On R-fuzzy Soft δ-Open Sets and Applications via Fuzzy Soft Topologies

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On R-fuzzy Soft δ-Open Sets and Applications via Fuzzy Soft Topologies

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Submitted:

21 January 2024

Posted:

24 January 2024

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Abstract

In this paper, we first define the concepts of r-fuzzy soft α-open (semi-open and δ-open) sets on fuzzy soft topological spaces based on the article Aygunoglu et al. (Hacet. J. Math. Stat., 43 (2014), 193-208), and the relations of these sets with each other are established. In addition, we introduce the concepts of fuzzy soft δ-closure (δ-interior) operators, and study some properties of them. Also, the concept of r-fuzzy soft δ-connected sets is introduced and studied with help of fuzzy soft δ-closure operators. Thereafter, we define the concepts of fuzzy soft α-continuous (β-continuous, semi-continuous, pre-continuous and δ-continuous) functions, which are weaker forms of fuzzy soft continuity, and some properties of these functions along with their mutual relationships are discussed. Moreover, a decomposition of fuzzy soft α-continuity and a decomposition of fuzzy soft semi-continuity is obtained. Finally, as a weaker form of a fuzzy soft continuity, the concepts of fuzzy soft almost (weakly) continuous functions are defined, and some properties are specified. Additionally, we explore the notion of continuity in a very general setting called, fuzzy soft (L, M, N, O)-continuous functions and a historical justification is introduced.

Keywords:

Subject: Computer Science and Mathematics - Geometry and Topology

MSC: 54A05; 54A40; 54C05; 54C08; 54D05

1. Introduction

In [1], the author initiated a novel concept of soft set theory, which is a completely new approach for modeling uncertainty and vagueness. He showed many applications of these theory in solving several practical problems in engineering, economics, and medical science, social science etc. Akdag and Ozkan [2] introduced and studied the concept of soft

α

-open sets on soft topological spaces. Also, the concept of soft

β

-open sets was introduced and studied by the authors of [3,4]. Moreover, the concepts of 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 specified its main properties. Also, Al-shami et al. [8] initiated the concept of weakly soft

β

-open sets and obtained 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 recent years, many authors have contributed to soft set theory in the different fields such as topology, algebra, see e.g. [11-15].

Maji et al. [16] introduced the concept of fuzzy soft sets which combines soft sets [1] and fuzzy sets [17]. The concept of fuzzy soft topology is introduced and some of its properties such as fuzzy soft continuity, interior fuzzy soft set, closure fuzzy soft set and fuzzy soft subspace topology is obtained in [18,19] based on fuzzy topologies in Šostaks sense [20]. A new approach to studying separation and regularity axioms via fuzzy soft sets was introduced by the author of [21,22] based on the paper Aygünoǧlu et al. [18]. The concept of r-fuzzy soft regularly open sets was introduced by Çetkin and Aygün [23]. Moreover, the concepts of r-fuzzy soft

β

-open (resp. pre-open) sets were also introduced by Taha [24].

The organization of this paper is as follows:

• In Section 2, we define new types of fuzzy soft sets on fuzzy soft topological spaces based on the paper Aygünoǧlu et al. [18]. Also, the relations of these sets with each other are established with the help of some examples. Moreover, the concept of r-fuzzy soft

δ

-connected sets is introduced and characterized with help of fuzzy soft

δ

-closure operators.

• In Section 3, we introduce the concepts of fuzzy soft

α

-continuous (

β

-continuous, semi-continuous, pre-continuous and

δ

-continuous) functions, and the relations of these functions with each other are investigated. Also, a decomposition of fuzzy soft semi-continuity and a decomposition of fuzzy soft

α

-continuity is given.

• In Section 4, as a weaker form of fuzzy soft continuity [18], the concepts of fuzzy soft almost (weakly) continuous functions are introduced, and some properties are obtained. Also, we show that fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft weakly continuity, but the converse may not be true. In the end, we explore the notion of continuity in a very general setting called, fuzzy soft

(L, M, N, O)

-continuous functions and a historical justification is given.

• Finally, we close this paper with some conclusions and make a plan to suggest some future works in Section 5.

Throughout this article, nonempty sets will be denoted by U, V etc. E is the set of all parameters for U and

A \subseteq E

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

I^{U}

(where

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

), and for

t \in I,

\underset{̲}{t} (u) = t

, for all

u \in U .

The following definitions will be used in the next sections:

Definition 1.1.

[18, 25, 26]

A fuzzy soft set

f_{A}

on U is a function from E to

I^{U}

such that

f_{A} (e)

is a fuzzy set on U, for each

e \in A

and

f_{A} (e) = \underset{̲}{0}

, if

e \notin A

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

\tilde{(U, E)}

Definition 1.2.

[27]

A fuzzy soft point

e_{u_{t}}

on U is a fuzzy soft set defined as follows:

e_{u_{t}} (k) = \{\begin{matrix} u_{t}, & if & k = e, \\ \underset{̲}{0}, & if & k \in E - {e}, \end{matrix}

where

u_{t}

is a fuzzy point on U.

e_{u_{t}}

is said to belong to a fuzzy soft set

f_{A}

, denoted by

e_{u_{t}} \tilde{\in} f_{A}

, if

t \leq f_{A} (e) (u)

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

\tilde{P_{t} (U)}

Definition 1.3.

[28]

A fuzzy soft point

e_{u_{t}} \in \tilde{P_{t} (U)}

is called a soft quasi-coincident with

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

and denoted by

e_{u_{t}} \tilde{q} f_{A}

, if

t + f_{A} (e) (u) > 1

. A fuzzy soft set

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

is called a soft quasi-coincident with

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

and denoted by

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

, if there is

e \in E

and

u \in U

such that

f_{A} (e) (u) + g_{B} (e) (u) > 1

. If

f_{A}

is not soft quasi-coincident with

g_{B}

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

Definition 1.4.

[18]

A function

τ : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

is called a fuzzy soft topology on U if it satisfies the following conditions for every

e \in E

(i)

τ_{e} (Φ) = τ_{e} (\tilde{E}) = 1,

(ii)

τ_{e} (f_{A} ⊓ g_{B}) \geq τ_{e} (f_{A}) \land τ_{e} (g_{B}),

for every

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

(iii)

τ_{e} (⨆_{δ \in Δ} {(f_{A})}_{δ}) \geq ⋀_{δ \in Δ} τ_{e} ({(f_{A})}_{δ}),

for every

{(f_{A})}_{δ} \in \tilde{(U, E)}, δ \in Δ .

Then,

(U, τ_{E})

is called a fuzzy soft topological space (briefly, FSTS) in Šostaks sense [20].

Definition 1.5.

[18]

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs. A fuzzy soft function

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

is said to be a fuzzy soft continuous if,

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

for every

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

e \in E

and

(k = ψ (e)) \in F

Definition 1.6.

[19, 23]

In a FSTS

(U, τ_{E})

, for each

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

e \in E

and

r \in I_{0}

, we define the fuzzy soft operators

C_{τ}

and

I_{τ}

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

as follows:

C_{τ} (e, f_{A}, r) = ⊓ {g_{B} \in \tilde{(U, E)} : f_{A} ⊑ g_{B}, τ_{e} (g_{B}^{c}) \geq r}

I_{τ} (e, f_{A}, r) = ⊔ {g_{B} \in \tilde{(U, E)} : g_{B} ⊑ f_{A}, τ_{e} (g_{B}) \geq r} .

Definition 1.7.

Let

(U, τ_{E})

be a FSTS and

r \in I_{0}

. A fuzzy soft set

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

is said to be an r-fuzzy soft regularly open [23] (resp. pre-open [24] and

β

-open [24]) if,

f_{A} = I_{τ} (e, C_{τ} (e, f_{A}, r), r)

(resp.

f_{A} ⊑ I_{τ} (e, C_{τ} (e, f_{A}, r), r)

and

f_{A} ⊑ C_{τ} (e, I_{τ} (e, C_{τ} (e, f_{A}, r), r), r)

) for every

e \in E

Lemma 1.1.

[24]

Every r-fuzzy soft regularly open set is r-fuzzy soft pre-open.

In general, the converse of Lemma 1.1 is not true, as shown by Example 1.1.

Example 1.1.

[24]

Let

U = {u_{1}, u_{2}}

E = {e, k}

and define

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

as follows:

g_{E} = {(e, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}}), (k, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}})}

f_{E} = {(e, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.2}}), (k, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.2}})}

. Define fuzzy soft topology

τ_{E} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

τ_{e} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, if m_{E} = g_{E}, \\ \frac{1}{3}, if m_{E} = f_{E}, \\ \frac{1}{3}, if m_{E} = g_{E} ⊓ f_{E}, \\ \frac{1}{4}, if m_{E} = g_{E} ⊔ f_{E}, \\ 0, otherwise, \end{matrix}

τ_{k} (m_{E}) =

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

Then,

f_{E}

\frac{1}{4}

-fuzzy soft pre-open set, but it is not

\frac{1}{4}

-fuzzy soft regularly open set.

The basic definitions and results which we need next sections are found in [18,19].

2. Some Properties of r-Fuzzy Soft $δ$ -Open Sets

Here, we are going to give the concepts of r-fuzzy soft

δ

-open (semi-open and

α

-open) sets on fuzzy soft topological space

(U, τ_{E})

. Some properties of these sets along with their mutual relationships are investigated with the help of some examples. Additionally, the concept of an r-fuzzy soft

δ

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

δ

-closure operators.

Definition 2.1.

Let

(U, τ_{E})

be a FSTS. A fuzzy soft set

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

is said to be an r-fuzzy soft

δ

-open (resp. semi-open and

α

-open) if

I_{τ} (e, C_{τ} (e, f_{A}, r), r) ⊑ C_{τ} (e, I_{τ} (e, f_{A}, r), r)

(resp.

f_{A} ⊑ C_{τ} (e, I_{τ} (e, f_{A}, r), r)

and

f_{A} ⊑ I_{τ} (e, C_{τ} (e, I_{τ} (e, f_{A}, r), r), r)

) for every

e \in E

and

r \in I_{0}

Remark 2.1.

The concept of an r-fuzzy soft

δ

-open set and r-fuzzy soft

β

-open set [24] are independent concepts, as shown by Examples 2.1 and 2.2.

Example 2.1.

Let

U = {u_{1}, u_{2}}

E = {e, k}

and define

h_{E}, g_{E}, f_{E} \in \tilde{(U, E)}

as follows:

h_{E} = {(e, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}}), (k, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}})}

g_{E} = {(e, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}}), (k, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}})}

f_{E} = {(e, {\frac{u_{1}}{0.8}, \frac{u_{2}}{0.7}}), (k, {\frac{u_{1}}{0.8}, \frac{u_{2}}{0.7}})}

. Define fuzzy soft topology

τ_{E} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

τ_{e} (m_{E}) =

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

τ_{k} (m_{E}) =

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

Then,

h_{E}

\frac{1}{3}

-fuzzy soft

β

-open set, but it is neither

\frac{1}{3}

-fuzzy soft

δ

-open nor

\frac{1}{3}

-fuzzy soft semi-open.

Example 2.2.

Let

U = {u_{1}, u_{2}, u_{3}}

E = {e, k}

and define

h_{E}, g_{E}, f_{E} \in \tilde{(U, E)}

as follows:

h_{E} = {(e, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{1}}), (k, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{1}})}

g_{E} = {(e, {\frac{u_{1}}{0}, \frac{u_{2}}{0}, \frac{u_{3}}{1}}), (k, {\frac{u_{1}}{0}, \frac{u_{2}}{0}, \frac{u_{3}}{1}})}

f_{E} = {(e, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{0}}), (k, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{0}})}

. Define fuzzy soft topology

τ_{E} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

τ_{e} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, if m_{E} = g_{E}, \\ \frac{1}{2}, if m_{E} = f_{E}, \\ \frac{1}{3}, if m_{E} = h_{E}, \\ 0, otherwise, \end{matrix}

τ_{k} (m_{E}) =

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

Then,

h_{E}^{c}

\frac{1}{4}

-fuzzy soft

δ

-open set, but it is neither

\frac{1}{4}

-fuzzy soft

β

-open nor

\frac{1}{4}

-fuzzy soft semi-open.

Remark 2.2.

The complement of an r-fuzzy soft

δ

-open (resp. semi-open,

α

-open and

β

-open) set is said to be an r-fuzzy soft

δ

-closed (resp. semi-closed,

α

-closed and

β

-closed).

Proposition 2.1.

Let

(U, τ_{E})

be a FSTS,

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

e \in E

and

r \in I_{0}

. The following statements are equivalent:

(i)

f_{A}

is r-fuzzy soft semi-open.

(ii)

f_{A}

is r-fuzzy soft

δ

-open and r-fuzzy soft

β

-open.

Proof.

(i) ⇒ (ii) Let

f_{A}

be an r-fuzzy soft semi-open, then

f_{A} ⊑ C_{τ} (e, I_{τ} (e, f_{A}, r), r) ⊑ C_{τ} (e, I_{τ} (C_{τ} (e, f_{A}, r), r), r) .

This shows that

f_{A}

is r-fuzzy soft

β

-open. Moreover,

I_{τ} (e, C_{τ} (e, f_{A}, r), r) ⊑ C_{τ} (e, f_{A}, r) ⊑ C_{τ} (e, C_{τ} (e, I_{τ} (e, f_{A}, r), r), r) = C_{τ} (e, I_{τ} (e, f_{A}, r), r) .

Therefore,

f_{A}

is r-fuzzy soft

δ

-open.

(ii) ⇒ (i) Let

f_{A}

be an r-fuzzy soft

δ

-open and r-fuzzy soft

β

-open, then

I_{τ} (e, C_{τ} (e, f_{A}, r), r)

⊑ C_{τ} (e, I_{τ} (e, f_{A}, r), r)

and

f_{A} ⊑ C_{τ} (e, I_{τ} (e, C_{τ} (e, f_{A}, r), r), r)

. Thus,

f_{A} ⊑ C_{τ} (e, I_{τ} (e, C_{τ} (e, f_{A}, r), r), r) ⊑ C_{τ} (e, C_{τ} (e, I_{τ} (e, f_{A}, r), r), r) = C_{τ} (e, I_{τ} (e, f_{A}, r), r) .

This shows that

f_{A}

is r-fuzzy soft semi-open.

□

Proposition 2.2.

Let

(U, τ_{E})

be a FSTS,

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

e \in E

and

r \in I_{0}

. The following statements are equivalent:

(i)

f_{A}

is r-fuzzy soft

α

-open.

(ii)

f_{A}

is r-fuzzy soft

δ

-open and r-fuzzy soft pre-open.

Proof. (i) ⇒ (ii) From Proposition 2.1 the proof is straightforward.

(ii) ⇒ (i) Let

f_{A}

be an r-fuzzy soft pre-open and r-fuzzy soft

δ

-open. Then,

f_{A} ⊑ I_{τ} (e, C_{τ} (e, f_{A}, r), r) ⊑ I_{τ} (e, C_{τ} (e, I_{τ} (e, f_{A}, r), r), r) .

This shows that

f_{A}

is r-fuzzy soft

α

-open.

□

Remark 2.3.

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

α - o p e n s e t

⇓ ⇓

p r e - o p e n s e t s e m i - o p e n s e t \Rightarrow δ - o p e n s e t

⇓ ⇓

β - o p e n s e t

Remark 2.4.

In general, the converses of the above relationships are not true, as shown by Examples 2.1, 2.2, 2.3, 2.4 and 2.5.

Example 2.3.

Let

U = {u_{1}, u_{2}}

E = {e, k}

and define

g_{E}, f_{E}, h_{E} \in \tilde{(U, E)}

as follows:

g_{E} = {(e, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}}), (k, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}})}

f_{E} = {(e, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.2}}), (k, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.2}})}

h_{E} = {(e, {\frac{u_{1}}{0.7}, \frac{u_{2}}{0.5}}), (k, {\frac{u_{1}}{0.7}, \frac{u_{2}}{0.5}})}

. Define fuzzy soft topology

τ_{E} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

τ_{e} (m_{E}) =

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

τ_{k} (m_{E}) =

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

Then,

h_{E}

\frac{1}{3}

-fuzzy soft semi-open set, but it is neither

\frac{1}{3}

-fuzzy soft

α

-open nor

\frac{1}{3}

-fuzzy soft pre-open.

Example 2.4.

Let

U = {u_{1}, u_{2}, u_{3}}

E = {e, k}

and define

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

as follows:

g_{E} = {(e, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}, \frac{u_{3}}{0.2}}), (k, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}, \frac{u_{3}}{0.2}})}

f_{E} = {(e, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}, \frac{u_{3}}{0.8}}), (k, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}, \frac{u_{3}}{0.8}})}

. Define fuzzy soft topology

τ_{E} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

τ_{e} (m_{E}) =

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

τ_{k} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{3}, if m_{E} = g_{E}, \\ 0, otherwise . \end{matrix}

Then,

f_{E}

\frac{1}{3}

-fuzzy soft

β

-open set, but it is not

\frac{1}{3}

-fuzzy soft pre-open.

Example 2.5.

Let

U = {u_{1}, u_{2}}

E = {e, k}

and define

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

as follows:

g_{E} = {(e, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}}), (k, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}})}

f_{E} = {(e, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}}), (k, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}})}

. Define fuzzy soft topology

τ_{E} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

τ_{e} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, if m_{E} = g_{E}, \\ 0, otherwise, \end{matrix}

τ_{k} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{2}, if m_{E} = g_{E}, \\ 0, otherwise . \end{matrix}

Then,

f_{E}

\frac{1}{4}

-fuzzy soft pre-open set, but it is neither

\frac{1}{4}

-fuzzy soft

α

-open nor

\frac{1}{4}

-fuzzy soft semi-open.

Theorem 2.1.

Let

(U, τ_{E})

be a FSTS,

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

e \in E

and

r \in I_{0}

. If

f_{A}

is r-fuzzy soft

δ

-open set such that

f_{A} ⊑ g_{B} ⊑ C_{τ} (e, f_{A}, r)

, then

g_{B}

is also r-fuzzy soft

δ

-open.

Proof.

Suppose that

f_{A}

is r-fuzzy soft

δ

-open and

f_{A} ⊑ g_{B} ⊑ C_{τ} (e, f_{A}, r)

. Then,

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

Since

g_{B} ⊑ C_{τ} (e, f_{A}, r)

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

. This shows that

g_{B}

is r-fuzzy soft

δ

-open.

□

Definition 2.2.

In a FSTS

(U, τ_{E})

, for each

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

e \in E

and

r \in I_{0}

, we define a fuzzy soft

δ

-closure operator

δ C_{τ}

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

as follows:

δ C_{τ} (e, f_{A}, r) = ⊓ {g_{B} \in \tilde{(U, E)} : f_{A} ⊑ g_{B}, g_{B} is r - fuzzy soft δ - closed} .

Theorem 2.2.

In a FSTS

(U, τ_{E})

, for each

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

e \in E

and

r \in I_{0}

, the operator

δ C_{τ}

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

satisfies the following properties.

(1)

δ C_{τ} (e, Φ, r) = Φ

(2)

f_{A} ⊑ δ C_{τ} (e, f_{A}, r) ⊑ C_{τ} (e, f_{A}, r)

(3)

δ C_{τ} (e, f_{A}, r) ⊑ δ C_{τ} (e, g_{B}, r)

if,

f_{A} ⊑ g_{B}

(4)

δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r) = δ C_{τ} (e, f_{A}, r)

(5)

δ C_{τ} (e, f_{A} ⊔ g_{B}, r) ⊒ δ C_{τ} (e, f_{A}, r) ⊔ δ C_{τ} (e, g_{B}, r)

(6)

δ C_{τ} (e, f_{A}, r) = f_{A}

iff

f_{A}

is r-fuzzy soft

δ

-closed.

(7)

δ C_{τ} (e, C_{τ} (e, f_{A}, r), r) = C_{τ} (e, f_{A}, r)

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

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

δ C_{τ} (e, f_{A}, r) ⊑ δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r)

. Now we show that

δ C_{τ} (e, f_{A}, r) ⊒ δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r)

. Suppose that

δ C_{τ} (e, f_{A}, r)

is not contain

δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r)

. Then, there is

u \in U

and

t \in (0, 1)

such that

δ C_{τ} (e, f_{A}, r) (e) (u) < t < δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r) (e) (u) . (A)

Since

δ C_{τ} (e, f_{A}, r) (e) (u) < t

, by the definition of

δ C_{τ}

, there is

g_{B}

is r-fuzzy soft

δ

-closed with

f_{A} ⊑ g_{B}

such that

δ C_{τ} (e, f_{A}, r) (e) (u) \leq g_{B} (e) (u) < t

. Since

f_{A} ⊑ g_{B}

, then

δ C_{τ} (e, f_{A}, r) ⊑ g_{B}

. Again, by the definition of

δ C_{τ}

, we have

δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r) ⊑ g_{B}

. Hence,

δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r) (e) (u) \leq g_{B} (e) (u) < t

, it is a contradiction for

(A)

. Thus,

δ C_{τ} (e, f_{A}, r) ⊒ δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r)

. Then,

δ C_{τ} (e, δ C_{τ} (e, f_{A}, r), r) = δ C_{τ} (e, f_{A}, r)

(5) Since

f_{A}

and

g_{B}

⊑ f_{A} ⊔ g_{B}

, hence by (3),

δ C_{τ} (e, f_{A}, r) ⊑ δ C_{τ} (e, f_{A} ⊔ g_{B}, r)

and

δ C_{τ} (e, g_{B}, r) ⊑ δ C_{τ} (e, f_{A} ⊔ g_{B}, r)

. Thus,

δ C_{τ} (e, f_{A} ⊔ g_{B}, r) ⊒ δ C_{τ} (e, f_{A}, r) ⊔ δ C_{τ} (e, g_{B}, r)

(7) From (6) and

C_{τ} (e, f_{A}, r)

is r-fuzzy soft

δ

-closed set, hence

δ C_{τ} (e, C_{τ} (e, f_{A}, r), r) = C_{τ} (e, f_{A}, r)

□

Theorem 2.3.

In a FSTS

(U, τ_{E})

, for each

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

e \in E

and

r \in I_{0}

, we define a fuzzy soft

δ

-interior operator

δ I_{τ}

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

as follows:

δ I_{τ} (e, f_{A}, r) = ⊔ {g_{B} \in \tilde{(U, E)} : g_{B} ⊑ f_{A}, g_{B} is r - fuzzy soft δ - open} .

Then, for each

f_{A}

and

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

, the operator

δ I_{τ}

satisfies the following properties.

(1)

δ I_{τ} (e, \tilde{E}, r) = \tilde{E}

(2)

I_{τ} (e, f_{A}, r) ⊑ δ I_{τ} (e, f_{A}, r) ⊑ f_{A}

(3)

δ I_{τ} (e, f_{A}, r) ⊑ δ I_{τ} (e, g_{B}, r)

if,

f_{A} ⊑ g_{B}

(4)

δ I_{τ} (e, δ I_{τ} (e, f_{A}, r), r) = δ I_{τ} (e, f_{A}, r)

(5)

δ I_{τ} (e, f_{A}, r) ⊓ δ I_{τ} (e, g_{B}, r) ⊒ δ I_{τ} (e, f_{A} ⊓ g_{B}, r)

(6)

δ I_{τ} (e, f_{A}, r) = f_{A}

iff

f_{A}

is r-fuzzy soft

δ

-open.

(7)

δ I_{τ} (e, f_{A}^{c}, r) = {(δ C_{τ} (e, f_{A}, r))}^{c}

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

δ I_{τ}

(4) and (5) are easily proved by a similar way in Theorem 2.2.

(7) For each

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

e \in E

and

r \in I_{0}

, we have

δ I_{τ} (e, f_{A}^{c}, r) = ⊔ {g_{B} \in \tilde{(U, E)} : g_{B} ⊑ f_{A}^{c}, g_{B} is r - fuzzy soft δ - open}

= [

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

{(δ C_{τ} (e, f_{A}, r))}^{c}

□

Definition 2.3.

Let

(U, τ_{E})

be a FSTS,

r \in I_{0}

and

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

. Then, we have:

(1) Two fuzzy soft sets

f_{A}

and

g_{B}

are called r-fuzzy soft

δ

-separated iff

g_{B} / \tilde{q} δ C_{τ} (e, f_{A}, r)

and

f_{A} / \tilde{q} δ C_{τ} (e, g_{B}, r)

for each

e \in E

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

δ

-separated sets is called an r-fuzzy soft

δ

-connected.

Theorem 2.4.

In a FSTS

(U, τ_{E})

, we have:

(1) If

f_{A}

and

g_{B}

\in \tilde{(U, E)}

are r-fuzzy soft

δ

-separated and

h_{C}

t_{D}

\in \tilde{(U, E)}

such that

h_{C} ⊑ f_{A}

and

t_{D} ⊑ g_{B}

, then

h_{C}

and

t_{D}

are r-fuzzy soft

δ

-separated.

(2) If

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

and either both are r-fuzzy soft

δ

-open or both r-fuzzy soft

δ

-closed, then

f_{A}

and

g_{B}

are r-fuzzy soft

δ

-separated.

(3) If

f_{A}

and

g_{B}

are either both r-fuzzy soft

δ

-open or both r-fuzzy soft

δ

-closed, then

f_{A} ⊓ g_{B}^{c}

and

g_{B} ⊓ f_{A}^{c}

are r-fuzzy soft

δ

-separated.

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

(3) Let

f_{A}

and

g_{B}

be an r-fuzzy soft

δ

-open. Since

f_{A} ⊓ g_{B}^{c} ⊑ g_{B}^{c}

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

and hence

δ C_{τ} (e, f_{A} ⊓ g_{B}^{c}, r) / \tilde{q} g_{B}

. Then,

δ C_{τ} (e, f_{A} ⊓ g_{B}^{c}, r) / \tilde{q} (g_{B} ⊓ f_{A}^{c}) .

Again, since

g_{B} ⊓ f_{A}^{c} ⊑ f_{A}^{c}

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

and hence

δ C_{τ} (e, g_{B}

⊓ f_{A}^{c}, r) / \tilde{q} f_{A}

. Then,

δ C_{τ} (e, g_{B} ⊓ f_{A}^{c}, r) / \tilde{q} (f_{A} ⊓ g_{B}^{c}) .

Thus,

f_{A} ⊓ g_{B}^{c}

and

g_{B} ⊓ f_{A}^{c}

are r-fuzzy soft

δ

-separated. The other case follows similar lines.

□

Theorem 2.5.

In a FSTS

(U, τ_{E})

, then

f_{A}

g_{B}

\in \tilde{(U, E)}

are r-fuzzy soft

δ

-separated iff there exist two r-fuzzy soft

δ

-open sets

h_{C}

and

t_{D}

such that

f_{A} ⊑ h_{C}

g_{B} ⊑ t_{D}

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

and

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

Proof. (⇒) Let

f_{A}

and

g_{B}

\in \tilde{(U, E)}

be an r-fuzzy soft

δ

-separated,

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

and

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

, where

t_{D}

and

h_{C}

are r-fuzzy soft

δ

-open, then

t_{D} / \tilde{q} δ C_{τ} (e, f_{A}, r)

and

h_{C} / \tilde{q} δ C_{τ} (e, g_{B}, r)

. Thus,

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

and

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

. Hence, we obtain the required result.

(⇐) Let

h_{C}

and

t_{D}

be an r-fuzzy soft

δ

-open such that

g_{B} ⊑ t_{D}

f_{A} ⊑ h_{C}

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

and

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

. Then,

g_{B} ⊑ h_{C}^{c}

and

f_{A} ⊑ t_{D}^{c}

. Hence,

δ C_{τ} (e, g_{B}, r) ⊑ h_{C}^{c}

and

δ C_{τ} (e, f_{A}, r) ⊑ t_{D}^{c}

. Then,

δ C_{τ} (e, g_{B}, r) / \tilde{q} f_{A}

and

δ C_{τ} (e, f_{A}, r) / \tilde{q} g_{B}

. Thus,

f_{A}

and

g_{B}

are r- fuzzy soft

δ

-separated. Hence, we obtain the required result.

□

Theorem 2.6.

In a FSTS

(U, τ_{E})

, if

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

is r-fuzzy soft

δ

-connected such that

g_{B} ⊑ f_{A} ⊑ δ C_{τ} (e, g_{B}, r)

, then

f_{A}

is r-fuzzy soft

δ

-connected.

Proof.

Suppose that

f_{A}

is not r-fuzzy soft

δ

-connected, then there is r-fuzzy soft

δ

-separated sets

h_{C}^{*}

and

t_{D}^{*}

\in \tilde{(U, E)}

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 2.4(1),

h_{C}

and

t_{D}

are r-fuzzy soft

δ

-separated, it is a contradiction. Thus,

f_{A}

is r-fuzzy soft

δ

-connected, as required.

□

3. New Types of Fuzzy Soft Continuity

Here, we introduce the concepts of fuzzy soft

δ

-continuous (

β

-continuous, semi-continuous, pre-continuous and

α

-continuous) functions, which are weaker forms of fuzzy soft continuity on fuzzy soft topological spaces in Šostaks sense. Also, we study several relationships related to fuzzy soft

δ

-continuity with the help of some problems. In addition, a decomposition of fuzzy soft semi-continuity and a decomposition of fuzzy soft

α

-continuity is obtained.

Definition 3.1.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs. A fuzzy soft function

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

is said to be a fuzzy soft

δ

-continuous (resp.

β

-continuous, semi-continuous, pre-continuous,

α

-continuous) if,

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

δ

-open (resp.

β

-open, semi-open, pre-open,

α

-open) set for every

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

with

τ_{k}^{*} (g_{B}) \geq r

e \in E

(k = ψ (e)) \in F

and

r \in I_{o}

Remark 3.1.

Fuzzy soft

δ

-continuity and fuzzy soft

β

-continuity are independent concepts, as shown by Examples 3.1 and 3.2.

Example 3.1.

Let

U = {u_{1}, u_{2}}

E = {e_{1}, e_{2}}

and define

h_{E}, g_{E}, f_{E} \in \tilde{(U, E)}

as follows:

h_{E} = {(e_{1}, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}}), (e_{2}, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}})}

g_{E} = {(e_{1}, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}}), (e_{2}, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}})}

f_{E} = {(e_{1}, {\frac{u_{1}}{0.8}, \frac{u_{2}}{0.7}}), (e_{2}, {\frac{u_{1}}{0.8}, \frac{u_{2}}{0.7}})}

. Define fuzzy soft topologies

τ_{E}, τ_{E}^{*} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

\forall e \in E

τ_{e} (m_{E}) =

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

τ_{e}^{*} (m_{E}) =

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

Then, the identity fuzzy soft function

φ_{ψ} : (U, τ_{E}) ⟶ (U, τ_{E}^{*})

is fuzzy soft

β

-continuous, but it is neither fuzzy soft

δ

-continuous nor fuzzy soft semi-continuous.

Example 3.2.

Let

U = {u_{1}, u_{2}, u_{3}}

E = {e_{1}, e_{2}}

and define

h_{E}, g_{E}, f_{E} \in \tilde{(U, E)}

as follows:

h_{E} = {(e_{1}, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{1}}), (e_{2}, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{1}})}

g_{E} = {(e_{1}, {\frac{u_{1}}{0}, \frac{u_{2}}{0}, \frac{u_{3}}{1}}), (e_{2}, {\frac{u_{1}}{0}, \frac{u_{2}}{0}, \frac{u_{3}}{1}})}

f_{E} = {(e_{1}, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{0}}), (e_{2}, {\frac{u_{1}}{0}, \frac{u_{2}}{1}, \frac{u_{3}}{0}})}

. Define fuzzy soft topologies

τ_{E}, τ_{E}^{*} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

\forall e \in E

τ_{e} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, if m_{E} = g_{E}, \\ \frac{1}{2}, if m_{E} = f_{E}, \\ \frac{1}{3}, if m_{E} = h_{E}, \\ 0, otherwise, \end{matrix}

τ_{e}^{*} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, if m_{E} = h_{E}^{c}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (U, τ_{E}) ⟶ (U, τ_{E}^{*})

is fuzzy soft

δ

-continuous, but it is neither fuzzy soft

β

-continuous nor fuzzy soft semi-continuous.

Now, we have the following decomposition of fuzzy soft semi-continuity and decomposition of fuzzy soft

α

-continuity, according to Propositions 2.1 and 2.2.

Proposition 3.1.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs.

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

is fuzzy soft semi-continuous function iff it is both fuzzy soft

δ

-continuous and fuzzy soft

β

-continuous.

Proof.

The proof is obvious by Proposition 2.1. □

Proposition 3.2.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs.

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

is fuzzy soft

α

-continuous function iff it is both fuzzy soft

δ

-continuous and fuzzy soft pre-continuous.

Proof.

The proof is obvious by Proposition 2.2. □

Remark 3.2.

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

c o n t i n u i t y

⇓

α - c o n t i n u i t y

⇓ ⇓

p r e - c o n t i n u i t y s e m i - c o n t i n u i t y \Rightarrow δ - c o n t i n u i t y

⇓ ⇓

β - c o n t i n u i t y

Remark 3.3.

In general, the converses of the above relationships are not true, as shown by Examples 3.1, 3.2, 3.3, 3.4 and 3.5.

Example 3.3.

Let

U = {u_{1}, u_{2}}

E = {e_{1}, e_{2}}

and define

g_{E}, f_{E}, h_{E} \in \tilde{(U, E)}

as follows:

g_{E} = {(e_{1}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}}), (e_{2}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}})}

f_{E} = {(e_{1}, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.2}}), (e_{2}, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.2}})}

h_{E} = {(e_{1}, {\frac{u_{1}}{0.7}, \frac{u_{2}}{0.5}}), (e_{2}, {\frac{u_{1}}{0.7}, \frac{u_{2}}{0.5}})}

. Define fuzzy soft topologies

τ_{E}, τ_{E}^{*} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

\forall e \in E

τ_{e} (m_{E}) =

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

τ_{e}^{*} (m_{E}) =

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

Then, the identity fuzzy soft function

φ_{ψ} : (U, τ_{E}) ⟶ (U, τ_{E}^{*})

is fuzzy soft semi-continuous, but it is neither fuzzy soft

α

-continuous nor fuzzy soft pre-continuous.

Example 3.4.

Let

U = {u_{1}, u_{2}, u_{3}}

E = {e_{1}, e_{2}}

and define

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

as follows:

g_{E} = {(e_{1}, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}, \frac{u_{3}}{0.2}}), (e_{2}, {\frac{u_{1}}{0.2}, \frac{u_{2}}{0.3}, \frac{u_{3}}{0.2}})}

f_{E} = {(e_{1}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}, \frac{u_{3}}{0.8}}), (e_{2}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}, \frac{u_{3}}{0.8}})}

. Define fuzzy soft topologies

τ_{E}, τ_{E}^{*} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

\forall e \in E

τ_{e} (m_{E}) =

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

τ_{e}^{*} (m_{E}) =

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

Then, the identity fuzzy soft function

φ_{ψ} : (U, τ_{E}) ⟶ (U, τ_{E}^{*})

is fuzzy soft

β

-continuous, but it is not fuzzy soft pre-continuous.

Example 3.5.

Let

U = {u_{1}, u_{2}}

E = {e_{1}, e_{2}}

and define

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

as follows:

g_{E} = {(e_{1}, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}}), (e_{2}, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}})}

f_{E} = {(e_{1}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}}), (e_{2}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}})}

. Define fuzzy soft topologies

τ_{E}, τ_{E}^{*} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

\forall e \in E

τ_{e} (m_{E}) =

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

τ_{e}^{*} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, if m_{E} = f_{E}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (U, τ_{E}) ⟶ (U, τ_{E}^{*})

is fuzzy soft pre-continuous, but it is neither fuzzy soft

α

-continuous nor fuzzy soft semi-continuous.

Theorem 3.1.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs, and

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

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

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

e \in E

(k = ψ (e)) \in F

and

r \in I_{\circ}

(i)

φ_{ψ}

is fuzzy soft

β

-continuous.

(ii)

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

, if

τ_{k}^{*} (g_{B}^{c}) \geq r

(iii)

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

(iv)

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

Proof. (i) ⇒ (ii) Let

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

with

τ_{k}^{*} (g_{B}^{c}) \geq r

. Then by Definition 3.1,

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

Thus,

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

(ii) ⇒ (iii) Obvious.

(iii) ⇒ (iv) Since

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

and

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

. Then,

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

, for each

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

(iv) ⇒ (i) Let

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

with

τ_{k}^{*} (g_{B}) \geq r

. Then by (iv) and

g_{B} = I_{τ^{*}} (k, g_{B}, r)

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

. Thus,

φ_{ψ}

is fuzzy soft

β

-continuous.

The following theorem is similarly proved as in Theorem 3.1. □

Theorem 3.2.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs, and

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

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

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

e \in E

(k = ψ (e)) \in F

and

r \in I_{\circ}

(i)

φ_{ψ}

is fuzzy soft

δ

-continuous.

(ii)

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

, if

τ_{k}^{*} (g_{B}^{c}) \geq r

(iii)

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

(iv)

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

Proposition 3.3.

Let

(U, τ_{E})

(V, τ_{F}^{*})

and

(W, γ_{H})

be a FSTSs, and

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

φ_{ψ^{*}}^{*} : \tilde{(V, F)} ⟶ \tilde{(W, H)}

be two fuzzy soft functions. Then, the composition

φ_{ψ^{*}}^{*} \circ φ_{ψ}

is fuzzy soft

δ

-continuous (resp.

β

-continuous) if,

φ_{ψ}

is fuzzy soft

δ

-continuous (resp.

β

-continuous) and

φ_{ψ^{*}}^{*}

is fuzzy soft continuous.

Proof.

Obvious. □

4. Some Weaker Forms of Fuzzy Soft Continuity

Here, as a weaker form of fuzzy soft continuity [18], the concepts of fuzzy soft almost (weakly) continuous functions are introduced, and some properties are obtained. Furthermore, we show that fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft weakly continuity, but the converse may not be true. Finally, we introduce the notion of continuity in a very general setting called, fuzzy soft

(L, M, N, O)

-continuous functions.

Definition 4.1.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs. A fuzzy soft function

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

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

e_{u_{t}} \in \tilde{P_{t} (U)}

and each

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

with

τ_{k}^{*} (g_{B}) \geq r

containing

φ_{ψ} (e_{u_{t}})

, there is

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

with

τ_{e} (f_{A}) \geq r

containing

e_{u_{t}}

such that

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

(resp.

φ_{ψ} (f_{A}) ⊑ C_{τ^{*}} (k, g_{B}, r)

Theorem 4.1.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs, and

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

be a fuzzy soft function. Suppose that one of the following holds for every

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

e \in E

(k = ψ (e)) \in F

and

r \in I_{\circ}

(i) If

τ_{k}^{*} (g_{B}) \geq r

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

(ii)

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

, if

τ_{k}^{*} (g_{B}^{c}) \geq r

Then,

φ_{ψ}

is fuzzy soft almost continuous.

Proof. (i) ⇒ (ii) Let

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

with

τ_{k}^{*} (g_{B}^{c}) \geq r

. From (i), it follows

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

= I_{τ} (e, φ_{ψ}^{- 1} ({(C_{τ^{*}} (k, I_{τ^{*}} (k, g_{B}, r), r))}^{c}), r) = I_{τ} (e, {(φ_{ψ}^{- 1} (C_{τ^{*}} (k, I_{τ^{*}} (k, g_{B}, r), r)))}^{c}, r) = {(C_{τ} (e, φ_{ψ}^{- 1} (C_{τ^{*}} (k, I_{τ^{*}} (k, g_{B}, r), r)), r))}^{c} .

Hence,

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

. Similarly, we get (ii) ⇒ (i).

Suppose that (i) holds. Let

e_{u_{t}} \in \tilde{P_{t} (U)}

and

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

with

τ_{k}^{*} (g_{B}) \geq r

containing

φ_{ψ} (e_{u_{t}})

. Then, by (i),

e_{u_{t}} \tilde{\in} I_{τ} (e, φ_{ψ}^{- 1} (I_{τ^{*}} (k, C_{τ^{*}} (k, g_{B}, r), r)), r),

and so there is

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

with

τ_{e} (f_{A}) \geq r

containing

e_{u_{t}}

such that

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

. Hence,

φ_{ψ} (f_{A})

⊑ I_{τ^{*}} (k, C_{τ^{*}} (k, g_{B}, r), r)

. Then,

φ_{ψ}

is fuzzy soft almost continuous.

□

Lemma 4.1.

Every fuzzy soft continuous function [18] is fuzzy soft almost continuous.

Proof.

It follows from Definitions 1.4 and 4.1. □

Remark 4.1.

In general, the converse of Lemma 4.1 is not true, as shown by Example 4.1.

Example 4.1.

Let

U = {u_{1}, u_{2}}

E = {e_{1}, e_{2}}

and define

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

as follows:

g_{E} = {(e_{1}, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}}), (e_{2}, {\frac{u_{1}}{0.4}, \frac{u_{2}}{0.5}})}

f_{E} = {(e_{1}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}}), (e_{2}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0.4}})}

. Define fuzzy soft topologies

τ_{E}, τ_{E}^{*} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

\forall e \in E

τ_{e} (m_{E}) =

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

τ_{e}^{*} (m_{E}) =

\{\begin{matrix} 1, if m_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, if m_{E} \in {f_{E}, g_{E}}, \\ 0, otherwise . \end{matrix}

Then, the identity fuzzy soft function

φ_{ψ} : (U, τ_{E}) ⟶ (U, τ_{E}^{*})

is fuzzy soft almost continuous, but it is not fuzzy soft continuous.

Theorem 4.2.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs, and

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

be a fuzzy soft function. Suppose that one of the following holds for every

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

e \in E

(k = ψ (e)) \in F

and

r \in I_{\circ}

(i)

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

, if

τ_{k}^{*} (g_{B}) \geq r

(ii)

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

, if

τ_{k}^{*} (g_{B}^{c}) \geq r

Then,

φ_{ψ}

is fuzzy soft weakly continuous.

Proof. (i) ⇒ (ii) Let

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

with

τ_{k}^{*} (g_{B}^{c}) \geq r

. From (i), it follows

φ_{ψ}^{- 1} (g_{B}^{c}) ⊑ I_{τ} (e, φ_{ψ}^{- 1} (C_{τ^{*}} (k, g_{B}^{c}, r)), r) = I_{τ} (e, φ_{ψ}^{- 1} ({(I_{τ^{*}} (k, g_{B}, r))}^{c}), r) = I_{τ} (e, {(φ_{ψ}^{- 1} (I_{τ^{*}} (k, g_{B}, r)))}^{c}, r) = {(C_{τ} (e, φ_{ψ}^{- 1} (I_{τ^{*}} (k, g_{B}, r)), r))}^{c} .

Hence,

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

. Similarly, we get (ii) ⇒ (i).

Suppose that (i) holds. Let

e_{u_{t}} \in \tilde{P_{t} (U)}

and

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

with

τ_{k}^{*} (g_{B}) \geq r

containing

φ_{ψ} (e_{u_{t}})

. Then, by (i),

e_{u_{t}} \tilde{\in} I_{τ} (e, φ_{ψ}^{- 1} (C_{τ^{*}} (k, g_{B}, r)), r),

and so there is

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

with

τ_{e} (f_{A}) \geq r

containing

e_{u_{t}}

such that

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

. Thus,

φ_{ψ} (f_{A}) ⊑ C_{τ^{*}} (k, g_{B}, r)

. Hence,

φ_{ψ}

is fuzzy soft weakly continuous. □

Lemma 4.2.

Every fuzzy soft almost continuous function is fuzzy soft weakly continuous.

Proof.

It follows from Definition 4.1. □

Remark 4.2.

In general, the converse of Lemma 4.2 is not true, as shown by Example 4.2.

Example 4.2.

Let

U = {u_{1}, u_{2}, u_{3}}

E = {e_{1}, e_{2}}

and define

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

as follows:

g_{E} = {(e_{1}, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.6}, \frac{u_{3}}{0.5}}), (e_{2}, {\frac{u_{1}}{0.6}, \frac{u_{2}}{0.6}, \frac{u_{3}}{0.5}})}

f_{E} = {(e_{1}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0}, \frac{u_{3}}{0.5}}), (e_{2}, {\frac{u_{1}}{0.3}, \frac{u_{2}}{0}, \frac{u_{3}}{0.5}})}

. Define fuzzy soft topologies

τ_{E}, τ_{E}^{*} : E ⟶ {[0, 1]}^{\tilde{(U, E)}}

as follows:

\forall e \in E

τ_{e} (m_{E}) =

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

τ_{e}^{*} (m_{E}) =

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

Then, the identity fuzzy soft function

φ_{ψ} : (U, τ_{E}) ⟶ (U, τ_{E}^{*})

is fuzzy soft weakly continuous, but it is not fuzzy soft almost continuous.

Remark 4.3.

From the previous results, we have: Fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft weakly continuity.

In [29], the difference between

f_{A}

and

g_{B}

is a fuzzy soft set defined as follows:

(f_{A} \bar{⊓} g_{B}) (e) = \{\begin{matrix} \underset{̲}{0}, & i f f_{A} (e) \leq g_{B} (e), \\ f_{A} (e) \land {(g_{B} (e))}^{c}, & o t h e r w i s e, \end{matrix} \forall e \in E .

Let

L

and

M

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

be operators on

\tilde{(U, E)}

, and

N

and

O

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

be operators on

\tilde{(V, F)}

Definition 4.2.

Let

(U, τ_{E})

and

(V, τ_{F}^{*})

be a FSTSs.

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

is said to be a fuzzy soft

(L, M, N, O)

-continuous function if,

L [e, φ_{ψ}^{- 1} (O (k, g_{B}, r)), r] \bar{⊓} M [e, φ_{ψ}^{- 1} (N (k, g_{B}, r)), r]

= Φ

for each

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

with

τ_{k}^{*} (g_{B}) \geq r

e \in E

and

(k = ψ (e)) \in F

In (2014), Aygünoǧlu et al. [18] defined the concept of fuzzy soft continuous functions:

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

, for each

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

e \in E

and

(k = ψ (e)) \in F

. We can see that Definition 4.2 generalizes the concept of fuzzy soft continuous functions, when we choose

L

= identity operator,

M

= interior operator,

N

= identity operator and

O

= identity operator.

A historical justification of Definition 4.2:

(1) In Section 3, we introduced the concept of fuzzy soft

δ

-continuous functions:

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

, for each

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

with

τ_{k}^{*} (g_{B}) \geq r

. Here,

L

= interior closure operator,

M

= closure interior operator,

N

= identity operator and

O

= identity operator.

(2) In Section 3, we introduced the concept of fuzzy soft

β

-continuous functions:

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

, for each

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

with

τ_{k}^{*} (g_{B}) \geq r

. Here,

L

= identity operator,

M

= closure interior closure operator,

N

= identity operator and

O

= identity operator.

(3) In Section 3, we introduced the concept of fuzzy soft semi-continuous functions:

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

, for each

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

with

τ_{k}^{*} (g_{B}) \geq r

. Here,

L

= identity operator,

M

= closure interior operator,

N

= identity operator and

O

= identity operator.

(4) In Section 3, we introduced the concept of fuzzy soft pre-continuous functions:

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

, for each

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

with

τ_{k}^{*} (g_{B}) \geq r

. Here,

L

= identity operator,

M

= interior closure operator,

N

= identity operator and

O

= identity operator.

(5) In Section 3, we introduced the concept of fuzzy soft

α

-continuous functions:

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

, for each

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

with

τ_{k}^{*} (g_{B}) \geq r

. Here,

L

= identity operator,

M

= interior closure interior operator,

N

= identity operator and

O

= identity operator.

(6) In Section 4, we introduced the concept of fuzzy soft almost continuous functions:

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

, for each

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

with

τ_{k}^{*} (g_{B}) \geq r

. Here,

L

= identity operator,

M

= interior operator,

N

= interior closure operator and

O

= identity operator.

(7) In Section 4, we introduced the concept of fuzzy soft weakly continuous functions:

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

, for each

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

with

τ_{k}^{*} (g_{B}) \geq r

. Here,

L

= identity operator,

M

= interior operator,

N

= closure operator and

O

= identity operator.

5. Conclusion and Future Work

This article is lay out as follows:

(1)

In Section 2, some new types of a fuzzy soft set called an r-fuzzy soft

δ

-open (semi-open and

α

-open) set are introduced on fuzzy soft topological space based on the paper Aygünoǧlu et al. [18]. Also, we have the following relationships, but the converses are not true.

α - o p e n s e t

⇓ ⇓

p r e - o p e n s e t s e m i - o p e n s e t \Rightarrow δ - o p e n s e t

⇓ ⇓

β - o p e n s e t

(2)

In Section 3, we introduce the concepts of fuzzy soft

δ

-continuous (

β

-continuous, semi-continuous, pre-continuous and

α

-continuous) functions, and the relations of these functions with each other are investigated with the help of some illustrative examples. Moreover, a decomposition of fuzzy soft semi-continuity and a decomposition of fuzzy soft

α

-continuity is given.

(3)

In Section 4, as a weaker form of fuzzy soft continuity [18], the concepts of fuzzy soft almost (weakly) continuous functions are introduced, and some properties are obtained. Also, we show that fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft weakly continuity, but the converse may not be true. Finally, we explore the notion of continuity in a very general setting namely, fuzzy soft

(L, M, N, O)

-continuous functions. Then, we have the following results:

• Fuzzy soft

(i d_{U}, I_{τ}, i d_{V}, i d_{V})

-continuous function is a fuzzy soft continuous function [18].

• Fuzzy soft

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

-continuous function is a fuzzy soft

δ

-continuous function.

• Fuzzy soft

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

-continuous function is a fuzzy soft

β

-continuous function.

• Fuzzy soft

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

-continuous function is a fuzzy soft semi-continuous function.

• Fuzzy soft

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

-continuous function is a fuzzy soft pre-continuous function.

• Fuzzy soft

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

-continuous function is a fuzzy soft

α

-continuous function.

• Fuzzy soft

(i d_{U}, I_{τ}, I_{τ^{*}} (C_{τ^{*}}), i d_{V})

-continuous function is a fuzzy soft almost continuous function.

• Fuzzy soft

(i d_{U}, I_{τ}, C_{τ^{*}}, i d_{V})

-continuous function is a fuzzy soft weakly continuous function.

In upcoming articles, we will use the r-fuzzy soft

δ

-open sets to introduce some new separation axioms and to define the concept of

δ

-compact spaces on fuzzy soft topological space based on the paper Aygünoǧlu et al. [18].

Author Contributions

Methodology, Islam M. Taha; Formal Analysis, Ibtesam Alshammari, Mesfer H. Alqahtani and Islam M. Taha; Investigation, Ibtesam Alshammari, Mesfer H. Alqahtani and Islam M. Taha; Writing Original Draft Preparation, Islam M. Taha; Writing Review and Editing, Ibtesam Alshammari, Mesfer H. Alqahtani and Islam M. Taha; Funding Acquisition, Ibtesam Alshammari, Mesfer H. Alqahtani and Islam M. Taha.

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 R-fuzzy Soft δ-Open Sets and Applications via Fuzzy Soft Topologies

Abstract

1. Introduction

2. Some Properties of r-Fuzzy Soft δ -Open Sets

3. New Types of Fuzzy Soft Continuity

4. Some Weaker Forms of Fuzzy Soft Continuity

5. Conclusion and Future Work

Author Contributions

Data Availability Statement

Conflicts of Interest

References

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2. Some Properties of r-Fuzzy Soft $δ$ -Open Sets