On fuzzy soft $\beta$-continuity and $\beta$-irresoluteness: some new results

1.   Introduction and preliminaries

The theory of soft set was pioneered by Molodtsov ^[1], which is a completely novel approach for modeling uncertainty and vagueness. He demonstrated some applications of this theory in solving some practical problems in engineering, economics, medical science, social science, etc. The concept of soft sets was used to define soft topological spaces in ^[2]. The study in ^[2] was particularly important in the development of the field of soft topology (for more details, see ^[3,4,5,6]). Generalizations of soft open sets play an effective role in soft topology through their use to improve on some known results or to open the door to redefine and investigate some of the soft topological concepts such as soft continuity ^[7], soft connectedness ^[8,9], soft separation axioms ^[10,11], etc. Akdag and Ozkan ^[12] initiated and studied the concept of soft $\alpha$-open sets on soft topological spaces. The concept of soft $\beta$-open sets was studied by the authors of ^[13,14], and some properties of soft $\beta$-continuity were investigated. Also, the concepts of somewhere dense and Q-sets were defined and studied by the authors of ^[15,16]. Al-shami et al. ^[17] initiated the concept of weakly soft semi-open sets and studied its main properties. Also, Al-shami et al. ^[18] defined and studied the concept of weakly soft $\beta$-open sets. Kaur et al. ^[19] introduced a new approach to studying soft continuous mappings using an induced mapping based on soft sets. Al Ghour and Al-Mufarrij ^[20] defined new concepts of mappings over soft topological spaces: soft somewhat-$r$-continuity and soft somewhat-$r$-openness. Ameen et al. ^[21] explored more properties of soft somewhere dense continuity.

The notion of fuzzy soft sets was introduced by Maji et al. ^[22], which combines fuzzy sets ^[23] and soft sets ^[1]. Based on fuzzy topologies in the sense of Šostak ^[24], the notion of fuzzy soft topology is defined and some properties such as fuzzy soft continuity, fuzzy soft interior (closure) set, and fuzzy soft subspace are introduced in ^[25,26]. The notion of $r$-fuzzy soft regularly open sets was defined and studied by Çetkin and Aygün ^[27]. In addition, the notions of $r$-fuzzy soft $\beta$-open (resp., pre-open) sets were introduced by Taha ^[28]. A new approach to studying separation and regularity axioms via fuzzy soft sets was introduced by the author of ^[29,30] based on the paper by Aygünoǧlu et al. ^[25].

The main contribution of this study is arranged as follows:

● In Section 2, we are going to present the notions of fuzzy soft $\beta$-closure ($\beta$-interior) operators in fuzzy soft topological spaces based on the article by Aygünoǧlu et al. ^[25] and study some properties of them. Also, the concept of $r$-fuzzy soft $\beta$-connected sets was defined and studied.

● In Section 3, we investigate some properties of a fuzzy soft $\beta$-continuity. Moreover, we explore and study the notions of fuzzy soft almost (weakly) $\beta$-continuous functions, which are weaker forms of fuzzy soft $\beta$-continuous functions. Also, we show that fuzzy soft $\beta$-continuity $\Rightarrow$ fuzzy soft almost $\beta$-continuity $\Rightarrow$ fuzzy soft weakly $\beta$-continuity, but the converse may not be true.

● In Section 4, we introduce the notions of fuzzy soft $\beta$-irresolute (resp., strongly $\beta$-irresolute, $\beta$-irresolute open, $\beta$-irresolute closed, and $\beta$-irresolute homeomorphism) functions between two fuzzy soft topological spaces $(V, \eta_F)$ and $(U, \tau_E)$ and investigate some properties of these functions. Additionally, the relationships between these classes of functions are considered with the help of some examples.

● In the end, we close this study with some conclusions and open a door to suggest some future papers in Section 5.

In this study, nonempty sets will be denoted by $V$, $U$, etc. $F$ is the set of all parameters for $V$ and $B \subseteq F$. The family of all fuzzy sets on $V$ is denoted by $I^{V}$ (where $I_{\circ} = (0, 1], \; I = [0, 1]$), and for $t\in I,$ $\underline{t}(v) = t$, for all $v\in V.$ The following notions and results will be used in the next sections:

Definition 1.1. ^[25,31,32] A fuzzy soft set $g_B$ on $V$ is a function from $F$ to $I^V$ such that $g_B(k)$ is a fuzzy set on $V$, for each $k\in B$ and $g_B(k) = \underline{0}$, if $k\not\in B$. The family of all fuzzy soft sets on $V$ is denoted by $\widetilde{(V, F)}$. In ^[33], the difference between two fuzzy soft sets $g_B$ and $f_A$ is a fuzzy soft set, defined as follows, for each $k\in F$:

$(g_B \; \overline{\sqcap}\; f_A)(k) = \left\{ \begin{array}{rcl} \underline{0}, & \mbox{if}\; g_B(k) \le f_A(k), \\ g_B(k) \wedge (f_A(k))^c, & \mbox{otherwise.} \end{array}\right.$

Definition 1.2. ^[34] A fuzzy soft point $k_{v_t}$ on $V$ is a fuzzy soft set, defined as follows:

where $v_{t}$ is a fuzzy point on $V$. $k_{v_t}$ is said to belong to a fuzzy soft set $g_B$, denoted by $k_{v_t}\tilde{\in} g_B$, if $t\le g_B(k)(v)$. The family of all fuzzy soft points on $V$ is denoted by $\widetilde{P_t(V)}$.

Definition 1.3. ^[35] A fuzzy soft point $k_{v_t} \in \widetilde{P_t(V)}$ is called a soft quasi-coincident with $g_B\in \widetilde{(V, F)}$, denoted by $k_{v_t} \; \nabla\; g_B$, if $t+g_B(k)(v) > 1$. A fuzzy soft set $f_A\in \widetilde{(V, F)}$ is called a soft quasi-coincident with $g_B\in \widetilde{(V, F)}$, denoted by $f_A \; \nabla\; g_B$, if there is $k\in F$ and $v\in V$ such that, $f_A(k)(v)+g_B(k)(v) > 1$, if $f_A$ is not soft quasi-coincident with $g_B$, $f_A \; \overline{\nabla}\; g_B$.

Definition 1.4. ^[25] A function $\eta: F \longrightarrow [0, 1]^{\widetilde{ {(V, F)}}}$ is said to be a fuzzy soft topology on $V$ if it satisfies the following, for each $k\in F$:

(1) $\eta_{k}(\Phi) = \eta_{k}(\widetilde{F}) = 1,$

(2) $\eta_{k}(g_B \sqcap f_A)\geq \eta_{k}(g_B)\wedge \eta_{k} (f_A),$ for each $g_{B}, f_{A}\in \widetilde{(V, F)},$

(3) $\eta_{k}(\sqcup_{\delta\in \Delta}(g_B)_\delta)\ge \wedge_{\delta\in\Delta}\eta_{k}((g_B)_\delta),$ for each $(g_B)_\delta\in \widetilde{(V, F)}, \delta\in \Delta.$

Thus, $(V, \eta_F)$ is said to be a fuzzy soft topological space (briefly, FSTS) based on the sense of Šostak ^[24].

Definition 1.5. ^[25] Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be an FSTSs. A fuzzy soft function $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be fuzzy soft continuous if $\eta_{k}(\varphi_\psi^{-1}(g_B))\ge \tau_{e}(g_B)$ for each $g_B \in \widetilde{(U, E)}$, $k\in F$, and $(e = \psi(k))\in E$.

Definition 1.6. ^[26,27] In an FSTS $(V, \eta_{F})$, for each $g_B \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$, we define the fuzzy soft operators $C_{\eta}$ and $I_{\eta}$ $: F \times\widetilde{(V, F)}\times I_{\circ}\rightarrow \widetilde{(V, F)}$ as follows:

$C_{\eta}(k, g_B, r) = \sqcap \; \{f_A \in \widetilde{(V, F)} : g_B \sqsubseteq f_A, \; \eta_{k}(f^c_{A})\ge r \}$,

$I_{\eta}(k, g_B, r) = \sqcup \; \{f_A \in \widetilde{(V, F)} : f_A \sqsubseteq g_B, \; \eta_{k}(f_A)\ge r \}.$

Definition 1.7. Let $(V, \eta_{F})$ be an FSTS and $r \in I_{0}$. A fuzzy soft set $g_B \in \widetilde{(V, F)}$ is said to be $r$-fuzzy soft $\beta$-open ^[28] (resp., pre-open ^[28], semi-open ^[36], and regularly open ^[27]) if $g_B \sqsubseteq C_{\eta }(k, I_{\eta }(k, C_{\eta }(k, g_B, r), r), r)$ (resp., $g_B \sqsubseteq I_{\eta }(k, C_{\eta }(k, g_B, r), r)$, $g_B \sqsubseteq C_{\eta }(k, I_{\eta }(k, g_B, r), r)$, and $g_B = I_{\eta }(k, C_{\eta }(k, g_B, r), r)$) for each $k\in F$.

Definition 1.8. ^[27] Let $(V, \eta_{F})$ be an FSTS and $r \in I_{0}$. A fuzzy soft set $g_B \in \widetilde{(V, F)}$ is said to be an $r$-fuzzy soft regularly closed if $g_B = C_{\eta }(k, I_{\eta }(k, g_B, r), r)$ for each $k\in F$.

Remark 1.1. ^[28] From the previous definition, we can summarize the relationships among different types of fuzzy soft sets as in the next diagram.

Definition 1.9. ^[36] Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be an FSTSs. A fuzzy soft function $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be a fuzzy soft almost (resp., weakly) continuous if, for each $k_{v_t}\in\widetilde{P_t(V)}$ and each $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$ containing $\varphi_\psi(k_{v_t})$, there is $f_A \in \widetilde{(V, F)}$ with $\eta_{k}(f_A)\ge r$ containing, $k_{v_t}$ such that $\varphi_\psi(f_A)\sqsubseteq I_{\tau}(e, C_{\tau}(e, g_B, r), r)$ (resp., $\varphi_\psi(f_A)\sqsubseteq C_{\tau}(e, g_B, r)$).

Remark 1.2. ^[36] From Definitions 1.5 and 1.9, we have: fuzzy soft continuity $\Rightarrow$ fuzzy soft almost continuity $\Rightarrow$ fuzzy soft weakly continuity.

The basic results and definitions that we need in the next sections are found in ^[25,26].

2.   Some properties of $r$-fuzzy soft $\beta$-closed sets

Here, we introduce the concept of $r$-fuzzy soft $\beta$-closed sets in FSTSs based on the sense of Šostak ^[24] and investigate some properties of them. Also, we define and study the concepts of fuzzy soft $\beta$-closure ($\beta$-interior) operators. Moreover, the concept of $r$-fuzzy soft $\beta$-connected sets is defined and characterized.

Definition 2.1. Let $(V, \eta_{F})$ be an FSTS. A fuzzy soft set $g_B \in \widetilde{(V, F)}$ is said to be an $r$-fuzzy soft $\beta$-closed if $I_{\eta }(k, C_{\eta }(k, I_{\eta }(k, g_B, r), r), r) \sqsubseteq g_B$ for each $k\in F$ and $r \in I_{0}$.

Proposition 2.1. Let $(V, \eta_{F})$ be an FSTS, $g_B \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$, then we have

(1) $g_B$ is an $r$-fuzzy soft $\beta$-closed set if $g^c_{B}$ is $r$-fuzzy soft $\beta$-open ^[28].

(2) Any intersection of $r$-fuzzy soft $\beta$-closed sets is $r$-fuzzy soft $\beta$-closed.

(3) Any union of $r$-fuzzy soft $\beta$-open sets is $r$-fuzzy soft $\beta$-open.

Proof. Follows from Definitions 1.7 and 2.1.

Proposition 2.2. Let $(V, \eta_{F})$ be an FSTS, $g_B, f_A \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$. If $g_B$ is an $r$-fuzzy soft pre-open set such that $g_B\sqsubseteq f_A\sqsubseteq C_{\eta}(k, I_{\eta}(k, g_B, r), r)$, $f_A$ is $r$-fuzzy soft $\beta$-open.

Proof. Since $g_B$ is $r$-fuzzy soft pre-open and $g_B\sqsubseteq f_A$, then $g_B \sqsubseteq I_{\eta}(k, C_{\eta}(k, g_B, r), r)\sqsubseteq I_{\eta}(k, C_{\eta}(k, f_A, r), r).$ Since $f_A\sqsubseteq C_{\eta}(k, I_{\eta}(k, g_B, r), r)$, then $f_A \sqsubseteq C_{\eta}(k, I_{\eta}(k, I_{\eta}(k, C_{\eta}(k, f_A, r), r), r), r) = C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, f_A, r), r), r)$, so $f_A$ is $r$-fuzzy soft $\beta$-open.

Proposition 2.3. Let $(V, \eta_{F})$ be an FSTS, $g_B, f_A \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$. If $g_B$ is an $r$-fuzzy soft pre-closed set such that $g_B\sqsupseteq f_A\sqsupseteq I_{\eta}(k, C_{\eta}(k, g_B, r), r)$, $f_A$ is $r$-fuzzy soft $\beta$-closed.

Proof. Easily proved by a similar way in Proposition 2.2.

Definition 2.2. In an FSTS $(V, \eta_{F})$, for each $g_B \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$, we define a fuzzy soft operator $\beta C_{\eta}$ $: F \times\widetilde{(V, F)}\times I_{\circ}\rightarrow \widetilde{(V, F)}$ as follows: $\beta C_{\eta}(k, g_B, r) = \sqcap \; \{f_A \in \widetilde{(V, F)} : g_B \sqsubseteq f_A, \; f_A \; \text{is} \; r\text{-fuzzy soft} \; \beta\text{-closed}\}.$

Theorem 2.1. In an FSTS $(V, \eta_{F})$, for each $f_A, g_B \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$, the operator $\beta C_{\eta}$ $: F \times\widetilde{(V, F)}\times I_{\circ}\rightarrow \widetilde{(V, F)}$ satisfies the following properties:

(1) $\beta C_{\eta}(k, \Phi, r) = \Phi$.

(2) $g_B \sqsubseteq \beta C_{\eta}(k, g_B, r) \sqsubseteq C_{\eta}(k, g_B, r)$.

(3) $\beta C_{\eta}(k, g_B, r) \sqsubseteq \beta C_{\eta}(k, f_A, r)$, if $g_B \sqsubseteq f_A$.

(4) $\beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r) = \beta C_{\eta}(k, g_B, r)$.

(5) $\beta C_{\eta}(k, g_B \sqcup f_A, r) \sqsupseteq \beta C_{\eta}(k, g_B, r) \sqcup \beta C_{\eta}(k, f_A, r)$.

(6) $g_B$ is $r$-fuzzy soft $\beta$-closed if $\beta C_{\eta}(k, g_B, r) = g_B$.

(7) $\beta C_{\eta}(k, C_{\eta}(k, g_B, r), r) = C_{\eta}(k, g_B, r)$.

Proof. (1), (2), (3), and (6) are easily proved from the definition of $\beta C_{\eta}$.

(4) From (2) and (3), $\beta C_{\eta}(k, g_B, r)\sqsubseteq \beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r)$. Now, we show that $\beta C_{\eta}(k, g_B, r)\sqsupseteq \beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r)$. Suppose that $\beta C_{\eta}(k, g_B, r)$ does not contain $\beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r)$, then there is $v \in V$ and $t \in (0, 1)$ such that $\beta C_{\eta}(k, g_B, r)(k)(v) < t < \beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r)(k)(v). \quad \quad \quad \quad (A)$

Since $\beta C_{\eta}(k, g_B, r)(k)(v) < t$, by the definition of $\beta C_{\eta}$, there is $h_F$ as an $r$-fuzzy soft $\beta$-closed with $g_B\sqsubseteq h_F$ such that $\beta C_{\eta}(k, g_B, r)(k)(v)\le h_F(k)(v) < t$. Since $g_B\sqsubseteq h_F$, we have $\beta C_{\eta}(k, g_B, r)\sqsubseteq h_F$. Again, by the definition of $\beta C_{\eta}$, $\beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r)\sqsubseteq h_F$. Hence, $\beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r)(k)(v)\le h_F(k)(v) < t$, which is a contradiction for $(A)$. Thus, $\beta C_{\eta}(k, g_B, r)\sqsupseteq \beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r)$, then $\beta C_{\eta}(k, \beta C_{\eta}(k, g_B, r), r) = \beta C_{\eta}(k, g_B, r).$

(5) Since $g_B \sqsubseteq g_B \sqcup f_A$ and $f_A \sqsubseteq g_B \sqcup f_A$, then by (3) we have $\beta C_{\eta}(k, g_B, r)\sqsubseteq \beta C_{\eta}(k, g_B \sqcup f_A, r)$ and $\beta C_{\eta}(k, f_A, r)\sqsubseteq \beta C_{\eta}(k, g_B \sqcup f_A, r)$. Thus, $\beta C_{\eta}(k, g_B \sqcup f_A, r) \sqsupseteq \beta C_{\eta}(k, g_B, r) \sqcup \beta C_{\eta}(k, f_A, r)$.

(7) Since $C_{\eta}(k, g_B, r)$ is an $r$-fuzzy soft $\beta$-closed set, then by (6) we have $\beta C_{\eta}(k, C_{\eta}(k, g_B, r), r) = C_{\eta}(k, g_B, r)$.

Definition 2.3. Let $(V, \eta_{F})$ be an FSTS, $r \in I_{0}$, and $g_B, f_A\in \widetilde{(V, F)}$, then we have:

(1) Two fuzzy soft sets $g_B$ and $f_A$ are called $r$-fuzzy soft $\beta$-separated if $g_B \; \overline{\nabla} \; \beta C_{\eta}(k, f_A, r)$ and $f_A \; \overline{\nabla} \; \beta C_{\eta}(k, g_B, r)$ for each $k\in F$.

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

Theorem 2.2. In an FSTS $(V, \eta_{F})$, we have:

(1) If $f_A$ and $g_B$ $\in \widetilde{(V, F)}$ are $r$-fuzzy soft $\beta$-separated and $h_C$, $t_D$ $\in \widetilde{(V, F)}$ such that $h_C\sqsubseteq f_A$ and $t_D\sqsubseteq g_B$, then $h_C$ and $t_D$ are $r$-fuzzy soft $\beta$-separated.

(2) If $f_A \; \overline{\nabla} \; g_B$ are either both $r$-fuzzy soft $\beta$-open or both are $r$-fuzzy soft $\beta$-closed, then $f_A$ and $g_B$ are $r$-fuzzy soft $\beta$-separated.

(3) If $f_A$ and $g_B$ are either both $r$-fuzzy soft $\beta$-open or both $r$-fuzzy soft $\beta$-closed, then $f_A\sqcap g^c_{B}$ and $g_B\sqcap f^c_{A}$ are $r$-fuzzy soft $\beta$-separated.

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

(3) Let $f_A$ and $g_B$ be an $r$-fuzzy soft $\beta$-open. Since $f_A\; \sqcap \; g^c_{B}\sqsubseteq g^c_{B}$, $\beta C_{\eta}(k, f_A\sqcap g^c_{B}, r)\sqsubseteq g^c_{B}$, and $\beta C_{\eta}(k, f_A\sqcap g^c_{B}, r) \; \overline{\nabla} \; g_B$, then $\beta C_{\eta}(k, f_A\sqcap g^c_{B}\; , r) \overline{\nabla} \; (g_B\sqcap f^c_{A}).$ Again, since $g_B\sqcap f^c_{A} \sqsubseteq f^c_{A}$, $\beta C_{\eta}(k, g_B\sqcap f^c_{A}, r)\sqsubseteq f^c_{A}$, and $\beta C_{\eta}(k, g_B$ $\sqcap f^c_{A}, r) \; \overline{\nabla} \; f_A$, then $\beta C_{\eta}(k, g_B\sqcap f^c_{A}, r) \; \overline{\nabla} \; (f_A\sqcap g^c_{B}).$ Thus, $f_A\sqcap g^c_{B}$ and $g_B\sqcap f^c_{A}$ are $r$-fuzzy soft $\beta$-separated. The other case follows similar lines.

Theorem 2.3. In an FSTS $(V, \eta_{F})$, $f_A$, $g_B$ $\in \widetilde{(V, F)}$ are $r$-fuzzy soft $\beta$-separated if there exist two $r$-fuzzy soft $\beta$-open sets $h_C$ and $t_D$ such that $f_A \sqsubseteq h_C$, $g_B \sqsubseteq t_D$, $f_A \; \overline{\nabla} \; t_D$, and $g_B \; \overline{\nabla} \; h_C$.

Proof. ($\Rightarrow$) Let $f_A$ and $g_B$ $\in \widetilde{(V, F)}$ be $r$-fuzzy soft $\beta$-separated, $f_A \sqsubseteq (\beta C_{\eta}(k, g_B, r))^c = h_C$ and $g_B \sqsubseteq (\beta C_{\eta}(k, f_A, r))^c = t_D$, where $t_D$ and $h_C$ are $r$-fuzzy soft $\beta$-open, then $t_D \; \overline{\nabla}\; \beta C_{\eta}(k, f_A, r)$ and $h_C \; \overline{\nabla}\; \beta C_{\eta}(k, g_B, r)$. Thus, $g_B \; \overline{\nabla} \; h_C$ and $f_A \; \overline{\nabla}\; \; t_D$. Hence, we obtain the required result.

($\Leftarrow$) Let $h_C$ and $t_D$ be $r$-fuzzy soft $\beta$-open such that $g_B\sqsubseteq t_D$, $f_A\sqsubseteq h_C$, $g_B \; \overline{\nabla} \; h_C$, and $f_A \; \overline{\nabla} \; t_D$, then $g_B\sqsubseteq h^c_{C}$ and $f_A\sqsubseteq t^c_{D}$. Hence, $\beta C_{\eta}(k, g_B, r)\sqsubseteq h^c_{C}$ and $\beta C_{\eta}(k, f_A, r)\sqsubseteq t^c_{D}$, then $\beta C_{\eta}(k, g_B, r) \; \overline{\nabla} \; f_A$ and $\beta C_{\eta}(k, f_A, r) \; \overline{\nabla} \; g_B$. Thus, $g_B$ and $f_A$ are $r$-fuzzy soft $\beta$-separated. Hence, we obtain the required result.

Theorem 2.4. In an FSTS $(V, \eta_{F})$, if $g_B \in \widetilde{(V, F)}$ is $r$-fuzzy soft $\beta$-connected such that $g_B \sqsubseteq f_A \sqsubseteq \beta C_{\eta}(k, g_B, r)$, then $f_A$ is $r$-fuzzy soft $\beta$-connected.

Proof. Suppose that $f_A$ is not $r$-fuzzy soft $\beta$-connected, then there is $r$-fuzzy soft $\beta$-separated sets $h^*_{C}$ and $t^*_{D}$ $\in \widetilde{(V, F)}$ such that $f_A = h^*_{C} \sqcup t^*_{D}$. Let $h_C = g_B \sqcap h^*_{C}$ and $t_D = g_B \sqcap t^*_{D}$, then $g_B = t_D \sqcup h_C$. Since $h_C \sqsubseteq h^*_{C}$ and $t_D \sqsubseteq t^*_{D}$, by Theorem 2.2(1), $h_C$ and $t_D$ are $r$-fuzzy soft $\beta$-separated, which is a contradiction. Thus, $f_A$ is $r$-fuzzy soft $\beta$-connected, as required.

Theorem 2.5. In an FSTS $(V, \eta_{F})$, for each $g_B \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$, we define a fuzzy soft operator $\beta I_{\eta}$ $: F \times\widetilde{(V, F)}\times I_{\circ}\rightarrow \widetilde{(V, F)}$ as follows: $\beta I_{\eta}(k, g_B, r) = \sqcup \; \{f_A \in \widetilde{(V, F)} : f_A \sqsubseteq g_B, \mathit{\text{and}}\; f_A \; \mathit{\text{is}} \; r\mathit{\text{-fuzzy soft}} \; \beta\mathit{\text{-open}}\}.$ For each $f_A, g_B \in \widetilde{(V, F)}$, the operator $\beta I_{\eta}$ satisfies the following properties:

(1) $\beta I_{\eta}(k, \widetilde{F}, r) = \widetilde{F}$.

(2) $I_{\eta}(k, g_B, r) \sqsubseteq \beta I_{\eta}(k, g_B, r) \sqsubseteq g_B$.

(3) $\beta I_{\eta}(k, g_B, r) \sqsubseteq \beta I_{\eta}(k, f_A, r)$, if $g_B \sqsubseteq f_A$.

(4) $\beta I_{\eta}(k, \beta I_{\eta}(k, g_B, r), r) = \beta I_{\eta}(k, g_B, r)$.

(5) $\beta I_{\eta}(k, g_B, r) \sqcap \beta I_{\eta}(k, f_A, r) \sqsupseteq \beta I_{\eta}(k, g_B \sqcap f_A, r)$.

(6) $g_B$ is $r$-fuzzy soft $\beta$-open if $\beta I_{\eta}(k, g_B, r) = g_B$.

(7) $\beta I_{\eta}(k, g_B^c, r) = (\beta C_{\eta}(k, g_B, r))^c$.

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

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

(7) For each $g_B \in \widetilde{(V, F)}$, $k\in F$, and $r \in I_{0}$, we have $\beta I_{\eta}(k, g^c_{B}, r) = \sqcup\{f_A \in \widetilde{(V, F)}\; : \; f_A \sqsubseteq g^c_{B}, \; f_A \; \text{is} \; r\text{-fuzzy soft} \; \beta\text{-open}\}$ = [$\sqcap\{f^c_{A} \in \widetilde{(V, F)}\; : \; g_B \sqsubseteq f^c_{A}, \; \text{and} \; f^c_{A} \; \text{is} \; r\text{-fuzzy soft} \; \beta\text{-closed}\}]^c$ = $(\beta C_{\eta}(k, g_B, r))^c$.

3.   Weaker forms of fuzzy soft $\beta$-continuity

Here, we investigate some properties of fuzzy soft $\beta$-continuity. As a weaker form of fuzzy soft $\beta$-continuity, the concepts of fuzzy soft almost (weakly) $\beta$-continuous functions are introduced and some properties are given.

Definition 3.1. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs. A fuzzy soft function $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be a fuzzy soft $\beta$-continuous if $\varphi_\psi^{-1}(g_B)$ is an $r$-fuzzy soft $\beta$-closed set for each $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g^c_B)\ge r$, $k\in F$, $(e = \psi(k))\in E$, and $r\in I_{o}$.

Theorem 3.1. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a fuzzy soft function. The following statements are equivalent for each $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E,$ and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft $\beta$-continuous.

(2) For each $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_{B})\ge r$, $\varphi_\psi^{-1}(g_B)$ is $r$-fuzzy soft $\beta$-open.

(3) $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$.

(4) $\varphi_\psi^{-1}(I_{\tau}(e, g_B, r))\sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(g_B), r)$.

(5) $I_{\eta}(k, C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$.

Proof. (1) $\Leftrightarrow$ (2) Follows from Proposition 2.1 and $\varphi_\psi^{-1}(g^c_{B}) = (\varphi_\psi^{-1}(g_B))^c$.

(1) $\Rightarrow$ (3) Let $g_B \in \widetilde{(U, E)}$; hence, by (1), $\varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$ is $r$-fuzzy soft $\beta$-closed, then we obtain $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$.

(3) $\Leftrightarrow$ (4) Follows from Theorem 2.5.

(3) $\Rightarrow$ (5) Let $g_B \in \widetilde{(U, E)}$; hence, by (3), we obtain $I_{\eta}(k, C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r), r) \sqsubseteq \beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}( C_{\tau}(e, g_B, r))$.

(5) $\Rightarrow$ (2) Let $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$. By (5), we obtain $(\varphi_\psi^{-1}(g_B))^c = \varphi_\psi^{-1}(g^c_{B}) \sqsupseteq I_{\eta}(k, C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g^c_{B}), r), r), r) = (C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_{B}), r), r), r))^c$, then $\varphi_\psi^{-1}(g_{B})\sqsubseteq C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_{B}), r), r), r)$, so $\varphi_\psi^{-1}(g_{B})$ is $r$-fuzzy soft $\beta$-open.

Lemma 3.1. Every fuzzy soft continuous function ^[25] is fuzzy soft $\beta$-continuous.

Proof. Follows from Definition 2.5 and Theorem 3.1.

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

Example 3.1. Let $V = \{v_1, v_2, v_3\}$, $F = \{k_1, k_2\}$, and define $g_F, h_F\in\widetilde{(V, F)}$ as follows: $g_F = \{(k_1, \{\frac {v_1}{0.2}, \frac {v_2}{0.3}, \frac {v_3}{0.2}\}), (k_2, \{\frac {v_1}{0.2}, \frac {v_2}{0.3}, \frac {v_3}{0.2}\})\}$, $h_F = \{(k_1, \{\frac {v_1}{0.3}, \frac {v_2}{0.4}, \frac {v_3}{0.8}\}), (k_2, \{\frac {v_1}{0.3}, \frac {v_2}{0.4}, \frac {v_3}{0.8}\})\}$. Define fuzzy soft topologies $\eta_F, \tau_{F}: F \longrightarrow [0, 1]^{\widetilde{ {(V, F)}}}$ as follows: $\forall k\in F$,

$\eta_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{2}, \quad\text{if}\quad n_F = g_F, \\ 0, \quad\text{otherwise, } \end{cases}$

$\tau_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{3}, \quad\text{if}\quad n_F = h_F, \\ 0, \quad\text{otherwise.} \end{cases}$

Thus, the identity fuzzy soft function $\varphi_\psi:(V, \eta_{F})\longrightarrow (V, \tau_{F})$ is fuzzy soft $\beta$-continuous, but it is not fuzzy soft continuous.

Definition 3.2. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs. A fuzzy soft function $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be fuzzy soft almost (resp., weakly) $\beta$-continuous if for each $k_{v_t}\in\widetilde{P_t(V)}$ and each $f_A \in \widetilde{(U, E)}$ with $\tau_{e}(f_A)\ge r$ containing $\varphi_\psi(k_{v_t})$, there is $g_B \in \widetilde{(V, F)}$ that is an $r$-fuzzy soft $\beta$-open set containing $k_{v_t}$, such that $\varphi_\psi(g_B)\sqsubseteq I_{\tau}(e, C_{\tau}(e, f_A, r), r)$ (resp., $\varphi_\psi(g_B)\sqsubseteq C_{\tau}(e, f_A, r)$), $k\in F$, $(e = \psi(k))\in E$, and $r \in I_{\circ}$.

Lemma 3.2. (1) Every fuzzy soft $\beta$-continuous function is fuzzy soft almost $\beta$-continuous.

(2) Every fuzzy soft almost $\beta$-continuous function is fuzzy soft weakly $\beta$-continuous.

Proof. Follows from Definition 3.2 and Theorem 3.1.

Remark 3.2. The converse of Lemma 2 is not true, as shown by Examples 3.2 and 3.3.

Example 3.2. Let $V = \{v_1, v_2, v_3\}$, $F = \{k_1, k_2\}$, and define $f_F, g_F, h_F$ $\in\widetilde{(V, F)}$ as follows: $f_F = \{(k_1, \{\frac {v_1}{0.5}, \frac {v_2}{0.5}, \frac {v_3}{0.5}\}), (k_2, \{\frac {v_1}{0.5}, \frac {v_2}{0.5}, \frac {v_3}{0.5}\})\}$, $g_F = \{(k_1, \{\frac {v_1}{0.4}, \frac {v_2}{0.4}, \frac {v_3}{0.4}\}), (k_2, \{\frac {v_1}{0.4}, \frac {v_2}{0.4}, \frac {v_3}{0.4}\})\}$, $h_F = \{(k_1, \{\frac {v_1}{0.5}, \frac {v_2}{0.5}, \frac {v_3}{0.6}\}), (k_2, \{\frac {v_1}{0.5}, \frac {v_2}{0.5}, \frac {v_3}{0.6}\})\}$. Define fuzzy soft topologies $\eta_F, \tau_{F}: F \longrightarrow [0, 1]^{\widetilde{ {(V, F)}}}$ as follows: $\forall k\in F$,

$\eta_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{2}{3}, \quad\text{if}\quad n_F = f_F, \\ \frac{1}{2}, \quad\text{if}\quad n_F = g_F, \\ 0, \quad\text{otherwise, } \end{cases}$

$\tau_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{2}{3}, \quad\text{if}\quad n_F = f_F, \\ \frac{1}{3}, \quad\text{if}\quad n_F = h_F, \\ 0, \quad\text{otherwise.} \end{cases}$

Thus, the identity fuzzy soft function $\varphi_\psi:(V, \eta_{F})\longrightarrow (V, \tau_{F})$ is fuzzy soft almost $\beta$-continuous, but it is not fuzzy soft $\beta$-continuous.

Example 33. Let $V = \{v_1, v_2, v_3\}$, $F = \{k_1, k_2\}$, and define $g_F, h_F\in\widetilde{(V, F)}$ as follows: $g_F = \{(k_1, \{\frac {v_1}{0.6}, \frac {v_2}{0.2}, \frac {v_3}{0.4}\}), (k_2, \{\frac {v_1}{0.6}, \frac {v_2}{0.2}, \frac {v_3}{0.4}\})\}$, $h_F = \{(k_1, \{\frac {v_1}{0.3}, \frac {v_2}{0.2}, \frac {v_3}{0.5}\}), (k_2, \{\frac {v_1}{0.3}, \frac {v_2}{0.2}, \frac {v_3}{0.5}\})\}$. Define fuzzy soft topologies $\eta_F, \tau_{F}: F \longrightarrow [0, 1]^{\widetilde{ {(V, F)}}}$ as follows: $\forall k\in F$,

$\eta_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{2}, \quad\text{if}\quad n_F = g_F, \\ 0, \quad\text{otherwise, } \end{cases}$

$\tau_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{3}, \quad\text{if}\quad n_F = h_F, \\ 0, \quad\text{otherwise.} \end{cases}$

Thus, the identity fuzzy soft function $\varphi_\psi:(V, \eta_{F})\longrightarrow (V, \tau_{F})$ is fuzzy soft weakly $\beta$-continuous, but it is not fuzzy soft almost $\beta$-continuous.

Lemma 3.3. (1) Every fuzzy soft almost continuous function is fuzzy soft almost $\beta$-continuous.

(2) Every fuzzy soft weakly continuous function is fuzzy soft weakly $\beta$-continuous.

Proof. Follows from Definitions 1.9 and 3.2.

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.

Theorem 3.2. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a fuzzy soft function. The following statements are equivalent for each $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E,$ and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft almost $\beta$-continuous.

(2) $\varphi_\psi^{-1}(g_B)$ is $r$-fuzzy soft $\beta$-open, for each $g_B$ is $r$-fuzzy soft regularly open.

(3) $\varphi_\psi^{-1}(g_B)$ is $r$-fuzzy soft $\beta$-closed, for each $g_B$ is $r$-fuzzy soft regularly closed.

(4) $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$, for each $g_B$ is $r$-fuzzy soft $\beta$-open.

(5) $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$, for each $g_B$ is $r$-fuzzy soft semi-open.

(6) $\beta I_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)), r) \sqsupseteq \varphi_\psi^{-1}(g_B)$, for each $g_B$ with $\tau_{e}(g_B)\ge r$.

Proof. (1) $\Rightarrow$ (2) Let $k_{v_t}\in\widetilde{P_t(V)}$ and $g_B \in \widetilde{(U, E)}$ be an $r$-fuzzy soft regularly open set containing $\varphi_\psi(k_{v_t})$. Hence, by (1), $f_A \in \widetilde{(V, F)}$ is an $r$-fuzzy soft $\beta$-open set containing $k_{v_t}$ such that $\varphi_\psi(f_A)\sqsubseteq I_{\tau}(e, C_{\tau}(e, g_B, r), r)$.

Thus, $f_A\sqsubseteq \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)) = \varphi_\psi^{-1}(g_B)$ and $k_{v_t}\tilde{\in}f_A\sqsubseteq\varphi_\psi^{-1}(g_B)$, then $k_{v_t}\tilde{\in} C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r), r)$ and $\varphi_\psi^{-1}(g_B) \sqsubseteq C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r), r)$. Therefore, $\varphi_\psi^{-1}(g_B)$ is an $r$-fuzzy soft $\beta$-open set.

(2) $\Rightarrow$ (3) Let $g_B$ be an $r$-fuzzy soft regularly closed set. Hence, by (2), $\varphi_\psi^{-1}(g^c_{B}) = (\varphi_\psi^{-1}(g_B))^c$ is an $r$-fuzzy soft $\beta$-open set, then $\varphi_\psi^{-1}(g_B)$ is an $r$-fuzzy soft $\beta$-closed set.

(3) $\Rightarrow$ (4) Let $g_B$ be an $r$-fuzzy soft $\beta$-open set. Since $C_{\tau}(e, g_B, r)$ is an $r$-fuzzy soft regularly closed set, by (3), $\varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$ is an $r$-fuzzy soft $\beta$-closed set. Since $\varphi_\psi^{-1}(g_B) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$, then we have $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$.

(4) $\Rightarrow$ (5) This is obvious from every $r$-fuzzy soft semi-open set that is an $r$-fuzzy soft $\beta$-open set.

(5) $\Rightarrow$ (3) Let $g_B$ be an $r$-fuzzy soft regularly closed set. Hence, $g_B$ is an $r$-fuzzy soft semi-open set, then by (5), $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r)) = \varphi_\psi^{-1}(g_B)$. Therefore, $\varphi_\psi^{-1}(g_B)$ is an $r$-fuzzy soft $\beta$-closed set.

(3) $\Rightarrow$ (6) Let $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$ and $k_{v_t}\tilde{\in} \varphi_\psi^{-1}(g_B)$, then we have $k_{v_t}\tilde{\in}\varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r))$. Since $[I_{\tau}(e, C_{\tau}(e, g_B, r), r)]^c$ is an $r$-fuzzy soft regularly closed set, by (3), $\varphi_\psi^{-1}([I_{\tau}(e, C_{\tau}(e, g_B, r), r)]^c)$ is an $r$-fuzzy soft $\beta$-closed set. Thus, $\varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r))$ is an $r$-fuzzy soft $\beta$-open set and $k_{v_t}\tilde{\in}\beta I_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)), r)$, then $\varphi_\psi^{-1}(g_B) \sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)), r)$.

(6) $\Rightarrow$ (1) Let $k_{v_t}\in\widetilde{P_t(V)}$ and $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$ containing $\varphi_\psi(k_{v_t})$; hence, by (6), $\varphi_\psi^{-1}(g_B) \sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)), r)$.

Since $k_{v_t}\tilde{\in} \varphi_\psi^{-1}(g_B)$, then we obtain $k_{v_t}\tilde{\in} \beta I_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)), r) = f_A$ (say). Hence, $f_A \in \widetilde{(V, F)}$ is an $r$-fuzzy soft $\beta$-open set containing $k_{v_t}$ such that $\varphi_\psi(f_A)\sqsubseteq I_{\tau}(e, C_{\tau}(e, g_B, r), r)$. Therefore, $\varphi_\psi$ is fuzzy soft almost $\beta$-continuous.

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

Theorem 3.3. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a fuzzy soft function. The following statements are equivalent for each $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E,$ and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft weakly $\beta$-continuous.

(2) $C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(C_{\tau}(e, g_B, r)), r), r), r) \sqsupseteq \varphi_\psi^{-1}(g_B)$, if $\tau_{e}(g_{B})\ge r$.

(3) $I_{\eta}(k, C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, g_B, r)), r), r), r) \sqsubseteq \varphi_\psi^{-1}(g_B)$, if $\tau_{e}(g^c_{B})\ge r$.

(4) $\beta C_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, g_B, r)), r) \sqsubseteq \varphi_\psi^{-1}(g_B)$, if $\tau_{e}(g^c_{B})\ge r$.

(5) $\beta C_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)), r) \sqsubseteq \varphi_\psi^{-1}(C_{\tau}(e, g_B, r))$.

(6) $\beta I_{\eta}(k, \varphi_\psi^{-1}(C_{\tau}(e, I_{\tau}(e, g_B, r), r)), r) \sqsupseteq \varphi_\psi^{-1}(I_{\tau}(e, g_B, r))$.

(7) $\varphi_\psi^{-1}(g_B) \sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(C_{\tau}(e, g_B, r)), r)$, if $\tau_{e}(g_B)\ge r$.

Let $\mathcal{P}$ and $\mathcal{Q}$ $: F \times\widetilde{(V, F)}\times I_{\circ}\rightarrow \widetilde{(V, F)}$ be operators on $\widetilde{(V, F)}$, and $\mathcal{R}$ and $\mathcal{S}$ $: E \times\widetilde{(U, E)}\times I_{\circ}\rightarrow \widetilde{(U, E)}$ be operators on $\widetilde{(U, E)}$.

Definition 3.3. ^[36] Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs. $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be a fuzzy soft $(\mathcal{P}, \mathcal{Q}, \mathcal{R}, \mathcal{S})$-continuous function if $\mathcal{P} [k, \varphi_\psi^{-1}(\mathcal{S}(e, g_B, r)), r] \; \overline{\sqcap}\; \mathcal{Q} [k, \varphi_\psi^{-1}(\mathcal{R} (e, g_B, r)), r]$ $= \Phi$ for each $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$, $k\in F$, and $(e = \psi(k))\in E$.

In (2014), Aygünoǧlu et al. ^[25] defined the notion of fuzzy soft continuous functions: $\eta_{k}(\varphi_\psi^{-1}(f_A))\ge \tau_{e}(f_A)$, for each $f_A \in \widetilde{(U, E)}$, $k\in F$, and $(e = \psi(k))\in E$. We can see that Definition 3.3 generalizes the concept of fuzzy soft continuous functions when we choose $\mathcal{P}$ = identity operator, $\mathcal{Q}$ = interior operator, $\mathcal{R}$ = identity operator, and $\mathcal{S}$ = identity operator.

A historical justification of Definition 3.3:

(1) In Section 3, we introduced the notion of fuzzy soft $\beta$-continuous functions: $\varphi_\psi^{-1}(g_B) \sqsubseteq C_{\eta }(k, I_{\eta }(k, C_{\eta }(k, \varphi_\psi^{-1}(g_B), r), r), r)$, for each $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$. Here, $\mathcal{P}$ = identity operator, $\mathcal{Q}$ = closure interior closure operator, $\mathcal{R}$ = identity operator, and $\mathcal{S}$ = identity operator.

(2) In Section 3, we introduced the notion of fuzzy soft almost $\beta$-continuous functions: $\varphi_\psi^{-1}(g_B) \sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(I_{\tau}(e, C_{\tau}(e, g_B, r), r)), r)$, for each $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$. Here, $\mathcal{P}$ = identity operator, $\mathcal{Q}$ = $\beta$-interior operator, $\mathcal{R}$ = interior closure operator, and $\mathcal{S}$ = identity operator.

(3) In Section 3, we introduced the notion of fuzzy soft weakly $\beta$-continuous functions: $\varphi_\psi^{-1}(g_B) \sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(C_{\tau}(e, g_B, r)), r)$, for each $g_B \in \widetilde{(U, E)}$ with $\tau_{e}(g_B)\ge r$. Here, $\mathcal{P}$ = identity operator, $\mathcal{Q}$ = $\beta$-interior operator, $\mathcal{R}$ = closure operator, and $\mathcal{S}$ = identity operator.

4.   Fuzzy soft $\beta$-irresoluteness

Here, we introduce the concepts of fuzzy soft $\beta$-irresolute (resp., strongly $\beta$-irresolute, $\beta$-irresolute open, $\beta$-irresolute closed, and $\beta$-irresolute homeomorphism) functions between two FSTSs $(V, \eta_{F})$ and $(U, \tau_{E})$ and study some of its features. Also, we show that fuzzy soft strongly $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-continuity, but the converse may not be true.

Definition 4.1. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs. A fuzzy soft function $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be a fuzzy soft $\beta$-irresolute (resp., strongly $\beta$-irresolute) if $\varphi_\psi^{-1}(g_B)$ is an $r$-fuzzy soft $\beta$-open (resp., semi-open) set for each $g_B \in \widetilde{(U, E)}$ $r$-fuzzy soft $\beta$-open set and $r\in I_{o}$.

Lemma 4.1. (1) Every fuzzy soft strongly $\beta$-irresolute function is fuzzy soft $\beta$-irresolute.

(2) Every fuzzy soft $\beta$-irresolute function is fuzzy soft $\beta$-continuous.

Proof. Follows from Definition 4.1 and Theorem 3.1.

Remark 4.1. The converse of Lemma 4.1 is not true, as shown by Examples 4.1 and 4.2.

Example 4.1. Let $V = \{v_1, v_2, v_3\}$, $F = \{k_1, k_2\}$, and define $f_F, g_F\in\widetilde{(V, F)}$ as follows: $f_F = \{(k_1, \{\frac {v_1}{0.3}, \frac {v_2}{0.3}, \frac {v_3}{0.3}\}), (k_2, \{\frac {v_1}{0.3}, \frac {v_2}{0.3}, \frac {v_3}{0.3}\})\}$, $g_F = \{(k_1, \{\frac {v_1}{0.4}, \frac {v_2}{0.4}, \frac {v_3}{0.4}\}), (k_2, \{\frac {v_1}{0.4}, \frac {v_2}{0.4}, \frac {v_3}{0.4}\})\}$. Define fuzzy soft topologies $\eta_F, \tau_{F}: F \longrightarrow [0, 1]^{\widetilde{ {(V, F)}}}$ as follows: $\forall k\in F$,

$\eta_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{2}, \quad\text{if}\quad n_F = g_F, \\ 0, \quad\text{otherwise, } \end{cases}$

$\tau_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{2}, \quad\text{if}\quad n_F = f_F, \\ 0, \quad\text{otherwise.} \end{cases}$

Thus, the identity fuzzy soft function $\varphi_\psi:(V, \eta_{F})\longrightarrow (V, \tau_{F})$ is fuzzy soft $\beta$-irresolute, but it is not fuzzy soft strongly $\beta$-irresolute.

Example 4.2. Let $V = \{v_1, v_2, v_3\}$, $F = \{k_1, k_2\}$, and define $f_F, g_F\in\widetilde{(V, F)}$ as follows: $f_F = \{(k_1, \{\frac {v_1}{0.5}, \frac {v_2}{0.5}, \frac {v_3}{0.5}\}), (k_2, \{\frac {v_1}{0.5}, \frac {v_2}{0.5}, \frac {v_3}{0.5}\})\}$, $g_F = \{(k_1, \{\frac {v_1}{0.4}, \frac {v_2}{0.4}, \frac {v_3}{0.4}\}), (k_2, \{\frac {v_1}{0.4}, \frac {v_2}{0.4}, \frac {v_3}{0.4}\})\}$. Define fuzzy soft topologies $\eta_F, \tau_{F}: F \longrightarrow [0, 1]^{\widetilde{ {(V, F)}}}$ as follows: $\forall k\in F$,

$\eta_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{3}, \quad\text{if}\quad n_F = f_F, \\ \frac{1}{2}, \quad\text{if}\quad n_F = g_F, \\ 0, \quad\text{otherwise, } \end{cases}$

$\tau_{k}(n_F) =$ $\begin{cases} 1, \quad\text{if}\quad n_F \in\{\Phi, \widetilde{F}\}, \\ \frac{1}{3}, \quad\text{if}\quad n_F = f_F, \\ 0, \quad\text{otherwise.} \end{cases}$

Thus, the identity fuzzy soft function $\varphi_\psi:(V, \eta_{F})\longrightarrow (V, \tau_{F})$ is fuzzy soft $\beta$-continuous, but it is not fuzzy soft $\beta$-irresolute.

Remark 4.2. From the previous results, we have: fuzzy soft strongly $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-continuity.

Theorem 4.1. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a fuzzy soft function. The following statements are equivalent for each $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E,$ and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft $\beta$-irresolute.

(2) For each $g_B$ $r$-fuzzy soft $\beta$-closed, $\varphi_\psi^{-1}(g_B)$ is $r$-fuzzy soft $\beta$-closed.

(3) $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$.

(4) $\varphi_\psi^{-1}(\beta I_{\tau}(e, g_B, r))\sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(g_B), r)$.

(5) $I_{\eta}(k, C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r), r) \sqsubseteq \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$.

Proof. (1) $\Leftrightarrow$ (2) Follows from Proposition 2.1 and $\varphi_\psi^{-1}(g^c_{B}) = (\varphi_\psi^{-1}(g_B))^c$.

(2) $\Rightarrow$ (3) Let $g_B \in \widetilde{(U, E)}$. Hence, by (2), $\varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$ is $r$-fuzzy soft $\beta$-closed, then we obtain $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$.

(3) $\Leftrightarrow$ (4) Follows from Theorem 2.5.

(3) $\Rightarrow$ (5) Let $g_B \in \widetilde{(U, E)}$. Hence, by (3), we obtain $I_{\eta}(k, C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r), r) \sqsubseteq \beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$.

(5) $\Rightarrow$ (1) Let $g_B \in \widetilde{(U, E)}$ be an $r$-fuzzy soft $\beta$-open. Hence, by (5), we obtain $(\varphi_\psi^{-1}(g_B))^c = \varphi_\psi^{-1}(g^c_{B}) \sqsupseteq I_{\eta}(k, C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g^c_{B}), r), r), r) = (C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_{B}), r), r), r))^c$, then $\varphi_\psi^{-1}(g_{B})\sqsubseteq C_{\eta}(k, I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_{B}), r), r), r)$, so $\varphi_\psi^{-1}(g_{B})$ is $r$-fuzzy soft $\beta$-open. Hence, $\varphi_\psi$ is fuzzy soft $\beta$-irresolute.

Theorem 4.2. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a fuzzy soft function. The following statements are equivalent for each $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E,$ and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft strongly $\beta$-irresolute.

(2) For each $g_B$ $r$-fuzzy soft $\beta$-closed, $\varphi_\psi^{-1}(g_B)$ is $r$-fuzzy soft semi-closed.

(3) $I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r) \sqsubseteq \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$.

Proof. (1) $\Leftrightarrow$ (2) Follows from Proposition 2.1 and $\varphi_\psi^{-1}(g^c_{B}) = (\varphi_\psi^{-1}(g_B))^c$.

(2) $\Rightarrow$ (3) Let $g_B \in \widetilde{(U, E)}$. Hence, by (2), $\varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$ is $r$-fuzzy soft semi-closed, then we obtain $I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g_B), r), r) \sqsubseteq \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$.

(3) $\Rightarrow$ (1) Let $g_B \in \widetilde{(U, E)}$ be an $r$-fuzzy soft $\beta$-open. Hence, by (3), we obtain $(\varphi_\psi^{-1}(g_B))^c = \varphi_\psi^{-1}(g^c_{B}) \sqsupseteq I_{\eta}(k, C_{\eta}(k, \varphi_\psi^{-1}(g^c_{B}), r), r) = (C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g_{B}), r), r))^c$, then $\varphi_\psi^{-1}(g_{B})\sqsubseteq C_{\eta}(k, I_{\eta}(k, \varphi_\psi^{-1}(g_{B}), r), r)$, so $\varphi_\psi^{-1}(g_{B})$ is $r$-fuzzy soft semi-open. Hence, $\varphi_\psi$ is fuzzy soft strongly $\beta$-irresolute.

Proposition 4.1. Let $(V, \eta_{F})$, $(U, \tau_{E})$, and $(W, \gamma_{H})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$, $\varphi_{\psi^*}^*:\widetilde{(U, E)}\longrightarrow \widetilde{(W, H)}$ be two fuzzy soft functions, then the composition $\varphi_{\psi^*}^*\circ \varphi_\psi$ is fuzzy soft $\beta$-irresolute (resp., strongly $\beta$-irresolute and $\beta$-continuous) if $\varphi_\psi$ is fuzzy soft $\beta$-irresolute (resp., strongly $\beta$-irresolute and $\beta$-irresolute) and $\varphi_{\psi^*}^*$ is fuzzy soft $\beta$-irresolute (resp., $\beta$-irresolute and $\beta$-continuous).

Proof. Follows from Definition 4.1 and Theorem 3.1.

Proposition 4.2. Let $(V, \eta_{F})$, $(U, \tau_{E})$, and $(W, \gamma_{H})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$, $\varphi_{\psi^*}^*:\widetilde{(U, E)}\longrightarrow \widetilde{(W, H)}$ be two fuzzy soft functions, then the composition $\varphi_{\psi^*}^*\circ \varphi_\psi$ is fuzzy soft almost $\beta$-continuous if $\varphi_\psi$ is fuzzy soft $\beta$-continuous (resp., $\beta$-irresolute and $\beta$-continuous) and $\varphi_{\psi^*}^*$ is fuzzy soft almost continuous (resp., almost $\beta$-continuous and continuous).

Proof. Follows from the above definitions.

Definition 4.2. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $r\in I_{o}$. A fuzzy soft function $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be a fuzzy soft $\beta$-irresolute open (resp., closed) if $\varphi_\psi(f_A)$ is an $r$-fuzzy soft $\beta$-open (resp., $\beta$-closed) set for each $f_A \in \widetilde{(V, F)}$ which is an $r$-fuzzy soft $\beta$-open (resp., $\beta$-closed) set.

Theorem 4.3. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a fuzzy soft function. The following statements are equivalent for each $f_A \in \widetilde{(V, F)}$, $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E$, and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft $\beta$-irresolute open.

(2) $\varphi_\psi(\beta I_{\eta}(k, f_A, r))\sqsubseteq \beta I_{\tau}(e, \varphi_\psi(f_A), r)$.

(3) $\beta I_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(\beta I_{\tau}(e, g_B, r))$.

(4) For each $g_B$ and each $f_A$ $r$-fuzzy soft $\beta$-closed with $\varphi_\psi^{-1}(g_B) \sqsubseteq f_A$, $h_C \in \widetilde{(U, E)}$ is $r$-fuzzy soft $\beta$-closed with $g_B \sqsubseteq h_C$ such that $\varphi_\psi^{-1}(h_C) \sqsubseteq f_A$.

Proof. (1) $\Rightarrow$ (2) Since $\varphi_\psi(\beta I_{\eta}(k, f_A, r))\sqsubseteq \varphi_\psi(f_A)$, by (1), $\varphi_\psi(\beta I_{\eta}(k, f_A, r))$ is $r$-fuzzy soft $\beta$-open. Hence, $\varphi_\psi(\beta I_{\eta}(k, f_A, r))\sqsubseteq \beta I_{\tau}(e, \varphi_\psi(f_A), r)$.

(2) $\Rightarrow$ (3) Put $f_A = \varphi_\psi^{-1}(g_B)$. By (2), $\varphi_\psi(\beta I_{\eta}(k, \varphi_\psi^{-1}(g_B), r))\sqsubseteq \beta I_{\tau}(e, \varphi_\psi(\varphi_\psi^{-1}(g_B)), r) \sqsubseteq \beta I_{\tau}(e, g_B, r)$, then $\beta I_{\eta}(k, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(\beta I_{\tau}(e, g_B, r))$.

(3) $\Rightarrow$ (4) Let $g_B \in \widetilde{(U, E)}$ and $f_A \in \widetilde{(V, F)}$ be an $r$-fuzzy soft $\beta$-closed with $\varphi_\psi^{-1}(g_B) \sqsubseteq f_A$. Since $f^c_A\sqsubseteq \varphi_\psi^{-1}(g^c_B)$, $f^c_A = \beta I_{\eta}(k, f^c_A, r)\sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(g^c_B), r)$. Hence, by (3), $f^c_A \sqsubseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(g^c_B), r)\sqsubseteq \varphi_\psi^{-1}(\beta I_{\tau}(e, g^c_B, r))$. Thus, $f_A \sqsupseteq (\varphi_\psi^{-1}(\beta I_{\tau}(e, g^c_B, r)))^c = \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r)).$ Hence, $\beta C_{\tau}(e, g_B, r) \in \widetilde{(U, E)}$ is $r$-fuzzy soft $\beta$-closed with $g_B \sqsubseteq \beta C_{\tau}(e, g_B, r)$ such that $\varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r)) \sqsubseteq f_A$.

(4) $\Rightarrow$ (1) Let $w_D \in \widetilde{(V, F)}$ be $r$-fuzzy soft $\beta$-open. Put $g_B = (\varphi_\psi(w_D))^c$ and $f_A = w^c_D$, $\varphi_\psi^{-1}(g_B) = \varphi_\psi^{-1}((\varphi_\psi(w_D))^c)\sqsubseteq f_A$. Hence, by (4), $h_C \in \widetilde{(U, E)}$ is $r$-fuzzy soft $\beta$-closed with $g_B \sqsubseteq h_C$ such that $\varphi_\psi^{-1}(h_C) \sqsubseteq f_A = w^c_D$. Thus, $\varphi_\psi(w_D) \sqsubseteq \varphi_\psi(\varphi_\psi^{-1}(h^c_C)) \sqsubseteq h^c_C$. On the other hand, since $g_B \sqsubseteq h_C$, $\varphi_\psi(w_D) = g^c_B \sqsupseteq h^c_C$. Thus, $\varphi_\psi(w_D) = h^c_C$, so $\varphi_\psi(w_D)$ is $r$-fuzzy soft $\beta$-open, then $\varphi_\psi$ is fuzzy soft $\beta$-irresolute open.

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

Theorem 4.4. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a bijective fuzzy soft function. The following statements are equivalent for each $f_A \in \widetilde{(V, F)}$, $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E$, and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft $\beta$-irresolute closed.

(2) $\beta C_{\tau}(e, \varphi_\psi(f_A), r) \sqsubseteq \varphi_\psi(\beta C_{\eta}(k, f_A, r))$.

(3) $\varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r)) \sqsubseteq \beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r)$.

(4) For each $g_B$ and each $f_A$ $r$-fuzzy soft $\beta$-open with $\varphi_\psi^{-1}(g_B) \sqsubseteq f_A$, $h_C \in \widetilde{(U, E)}$ is $r$-fuzzy soft $\beta$-open with $g_B \sqsubseteq h_C$ such that $\varphi_\psi^{-1}(h_C) \sqsubseteq f_A$.

Proposition 4.3. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a bijective fuzzy soft function. $\varphi_\psi$ is fuzzy soft $\beta$-irresolute closed if $\varphi_\psi$ is fuzzy soft $\beta$-irresolute open.

Proof. It is easily proved from:

$\varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r)) \sqsubseteq \beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r)$

$\Longleftrightarrow\varphi_\psi^{-1}(\beta I_{\tau}(e, g^c_{B}, r)) \sqsupseteq \beta I_{\eta}(k, \varphi_\psi^{-1}(g^c_{B}), r)$.

Definition 4.3. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs. A fuzzy soft function $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ is said to be a fuzzy soft $\beta$-irresolute homeomorphism if $\varphi_\psi$ is bijective and both of $\varphi_\psi$ and $\varphi_\psi^{-1}$ are $\beta$-irresolute.

From the above theorems, we obtain the following corollary.

Corollary 4.1. Let $(V, \eta_{F})$ and $(U, \tau_{E})$ be FSTSs and $\varphi_\psi:\widetilde{(V, F)}\longrightarrow \widetilde{(U, E)}$ be a bijective fuzzy soft function. The following statements are equivalent for each $f_A \in \widetilde{(V, F)}$, $g_B \in \widetilde{(U, E)}$, $k\in F$, $(e = \psi(k))\in E$, and $r \in I_{\circ}$:

(1) $\varphi_\psi$ is fuzzy soft $\beta$-irresolute homeomorphism.

(2) $\varphi_\psi$ is fuzzy soft $\beta$-irresolute and fuzzy soft $\beta$-irresolute open.

(3) $\varphi_\psi$ is fuzzy soft $\beta$-irresolute and fuzzy soft $\beta$-irresolute closed.

(4) $\varphi_\psi(\beta I_{\eta}(k, f_A, r)) = \beta I_{\tau}(e, \varphi_\psi(f_A), r)$.

(5) $\varphi_\psi(\beta C_{\eta}(k, f_A, r)) = \beta C_{\tau}(e, \varphi_\psi(f_A), r)$.

(6) $\beta I_{\eta}(k, \varphi_\psi^{-1}(g_B), r) = \varphi_\psi^{-1}(\beta I_{\tau}(e, g_B, r))$.

(7) $\beta C_{\eta}(k, \varphi_\psi^{-1}(g_B), r) = \varphi_\psi^{-1}(\beta C_{\tau}(e, g_B, r))$.

5.   Conclusions and future work

This article is laid out as follows:

$({\bf 1})$ In Section 2, fuzzy soft $\beta$-closure ($\beta$-interior) operators are introduced and studied in FSTSs based on the article by Aygünoǧlu et al. ^[25]. Moreover, the concept of $r$-fuzzy soft $\beta$-connected sets is defined and characterized.

$({\bf 2})$ In Section 3, some properties of a fuzzy soft $\beta$-continuity are investigated. As a weaker form of the notion of fuzzy soft $\beta$-continuous functions, the notions of fuzzy soft almost (weakly) $\beta$-continuous functions are introduced and some properties are obtained. Also, we show that fuzzy soft $\beta$-continuity $\Rightarrow$ fuzzy soft almost $\beta$-continuity $\Rightarrow$ fuzzy soft weakly $\beta$-continuity, but the converse may not be true. Furthermore, we have the following results:

● Fuzzy soft $(id_{V}, C_{\eta}(I_{\eta}(C_{\eta})), id_{U}, id_{U})$-continuous function is fuzzy soft $\beta$-continuous.

● Fuzzy soft $(id_{V}, \beta I_{\eta}, I_{\tau}(C_{\tau}), id_{U})$-continuous function is fuzzy soft almost $\beta$-continuous.

● Fuzzy soft $(id_{V}, \beta I_{\eta}, C_{\tau}, id_{U})$-continuous function is fuzzy soft weakly $\beta$-continuous.

$({\bf 3})$ In Section 4, the notions of fuzzy soft $\beta$-irresolute (resp., strongly $\beta$-irresolute, $\beta$-irresolute open, $\beta$-irresolute closed, and $\beta$-irresolute homeomorphism) functions are introduced between two FSTSs $(V, \eta_F)$ and $(U, \tau_E)$, and some properties of these functions are investigated. Additionally, we show that fuzzy soft strongly $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-irresolute $\Rightarrow$ fuzzy soft $\beta$-continuity, but the converse may not be true.

In upcoming manuscripts, we shall investigate the notions given here in the frames of fuzzy soft $r$-minimal structures ^[28]. Also, we will use the fuzzy soft $\beta$-closure operator to introduce some new separation axioms on FSTS based on the article by Aygünoǧlu et al. ^[25].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no conflicts of interest.