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Generalized Fuzzy Ideal Closed Sets on Fuzzy Topological Spaces in Sostak Sense
International Journal of Fuzzy Logic and Intelligent Systems 2018;18(3):161-166
Published online September 25, 2018
© 2018 Korean Institute of Intelligent Systems.

Yasser M. Saber, and Fahad Alsharari

1Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut Egypt, 2Department of Mathematics, College of Science and Human Studies of Hotat Sudair, Majmaah University, Majmaah, Saudi Arabia
Correspondence to: Yasser M. Saber (m.ah75@yahoo.com)
Received April 4, 2018; Revised August 21, 2018; Accepted September 18, 2018.
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 non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum paretical physics in connection with string theory and E-infinity space time theory. In this paper, we define concept r-generalized fuzzy ideal closed sets with respect to an fuzzy ideal topological space in Sostak sense. We investigate some properties of them, we investigate the relationships between r-generalized fuzzy ideal closed sets with respect to an ideal and r-fuzzy separated

Keywords : r-generalized fuzzy closed sets, r-generalized fuzzy closed sets with respect to an fuzzy ideal topological space in Sostak sense, r-fuzzy separated
1. Introduction

Sostak [1], introduce the fundamental concept of fuzzy topological structure as an extension of both crisp topology and Chang’s fuzzy topology [2], in the sense that not only the object were fuzzified, but also the axiomatics. Chattopdhyay et al. [3] and El Naschie [4] have redefined the similar concept. In [5], the author gave a similar definition namely “Smooth fuzzy topology”. We must point out that; the concept of fuzzy topological spaces, has been a significant concept in string theory and E-infinity theory pertaining to quantum particular physics ever since El-Naschie [614]. After that several authors [1517] have introduced the smooth definition and studied smooth fuzzy ideai topological spaces being unaware of Sostak works.

Throughout this paper, let X be a nonempty set I = [0, 1] and I0 = (0, 1]. For αI, ᾱ(x) = α for all xX. The family of all fuzzy sets on X denoted by IX. For two fuzzy sets we write λqμ to mean that λ is quasi-coincident (q-coincident, for short) with μ, i.e, there exists at least one point xX such that λ(x) + μ(x) > 1. Negation of such a statement is denoted as λq̄μ.

Definition 1.1 ( [1])

A mapping τ : IXI is called a fuzzy topology on X if it satisfies the following conditions:

  1. (O1) τ (0̄) = τ (1̄) = 1̄.

  2. (O2) τ (∧i∈Γμi) ≥ ∨i∈Γτ (μi), for any {μi}i∈ΓIX.

  3. (O3) τ (μ1μ2) ≥ τ (μ1) ∧ τ (μ2), for any μ1, μ2IX.

The pair (X, τ) is called a fuzzy topological space (for short, fts).

Definition 1.2 ( [18])

Let (X, τ) be a fts, λ, μIX and rI0.

  1. A fuzzy set λ is called r-generalized fuzzy closed (for short, r-gfc) if Cτ (λ, r) ≤ μ whenever λμ and τ (μ) ≥ r.

  2. A fuzzy set λ is called r-generalized fuzzy open (for short, r-gfo) if Iτ (λ, r) ≥ μ whenever λμ and τ (1̄−μ) ≥ r.

Definition 1.3 ( [1, 5, 15, 16])

A mapping I : IXI is called fuzzy ideal on X iff:

  1. I(0) = 1, I(1) = 0.

  2. If λμ, then I(λ) ≥ I(μ), for each λ, μIX.

  3. For each λ, μIX, I(λμ) ≥ I(λ) ∧ I(μ) [finite additivity].

Lemma 1.4

Let (X, τ, ) be a fits. The simplest fuzzy ideal on X are 0, 1 : IXI where

I1(λ)={1,if λ=0_,0,otherwise.I0(λ)={0,if λ=1_,1,otherwise.

If we take = 0, for each we have Ar*=Cτ(A,r).

If we take = 1, for each we have Ar*=0_, where, 1 ∉ Θ be a subset of IX.

Definition 1.5 ( [19])

Let (X, τ, I) be a fuzzy ideal topological space. Let μ, λIX, the r-fuzzy open local function μr* of μ is the union of all fuzzy points xt such that if ρQ(xt, r) and I(λ) ≥ r then there is at least one yX for which ρ(y) + μ(y) − 1 > λ(y).

Theorem 1.6 ( [4])

Let (X, τ) be a fts. Then for each rI0, λIX we define an operator Cτ : IX × I0IX as follows:

Cτ(λ,r)={μIX:λμ,τ(1¯-μ)r}.

For λ, μIX and r, sI0, the operator Cτ satisfies the following conditions:

  1. Cτ (0̄, r) = 0̄.

  2. λCτ (λ, r).

  3. Cτ (λ, r) ∨ Cτ (μ, r) = Cτ (λμ, r).

  4. Cτ (λ, r) ≤ Cτ (λ, s) if rs.

  5. Cτ (Cτ (λ, r), r) = Cτ (λ, r).

Theorem 1.7 ( [20])

Let (X, τ) be a fts. Then for each rI0, λIX we define an operator Iτ : IX × I0IX as follows:

Iτ(λ,r)={μIX:λμ,τ(μ)r}.

For λ, μIX and r, sI0, the operator Iτ satisfies the following conditions:

  1. Iτ (1̄ − λ, r) = 1̄ − Cτ (λ, r) and Cτ (1̄ − λ, r) = 1̄ − Iτ (λ, r).

  2. Iτ (1̄, r) = 1̄.

  3. λIτ (λ, r).

  4. Iτ (λ, r) ∧ Iτ (μ, r) = Iτ (λμ, r).

  5. Iτ (λ, r) ≤ Iτ (λ, s) if rs.

  6. Iτ (Iτ (λ, r), r) = Iτ (λ, r).

2. r-generalized Fuzzy Closed Sets with Respect to an Ideal

Definition 2.1

Let (X, τ, I) be fuzzy ideal topological space, μIX and rI0. A fuzzy set μ is called r-generalized fuzzy closed with respect to an ideal (briefly, r-gfIc) if I(Cτ (μ, r) λ) ≥ r, whenever μλ and τ (λ) ≥ r.

Lemma 2.2

Every r-gfc set is r-gfIc.

Proof

Let μλ and τ (λ) ≥ r. Since μ is r-gfc set, then Cτ (μ, r) ≤ λ, this implies that Cτ (μ, r)1λ, implies Cτ (μ, r)(x) + (1λ)(x) ≤ 1, then Cτ (μ, r)(x) − λ(x) ≤ 0. Thus, I(Cτ (μ, r)λ) ≥ r.

Example 2.3

The converse Lemma 2.2 is not true. Let X = {a, b, c} be a set and α, β, γIX are defined as follows:

α(a)=0.2,α(b)=0.4;α(c)=0.7β(a)=0.7,β(b)=0.6;β(c)=0.8γ(a)=0.6;γ(b)=0.4,γ(c)=0.7.

We define fuzzy topology and fuzzy ideal τ, I : IXI as follows:

τ(λ)={1,if ν=1_,0_,12,if ν=α,12,if ν=β,0,otherwise.I(λ)={1,if ν=0_,12,if ν=0.3_,12,if 0_<ν<0_.3,0,otherwise,

For r=13, 1γ is r-gfIc set, where

1_-γβ,   τ(β)13,Cτ(1_-γ,13)=1_-αβ=a0.3.

Therefore, I((Cτ(1_-γ,13)α),13)13.

But 1γ is not r-gfc set because

1_-γβ,   τ(β)13,   (Cτ(1_-γ,   13)=1_-α)β.

Theorem 2.4

Let (X, τ, I) be an fuzzy ideal topological space, μ, λIX and rI0. If μ and λ are r-gfIc sets, then μλ is r-gfIc.

Proof

Suppose μ and λ are r-gfIc sets. If μλρ and τ (ρ) ≥ r, then μρ and λρ. By assumption, I(Cτ (μ, r) ρ) ≥ r and I(Cτ (λ, r) ρ) ≥ r and hence

I(Cτ(μλ,r)ρ=Cτ(μ,r)ρCτ(λ,r)ρ)r.

Therefore, μλ is r-gfIc.

Remark 2.5

The intersection of two r-gfIc sets need not be an r-gfIc set as shown by the following example.

Example 2.6

Let X = {a, b, c} be a set and α, β, γIX are defined as follows:

α(a)=0.8,α(b)=0.4;α(c)=0.7β(a)=0.6,β(b)=0.5;β(c)=0.8γ(a)=0.6;γ(b)=0.4,γ(c)=0.7.

We define fuzzy topology and fuzzy ideal τ, I : IXI as follows:

τ(λ)={1,if ν=1_,0_,12,if ν=γ,0,otherwise,I(λ)={1,if ν=0_,12,if ν=0.3_,12,if 0_<ν<0_.3,0,otherwise.

For r=13, βγ is r-gfIc set. But βγ = γ is not r-gfIc set because γγ, τ(γ) ≥ r, Cτ(γ,13)=1_γ=1_. Therefore I((Cτ(γ,13)γ),13)<13.

Theorem 2.7

Let (X, τ, I) be an fuzzy ideal topological space, μ, λIX and rI0. If μ is r-gfIc set and μλCτ (μ, r), then λ are r-gfIc.

Proof

Let μ is r-gfIc set and μλCτ (μ, r). Suppose λρ and τ (ρ) ≥ r. Then μρ. Since μ is r-gfIc, we have I(Cτ (μ, r) ρ) ≥ r. Now λCτ (μ, r) implies that

Cτ(λ,r)ρCτ(μ,r)ρ,

and hence, I(Cτ (λ, r) ρ) ≥ r. Therefore, λ is r-gfIc set.

Definition 2.8

Let (X, τ, I) be fuzzy ideal topological space, μIX and rI0. A fuzzy set μ is called r-fuzzy generalized open with respect to an ideal I (briefly, r-gfIo) if 1μ is r-gfIc set.

Theorem 2.9

Let (X, τ, I) be an fuzzy ideal topological space, μ, λ, ρIX and rI0. If μ is r-gfIo sets if and only if λ ρIτ (μ, r) for some I(ρ) ≥ r, whenever λμ and τ (1λ) ≥ r.

Proof

Suppose that μ is r-gfIo sets. Suppose λμ and τ (1λ) ≥ r. We have 1λ1μ. By assumption,

Cτ(1_-μ,r)1_-λρ.

For some I(ρ) ≥ r. This implies

1_-((1_-λ)ρ)1_-Cτ(1_-μ),

and hence, λ ρIτ (μ, r).

Conversely, assume that λμ and τ (1λ) ≥ r imply λρIτ (μ, r) for some I(ρ) ≥ r. Consider τ (ω) ≥ r such that 1μω. Then 1ωμ. By assumption,

1_-ωρIτ(μ,r)=1_-Cτ(1_-μ,r)

For some I(ρ) ≥ r. This gives that

1_-(ωρ)1_-Cτ(1_-μ,r).

Therefore, Cτ (1μ, r) ≤ ωρ, for some I(ρ) ≥ r. This show that I(Cτ (1μ, r) ω) ≥ r. Hence, 1μ is r-gfIc set.

Recall that the sets μ and λ are fuzzy separated if Cτ (μ, r) and μqCτ (λ, r).

Theorem 2.10

Let (X, τ, I) be an fuzzy ideal topological space, μ, λ, ∈ IX and rI0. If μ and λ are fuzzy separated and r-gfIo sets, then μλ is r-gfIo.

Proof

Suppose μ and λ are fuzzy separated and r-gfIo sets and ρμλ, and τ (1ρ) ≥ r. Then ρCτ (μ, r) ≤ μ and ρCτ (λ, r) ≤ λ. By assumption,

ρCτ(μ,r)ν1Iτ(μ,r),ρCτ(λ,r)ν2Iτ(λ,r),

for some I((ν1, ν2), r) ≥ r. This means

I(ρCτ(μ,r)Iτ(μ,r),r)r,

and

I(ρCτ(λ,r)Iτ(λ,r),r)r.

Thus,

I((ρCτ(μ,r)Iτ(μ,r))(ρCτ(λ,r))Iτ(λ,r),r)r.

Therefore,

I(ρ(Cτ(μ,r)Cτ(λ,r))(Iτ(μ,r)Iτ(λ,r)),r)r.

But

ρ=ρ(μλ)ρ(Cτ(μλ,r)),

and we have

I(ρIτ(μλ,r)(ρCτ(μλ,r))Iτ(μλ,r)(ρCτ(μλ,r))(Iτ(μ,r)Iτ(λ,r)))r.

Hence, ρ νIntτ (μλ, r) for some I(ν) ≥ r. This proves that μλ is r-gfIo.

Corollary 2.11

Let (X, τ, I) be an fuzzy ideal topological space, μ, λ, ∈ IX and rI0. If μ and λ are r-gfIo sets, 1μ and 1λ are fuzzy separated. Then μλ is r-gfIc.

Proof

Obvious.

Corollary 2.12

Let (X, τ, I) be an fuzzy ideal topological space, μ, λ, ∈ IX and rI0. If μ and λ are r-gfIo sets, then μλ is r-gfIo.

Proof

Obvious.

Theorem 2.13

Let (X, τ, I) be an fuzzy ideal topological space, μ, λ, ∈ IX and rI0. If and μλ, and μ r-gfIo relative to λ and λ is r-gfIo relative to X, then μ r-gfIo relative to X.

Proof

Suppose that μλ, μ is r-gfIo relative to λ and λ is r-gfIo relative to X. Let ρμ and τ (1ρ) ≥ r. Since μ is r-gfIo relative to λ. By Theorem 2.9, ρν1Iλ(μ, r) for some I(ν1) ≥ r. This implies that there exists τ (ω1) ≥ r such that

ρν1ω1λμ,

for some I(ν1) ≥ r. Let ρλ and τ (1ρ) ≥ r. Since λ is r-gfIo, we have

ρν2Intτ(λ,r)

for some I(ν2) ≥ r. This implies that there exists τ (ω2) ≥ r such that

ρν2ω2λ,

for some I(ν2) ≥ r. Now

ρ(ν1ν2)=(ρν1)(ρν2)ω1ω2ω1λμ.

This implies that ρ(ν1ν2) ≤ Iλ(μ, r) for some I(ν1ν2) ≥ r. Thus, μ r-gfIo relative to X.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Acknowledgements

The authors are grateful for the support by Faculty of Science and Humanities, Majmaah University.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.


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Biographies

Yasser M. Saber received B.Sc., M.Sc., and Ph.D., in Faculty of Science, Al-Azhar University, Egypt, in 1998, 2006, and 2010, respectively. From 1998 to 2006, he has worked as a Demonstrator in Faculty of Science, Al-Azhar University. From 2006 to 2010, he has worked as Assistant Lecturer in Faculty of Science, Al-Azhar University, Egypt. From 2011 until now he has been worked as an Assistant Professor in Faculty of Science, Al-Azhar University, Egypt. From 2014 until now, he has been worked as an Assistant Professor in College of Science and Human studies of Hotat Sudair, Majmaah University, Majmaah. His research interests general topology, fuzzy topology and Applications of general and fuzzy topology.

E-mail: m.ah75@yahoo.com


Fahad Alsharari I received his B.Sc. in mathematics in 2006 from Jouf University and his M.Sc. and Ph.D. from Faculty of Science and Technology, National University of Malaysia, Malaysia in 2011 and 2016, respectively. From 2017 until now, he has been worked as an Assistant Professor in College of Science and Human studies of Hotat Sudair, Majmaah University, Majmaah. His research interests general topology, fuzzy topology and Applications of general and fuzzy topology.

E-mail: f.alsharari@mu.edu.sa.