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:

17 December 2023

Posted:

18 December 2023

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Abstract

In this work, we introduce the concepts of r-fuzzy soft δ-open (resp. semi-open and α-open) sets on fuzzy soft topological space, the relations of these sets with each other are established. Next, we introduce the concepts of fuzzy soft δ-continuous (resp. β-continuous, semi-continuous, pre-continuous and α-continuous) functions, and some properties of these functions along with their mutual relationships are investigated. Also, a decomposition of fuzzy soft semi-continuity and a decomposition of fuzzy soft α-continuity are given. Finally, as a weaker form of fuzzy soft continuity by Aygunoglu et al. (2014), the concepts of fuzzy soft almost (resp. weakly) continuous functions are introduced, and some properties are specified. Moreover, we explore the notion of continuity in a very general setting namely fuzzy soft (L, M, N, O)-continuous functions, and a historical justification are obtained.

Keywords:

Subject: Computer Science and Mathematics - Geometry and Topology

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 this theory in solving several practical problems in engineering, economics, medical science, social science, etc. Maji et al. [2] introduced the concept of fuzzy soft sets which combines soft sets [1] and fuzzy sets [3]. After that, many authors have contributed to soft set (fuzzy soft set) theory in the different fields such as topology, algebra and etc., see [4–14].

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 are obtained in [15,16] based on fuzzy topologies in Šostaks sense [17]. The concept of r-fuzzy soft regularly open sets was introduced by Çetkin and Aygün [16] based on the paper Aygünoǧlu et al. [15]. Also, the concepts of r-fuzzy soft

β

-open (resp. pre-open) sets was also considered by Taha [18].

The organization of this paper is as follows:

(i) In Section 2, we define new types of fuzzy soft sets on fuzzy soft topological space based on the paper Aygünoǧlu et al. [15]. Also, the relations of these sets with each other are established with the help of some examples.

(ii) In Section 3, we introduce the concepts of fuzzy soft

δ

-continuous (resp.

β

-continuous, semi-continuous, pre-continuous and

α

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

α

-continuity are given.

(iii) In Section 4, as a weaker form of fuzzy soft continuity [15], the concepts of fuzzy soft almost (resp. 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 need not be true in general. In the end, we explore the notion of continuity in a very general setting namely fuzzy soft

(L, M, N, O)

-continuous functions, and a historical justification are given.

(iv) Finally, we close this paper with some conclusions and make a plan for 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.

[2, 4, 15]

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.

[19]

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}, & i f & k = e, \\ \underset{̲}{0}, & i f & 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.

[15]

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

for every

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

Then,

(U, τ_{E})

is called a fuzzy soft topological space (briefly, FSTS) in Šostaks sense [17].

Definition 1.4.

[15]

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 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 1.5.

[16, 18]

Let

(U, τ_{E})

be a FSTS. A fuzzy soft set

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

is said to be r-fuzzy soft regularly open (resp. pre-open and

β

-open) 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

and

r \in I_{0}

Lemma 1.1.

[18]

Every r-fuzzy soft regularly open set [16] is r-fuzzy soft pre-open [18].

The converse of Lemma 1.1 is not true, in general, as shown by Example 1.1.

Example 1.1.

[18]

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.

In a fuzzy soft topological space

(U, τ_{E})

[16], the interior of

f_{A}

and the closure of

f_{A}

will be denoted by

I_{τ} (e, f_{A}, r), r)

and

C_{τ} (e, f_{A}, r), r)

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

2. On r-Fuzzy Soft $δ$ -Open Sets

In this section, we are going to give the concepts of r-fuzzy soft

δ

-open (resp. 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.

Definition 2.1.

Let

(U, τ_{E})

be a FSTS. A fuzzy soft set

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

is said to be 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 r-fuzzy soft

δ

-open set and r-fuzzy soft

β

-open set [18] are independent concepts, as shown by Example 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 r-fuzzy soft

δ

-open (resp. semi-open,

α

-open and

β

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

δ

-open and r-fuzzy soft

β

-open,

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 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.

□

Then, we have the following implications:

α - o p e n

↓ ↓

p r e - o p e n s e m i - o p e n ⟶ δ - o p e n

↓ ↓

β - o p e n

Moreover, the converses of the above relationships are not true, in general, as shown by Example 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.

□

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

In this section, we introduce the concepts of fuzzy soft

δ

-continuous (resp.

β

-continuous, semi-continuous, pre-continuous and

α

-continuous) functions on fuzzy soft topological space in Šostaks sense. Also, we study several relationships related to fuzzy soft

δ

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

α

-continuity are 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 fuzzy soft

δ

-continuous (resp.

β

-continuous, semi-continuous, pre-continuous and

α

-continuous) if

φ_{ψ}^{- 1} (g_{B})

is r-fuzzy soft

δ

-open (resp.

β

-open, semi-open, pre-open and

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 Example 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. □

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)

Then, we have the following implications:

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 ⟶ δ - c o n t i n u i t y

↓ ↓

β - c o n t i n u i t y

Moreover, the converses of the above relationships are not true, in general, as shown by Example 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.

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

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. New Types of Fuzzy Soft Functions

In this section, we introduce a weaker form of fuzzy soft continuous function [15] called fuzzy soft almost (resp. weakly) continuous function and study some properties of these functions. Also, we show that fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft weakly continuity, but the converse need not be true. Finally, we introduce a concept 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 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 [15] is fuzzy soft almost continuous.

Proof.

It follows from Theorem 4.1. □

The converse of the above Lemma is not true, in general, 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. □

The converse of the above Lemma is not true, in general, 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.

Then, we have the following implications:

fuzzy soft continuity

⇓

fuzzy soft almost continuity

⇓

fuzzy soft weakly continuity .

In [20], 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 I^{U}

be operators on

\tilde{(U, E)}

, and

N

and

O

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

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 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. [15] defined the concept of fuzzy soft continuous functions:

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

. 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:

I. In Section 2, some new types of fuzzy soft sets namely r-fuzzy soft

δ

-open (resp. semi-open and

α

-open) sets are introduced on fuzzy soft topological space based on the paper Aygünoǧlu et al. [15]. Also, we have the following relationships:

α - o p e n

↓ ↓

p r e - o p e n s e m i - o p e n ⟶ δ - o p e n

↓ ↓

β - o p e n

but the converse need not be true in general.

II. In Section 3, we introduce the concepts of fuzzy soft

δ

-continuous (resp.

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 are given.

III. In Section 4, as a weaker form of fuzzy soft continuity [15], the concepts of fuzzy soft almost (resp. 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 need not be true in general. 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:

(1) Fuzzy soft

(i d_{U}, I_{τ}, i d_{V}, i d_{V})

-continuous function is a fuzzy soft continuous function [15].

(2) 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.

(3) 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.

(4) Fuzzy soft

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

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

(5) Fuzzy soft

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

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

(6) 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.

(7) Fuzzy soft

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

-continuous function is a fuzzy soft almost continuous function.

(8) 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. [15].

Data Availability Statement

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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

Abstract

1. Introduction

2. On r-Fuzzy Soft δ -Open Sets

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

4. New Types of Fuzzy Soft Functions

5. Conclusion and Future Work

Data Availability Statement

Conflicts of Interest

References

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2. On r-Fuzzy Soft $δ$ -Open Sets

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