuity ⇒ fuzzy soft almost $\alpha$-continuity ⇒ fuzzy soft weakly $\alpha$-continuity.

Definition 3.1.

[26] Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs and $r\in I_o$. A fuzzy soft mapping ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ is called a fuzzy soft $\alpha$-continuous if, ${\phi }_{\psi }^{-1}\left(h_C\right)$ is r-fuzzy soft $\alpha$-open set for each $h_C\in \widetilde{(V,F)}$ with ${\eta }_k\left(h_C\right)\ge r$, $n\in N$, $(k=\psi (n\left)\right)\in F$.

Theorem 3.1.

Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs, and ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ be a fuzzy soft mapping. The following statements are equivalent for each $f_A\in \widetilde{(V,F)}$, $n\in N$, $(k=\psi (n\left)\right)\in F$ and $r\in I_{\circ }$:

(1) ${\phi }_{\psi }$ is fuzzy soft $\alpha$-continuous.

(2) For each $f_A$ with ${\eta }_k\left(f_A^c\right)\ge r$, ${\phi }_{\psi }^{-1}\left(f_A\right)$ is r-fuzzy soft $\alpha$-closed.

(3) $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$.

(4) ${\phi }_{\psi }^{-1}\left(I_{\eta }(k,f_A,r)\right)⊑\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)$.

(5) $C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r),r),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$.

Proof. (1) ⇔ (2) Follows from Remark 2.1 and ${\phi }_{\psi }^{-1}\left(f_A^c\right)={\left({\phi }_{\psi }^{-1}\left(f_A\right)\right)}^c$.

(2) ⇒ (3) Let $f_A\in \widetilde{(V,F)}$, hence by (2), ${\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$ is r-fuzzy soft $\alpha$-closed. Then, we obtain $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$.

(3) ⇔ (4) Follows from Theorem 2.2(7).

(3) ⇒ (5) Let $f_A\in \widetilde{(V,F)}$, hence by (3), we obtain $C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r),r),r)⊑\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$.

(5) ⇒ (1) Let $f_A\in \widetilde{(V,F)}$ with ${\eta }_k\left(f_A\right)\ge r$, hence by (3), we obtain ${\left({\phi }_{\psi }^{-1}\left(f_A\right)\right)}^c={\phi }_{\psi }^{-1}\left(f_A^c\right)⊒C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A^c\right),r),r),r)={\left(I_{\tau }(n,C_{\tau }(n,I_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r),r),r)\right)}^c$. Then, ${\phi }_{\psi }^{-1}\left(f_A\right)⊑I_{\tau }(n,C_{\tau }(n,I_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r),r),r)$, so ${\phi }_{\psi }^{-1}\left(f_A\right)$ is r-fuzzy soft $\alpha$-open. Hence, ${\phi }_{\psi }$ is fuzzy soft $\alpha$-continuous.

□

Lemma 3.1.

Every fuzzy soft continuous mapping [14] is fuzzy soft $\alpha$-continuous.

Proof. 

Follows from Definitions 1.6 and 3.1.

□

Remark 3.1.

The converse of Lemma 3.1 is not true, as shown by Example 3.1.

Example 3.1.

Let $W=\{w_1,w_2,w_3\}$, $N=\{n_1,n_2\}$ and define $f_N,g_N,h_N\in$$\widetilde{(W,N)}$ as: $f_N=\{(n_1,\{\frac{w_1}{0.4},\frac{w_2}{0.5},\frac{w_3}{0.5}\}),(n_2,\{\frac{w_1}{0.4},\frac{w_2}{0.5},\frac{w_3}{0.5}\})\}$, $g_N=\{(n_1,\{\frac{w_1}{0.3},\frac{w_2}{0.3},\frac{w_3}{0.4}\}),(n_2,\{\frac{w_1}{0.3},\frac{w_2}{0.3},\frac{w_3}{0.4}\})\}$, $h_N=\{(n_1,\{\frac{w_1}{0.3},\frac{w_2}{0.4},\frac{w_3}{0.4}\}),(n_2,\{\frac{w_1}{0.3},\frac{w_2}{0.4},\frac{w_3}{0.4}\})\}$. Define fuzzy soft topologies ${\tau }_N,{\eta }_N:N⟶{[0,1]}^{\widetilde{(W,N)}}$ as follows: $\forall n\in N$,

${\tau }_n\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=f_N,\hfill \\ \frac{2}{3},\mathrm{if}{t}_N=g_N,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$

${\eta }_n\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=f_N,\hfill \\ \frac{1}{3},\mathrm{if}{t}_N=h_N,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$

Then, the identity fuzzy soft mapping ${\phi }_{\psi }:(W,{\tau }_N)⟶(W,{\eta }_N)$ is fuzzy soft $\alpha$-continuous, but it is not fuzzy soft continuous.

Definition 3.2.

Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs. A fuzzy soft mapping ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ is called fuzzy soft almost (resp. weakly) $\alpha$-continuous if, for each $n_{w_s}\in \widetilde{P_s\left(W\right)}$ and each $g_B\in \widetilde{(V,F)}$ with ${\eta }_k\left(g_B\right)\ge r$ containing ${\phi }_{\psi }\left(n_{w_s}\right)$, there is $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-open set containing $n_{w_s}$ such that ${\phi }_{\psi }\left(h_C\right)⊑I_{\eta }(k,C_{\eta }(k,g_B,r),r)$ (resp. ${\phi }_{\psi }\left(h_C\right)⊑C_{\eta }(k,g_B,r)$), $n\in N$, $(k=\psi (n\left)\right)\in F$ and $r\in I_{\circ }$.

Lemma 3.2. (1) Every fuzzy soft $\alpha$-continuous mapping is fuzzy soft almost $\alpha$-continuous.

(2) Every fuzzy soft almost $\alpha$-continuous mapping is fuzzy soft weakly $\alpha$-continuous.

Proof. 

Follows from Definitions 3.1 and 3.2. □

Remark 3.2.

The converse of Lemma 3.2 is not true, as shown by Examples 3.2 and 3.3.

Example 3.2.

Let $W=\{w_1,w_2,w_3\}$, $N=\{n_1,n_2\}$ and define $g_N,h_N\in$$\widetilde{(W,N)}$ as follows: $g_N=\{(n_1,\{\frac{w_1}{0.5},\frac{w_2}{0.5},\frac{w_3}{0.4}\}),(n_2,\{\frac{w_1}{0.5},\frac{w_2}{0.5},\frac{w_3}{0.4}\})\}$, $h_N=\{(n_1,\{\frac{w_1}{0.3},\frac{w_2}{0.3},\frac{w_3}{0.4}\}),(n_2,\{\frac{w_1}{0.3},\frac{w_2}{0.3},\frac{w_3}{0.4}\})\}$. Define fuzzy soft topologies ${\tau }_N,{\eta }_N:N⟶{[0,1]}^{\widetilde{(W,N)}}$ as follows: $\forall n\in N$,

${\tau }_n\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=g_N,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$

${\eta }_n\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{2},\mathrm{if}{t}_N=g_N,\hfill \\ \frac{1}{3},\mathrm{if}{t}_N=h_N,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$

Then, the identity fuzzy soft mapping ${\phi }_{\psi }:(W,{\tau }_N)⟶(W,{\eta }_N)$ is fuzzy soft almost $\alpha$-continuous, but it is not fuzzy soft $\alpha$-continuous.

Example 3.3.

Let $W=\{w_1,w_2,w_3\}$, $N=\{n_1,n_2\}$ and define $g_N,h_N\in$$\widetilde{(W,N)}$ as follows: $g_N=\{(n_1,\{\frac{w_1}{0.5},\frac{w_2}{0.5},\frac{w_3}{0.5}\}),(n_2,\{\frac{w_1}{0.5},\frac{w_2}{0.5},\frac{w_3}{0.5}\})\}$, $h_N=\{(n_1,\{\frac{w_1}{0.3},\frac{w_2}{0},\frac{w_3}{0.5}\}),(n_2,\{\frac{w_1}{0.3},\frac{w_2}{0},\frac{w_3}{0.5}\})\}$. Define fuzzy soft topologies ${\tau }_N,{\eta }_N:N⟶{[0,1]}^{\widetilde{(W,N)}}$ as follows: $\forall n\in N$,

${\tau }_n\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{2}{3},\mathrm{if}{t}_N=g_N,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$

${\eta }_n\left(t_N\right)=$$\left\{\begin{array}{c}1,\mathrm{if}{t}_N\in \{\Phi ,\tilde{N}\},\hfill \\ \frac{1}{3},\mathrm{if}{t}_N=h_N,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$

Then, the identity fuzzy soft mapping ${\phi }_{\psi }:(W,{\tau }_N)⟶(W,{\eta }_N)$ is fuzzy soft weakly $\alpha$-continuous, but it is not fuzzy soft almost $\alpha$-continuous.

Theorem 3.2.

Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs, and ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ be a fuzzy soft mapping. The following statements are equivalent for each $f_A\in \widetilde{(V,F)}$, $n\in N$, $(k=\psi (n\left)\right)\in F$ and $r\in I_{\circ }$:

(1) ${\phi }_{\psi }$ is fuzzy soft almost $\alpha$-continuous.

(2) ${\phi }_{\psi }^{-1}\left(f_A\right)$ is r-fuzzy soft $\alpha$-open, for each $f_A$ is r-fuzzy soft regularly open.

(3) ${\phi }_{\psi }^{-1}\left(f_A\right)$ is r-fuzzy soft $\alpha$-closed, for each $f_A$ is r-fuzzy soft regularly closed.

(4) $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$, for each $f_A$ is r-fuzzy soft $\beta$-open.

(5) $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$, for each $f_A$ is r-fuzzy soft semi-open.

(6) $\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right),r)⊒{\phi }_{\psi }^{-1}\left(f_A\right)$, for each $f_A$ with ${\eta }_k\left(f_A\right)\ge r$.

Proof. (1) ⇒ (2) Let $n_{w_s}\in \widetilde{P_s\left(W\right)}$ and $f_A\in \widetilde{(V,F)}$ be an r-fuzzy soft regularly open set containing ${\phi }_{\psi }\left(n_{w_s}\right)$, hence by (1), there is $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-open set containing $n_{w_s}$ such that ${\phi }_{\psi }\left(h_C\right)⊑I_{\eta }(k,C_{\eta }(k,f_A,r),r)$.

Thus, $h_C⊑{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right)={\phi }_{\psi }^{-1}\left(f_A\right)$ and $n_{w_s}\tilde{\in }{h}_C⊑{\phi }_{\psi }^{-1}\left(f_A\right)$. Then, $n_{w_s}\tilde{\in }{I}_{\tau }(n,C_{\tau }(n,I_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r),r),r)$ and ${\phi }_{\psi }^{-1}\left(f_A\right)⊑I_{\tau }(n,C_{\tau }(n,I_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r),r),r)$. Therefore, ${\phi }_{\psi }^{-1}\left(f_A\right)$ is r-fuzzy soft $\alpha$-open set.

(2) ⇒ (3) Let $f_A$ be an r-fuzzy soft regularly closed set, hence by (2), ${\phi }_{\psi }^{-1}\left(f_A^c\right)={\left({\phi }_{\psi }^{-1}\left(f_A\right)\right)}^c$ is r-fuzzy soft $\alpha$-open set. Then, ${\phi }_{\psi }^{-1}\left(f_A\right)$ is r-fuzzy soft $\alpha$-closed set.

(3) ⇒ (4) Let $f_A$ be an r-fuzzy soft $\beta$-open set. Since $C_{\eta }(k,f_A,r)$ is r-fuzzy soft regularly closed set, hence by (3), ${\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$ is r-fuzzy soft $\alpha$-closed set. Since ${\phi }_{\psi }^{-1}\left(f_A\right)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$, then we have $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$.

(4) ⇒ (5) This is obvious from each r-fuzzy soft semi-open set is r-fuzzy soft $\beta$-open.

(5) ⇒ (3) Let $f_A$ be an r-fuzzy soft regularly closed set, hence $f_A$ is r-fuzzy soft semi-open. Then by (5), $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(f_A\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)={\phi }_{\psi }^{-1}\left(f_A\right)$. Therefore, ${\phi }_{\psi }^{-1}\left(f_A\right)$ is r-fuzzy soft $\alpha$-closed set.

(3) ⇒ (6) Let $f_A\in \widetilde{(V,F)}$ with ${\eta }_k\left(f_A\right)\ge r$ and $n_{w_s}\tilde{\in }{\phi }_{\psi }^{-1}\left(f_A\right)$, then we have $n_{w_s}\tilde{\in }{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right)$. Since ${\left[I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right]}^c$ is r-fuzzy soft regularly closed set, hence by (3), ${\phi }_{\psi }^{-1}\left({\left[I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right]}^c\right)$ is r-fuzzy soft $\alpha$-closed set. Thus, ${\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right)$ is r-fuzzy soft $\alpha$-open set and $n_{w_s}\tilde{\in }\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right),r)$. Then, ${\phi }_{\psi }^{-1}\left(f_A\right)⊑\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right),r)$.

(6) ⇒ (1) Let $n_{w_s}\in \widetilde{P_s\left(W\right)}$ and $f_A\in \widetilde{(V,F)}$ with ${\eta }_k\left(f_A\right)\ge r$ containing ${\phi }_{\psi }\left(n_{w_s}\right)$, hence by (6), ${\phi }_{\psi }^{-1}\left(f_A\right)⊑\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right),r)$.

Since $n_{w_s}\tilde{\in }{\phi }_{\psi }^{-1}\left(f_A\right)$, then we obtain $n_{w_s}\tilde{\in }\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right),r)=h_C$ (say). Hence, there is $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-open set containing $n_{w_s}$ such that ${\phi }_{\psi }\left(h_C\right)⊑I_{\eta }(k,C_{\eta }(k,f_A,r),r)$. Therefore, ${\phi }_{\psi }$ is fuzzy soft almost $\alpha$-continuous.

□

In a similar way, we can prove the following theorem.

Theorem 3.3.

Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs, and ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ be a fuzzy soft mapping. The following statements are equivalent for each $f_A\in \widetilde{(V,F)}$, $n\in N$, $(k=\psi (n\left)\right)\in F$ and $r\in I_{\circ }$:

(1) ${\phi }_{\psi }$ is fuzzy soft weakly $\alpha$-continuous.

(2) $I_{\tau }(n,C_{\tau }(n,I_{\tau }(n,{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right),r),r),r)⊒{\phi }_{\psi }^{-1}\left(f_A\right)$, if ${\eta }_k\left(f_A\right)\ge r$.

(3) $C_{\tau }(n,I_{\tau }(n,C_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,f_A,r)\right),r),r),r)⊑{\phi }_{\psi }^{-1}\left(f_A\right)$, if ${\eta }_k\left(f_A^c\right)\ge r$.

(4) $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,f_A,r)\right),r)⊑{\phi }_{\psi }^{-1}\left(f_A\right)$, if ${\eta }_k\left(f_A^c\right)\ge r$.

(5) $\alpha C_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,f_A,r),r)\right),r)⊑{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right)$.

(6) $\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(C_{\eta }(k,I_{\eta }(k,f_A,r),r)\right),r)⊒{\phi }_{\psi }^{-1}\left(I_{\eta }(k,f_A,r)\right)$.

(7) ${\phi }_{\psi }^{-1}\left(f_A\right)⊑\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(C_{\eta }(k,f_A,r)\right),r)$, if ${\eta }_k\left(f_A\right)\ge r$.

Remark 3.3.

From the previous definitions and results, we can summarize the relationships among different types of fuzzy soft continuity as in the next diagram.

$$\mathrm{continuity}\Rightarrow \alpha -\mathrm{continuity}$$

$$\Downarrow \Downarrow$$

$$\mathrm{almost}\mathrm{continuity}\Rightarrow \mathrm{almost}\alpha -\mathrm{continuity}$$

$$\Downarrow \Downarrow$$

$$\mathrm{weakly}\mathrm{continuity}\Rightarrow \mathrm{weakly}\alpha -\mathrm{continuity}$$

Proposition 3.1.

Let $(W,{\tau }_N)$, $(V,{\eta }_F)$ and $(U,{\gamma }_E)$ be a FSTSs, and ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$, ${\phi }_{{\psi }^{*}}^{*}:\widetilde{(V,F)}⟶\widetilde{(U,E)}$ be two fuzzy soft functions. Then, the composition ${\phi }_{{\psi }^{*}}^{*}\circ {\phi }_{\psi }$ is fuzzy soft almost $\alpha$-continuous if, ${\phi }_{\psi }$ is fuzzy soft $\alpha$-continuous and ${\phi }_{{\psi }^{*}}^{*}$ is fuzzy soft almost continuous (resp. continuous).

Proof. 

The proof is obvious. □

Let $\mathcal{H}$ and $\mathcal{I}$$:N\times \widetilde{(W,N)}\times I_{\circ }\to \widetilde{(W,N)}$ be operators on $\widetilde{(W,N)}$, and $\mathcal{J}$ and $\mathcal{K}$$:F\times \widetilde{(V,F)}\times I_{\circ }\to \widetilde{(V,F)}$ be operators on $\widetilde{(V,F)}$.

Definition 3.3.

[26] Let $(W,{\tau }_N)$ and $(V,{\eta }_F)$ be a FSTSs. ${\phi }_{\psi }:\widetilde{(W,N)}⟶\widetilde{(V,F)}$ is said to be a fuzzy soft $(\mathcal{H},\mathcal{I},\mathcal{J},\mathcal{K})$-continuous mapping if, $\mathcal{H}[n,{\phi }_{\psi }^{-1}\left(\mathcal{K}(k,h_C,r)\right),r]\overline{\sqcap }\mathcal{I}[n,{\phi }_{\psi }^{-1}\left(\mathcal{J}(k,h_C,r)\right),r]$$=\Phi$ for each $h_C\in \widetilde{(V,F)}$ with ${\eta }_k\left(h_C\right)\ge r$, $n\in N$ and $(k=\psi (n\left)\right)\in F$.

In (2023), Alshammari et al. [26] defined the notion of fuzzy soft $\alpha$-continuous mappings: ${\phi }_{\psi }^{-1}\left(h_C\right)⊑I_{\tau }(n,C_{\tau }(n,I_{\tau }(n,{\phi }_{\psi }^{-1}\left(h_C\right),r),r),r)$, for each $h_C\in \widetilde{(V,F)}$ with ${\eta }_k\left(h_C\right)\ge r$. We can see that Definition 3.3 generalizes the concept of fuzzy soft continuous functions, when we choose $\mathcal{H}$ = identity operator, $\mathcal{I}$ = interior closure interior operator, $\mathcal{J}$ = identity operator and $\mathcal{K}$ = identity operator.

A historical justification of Definition 3.3:

(1) In Section 3, we obtained the notion of fuzzy soft almost $\alpha$-continuous mappings: ${\phi }_{\psi }^{-1}\left(h_C\right)⊑\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(I_{\eta }(k,C_{\eta }(k,h_C,r),r)\right),r)$, for each $h_C\in \widetilde{(V,F)}$ with ${\eta }_k\left(h_C\right)\ge r$. Here, $\mathcal{H}$ = identity operator, $\mathcal{I}$ = $\alpha$-interior operator, $\mathcal{J}$ = interior closure operator and $\mathcal{K}$ = identity operator.

(2) In Section 3, we obtained the notion of fuzzy soft weakly $\alpha$-continuous mappings: ${\phi }_{\psi }^{-1}\left(h_C\right)⊑\alpha I_{\tau }(n,{\phi }_{\psi }^{-1}\left(C_{\eta }(k,h_C,r)\right),r)$, for each $h_C\in \widetilde{(V,F)}$ with ${\eta }_k\left(h_C\right)\ge r$. Here, $\mathcal{H}$ = identity operator, $\mathcal{I}$ = $\alpha$-interior operator, $\mathcal{J}$ = closure operator and $\mathcal{K}$ = identity operator.

4. New Types of Fuzzy Soft Compactness

Here, several types of fuzzy soft compactness via r-fuzzy soft $\alpha$-open sets are given, and the relationships between them are studied.

Definition 4.1.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. Then, $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft compact iff for every family ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(W,N)}|{\tau }_n\left({\left(g_B\right)}_{\delta }\right)\ge r\mathrm{for}\mathrm{each}n\in N\}}_{\delta \in \Delta }$ such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{\left(g_B\right)}_{\delta }$.

Definition 4.2.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. Then, $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft $\alpha$-compact iff for every family ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(W,N)}|{\left(g_B\right)}_{\delta }\mathrm{is}r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}}_{\delta \in \Delta }$ such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{\left(g_B\right)}_{\delta }$.

Lemma 4.1.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-compact, then $h_C$ is r-fuzzy soft compact.

Proof. 

Follows from Definitions 4.1 and 4.2.

□

Theorem 4.1.

Let ${\phi }_{\psi }:(W,{\tau }_N)⟶(V,{\eta }_F)$ be a fuzzy soft $\alpha$-continuous mapping. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-compact, then ${\phi }_{\psi }\left(h_C\right)$ is r-fuzzy soft compact.

Proof. 

Let ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(V,F)}|{\eta }_k\left({\left(g_B\right)}_{\delta }\right)\ge r\}}_{\delta \in \Delta }$ with ${\phi }_{\psi }\left(h_C\right)⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$ for each $k\in F$. Then, $\{{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)\in \widetilde{(W,N)}|{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)$ is ${r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}}_{\delta \in \Delta }$ (by ${\phi }_{\psi }$ is fuzzy soft $\alpha$-continuous) such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)$. Since $h_C$ is r-fuzzy soft $\alpha$-compact, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)$. Then, ${\phi }_{\psi }\left(h_C\right)⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{\left(g_B\right)}_{\delta }$. Hence, the proof is completed. □

Definition 4.3.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. Then, $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft almost compact iff for every family ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(W,N)}|{\tau }_n\left({\left(g_B\right)}_{\delta }\right)\ge r\}}_{\delta \in \Delta }$ such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{C}_{\tau }(n,{\left(g_B\right)}_{\delta },r)$ for each $n\in N$.

Definition 4.4.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. Then, $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft almost $\alpha$-compact iff for every family ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(W,N)}|{\left(g_B\right)}_{\delta }\mathrm{is}r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}}_{\delta \in \Delta }$ such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{C}_{\tau }(n,{\left(g_B\right)}_{\delta },r)$ for each $n\in N$.

Lemma 4.2.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft almost $\alpha$-compact, then $h_C$ is r-fuzzy soft almost compact.

Proof. 

Follows from Definitions 4.3 and 4.4.

□

Lemma 4.3.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-compact (resp. compact), then $h_C$ is r-fuzzy soft almost $\alpha$-compact (resp. almost compact).

Proof. 

Follows from Definitions 4.1–4.4.

□

Theorem 4.2.

Let ${\phi }_{\psi }:(W,{\tau }_N)⟶(V,{\eta }_F)$ be a fuzzy soft continuous mapping. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft almost $\alpha$-compact, then ${\phi }_{\psi }\left(h_C\right)$ is r-fuzzy soft almost compact.

Proof. 

Let ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(V,F)}|{\eta }_k\left({\left(g_B\right)}_{\delta }\right)\ge r\}}_{\delta \in \Delta }$ with ${\phi }_{\psi }\left(h_C\right)⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$ for each $k\in F$. Then, $\{{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)\in \widetilde{(W,N)}|{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)$ is ${r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}}_{\delta \in \Delta }$ (by ${\phi }_{\psi }$ is fuzzy soft $\alpha$-continuous) such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)$. Since $h_C$ is r-fuzzy soft almost $\alpha$-compact, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{C}_{\tau }(n,{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right),r)$. Since ${\phi }_{\psi }$ is fuzzy soft continuous mapping, it follows

$${\bigsqcup }_{\delta \in {\Delta }_{\circ }}{C}_{\tau }(n,{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right),r)⊑$$

$${\bigsqcup }_{\delta \in {\Delta }_{\circ }}{\phi }_{\psi }^{-1}\left(C_{\eta }(k,{\left(g_B\right)}_{\delta },r)\right)=$$

$${\phi }_{\psi }^{-1}\left({\bigsqcup }_{\delta \in {\Delta }_{\circ }}{C}_{\eta }(k,{\left(g_B\right)}_{\delta },r)\right).$$

Then, ${\phi }_{\psi }\left(h_C\right)⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{C}_{\eta }(k,{\left(g_B\right)}_{\delta },r)$. Hence, the proof is completed. □

Definition 4.5.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. Then, $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft nearly compact iff for every family ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(W,N)}|{\tau }_n\left({\left(g_B\right)}_{\delta }\right)\ge r\}}_{\delta \in \Delta }$ such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{I}_{\tau }(n,C_{\tau }(n,{\left(g_B\right)}_{\delta },r),r)$ for each $n\in N$.

Definition 4.6.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. Then, $h_C\in \widetilde{(W,N)}$ is called an r-fuzzy soft nearly $\alpha$-compact iff for every family ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(W,N)}|{\left(g_B\right)}_{\delta }\mathrm{is}r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}}_{\delta \in \Delta }$ such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{I}_{\tau }(n,C_{\tau }(n,{\left(g_B\right)}_{\delta },r),r)$ for each $n\in N$.

Lemma 4.4.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft nearly $\alpha$-compact, then $h_C$ is r-fuzzy soft nearly compact.

Proof. 

Follows from Definitions 4.5 and 4.6.

□

Lemma 4.5.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft $\alpha$-compact (resp. compact), then $h_C$ is r-fuzzy soft nearly $\alpha$-compact (resp. nearly compact).

Proof. 

Follows from Definitions 4.1, 4.2, 4.5 and 4.6.

□

Theorem 4.3.

Let ${\phi }_{\psi }:(W,{\tau }_N)⟶(V,{\eta }_F)$ be a fuzzy soft continuous and fuzzy soft open mapping. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft nearly $\alpha$-compact, then ${\phi }_{\psi }\left(h_C\right)$ is r-fuzzy soft nearly compact.

Proof. 

Let ${\{{\left(g_B\right)}_{\delta }\in \widetilde{(V,F)}|{\eta }_k\left({\left(g_B\right)}_{\delta }\right)\ge r\}}_{\delta \in \Delta }$ with ${\phi }_{\psi }\left(h_C\right)⊑{\bigsqcup }_{\delta \in \Delta }{\left(g_B\right)}_{\delta }$ for each $k\in F$. Then, $\{{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)\in \widetilde{(W,N)}|{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)$ is ${r-\mathrm{fuzzy}\mathrm{soft}\alpha -\mathrm{open}\}}_{\delta \in \Delta }$ (by ${\phi }_{\psi }$ is fuzzy soft $\alpha$-continuous) such that $h_C⊑{\bigsqcup }_{\delta \in \Delta }{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right)$. Since $h_C$ is r-fuzzy soft nearly $\alpha$-compact, there is a finite subset ${\Delta }_{\circ }$ of $\Delta$ such that $h_C⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{I}_{\tau }(n,C_{\tau }(n,{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right),r),r)$. Since ${\phi }_{\psi }$ is fuzzy soft continuous and fuzzy soft open mapping, it follows

$${\phi }_{\psi }\left(h_C\right)⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{\phi }_{\psi }\left(I_{\tau }(n,C_{\tau }(n,{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right),r),r)\right)$$

$$⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{I}_{\eta }(k,{\phi }_{\psi }\left(C_{\tau }(n,{\phi }_{\psi }^{-1}\left({\left(g_B\right)}_{\delta }\right),r)\right),r)$$

$$⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{I}_{\eta }(k,{\phi }_{\psi }\left({\phi }_{\psi }^{-1}\left(C_{\eta }(k,{\left(g_B\right)}_{\delta },r)\right)\right),r)$$

$$⊑{\bigsqcup }_{\delta \in {\Delta }_{\circ }}{I}_{\eta }(k,C_{\eta }(k,{\left(g_B\right)}_{\delta },r),r).$$

Hence, the proof is completed. □

Lemma 4.6.

Let $(W,{\tau }_N)$ be a FSTS and $r\in I_{\circ }$. If $h_C\in \widetilde{(W,N)}$ is r-fuzzy soft nearly $\alpha$-compact (resp. nearly compact), then $h_C$ is r-fuzzy soft almost $\alpha$-compact (resp. almost compact).

Proof. 

Follows from Definitions 4.3, 4.4, 4.5 and 4.6.

□

Remark 4.1.

From the previous definitions and results, we can summarize the relationships among different types of fuzzy soft compactness as in the next diagram.

$$\alpha -\mathrm{compactness}\Rightarrow \mathrm{compactness}$$

$$\Downarrow \Downarrow$$

$$\mathrm{nearly}\alpha -\mathrm{compactness}\Rightarrow \mathrm{nearly}\mathrm{compactness}$$

$$\Downarrow \Downarrow$$

$$\mathrm{almost}\alpha -\mathrm{compactness}\Rightarrow \mathrm{almost}\mathrm{compactness}$$

5. Conclusions and Future Work

The main achievements of this study are:

$\left(\mathbf{1}\right)$ In Section 2, the concepts of fuzzy soft $\alpha$-closure ($\alpha$-interior) operators are introduced in fuzzy soft topological spaces based on the paper Aygünoǧlu et al. [14], and some of their basic properties have been investigated. Furthermore, the notion of r-fuzzy soft $\alpha$-connected sets is defined and studied with help of fuzzy soft $\alpha$-closure operators.

$\left(\mathbf{2}\right)$ In Section 3, some properties of fuzzy soft $\alpha$-continuous mappings are obtained between two fuzzy soft topological spaces $(W,{\tau }_N)$ and $(V,{\eta }_F)$. Moreover, as a weaker forms of the notion of fuzzy soft $\alpha$-continuous mappings, the notions of fuzzy soft almost (weakly) $\alpha$-continuous mappings are introduced, and some of their basic properties and characterizations have been investigated. Also, we show that fuzzy soft $\alpha$-continuity ⇒ fuzzy soft almost $\alpha$-continuity ⇒ fuzzy soft weakly $\alpha$-continuity, and we have the following results:

• Fuzzy soft $(i{d}_W,I_{\tau }\left(C_{\tau }\left(I_{\tau }\right)\right),i{d}_V,i{d}_V)$-continuous mapping is fuzzy soft $\alpha$-continuous.

• Fuzzy soft $(i{d}_W,\alpha I_{\tau },I_{\eta }\left(C_{\eta }\right),i{d}_V)$-continuous mapping is fuzzy soft almost $\alpha$-continuous.

• Fuzzy soft $(i{d}_W,\alpha I_{\tau },C_{\eta },i{d}_V)$-continuous mapping is fuzzy soft weakly $\alpha$-continuous.

$\left(\mathbf{3}\right)$ In Section 4, several types of soft compactness via r-fuzzy soft $\alpha$-open sets are explored, and the relationships between them are studied.

In upcoming manuscripts, we will use the fuzzy soft $\alpha$-closure operator to define some new separation axioms on fuzzy soft topological space based on the paper Aygünoǧlu et al. [14]. Additionally, we shall discuss some of the notions given here in the frames of fuzzy soft r-minimal structures [20].