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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

© The Korean Institute of Intelligent Systems

Compactness on Fuzzy Soft r-Minimal Spaces

Islam M. Taha1,2

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.

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

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 : EP(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 [2,425]). 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.

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

Definition 2.1 [2,4,26]

A fuzzy soft set fA over X is a mapping from E to IX such that fA(e) is a fuzzy set on X for each eA and fA(e) = 0, if eA, where 0 is a zero function on X. The fuzzy set fA(e) for each eE is called an element of the fuzzy soft set fA. 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 ext over X is a fuzzy soft set defined as follows: ext(k) = xt, if k = e and ext (k) = 0, if kE −{e}, where xt is a fuzzy point. A fuzzy soft point ext is said to belong to a fuzzy soft set fA, denoted by ext ∊̃ fA if t < fA(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 eE.

  • τe(Φ) = τe( ) = 1.

  • τe(fAgB)τe(fA)τe(gB),fA,gB(X,E)˜.

  • τe(iΔ(fA)i)iΔτe((fA)i),(fA)i(X,E)˜,iΔ.

Then, the pair (X, τE) 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].

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 rI0. 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 eE contains Φ and . Then, (X, M̃) is called a fuzzy soft r-minimal space (simply, r-FMS˜). Every member of e,r is called a fuzzy soft r-minimal open set. A fuzzy soft set fA is called a fuzzy soft r-minimal closed set if the complement of fA is a fuzzy soft r-minimal open set.

Example 3.1

Let (X, τE) be an FSTS. It is then easy to see that for each eE and rI0, a fuzzy soft mapping 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, τE) be an FSTS. For fA(X,E)˜ and rI0, fA is called r-fuzzy soft regularly open if fA = Iτ (e,Cτ (e, fA, r), r).

Definition 3.3

Let (X, τE) be an FSTS. For fA(X,E)˜ and rI0.

(1) fA is called r-fuzzy soft preopen if fAIτ (e, Cτ (e, fA, r), r).

(2) fA is called r-fuzzy soft β-open if fACτ (e, Iτ (e, Cτ (e, fA, 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 fE 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=fEgE,14,if wE=fEgE,0,otherwise,τe2(wE)={1,if   wE{Φ,E˜},14,if   wE=fE,12,if   wE=gE,12,if wE=fEgE,14,if wE=fEgE,0,otherwise.

Then, gE 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, τE) be an FSTS and rI0. 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˜, eE, and rI0. The fuzzy soft r-minimal interior and fuzzy soft r-minimal closure of fA, denoted by Im(e, fA, r) and Cm(e, fA, r), respectively, are defined as Im(e,fA,r)={gB(X,E˜):gBfA,gBM˜e,r};Cm(e,fA,r)={gB(X,E)˜:fAgB,gBcM˜e,r}.

Theorem 3.1

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

(1) Im(e, fA, r) ⊑ fA and if fAe,r, Im(e, fA, r) = fA.

(2) fACm(e, fA, r) and if fAcM˜e,r, Cm(e, fA, r) = fA.

(3) If fAgB, Im(e, fA, r) ⊑ Im(e, gB, r) and Cm(e, fA, r) ⊑ Cm(e, gB, r).

(4) Im(e, fAgB, r) = Im(e, fA, r) ⊓ Im(e, gB, r).

(5) Cm(e, fAgB, r) = Cm(e, fA, r) ⊔ Cm(e, gB, r).

(6) Im(e, Im(e, fA, r), r) = Im(e, fA, r) and Cm(e, Cm(e, fA, r), r) = Cm(e, fA, 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 Cm(e, fA, r) ⊑ Cm(e, fAgB, r) and Cm(e, gB, r) ⊑ Cm(e, fAgB, r), we obtain Cm(e, fA, r) ⊔ Cm(e, gB, r) ⊑ Cm(e, fAgB, r). For hCc,kDcMe,r,hCckDc does not always belong to e,r. Thus, we obtain the following:

Cm(e,fA,r)Cm(e,gB,r)=({hC(X,E)˜hCcM˜e,r,fAhC})({kD(X,E)˜kDcM˜e,r,gBkD})={hCkD(X,E)˜fAhC,gBkD}{wE(X,E)˜wEcM˜e,r,fAgBwE}=Cm(e,fAgB,r).

Hence, we obtain Cm(e, fAgB, r) = Cm(e, fA, r) ⊔ Cm(e, gB, r).

Definition 3.5

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

Theorem 3.2

Let ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, fA(X,E)˜,gB(Y,F)˜, eE, and rIo.

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

(2) (ϕψ-1(gB))cM˜e,r for every gBcM˜k,r*.

(3) ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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 ext over X and each gBM˜k,r* containing ϕψ(ext ), there exists fAe,r containing ext such that ϕψ(fA) ⊑ gB.

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)˜gBcM˜k,r*,ϕψ(fA)gB})={ϕψ-1(gB)(X,E)˜gBcM˜k,r*,fAϕψ-1(gB)}{ϕψ-1(gB)(ϕψ-1(gB))cM˜e,rfAϕψ-1(gB)}=Cm(e,fA,r).

Hence ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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 extPt(X)˜ and gBM˜k,r* with ϕψ(ext )∊̃gB. From (5), it follows that ext˜Im(e,ϕψ-1(gB),r). Thus, there exists fAe,r such that ext˜fAϕψ-1(gB). Hence, we obtain (6).

Lemma 3.1

Let ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft r-minimal continuous mapping. The following are then equivalent: For eE and rIo,

  • (1) ϕψ(fA) ⊑ Cm* (k,ϕψ(fA), r) for each fAcM˜e,r;

  • (2) ϕψ-1(gB)=Cm(e,ϕψ-1(gB),r) for each gBcM˜k,r*;

  • (3) ϕψ-1(gB)=Im(e,ϕψ-1(gB),r) for each gBM˜k,r*.

Proof

The proof is obvious.

Definition 3.6

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

M˜e(jJ(fA)j)jJM˜e((fA)j).

Theorem 3.3

Let (X, M̃) be an r-FMS˜ with the property (P). Then, the following statements hold: (1) Im(e, fA, r) = fA iff fAe,r for fA(X,E)˜, and (2) Cm(e, fA, r) = fA if fAcM˜e,r for fA(X,E)˜.

Proof

(1) From Theorem 3.1, it is sufficient to show that if Im(e, fA, r) = fA then fAe,r. Let Im(e, fA, r) = fA, eE, and rIo. Then, M˜e(fA)=M˜e({gB(X,E)˜:gBfA,gBM˜e,r})M˜e(gB)r. Hence, fAe,r.

(2) From Theorem 3.1, it is sufficient to show that if Cm(e, fA, r) = fA, then fAcM˜e,r. Let Cm(e, fA, r) = fA, eE and rIo. Then, from Theorem 3.1, it follows that fAc=(Cm(e,fA,r))c=Im(e,fAc,r). Hence by (1), fAcM˜e,r.

Lemma 3.2

Let ϕψ : (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)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal continuous.

  • (2) (ϕψ-1(gB))cM˜e,r for every (gB)cM˜k,r*.

  • (3) ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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 ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, eE and rIo. Then, ϕψ is called

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

(2) fuzzy soft r-minimal closed if (ϕψ(fA))cM˜k,r* for every fAcM˜e,r.

Theorem 3.4

Let ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, fA(X,E)˜,gB(Y,F)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal open.

  • (2) ϕψ(Im(e, fA, r)) ⊑ Im* (k,ϕψ(fA), 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)˜:hCfA,hCM˜e,r}){ϕψ(hC):ϕψ(hC)ϕψ(fA),ϕψ(hC)M˜k,r*}Im*(k,ϕψ(fA),r).

Hence, ϕψ(Im(e, fA, r)) ⊑ Im* (k,ϕψ(fA), 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 ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft r-minimal open. Then, ϕψ(fA) = Im* (k,ϕψ(fA), r) for every fAe,r.

Proof

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

Lemma 3.3

Let ϕψ : (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)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal open.

  • (2) ϕψ(Im(e, fA, r)) ⊑ Im* (k,ϕψ(fA), 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 ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, fA(X,E)˜,gB(Y,F)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal closed.

  • (2) ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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)˜:fAhC,hCcM˜e,r}){ϕψ(hC):ϕψ(fA)ϕψ(hC),(ϕψ(hC))cM˜k,r*}Cm*(k,ϕψ(fA).

Hence, ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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).

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)˜, eE, and rIo. Then, gB is called a fuzzy soft r-minimal compact iff for every family {(fA)i(X,E)˜(fA)iM˜e,r}iΓ such that gB ⊑⊔i∈Γ(fA)i, there exists a finite subset Γo of Γ such that gB ⊑⊔i∈Γo (fA)i.

Theorem 4.1

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

Proof

Let {(gB)i(Y,F)˜(gB)iM˜k,r*}iΓ with ϕψ(fA) ⊑⊔i∈Γ(gB)i. Then, {ϕψ-1((gB)i)(X,E)˜ϕψ-1((gB)i)M˜e,r}iΓ (by ϕψ is fuzzy soft r-minimal continuous) such that fAiΓϕψ-1((gB)i). Because fA is fuzzy soft r-minimal compact, there exists a finite subset Γo of Γ such that fAiΓoϕψ-1((gB)i). Hence, ϕψ(fA) ⊑⊔i∈Γo (gB)i.

Definition 4.2

Let (X, M̃) be an r-FMS˜,gB(X,E)˜, eE, and rIo. Then, gB is called fuzzy soft r-minimal almost compact iff for every family {(fA)i(X,E)˜(fA)iM˜e,r}iΓ such that gB ⊑⊔i∈Γ(fA)i, there exists a finite subset Γo of Γ such that gB ⊑⊔i∈Γo Cm(e, (fA)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, nN − {1}, and E = {e1, e2} be the parameter set of X. Define fEn and gE1(X,E)˜ as follows ∀ eE:

fEn(e)(x)={0.8,ifx=0nx,if0<x1n,1,if1n<x1,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: ∀ eE:

M˜e(wE)={45,   if   wE{Φ,E˜}nn+1,   if   wEfEn,23,   if   wEgE1,0,   otherwise.

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

Theorem 4.2

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

Proof

Let {(gB)i(Y,F)˜(gB)iM˜k,r*}iΓ with ϕψ(fA) ⊑⊔i∈Γ(gB)i. Then, {ϕψ-1((gB)i)(X,E)˜ϕψ-1((gB)i)M˜e,r}iΓ (by ϕψ is fuzzy soft r-minimal continuous) such that fAiΓϕψ-1((gB)i). Because fA is almost fuzzy soft r-minimal compact, a finite subset Γo exists such that fAiΓoCm(e,ϕψ-1((gB)i),r). From Theorem 3.2(4), it follows that

iΓoCm(e,ϕψ-1((gB)i),r)iΓoϕψ-1(Cm*(k,(gB)i,r)=ϕψ-1(iΓoCm*(k,(gB)i,r)).

Hence, ϕψ(fA) ⊑⊔i∈Γ0Cm* (k, (gB)i, r).

Definition 4.3

Let (X, M̃) be an r-FMS˜,gB(X,E)˜, eE, and rIo. Then, gB is called nearly fuzzy soft r-minimal compact iff for every family {(fA)i(X,E)˜(fA)iM˜e,r}iΓ such that gB ⊑⊔i∈Γ(fA)i, there exists a finite subset Γo of Γ in which gB⊑⊔i∈Γo Im(e,Cm(e, (fA)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 fAe,r from Theorem 3.1, it follows that fA = Im(e, fA, r) ⊑ Im(e,Cm(e, fA, 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 fEn, fE, and gE(X,E)˜ as follows ∀ eE:

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

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

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 ϕψ : (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 ϕψ(fA) is nearly fuzzy soft r-minimal compact.

Proof

Let {(gB)i(Y,F)˜(gB)iM˜k,r*}iΓ with ϕψ(fA) ⊑⊔i∈Γ(gB)i. Then, {ϕψ-1((gB)i)(X,E)˜ϕψ-1((gB)i)M˜e,r}iΓ (by ϕψ is fuzzy soft r-minimal continuous) such that fAiΓϕψ-1((gB)i). Because fA is nearly fuzzy soft r-minimal compact, there exists a finite subset Γo of Γ such that fAiΓoIm(e,Cm(e,ϕψ-1((gB)i),r),r).

From Theorems 3.2, and 3.4, it follows that

ϕψ(fA)iΓoϕψ(Im(e,Cm(e,ϕψ-1((gB)i),r),r))iΓoIm*(k,ϕψ(Cm(e,ϕψ-1((gB)i),r)),r)iΓoIm*(k,ϕψ(ϕψ-1(Cm*(k,(gB)i,r))),r)iΓoIm*(k,Cm*(k,(gB)i,r),r).

Hence

ϕψ(fA)iΓoIm*(k,Cm*(k,(gB)i,r),r).

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.

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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

Copyright © The Korean Institute of Intelligent Systems.

Compactness on Fuzzy Soft r-Minimal Spaces

Islam M. Taha1,2

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

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

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 : EP(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 [2,425]). 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

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

Definition 2.1 [2,4,26]

A fuzzy soft set fA over X is a mapping from E to IX such that fA(e) is a fuzzy set on X for each eA and fA(e) = 0, if eA, where 0 is a zero function on X. The fuzzy set fA(e) for each eE is called an element of the fuzzy soft set fA. 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 ext over X is a fuzzy soft set defined as follows: ext(k) = xt, if k = e and ext (k) = 0, if kE −{e}, where xt is a fuzzy point. A fuzzy soft point ext is said to belong to a fuzzy soft set fA, denoted by ext ∊̃ fA if t < fA(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 eE.

  • τe(Φ) = τe( ) = 1.

  • τe(fAgB)τe(fA)τe(gB),fA,gB(X,E)˜.

  • τe(iΔ(fA)i)iΔτe((fA)i),(fA)i(X,E)˜,iΔ.

Then, the pair (X, τE) 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

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 rI0. 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 eE contains Φ and . Then, (X, M̃) is called a fuzzy soft r-minimal space (simply, r-FMS˜). Every member of e,r is called a fuzzy soft r-minimal open set. A fuzzy soft set fA is called a fuzzy soft r-minimal closed set if the complement of fA is a fuzzy soft r-minimal open set.

Example 3.1

Let (X, τE) be an FSTS. It is then easy to see that for each eE and rI0, a fuzzy soft mapping 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, τE) be an FSTS. For fA(X,E)˜ and rI0, fA is called r-fuzzy soft regularly open if fA = Iτ (e,Cτ (e, fA, r), r).

Definition 3.3

Let (X, τE) be an FSTS. For fA(X,E)˜ and rI0.

(1) fA is called r-fuzzy soft preopen if fAIτ (e, Cτ (e, fA, r), r).

(2) fA is called r-fuzzy soft β-open if fACτ (e, Iτ (e, Cτ (e, fA, 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 fE 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=fEgE,14,if wE=fEgE,0,otherwise,τe2(wE)={1,if   wE{Φ,E˜},14,if   wE=fE,12,if   wE=gE,12,if wE=fEgE,14,if wE=fEgE,0,otherwise.

Then, gE 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, τE) be an FSTS and rI0. 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˜, eE, and rI0. The fuzzy soft r-minimal interior and fuzzy soft r-minimal closure of fA, denoted by Im(e, fA, r) and Cm(e, fA, r), respectively, are defined as Im(e,fA,r)={gB(X,E˜):gBfA,gBM˜e,r};Cm(e,fA,r)={gB(X,E)˜:fAgB,gBcM˜e,r}.

Theorem 3.1

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

(1) Im(e, fA, r) ⊑ fA and if fAe,r, Im(e, fA, r) = fA.

(2) fACm(e, fA, r) and if fAcM˜e,r, Cm(e, fA, r) = fA.

(3) If fAgB, Im(e, fA, r) ⊑ Im(e, gB, r) and Cm(e, fA, r) ⊑ Cm(e, gB, r).

(4) Im(e, fAgB, r) = Im(e, fA, r) ⊓ Im(e, gB, r).

(5) Cm(e, fAgB, r) = Cm(e, fA, r) ⊔ Cm(e, gB, r).

(6) Im(e, Im(e, fA, r), r) = Im(e, fA, r) and Cm(e, Cm(e, fA, r), r) = Cm(e, fA, 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 Cm(e, fA, r) ⊑ Cm(e, fAgB, r) and Cm(e, gB, r) ⊑ Cm(e, fAgB, r), we obtain Cm(e, fA, r) ⊔ Cm(e, gB, r) ⊑ Cm(e, fAgB, r). For hCc,kDcMe,r,hCckDc does not always belong to e,r. Thus, we obtain the following:

Cm(e,fA,r)Cm(e,gB,r)=({hC(X,E)˜hCcM˜e,r,fAhC})({kD(X,E)˜kDcM˜e,r,gBkD})={hCkD(X,E)˜fAhC,gBkD}{wE(X,E)˜wEcM˜e,r,fAgBwE}=Cm(e,fAgB,r).

Hence, we obtain Cm(e, fAgB, r) = Cm(e, fA, r) ⊔ Cm(e, gB, r).

Definition 3.5

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

Theorem 3.2

Let ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, fA(X,E)˜,gB(Y,F)˜, eE, and rIo.

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

(2) (ϕψ-1(gB))cM˜e,r for every gBcM˜k,r*.

(3) ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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 ext over X and each gBM˜k,r* containing ϕψ(ext ), there exists fAe,r containing ext such that ϕψ(fA) ⊑ gB.

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)˜gBcM˜k,r*,ϕψ(fA)gB})={ϕψ-1(gB)(X,E)˜gBcM˜k,r*,fAϕψ-1(gB)}{ϕψ-1(gB)(ϕψ-1(gB))cM˜e,rfAϕψ-1(gB)}=Cm(e,fA,r).

Hence ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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 extPt(X)˜ and gBM˜k,r* with ϕψ(ext )∊̃gB. From (5), it follows that ext˜Im(e,ϕψ-1(gB),r). Thus, there exists fAe,r such that ext˜fAϕψ-1(gB). Hence, we obtain (6).

Lemma 3.1

Let ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft r-minimal continuous mapping. The following are then equivalent: For eE and rIo,

  • (1) ϕψ(fA) ⊑ Cm* (k,ϕψ(fA), r) for each fAcM˜e,r;

  • (2) ϕψ-1(gB)=Cm(e,ϕψ-1(gB),r) for each gBcM˜k,r*;

  • (3) ϕψ-1(gB)=Im(e,ϕψ-1(gB),r) for each gBM˜k,r*.

Proof

The proof is obvious.

Definition 3.6

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

M˜e(jJ(fA)j)jJM˜e((fA)j).

Theorem 3.3

Let (X, M̃) be an r-FMS˜ with the property (P). Then, the following statements hold: (1) Im(e, fA, r) = fA iff fAe,r for fA(X,E)˜, and (2) Cm(e, fA, r) = fA if fAcM˜e,r for fA(X,E)˜.

Proof

(1) From Theorem 3.1, it is sufficient to show that if Im(e, fA, r) = fA then fAe,r. Let Im(e, fA, r) = fA, eE, and rIo. Then, M˜e(fA)=M˜e({gB(X,E)˜:gBfA,gBM˜e,r})M˜e(gB)r. Hence, fAe,r.

(2) From Theorem 3.1, it is sufficient to show that if Cm(e, fA, r) = fA, then fAcM˜e,r. Let Cm(e, fA, r) = fA, eE and rIo. Then, from Theorem 3.1, it follows that fAc=(Cm(e,fA,r))c=Im(e,fAc,r). Hence by (1), fAcM˜e,r.

Lemma 3.2

Let ϕψ : (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)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal continuous.

  • (2) (ϕψ-1(gB))cM˜e,r for every (gB)cM˜k,r*.

  • (3) ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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 ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, eE and rIo. Then, ϕψ is called

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

(2) fuzzy soft r-minimal closed if (ϕψ(fA))cM˜k,r* for every fAcM˜e,r.

Theorem 3.4

Let ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, fA(X,E)˜,gB(Y,F)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal open.

  • (2) ϕψ(Im(e, fA, r)) ⊑ Im* (k,ϕψ(fA), 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)˜:hCfA,hCM˜e,r}){ϕψ(hC):ϕψ(hC)ϕψ(fA),ϕψ(hC)M˜k,r*}Im*(k,ϕψ(fA),r).

Hence, ϕψ(Im(e, fA, r)) ⊑ Im* (k,ϕψ(fA), 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 ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft r-minimal open. Then, ϕψ(fA) = Im* (k,ϕψ(fA), r) for every fAe,r.

Proof

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

Lemma 3.3

Let ϕψ : (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)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal open.

  • (2) ϕψ(Im(e, fA, r)) ⊑ Im* (k,ϕψ(fA), 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 ϕψ : (X, M̃) → (Y, M̃*) be a fuzzy soft mapping, fA(X,E)˜,gB(Y,F)˜, eE, and rIo.

  • (1) ϕψ is fuzzy soft r-minimal closed.

  • (2) ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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)˜:fAhC,hCcM˜e,r}){ϕψ(hC):ϕψ(fA)ϕψ(hC),(ϕψ(hC))cM˜k,r*}Cm*(k,ϕψ(fA).

Hence, ϕψ(Cm(e, fA, r)) ⊑ Cm* (k,ϕψ(fA), 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

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)˜, eE, and rIo. Then, gB is called a fuzzy soft r-minimal compact iff for every family {(fA)i(X,E)˜(fA)iM˜e,r}iΓ such that gB ⊑⊔i∈Γ(fA)i, there exists a finite subset Γo of Γ such that gB ⊑⊔i∈Γo (fA)i.

Theorem 4.1

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

Proof

Let {(gB)i(Y,F)˜(gB)iM˜k,r*}iΓ with ϕψ(fA) ⊑⊔i∈Γ(gB)i. Then, {ϕψ-1((gB)i)(X,E)˜ϕψ-1((gB)i)M˜e,r}iΓ (by ϕψ is fuzzy soft r-minimal continuous) such that fAiΓϕψ-1((gB)i). Because fA is fuzzy soft r-minimal compact, there exists a finite subset Γo of Γ such that fAiΓoϕψ-1((gB)i). Hence, ϕψ(fA) ⊑⊔i∈Γo (gB)i.

Definition 4.2

Let (X, M̃) be an r-FMS˜,gB(X,E)˜, eE, and rIo. Then, gB is called fuzzy soft r-minimal almost compact iff for every family {(fA)i(X,E)˜(fA)iM˜e,r}iΓ such that gB ⊑⊔i∈Γ(fA)i, there exists a finite subset Γo of Γ such that gB ⊑⊔i∈Γo Cm(e, (fA)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, nN − {1}, and E = {e1, e2} be the parameter set of X. Define fEn and gE1(X,E)˜ as follows ∀ eE:

fEn(e)(x)={0.8,ifx=0nx,if0<x1n,1,if1n<x1,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: ∀ eE:

M˜e(wE)={45,   if   wE{Φ,E˜}nn+1,   if   wEfEn,23,   if   wEgE1,0,   otherwise.

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

Theorem 4.2

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

Proof

Let {(gB)i(Y,F)˜(gB)iM˜k,r*}iΓ with ϕψ(fA) ⊑⊔i∈Γ(gB)i. Then, {ϕψ-1((gB)i)(X,E)˜ϕψ-1((gB)i)M˜e,r}iΓ (by ϕψ is fuzzy soft r-minimal continuous) such that fAiΓϕψ-1((gB)i). Because fA is almost fuzzy soft r-minimal compact, a finite subset Γo exists such that fAiΓoCm(e,ϕψ-1((gB)i),r). From Theorem 3.2(4), it follows that

iΓoCm(e,ϕψ-1((gB)i),r)iΓoϕψ-1(Cm*(k,(gB)i,r)=ϕψ-1(iΓoCm*(k,(gB)i,r)).

Hence, ϕψ(fA) ⊑⊔i∈Γ0Cm* (k, (gB)i, r).

Definition 4.3

Let (X, M̃) be an r-FMS˜,gB(X,E)˜, eE, and rIo. Then, gB is called nearly fuzzy soft r-minimal compact iff for every family {(fA)i(X,E)˜(fA)iM˜e,r}iΓ such that gB ⊑⊔i∈Γ(fA)i, there exists a finite subset Γo of Γ in which gB⊑⊔i∈Γo Im(e,Cm(e, (fA)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 fAe,r from Theorem 3.1, it follows that fA = Im(e, fA, r) ⊑ Im(e,Cm(e, fA, 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 fEn, fE, and gE(X,E)˜ as follows ∀ eE:

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

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

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 ϕψ : (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 ϕψ(fA) is nearly fuzzy soft r-minimal compact.

Proof

Let {(gB)i(Y,F)˜(gB)iM˜k,r*}iΓ with ϕψ(fA) ⊑⊔i∈Γ(gB)i. Then, {ϕψ-1((gB)i)(X,E)˜ϕψ-1((gB)i)M˜e,r}iΓ (by ϕψ is fuzzy soft r-minimal continuous) such that fAiΓϕψ-1((gB)i). Because fA is nearly fuzzy soft r-minimal compact, there exists a finite subset Γo of Γ such that fAiΓoIm(e,Cm(e,ϕψ-1((gB)i),r),r).

From Theorems 3.2, and 3.4, it follows that

ϕψ(fA)iΓoϕψ(Im(e,Cm(e,ϕψ-1((gB)i),r),r))iΓoIm*(k,ϕψ(Cm(e,ϕψ-1((gB)i),r)),r)iΓoIm*(k,ϕψ(ϕψ-1(Cm*(k,(gB)i,r))),r)iΓoIm*(k,Cm*(k,(gB)i,r),r).

Hence

ϕψ(fA)iΓoIm*(k,Cm*(k,(gB)i,r),r).

5. Conclusion

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.

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