Generalized Fuzzy Ideal Closed Sets on Fuzzy Topological Spaces in Sostak Sense

Yasser M. Saber, and Fahad Alsharari

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 [6–14]. After that several authors [15–17] 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*I*_{0}= (0, 1]. For*α*∈*I*,*ᾱ*(*x*) =*α*for all*x*∈*X*. The family of all fuzzy sets on*X*denoted by*I*. For two fuzzy sets we write^{X}*λqμ*to mean that*λ*is quasi-coincident (q-coincident, for short) with*μ*, i.e, there exists at least one point*x*∈*X*such that*λ*(*x*) +*μ*(*x*) > 1. Negation of such a statement is denoted as*λq̄μ*.### Definition 1.1 ( [1])

A mapping

*τ*:*I*→^{X}*I*is called a fuzzy topology on*X*if it satisfies the following conditions:(O1)

*τ*(0̄) =*τ*(1̄) = 1̄.(O2)

*τ*(∧_{i}_{∈Γ}*μ*) ≥ ∨_{i}_{i}_{∈Γ}*τ*(*μ*), for any {_{i}*μ*}_{i}_{i}_{∈Γ}∈*I*.^{X}(O3)

*τ*(*μ*_{1}∧*μ*_{2}) ≥*τ*(*μ*_{1}) ∧*τ*(*μ*_{2}), for any*μ*_{1},*μ*_{2}∈*I*.^{X}

The pair (

*X*,*τ*) is called a fuzzy topological space (for short, fts).### Definition 1.2 ( [18])

Let (

*X*,*τ*) be a fts,*λ*,*μ*∈*I*and^{X}*r*∈*I*_{0}.A fuzzy set

*λ*is called r-generalized fuzzy closed (for short, r-gfc) if*C*(_{τ}*λ*,*r*) ≤*μ*whenever*λ*≤*μ*and*τ*(*μ*) ≥*r*.A fuzzy set

*λ*is called r-generalized fuzzy open (for short, r-gfo) if*I*(_{τ}*λ*,*r*) ≥*μ*whenever*λ*≥*μ*and*τ*(1̄−*μ*) ≥*r*.

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

A mapping

I :*I*→^{X}*I*is called fuzzy ideal on X iff:I (0 ) = 1,I (1 ) = 0.If

*λ*≤*μ*, thenI (*λ*) ≥I (*μ*), for each*λ*,*μ*∈*I*.^{X}For each

*λ*,*μ*∈*I*,^{X}I (*λ*∨*μ*) ≥I (*λ*) ∧I (*μ*) [finite additivity].

### Lemma 1.4

Let (

*X*,*τ*,*ℐ*) be a fits. The simplest fuzzy ideal on*X*are*ℐ*^{0},*ℐ*^{1}:*I*→^{X}*I*where$$\begin{array}{l}{\mathcal{I}}^{1}(\lambda )=\{\begin{array}{ll}1,\hfill & \text{if\hspace{0.17em}}\lambda =\underset{\_}{0},\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}\\ {\mathcal{I}}^{0}(\lambda )=\{\begin{array}{ll}0,\hfill & \text{if\hspace{0.17em}}\lambda =\underset{\_}{1},\hfill \\ 1,\hfill & \text{otherwise.}\hfill \end{array}\end{array}$$ If we take

*ℐ*=*ℐ*^{0}, for each we have${\mathcal{A}}_{r}^{*}={C}_{\tau}(\mathcal{A},r)$ .If we take

*ℐ*=*ℐ*^{1}, for each we have${\mathcal{A}}_{r}^{*}=\underset{\_}{0}$ , where,1 ∉ Θ*′*be a subset of*I*.^{X}### Definition 1.5 ( [19])

Let (

*X*,*τ*,I ) be a fuzzy ideal topological space. Let*μ*,*λ*∈*I*, the r-fuzzy open local function^{X}${\mu}_{r}^{*}$ of*μ*is the union of all fuzzy points*x*such that if_{t}*ρ*∈*Q*(*x*,_{t}*r*) andI (*λ*) ≥*r*then there is at least one*y*∈*X*for which*ρ*(*y*) +*μ*(*y*) − 1 >*λ*(*y*).### Theorem 1.6 ( [4])

Let (

*X*,*τ*) be a fts. Then for each*r*∈*I*_{0},*λ*∈*I*we define an operator^{X}*C*:_{τ}*I*^{X}*× I*_{0}→*I*as follows:^{X}$${C}_{\tau}(\lambda ,r)=\wedge \{\mu \in {I}^{X}:\lambda \le \mu ,\hspace{0.17em}\tau (\overline{1}-\mu )\ge r\}.$$ For

*λ*,*μ*∈*I*and^{X}*r*,*s*∈*I*_{0}, the operator*C*satisfies the following conditions:_{τ}*C*(0̄,_{τ}*r*) = 0̄.*λ*≤*C*(_{τ}*λ*,*r*).*C*(_{τ}*λ*,*r*) ∨*C*(_{τ}*μ*,*r*) =*C*(_{τ}*λ*∨*μ*,*r*).*C*(_{τ}*λ*,*r*) ≤*C*(_{τ}*λ*,*s*) if*r*≤*s*.*C*(_{τ}*C*(_{τ}*λ*,*r*),*r*) =*C*(_{τ}*λ*,*r*).

### Theorem 1.7 ( [20])

Let (

*X*,*τ*) be a fts. Then for each*r*∈*I*_{0},*λ*∈*I*we define an operator^{X}*I*:_{τ}*I*^{X}*× I*_{0}→*I*as follows:^{X}$${I}_{\tau}(\lambda ,r)=\vee \{\mu \in {I}^{X}:\lambda \ge \mu ,\hspace{0.17em}\tau (\mu )\ge r\}.$$ For

*λ*,*μ*∈*I*and^{X}*r*,*s*∈*I*_{0}, the operator*I*satisfies the following conditions:_{τ}*I*(1̄ −_{τ}*λ*,*r*) = 1̄ −*C*(_{τ}*λ*,*r*) and*C*(1̄ −_{τ}*λ*,*r*) = 1̄ −*I*(_{τ}*λ*,*r*).*I*(1̄,_{τ}*r*) = 1̄.*λ*≥*I*(_{τ}*λ*,*r*).*I*(_{τ}*λ*,*r*) ∧*I*(_{τ}*μ*,*r*) =*I*(_{τ}*λ*∧*μ*,*r*).*I*(_{τ}*λ*,*r*) ≤*I*(_{τ}*λ*,*s*) if*r*≥*s*.*I*(_{τ}*I*(_{τ}*λ*,*r*),*r*) =*I*(_{τ}*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,*μ*∈*I*and^{X}*r*∈*I*_{0}. A fuzzy set*μ*is called r-generalized fuzzy closed with respect to an ideal (briefly, r-gfIc) ifI (*C*(_{τ}*μ*,*r*)*λ*) ≥*r*, whenever*μ*≤*λ*and*τ*(*λ*) ≥*r*.### Lemma 2.2

Every r-gfc set is r-gfIc.

**Proof**Let

*μ*≤*λ*and*τ*(*λ*) ≥*r*. Since*μ*is r-gfc set, then*C*(_{τ}*μ*,*r*) ≤*λ*, this implies that*C*(_{τ}*μ*,*r*)*q̄*1 −*λ*, implies*C*(_{τ}*μ*,*r*)(*x*) + (1 −*λ*)(*x*) ≤ 1, then*C*(_{τ}*μ*,*r*)(*x*) −*λ*(*x*) ≤ 0. Thus,I (*C*(_{τ}*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*,*γ*∈*I*are defined as follows:^{X}$$\begin{array}{l}\alpha (a)=0.2,\hspace{0.17em}\alpha (b)=0.4;\hspace{0.17em}\alpha (c)=0.7\\ \beta (a)=0.7,\hspace{0.17em}\beta (b)=0.6;\hspace{0.17em}\beta (c)=0.8\\ \gamma (a)=0.6;\hspace{0.17em}\gamma (b)=0.4,\hspace{0.17em}\gamma (c)=\mathrm{0.7.}\end{array}$$ We define fuzzy topology and fuzzy ideal

*τ*,I :*I*→^{X}*I*as follows:$$\begin{array}{l}\tau (\lambda )=\{\begin{array}{ll}1,\hfill & \text{if\hspace{0.17em}}\nu =\underset{\_}{1},\underset{\_}{0},\hfill \\ \frac{1}{2},\hfill & \text{if\hspace{0.17em}}\nu =\alpha ,\hfill \\ \frac{1}{2},\hfill & \text{if\hspace{0.17em}}\nu =\beta ,\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}\\ \mathbb{I}(\lambda )=\{\begin{array}{ll}1,\hfill & \text{if\hspace{0.17em}}\nu =\underset{\_}{0},\hfill \\ \frac{1}{2},\hfill & \text{if\hspace{0.17em}}\nu =\underset{\_}{0.3},\hfill \\ \frac{1}{2},\hfill & \text{if\hspace{0.17em}}\underset{\_}{0}<\nu <\underset{\_}{0}.3,\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}\end{array}$$ For

$r={\scriptstyle \frac{1}{3}}$ ,1 −*γ*is r-gfIc set, where$$\begin{array}{l}\underset{\_}{1}-\gamma \le \beta ,\mathrm{\hspace{0.17em}\u200a\u200a}\tau (\beta )\ge \frac{1}{3},\\ {C}_{\tau}(\underset{\_}{1}-\gamma ,\frac{1}{3})=\underset{\_}{1}-\alpha \beta ={a}_{0.3}.\end{array}$$ Therefore,

$\mathbf{I}(({C}_{\tau}(\underset{\_}{1}-\gamma ,{\scriptstyle \frac{1}{3}})\alpha ),{\scriptstyle \frac{1}{3}})\ge {\scriptstyle \frac{1}{3}}$ .But

1 −*γ*is not r-gfc set because$$\underset{\_}{1}-\gamma \le \beta ,\mathrm{\hspace{0.17em}\u200a\u200a}\tau (\beta )\ge \frac{1}{3},\mathrm{\hspace{0.17em}\u200a\u200a}({C}_{\tau}(\underset{\_}{1}-\gamma ,\mathrm{\hspace{0.17em}\u200a\u200a}\frac{1}{3})=\underset{\_}{1}-\alpha )\ge \beta .$$ ### Theorem 2.4

Let (

*X*,*τ*,I ) be an fuzzy ideal topological space,*μ*,*λ*∈*I*and^{X}*r*∈*I*_{0}. If*μ*and*λ*are r-gfIc sets, then*μ*∨*λ*is r-gfIc.**Proof**Suppose

*μ*and*λ*are r-gfIc sets. If*μ*∨*λ*≤*ρ*and*τ*(*ρ*) ≥*r*, then*μ*≤*ρ*and*λ*≤*ρ*. By assumption,I (*C*(_{τ}*μ*,*r*)*ρ*) ≥*r*andI (*C*(_{τ}*λ*,*r*)*ρ*) ≥*r*and hence$$\mathbf{I}({C}_{\tau}(\mu \vee \lambda ,r)\rho ={C}_{\tau}(\mu ,r)\rho \vee {C}_{\tau}(\lambda ,r)\rho )\ge r.$$ Therefore,

*μ*∨*λ*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*,*γ*∈*I*are defined as follows:^{X}$$\begin{array}{l}\alpha (a)=0.8,\hspace{0.17em}\alpha (b)=0.4;\hspace{0.17em}\alpha (c)=0.7\\ \beta (a)=0.6,\hspace{0.17em}\beta (b)=0.5;\hspace{0.17em}\beta (c)=0.8\\ \gamma (a)=0.6;\hspace{0.17em}\gamma (b)=0.4,\hspace{0.17em}\gamma (c)=\mathrm{0.7.}\end{array}$$ We define fuzzy topology and fuzzy ideal

*τ*,I :*I*→^{X}*I*as follows:$$\begin{array}{l}\tau (\lambda )=\{\begin{array}{ll}1,\hfill & \text{if\hspace{0.17em}}\nu =\underset{\_}{1},\underset{\_}{0},\hfill \\ \frac{1}{2},\hfill & \text{if\hspace{0.17em}}\nu =\gamma ,\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}\\ \mathbb{I}(\lambda )=\{\begin{array}{ll}1,\hfill & \text{if\hspace{0.17em}}\nu =\underset{\_}{0},\hfill \\ \frac{1}{2},\hfill & \text{if\hspace{0.17em}}\nu =\underset{\_}{0.3},\hfill \\ \frac{1}{2},\hfill & \text{if\hspace{0.17em}}\underset{\_}{0}<\nu <\underset{\_}{0}.3,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\end{array}$$ For

$r={\scriptstyle \frac{1}{3}}$ ,*β*∧*γ*is r-gfIc set. But*β*∧*γ*=*γ*is not r-gfIc set because*γ*≤*γ*,*τ*(*γ*) ≥*r*,${C}_{\tau}(\gamma ,{\scriptstyle \frac{1}{3}})=\underset{\_}{1}\gamma =\underset{\_}{1}$ . Therefore$\mathbf{I}(({C}_{\tau}(\gamma ,{\scriptstyle \frac{1}{3}})\gamma ),{\scriptstyle \frac{1}{3}})<{\scriptstyle \frac{1}{3}}$ .### Theorem 2.7

Let (

*X*,*τ*,I ) be an fuzzy ideal topological space,*μ*,*λ*∈*I*and^{X}*r*∈*I*_{0}. If*μ*is r-gfIc set and*μ*≤*λ*≤*C*(_{τ}*μ*,*r*), then*λ*are r-gfIc.**Proof**Let

*μ*is r-gfIc set and*μ*≤*λ*≤*C*(_{τ}*μ*,*r*). Suppose*λ*≤*ρ*and*τ*(*ρ*) ≥*r*. Then*μ*≤*ρ*. Since*μ*is r-gfIc, we haveI (*C*(_{τ}*μ*,*r*)*ρ*) ≥*r*. Now*λ*≤*C*(_{τ}*μ*,*r*) implies that$${C}_{\tau}(\lambda ,r)\rho \le {C}_{\tau}(\mu ,r)\rho ,$$ and hence,

I (*C*(_{τ}*λ*,*r*)*ρ*) ≥*r*. Therefore,*λ*is r-gfIc set.### Definition 2.8

Let (

*X*,*τ*,I ) be fuzzy ideal topological space,*μ*∈*I*and^{X}*r*∈*I*_{0}. A fuzzy set*μ*is called r-fuzzy generalized open with respect to an idealI (briefly, r-gfIo) if1 −*μ*is r-gfIc set.### Theorem 2.9

Let (

*X*,*τ*,I ) be an fuzzy ideal topological space,*μ*,*λ*,*ρ*∈*I*and^{X}*r*∈*I*_{0}. If*μ*is r-gfIo sets if and only if*λ**ρ*≤*I*(_{τ}*μ*,*r*) for someI (*ρ*) ≥*r*, whenever*λ*≤*μ*and*τ*(1 −*λ*) ≥*r*.**Proof**Suppose that

*μ*is r-gfIo sets. Suppose*λ*≤*μ*and*τ*(1 −*λ*) ≥*r*. We have1 −*λ*≥1 −*μ*. By assumption,$${C}_{\tau}(\underset{\_}{1}-\mu ,r)\le \underset{\_}{1}-\lambda \vee \rho .$$ For some

I (*ρ*) ≥*r*. This implies$$\underset{\_}{1}-((\underset{\_}{1}-\lambda )\vee \rho )\le \underset{\_}{1}-{C}_{\tau}(\underset{\_}{1}-\mu ),$$ and hence,

*λ**ρ*≤*I*(_{τ}*μ*,*r*).Conversely, assume that

*λ*≤*μ*and*τ*(1 −*λ*) ≥*r*imply*λ**ρ*≤*I*(_{τ}*μ*,*r*) for someI (*ρ*) ≥*r*. Consider*τ*(*ω*) ≥*r*such that1 −*μ*≤*ω*. Then1 −*ω*≤*μ*. By assumption,$$\underset{\_}{1}-\omega \rho \le {I}_{\tau}(\mu ,r)=\underset{\_}{1}-{C}_{\tau}(\underset{\_}{1}-\mu ,r)$$ For some

I (*ρ*) ≥*r*. This gives that$$\underset{\_}{1}-(\omega \vee \rho )\le \underset{\_}{1}-{C}_{\tau}(\underset{\_}{1}-\mu ,r).$$ Therefore,

*C*(_{τ}1 −*μ*,*r*) ≤*ω*∨*ρ*, for someI (*ρ*) ≥*r*. This show thatI (*C*(_{τ}1 −*μ*,*r*)*ω*) ≥*r*. Hence,1 −*μ*is r-gfIc set.Recall that the sets

*μ*and*λ*are fuzzy separated if*C*(_{τ}*μ*,*r*)*qλ*and*μqC*(_{τ}*λ*,*r*).### Theorem 2.10

Let (

*X*,*τ*,I ) be an fuzzy ideal topological space,*μ*,*λ*, ∈*I*and^{X}*r*∈*I*_{0}. If*μ*and*λ*are fuzzy separated and r-gfIo sets, then*μ*∨*λ*is r-gfIo.**Proof**Suppose

*μ*and*λ*are fuzzy separated and r-gfIo sets and*ρ*≤*μ*∨*λ*, and*τ*(1 −*ρ*) ≥*r*. Then*ρ*∧*C*(_{τ}*μ*,*r*) ≤*μ*and*ρ*∧*C*(_{τ}*λ*,*r*) ≤*λ*. By assumption,$$\begin{array}{l}\rho \wedge {C}_{\tau}(\mu ,r){\nu}_{1}\le {I}_{\tau}(\mu ,r),\\ \rho \wedge {C}_{\tau}(\lambda ,r){\nu}_{2}\le {I}_{\tau}(\lambda ,r),\end{array}$$ for some

I ((*ν*_{1},*ν*_{2}),*r*) ≥*r*. This means$$\mathbf{I}(\rho \wedge {C}_{\tau}(\mu ,r){I}_{\tau}(\mu ,r),r)\ge r,$$ and

$$\mathbf{I}(\rho \wedge {C}_{\tau}(\lambda ,r){I}_{\tau}(\lambda ,r),r)\ge r.$$ Thus,

$$\mathbf{I}((\rho \wedge {C}_{\tau}(\mu ,r){I}_{\tau}(\mu ,r))\vee (\rho \wedge {C}_{\tau}(\lambda ,r)){I}_{\tau}(\lambda ,r),r)\ge r.$$ Therefore,

$$\mathbf{I}(\rho \wedge ({C}_{\tau}(\mu ,r)\vee {C}_{\tau}(\lambda ,r))({I}_{\tau}(\mu ,r)\vee {I}_{\tau}(\lambda ,r)),r)\ge r.$$ But

$$\rho =\rho \wedge (\mu \vee \lambda )\le \rho \wedge ({C}_{\tau}(\mu \vee \lambda ,r)),$$ and we have

$$\begin{array}{l}\mathbf{I}(\rho {I}_{\tau}(\mu \vee \lambda ,r)\le (\rho \wedge {C}_{\tau}(\mu \vee \lambda ,r)){I}_{\tau}(\mu \vee \lambda ,r)\\ \le (\rho \wedge {C}_{\tau}(\mu \vee \lambda ,r))({I}_{\tau}(\mu ,r)\vee {I}_{\tau}(\lambda ,r)))\ge r.\end{array}$$ Hence,

*ρ**ν*≤*Int*(_{τ}*μ*∨*λ*,*r*) for someI (*ν*) ≥*r*. This proves that*μ*∨*λ*is r-gfIo.### Corollary 2.11

Let (

*X*,*τ*,I ) be an fuzzy ideal topological space,*μ*,*λ*, ∈*I*and^{X}*r*∈*I*_{0}. If*μ*and*λ*are r-gfIo sets,1 −*μ*and1 −*λ*are fuzzy separated. Then*μ*∧*λ*is r-gfIc.**Proof**Obvious.

### Corollary 2.12

Let (

*X*,*τ*,I ) be an fuzzy ideal topological space,*μ*,*λ*, ∈*I*and^{X}*r*∈*I*_{0}. If*μ*and*λ*are r-gfIo sets, then*μ*∧*λ*is r-gfIo.**Proof**Obvious.

### Theorem 2.13

Let (

*X*,*τ*,I ) be an fuzzy ideal topological space,*μ*,*λ*, ∈*I*and^{X}*r*∈*I*_{0}. If and*μ*≤*λ*, 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

*μ*≤*λ*,*μ*is r-gfIo relative to*λ*and*λ*is r-gfIo relative to*X*. Let*ρ*≤*μ*and*τ*(1 −*ρ*) ≥*r*. Since*μ*is r-gfIo relative to*λ*. By Theorem 2.9,*ρ**ν*_{1}≤*I*(_{λ}*μ*,*r*) for someI (*ν*_{1}) ≥*r*. This implies that there exists*τ*(*ω*_{1}) ≥*r*such that$$\rho {\nu}_{1}\le {\omega}_{1}\wedge \lambda \le \mu ,$$ for some

I (*ν*_{1}) ≥*r*. Let*ρ*≤*λ*and*τ*(1 −*ρ*) ≥*r*. Since*λ*is r-gfIo, we have$$\rho {\nu}_{2}\le {Int}_{\tau}(\lambda ,r)$$ for some

I (*ν*_{2}) ≥*r*. This implies that there exists*τ*(*ω*_{2}) ≥*r*such that$$\rho {\nu}_{2}\le {\omega}_{2}\le \lambda ,$$ for some

I (*ν*_{2}) ≥*r*. Now$$\rho ({\nu}_{1}\vee {\nu}_{2})=(\rho {\nu}_{1})\wedge (\rho {\nu}_{2})\le {\omega}_{1}\wedge {\omega}_{2}\le {\omega}_{1}\wedge \lambda \le \mu .$$ This implies that

*ρ*(*ν*_{1}∨*ν*_{2}) ≤*I*(_{λ}*μ*,*r*) for someI (*ν*_{1}∨*ν*_{2}) ≥*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.