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

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

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Submitted:

12 February 2024

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13 February 2024

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Abstract

In this manuscript, we first introduce some properties of r-fuzzy soft α-open sets in fuzzy soft topological spaces based on the manuscript Aygunoglu et al. (Hacet. J. Math. Stat. 2014, 43, 193-208). In addition, we define the concepts of fuzzy soft α-closure (α-interior) operators, and investigate some properties of them. Furthermore, the concept of r-fuzzy soft α-connected sets is defined and characterized with help of fuzzy soft α-closure operators. Thereafter, we introduce and study the concepts of fuzzy soft almost (weakly) α-continuous mappings, which are weaker forms of fuzzy soft α-continuous mappings, and we discuss some properties of fuzzy soft α-continuity. Moreover, we establish that fuzzy soft α-continuity → fuzzy soft almost α-continuity → fuzzy soft weakly α-continuity, but the converse may not be true. It is also we show that the composition is fuzzy soft almost α-continuous mapping if, is fuzzy soft α-continuous mapping and is fuzzy soft almost continuous mapping. Finally, some new types of compactness via r-fuzzy soft α-open sets are defined, and the relationships between them are examined.

Keywords:

Subject: Computer Science and Mathematics - Geometry and Topology

MSC: 54A05; 54A40; 54C05; 54C10; 54D30

1. Introduction

Theory of soft sets was first introduced by Molodtsov [1], which is a completely new approach for modeling uncertainty and vagueness. He demonstrated many applications of these theory in solving several practical problems in mathematics, engineering, economics, and social science etc. Akdag and Ozkan [2] defined the concept of soft

α

-open sets on soft topological spaces, and some properties are specified. The concept of soft

β

-open sets was defined and studied by the authors of [3,4]. Also, the concepts of soft semi-open, somewhere dense and Q-sets were studied by the authors of [5,6]. Al-shami et al. [7] introduced the concept of weakly soft semi-open sets and obtained its main properties. Moreover, Al-shami et al. [8] initiated the concept of weakly soft

β

-open sets and examined weakly soft

β

-continuity. Kaur et al. [9] introduced a new approach to studying soft continuous mappings using an induced mapping based on soft sets. Al Ghour and Al-Mufarrij [10] defined two new concepts of mappings over soft topological spaces: soft somewhat-r-continuity and soft somewhat-r-openness. In 2024, Ameen et al. [11] explored more properties of soft somewhere dense continuous mappings.

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

β

-open) sets were defined by Taha [20].

In our study, the layout is designed as follows.

• Firstly, we introduce the concepts of fuzzy soft

α

-closure (

α

-interior) operators in fuzzy soft topological space

(W, τ_{N})

based on the paper Aygünoǧlu et al. [14], and examine some of its properties. Also, the concept of r-fuzzy soft

α

-connected sets is introduced and studied.

• Secondly, we are going to investigate some properties of fuzzy soft

α

-continuous mappings between two fuzzy soft topological spaces

(W, τ_{N})

and

(V, η_{F})

. Moreover, we define and study the concepts of fuzzy soft weakly (almost)

α

-continuous mappings, which are weaker forms of fuzzy soft

α

-continuous mappings. Also, the relationships between these classes of mappings are investigated with the help of some examples.

• Finally, several types of fuzzy soft compactness via r-fuzzy soft

α

-open sets are defined, and the relationships between them are specified.

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

In this work, nonempty sets will be denoted by W, V etc. N is the set of all parameters for W and

C \subseteq N

. The family of all fuzzy sets on W is denoted by

I^{W}

(where

I_{\circ} = (0, 1], I = [0, 1]

), and for

s \in I,

\underset{̲}{s} (w) = s

, for all

w \in W .

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

Definition 1.1.

[14,21,22] A fuzzy soft set

h_{C}

on W is a mapping from N to

I^{W}

such that

h_{C} (n)

is a fuzzy set on W, for each

n \in C

and

h_{C} (n) = \underset{̲}{0}

, if

n \notin C

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

\tilde{(W, N)}

Definition 1.2.

[23] The difference between two fuzzy soft sets

h_{C}

and

g_{B}

is a fuzzy soft set defined as follows, for each

n \in N

(h_{C} \bar{⊓} g_{B}) (n) = \{\begin{matrix} \underset{̲}{0}, & i f h_{C} (n) \leq g_{B} (n), \\ h_{C} (n) \land {(g_{B} (n))}^{c}, & o t h e r w i s e . \end{matrix}

Definition 1.3.

[24] A fuzzy soft point

n_{w_{s}}

on W is a fuzzy soft set defined as follows:

n_{w_{s}} (k) = \{\begin{matrix} w_{s}, & i f & k = n, \\ \underset{̲}{0}, & i f & k \in N - {n}, \end{matrix}

where

w_{s}

is a fuzzy point on W.

n_{w_{s}}

is called belong to a fuzzy soft set

f_{A}

, denoted by

n_{w_{s}} \tilde{\in} f_{A}

, if

s \leq f_{A} (n) (w)

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

\tilde{P_{s} (W)}

Definition 1.4.

[25] A fuzzy soft point

n_{w_{s}} \in \tilde{P_{s} (W)}

is called a soft quasi-coincident with

h_{C} \in \tilde{(W, N)}

and denoted by

n_{w_{s}} \tilde{q} h_{C}

, if

s + h_{C} (n) (w) > 1

. A fuzzy soft set

h_{C} \in \tilde{(W, N)}

is called a soft quasi-coincident with

g_{B} \in \tilde{(W, N)}

and denoted by

h_{C} \tilde{q} g_{B}

, if there is

n \in N

and

w \in W

such that

h_{C} (n) (w) + g_{B} (n) (w) > 1

. If

h_{C}

is not soft quasi-coincident with

g_{B}

h_{C} \neg \tilde{q} g_{B}

Definition 1.5.

[14] A mapping

τ : N ⟶ {[0, 1]}^{\tilde{(W, N)}}

is called a fuzzy soft topology on W if it satisfies the following, for each

n \in N

(1)

τ_{n} (Φ) = τ_{n} (\tilde{N}) = 1,

(2)

τ_{n} (h_{C} ⊓ g_{B}) \geq τ_{n} (h_{C}) \land τ_{n} (g_{B}),

for each

h_{C}, g_{B} \in \tilde{(W, N)},

(3)

τ_{n} (⊔_{δ \in Δ} {(h_{C})}_{δ}) \geq \land_{δ \in Δ} τ_{n} ({(h_{C})}_{δ}),

for each

{(h_{C})}_{δ} \in \tilde{(W, N)}, δ \in Δ .

Then,

(W, τ_{N})

is called a fuzzy soft topological space (briefly, FSTS) in the sense of Šostak.

Definition 1.6.

[14] Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs. A fuzzy soft mapping

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

is called a fuzzy soft continuous if,

τ_{n} (φ_{ψ}^{- 1} (h_{C})) \geq η_{k} (h_{C})

for each

h_{C} \in \tilde{(V, F)}

n \in N

and

(k = ψ (n)) \in F

Definition 1.7.

[15,19] In a FSTS

(W, τ_{N})

, for each

h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

, we define the fuzzy soft operators

C_{τ}

and

I_{τ}

: N \times \tilde{(W, N)} \times I_{\circ} \to \tilde{(W, N)}

as follows:

C_{τ} (n, h_{C}, r) = ⊓ {g_{B} \in \tilde{(W, N)} : h_{C} ⊑ g_{B}, τ_{n} (g_{B}^{c}) \geq r}

I_{τ} (n, h_{C}, r) = ⊔ {g_{B} \in \tilde{(W, N)} : g_{B} ⊑ h_{C}, τ_{n} (g_{B}) \geq r} .

Definition 1.8.

Let

(W, τ_{N})

be a FSTS and

r \in I_{0}

. A fuzzy soft set

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft regularly open [19] (resp.

β

-open [20], pre-open [20],

α

-open [26] and semi-open [26]) if,

h_{C} = I_{τ} (n, C_{τ} (n, h_{C}, r), r)

(resp.

h_{C} ⊑ C_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r)

h_{C} ⊑ I_{τ} (n, C_{τ} (n, h_{C}, r), r)

h_{C} ⊑ I_{τ} (n, C_{τ} (n, I_{τ} (n, h_{C}, r), r), r)

and

h_{C} ⊑ C_{τ} (n, I_{τ} (n, h_{C}, r), r)

) for each

n \in N

Definition 1.9.

[19] Let

(W, τ_{N})

be a FSTS and

r \in I_{0}

. A fuzzy soft set

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft regularly closed if,

h_{C} = C_{τ} (n, I_{τ} (n, h_{C}, r), r)

for each

n \in N

Definition 1.10.

[26] Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs and

r \in I_{0}

. A fuzzy soft mapping

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

is called a fuzzy soft almost (resp. weakly) continuous if, for any

n_{w_{s}} \in \tilde{P_{s} (W)}

and any

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

with

η_{k} (f_{A}) \geq r

containing

φ_{ψ} (n_{w_{s}})

, there is

h_{C} \in \tilde{(W, N)}

with

τ_{n} (h_{C}) \geq r

containing

n_{w_{s}}

such that

φ_{ψ} (h_{C}) ⊑ I_{η} (k, C_{η} (k, f_{A}, r), r)

(resp.

φ_{ψ} (h_{C}) ⊑ C_{η} (k, f_{A}, r)

Remark 1.1.

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

Lemma 1.1.

Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs and

r \in I_{0}

. A fuzzy soft mapping

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

is fuzzy soft almost continuous if,

τ_{n} (φ_{ψ}^{- 1} (h_{C})) \geq r

for each

h_{C} \in \tilde{(V, F)}

is r-fuzzy soft regularly open,

n \in N

and

(k = ψ (n)) \in F

Proof.

Easily proved from Definition 1.10.

□

Definition 1.11.

Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs. A fuzzy soft mapping

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

is called a fuzzy soft open if,

η_{k} (φ_{ψ} (h_{C})) \geq τ_{n} (h_{C})

for each

h_{C} \in \tilde{(W, N)}

n \in N

and

(k = ψ (n)) \in F

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

2. On r-Fuzzy Soft $α$ -Open Sets

Here, we introduce and discuss the notions of fuzzy soft

α

-closure (

α

-interior) operators in fuzzy soft topological spaces based on the paper Aygünoǧlu et al. [14]. Also, the notion of r-fuzzy soft

α

-connected sets is defined and studied with help of fuzzy soft

α

-closure operators.

Definition 2.1.

Let

(W, τ_{N})

be a FSTS and

r \in I_{0}

. A fuzzy soft set

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft

α

-closed (resp. semi-closed,

β

-closed and pre-closed) if,

C_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r) ⊑ h_{C}

(resp.

I_{τ} (n, C_{τ} (n, h_{C}, r), r) ⊑ h_{C}

I_{τ} (n, C_{τ} (n, I_{τ} (n, h_{C}, r), r), r) ⊑ h_{C}

and

C_{τ} (n, I_{τ} (n, h_{C}, r), r) ⊑ h_{C}

) for each

n \in N

Remark 2.1.

The complement of an r-fuzzy soft

α

-open [26] (resp. semi-open [26],

β

-open [20] and pre-open [20]) set is an r-fuzzy soft

α

-closed (resp. semi-closed,

β

-closed and pre-closed) set.

Lemma 2.1.

Let

(W, τ_{N})

be a FSTS and

r \in I_{0}

. Then, any intersection (resp. union) of r-fuzzy soft

α

-closed (resp.

α

-open) sets is an r-fuzzy soft

α

-closed (resp.

α

-open) set.

Proof.

Easily proved from Definitions 1.8 and 2.1. □

Proposition 2.1.

Let

(W, τ_{N})

be a FSTS,

h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

. Then, the following statements are equivalent.

(1)

h_{C}

is r-fuzzy soft

α

-closed.

(2)

h_{C}

is r-fuzzy soft semi-closed and r-fuzzy soft pre-closed.

Proof. (1) ⇒ (2) Let

h_{C}

be an r-fuzzy soft

α

-closed,

h_{C} ⊒ C_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r) ⊒ I_{τ} (n, C_{τ} (n, h_{C}, r), r)

. This shows that

h_{C}

is r-fuzzy soft semi-closed.

Since

h_{C} ⊒ C_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r)

and

C_{τ} (n, h_{C}, r) ⊒ h_{C}

, then

h_{C} ⊒ C_{τ} (n, I_{τ} (n, h_{C}, r), r)

. Therefore,

h_{C}

is r-fuzzy soft pre-closed

(2) ⇒ (1) Let

h_{C}

be an r-fuzzy soft semi-closed and r-fuzzy soft pre-closed, then

h_{C} ⊒ C_{τ} (n, I_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r), r) = C_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r)

. This shows that

h_{C}

is r-fuzzy soft

α

-closed.

□

Proposition 2.2.

Let

(W, τ_{N})

be a FSTS,

g_{B}, h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

. If

g_{B}

is r-fuzzy soft semi-closed set such that

g_{B} ⊒ h_{C} ⊒ C_{τ} (n, I_{τ} (n, g_{B}, r), r)

h_{C}

is r-fuzzy soft

α

-closed.

Proof.

Let

g_{B}

be an r-fuzzy soft semi-closed and

g_{B} ⊒ h_{C}

, then

g_{B} ⊒ I_{τ} (n, C_{τ} (n, g_{B}, r), r) ⊒ I_{τ} (n, C_{τ} (n, h_{C}, r), r) .

Let

h_{C} ⊒ C_{τ} (n, I_{τ} (n, g_{B}, r), r)

, then

h_{C} ⊒ C_{τ} (n, I_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r), r) = C_{τ} (n, I_{τ} (n, C_{τ} (n, h_{C}, r), r), r)

. Therefore,

h_{C}

is r-fuzzy soft

α

-closed.

□

Lemma 2.2.

Let

(W, τ_{N})

be a FSTS,

g_{B}, h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

. If

g_{B}

is r-fuzzy soft

α

-closed set such that

g_{B} ⊒ h_{C} ⊒ C_{τ} (n, I_{τ} (n, g_{B}, r), r)

h_{C}

is r-fuzzy soft

α

-closed.

Proof.

It is easily proved from every r-fuzzy soft

α

-closed set is r-fuzzy soft semi-closed set.

□

Remark 2.2.

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

α - c l o s e d s e t

⇓ ⇓

s e m i - c l o s e d s e t p r e - c l o s e d s e t

⇓ ⇓

β - c l o s e d s e t

Remark 2.3.

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

Example 2.1.

Let

W = {w_{1}, w_{2}}

N = {n_{1}, n_{2}}

and define

f_{N}, g_{N}, h_{N} \in \tilde{(W, N)}

as follows:

f_{N} = {(n_{1}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.4}}), (n_{2}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.4}})}

g_{N} = {(n_{1}, {\frac{w_{1}}{0.6}, \frac{w_{2}}{0.2}}), (n_{2}, {\frac{w_{1}}{0.6}, \frac{w_{2}}{0.2}})}

h_{N} = {(n_{1}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.5}}), (n_{2}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.5}})}

. Define fuzzy soft topology

τ_{N} : N ⟶ {[0, 1]}^{\tilde{(W, N)}}

as follows:

τ_{n_{1}} (t_{N}) =

\{\begin{matrix} 1, if t_{N} \in {Φ, \tilde{N}}, \\ \frac{1}{2}, if t_{N} = f_{N}, \\ \frac{2}{3}, if t_{N} = g_{N}, \\ \frac{2}{3}, if t_{N} = f_{N} ⊓ g_{N}, \\ \frac{1}{2}, if t_{N} = f_{N} ⊔ g_{N}, \\ 0, otherwise, \end{matrix}

τ_{n_{2}} (t_{N}) =

\{\begin{matrix} 1, if t_{N} \in {Φ, \tilde{N}}, \\ \frac{1}{4}, if t_{N} = f_{N}, \\ \frac{1}{2}, if t_{N} = g_{N}, \\ \frac{1}{2}, if t_{N} = f_{N} ⊓ g_{N}, \\ \frac{1}{4}, if t_{N} = f_{N} ⊔ g_{N}, \\ 0, otherwise . \end{matrix}

Then,

h_{N}

\frac{1}{4}

-fuzzy soft semi-closed and

\frac{1}{4}

-fuzzy soft

β

-closed, but it is neither

\frac{1}{4}

-fuzzy soft

α

-closed nor

\frac{1}{4}

-fuzzy soft pre-closed.

Example 2.2.

Let

W = {w_{1}, w_{2}}

N = {n_{1}, n_{2}}

and define

f_{N}, h_{N} \in \tilde{(W, N)}

as follows:

f_{N} = {(n_{1}, {\frac{w_{1}}{0.4}, \frac{w_{2}}{0.5}}), (n_{2}, {\frac{w_{1}}{0.4}, \frac{w_{2}}{0.5}})}

h_{N} = {(n_{1}, {\frac{w_{1}}{0.7}, \frac{w_{2}}{0.6}}), (n_{2}, {\frac{w_{1}}{0.7}, \frac{w_{2}}{0.6}})}

. Define fuzzy soft topology

τ_{N} : N ⟶ {[0, 1]}^{\tilde{(W, N)}}

as follows:

τ_{n_{1}} (t_{N}) =

\{\begin{matrix} 1, if t_{N} \in {Φ, \tilde{N}}, \\ \frac{1}{3}, if t_{N} = f_{N}, \\ 0, otherwise, \end{matrix}

τ_{n_{2}} (t_{N}) =

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

Then,

h_{N}

\frac{1}{3}

-fuzzy soft pre-closed and

\frac{1}{3}

-fuzzy soft

β

-closed, but it is neither

\frac{1}{3}

-fuzzy soft semi-closed nor

\frac{1}{3}

-fuzzy soft

α

-closed.

Definition 2.2.

In a FSTS

(W, τ_{N})

, for each

h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

, we define a fuzzy soft

α

-closure operator

α C_{τ}

: N \times \tilde{(W, N)} \times I_{\circ} \to \tilde{(W, N)}

as follows:

α C_{τ} (n, h_{C}, r) = ⊓ {f_{A} \in \tilde{(W, N)} : h_{C} ⊑ f_{A}, f_{A} is r - fuzzy soft α - closed} .

Theorem 2.1.

In a FSTS

(W, τ_{N})

, for each

g_{B}, h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

, the operator

α C_{τ}

: N \times \tilde{(W, N)} \times I_{\circ} \to \tilde{(W, N)}

satisfies the following properties.

(1)

α C_{τ} (n, Φ, r) = Φ

(2)

h_{C} ⊑ α C_{τ} (n, h_{C}, r) ⊑ C_{τ} (n, h_{C}, r)

(3)

α C_{τ} (n, h_{C}, r) ⊑ α C_{τ} (n, g_{B}, r)

if,

h_{C} ⊑ g_{B}

(4)

α C_{τ} (n, α C_{τ} (n, h_{C}, r), r) = α C_{τ} (n, h_{C}, r)

(5)

α C_{τ} (n, h_{C} ⊔ g_{B}, r) ⊒ α C_{τ} (n, h_{C}, r) ⊔ α C_{τ} (n, g_{B}, r)

(6)

α C_{τ} (n, h_{C}, r) = h_{C}

iff

h_{C}

is r-fuzzy soft

α

-closed.

(7)

α C_{τ} (n, C_{τ} (n, h_{C}, r), r) = C_{τ} (n, h_{C}, r)

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

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

α C_{τ} (n, h_{C}, r) ⊑ α C_{τ} (n, α C_{τ} (n, h_{C}, r), r)

. Now we show that

α C_{τ} (n, h_{C}, r) ⊒ α C_{τ} (n, α C_{τ} (n, h_{C}, r), r)

. Suppose that

α C_{τ} (n, h_{C}, r)

is not contain

α C_{τ} (n, α C_{τ} (n, h_{C}, r), r)

. Then, there is

w \in W

and

s \in (0, 1)

such that

α C_{τ} (n, h_{C}, r) (n) (w) < s < α C_{τ} (n, α C_{τ} (n, h_{C}, r), r) (n) (w) . (A)

Since

α C_{τ} (n, h_{C}, r) (n) (w) < s

, by the definition of

α C_{τ}

, there is

g_{B}

is r-fuzzy soft

α

-closed with

h_{C} ⊑ g_{B}

such that

α C_{τ} (n, h_{C}, r) (n) (w) \leq g_{B} (n) (w) < s

. Since

h_{C} ⊑ g_{B}

, we have

α C_{τ} (n, h_{C}, r) ⊑ g_{B}

. Again, by the definition of

α C_{τ}

, we have

α C_{τ} (n, α C_{τ} (n, h_{C}, r), r) ⊑ g_{B}

. Hence

α C_{τ} (n, α C_{τ} (n, h_{C}, r), r) (n) (w) \leq g_{B} (n) (w) < s

, it is a contradiction for

(A)

. Thus,

α C_{τ} (n, h_{C}, r) ⊒ α C_{τ} (n, α C_{τ} (n, h_{C}, r), r)

. Then,

α C_{τ} (n, α C_{τ} (n, h_{C}, r), r) = α C_{τ} (n, h_{C}, r)

(5) Since

h_{C}

and

g_{B}

⊑ h_{C} ⊔ g_{B}

, hence by (3),

α C_{τ} (n, h_{C}, r) ⊑ α C_{τ} (n, h_{C} ⊔ g_{B}, r)

and

α C_{τ} (n, g_{B}, r) ⊑ α C_{τ} (n, h_{C} ⊔ g_{B}, r)

. Thus,

α C_{τ} (n, h_{C} ⊔ g_{B}, r) ⊒ α C_{τ} (n, h_{C}, r) ⊔ α C_{τ} (n, g_{B}, r)

(7) From (6) and

C_{τ} (n, h_{C}, r)

is r-fuzzy soft

α

-closed set, hence

α C_{τ} (n, C_{τ} (n, h_{C}, r), r) = C_{τ} (n, h_{C}, r)

□

Theorem 2.2.

In a FSTS

(W, τ_{N})

, for each

h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

, we define a fuzzy soft

α

-interior operator

α I_{τ}

: N \times \tilde{(W, N)} \times I_{\circ} \to \tilde{(W, N)}

as follows:

α I_{τ} (n, h_{C}, r) = ⊔ {f_{A} \in \tilde{(W, N)} : f_{A} ⊑ h_{C}, f_{A} is r - fuzzy soft α - open} .

Then, for each

g_{B}

and

h_{C} \in \tilde{(W, N)}

, the operator

α I_{τ}

satisfies the following properties.

(1)

α I_{τ} (n, \tilde{N}, r) = \tilde{N}

(2)

I_{τ} (n, h_{C}, r) ⊑ α I_{τ} (n, h_{C}, r) ⊑ h_{C}

(3)

α I_{τ} (n, h_{C}, r) ⊑ α I_{τ} (n, g_{B}, r)

if,

h_{C} ⊑ g_{B}

(4)

α I_{τ} (n, α I_{τ} (n, h_{C}, r), r) = α I_{τ} (n, h_{C}, r)

(5)

α I_{τ} (n, h_{C}, r) ⊓ α I_{τ} (n, g_{B}, r) ⊒ α I_{τ} (n, h_{C} ⊓ g_{B}, r)

(6)

α I_{τ} (n, h_{C}, r) = h_{C}

iff

h_{C}

is r-fuzzy soft

α

-open.

(7)

α I_{τ} (n, h_{C}^{c}, r) = {(α C_{τ} (n, h_{C}, r))}^{c}

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

α I_{τ}

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

(7) For each

h_{C} \in \tilde{(W, N)}

n \in N

and

r \in I_{0}

, we have

α I_{τ} (n, h_{C}^{c}, r) = ⊔ {f_{A} \in \tilde{(W, N)} : f_{A} ⊑ h_{C}^{c}, f_{A} is r - fuzzy soft α - open}

= [

⊓ {f_{A}^{c} \in \tilde{(W, N)} : h_{C} ⊑ f_{A}^{c}, f_{A}^{c} is r - fuzzy soft α - closed}]^{c}

{(α C_{τ} (n, h_{C}, r))}^{c}

□

Definition 2.3.

Let

(W, τ_{N})

be a FSTS,

r \in I_{0}

and

g_{B}, h_{C} \in \tilde{(W, N)}

. Then, we have:

(1) Two fuzzy soft sets

g_{B}

and

h_{C}

are called r-fuzzy soft

α

-separated iff

g_{B} \neg \tilde{q} α C_{τ} (n, h_{C}, r)

and

h_{C} \neg \tilde{q} α C_{τ} (n, g_{B}, r)

for each

n \in N

(2) Any fuzzy soft set which cannot be expressed as the union of two r-fuzzy soft

α

-separated sets is called an r-fuzzy soft

α

-connected.

Theorem 2.3.

In a FSTS

(W, τ_{N})

, we have:

(1) If

f_{A}

and

g_{B}

\in \tilde{(W, N)}

are r-fuzzy soft

α

-separated and

h_{C}

t_{D}

\in \tilde{(W, N)}

such that

h_{C} ⊑ f_{A}

and

t_{D} ⊑ g_{B}

, then

h_{C}

and

t_{D}

are r-fuzzy soft

α

-separated.

(2) If

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

and either both are r-fuzzy soft

α

-open or both r-fuzzy soft

α

-closed, then

f_{A}

and

g_{B}

are r-fuzzy soft

α

-separated.

(3) If

f_{A}

and

g_{B}

are either both r-fuzzy soft

α

-open or both r-fuzzy soft

α

-closed, then

f_{A} ⊓ g_{B}^{c}

and

g_{B} ⊓ f_{A}^{c}

are r-fuzzy soft

α

-separated.

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

(3) Let

f_{A}

and

g_{B}

be an r-fuzzy soft

α

-open. Since

f_{A} ⊓ g_{B}^{c} ⊑ g_{B}^{c}

α C_{τ} (n, f_{A} ⊓ g_{B}^{c}, r) ⊑ g_{B}^{c}

and hence

α C_{τ} (n, f_{A} ⊓ g_{B}^{c}, r) \neg \tilde{q} g_{B}

. Then,

α C_{τ} (n, f_{A} ⊓ g_{B}^{c}, r) \neg \tilde{q} (g_{B} ⊓ f_{A}^{c}) .

Again, since

g_{B} ⊓ f_{A}^{c} ⊑ f_{A}^{c}

α C_{τ} (n, g_{B} ⊓ f_{A}^{c}, r) ⊑ f_{A}^{c}

and hence

α C_{τ} (n, g_{B}

⊓ f_{A}^{c}, r) \neg \tilde{q} f_{A}

. Then,

α C_{τ} (n, 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

α

-separated. The other case follows similar lines.

□

Theorem 2.4.

In a FSTS

(W, τ_{N})

, then

f_{A}

g_{B}

\in \tilde{(W, N)}

are r-fuzzy soft

α

-separated iff there exist two r-fuzzy soft

α

-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{(W, N)}

be an r-fuzzy soft

α

-separated,

f_{A} ⊑ {(α C_{τ} (n, g_{B}, r))}^{c} = h_{C}

and

g_{B} ⊑ {(α C_{τ} (n, f_{A}, r))}^{c} = t_{D}

, where

t_{D}

and

h_{C}

are r-fuzzy soft

α

-open, then

t_{D} \neg \tilde{q} α C_{τ} (n, f_{A}, r)

and

h_{C} \neg \tilde{q} α C_{τ} (n, g_{B}, r)

. Thus,

g_{B} \neg \tilde{q} h_{C}

and

f_{A} \neg \tilde{q} t_{D}

. Hence, we obtain the required result.

(⇐) Let

h_{C}

and

t_{D}

be an r-fuzzy soft

α

-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,

α C_{τ} (n, g_{B}, r) ⊑ h_{C}^{c}

and

α C_{τ} (n, f_{A}, r) ⊑ t_{D}^{c}

. Then,

α C_{τ} (n, g_{B}, r) \neg \tilde{q} f_{A}

and

α C_{τ} (n, f_{A}, r) \neg \tilde{q} g_{B}

. Thus,

g_{B}

and

f_{A}

are r- fuzzy soft

α

-separated. Hence, we obtain the required result.

□

Theorem 2.5.

In a FSTS

(W, τ_{N})

, if

g_{B} \in \tilde{(W, N)}

is r-fuzzy soft

α

-connected such that

g_{B} ⊑ f_{A} ⊑ α C_{τ} (n, g_{B}, r)

, then

f_{A}

is r-fuzzy soft

α

-connected.

Proof.

Suppose that

f_{A}

is not r-fuzzy soft

α

-connected, then there is r-fuzzy soft

α

-separated sets

h_{C}^{*}

and

t_{D}^{*}

\in \tilde{(W, N)}

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 2.3(1),

h_{C}

and

t_{D}

are r-fuzzy soft

α

-separated, it is a contradiction. Thus,

f_{A}

is r-fuzzy soft

α

-connected, as required.

□

3. Continuity via r-Fuzzy Soft $α$ -Open Sets

Here, we investigate some properties of fuzzy soft

α

-continuous mappings. Additionally, we introduce and study the notions of fuzzy soft almost (weakly)

α

-continuous mappings, which are weaker forms of fuzzy soft

α

-continuous mappings. Also, we show that fuzzy soft

α

-continuity ⇒ fuzzy soft almost

α

-continuity ⇒ fuzzy soft weakly

α

-continuity.

Definition 3.1.

[26] Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs and

r \in I_{o}

. A fuzzy soft mapping

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

is called a fuzzy soft

α

-continuous if,

φ_{ψ}^{- 1} (h_{C})

is r-fuzzy soft

α

-open set for each

h_{C} \in \tilde{(V, F)}

with

η_{k} (h_{C}) \geq r

n \in N

(k = ψ (n)) \in F

Theorem 3.1.

Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs, and

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

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

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

n \in N

(k = ψ (n)) \in F

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft

α

-continuous.

(2) For each

f_{A}

with

η_{k} (f_{A}^{c}) \geq r

φ_{ψ}^{- 1} (f_{A})

is r-fuzzy soft

α

-closed.

(3)

α C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

(4)

φ_{ψ}^{- 1} (I_{η} (k, f_{A}, r)) ⊑ α I_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r)

(5)

C_{τ} (n, I_{τ} (n, C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r), r), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

Proof. (1) ⇔ (2) Follows from Remark 2.1 and

φ_{ψ}^{- 1} (f_{A}^{c}) = {(φ_{ψ}^{- 1} (f_{A}))}^{c}

(2) ⇒ (3) Let

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

, hence by (2),

φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

is r-fuzzy soft

α

-closed. Then, we obtain

α C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

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

(3) ⇒ (5) Let

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

, hence by (3), we obtain

C_{τ} (n, I_{τ} (n, C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r), r), r) ⊑ α C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

(5) ⇒ (1) Let

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

with

η_{k} (f_{A}) \geq r

, hence by (3), we obtain

{(φ_{ψ}^{- 1} (f_{A}))}^{c} = φ_{ψ}^{- 1} (f_{A}^{c}) ⊒ C_{τ} (n, I_{τ} (n, C_{τ} (n, φ_{ψ}^{- 1} (f_{A}^{c}), r), r), r) = {(I_{τ} (n, C_{τ} (n, I_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r), r), r))}^{c}

. Then,

φ_{ψ}^{- 1} (f_{A}) ⊑ I_{τ} (n, C_{τ} (n, I_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r), r), r)

, so

φ_{ψ}^{- 1} (f_{A})

is r-fuzzy soft

α

-open. Hence,

φ_{ψ}

is fuzzy soft

α

-continuous.

□

Lemma 3.1.

Every fuzzy soft continuous mapping [14] is fuzzy soft

α

-continuous.

Proof.

Follows from Definitions 1.6 and 3.1.

□

Remark 3.1.

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

Example 3.1.

Let

W = {w_{1}, w_{2}, w_{3}}

N = {n_{1}, n_{2}}

and define

f_{N}, g_{N}, h_{N} \in

\tilde{(W, N)}

as:

f_{N} = {(n_{1}, {\frac{w_{1}}{0.4}, \frac{w_{2}}{0.5}, \frac{w_{3}}{0.5}}), (n_{2}, {\frac{w_{1}}{0.4}, \frac{w_{2}}{0.5}, \frac{w_{3}}{0.5}})}

g_{N} = {(n_{1}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.3}, \frac{w_{3}}{0.4}}), (n_{2}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.3}, \frac{w_{3}}{0.4}})}

h_{N} = {(n_{1}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.4}, \frac{w_{3}}{0.4}}), (n_{2}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.4}, \frac{w_{3}}{0.4}})}

. Define fuzzy soft topologies

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

as follows:

\forall n \in N

τ_{n} (t_{N}) =

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

η_{n} (t_{N}) =

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

Then, the identity fuzzy soft mapping

φ_{ψ} : (W, τ_{N}) ⟶ (W, η_{N})

is fuzzy soft

α

-continuous, but it is not fuzzy soft continuous.

Definition 3.2.

Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs. A fuzzy soft mapping

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

is called fuzzy soft almost (resp. weakly)

α

-continuous if, for each

n_{w_{s}} \in \tilde{P_{s} (W)}

and each

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

with

η_{k} (g_{B}) \geq r

containing

φ_{ψ} (n_{w_{s}})

, there is

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft

α

-open set containing

n_{w_{s}}

such that

φ_{ψ} (h_{C}) ⊑ I_{η} (k, C_{η} (k, g_{B}, r), r)

(resp.

φ_{ψ} (h_{C}) ⊑ C_{η} (k, g_{B}, r)

n \in N

(k = ψ (n)) \in F

and

r \in I_{\circ}

Lemma 3.2. (1) Every fuzzy soft

α

-continuous mapping is fuzzy soft almost

α

-continuous.

(2) Every fuzzy soft almost

α

-continuous mapping is fuzzy soft weakly

α

-continuous.

Proof.

Follows from Definitions 3.1 and 3.2. □

Remark 3.2.

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

Example 3.2.

Let

W = {w_{1}, w_{2}, w_{3}}

N = {n_{1}, n_{2}}

and define

g_{N}, h_{N} \in

\tilde{(W, N)}

as follows:

g_{N} = {(n_{1}, {\frac{w_{1}}{0.5}, \frac{w_{2}}{0.5}, \frac{w_{3}}{0.4}}), (n_{2}, {\frac{w_{1}}{0.5}, \frac{w_{2}}{0.5}, \frac{w_{3}}{0.4}})}

h_{N} = {(n_{1}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.3}, \frac{w_{3}}{0.4}}), (n_{2}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0.3}, \frac{w_{3}}{0.4}})}

. Define fuzzy soft topologies

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

as follows:

\forall n \in N

τ_{n} (t_{N}) =

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

η_{n} (t_{N}) =

\{\begin{matrix} 1, if t_{N} \in {Φ, \tilde{N}}, \\ \frac{1}{2}, if t_{N} = g_{N}, \\ \frac{1}{3}, if t_{N} = h_{N}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft mapping

φ_{ψ} : (W, τ_{N}) ⟶ (W, η_{N})

is fuzzy soft almost

α

-continuous, but it is not fuzzy soft

α

-continuous.

Example 3.3.

Let

W = {w_{1}, w_{2}, w_{3}}

N = {n_{1}, n_{2}}

and define

g_{N}, h_{N} \in

\tilde{(W, N)}

as follows:

g_{N} = {(n_{1}, {\frac{w_{1}}{0.5}, \frac{w_{2}}{0.5}, \frac{w_{3}}{0.5}}), (n_{2}, {\frac{w_{1}}{0.5}, \frac{w_{2}}{0.5}, \frac{w_{3}}{0.5}})}

h_{N} = {(n_{1}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0}, \frac{w_{3}}{0.5}}), (n_{2}, {\frac{w_{1}}{0.3}, \frac{w_{2}}{0}, \frac{w_{3}}{0.5}})}

. Define fuzzy soft topologies

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

as follows:

\forall n \in N

τ_{n} (t_{N}) =

\{\begin{matrix} 1, if t_{N} \in {Φ, \tilde{N}}, \\ \frac{2}{3}, if t_{N} = g_{N}, \\ 0, otherwise, \end{matrix}

η_{n} (t_{N}) =

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

Then, the identity fuzzy soft mapping

φ_{ψ} : (W, τ_{N}) ⟶ (W, η_{N})

is fuzzy soft weakly

α

-continuous, but it is not fuzzy soft almost

α

-continuous.

Theorem 3.2.

Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs, and

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

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

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

n \in N

(k = ψ (n)) \in F

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft almost

α

-continuous.

(2)

φ_{ψ}^{- 1} (f_{A})

is r-fuzzy soft

α

-open, for each

f_{A}

is r-fuzzy soft regularly open.

(3)

φ_{ψ}^{- 1} (f_{A})

is r-fuzzy soft

α

-closed, for each

f_{A}

is r-fuzzy soft regularly closed.

(4)

α C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

, for each

f_{A}

is r-fuzzy soft

β

-open.

(5)

α C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

, for each

f_{A}

is r-fuzzy soft semi-open.

(6)

α I_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r)), r) ⊒ φ_{ψ}^{- 1} (f_{A})

, for each

f_{A}

with

η_{k} (f_{A}) \geq r

Proof. (1) ⇒ (2) Let

n_{w_{s}} \in \tilde{P_{s} (W)}

and

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

be an r-fuzzy soft regularly open set containing

φ_{ψ} (n_{w_{s}})

, hence by (1), there is

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft

α

-open set containing

n_{w_{s}}

such that

φ_{ψ} (h_{C}) ⊑ I_{η} (k, C_{η} (k, f_{A}, r), r)

Thus,

h_{C} ⊑ φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r)) = φ_{ψ}^{- 1} (f_{A})

and

n_{w_{s}} \tilde{\in} h_{C} ⊑ φ_{ψ}^{- 1} (f_{A})

. Then,

n_{w_{s}} \tilde{\in} I_{τ} (n, C_{τ} (n, I_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r), r), r)

and

φ_{ψ}^{- 1} (f_{A}) ⊑ I_{τ} (n, C_{τ} (n, I_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r), r), r)

. Therefore,

φ_{ψ}^{- 1} (f_{A})

is r-fuzzy soft

α

-open set.

(2) ⇒ (3) Let

f_{A}

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

φ_{ψ}^{- 1} (f_{A}^{c}) = {(φ_{ψ}^{- 1} (f_{A}))}^{c}

is r-fuzzy soft

α

-open set. Then,

φ_{ψ}^{- 1} (f_{A})

is r-fuzzy soft

α

-closed set.

(3) ⇒ (4) Let

f_{A}

be an r-fuzzy soft

β

-open set. Since

C_{η} (k, f_{A}, r)

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

φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

is r-fuzzy soft

α

-closed set. Since

φ_{ψ}^{- 1} (f_{A}) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

, then we have

α C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

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

β

-open.

(5) ⇒ (3) Let

f_{A}

be an r-fuzzy soft regularly closed set, hence

f_{A}

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

α C_{τ} (n, φ_{ψ}^{- 1} (f_{A}), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r)) = φ_{ψ}^{- 1} (f_{A})

. Therefore,

φ_{ψ}^{- 1} (f_{A})

is r-fuzzy soft

α

-closed set.

(3) ⇒ (6) Let

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

with

η_{k} (f_{A}) \geq r

and

n_{w_{s}} \tilde{\in} φ_{ψ}^{- 1} (f_{A})

, then we have

n_{w_{s}} \tilde{\in} φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r))

. Since

{[I_{η} (k, C_{η} (k, f_{A}, r), r)]}^{c}

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

φ_{ψ}^{- 1} ({[I_{η} (k, C_{η} (k, f_{A}, r), r)]}^{c})

is r-fuzzy soft

α

-closed set. Thus,

φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r))

is r-fuzzy soft

α

-open set and

n_{w_{s}} \tilde{\in} α I_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r)), r)

. Then,

φ_{ψ}^{- 1} (f_{A}) ⊑ α I_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r)), r)

(6) ⇒ (1) Let

n_{w_{s}} \in \tilde{P_{s} (W)}

and

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

with

η_{k} (f_{A}) \geq r

containing

φ_{ψ} (n_{w_{s}})

, hence by (6),

φ_{ψ}^{- 1} (f_{A}) ⊑ α I_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r)), r)

Since

n_{w_{s}} \tilde{\in} φ_{ψ}^{- 1} (f_{A})

, then we obtain

n_{w_{s}} \tilde{\in} α I_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r)), r) = h_{C}

(say). Hence, there is

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft

α

-open set containing

n_{w_{s}}

such that

φ_{ψ} (h_{C}) ⊑ I_{η} (k, C_{η} (k, f_{A}, r), r)

. Therefore,

φ_{ψ}

is fuzzy soft almost

α

-continuous.

□

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

Theorem 3.3.

Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs, and

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

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

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

n \in N

(k = ψ (n)) \in F

and

r \in I_{\circ}

(1)

φ_{ψ}

is fuzzy soft weakly

α

-continuous.

(2)

I_{τ} (n, C_{τ} (n, I_{τ} (n, φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r)), r), r), r) ⊒ φ_{ψ}^{- 1} (f_{A})

, if

η_{k} (f_{A}) \geq r

(3)

C_{τ} (n, I_{τ} (n, C_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, f_{A}, r)), r), r), r) ⊑ φ_{ψ}^{- 1} (f_{A})

, if

η_{k} (f_{A}^{c}) \geq r

(4)

α C_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, f_{A}, r)), r) ⊑ φ_{ψ}^{- 1} (f_{A})

, if

η_{k} (f_{A}^{c}) \geq r

(5)

α C_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, f_{A}, r), r)), r) ⊑ φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r))

(6)

α I_{τ} (n, φ_{ψ}^{- 1} (C_{η} (k, I_{η} (k, f_{A}, r), r)), r) ⊒ φ_{ψ}^{- 1} (I_{η} (k, f_{A}, r))

(7)

φ_{ψ}^{- 1} (f_{A}) ⊑ α I_{τ} (n, φ_{ψ}^{- 1} (C_{η} (k, f_{A}, r)), r)

, if

η_{k} (f_{A}) \geq r

Remark 3.3.

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

continuity \Rightarrow α - continuity

⇓ ⇓

almost continuity \Rightarrow almost α - continuity

⇓ ⇓

weakly continuity \Rightarrow weakly α - continuity

Proposition 3.1.

Let

(W, τ_{N})

(V, η_{F})

and

(U, γ_{E})

be a FSTSs, and

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

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

be two fuzzy soft functions. Then, the composition

φ_{ψ^{*}}^{*} \circ φ_{ψ}

is fuzzy soft almost

α

-continuous if,

φ_{ψ}

is fuzzy soft

α

-continuous and

φ_{ψ^{*}}^{*}

is fuzzy soft almost continuous (resp. continuous).

Proof.

The proof is obvious. □

Let

H

and

I

: N \times \tilde{(W, N)} \times I_{\circ} \to \tilde{(W, N)}

be operators on

\tilde{(W, N)}

, and

J

and

K

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

be operators on

\tilde{(V, F)}

Definition 3.3.

[26] Let

(W, τ_{N})

and

(V, η_{F})

be a FSTSs.

φ_{ψ} : \tilde{(W, N)} ⟶ \tilde{(V, F)}

is said to be a fuzzy soft

(H, I, J, K)

-continuous mapping if,

H [n, φ_{ψ}^{- 1} (K (k, h_{C}, r)), r] \bar{⊓} I [n, φ_{ψ}^{- 1} (J (k, h_{C}, r)), r]

= Φ

for each

h_{C} \in \tilde{(V, F)}

with

η_{k} (h_{C}) \geq r

n \in N

and

(k = ψ (n)) \in F

In (2023), Alshammari et al. [26] defined the notion of fuzzy soft

α

-continuous mappings:

φ_{ψ}^{- 1} (h_{C}) ⊑ I_{τ} (n, C_{τ} (n, I_{τ} (n, φ_{ψ}^{- 1} (h_{C}), r), r), r)

, for each

h_{C} \in \tilde{(V, F)}

with

η_{k} (h_{C}) \geq r

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

H

= identity operator,

I

= interior closure interior operator,

J

= identity operator and

K

= identity operator.

A historical justification of Definition 3.3:

(1) In Section 3, we obtained the notion of fuzzy soft almost

α

-continuous mappings:

φ_{ψ}^{- 1} (h_{C}) ⊑ α I_{τ} (n, φ_{ψ}^{- 1} (I_{η} (k, C_{η} (k, h_{C}, r), r)), r)

, for each

h_{C} \in \tilde{(V, F)}

with

η_{k} (h_{C}) \geq r

. Here,

H

= identity operator,

I

α

-interior operator,

J

= interior closure operator and

K

= identity operator.

(2) In Section 3, we obtained the notion of fuzzy soft weakly

α

-continuous mappings:

φ_{ψ}^{- 1} (h_{C}) ⊑ α I_{τ} (n, φ_{ψ}^{- 1} (C_{η} (k, h_{C}, r)), r)

, for each

h_{C} \in \tilde{(V, F)}

with

η_{k} (h_{C}) \geq r

. Here,

H

= identity operator,

I

α

-interior operator,

J

= closure operator and

K

= identity operator.

4. New Types of Fuzzy Soft Compactness

Here, several types of fuzzy soft compactness via r-fuzzy soft

α

-open sets are given, and the relationships between them are studied.

Definition 4.1.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. Then,

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft compact iff for every family

{{(g_{B})}_{δ} \in \tilde{(W, N)} | τ_{n} ({(g_{B})}_{δ}) \geq r for each n \in N}_{δ \in Δ}

such that

h_{C} ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} {(g_{B})}_{δ}

Definition 4.2.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. Then,

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft

α

-compact iff for every family

{{(g_{B})}_{δ} \in \tilde{(W, N)} | {(g_{B})}_{δ} is r - fuzzy soft α - open}_{δ \in Δ}

such that

h_{C} ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} {(g_{B})}_{δ}

Lemma 4.1.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft

α

-compact, then

h_{C}

is r-fuzzy soft compact.

Proof.

Follows from Definitions 4.1 and 4.2.

□

Theorem 4.1.

Let

φ_{ψ} : (W, τ_{N}) ⟶ (V, η_{F})

be a fuzzy soft

α

-continuous mapping. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft

α

-compact, then

φ_{ψ} (h_{C})

is r-fuzzy soft compact.

Proof.

Let

{{(g_{B})}_{δ} \in \tilde{(V, F)} | η_{k} ({(g_{B})}_{δ}) \geq r}_{δ \in Δ}

with

φ_{ψ} (h_{C}) ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

for each

k \in F

. Then,

{φ_{ψ}^{- 1} ({(g_{B})}_{δ}) \in \tilde{(W, N)} | φ_{ψ}^{- 1} ({(g_{B})}_{δ})

{r - fuzzy soft α - open}}_{δ \in Δ}

(by

φ_{ψ}

is fuzzy soft

α

-continuous) such that

h_{C} ⊑ ⊔_{δ \in Δ} φ_{ψ}^{- 1} ({(g_{B})}_{δ})

. Since

h_{C}

is r-fuzzy soft

α

-compact, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} φ_{ψ}^{- 1} ({(g_{B})}_{δ})

. Then,

φ_{ψ} (h_{C}) ⊑ ⊔_{δ \in Δ_{\circ}} {(g_{B})}_{δ}

. Hence, the proof is completed. □

Definition 4.3.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. Then,

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft almost compact iff for every family

{{(g_{B})}_{δ} \in \tilde{(W, N)} | τ_{n} ({(g_{B})}_{δ}) \geq r}_{δ \in Δ}

such that

h_{C} ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} C_{τ} (n, {(g_{B})}_{δ}, r)

for each

n \in N

Definition 4.4.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. Then,

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft almost

α

-compact iff for every family

{{(g_{B})}_{δ} \in \tilde{(W, N)} | {(g_{B})}_{δ} is r - fuzzy soft α - open}_{δ \in Δ}

such that

h_{C} ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} C_{τ} (n, {(g_{B})}_{δ}, r)

for each

n \in N

Lemma 4.2.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft almost

α

-compact, then

h_{C}

is r-fuzzy soft almost compact.

Proof.

Follows from Definitions 4.3 and 4.4.

□

Lemma 4.3.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft

α

-compact (resp. compact), then

h_{C}

is r-fuzzy soft almost

α

-compact (resp. almost compact).

Proof.

Follows from Definitions 4.1–4.4.

□

Theorem 4.2.

Let

φ_{ψ} : (W, τ_{N}) ⟶ (V, η_{F})

be a fuzzy soft continuous mapping. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft almost

α

-compact, then

φ_{ψ} (h_{C})

is r-fuzzy soft almost compact.

Proof.

Let

{{(g_{B})}_{δ} \in \tilde{(V, F)} | η_{k} ({(g_{B})}_{δ}) \geq r}_{δ \in Δ}

with

φ_{ψ} (h_{C}) ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

for each

k \in F

. Then,

{φ_{ψ}^{- 1} ({(g_{B})}_{δ}) \in \tilde{(W, N)} | φ_{ψ}^{- 1} ({(g_{B})}_{δ})

{r - fuzzy soft α - open}}_{δ \in Δ}

(by

φ_{ψ}

is fuzzy soft

α

-continuous) such that

h_{C} ⊑ ⊔_{δ \in Δ} φ_{ψ}^{- 1} ({(g_{B})}_{δ})

. Since

h_{C}

is r-fuzzy soft almost

α

-compact, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} C_{τ} (n, φ_{ψ}^{- 1} ({(g_{B})}_{δ}), r)

. Since

φ_{ψ}

is fuzzy soft continuous mapping, it follows

⊔_{δ \in Δ_{\circ}} C_{τ} (n, φ_{ψ}^{- 1} ({(g_{B})}_{δ}), r) ⊑

⊔_{δ \in Δ_{\circ}} φ_{ψ}^{- 1} (C_{η} (k, {(g_{B})}_{δ}, r)) =

φ_{ψ}^{- 1} (⊔_{δ \in Δ_{\circ}} C_{η} (k, {(g_{B})}_{δ}, r)) .

Then,

φ_{ψ} (h_{C}) ⊑ ⊔_{δ \in Δ_{\circ}} C_{η} (k, {(g_{B})}_{δ}, r)

. Hence, the proof is completed. □

Definition 4.5.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. Then,

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft nearly compact iff for every family

{{(g_{B})}_{δ} \in \tilde{(W, N)} | τ_{n} ({(g_{B})}_{δ}) \geq r}_{δ \in Δ}

such that

h_{C} ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} I_{τ} (n, C_{τ} (n, {(g_{B})}_{δ}, r), r)

for each

n \in N

Definition 4.6.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. Then,

h_{C} \in \tilde{(W, N)}

is called an r-fuzzy soft nearly

α

-compact iff for every family

{{(g_{B})}_{δ} \in \tilde{(W, N)} | {(g_{B})}_{δ} is r - fuzzy soft α - open}_{δ \in Δ}

such that

h_{C} ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} I_{τ} (n, C_{τ} (n, {(g_{B})}_{δ}, r), r)

for each

n \in N

Lemma 4.4.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft nearly

α

-compact, then

h_{C}

is r-fuzzy soft nearly compact.

Proof.

Follows from Definitions 4.5 and 4.6.

□

Lemma 4.5.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft

α

-compact (resp. compact), then

h_{C}

is r-fuzzy soft nearly

α

-compact (resp. nearly compact).

Proof.

Follows from Definitions 4.1, 4.2, 4.5 and 4.6.

□

Theorem 4.3.

Let

φ_{ψ} : (W, τ_{N}) ⟶ (V, η_{F})

be a fuzzy soft continuous and fuzzy soft open mapping. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft nearly

α

-compact, then

φ_{ψ} (h_{C})

is r-fuzzy soft nearly compact.

Proof.

Let

{{(g_{B})}_{δ} \in \tilde{(V, F)} | η_{k} ({(g_{B})}_{δ}) \geq r}_{δ \in Δ}

with

φ_{ψ} (h_{C}) ⊑ ⊔_{δ \in Δ} {(g_{B})}_{δ}

for each

k \in F

. Then,

{φ_{ψ}^{- 1} ({(g_{B})}_{δ}) \in \tilde{(W, N)} | φ_{ψ}^{- 1} ({(g_{B})}_{δ})

{r - fuzzy soft α - open}}_{δ \in Δ}

(by

φ_{ψ}

is fuzzy soft

α

-continuous) such that

h_{C} ⊑ ⊔_{δ \in Δ} φ_{ψ}^{- 1} ({(g_{B})}_{δ})

. Since

h_{C}

is r-fuzzy soft nearly

α

-compact, there is a finite subset

Δ_{\circ}

Δ

such that

h_{C} ⊑ ⊔_{δ \in Δ_{\circ}} I_{τ} (n, C_{τ} (n, φ_{ψ}^{- 1} ({(g_{B})}_{δ}), r), r)

. Since

φ_{ψ}

is fuzzy soft continuous and fuzzy soft open mapping, it follows

φ_{ψ} (h_{C}) ⊑ ⊔_{δ \in Δ_{\circ}} φ_{ψ} (I_{τ} (n, C_{τ} (n, φ_{ψ}^{- 1} ({(g_{B})}_{δ}), r), r))

⊑ ⊔_{δ \in Δ_{\circ}} I_{η} (k, φ_{ψ} (C_{τ} (n, φ_{ψ}^{- 1} ({(g_{B})}_{δ}), r)), r)

⊑ ⊔_{δ \in Δ_{\circ}} I_{η} (k, φ_{ψ} (φ_{ψ}^{- 1} (C_{η} (k, {(g_{B})}_{δ}, r))), r)

⊑ ⊔_{δ \in Δ_{\circ}} I_{η} (k, C_{η} (k, {(g_{B})}_{δ}, r), r) .

Hence, the proof is completed. □

Lemma 4.6.

Let

(W, τ_{N})

be a FSTS and

r \in I_{\circ}

. If

h_{C} \in \tilde{(W, N)}

is r-fuzzy soft nearly

α

-compact (resp. nearly compact), then

h_{C}

is r-fuzzy soft almost

α

-compact (resp. almost compact).

Proof.

Follows from Definitions 4.3, 4.4, 4.5 and 4.6.

□

Remark 4.1.

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

α - compactness \Rightarrow compactness

⇓ ⇓

nearly α - compactness \Rightarrow nearly compactness

⇓ ⇓

almost α - compactness \Rightarrow almost compactness

5. Conclusions and Future Work

The main achievements of this study are:

(1)

In Section 2, the concepts of fuzzy soft

α

-closure (

α

-interior) operators are introduced in fuzzy soft topological spaces based on the paper Aygünoǧlu et al. [14], and some of their basic properties have been investigated. Furthermore, the notion of r-fuzzy soft

α

-connected sets is defined and studied with help of fuzzy soft

α

-closure operators.

(2)

In Section 3, some properties of fuzzy soft

α

-continuous mappings are obtained between two fuzzy soft topological spaces

(W, τ_{N})

and

(V, η_{F})

. Moreover, as a weaker forms of the notion of fuzzy soft

α

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

α

-continuous mappings are introduced, and some of their basic properties and characterizations have been investigated. Also, we show that fuzzy soft

α

-continuity ⇒ fuzzy soft almost

α

-continuity ⇒ fuzzy soft weakly

α

-continuity, and we have the following results:

• Fuzzy soft

(i d_{W}, I_{τ} (C_{τ} (I_{τ})), i d_{V}, i d_{V})

-continuous mapping is fuzzy soft

α

-continuous.

• Fuzzy soft

(i d_{W}, α I_{τ}, I_{η} (C_{η}), i d_{V})

-continuous mapping is fuzzy soft almost

α

-continuous.

• Fuzzy soft

(i d_{W}, α I_{τ}, C_{η}, i d_{V})

-continuous mapping is fuzzy soft weakly

α

-continuous.

(3)

In Section 4, several types of soft compactness via r-fuzzy soft

α

-open sets are explored, and the relationships between them are studied.

In upcoming manuscripts, we will use the fuzzy soft

α

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

Author Contributions

Methodology, Islam M. Taha; Formal Analysis, Islam M. Taha and Wafa Alqurashi; Investigation, Islam M. Taha and Wafa Alqurashi; Writing Original Draft Preparation, Islam M. Taha and Wafa Alqurashi; Writing Review and Editing, Islam M. Taha and Wafa Alqurashi; Funding Acquisition, Islam M. Taha and Wafa Alqurashi.

Data Availability Statement

No data were used to support this study.

Acknowledgments

In this section you can acknowledge any support given which is not covered by the author contribution or funding sections. This may include administrative and technical support, or donations in kind (e.g., materials used for experiments).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Use of AI tools declaration

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

References

Molodtsov, D. Soft set theory-first results. Comput. Math. Appl. 1999, 37, 19–31. [Google Scholar] [CrossRef]
Akdag, M.; Ozkan, A. Soft α-open sets and soft α-continuous functions. Abst. Appl. Anal. 2014, 2014, 891341. [Google Scholar] [CrossRef]
Akdag, M.; Ozkan, A. On soft β-open sets and soft β-continuous functions. Sci. World J. 2014, 2014, 843456. [Google Scholar] [CrossRef]
El-Sheikh, S.A.; Hosny, R.A.; Abd El-latif, A.M. Characterizations of β-soft separation axioms in soft topological spaces. Inf. Sci. Lett. 2015, 4, 125–133. [Google Scholar]
Al-shami, T.M. Soft somewhere dense sets on soft topological spaces. Commun. Korean Math. Soc. 2018, 33, 1341–1356. [Google Scholar]
Al-Ghour, S. Boolean algebra of soft Q-Sets in soft topological spaces. Appl. Comput. Intell. Soft Comput. 2022, 2022, 5200590. [Google Scholar] [CrossRef]
Al-shami, T.M.; Mhemdi, A.; Abu-Gdairid, R. A Novel framework for generalizations of soft open sets and its applications via soft topologies. Mathematics 2023, 11, 840. [Google Scholar] [CrossRef]
Al-shami, T.M.; Arar, M.; Abu-Gdairi, R.; Ameen, Z.A. On weakly soft β-open sets and weakly soft β-continuity. J. Inte. Fuzzy Syst. 2023, 45, 6351–6363. [Google Scholar] [CrossRef]
Kaur, S.; Al-shami, T.M.; Ozkan, A.; Hosny, M. A new approach to soft continuity. Mathematics 2023, 11, 3164. [Google Scholar] [CrossRef]
Al-Ghour, S.; Al-Mufarrij, J. Between soft complete continuity and soft somewhat-continuity. Symmetry 2023, 15, 1–14. [Google Scholar]
Ameen, Z.A.; Abu-Gdairi, R.; Al-shami, T.M.; Asaad, B.A.; Arar, M. Further properties of soft somewhere dense continuous functions and soft Baire spaces. J. Math. Computer Sci. 2024, 32, 54–63. [Google Scholar] [CrossRef]
Maji, P.K.; Biswas, R.; Roy, A.R. Fuzzy soft sets. J. Fuzzy Math. 2001, 9, 589–602. [Google Scholar]
Zadeh, L.A. Fuzzy Sets. Inform. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
Aygünoǧlu, A.; Çetkin, V.; Aygün, H. An introduction to fuzzy soft topological spaces. Hacet. J. Math. Stat. 2014, 43, 193–208. [Google Scholar] [CrossRef]
Çetkin, V.; Aygünoǧlu, A.; Aygün, H. On soft fuzzy closure and interior operators. Util. Math. 2016, 99, 341–367. [Google Scholar]
Šostak, A.P. On a fuzzy topological structure. In: Proceedings of the 13th winter school on abstract analysis, Section of topology, Palermo: Circolo Matematico di Palermo 1985, 89–103.
Taha, I.M. A new approach to separation and regularity axioms via fuzzy soft sets. Ann. Fuzzy Math. Inform. 2020, 20, 115–123. [Google Scholar]
Taha, I.M. Some new separation axioms in fuzzy soft topological spaces. Filomat 2021, 35, 1775–1783. [Google Scholar] [CrossRef]
Çetkin, V.; Aygün, H. Fuzzy soft semiregularization spaces. Ann. Fuzzy Math. Inform. 2014, 7, 687–697. [Google Scholar]
Taha, I.M. Compactness on fuzzy soft r-minimal spaces. Int. J. Fuzzy Logic Intell. Syst. 2021, 21, 251–258. [Google Scholar] [CrossRef]
Ahmad, B.; Kharal, A. On fuzzy soft sets. Adv. Fuzzy Syst. 2009, 2009, 586507. [Google Scholar] [CrossRef]
Çaǧman, N.; Enginoǧlu, S.; Çitak, F. Fuzzy soft set theory and its applications. Iran. J. Fuzzy Syst. 2011, 8, 137–147. [Google Scholar]
Taha, I.M. Some new results on fuzzy soft r-minimal spaces. AIMS Math. 2022, 7, 12458–12470. [Google Scholar] [CrossRef]
Mishra, S.; Srivastava, R. Hausdorff fuzzy soft topological spaces. Ann. Fuzzy Math. Inform. 2015, 9, 247–260. [Google Scholar]
Atmaca, S.; Zorlutuna, I. On fuzzy soft topological spaces. Ann. Fuzzy Math. Inform. 2013, 5, 377–386. [Google Scholar]
Alshammari, I.; Alqahtani, M.H.; Taha, I.M. On r-fuzzy soft δ-open sets and applications via fuzzy soft topologies. Preprints 2023, 2023121240. [Google Scholar]

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

Abstract

1. Introduction

2. On r-Fuzzy Soft α -Open Sets

3. Continuity via r-Fuzzy Soft α -Open Sets

4. New Types of Fuzzy Soft Compactness

5. Conclusions and Future Work

Author Contributions

Data Availability Statement

Acknowledgments

Conflicts of Interest

Use of AI tools declaration

References

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2. On r-Fuzzy Soft $α$ -Open Sets

3. Continuity via r-Fuzzy Soft $α$ -Open Sets