On Fuzzy Soft Beta-Continuity and Beta-Irresoluteness: Properties and Applications

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On Fuzzy Soft Beta-Continuity and Beta-Irresoluteness: Properties and Applications

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Submitted:

19 January 2024

Posted:

22 January 2024

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Abstract

In the present study. At first, some properties of r-fuzzy soft β-closed sets are introduced on fuzzy soft topological spaces based on the paper Aygunoglu et al. (Hacet. J. Math. Stat. 2014, 43, 193-208). Also, we define the closure and interior operators with respect to the classes of r-fuzzy soft β-closed and r-fuzzy soft β-open sets, and investigate some of their properties. In addition, the concept of r-fuzzy soft β-connected sets is introduced and characterized with help of fuzzy soft β-closure operators. Thereafter, some properties of a fuzzy soft β-continuity are studied. Furthermore, we introduce and study the concepts of fuzzy soft almost (weakly) β-continuous functions, which are weaker forms of a fuzzy soft β-continuity. Also, the relationships between these classes of functions are specified with the help of some illustrative examples. Finally, we explore new types of fuzzy soft functions called, fuzzy soft β-irresolute (strongly β-irresolute, β-irresolute open, β-irresolute closed and β-irresolute homeomorphism) functions, and discuss some properties of them. Also, we show that fuzzy soft strongly β-irresolute → fuzzy soft β-irresolute → fuzzy soft β-continuity, but the converse may not be true.

Keywords:

1. Introduction

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 these theory in solving some practical problems in engineering, economics, and medical science, social science etc. Akdag and Ozkan [2] initiated and studied the concept of soft

α

-open sets on soft topological spaces. The concept of soft

β

-open sets was studied by the authors of [3,4], and some properties of soft

β

-continuity are investigated. Also, the concepts of somewhere dense and Q-sets were defined and studied by the authors of [5,6]. Al-shami et al. [7] initiated the concept of weakly soft semi-open sets and studied its main properties. Also, Al-shami et al. [8] defined and studied the concept of weakly soft

β

-open sets. 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 new concepts of mappings over soft topological spaces: soft somewhat-r-continuity and soft somewhat-r-openness. Ameen et al. [11] explored more properties of soft somewhere dense continuity.

The notion of fuzzy soft sets was introduced by Maji et al. [12], which combines fuzzy sets [13] and soft sets [1]. Based on fuzzy topologies in the sense of Šostak [14], 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 is introduced in [15,16]. 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. [15]. The notion of r-fuzzy soft regularly open sets was defined and studied by Çetkin and Aygün [19]. In addition, the notions of r-fuzzy soft

β

-open (resp. pre-open) sets were introduced by Taha [20].

The main contribution of this study is arranged as follows:

• In Section 2, we are going to present the notions of fuzzy soft

β

-closure (

β

-interior) operators on fuzzy soft topological space based on the article Aygünoǧlu et al. [15], and study some properties of them. Also, the concept of r-fuzzy soft

β

-connected sets is defined and studied.

• In Section 3, we investigate some properties of a fuzzy soft

β

-continuity. Moreover, we explore and study the notions of fuzzy soft almost (weakly)

β

-continuous functions, which are weaker forms of fuzzy soft

β

-continuous functions. Also, we show that fuzzy soft

β

-continuity ⇒ fuzzy soft almost

β

-continuity ⇒ fuzzy soft weakly

β

-continuity, but the converse may not be true.

• In Section 4, we introduce the notions of fuzzy soft

β

-irresolute (resp. strongly

β

-irresolute,

β

-irresolute open,

β

-irresolute closed and

β

-irresolute homeomorphism) functions between two fuzzy soft topological spaces

(V, η_{F})

and

(U, τ_{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,

\underset{̲}{t} (v) = t

, for all

v \in V .

The following notions and results will be used in the next sections:

Definition 1.

[15, 21, 22]

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) = \underset{̲}{0}

, if

k \notin B

. The family of all fuzzy soft sets on V is denoted by

\tilde{(V, F)}

. In [23], 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} \bar{⊓} f_{A}) (k) = \{\begin{matrix} \underset{̲}{0}, & i f g_{B} (k) \leq f_{A} (k), \\ g_{B} (k) \land {(f_{A} (k))}^{c}, & otherwise . \end{matrix}

Definition 2.

[24]

A fuzzy soft point

k_{v_{t}}

on V is a fuzzy soft set defined as follows:

k_{v_{t}} (e) = \{\begin{matrix} v_{t}, & i f & e = k, \\ \underset{̲}{0}, & i f & e \in F - {k}, \end{matrix}

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 \leq g_{B} (k) (v)

. The family of all fuzzy soft points on V is denoted by

\tilde{P_{t} (V)}

Definition 3.

[25]

A fuzzy soft point

k_{v_{t}} \in \tilde{P_{t} (V)}

is called a soft quasi-coincident with

g_{B} \in \tilde{(V, F)}

, denoted by

k_{v_{t}} \tilde{q} g_{B}

, if

t + g_{B} (k) (v) > 1

. A fuzzy soft set

f_{A} \in \tilde{(V, F)}

is called a soft quasi-coincident with

g_{B} \in \tilde{(V, F)}

, denoted by

f_{A} \tilde{q} 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} / \tilde{q} g_{B}

Definition 4.

[15]

A function

η : F ⟶ {[0, 1]}^{\tilde{(V, F)}}

is said to be a fuzzy soft topology on V if it satisfies the following, for each

k \in F

(1)

η_{k} (Φ) = η_{k} (\tilde{F}) = 1,

(2)

η_{k} (g_{B} ⊓ f_{A}) \geq η_{k} (g_{B}) \land η_{k} (f_{A}),

for each

g_{B}, f_{A} \in \tilde{(V, F)},

(3)

η_{k} (⊔_{δ \in Δ} {(g_{B})}_{δ}) \geq \land_{δ \in Δ} η_{k} ({(g_{B})}_{δ}),

for each

{(g_{B})}_{δ} \in \tilde{(V, F)}, δ \in Δ .

Then,

(V, η_{F})

is said to be a fuzzy soft topological space (briefly, FSTS) based on the sense of Šostak [14].

Definition 5.

[15]

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs. A fuzzy soft function

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft continuous if,

η_{k} (φ_{ψ}^{- 1} (g_{B})) \geq τ_{e} (g_{B})

for each

g_{B} \in \tilde{(U, E)}

k \in F

and

(e = ψ (k)) \in E

Definition 6.

[16, 19]

In a FSTS

(V, η_{F})

, for each

g_{B} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

, we define the fuzzy soft operators

C_{η}

and

I_{η}

: F \times \tilde{(V, F)} \times I_{\circ} \to \tilde{(V, F)}

as follows:

C_{η} (k, g_{B}, r) = ⊓ {f_{A} \in \tilde{(V, F)} : g_{B} ⊑ f_{A}, η_{k} (f_{A}^{c}) \geq r}

I_{η} (k, g_{B}, r) = ⊔ {f_{A} \in \tilde{(V, F)} : f_{A} ⊑ g_{B}, η_{k} (f_{A}) \geq r} .

Definition 7.

Let

(V, η_{F})

be a FSTS and

r \in I_{0}

. A fuzzy soft set

g_{B} \in \tilde{(V, F)}

is said to be an r-fuzzy soft

β

-open [20] (resp. pre-open [20], semi-open [26] and regularly open [19]) if,

g_{B} ⊑ C_{η} (k, I_{η} (k, C_{η} (k, g_{B}, r), r), r)

(resp.

g_{B} ⊑ I_{η} (k, C_{η} (k, g_{B}, r), r)

g_{B} ⊑ C_{η} (k, I_{η} (k, g_{B}, r), r)

and

g_{B} = I_{η} (k, C_{η} (k, g_{B}, r), r)

) for each

k \in F

Definition 8.

[19]

Let

(V, η_{F})

be a FSTS and

r \in I_{0}

. A fuzzy soft set

g_{B} \in \tilde{(V, F)}

is said to be an r-fuzzy soft regularly closed if,

g_{B} = C_{η} (k, I_{η} (k, g_{B}, r), r)

for each

k \in F

Definition 9.

[20]

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

r e g u l a r l y o p e n s e t

⇓

p r e - o p e n s e t \Rightarrow β - o p e n s e t

Definition 10.

[26]

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs. A fuzzy soft function

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft almost (resp. weakly) continuous if, for each

k_{v_{t}} \in \tilde{P_{t} (V)}

and each

g_{B} \in \tilde{(U, E)}

with

τ_{e} (g_{B}) \geq r

containing

φ_{ψ} (k_{v_{t}})

, there is

f_{A} \in \tilde{(V, F)}

with

η_{k} (f_{A}) \geq r

containing

k_{v_{t}}

such that

φ_{ψ} (f_{A}) ⊑ I_{τ} (e, C_{τ} (e, g_{B}, r), r)

(resp.

φ_{ψ} (f_{A}) ⊑ C_{τ} (e, g_{B}, r)

Definition 11.

[26]

From Definitions 1.5 and 1.9, we have: Fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft weakly continuity.

The basic results and definitions which we need in the next sections are found in [15,16].

2. Some Properties of r-Fuzzy Soft $β$ -Closed Sets

Here, we introduce the concept of r-fuzzy soft

β

-closed sets on fuzzy soft topological space based on the sense of Šostak [14], and investigate some properties of them. Also, we define and study the concepts of fuzzy soft

β

-closure (

β

-interior) operators. Moreover, the concept of r-fuzzy soft

β

-connected sets is defined and characterized.

Definition 12.

Let

(V, η_{F})

be a FSTS. A fuzzy soft set

g_{B} \in \tilde{(V, F)}

is said to be an r-fuzzy soft

β

-closed if,

I_{η} (k, C_{η} (k, I_{η} (k, g_{B}, r), r), r) ⊑ g_{B}

for each

k \in F

and

r \in I_{0}

Proposition 1.

Let

(V, η_{F})

be a FSTS,

g_{B} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

. Then, we have

(1)

g_{B}

is r-fuzzy soft

β

-closed set iff

g_{B}^{c}

is r-fuzzy soft

β

-open [20].

(2) Any intersection of r-fuzzy soft

β

-closed sets is r-fuzzy soft

β

-closed.

(3) Any union of r-fuzzy soft

β

-open sets is r-fuzzy soft

β

-open.

Proof.

Follows from Definitions 1.7 and 2.1.

□

Proposition 2.

Let

(V, η_{F})

be a FSTS,

g_{B}, f_{A} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

. If

g_{B}

is r-fuzzy soft pre-open set such that

g_{B} ⊑ f_{A} ⊑ C_{η} (k, I_{η} (k, g_{B}, r), r)

f_{A}

is r-fuzzy soft

β

-open.

Proof.

Since

g_{B}

is r-fuzzy soft pre-open and

g_{B} ⊑ f_{A}

, then

g_{B} ⊑ I_{η} (k, C_{η} (k, g_{B}, r), r) ⊑ I_{η} (k, C_{η} (k, f_{A}, r), r) .

Since

f_{A} ⊑ C_{η} (k, I_{η} (k, g_{B}, r), r)

, then

f_{A} ⊑ C_{η} (k, I_{η} (k, I_{η} (k, C_{η} (k, f_{A}, r), r), r), r) = C_{η} (k, I_{η} (k, C_{η} (k, f_{A}, r), r), r)

, so

f_{A}

is r-fuzzy soft

β

-open.

□

Proposition 3.

Let

(V, η_{F})

be a FSTS,

g_{B}, f_{A} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

. If

g_{B}

is r-fuzzy soft pre-closed set such that

g_{B} ⊒ f_{A} ⊒ I_{η} (k, C_{η} (k, g_{B}, r), r)

f_{A}

is r-fuzzy soft

β

-closed.

Proof.

Easily proved by a similar way in Proposition 2.2.

□

Definition 13.

In a FSTS

(V, η_{F})

, for each

g_{B} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

, we define a fuzzy soft operator

β C_{η}

: F \times \tilde{(V, F)} \times I_{\circ} \to \tilde{(V, F)}

as follows:

β C_{η} (k, g_{B}, r) = ⊓ {f_{A} \in \tilde{(V, F)} : g_{B} ⊑ f_{A}, f_{A} is r - fuzzy soft β - closed} .

Theorem 1.

In a FSTS

(V, η_{F})

, for each

f_{A}, g_{B} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

, the operator

β C_{η}

: F \times \tilde{(V, F)} \times I_{\circ} \to \tilde{(V, F)}

satisfies the following properties:

(1)

β C_{η} (k, Φ, r) = Φ

(2)

g_{B} ⊑ β C_{η} (k, g_{B}, r) ⊑ C_{η} (k, g_{B}, r)

(3)

β C_{η} (k, g_{B}, r) ⊑ β C_{η} (k, f_{A}, r)

, if

g_{B} ⊑ f_{A}

(4)

β C_{η} (k, β C_{η} (k, g_{B}, r), r) = β C_{η} (k, g_{B}, r)

(5)

β C_{η} (k, g_{B} ⊔ f_{A}, r) ⊒ β C_{η} (k, g_{B}, r) ⊔ β C_{η} (k, f_{A}, r)

(6)

g_{B}

is r-fuzzy soft

β

-closed iff

β C_{η} (k, g_{B}, r) = g_{B}

(7)

β C_{η} (k, C_{η} (k, g_{B}, r), r) = C_{η} (k, g_{B}, r)

Proof. (1), (2), (3) and (6) are easily proved from the definition of

β C_{η}

(4) From (2) and (3),

β C_{η} (k, g_{B}, r) ⊑ β C_{η} (k, β C_{η} (k, g_{B}, r), r)

. Now we show that

β C_{η} (k, g_{B}, r) ⊒ β C_{η} (k, β C_{η} (k, g_{B}, r), r)

. Suppose that

β C_{η} (k, g_{B}, r)

is not contain

β C_{η} (k, β C_{η} (k, g_{B}, r), r)

. Then, there is

v \in V

and

t \in (0, 1)

such that

β C_{η} (k, g_{B}, r) (k) (v) < t < β C_{η} (k, β C_{η} (k, g_{B}, r), r) (k) (v) . (A)

Since

β C_{η} (k, g_{B}, r) (k) (v) < t

, by the definition of

β C_{η}

, there is

h_{F}

is r-fuzzy soft

β

-closed with

g_{B} ⊑ h_{F}

such that

β C_{η} (k, g_{B}, r) (k) (v) \leq h_{F} (k) (v) < t

. Since

g_{B} ⊑ h_{F}

, we have

β C_{η} (k, g_{B}, r) ⊑ h_{F}

. Again, by the definition of

β C_{η}

β C_{η} (k, β C_{η} (k, g_{B}, r), r) ⊑ h_{F}

. Hence

β C_{η} (k, β C_{η} (k, g_{B}, r), r) (k) (v) \leq h_{F} (k) (v) < t

, it is a contradiction for

(A)

. Thus,

β C_{η} (k, g_{B}, r) ⊒ β C_{η} (k, β C_{η} (k, g_{B}, r), r)

. Then,

β C_{η} (k, β C_{η} (k, g_{B}, r), r) = β C_{η} (k, g_{B}, r) .

(5) Since

g_{B}

and

f_{A}

⊑ g_{B} ⊔ f_{A}

, hence by (3),

β C_{η} (k, g_{B}, r) ⊑ β C_{η} (k, g_{B} ⊔ f_{A}, r)

and

β C_{η} (k, f_{A}, r) ⊑ β C_{η} (k, g_{B} ⊔ f_{A}, r)

. Thus,

β C_{η} (k, g_{B} ⊔ f_{A}, r) ⊒ β C_{η} (k, g_{B}, r) ⊔ β C_{η} (k, f_{A}, r)

(7) From (6) and

C_{η} (k, g_{B}, r)

is r-fuzzy soft

β

-closed set, hence

β C_{η} (k, C_{η} (k, g_{B}, r), r) = C_{η} (k, g_{B}, r)

□

Definition 14.

Let

(V, η_{F})

be a FSTS,

r \in I_{0}

and

g_{B}, f_{A} \in \tilde{(V, F)}

. Then, we have:

(1) Two fuzzy soft sets

g_{B}

and

f_{A}

are called r-fuzzy soft

β

-separated iff

g_{B} \neg \tilde{q} β C_{η} (k, f_{A}, r)

and

f_{A} \neg \tilde{q} β C_{η} (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

β

-separated sets is called an r-fuzzy soft

β

-connected.

Theorem 2.

In a FSTS

(V, η_{F})

, we have:

(1) If

f_{A}

and

g_{B}

\in \tilde{(V, F)}

are r-fuzzy soft

β

-separated and

h_{C}

t_{D}

\in \tilde{(V, F)}

such that

h_{C} ⊑ f_{A}

and

t_{D} ⊑ g_{B}

, then

h_{C}

and

t_{D}

are r-fuzzy soft

β

-separated.

(2) If

f_{A} \neg \tilde{q} g_{B}

and either both are r-fuzzy soft

β

-open or both r-fuzzy soft

β

-closed, then

f_{A}

and

g_{B}

are r-fuzzy soft

β

-separated.

(3) If

f_{A}

and

g_{B}

are either both r-fuzzy soft

β

-open or both r-fuzzy soft

β

-closed, then

f_{A} ⊓ g_{B}^{c}

and

g_{B} ⊓ f_{A}^{c}

are r-fuzzy soft

β

-separated.

Proof.

(1) and (2) are obvious.

(3) Let

f_{A}

and

g_{B}

be an r-fuzzy soft

β

-open. Since

f_{A} ⊓ g_{B}^{c} ⊑ g_{B}^{c}

β C_{η} (k, f_{A} ⊓ g_{B}^{c}, r) ⊑ g_{B}^{c}

and hence

β C_{η} (k, f_{A} ⊓ g_{B}^{c}, r) \neg \tilde{q} g_{B}

. Then,

β C_{η} (k, f_{A} ⊓ g_{B}^{c}, r) \neg \tilde{q} (g_{B} ⊓ f_{A}^{c}) .

Again, since

g_{B} ⊓ f_{A}^{c} ⊑ f_{A}^{c}

β C_{η} (k, g_{B} ⊓ f_{A}^{c}, r) ⊑ f_{A}^{c}

and hence

β C_{η} (k, g_{B}

⊓ f_{A}^{c}, r) \neg \tilde{q} f_{A}

. Then,

β C_{η} (k, g_{B} ⊓ f_{A}^{c}, r) \neg \tilde{q} (f_{A} ⊓ g_{B}^{c}) .

Thus,

f_{A} ⊓ g_{B}^{c}

and

g_{B} ⊓ f_{A}^{c}

are r-fuzzy soft

β

-separated. The other case follows similar lines.

□

Theorem 3.

In a FSTS

(V, η_{F})

, then

f_{A}

g_{B}

\in \tilde{(V, F)}

are r-fuzzy soft

β

-separated iff there exist two r-fuzzy soft

β

-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 \tilde{(V, F)}

be an r-fuzzy soft

β

-separated,

f_{A} ⊑ {(β C_{η} (k, g_{B}, r))}^{c} = h_{C}

and

g_{B} ⊑ {(β C_{η} (k, f_{A}, r))}^{c} = t_{D}

, where

t_{D}

and

h_{C}

are r-fuzzy soft

β

-open, then

t_{D} \neg \tilde{q} β C_{η} (k, f_{A}, r)

and

h_{C} \neg \tilde{q} β C_{η} (k, 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

β

-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,

β C_{η} (k, g_{B}, r) ⊑ h_{C}^{c}

and

β C_{η} (k, f_{A}, r) ⊑ t_{D}^{c}

. Then,

β C_{η} (k, g_{B}, r) \neg \tilde{q} f_{A}

and

β C_{η} (k, f_{A}, r) \neg \tilde{q} g_{B}

. Thus,

g_{B}

and

f_{A}

are r-fuzzy soft

β

-separated. Hence, we obtain the required result.

□

Theorem 4.

In a FSTS

(V, η_{F})

, if

g_{B} \in \tilde{(V, F)}

is r-fuzzy soft

β

-connected such that

g_{B} ⊑ f_{A} ⊑ β C_{η} (k, g_{B}, r)

, then

f_{A}

is r-fuzzy soft

β

-connected.

Proof.

Suppose that

f_{A}

is not r-fuzzy soft

β

-connected, then there is r-fuzzy soft

β

-separated sets

h_{C}^{*}

and

t_{D}^{*}

\in \tilde{(V, F)}

such that

f_{A} = h_{C}^{*} ⊔ t_{D}^{*}

. Let

h_{C} = g_{B} ⊓ h_{C}^{*}

and

t_{D} = g_{B} ⊓ t_{D}^{*}

, then

g_{B} = t_{D} ⊔ h_{C}

. Since

h_{C} ⊑ h_{C}^{*}

and

t_{D} ⊑ t_{D}^{*}

, hence by Theorem 2.2(1),

h_{C}

and

t_{D}

are r-fuzzy soft

β

-separated, it is a contradiction. Thus,

f_{A}

is r-fuzzy soft

β

-connected, as required.

□

Theorem 5.

In a FSTS

(V, η_{F})

, for each

g_{B} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

, we define a fuzzy soft operator

β I_{η}

: F \times \tilde{(V, F)} \times I_{\circ} \to \tilde{(V, F)}

as follows:

β I_{η} (k, g_{B}, r) = ⊔ {f_{A} \in \tilde{(V, F)} : f_{A} ⊑ g_{B}, f_{A} is r - fuzzy soft β - open} .

Then, for each

f_{A}, g_{B} \in \tilde{(V, F)}

, the operator

β I_{η}

satisfies the following properties:

(1)

β I_{η} (k, \tilde{F}, r) = \tilde{F}

(2)

I_{η} (k, g_{B}, r) ⊑ β I_{η} (k, g_{B}, r) ⊑ g_{B}

(3)

β I_{η} (k, g_{B}, r) ⊑ β I_{η} (k, f_{A}, r)

, if

g_{B} ⊑ f_{A}

(4)

β I_{η} (k, β I_{η} (k, g_{B}, r), r) = β I_{η} (k, g_{B}, r)

(5)

β I_{η} (k, g_{B}, r) ⊓ β I_{η} (k, f_{A}, r) ⊒ β I_{η} (k, g_{B} ⊓ f_{A}, r)

(6)

g_{B}

is r-fuzzy soft

β

-open iff

β I_{η} (k, g_{B}, r) = g_{B}

(7)

β I_{η} (k, g_{B}^{c}, r) = {(β C_{η} (k, g_{B}, r))}^{c}

Proof.

(1), (2), (3) and (6) are easily proved from the definition of

β I_{η}

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

(7) For each

g_{B} \in \tilde{(V, F)}

k \in F

and

r \in I_{0}

, we have

β I_{η} (k, g_{B}^{c}, r) = ⊔ {f_{A} \in \tilde{(V, F)} : f_{A} ⊑ g_{B}^{c}, f_{A} is r - fuzzy soft β - open}

= [

⊓ {f_{A}^{c} \in \tilde{(V, F)} : g_{B} ⊑ f_{A}^{c}, f_{A}^{c} is r - fuzzy soft β - closed}]^{c}

{(β C_{η} (k, g_{B}, r))}^{c}

□

3. Weaker Forms of Fuzzy Soft $β$ -Continuity

Here, we investigate some properties of fuzzy soft

β

-continuity. As a weaker form of fuzzy soft

β

-continuity, the concepts of fuzzy soft almost (weakly)

β

-continuous functions are introduced, and some properties are given. Also, we show that fuzzy soft

β

-continuity ⇒ fuzzy soft almost

β

-continuity ⇒ fuzzy soft weakly

β

-continuity, but the converse may not be true.

Definition 15.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs. A fuzzy soft function

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft

β

-continuous if,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-closed set for each

g_{B} \in \tilde{(U, E)}

with

τ_{e} (g_{B}^{c}) \geq r

k \in F

(e = ψ (k)) \in E

and

r \in I_{o}

Theorem 6.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a fuzzy soft function. The following statements are equivalent for each

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft

β

-continuous.

(2) For each

g_{B} \in \tilde{(U, E)}

with

τ_{e} (g_{B}) \geq r

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-open.

(3)

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

(4)

φ_{ψ}^{- 1} (I_{τ} (e, g_{B}, r)) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r)

(5)

I_{η} (k, C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

Proof.

(1) ⇔ (2) Follows from Proposition 2.1(1) and

φ_{ψ}^{- 1} (g_{B}^{c}) = {(φ_{ψ}^{- 1} (g_{B}))}^{c}

(1) ⇒ (3) Let

g_{B} \in \tilde{(U, E)}

, hence by (1),

φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

is r-fuzzy soft

β

-closed. Then, we obtain

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

(3) ⇔ (4) Follows from Theorem 2.5(7).

(3) ⇒ (5) Let

g_{B} \in \tilde{(U, E)}

, hence by (3), we obtain

I_{η} (k, C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r) ⊑ β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

(5) ⇒ (2) Let

g_{B} \in \tilde{(U, E)}

with

τ_{e} (g_{B}) \geq r

, hence by (5), we obtain

{(φ_{ψ}^{- 1} (g_{B}))}^{c} = φ_{ψ}^{- 1} (g_{B}^{c}) ⊒ I_{η} (k, C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}^{c}), r), r), r) = {(C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r))}^{c}

. Then,

φ_{ψ}^{- 1} (g_{B}) ⊑ C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r)

, so

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-open.

□

Lemma 1.

Every fuzzy soft continuous function [15] is fuzzy soft

β

-continuous.

Proof.

Follows from Definition 1.5 and Theorem 3.1.

□

Remark 1.

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

Example 1.

Let

V = {v_{1}, v_{2}, v_{3}}

F = {k_{1}, k_{2}}

and define

g_{F}, h_{F} \in \tilde{(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

η_{F}, τ_{F} : F ⟶ {[0, 1]}^{\tilde{(V, F)}}

as follows:

\forall k \in F

η_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{2}, if n_{F} = g_{F}, \\ 0, otherwise, \end{matrix}

τ_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{3}, if n_{F} = h_{F}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (V, η_{F}) ⟶ (V, τ_{F})

is fuzzy soft

β

-continuous, but it is not fuzzy soft continuous.

Definition 16.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs. A fuzzy soft function

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft almost (resp. weakly)

β

-continuous if, for each

k_{v_{t}} \in \tilde{P_{t} (V)}

and each

f_{A} \in \tilde{(U, E)}

with

τ_{e} (f_{A}) \geq r

containing

φ_{ψ} (k_{v_{t}})

, there is

g_{B} \in \tilde{(V, F)}

is r-fuzzy soft

β

-open set containing

k_{v_{t}}

such that

φ_{ψ} (g_{B}) ⊑ I_{τ} (e, C_{τ} (e, f_{A}, r), r)

(resp.

φ_{ψ} (g_{B}) ⊑ C_{τ} (e, f_{A}, r)

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

Lemma 2. (1) Every fuzzy soft

β

-continuous function is fuzzy soft almost

β

-continuous.

(2) Every fuzzy soft almost

β

-continuous function is fuzzy soft weakly

β

-continuous.

Proof.

Follows from Definition 3.2 and Theorem 3.1.

□

Remark 2.

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

Example 2.

Let

V = {v_{1}, v_{2}, v_{3}}

F = {k_{1}, k_{2}}

and define

f_{F}, g_{F}, h_{F}

\in \tilde{(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

η_{F}, τ_{F} : F ⟶ {[0, 1]}^{\tilde{(V, F)}}

as follows:

\forall k \in F

η_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{2}{3}, if n_{F} = f_{F}, \\ \frac{1}{2}, if n_{F} = g_{F}, \\ 0, otherwise, \end{matrix}

τ_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{2}{3}, if n_{F} = f_{F}, \\ \frac{1}{3}, if n_{F} = h_{F}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (V, η_{F}) ⟶ (V, τ_{F})

is fuzzy soft almost

β

-continuous, but it is not fuzzy soft

β

-continuous.

Example 3.

Let

V = {v_{1}, v_{2}, v_{3}}

F = {k_{1}, k_{2}}

and define

g_{F}, h_{F} \in \tilde{(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

η_{F}, τ_{F} : F ⟶ {[0, 1]}^{\tilde{(V, F)}}

as follows:

\forall k \in F

η_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{2}, if n_{F} = g_{F}, \\ 0, otherwise, \end{matrix}

τ_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{3}, if n_{F} = h_{F}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (V, η_{F}) ⟶ (V, τ_{F})

is fuzzy soft weakly

β

-continuous, but it is not fuzzy soft almost

β

-continuous.

Lemma 3. (1) Every fuzzy soft almost continuous function is fuzzy soft almost

β

-continuous.

(2) Every fuzzy soft weakly continuous function is fuzzy soft weakly

β

-continuous.

Proof.

Follows from Definitions 1.9 and 3.2.

□

Remark 3.

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

continuity \Rightarrow β - continuity

⇓ ⇓

almost continuity \Rightarrow almost β - continuity

⇓ ⇓

weakly continuity \Rightarrow weakly β - continuity

Theorem 7.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a fuzzy soft function. The following statements are equivalent for each

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft almost

β

-continuous.

(2)

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-open, for each

g_{B}

is r-fuzzy soft regularly open.

(3)

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-closed, for each

g_{B}

is r-fuzzy soft regularly closed.

(4)

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

, for each

g_{B}

is r-fuzzy soft

β

-open.

(5)

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

, for each

g_{B}

is r-fuzzy soft semi-open.

(6)

β I_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)), r) ⊒ φ_{ψ}^{- 1} (g_{B})

, for each

g_{B}

with

τ_{e} (g_{B}) \geq r

Proof.

(1) ⇒ (2) Let

k_{v_{t}} \in \tilde{P_{t} (V)}

and

g_{B} \in \tilde{(U, E)}

be an r-fuzzy soft regularly open set containing

φ_{ψ} (k_{v_{t}})

, hence by (1), there is

f_{A} \in \tilde{(V, F)}

is r-fuzzy soft

β

-open set containing

k_{v_{t}}

such that

φ_{ψ} (f_{A}) ⊑ I_{τ} (e, C_{τ} (e, g_{B}, r), r)

Thus,

f_{A} ⊑ φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)) = φ_{ψ}^{- 1} (g_{B})

and

k_{v_{t}} \tilde{\in} f_{A} ⊑ φ_{ψ}^{- 1} (g_{B})

. Then,

k_{v_{t}} \tilde{\in} C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r)

and

φ_{ψ}^{- 1} (g_{B}) ⊑ C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r)

. Therefore,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-open set.

(2) ⇒ (3) Let

g_{B}

be an r-fuzzy soft regularly closed set, hence by (2),

φ_{ψ}^{- 1} (g_{B}^{c}) = {(φ_{ψ}^{- 1} (g_{B}))}^{c}

is r-fuzzy soft

β

-open set. Then,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-closed set.

(3) ⇒ (4) Let

g_{B}

be an r-fuzzy soft

β

-open set. Since

C_{τ} (e, g_{B}, r)

is r-fuzzy soft regularly closed set, hence by (3),

φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

is r-fuzzy soft

β

-closed set. Since

φ_{ψ}^{- 1} (g_{B}) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

, then we have

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

(4) ⇒ (5) This is obvious from every r-fuzzy soft semi-open set is r-fuzzy soft

β

-open set.

(5) ⇒ (3) Let

g_{B}

be an r-fuzzy soft regularly closed set, hence

g_{B}

is r-fuzzy soft semi-open set. Then by (5),

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r)) = φ_{ψ}^{- 1} (g_{B})

. Therefore,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-closed set.

(3) ⇒ (6) Let

g_{B} \in \tilde{(U, E)}

with

τ_{e} (g_{B}) \geq r

and

k_{v_{t}} \tilde{\in} φ_{ψ}^{- 1} (g_{B})

, then we have

k_{v_{t}} \tilde{\in} φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r))

. Since

{[I_{τ} (e, C_{τ} (e, g_{B}, r), r)]}^{c}

is r-fuzzy soft regularly closed set, hence by (3),

φ_{ψ}^{- 1} ({[I_{τ} (e, C_{τ} (e, g_{B}, r), r)]}^{c})

is r-fuzzy soft

β

-closed set. Thus,

φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r))

is r-fuzzy soft

β

-open set and

k_{v_{t}} \tilde{\in} β I_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)), r)

. Then,

φ_{ψ}^{- 1} (g_{B}) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)), r)

(6) ⇒ (1) Let

k_{v_{t}} \in \tilde{P_{t} (V)}

and

g_{B} \in \tilde{(U, E)}

with

τ_{e} (g_{B}) \geq r

containing

φ_{ψ} (k_{v_{t}})

, hence by (6),

φ_{ψ}^{- 1} (g_{B}) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)), r)

Since

k_{v_{t}} \tilde{\in} φ_{ψ}^{- 1} (g_{B})

, then we obtain

k_{v_{t}} \tilde{\in} β I_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)), r) = f_{A}

(say). Hence, there is

f_{A} \in \tilde{(V, F)}

is r-fuzzy soft

β

-open set containing

k_{v_{t}}

such that

φ_{ψ} (f_{A}) ⊑ I_{τ} (e, C_{τ} (e, g_{B}, r), r)

. Therefore,

φ_{ψ}

is fuzzy soft almost

β

-continuous.

□

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

Theorem 8.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a fuzzy soft function. The following statements are equivalent for each

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft weakly

β

-continuous.

(2)

C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r)), r), r), r) ⊒ φ_{ψ}^{- 1} (g_{B})

, if

τ_{e} (g_{B}) \geq r

(3)

I_{η} (k, C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, g_{B}, r)), r), r), r) ⊑ φ_{ψ}^{- 1} (g_{B})

, if

τ_{e} (g_{B}^{c}) \geq r

(4)

β C_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, g_{B}, r)), r) ⊑ φ_{ψ}^{- 1} (g_{B})

, if

τ_{e} (g_{B}^{c}) \geq r

(5)

β C_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)), r) ⊑ φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r))

(6)

β I_{η} (k, φ_{ψ}^{- 1} (C_{τ} (e, I_{τ} (e, g_{B}, r), r)), r) ⊒ φ_{ψ}^{- 1} (I_{τ} (e, g_{B}, r))

(7)

φ_{ψ}^{- 1} (g_{B}) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r)), r)

, if

τ_{e} (g_{B}) \geq r

Let

P

and

Q

: F \times \tilde{(V, F)} \times I_{\circ} \to \tilde{(V, F)}

be operators on

\tilde{(V, F)}

, and

R

and

S

: E \times \tilde{(U, E)} \times I_{\circ} \to \tilde{(U, E)}

be operators on

\tilde{(U, E)}

Definition 17.

[26]

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs.

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft

(P, Q, R, S)

-continuous function if,

P [k, φ_{ψ}^{- 1} (S (e, g_{B}, r)), r] \bar{⊓} Q [k, φ_{ψ}^{- 1} (R (e, g_{B}, r)), r]

= Φ

for each

g_{B} \in \tilde{(U, E)}

with

τ_{e} (g_{B}) \geq r

k \in F

and

(e = ψ (k)) \in E

In (2014), Aygünoǧlu et al. [15] defined the notion of fuzzy soft continuous functions:

η_{k} (φ_{ψ}^{- 1} (f_{A})) \geq τ_{e} (f_{A})

, for each

f_{A} \in \tilde{(U, E)}

k \in F

and

(e = ψ (k)) \in E

. We can see that Definition 3.3 generalizes the concept of fuzzy soft continuous functions, when we choose

P

= identity operator,

Q

= interior operator,

R

= identity operator and

S

= identity operator.

A historical justification of Definition 3.3:

(1): (1) In Section 3, we introduced the notion of fuzzy soft $β$ -continuous functions: $φ_{ψ}^{- 1} (g_{B}) ⊑ C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r)$ , for each $g_{B} \in \tilde{(U, E)}$ with $τ_{e} (g_{B}) \geq r$ . Here, $P$ = identity operator, $Q$ = closure interior closure operator, $R$ = identity operator and $S$ = identity operator.
(2): (2) In Section 3, we introduced the notion of fuzzy soft almost $β$ -continuous functions: $φ_{ψ}^{- 1} (g_{B}) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (I_{τ} (e, C_{τ} (e, g_{B}, r), r)), r)$ , for each $g_{B} \in \tilde{(U, E)}$ with $τ_{e} (g_{B}) \geq r$ . Here, $P$ = identity operator, $Q$ = $β$ -interior operator, $R$ = interior closure operator and $S$ = identity operator.
(3): (3) In Section 3, we introduced the notion of fuzzy soft weakly $β$ -continuous functions: $φ_{ψ}^{- 1} (g_{B}) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (C_{τ} (e, g_{B}, r)), r)$ , for each $g_{B} \in \tilde{(U, E)}$ with $τ_{e} (g_{B}) \geq r$ . Here, $P$ = identity operator, $Q$ = $β$ -interior operator, $R$ = closure operator and $S$ = identity operator.

4. Fuzzy Soft $β$ -Irresoluteness

Here, we introduce the concepts of fuzzy soft

β

-irresolute (resp. strongly

β

-irresolute,

β

-irresolute open,

β

-irresolute closed and

β

-irresolute homeomorphism) functions between two fuzzy soft topological spaces

(V, η_{F})

and

(U, τ_{E})

, and study some of its features. Also, we show that fuzzy soft strongly

β

-irresolute ⇒ fuzzy soft

β

-irresolute ⇒ fuzzy soft

β

-continuity, but the converse may not be true.

Definition 18.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs. A fuzzy soft function

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft

β

-irresolute (resp. strongly

β

-irresolute) if,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-open (resp. semi-open) set for each

g_{B} \in \tilde{(U, E)}

is r-fuzzy soft

β

-open set and

r \in I_{o}

Lemma 4. (1) Every fuzzy soft strongly

β

-irresolute function is fuzzy soft

β

-irresolute.

(2) Every fuzzy soft

β

-irresolute function is fuzzy soft

β

-continuous.

Proof.

Follows from Definition 4.1 and Theorem 3.1.

□

Remark 4.

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

Example 4.

Let

V = {v_{1}, v_{2}, v_{3}}

F = {k_{1}, k_{2}}

and define

f_{F}, g_{F} \in \tilde{(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

η_{F}, τ_{F} : F ⟶ {[0, 1]}^{\tilde{(V, F)}}

as follows:

\forall k \in F

η_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{2}, if n_{F} = g_{F}, \\ 0, otherwise, \end{matrix}

τ_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{2}, if n_{F} = f_{F}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (V, η_{F}) ⟶ (V, τ_{F})

is fuzzy soft

β

-irresolute, but it is not fuzzy soft strongly

β

-irresolute.

Example 5.

Let

V = {v_{1}, v_{2}, v_{3}}

F = {k_{1}, k_{2}}

and define

f_{F}, g_{F} \in \tilde{(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

η_{F}, τ_{F} : F ⟶ {[0, 1]}^{\tilde{(V, F)}}

as follows:

\forall k \in F

η_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{3}, if n_{F} = f_{F}, \\ \frac{1}{2}, if n_{F} = g_{F}, \\ 0, otherwise, \end{matrix}

τ_{k} (n_{F}) =

\{\begin{matrix} 1, if n_{F} \in {Φ, \tilde{F}}, \\ \frac{1}{3}, if n_{F} = f_{F}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (V, η_{F}) ⟶ (V, τ_{F})

is fuzzy soft

β

-continuous, but it is not fuzzy soft

β

-irresolute.

Remark 5.

From the previous results, we have: Fuzzy soft strongly

β

-irresolute ⇒ fuzzy soft

β

-irresolute ⇒ fuzzy soft

β

-continuity.

Theorem 9.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a fuzzy soft function. The following statements are equivalent for each

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft

β

-irresolute.

(2) For each

g_{B}

is r-fuzzy soft

β

-closed,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-closed.

(3)

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

(4)

φ_{ψ}^{- 1} (β I_{τ} (e, g_{B}, r)) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r)

(5)

I_{η} (k, C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r) ⊑ φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

Proof.

(1) ⇔ (2) Follows from Proposition 2.1(1) and

φ_{ψ}^{- 1} (g_{B}^{c}) = {(φ_{ψ}^{- 1} (g_{B}))}^{c}

(2) ⇒ (3) Let

g_{B} \in \tilde{(U, E)}

, hence by (2),

φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

is r-fuzzy soft

β

-closed. Then, we obtain

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

(3) ⇔ (4) Follows from Theorem 2.5(7).

(3) ⇒ (5) Let

g_{B} \in \tilde{(U, E)}

, hence by (3), we obtain

I_{η} (k, C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r) ⊑ β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

(5) ⇒ (1) Let

g_{B} \in \tilde{(U, E)}

be an r-fuzzy soft

β

-open, hence by (5), we obtain

{(φ_{ψ}^{- 1} (g_{B}))}^{c} = φ_{ψ}^{- 1} (g_{B}^{c}) ⊒ I_{η} (k, C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}^{c}), r), r), r) = {(C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r))}^{c}

. Then,

φ_{ψ}^{- 1} (g_{B}) ⊑ C_{η} (k, I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r), r)

, so

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

β

-open. Hence,

φ_{ψ}

is fuzzy soft

β

-irresolute.

□

Theorem 10.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a fuzzy soft function. The following statements are equivalent for each

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft strongly

β

-irresolute.

(2) For each

g_{B}

is r-fuzzy soft

β

-closed,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft semi-closed.

(3)

I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r) ⊑ φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

Proof.

(1) ⇔ (2) Follows from Proposition 2.1(1) and

φ_{ψ}^{- 1} (g_{B}^{c}) = {(φ_{ψ}^{- 1} (g_{B}))}^{c}

(2) ⇒ (3) Let

g_{B} \in \tilde{(U, E)}

, hence by (2),

φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

is r-fuzzy soft semi-closed. Then, we obtain

I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r) ⊑ φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

(3) ⇒ (1) Let

g_{B} \in \tilde{(U, E)}

be an r-fuzzy soft

β

-open, hence by (3), we obtain

{(φ_{ψ}^{- 1} (g_{B}))}^{c} = φ_{ψ}^{- 1} (g_{B}^{c}) ⊒ I_{η} (k, C_{η} (k, φ_{ψ}^{- 1} (g_{B}^{c}), r), r) = {(C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r))}^{c}

. Then,

φ_{ψ}^{- 1} (g_{B}) ⊑ C_{η} (k, I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r), r)

, so

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft semi-open. Hence,

φ_{ψ}

is fuzzy soft strongly

β

-irresolute.

□

Proposition 4.

Let

(V, η_{F})

(U, τ_{E})

and

(W, γ_{H})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

φ_{ψ^{*}}^{*} : \tilde{(U, E)} ⟶ \tilde{(W, H)}

be two fuzzy soft functions. Then, the composition

φ_{ψ^{*}}^{*} \circ φ_{ψ}

is fuzzy soft

β

-irresolute (resp. strongly

β

-irresolute and

β

-continuous) if,

φ_{ψ}

is fuzzy soft

β

-irresolute (resp. strongly

β

-irresolute and

β

-irresolute) and

φ_{ψ^{*}}^{*}

is fuzzy soft

β

-irresolute (resp.

β

-irresolute and

β

-continuous).

Proof.

Follows from Definition 4.1 and Theorem 3.1.

□

Proposition 5.

Let

(V, η_{F})

(U, τ_{E})

and

(W, γ_{H})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

φ_{ψ^{*}}^{*} : \tilde{(U, E)} ⟶ \tilde{(W, H)}

be two fuzzy soft functions. Then, the composition

φ_{ψ^{*}}^{*} \circ φ_{ψ}

is fuzzy soft almost

β

-continuous if,

φ_{ψ}

is fuzzy soft

β

-continuous (resp.

β

-irresolute and

β

-continuous) and

φ_{ψ^{*}}^{*}

is fuzzy soft almost continuous (resp. almost

β

-continuous and continuous).

Proof.

Follows from the above definitions. □

Definition 19.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs and

r \in I_{o}

. A fuzzy soft function

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft

β

-irresolute open (resp. closed) if,

φ_{ψ} (f_{A})

is r-fuzzy soft

β

-open (resp.

β

-closed) set for each

f_{A} \in \tilde{(V, F)}

is r-fuzzy soft

β

-open (resp.

β

-closed) set.

Theorem 11.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a fuzzy soft function. The following statements are equivalent for each

f_{A} \in \tilde{(V, F)}

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft

β

-irresolute open.

(2)

φ_{ψ} (β I_{η} (k, f_{A}, r)) ⊑ β I_{τ} (e, φ_{ψ} (f_{A}), r)

(3)

β I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (β I_{τ} (e, g_{B}, r))

(4) For each

g_{B}

and each

f_{A}

is r-fuzzy soft

β

-closed with

φ_{ψ}^{- 1} (g_{B}) ⊑ f_{A}

, there is

h_{C} \in \tilde{(U, E)}

is r-fuzzy soft

β

-closed with

g_{B} ⊑ h_{C}

such that

φ_{ψ}^{- 1} (h_{C}) ⊑ f_{A}

Proof.

(1) ⇒ (2) Since

φ_{ψ} (β I_{η} (k, f_{A}, r)) ⊑ φ_{ψ} (f_{A})

, hence by (1),

φ_{ψ} (β I_{η} (k, f_{A}, r))

is r-fuzzy soft

β

-open. Hence,

φ_{ψ} (β I_{η} (k, f_{A}, r)) ⊑ β I_{τ} (e, φ_{ψ} (f_{A}), r)

(2) ⇒ (3) Put

f_{A} = φ_{ψ}^{- 1} (g_{B})

, hence by (2),

φ_{ψ} (β I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r)) ⊑ β I_{τ} (e, φ_{ψ} (φ_{ψ}^{- 1} (g_{B})), r) ⊑ β I_{τ} (e, g_{B}, r)

. Then,

β I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) ⊑ φ_{ψ}^{- 1} (β I_{τ} (e, g_{B}, r))

(3) ⇒ (4) Let

g_{B} \in \tilde{(U, E)}

and

f_{A} \in \tilde{(V, F)}

be an r-fuzzy soft

β

-closed with

φ_{ψ}^{- 1} (g_{B}) ⊑ f_{A}

. Since

f_{A}^{c} ⊑ φ_{ψ}^{- 1} (g_{B}^{c})

f_{A}^{c} = β I_{η} (k, f_{A}^{c}, r) ⊑ β I_{η} (k, φ_{ψ}^{- 1} (g_{B}^{c}), r)

. Hence by (3),

f_{A}^{c} ⊑ β I_{η} (k, φ_{ψ}^{- 1} (g_{B}^{c}), r) ⊑ φ_{ψ}^{- 1} (β I_{τ} (e, g_{B}^{c}, r))

. Thus,

f_{A} ⊒ {(φ_{ψ}^{- 1} (β I_{τ} (e, g_{B}^{c}, r)))}^{c} = φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r)) .

Hence, there is

β C_{τ} (e, g_{B}, r) \in \tilde{(U, E)}

is r-fuzzy soft

β

-closed with

g_{B} ⊑ β C_{τ} (e, g_{B}, r)

such that

φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r)) ⊑ f_{A}

(4) ⇒ (1) Let

w_{D} \in \tilde{(V, F)}

be an r-fuzzy soft

β

-open. Put

g_{B} = {(φ_{ψ} (w_{D}))}^{c}

and

f_{A} = w_{D}^{c}

φ_{ψ}^{- 1} (g_{B}) = φ_{ψ}^{- 1} ({(φ_{ψ} (w_{D}))}^{c}) ⊑ f_{A}

. Hence by (4), there is

h_{C} \in \tilde{(U, E)}

is r-fuzzy soft

β

-closed with

g_{B} ⊑ h_{C}

such that

φ_{ψ}^{- 1} (h_{C}) ⊑ f_{A} = w_{D}^{c}

. Thus,

φ_{ψ} (w_{D}) ⊑ φ_{ψ} (φ_{ψ}^{- 1} (h_{C}^{c})) ⊑ h_{C}^{c}

. On the other hand, since

g_{B} ⊑ h_{C}

φ_{ψ} (w_{D}) = g_{B}^{c} ⊒ h_{C}^{c}

. Thus,

φ_{ψ} (w_{D}) = h_{C}^{c}

, so

φ_{ψ} (w_{D})

is r-fuzzy soft

β

-open. Then,

φ_{ψ}

is fuzzy soft

β

-irresolute open. □

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

Theorem 12.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a bijective fuzzy soft function. The following statements are equivalent for each

f_{A} \in \tilde{(V, F)}

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft

β

-irresolute closed.

(2)

β C_{τ} (e, φ_{ψ} (f_{A}), r) ⊑ φ_{ψ} (β C_{η} (k, f_{A}, r))

(3)

φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r)) ⊑ β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r)

(4) For each

g_{B}

and each

f_{A}

is r-fuzzy soft

β

-open with

φ_{ψ}^{- 1} (g_{B}) ⊑ f_{A}

, there is

h_{C} \in \tilde{(U, E)}

is r-fuzzy soft

β

-open with

g_{B} ⊑ h_{C}

such that

φ_{ψ}^{- 1} (h_{C}) ⊑ f_{A}

Proposition 6.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a bijective fuzzy soft function.

φ_{ψ}

is fuzzy soft

β

-irresolute closed iff

φ_{ψ}

is fuzzy soft

β

-irresolute open.

Proof.

It is easily proved from;

φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r)) ⊑ β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r)

⟺ φ_{ψ}^{- 1} (β I_{τ} (e, g_{B}^{c}, r)) ⊒ β I_{η} (k, φ_{ψ}^{- 1} (g_{B}^{c}), r)

□

Definition 20.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs. A fuzzy soft function

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

is said to be a fuzzy soft

β

-irresolute homeomorphism if,

φ_{ψ}

is bijective, and both of

φ_{ψ}

and

φ_{ψ}^{- 1}

are

β

-irresolute.

From the above theorems, we obtain the following corollary.

Corollary 1.

Let

(V, η_{F})

and

(U, τ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(V, F)} ⟶ \tilde{(U, E)}

be a bijective fuzzy soft function. The following statements are equivalent for each

f_{A} \in \tilde{(V, F)}

g_{B} \in \tilde{(U, E)}

k \in F

(e = ψ (k)) \in E

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft

β

-irresolute homeomorphism.

(2)

φ_{ψ}

is fuzzy soft

β

-irresolute and fuzzy soft

β

-irresolute open.

(3)

φ_{ψ}

is fuzzy soft

β

-irresolute and fuzzy soft

β

-irresolute closed.

(4)

φ_{ψ} (β I_{η} (k, f_{A}, r)) = β I_{τ} (e, φ_{ψ} (f_{A}), r)

(5)

φ_{ψ} (β C_{η} (k, f_{A}, r)) = β C_{τ} (e, φ_{ψ} (f_{A}), r)

(6)

β I_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) = φ_{ψ}^{- 1} (β I_{τ} (e, g_{B}, r))

(7)

β C_{η} (k, φ_{ψ}^{- 1} (g_{B}), r) = φ_{ψ}^{- 1} (β C_{τ} (e, g_{B}, r))

5. Conclusions and Future Work

This manuscript is lay out as follows:

(1)

In Section 2, fuzzy soft

β

-closure (

β

-interior) operators are introduced and studied on fuzzy soft topological space based on the article Aygünoǧlu et al. [15]. Moreover, the concept of r-fuzzy soft

β

-connected sets is defined and characterized.

(2)

In Section 3, some properties of a fuzzy soft

β

-continuity are investigated. As a weaker forms of the notion of fuzzy soft

β

-continuous functions, the notions of fuzzy soft almost (weakly)

β

-continuous functions are introduced, and some properties are obtained. Also, we show that fuzzy soft

β

-continuity ⇒ fuzzy soft almost

β

-continuity ⇒ fuzzy soft weakly

β

-continuity, but the converse may not be true. Furthermore, we have the following results:

• Fuzzy soft

(i d_{V}, C_{η} (I_{η} (C_{η})), i d_{U}, i d_{U})

-continuous function is fuzzy soft

β

-continuous.

• Fuzzy soft

(i d_{V}, β I_{η}, I_{τ} (C_{τ}), i d_{U})

-continuous function is fuzzy soft almost

β

-continuous.

• Fuzzy soft

(i d_{V}, β I_{η}, C_{τ}, i d_{U})

-continuous function is fuzzy soft weakly

β

-continuous.

(3)

In Section 4, the notions of fuzzy soft

β

-irresolute (resp. strongly

β

-irresolute,

β

-irresolute open,

β

-irresolute closed and

β

-irresolute homeomorphism) functions are introduced between two fuzzy soft topological spaces

(V, η_{F})

and

(U, τ_{E})

, and some properties of these functions are investigated. Additionally, we shown that fuzzy soft strongly

β

-irresolute ⇒ fuzzy soft

β

-irresolute ⇒ fuzzy soft

β

-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 [20]. Also, we will use the fuzzy soft

β

-closure operator to introduce some new separation axioms on fuzzy soft topological space based on the article Aygünoǧlu et al. [15].

Data Availability Statement

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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On Fuzzy Soft Beta-Continuity and Beta-Irresoluteness: Properties and Applications

Abstract

1. Introduction

2. Some Properties of r-Fuzzy Soft β -Closed Sets

3. Weaker Forms of Fuzzy Soft β -Continuity

4. Fuzzy Soft β -Irresoluteness

5. Conclusions and Future Work

Data Availability Statement

Conflicts of Interest

References

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2. Some Properties of r-Fuzzy Soft $β$ -Closed Sets

3. Weaker Forms of Fuzzy Soft $β$ -Continuity

4. Fuzzy Soft $β$ -Irresoluteness