Compactness on Fuzzy Soft r-Minimal Spaces

Islam M. Taha

doi:10.5391/IJFIS.2021.21.3.251

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 251-258

Published online September 25, 2021

https://doi.org/10.5391/IJFIS.2021.21.3.251

Compactness on Fuzzy Soft r-Minimal Spaces

Islam M. Taha^1,2

¹Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia, Egypt
²Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

Correspondence to :
Islam M. Taha (imtaha2010@yahoo.com)

Received: February 10, 2021; Revised: June 5, 2021; Accepted: June 25, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

In this study, a new form of fuzzy r-minimal structure [28] called a fuzzy soft r-minimal structure, is defined, which is an extension of the fuzzy soft topology introduced by Aygunoglu et al. [26]. In addition, the concepts of fuzzy soft r-minimal continuity and fuzzy soft r-minimal compactness are introduced in fuzzy soft r-minimal spaces. Some interesting properties and characterizations are also discussed. Finally, several types of fuzzy soft r-minimal compactness are defined, and the relationships between them are characterized.

Keywords: Fuzzy soft set, Fuzzy soft topology, Fuzzy soft r-minimal structure, Fuzzy soft r-minimal space, Fuzzy soft r-minimal continuity, Fuzzy soft r-minimal compactness

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

In 1999, Molodtsov [1] proposed a completely new concept called a soft set theory to model uncertainty, which associates a set with a set of parameters. Using a soft set, we mean a pair (F,E), where E is a set interpreted as the set of parameters, and the mapping F : E → P(X) is referred to as the soft structure on X. After an introduction of the notion of soft sets, several researchers have improved this concept. Soft set theory has been applied to many different fields with great success. Maji et al. [2] introduced the concept of a fuzzy soft set that combines fuzzy sets [3] with soft sets [1]. Soft set and fuzzy soft set theories have a rich potential for several types of applications. Thus far, many spectacular and creative studies on the theories of soft sets and fuzzy soft sets have been conducted (see [2,4–25]). In addition, Aygunoglu et al. [26] studied the topological structure of fuzzy soft sets based on the approach by Sostak [27].

The concept of a fuzzy r-minimal structure was introduced by Yoo et al. [28], and is an extension of the fuzzy topology introduced by Sostak [27]. In addition, the concepts of fuzzy r-minimal continuity and fuzzy r-minimal compactness were introduced in [28] and [29]. Later, Abbas and Min [30] introduced the concept of a (r, s)-fuzzy minimal structure, which is an extension of the intuitionistic fuzzy topology introduced by Mondal and Samanta [31]. In addition, the concepts of (r, s)-fuzzy minimal continuity and (r, s)-fuzzy minimal compactness were introduced in [30].

The aim of this study is to introduce and study the concepts of a fuzzy soft r-minimal structure, fuzzy soft r-minimal space, fuzzy soft r-minimal continuity, and fuzzy soft r-minimal compactness based on the approach developed by Aygunoglu et al. [26]. In addition, their properties are investigated. Later, several types of fuzzy soft r-minimal compactness are defined, and the relationships among them are discussed.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

Throughout this paper, X refers to an initial universe, E is the set of all parameters for X and A ⊆ E, I^X is the set of all fuzzy sets on X (where I = [0, 1], I₀ = (0, 1]), and for α ∈ I, α(x) = α, for all x ∈ X.

Definition 2.1 [2,4,26]

A fuzzy soft set f_A over X is a mapping from E to I^X such that f_A(e) is a fuzzy set on X for each e ∈ A and f_A(e) = 0, if e ∉ A, where 0 is a zero function on X. The fuzzy set f_A(e) for each e ∈ E is called an element of the fuzzy soft set f_A. Here, (X,E)˜ denotes the collection of all fuzzy soft sets on X and is called a fuzzy soft universe [32].

Definition 2.2 [33]

A fuzzy soft point e_x_{_t} over X is a fuzzy soft set defined as follows: e_x_{_t}(k) = x_t, if k = e and e_x_{_t} (k) = 0, if k ∈ E −{e}, where x_t is a fuzzy point. A fuzzy soft point e_x_{_t} is said to belong to a fuzzy soft set f_A, denoted by e_xt ∊̃ f_A if t < f_A(e)(x).

Definition 2.3 [26]

A mapping τ:E→[0,1](X,E)˜ is called a fuzzy soft topology on X if it satisfies the following conditions for each e ∈ E.

τ_e(Φ) = τ_e( Ẽ) = 1.
τe(fA⊓gB)≥τe(fA)∧τe(gB),∀fA,gB∈(X,E)˜.
τe(⊔i∈Δ(fA)i)≥∧i∈Δτe((fA)i),∀(fA)i∈(X,E)˜,i∈Δ.

Then, the pair (X, τ_E) is called a fuzzy soft topological space (FSTS).

All definitions and properties of fuzzy soft sets and fuzzy soft topologies are found in [2,4,12,16,24,26].

3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets

Go to

Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

In this section, a new form of a fuzzy r-minimal structure called a fuzzy soft r-minimal structure is defined, which is an extension of the fuzzy soft topology introduced by Aygunoglu et al. [26]. In addition, the concepts of fuzzy soft r-minimal continuity and fuzzy soft r-minimal compactness are introduced in fuzzy soft r-minimal spaces. Some interesting properties and characterizations are discussed.

Definition 3.1

Let X be a nonempty set and r ∈ I₀. Fuzzy soft mapping M˜:E→[0,1](X,E)˜ on X is said to be a fuzzy soft r-minimal structure if the family M˜e,r={fA∈(X,E)˜∣M˜e(fA)≥r} for each e ∈ E contains Φ and Ẽ. Then, (X, M̃) is called a fuzzy soft r-minimal space (simply, r-FMS˜). Every member of M̃_e,r is called a fuzzy soft r-minimal open set. A fuzzy soft set f_A is called a fuzzy soft r-minimal closed set if the complement of f_A is a fuzzy soft r-minimal open set.

Example 3.1

Let (X, τ_E) be an FSTS. It is then easy to see that for each e ∈ E and r ∈ I₀, a fuzzy soft mapping M̃ defined by M˜e,r={fA∈(X,E)˜∣τe(fA)≥r} is a fuzzy soft r-minimal structure on X.

Definition 3.2. [12]

Let (X, τ_E) be an FSTS. For fA∈(X,E)˜ and r ∈ I₀, f_A is called r-fuzzy soft regularly open if f_A = I_τ (e,C_τ (e, f_A, r), r).

Definition 3.3

Let (X, τ_E) be an FSTS. For fA∈(X,E)˜ and r ∈ I₀.

(1) f_A is called r-fuzzy soft preopen if f_A ⊑ I_τ (e, C_τ (e, f_A, r), r).

(2) f_A is called r-fuzzy soft β-open if f_A ⊑ C_τ (e, I_τ (e, C_τ (e, f_A, r), r).

The following implications hold:

r-fuzzy soft regularly open ⇒ r-fuzzy soft preopen ⇒ r-fuzzy soft β-open.

In general, the converses are not true.

Example 3.2

Let X = {x, y} be a classical set and E = {e₁, e₂} be the parameter set of X. Define f_E and gE∈(X,E)˜ as follows: fE={(e1,{x0.3,y0.4}),(e2,{x0.3,y0.4})},gE={(e1,{x0.6,y0.2}),(e2,{x0.6,y0.2})}. The fuzzy soft topology τE:E→[0,1](X,E)˜ is as follows:

τe1(wE)={1,if wE∈{Φ,E˜},14,if wE=fE,13,if wE=gE,13,if wE=fE⊓gE,14,if wE=fE⊔gE,0,otherwise,τe2(wE)={1,if wE∈{Φ,E˜},14,if wE=fE,12,if wE=gE,12,if wE=fE⊓gE,14,if wE=fE⊔gE,0,otherwise.

Then, g_E is a 14-fuzzy soft preopen set and 14-fuzzy soft β-open set but is not a 14-fuzzy soft regularly open set.

Remark 1

Let (X, τ_E) be an FSTS and r ∈ I₀. We denote the set of all r-fuzzy soft regularly open (respectively, r-fuzzy soft preopen and r-fuzzy soft β-open) sets by r-FRO(X)˜ (respectively, r-FPO(X)˜ and r-FβO(X)˜. Then, r-FRO(X)˜,r-FPO(X)˜, and r-FβO(X)˜are all fuzzy soft r-minimal structures.

Definition 3.4

Let (X, M̃) be an r-FMS˜, e ∈ E, and r ∈ I₀. The fuzzy soft r-minimal interior and fuzzy soft r-minimal closure of f_A, denoted by I_m(e, f_A, r) and C_m(e, f_A, r), respectively, are defined as Im(e,fA,r)=⊔{gB∈(X,E˜):gB⊑fA,gB∈M˜e,r};Cm(e,fA,r)=⊓{gB∈(X,E)˜:fA⊑gB,gBc∈M˜e,r}.

Theorem 3.1

Let (X, M̃) be an r-FMS˜ and fA,gB∈(X,E)˜. Then, the following statements hold:

(1) I_m(e, f_A, r) ⊑ f_A and if f_A ∈ M̃_e,_r, I_m(e, f_A, r) = f_A.

(2) f_A ⊑ C_m(e, f_A, r) and if fAc∈M˜e,r, C_m(e, f_A, r) = f_A.

(3) If f_A ⊑ g_B, I_m(e, f_A, r) ⊑ I_m(e, g_B, r) and C_m(e, f_A, r) ⊑ C_m(e, g_B, r).

(4) I_m(e, f_A ⊓ g_B, r) = I_m(e, f_A, r) ⊓ I_m(e, g_B, r).

(5) C_m(e, f_A ⊔ g_B, r) = C_m(e, f_A, r) ⊔ C_m(e, g_B, r).

(6) I_m(e, I_m(e, f_A, r), r) = I_m(e, f_A, r) and C_m(e, C_m(e, f_A, r), r) = C_m(e, f_A, r).

(7) (Cm(e,fA,r))c=Im(e,fAc,r) and (Im(e,fA,r))c=Cm(e,fAc,r).

Proof

We prove (5) only because the others are obvious. For gB∈(X,E)˜, because C_m(e, f_A, r) ⊑ C_m(e, f_A ⊔ g_B, r) and C_m(e, g_B, r) ⊑ C_m(e, f_A ⊔ g_B, r), we obtain C_m(e, f_A, r) ⊔ C_m(e, g_B, r) ⊑ C_m(e, f_A ⊔ g_B, r). For hCc,kDc∈Me,r,hCc⊓kDc does not always belong to M̃_e,r. Thus, we obtain the following:

Cm(e,fA,r)⊔Cm(e,gB,r)=(⊓{hC∈(X,E)˜∣hCc∈M˜e,r,fA⊑hC})⊔(⊓{kD∈(X,E)˜∣kDc∈M˜e,r,gB⊑kD})=⊓{hC⊔kD∈(X,E)˜∣fA⊑hC,gB⊑kD}⊒⊓{wE∈(X,E)˜∣wEc∈M˜e,r,fA⊔gB⊑wE}=Cm(e,fA⊔gB,r).

Hence, we obtain C_m(e, f_A ⊔ g_B, r) = C_m(e, f_A, r) ⊔ C_m(e, g_B, r).

Definition 3.5

Let (X, M̃) and (Y, M̃^*) be an r-FMS˜. Then, a fuzzy soft mapping ϕ_ψ from (X,E)˜ into (Y,F)˜ is called fuzzy soft r-minimal continuous if ϕψ-1(gB)∈M˜e,r for every gB∈M˜k,r*, e ∈ E, and (ψ(e) = k) ∈ F.

Theorem 3.2

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, fA∈(X,E)˜,gB∈(Y,F)˜, e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal continuous.

(2) (ϕψ-1(gB))c∈M˜e,r for every gBc∈M˜k,r*.

(3) ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).

(4) Cm(e,ϕψ-1(gB),r)⊑ϕψ-1(Cm*(k,gB,r)).

(5) ϕψ-1(Im*(k,gB,r))⊑Im(e,ϕψ-1(gB),r).

(6) If for every fuzzy soft point e_x_{_t} over X and each gB∈M˜k,r* containing ϕ_ψ(e_x_{_t} ), there exists f_A ∈ M̃_e,r containing e_x_{_t} such that ϕ_ψ(f_A) ⊑ g_B.

Then, (1) ⇔ (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇒ (6).

Proof

(1) ⇔ (2) Obvious.

(2) ⇒ (3) For all fA∈(X,E)˜,

ϕψ-1(Cm*(k,ϕψ(fA),r))=ϕψ-1(⊓{gB∈(Y,F)˜∣gBc∈M˜k,r*,ϕψ(fA)⊑gB})=⊓{ϕψ-1(gB)∈(X,E)˜∣gBc∈M˜k,r*,fA⊑ϕψ-1(gB)}⊒⊓{ϕψ-1(gB)∣(ϕψ-1(gB))c∈M˜e,rfA⊑ϕψ-1(gB)}=Cm(e,fA,r).

Hence ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).

(3) ⇔ (4) Let gB∈(Y,F)˜. Then, from (3), it follows that ϕψ(Cm(e,ϕψ-1(gB),r))⊑Cm*(k,ϕψ(ϕψ-1(gB)),r)⊑Cm*(k,gB,r). Hence, we obtain Equation (4). Similarly, we obtain (4) ⇒ (3).

(4) ⇔ (5) From (7) of Theorem 3.1.

(5) ⇒ (6) Let ext∈Pt(X)˜ and gB∈M˜k,r* with ϕ_ψ(e_x_{_t} )∊̃g_B. From (5), it follows that ext∈˜Im(e,ϕψ-1(gB),r). Thus, there exists f_A ∈ M̃_e,r such that ext∈˜fA⊑ϕψ-1(gB). Hence, we obtain (6).

Lemma 3.1

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft r-minimal continuous mapping. The following are then equivalent: For e ∈ E and r ∈ I_o,

(1) ϕ_ψ(f_A) ⊑ C_m_^* (k,ϕ_ψ(f_A), r) for each fAc∈M˜e,r;
(2) ϕψ-1(gB)=Cm(e,ϕψ-1(gB),r) for each gBc∈M˜k,r*;
(3) ϕψ-1(gB)=Im(e,ϕψ-1(gB),r) for each gB∈M˜k,r*.

Proof

The proof is obvious.

Definition 3.6

Let X be a nonempty set and M˜:E→[0,1](X,E)˜. Then, M̃ is said to have a property (P) if for (fA)j∈(X,E)˜, j ∈ J, and e ∈ E

M˜e(⊔j∈J(fA)j)≥⊓j∈JM˜e((fA)j).

Theorem 3.3

Let (X, M̃) be an r-FMS˜ with the property (P). Then, the following statements hold: (1) I_m(e, f_A, r) = f_A iff f_A ∈ M̃_e,r for fA∈(X,E)˜, and (2) C_m(e, f_A, r) = f_A if fAc∈M˜e,r for fA∈(X,E)˜.

Proof

(1) From Theorem 3.1, it is sufficient to show that if I_m(e, f_A, r) = f_A then f_A ∈ M̃_e,r. Let I_m(e, f_A, r) = f_A, e ∈ E, and r ∈ I_o. Then, M˜e(fA)=M˜e(⊔{gB∈(X,E)˜:gB⊑fA,gB∈M˜e,r})≥⊓M˜e(gB)≥r. Hence, f_A ∈ M̃_e,r.

(2) From Theorem 3.1, it is sufficient to show that if C_m(e, f_A, r) = f_A, then fAc∈M˜e,r. Let C_m(e, f_A, r) = f_A, e ∈ E and r ∈ I_o. Then, from Theorem 3.1, it follows that fAc=(Cm(e,fA,r))c=Im(e,fAc,r). Hence by (1), fAc∈M˜e,r.

Lemma 3.2

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping and X have the property (P). Then, the following are equivalent: For fA∈(X,E)˜,gB∈(Y,F)˜, e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal continuous.
(2) (ϕψ-1(gB))c∈M˜e,r for every (gB)c∈M˜k,r*.
(3) ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).
(4) Cm(e,ϕψ-1(gB),r)⊑ϕψ-1(Cm*(k,gB,r)).
(5) ϕψ-1(Im*(k,gB,r))⊑Im(e,ϕψ-1(gB),r).

Proof

The proof follows from Theorems 3.2 and 3.3.

Definition 3.7

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, e ∈ E and r ∈ I_o. Then, ϕ_ψ is called

(1) fuzzy soft r-minimal open if ϕψ(fA)∈M˜k,r* for every f_A ∈ M̃_e,r, and

(2) fuzzy soft r-minimal closed if (ϕψ(fA))c∈M˜k,r* for every fAc∈M˜e,r.

Theorem 3.4

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, fA∈(X,E)˜,gB∈(Y,F)˜, e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal open.
(2) ϕ_ψ(I_m(e, f_A, r)) ⊑ I_m_^* (k,ϕ_ψ(f_A), r).
(3) Im(e,ϕψ-1(gB),r)⊑ϕψ-1(Im*(k,gB,r)).

Then, (1) ⇒ (2) ⇔ (3).

Proof

(1) ⇒ (2) For fA∈(X,E)˜;

ϕψ(Im(e,fA,r))=ϕψ(⊔{hC∈(X,E)˜:hC⊑fA,hC∈M˜e,r})⊑⊔{ϕψ(hC):ϕψ(hC)⊑ϕψ(fA),ϕψ(hC)∈M˜k,r*}Im*(k,ϕψ(fA),r).

Hence, ϕ_ψ(I_m(e, f_A, r)) ⊑ I_m_^* (k,ϕ_ψ(f_A), r).

(2) ⇔ (3) For gB∈(Y,F)˜, from (2), it follows that ϕψ(Im(e,ϕψ-1(gB),r))⊑Im*(k,ϕψ(ϕψ-1(gB)),r)⊑Im*(k,gB,r).

Hence, we obtain (3). Similarly, we obtain (3) ⇒ (2).

Theorem 3.5

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft r-minimal open. Then, ϕ_ψ(f_A) = I_m_^* (k,ϕ_ψ(f_A), r) for every f_A ∈ M̃_e,r.

Proof

From Theorem 3.4(2), the proof is obvious.

Lemma 3.3

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping and Y have the property (P). Then, the following are equivalent: For fA∈(X,E)˜,gB∈(Y,F)˜, e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal open.
(2) ϕ_ψ(I_m(e, f_A, r)) ⊑ I_m_^* (k,ϕ_ψ(f_A), r).
(3) Im(e,ϕψ-1(gB),r)⊑ϕψ-1(Im*(k,gB,r)).

Proof

The proof follows from Theorems 3.3 and 3.5.

Theorem 3.6

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, fA∈(X,E)˜,gB∈(Y,F)˜, e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal closed.
(2) ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).
(3) Cm(e,ϕψ-1(gB),r)⊑ϕψ-1(Cm*(k,gB,r)).

Then, (1) ⇒ (2) ⇔ (3).

Proof

(1) ⇒ (2) For fA∈(X,E)˜,

ϕψ(Cm(e,fA,r))=ϕψ(⊓{hC∈(X,E)˜:fA⊂hC,hCc∈M˜e,r})⊑⊓{ϕψ(hC):ϕψ(fA)⊑ϕψ(hC),(ϕψ(hC))c∈M˜k,r*}Cm*(k,ϕψ(fA).

Hence, ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).

(2) ⇔ (3) For gB∈(Y,F)˜, from (2), it follows that ϕψ(Cm(e,ϕψ-1(gB),r))⊑Cm*(k,ϕψ(ϕψ-1(gB)),r)⊑Cm*(k,gB,r). Hence, we obtain (3). Similarly, we obtain (3) ⇒ (2).

4. Several Types of Fuzzy Soft r-Minimal Compactness

Go to

Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

In this section, several types of fuzzy soft r-minimal compactness are defined, and the relationships between them are characterized.

Definition 4.1

Let (X, M̃) be an r-FMS˜,gB∈(X,E)˜, e ∈ E, and r ∈ I_o. Then, g_B is called a fuzzy soft r-minimal compact iff for every family {(fA)i∈(X,E)˜∣(fA)i∈M˜e,r}i∈Γ such that g_B ⊑⊔_i_∈Γ(f_A)_i, there exists a finite subset Γ_o of Γ such that g_B ⊑⊔_i_{∈Γ_o} (f_A)_i.

Theorem 4.1

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be fuzzy soft r-minimal continuous. If fA∈(X,E)˜ is fuzzy soft r-minimal compact, then ϕ_ψ(f_A) is fuzzy soft r-minimal compact.

Proof

Let {(gB)i∈(Y,F)˜∣(gB)i∈M˜k,r*}i∈Γ with ϕ_ψ(f_A) ⊑⊔_i_∈Γ(g_B)_i. Then, {ϕψ-1((gB)i)∈(X,E)˜∣ϕψ-1((gB)i)∈M˜e,r}i∈Γ (by ϕ_ψ is fuzzy soft r-minimal continuous) such that fA⊑⊔i∈Γϕψ-1((gB)i). Because f_A is fuzzy soft r-minimal compact, there exists a finite subset Γ_o of Γ such that fA⊑⊔i∈Γoϕψ-1((gB)i). Hence, ϕ_ψ(f_A) ⊑⊔_i_{∈Γ_o} (g_B)_i.

Definition 4.2

Let (X, M̃) be an r-FMS˜,gB∈(X,E)˜, e ∈ E, and r ∈ I_o. Then, g_B is called fuzzy soft r-minimal almost compact iff for every family {(fA)i∈(X,E)˜∣(fA)i∈M˜e,r}i∈Γ such that g_B ⊑⊔_i_∈Γ(f_A)_i, there exists a finite subset Γ_o of Γ such that g_B ⊑⊔_i_{∈Γ_o} C_m(e, (f_A)_i, r).

Lemma 4.1

Let (X, M̃) be an r-FMS˜. If fA∈(X,E)˜ is fuzzy soft r-minimal compact, it is then also almost fuzzy soft r-minimal compact.

Proof

Obvious.

Example 4.1

Let X = I, n ∈ N − {1}, and E = {e₁, e₂} be the parameter set of X. Define f_E_{_n} and gE1∈(X,E)˜ as follows ∀ e ∈ E:

fEn(e) (x)={0.8,ifx=0nx,if0<x≤1n,1,if1n<x≤1,gE1(e) (x)={1,if x=0,12,otherwise.

We define the fuzzy soft r-minimal structure M˜E:E→[0,1](X,E)˜ as follows: ∀ e ∈ E:

M˜e(wE)={45, if wE∈{Φ,E˜}nn+1, if wE≤fEn,23, if wE≤gE1,0, otherwise.

Then, X is almost fuzzy soft 12-minimal compact but not fuzzy soft 12-minimal compact.

Theorem 4.2

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be fuzzy soft r-minimal continuous. If fA∈(X,E)˜ is almost fuzzy soft r-minimal compact, ϕ_ψ(f_A) is almost fuzzy soft r-minimal compact.

Proof

Let {(gB)i∈(Y,F)˜∣(gB)i∈M˜k,r*}i∈Γ with ϕ_ψ(f_A) ⊑⊔_i_∈Γ(g_B)_i. Then, {ϕψ-1((gB)i)∈(X,E)˜∣ϕψ-1((gB)i)∈M˜e,r}i∈Γ (by ϕ_ψ is fuzzy soft r-minimal continuous) such that fA⊑⊔i∈Γϕψ-1((gB)i). Because f_A is almost fuzzy soft r-minimal compact, a finite subset Γ_o exists such that fA⊑⊔i∈ΓoCm(e,ϕψ-1((gB)i),r). From Theorem 3.2(4), it follows that

⊔i∈ΓoCm(e,ϕψ-1((gB)i),r)⊑⊔i∈Γoϕψ-1(Cm*(k,(gB)i,r)=ϕψ-1(⊔i∈ΓoCm*(k,(gB)i,r)).

Hence, ϕ_ψ(f_A) ⊑⊔_i∈Γ₀C_m_^* (k, (g_B)_i, r).

Definition 4.3

Let (X, M̃) be an r-FMS˜,gB∈(X,E)˜, e ∈ E, and r ∈ I_o. Then, g_B is called nearly fuzzy soft r-minimal compact iff for every family {(fA)i∈(X,E)˜∣(fA)i∈M˜e,r}i∈Γ such that g_B ⊑⊔_i_∈Γ(f_A)_i, there exists a finite subset Γ_o of Γ in which g_B⊑⊔_i_{∈Γ_o} I_m(e,C_m(e, (f_A)_i, r).

Lemma 4.2

Let (X, M̃) be an r-FMS˜. If fA∈(X,E)˜ is fuzzy soft r-minimal compact, then it is also nearly fuzzy soft r-minimal compact.

Proof

For any f_A ∈ M̃_e,r from Theorem 3.1, it follows that f_A = I_m(e, f_A, r) ⊑ I_m(e,C_m(e, f_A, r), r). Thus, we obtain the following result.

Example 4.2

Let X = I, 0 < n < 1 and E = {e₁, e₂} be the parameter set of X. Define f_E_{_n}, f_E, and gE∈(X,E)˜ as follows ∀ e ∈ E:

fEn(e) (x)={xn, if 0≤x≤n,1-x1-n, if n<x≤1,fE(e) (x)={1, if x=0,12, if 0<x≤1,gE(e) (x)={12, if 0≤x<1,1, if x=1.

We define the fuzzy soft r-minimal structure M˜E:E→[0,1](X,E)˜ as follows: ∀ e ∈ E:

M˜e(wE)={1, if we∈{fE,gE,Φ,E˜},max({1-n,n}), if wE=fEn,0, otherwise.

Then, X is nearly fuzzy soft 12-minimal compact but is not fully fuzzy soft 12-minimal compact.

Theorem 4.3

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be fuzzy soft r-minimal continuous and fuzzy soft r-minimal open mapping. If fA∈(X,E)˜ is nearly fuzzy soft r-minimal compact, then ϕ_ψ(f_A) is nearly fuzzy soft r-minimal compact.

Proof

Let {(gB)i∈(Y,F)˜∣(gB)i∈M˜k,r*}i∈Γ with ϕ_ψ(f_A) ⊑⊔_i_∈Γ(g_B)_i. Then, {ϕψ-1((gB)i)∈(X,E)˜∣ϕψ-1((gB)i)∈M˜e,r}i∈Γ (by ϕ_ψ is fuzzy soft r-minimal continuous) such that fA⊑⊔i∈Γϕψ-1((gB)i). Because f_A is nearly fuzzy soft r-minimal compact, there exists a finite subset Γ_o of Γ such that fA⊑⊔i∈ΓoIm(e,Cm(e,ϕψ-1((gB)i),r),r).

From Theorems 3.2, and 3.4, it follows that

ϕψ(fA)⊑⊔i∈Γoϕψ(Im(e,Cm(e,ϕψ-1((gB)i),r),r))⊑⊔i∈ΓoIm*(k,ϕψ(Cm(e,ϕψ-1((gB)i),r)),r)⊑⊔i∈ΓoIm*(k,ϕψ(ϕψ-1(Cm*(k,(gB)i,r))),r)⊑⊔i∈ΓoIm*(k,Cm*(k,(gB)i,r),r).

Hence

ϕψ(fA)⊑⊔i∈ΓoIm*(k,Cm*(k,(gB)i,r),r).

5. Conclusion

Go to

Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

Many researchers have studied fuzzy soft set theory, which is easily applied to many problems having uncertainties from social life. In this paper, we introduced and explored a new form of fuzzy r-minimal structure introduced by Yoo et al. [28] called a fuzzy soft r-minimal structure, which is an extension of the fuzzy soft topology introduced by Aygunoglu et al. [26]. First, the concepts of fuzzy soft r-minimal continuity and fuzzy soft r-minimal compactness were introduced and studied in fuzzy soft r-minimal spaces. Finally, several types of fuzzy soft r-minimal compactness were defined, and the following implications were found to hold: fuzzy soft r-minimal compact ⇒ fuzzy soft r-minimal nearly compact ⇒ fuzzy soft r-minimal almost compact. We hope that the findings of this study will help researchers enhance and promote further studies on fuzzy soft r-minimal spaces and carry out a general framework for their applications in other scientific areas.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

No potential conflicts of interest relevant to this article are reported.

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

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Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Ahmad, B, and Kharal, A (2009). On fuzzy soft sets. Advances in Fuzzy Systems. 2009. article no. 586507
Aktas, H, and Cagman, N (2007). Soft sets and soft groups. Information Sciences. 177, 2726-2735. https://doi.org/10.1016/j.ins.2006.12.008
Al-Shami, TM (2020). Comments on some results related to soft separation axioms. Afrika Matematika. 31, 1105-1119. https://doi.org/10.1007/s13370-020-00783-4
Al-Shami, TM (2021). Compactness on soft topological ordered spaces and its application on the information system. Journal of Mathematics. 2021. article no. 6699092
Al-shami, TM (2021). On soft separation axioms and their applications on decision-making problem. Mathematical Problems in Engineering. 2021. article no. 8876978
Atmaca, S, and Zorlutuna, I (2013). On fuzzy soft topological spaces. Annals of Fuzzy Mathematics and Informatics. 5, 377-386.
Aygunoglu, A, and Aygun, H (2009). Introduction to fuzzy soft groups. Computers & Mathematics with Applications. 58, 1279-1286. https://doi.org/10.1016/j.camwa.2009.07.047
Cagman, N, Enginoglu, S, and Citak, F (2011). Fuzzy soft set theory and its applications. Iranian Journal of Fuzzy Systems. 8, 137-147. https://dx.doi.org/10.22111/ijfs.2011.292
Cetkin, V, and Aygun, H (2014). Fuzzy soft semiregularization spaces. Annals of Fuzzy Mathematics and Informatics. 7, 687-697.
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Hussain, S, and Ahmad, B (2015). Soft separation axioms in soft topological spaces. Hacettepe Journal of Mathematics and Statistics. 44, 559-568.
Jun, YB (2008). Soft bck/bci-algebras. Computers & Mathematics with Applications. 56, 1408-1413. https://doi.org/10.1016/j.camwa.2008.02.035
Kharal, A, and Ahmad, B (2009). Mappings on fuzzy soft classes. Advances in Fuzzy Systems. 2009. article no. 407890
Park, JH, Kwun, YC, and Son, MJ (2011). A generalized intuitionistic fuzzy soft set theoretic approach to decision making problems. International Journal of Fuzzy Logic and Intelligent Systems. 11, 71-76. https://doi.org/10.5391/IJFIS.2011.11.2.071
Park, JH (2011). Operations on generalized intuitionistic fuzzy soft sets. International Journal of Fuzzy Logic and Intelligent Systems. 11, 184-189. https://doi.org/10.5391/IJFIS.2011.11.3.184
Riaz, M, Davvaz, B, Fakhar, A, and Firdous, A (2020). Hesitant fuzzy soft topology and its applications to multi-attribute group decision-making. Soft Computing. 24, 16269-16289. https://doi.org/10.1007/s00500-020-04938-0
Riaz, M, and Tehrim, ST (2020). On bipolar fuzzy soft topology with decision-making. Soft Computing. 24, 18259-18272. https://doi.org/10.1007/s00500-020-05342-4
Riaz, M, Karaaslan, F, Nawaz, I, and Sohail, M (2021). Soft multi-rough set topology with applications to multi-criteria decision-making problems. Soft Computing. 25, 799-815. https://doi.org/10.1007/s00500-020-05382-w
Roy, AR, and Maji, PK (2007). A fuzzy soft set theoretic approach to decision making problems. Journal of Computational and Applied Mathematics. 203, 412-418. https://doi.org/10.1016/j.cam.2006.04.008
Shabir, M, and Naz, M (2011). On soft topological spaces. Computers & Mathematics with Applications. 61, 1786-1799. https://doi.org/10.1016/j.camwa.2011.02.006
Varol, BP, and Aygun, H (2012). Fuzzy soft topology. Hacettepe Journal of Mathematics and Statistics. 41, 407-419.
Zorlutuna, I, Akdag, M, Min, WK, and Atmaca, S (2012). Remarks on soft topological spaces. Annals of Fuzzy Mathematics and Informatics. 3, 171-185.
Aygunoglu, A, Cetkin, V, and Aygun, H (2014). An introduction to fuzzy soft topological spaces. Hacettepe Journal of Mathematics and Statistics. 43, 197-208.
Sostak, AP . On a fuzzy topological structure., Proceedings of the 13th Winter School on Abstract Analysis (Circolo Matematico di Palermo), 1985, Srni, Czech Republic, pp.89-103.
Yoo, YH, Min, WK, and Kim, JI (2009). Fuzzy r-minimal structures and fuzzy r-minimal spaces. Far East Journal of Mathematical Sciences. 33, 193-205. https://doi.org/10.5391/IJFIS.2009.9.4.281
Kim, JI, Min, WK, and Yoo, YH (2009). Fuzzy r-compactness on fuzzy r-minimal spaces. International Journal of Fuzzy Logic and Intelligent Systems. 9, 281-284. https://doi.org/10.5391/IJFIS.2009.9.4.281
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Biography

Go to

Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion
Conflict of Interest
References
Biography

Islam M. Taha is an Assistant Professor of Mathematics at the Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia, Egypt. He has published original articles in the finest journal in the area of his study. His research interests include general topology, fuzzy topology and soft topology.

E-mail: imtaha2010@yahoo.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 251-258

Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.251

Compactness on Fuzzy Soft r-Minimal Spaces

Islam M. Taha^1,2

¹Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia, Egypt
²Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

Correspondence to:Islam M. Taha (imtaha2010@yahoo.com)

Received: February 10, 2021; Revised: June 5, 2021; Accepted: June 25, 2021

Abstract

Keywords: Fuzzy soft set, Fuzzy soft topology, Fuzzy soft r-minimal structure, Fuzzy soft r-minimal space, Fuzzy soft r-minimal continuity, Fuzzy soft r-minimal compactness

1. Introduction

2. Preliminaries

Definition 2.1 [2,4,26]

A fuzzy soft set f_A over X is a mapping from E to I^X such that f_A(e) is a fuzzy set on X for each e ∈ A and f_A(e) = 0, if e ∉ A, where 0 is a zero function on X. The fuzzy set f_A(e) for each e ∈ E is called an element of the fuzzy soft set f_A. Here, $\tilde{(X, E)}$ denotes the collection of all fuzzy soft sets on X and is called a fuzzy soft universe [32].

Definition 2.2 [33]

Definition 2.3 [26]

A mapping $τ : E \to {[0, 1]}^{\tilde{(X, E)}}$ is called a fuzzy soft topology on X if it satisfies the following conditions for each e ∈ E.

τ_e(Φ) = τ_e( Ẽ) = 1.
$τ_{e} (f_{A} ⊓ g_{B}) \geq τ_{e} (f_{A}) \land τ_{e} (g_{B}), \forall f_{A}, g_{B} \in \tilde{(X, E)}$ .
$τ_{e} (⊔_{i \in Δ} {(f_{A})}_{i}) \geq \land_{i \in Δ} τ_{e} ({(f_{A})}_{i}), \forall {(f_{A})}_{i} \in \tilde{(X, E)}, i \in Δ$ .

Then, the pair (X, τ_E) is called a fuzzy soft topological space (FSTS).

All definitions and properties of fuzzy soft sets and fuzzy soft topologies are found in [2,4,12,16,24,26].

3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets

Definition 3.1

Let X be a nonempty set and r ∈ I₀. Fuzzy soft mapping $\tilde{M} : E \to {[0, 1]}^{\tilde{(X, E)}}$ on X is said to be a fuzzy soft r-minimal structure if the family ${\tilde{M}}_{e, r} = {f_{A} \in \tilde{(X, E)} ∣ {\tilde{M}}_{e} (f_{A}) \geq r}$ for each e ∈ E contains Φ and Ẽ. Then, (X, M̃) is called a fuzzy soft r-minimal space (simply, $r - \tilde{F M S}$ ). Every member of M̃_e,r is called a fuzzy soft r-minimal open set. A fuzzy soft set f_A is called a fuzzy soft r-minimal closed set if the complement of f_A is a fuzzy soft r-minimal open set.

Example 3.1

Let (X, τ_E) be an FSTS. It is then easy to see that for each e ∈ E and r ∈ I₀, a fuzzy soft mapping M̃ defined by ${\tilde{M}}_{e, r} = {f_{A} \in \tilde{(X, E)} ∣ τ_{e} (f_{A}) \geq r}$ is a fuzzy soft r-minimal structure on X.

Definition 3.2. [12]

Let (X, τ_E) be an FSTS. For $f_{A} \in \tilde{(X, E)}$ and r ∈ I₀, f_A is called r-fuzzy soft regularly open if f_A = I_τ (e,C_τ (e, f_A, r), r).

Definition 3.3

Let (X, τ_E) be an FSTS. For $f_{A} \in \tilde{(X, E)}$ and r ∈ I₀.

(1) f_A is called r-fuzzy soft preopen if f_A ⊑ I_τ (e, C_τ (e, f_A, r), r).

(2) f_A is called r-fuzzy soft β-open if f_A ⊑ C_τ (e, I_τ (e, C_τ (e, f_A, r), r).

The following implications hold:

r-fuzzy soft regularly open ⇒ r-fuzzy soft preopen ⇒ r-fuzzy soft β-open.

In general, the converses are not true.

Example 3.2

Let X = {x, y} be a classical set and E = {e₁, e₂} be the parameter set of X. Define f_E and $g_{E} \in \tilde{(X, E)}$ as follows: $f_{E} = {(e_{1}, {\frac{x}{0.3}, \frac{y}{0.4}}), (e_{2}, {\frac{x}{0.3}, \frac{y}{0.4}})}, g_{E} = {(e_{1}, {\frac{x}{0.6}, \frac{y}{0.2}}), (e_{2}, {\frac{x}{0.6}, \frac{y}{0.2}})}$ . The fuzzy soft topology $τ_{E} : E \to {[0, 1]}^{\tilde{(X, E)}}$ is as follows:

\begin{array}{l} τ_{e_{1}} (w_{E}) = {\begin{array}{l} 1, & if w_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, & if w_{E} = f_{E}, \\ \frac{1}{3}, & if w_{E} = g_{E}, \\ \frac{1}{3}, & if w_{E} = f_{E} ⊓ g_{E}, \\ \frac{1}{4}, & if w_{E} = f_{E} ⊔ g_{E}, \\ 0, & otherwise, \end{array} \\ τ_{e_{2}} (w_{E}) = {\begin{array}{l} 1, & if w_{E} \in {Φ, \tilde{E}}, \\ \frac{1}{4}, & if w_{E} = f_{E}, \\ \frac{1}{2}, & if w_{E} = g_{E}, \\ \frac{1}{2}, & if w_{E} = f_{E} ⊓ g_{E}, \\ \frac{1}{4}, & if w_{E} = f_{E} ⊔ g_{E}, \\ 0, & otherwise . \end{array} \end{array}

Then, g_E is a $\frac{1}{4}$ -fuzzy soft preopen set and $\frac{1}{4}$ -fuzzy soft β-open set but is not a $\frac{1}{4}$ -fuzzy soft regularly open set.

Remark 1

Let (X, τ_E) be an FSTS and r ∈ I₀. We denote the set of all r-fuzzy soft regularly open (respectively, r-fuzzy soft preopen and r-fuzzy soft β-open) sets by $r - \tilde{F R O (X)}$ (respectively, $r - \tilde{F P O (X)}$ and $r - \tilde{F β O (X)}$ . Then, $r - \tilde{F R O (X)}, r - \tilde{F P O (X)}$ , and $r - \tilde{F β O (X)}$ are all fuzzy soft r-minimal structures.

Definition 3.4

Let (X, M̃) be an $r - \tilde{F M S}$ , e ∈ E, and r ∈ I₀. The fuzzy soft r-minimal interior and fuzzy soft r-minimal closure of f_A, denoted by I_m(e, f_A, r) and C_m(e, f_A, r), respectively, are defined as $I_{m} (e, f_{A}, r) = ⊔ {g_{B} \in (\tilde{X, E}) : g_{B} ⊑ f_{A}, g_{B} \in {\tilde{M}}_{e, r}}; C_{m} (e, f_{A}, r) = ⊓ {g_{B} \in \tilde{(X, E)} : f_{A} ⊑ g_{B}, g_{B}^{c} \in {\tilde{M}}_{e, r}}$ .

Theorem 3.1

Let (X, M̃) be an $r - \tilde{F M S}$ and $f_{A}, g_{B} \in \tilde{(X, E)}$ . Then, the following statements hold:

(1) I_m(e, f_A, r) ⊑ f_A and if f_A ∈ M̃_e,_r, I_m(e, f_A, r) = f_A.

(2) f_A ⊑ C_m(e, f_A, r) and if $f_{A}^{c} \in {\tilde{M}}_{e, r}$ , C_m(e, f_A, r) = f_A.

(3) If f_A ⊑ g_B, I_m(e, f_A, r) ⊑ I_m(e, g_B, r) and C_m(e, f_A, r) ⊑ C_m(e, g_B, r).

(4) I_m(e, f_A ⊓ g_B, r) = I_m(e, f_A, r) ⊓ I_m(e, g_B, r).

(5) C_m(e, f_A ⊔ g_B, r) = C_m(e, f_A, r) ⊔ C_m(e, g_B, r).

(6) I_m(e, I_m(e, f_A, r), r) = I_m(e, f_A, r) and C_m(e, C_m(e, f_A, r), r) = C_m(e, f_A, r).

(7) ${(C_{m} (e, f_{A}, r))}^{c} = I_{m} (e, f_{A}^{c}, r)$ and ${(I_{m} (e, f_{A}, r))}^{c} = C_{m} (e, f_{A}^{c}, r)$ .

Proof

We prove (5) only because the others are obvious. For $g_{B} \in \tilde{(X, E)}$ , because C_m(e, f_A, r) ⊑ C_m(e, f_A ⊔ g_B, r) and C_m(e, g_B, r) ⊑ C_m(e, f_A ⊔ g_B, r), we obtain C_m(e, f_A, r) ⊔ C_m(e, g_B, r) ⊑ C_m(e, f_A ⊔ g_B, r). For $h_{C}^{c}, k_{D}^{c} \in M_{e, r}, h_{C}^{c} ⊓ k_{D}^{c}$ does not always belong to M̃_e,r. Thus, we obtain the following:

\begin{array}{l} C_{m} (e, f_{A}, r) ⊔ C_{m} (e, g_{B}, r) \\ = (⊓ {h_{C} \in \tilde{(X, E)} ∣ h_{C}^{c} \in {\tilde{M}}_{e, r}, f_{A} ⊑ h_{C}}) ⊔ (⊓ {k_{D} \in \tilde{(X, E)} ∣ k_{D}^{c} \in {\tilde{M}}_{e, r}, g_{B} ⊑ k_{D}}) \\ = ⊓ {h_{C} ⊔ k_{D} \in \tilde{(X, E)} ∣ f_{A} ⊑ h_{C}, g_{B} ⊑ k_{D}} \\ ⊒ ⊓ {w_{E} \in \tilde{(X, E)} ∣ w_{E}^{c} \in {\tilde{M}}_{e, r}, f_{A} ⊔ g_{B} ⊑ w_{E}} \\ = C_{m} (e, f_{A} ⊔ g_{B}, r) . \end{array}

Hence, we obtain C_m(e, f_A ⊔ g_B, r) = C_m(e, f_A, r) ⊔ C_m(e, g_B, r).

Definition 3.5

Let (X, M̃) and (Y, M̃^*) be an $r - \tilde{F M S}$ . Then, a fuzzy soft mapping ϕ_ψ from $\tilde{(X, E)}$ into $\tilde{(Y, F)}$ is called fuzzy soft r-minimal continuous if $ϕ_{ψ}^{- 1} (g_{B}) \in {\tilde{M}}_{e, r}$ for every $g_{B} \in {\tilde{M}}_{k, r}^{*}$ , e ∈ E, and (ψ(e) = k) ∈ F.

Theorem 3.2

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, $f_{A} \in \tilde{(X, E)}, g_{B} \in \tilde{(Y, F)}$ , e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal continuous.

(2) ${(ϕ_{ψ}^{- 1} (g_{B}))}^{c} \in {\tilde{M}}_{e, r}$ for every $g_{B}^{c} \in {\tilde{M}}_{k, r}^{*}$ .

(3) ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).

(4) $C_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r) ⊑ ϕ_{ψ}^{- 1} (C_{m^{*}} (k, g_{B}, r))$ .

(5) $ϕ_{ψ}^{- 1} (I_{m^{*}} (k, g_{B}, r)) ⊑ I_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)$ .

(6) If for every fuzzy soft point e_x_{_t} over X and each $g_{B} \in {\tilde{M}}_{k, r}^{*}$ containing ϕ_ψ(e_x_{_t} ), there exists f_A ∈ M̃_e,r containing e_x_{_t} such that ϕ_ψ(f_A) ⊑ g_B.

Then, (1) ⇔ (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇒ (6).

Proof

(1) ⇔ (2) Obvious.

(2) ⇒ (3) For all $f_{A} \in \tilde{(X, E)}$ ,

\begin{array}{l} ϕ_{ψ}^{- 1} (C_{m^{*}} (k, ϕ_{ψ} (f_{A}), r)) \\ = ϕ_{ψ}^{- 1} (⊓ {g_{B} \in \tilde{(Y, F)} ∣ g_{B}^{c} \in {\tilde{M}}_{k, r}^{*}, ϕ_{ψ} (f_{A}) ⊑ g_{B}}) \\ = ⊓ {ϕ_{ψ}^{- 1} (g_{B}) \in \tilde{(X, E)} ∣ g_{B}^{c} \in {\tilde{M}}_{k, r}^{*}, f_{A} ⊑ ϕ_{ψ}^{- 1} (g_{B})} \\ ⊒ ⊓ {ϕ_{ψ}^{- 1} (g_{B}) ∣ {(ϕ_{ψ}^{- 1} (g_{B}))}^{c} \in {\tilde{M}}_{e, r} f_{A} ⊑ ϕ_{ψ}^{- 1} (g_{B})} \\ = C_{m} (e, f_{A}, r) . \end{array}

Hence ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).

(3) ⇔ (4) Let $g_{B} \in \tilde{(Y, F)}$ . Then, from (3), it follows that $ϕ_{ψ} (C_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)) ⊑ C_{m^{*}} (k, ϕ_{ψ} (ϕ_{ψ}^{- 1} (g_{B})), r) ⊑ C_{m^{*}} (k, g_{B}, r)$ . Hence, we obtain Equation (4). Similarly, we obtain (4) ⇒ (3).

(4) ⇔ (5) From (7) of Theorem 3.1.

(5) ⇒ (6) Let $e_{x_{t}} \in \tilde{P_{t} (X)}$ and $g_{B} \in {\tilde{M}}_{k, r}^{*}$ with ϕ_ψ(e_x_{_t} )∊̃g_B. From (5), it follows that $e_{x_{t}} \tilde{\in} I_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)$ . Thus, there exists f_A ∈ M̃_e,r such that $e_{x_{t}} \tilde{\in} f_{A} ⊑ ϕ_{ψ}^{- 1} (g_{B})$ . Hence, we obtain (6).

Lemma 3.1

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft r-minimal continuous mapping. The following are then equivalent: For e ∈ E and r ∈ I_o,

(1) ϕ_ψ(f_A) ⊑ C_m_^* (k,ϕ_ψ(f_A), r) for each $f_{A}^{c} \in {\tilde{M}}_{e, r}$ ;
(2) $ϕ_{ψ}^{- 1} (g_{B}) = C_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)$ for each $g_{B}^{c} \in {\tilde{M}}_{k, r}^{*}$ ;
(3) $ϕ_{ψ}^{- 1} (g_{B}) = I_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)$ for each $g_{B} \in {\tilde{M}}_{k, r}^{*}$ .

Proof

The proof is obvious.

Definition 3.6

Let X be a nonempty set and $\tilde{M} : E \to {[0, 1]}^{\tilde{(X, E)}}$ . Then, M̃ is said to have a property (P) if for ${(f_{A})}_{j} \in \tilde{(X, E)}$ , j ∈ J, and e ∈ E

{\tilde{M}}_{e} (⊔_{j \in J} {(f_{A})}_{j}) \geq ⊓_{j \in J} {\tilde{M}}_{e} ({(f_{A})}_{j}) .

Theorem 3.3

Let (X, M̃) be an $r - \tilde{F M S}$ with the property (P). Then, the following statements hold: (1) I_m(e, f_A, r) = f_A iff f_A ∈ M̃_e,r for $f_{A} \in \tilde{(X, E)}$ , and (2) C_m(e, f_A, r) = f_A if $f_{A}^{c} \in {\tilde{M}}_{e, r}$ for $f_{A} \in \tilde{(X, E)}$ .

Proof

(1) From Theorem 3.1, it is sufficient to show that if I_m(e, f_A, r) = f_A then f_A ∈ M̃_e,r. Let I_m(e, f_A, r) = f_A, e ∈ E, and r ∈ I_o. Then, ${\tilde{M}}_{e} (f_{A}) = {\tilde{M}}_{e} (⊔ {g_{B} \in \tilde{(X, E)} : g_{B} ⊑ f_{A}, g_{B} \in {\tilde{M}}_{e, r}}) \geq ⊓ {\tilde{M}}_{e} (g_{B}) \geq r$ . Hence, f_A ∈ M̃_e,r.

(2) From Theorem 3.1, it is sufficient to show that if C_m(e, f_A, r) = f_A, then $f_{A}^{c} \in {\tilde{M}}_{e, r}$ . Let C_m(e, f_A, r) = f_A, e ∈ E and r ∈ I_o. Then, from Theorem 3.1, it follows that $f_{A}^{c} = {(C_{m} (e, f_{A}, r))}^{c} = I_{m} (e, f_{A}^{c}, r)$ . Hence by (1), $f_{A}^{c} \in {\tilde{M}}_{e, r}$ .

Lemma 3.2

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping and X have the property (P). Then, the following are equivalent: For $f_{A} \in \tilde{(X, E)}, g_{B} \in \tilde{(Y, F)}$ , e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal continuous.
(2) ${(ϕ_{ψ}^{- 1} (g_{B}))}^{c} \in {\tilde{M}}_{e, r}$ for every ${(g_{B})}^{c} \in {\tilde{M}}_{k, r}^{*}$ .
(3) ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).
(4) $C_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r) ⊑ ϕ_{ψ}^{- 1} (C_{m^{*}} (k, g_{B}, r))$ .
(5) $ϕ_{ψ}^{- 1} (I_{m^{*}} (k, g_{B}, r)) ⊑ I_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)$ .

Proof

The proof follows from Theorems 3.2 and 3.3.

Definition 3.7

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, e ∈ E and r ∈ I_o. Then, ϕ_ψ is called

(1) fuzzy soft r-minimal open if $ϕ_{ψ} (f_{A}) \in {\tilde{M}}_{k, r}^{*}$ for every f_A ∈ M̃_e,r, and

(2) fuzzy soft r-minimal closed if ${(ϕ_{ψ} (f_{A}))}^{c} \in {\tilde{M}}_{k, r}^{*}$ for every $f_{A}^{c} \in {\tilde{M}}_{e, r}$ .

Theorem 3.4

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, $f_{A} \in \tilde{(X, E)}, g_{B} \in \tilde{(Y, F)}$ , e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal open.
(2) ϕ_ψ(I_m(e, f_A, r)) ⊑ I_m_^* (k,ϕ_ψ(f_A), r).
(3) $I_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r) ⊑ ϕ_{ψ}^{- 1} (I_{m^{*}} (k, g_{B}, r))$ .

Then, (1) ⇒ (2) ⇔ (3).

Proof

(1) ⇒ (2) For $f_{A} \in \tilde{(X, E)}$ ;

\begin{array}{l} ϕ_{ψ} (I_{m} (e, f_{A}, r)) \\ = ϕ_{ψ} (⊔ {h_{C} \in \tilde{(X, E)} : h_{C} ⊑ f_{A}, h_{C} \in {\tilde{M}}_{e, r}}) \\ ⊑ ⊔ {ϕ_{ψ} (h_{C}) : ϕ_{ψ} (h_{C}) ⊑ ϕ_{ψ} (f_{A}), ϕ_{ψ} (h_{C}) \in {\tilde{M}}_{k, r}^{*}} \\ I_{m^{*}} (k, ϕ_{ψ} (f_{A}), r) . \end{array}

Hence, ϕ_ψ(I_m(e, f_A, r)) ⊑ I_m_^* (k,ϕ_ψ(f_A), r).

(2) ⇔ (3) For $g_{B} \in \tilde{(Y, F)}$ , from (2), it follows that $ϕ_{ψ} (I_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)) ⊑ I_{m^{*}} (k, ϕ_{ψ} (ϕ_{ψ}^{- 1} (g_{B})), r) ⊑ I_{m^{*}} (k, g_{B}, r)$ .

Hence, we obtain (3). Similarly, we obtain (3) ⇒ (2).

Theorem 3.5

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft r-minimal open. Then, ϕ_ψ(f_A) = I_m_^* (k,ϕ_ψ(f_A), r) for every f_A ∈ M̃_e,r.

Proof

From Theorem 3.4(2), the proof is obvious.

Lemma 3.3

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping and Y have the property (P). Then, the following are equivalent: For $f_{A} \in \tilde{(X, E)}, g_{B} \in \tilde{(Y, F)}$ , e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal open.
(2) ϕ_ψ(I_m(e, f_A, r)) ⊑ I_m_^* (k,ϕ_ψ(f_A), r).
(3) $I_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r) ⊑ ϕ_{ψ}^{- 1} (I_{m^{*}} (k, g_{B}, r))$ .

Proof

The proof follows from Theorems 3.3 and 3.5.

Theorem 3.6

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be a fuzzy soft mapping, $f_{A} \in \tilde{(X, E)}, g_{B} \in \tilde{(Y, F)}$ , e ∈ E, and r ∈ I_o.

(1) ϕ_ψ is fuzzy soft r-minimal closed.
(2) ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).
(3) $C_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r) ⊑ ϕ_{ψ}^{- 1} (C_{m^{*}} (k, g_{B}, r))$ .

Then, (1) ⇒ (2) ⇔ (3).

Proof

(1) ⇒ (2) For $f_{A} \in \tilde{(X, E)}$ ,

\begin{array}{l} ϕ_{ψ} (C_{m} (e, f_{A}, r)) \\ = ϕ_{ψ} (⊓ {h_{C} \in \tilde{(X, E)} : f_{A} \subset h_{C}, h_{C}^{c} \in {\tilde{M}}_{e, r}}) \\ ⊑ ⊓ {ϕ_{ψ} (h_{C}) : ϕ_{ψ} (f_{A}) ⊑ ϕ_{ψ} (h_{C}), {(ϕ_{ψ} (h_{C}))}^{c} \in {\tilde{M}}_{k, r}^{*}} \\ C_{m^{*}} (k, ϕ_{ψ} (f_{A}) . \end{array}

Hence, ϕ_ψ(C_m(e, f_A, r)) ⊑ C_m_^* (k,ϕ_ψ(f_A), r).

(2) ⇔ (3) For $g_{B} \in \tilde{(Y, F)}$ , from (2), it follows that $ϕ_{ψ} (C_{m} (e, ϕ_{ψ}^{- 1} (g_{B}), r)) ⊑ C_{m^{*}} (k, ϕ_{ψ} (ϕ_{ψ}^{- 1} (g_{B})), r) ⊑ C_{m^{*}} (k, g_{B}, r)$ . Hence, we obtain (3). Similarly, we obtain (3) ⇒ (2).

4. Several Types of Fuzzy Soft r-Minimal Compactness

In this section, several types of fuzzy soft r-minimal compactness are defined, and the relationships between them are characterized.

Definition 4.1

Let (X, M̃) be an $r - \tilde{F M S}, g_{B} \in \tilde{(X, E)}$ , e ∈ E, and r ∈ I_o. Then, g_B is called a fuzzy soft r-minimal compact iff for every family ${{(f_{A})}_{i} \in \tilde{(X, E)} ∣ {(f_{A})}_{i} \in {\tilde{M}}_{e, r}}_{i \in Γ}$ such that g_B ⊑⊔_i_∈Γ(f_A)_i, there exists a finite subset Γ_o of Γ such that g_B ⊑⊔_i_{∈Γ_o} (f_A)_i.

Theorem 4.1

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be fuzzy soft r-minimal continuous. If $f_{A} \in \tilde{(X, E)}$ is fuzzy soft r-minimal compact, then ϕ_ψ(f_A) is fuzzy soft r-minimal compact.

Proof

Let ${{(g_{B})}_{i} \in \tilde{(Y, F)} ∣ {(g_{B})}_{i} \in {\tilde{M}}_{k, r}^{*}}_{i \in Γ}$ with ϕ_ψ(f_A) ⊑⊔_i_∈Γ(g_B)_i. Then, ${ϕ_{ψ}^{- 1} ({(g_{B})}_{i}) \in \tilde{(X, E)} ∣ ϕ_{ψ}^{- 1} ({(g_{B})}_{i}) \in {\tilde{M}}_{e, r}}_{i \in Γ}$ (by ϕ_ψ is fuzzy soft r-minimal continuous) such that $f_{A} ⊑ ⊔_{i \in Γ} ϕ_{ψ}^{- 1} ({(g_{B})}_{i})$ . Because f_A is fuzzy soft r-minimal compact, there exists a finite subset Γ_o of Γ such that $f_{A} ⊑ ⊔_{i \in Γ_{o}} ϕ_{ψ}^{- 1} ({(g_{B})}_{i})$ . Hence, ϕ_ψ(f_A) ⊑⊔_i_{∈Γ_o} (g_B)_i.

Definition 4.2

Let (X, M̃) be an $r - \tilde{F M S}, g_{B} \in \tilde{(X, E)}$ , e ∈ E, and r ∈ I_o. Then, g_B is called fuzzy soft r-minimal almost compact iff for every family ${{(f_{A})}_{i} \in \tilde{(X, E)} ∣ {(f_{A})}_{i} \in {\tilde{M}}_{e, r}}_{i \in Γ}$ such that g_B ⊑⊔_i_∈Γ(f_A)_i, there exists a finite subset Γ_o of Γ such that g_B ⊑⊔_i_{∈Γ_o} C_m(e, (f_A)_i, r).

Lemma 4.1

Let (X, M̃) be an $r - \tilde{F M S}$ . If $f_{A} \in \tilde{(X, E)}$ is fuzzy soft r-minimal compact, it is then also almost fuzzy soft r-minimal compact.

Proof

Obvious.

Example 4.1

Let X = I, n ∈ N − {1}, and E = {e₁, e₂} be the parameter set of X. Define f_E_{_n} and $g_{E_{1}} \in \tilde{(X, E)}$ as follows ∀ e ∈ E:

\begin{array}{l} f_{E_{n}} (e) (x) = {\begin{array}{l} 0.8, & if & x = 0 \\ n x, & if & 0 < x \leq \frac{1}{n}, \\ 1, & if & \frac{1}{n} < x \leq 1, \end{array} \\ g_{E_{1}} (e) (x) = {\begin{array}{l} 1, & if x = 0, \\ \frac{1}{2}, & otherwise . \end{array} \end{array}

We define the fuzzy soft r-minimal structure ${\tilde{M}}_{E} : E \to {[0, 1]}^{\tilde{(X, E)}}$ as follows: ∀ e ∈ E:

{\tilde{M}}_{e} (w_{E}) = {\begin{array}{l} \frac{4}{5}, if w_{E} \in {Φ, \tilde{E}} \\ \frac{n}{n + 1}, if w_{E} \leq f_{E_{n}}, \\ \frac{2}{3}, i f w_{E} \leq g_{E_{1}}, \\ 0, otherwise . \end{array}

Then, X is almost fuzzy soft $\frac{1}{2}$ -minimal compact but not fuzzy soft $\frac{1}{2}$ -minimal compact.

Theorem 4.2

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be fuzzy soft r-minimal continuous. If $f_{A} \in \tilde{(X, E)}$ is almost fuzzy soft r-minimal compact, ϕ_ψ(f_A) is almost fuzzy soft r-minimal compact.

Proof

Let ${{(g_{B})}_{i} \in \tilde{(Y, F)} ∣ {(g_{B})}_{i} \in {\tilde{M}}_{k, r}^{*}}_{i \in Γ}$ with ϕ_ψ(f_A) ⊑⊔_i_∈Γ(g_B)_i. Then, ${ϕ_{ψ}^{- 1} ({(g_{B})}_{i}) \in \tilde{(X, E)} ∣ ϕ_{ψ}^{- 1} ({(g_{B})}_{i}) \in {\tilde{M}}_{e, r}}_{i \in Γ}$ (by ϕ_ψ is fuzzy soft r-minimal continuous) such that $f_{A} ⊑ ⊔_{i \in Γ} ϕ_{ψ}^{- 1} ({(g_{B})}_{i})$ . Because f_A is almost fuzzy soft r-minimal compact, a finite subset Γ_o exists such that $f_{A} ⊑ ⊔_{i \in Γ_{o}} C_{m} (e, ϕ_{ψ}^{- 1} ({(g_{B})}_{i}), r)$ . From Theorem 3.2(4), it follows that

\begin{array}{l} \underset{i \in Γ_{o}}{⊔} C_{m} (e, ϕ_{ψ}^{- 1} ({(g_{B})}_{i}), r) \\ ⊑ \underset{i \in Γ_{o}}{⊔} ϕ_{ψ}^{- 1} (C_{m^{*}} (k, {(g_{B})}_{i}, r) \\ = ϕ_{ψ}^{- 1} (\underset{i \in Γ_{o}}{⊔} C_{m^{*}} (k, {(g_{B})}_{i}, r)) . \end{array}

Hence, ϕ_ψ(f_A) ⊑⊔_i∈Γ₀C_m_^* (k, (g_B)_i, r).

Definition 4.3

Let (X, M̃) be an $r - \tilde{F M S}, g_{B} \in \tilde{(X, E)}$ , e ∈ E, and r ∈ I_o. Then, g_B is called nearly fuzzy soft r-minimal compact iff for every family ${{(f_{A})}_{i} \in \tilde{(X, E)} ∣ {(f_{A})}_{i} \in {\tilde{M}}_{e, r}}_{i \in Γ}$ such that g_B ⊑⊔_i_∈Γ(f_A)_i, there exists a finite subset Γ_o of Γ in which g_B⊑⊔_i_{∈Γ_o} I_m(e,C_m(e, (f_A)_i, r).

Lemma 4.2

Let (X, M̃) be an $r - \tilde{F M S}$ . If $f_{A} \in \tilde{(X, E)}$ is fuzzy soft r-minimal compact, then it is also nearly fuzzy soft r-minimal compact.

Proof

For any f_A ∈ M̃_e,r from Theorem 3.1, it follows that f_A = I_m(e, f_A, r) ⊑ I_m(e,C_m(e, f_A, r), r). Thus, we obtain the following result.

Example 4.2

Let X = I, 0 < n < 1 and E = {e₁, e₂} be the parameter set of X. Define f_E_{_n}, f_E, and $g_{E} \in \tilde{(X, E)}$ as follows ∀ e ∈ E:

\begin{array}{l} f_{E_{n}} (e) (x) = {\begin{array}{l} \frac{x}{n}, if 0 \leq x \leq n, \\ \frac{1 - x}{1 - n}, if n < x \leq 1, \end{array} \\ f_{E} (e) (x) = {\begin{array}{l} 1, if x = 0, \\ \frac{1}{2}, if 0 < x \leq 1, \end{array} \\ g_{E} (e) (x) = {\begin{array}{l} \frac{1}{2}, if 0 \leq x < 1, \\ 1, if x = 1. \end{array} \end{array}

We define the fuzzy soft r-minimal structure ${\tilde{M}}_{E} : E \to {[0, 1]}^{\tilde{(X, E)}}$ as follows: ∀ e ∈ E:

{\tilde{M}}_{e} (w_{E}) = {\begin{array}{l} 1, if w_{e} \in {f_{E}, g_{E}, Φ, \tilde{E}}, \\ max ({1 - n, n}), if w_{E} = f_{E_{n}}, \\ 0, otherwise . \end{array}

Then, X is nearly fuzzy soft $\frac{1}{2}$ -minimal compact but is not fully fuzzy soft $\frac{1}{2}$ -minimal compact.

Theorem 4.3

Let ϕ_ψ : (X, M̃) → (Y, M̃^*) be fuzzy soft r-minimal continuous and fuzzy soft r-minimal open mapping. If $f_{A} \in \tilde{(X, E)}$ is nearly fuzzy soft r-minimal compact, then ϕ_ψ(f_A) is nearly fuzzy soft r-minimal compact.

Proof

Let ${{(g_{B})}_{i} \in \tilde{(Y, F)} ∣ {(g_{B})}_{i} \in {\tilde{M}}_{k, r}^{*}}_{i \in Γ}$ with ϕ_ψ(f_A) ⊑⊔_i_∈Γ(g_B)_i. Then, ${ϕ_{ψ}^{- 1} ({(g_{B})}_{i}) \in \tilde{(X, E)} ∣ ϕ_{ψ}^{- 1} ({(g_{B})}_{i}) \in {\tilde{M}}_{e, r}}_{i \in Γ}$ (by ϕ_ψ is fuzzy soft r-minimal continuous) such that $f_{A} ⊑ ⊔_{i \in Γ} ϕ_{ψ}^{- 1} ({(g_{B})}_{i})$ . Because f_A is nearly fuzzy soft r-minimal compact, there exists a finite subset Γ_o of Γ such that $f_{A} ⊑ \underset{i \in Γ_{o}}{⊔} I_{m} (e, C_{m} (e, ϕ_{ψ}^{- 1} ({(g_{B})}_{i}), r), r) .$

From Theorems 3.2, and 3.4, it follows that

\begin{array}{l} ϕ_{ψ} (f_{A}) ⊑ \underset{i \in Γ_{o}}{⊔} ϕ_{ψ} (I_{m} (e, C_{m} (e, ϕ_{ψ}^{- 1} ({(g_{B})}_{i}), r), r)) \\ ⊑ \underset{i \in Γ_{o}}{⊔} I_{m^{*}} (k, ϕ_{ψ} (C_{m} (e, ϕ_{ψ}^{- 1} ({(g_{B})}_{i}), r)), r) \\ ⊑ \underset{i \in Γ_{o}}{⊔} I_{m^{*}} (k, ϕ_{ψ} (ϕ_{ψ}^{- 1} (C_{m^{*}} (k, {(g_{B})}_{i}, r))), r) \\ ⊑ \underset{i \in Γ_{o}}{⊔} I_{m^{*}} (k, C_{m^{*}} (k, {(g_{B})}_{i}, r), r) . \end{array}

Hence

ϕ_{ψ} (f_{A}) ⊑ \underset{i \in Γ_{o}}{⊔} I_{m^{*}} (k, C_{m^{*}} (k, {(g_{B})}_{i}, r), r) .

5. Conclusion

Abstract
1. Introduction
2. Preliminaries
3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets
4. Several Types of Fuzzy Soft r-Minimal Compactness
5. Conclusion

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Compactness on Fuzzy Soft r-Minimal Spaces

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 [2,4,26]

Definition 2.2 [33]

Definition 2.3 [26]

3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets

Definition 3.1

Example 3.1

Definition 3.2. [12]

Definition 3.3

Example 3.2

Remark 1

Definition 3.4

Theorem 3.1

Definition 3.5

Theorem 3.2

Lemma 3.1

Definition 3.6

Theorem 3.3

Lemma 3.2

Definition 3.7

Theorem 3.4

Theorem 3.5

Lemma 3.3

Theorem 3.6

4. Several Types of Fuzzy Soft r-Minimal Compactness

Definition 4.1

Theorem 4.1

Definition 4.2

Lemma 4.1

Example 4.1

Theorem 4.2

Definition 4.3

Lemma 4.2

Example 4.2

Theorem 4.3

5. Conclusion

Conflict of Interest

References

Biography

Article

Original Article

Compactness on Fuzzy Soft r-Minimal Spaces

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 [2,4,26]

Definition 2.2 [33]

Definition 2.3 [26]

3. r-Minimal Structures and r-Minimal Spaces through Fuzzy Soft Sets

Definition 3.1

Example 3.1

Definition 3.2. [12]

Definition 3.3

Example 3.2

Remark 1

Definition 3.4

Theorem 3.1

Definition 3.5

Theorem 3.2

Lemma 3.1

Definition 3.6

Theorem 3.3

Lemma 3.2

Definition 3.7

Theorem 3.4

Theorem 3.5

Lemma 3.3

Theorem 3.6

4. Several Types of Fuzzy Soft r-Minimal Compactness

Definition 4.1

Theorem 4.1

Definition 4.2

Lemma 4.1

Example 4.1

Theorem 4.2

International Journal
of Fuzzy Logic and
Intelligent Systems