Nano Topology via Idealization

ARAFA A.NASEF 1, RADWAN ABU-GDAIRI2,∗

1Department of Physics and Engineering Mathematics, Faculty of Engineering,

Kafrelsheikh University, EGYPT

2Department of Mathematics,Faculty of Science,

Zarqa university, zarqa 13132, JORDAN

Abstract: - Examining some weak forms of open and closed sets in nano ideal topological spaces serves as the most

substantial objective of this work . Along with their connections to specific other kinds of nano open sets, several

new attributes and numerous essential features of these nano sets are researched. In light of the aforementioned

novel ideas. In this work we extend nano topological model to a nano ideal topological space. Also, we show that

the concept of nano local function in terms of nano ideal topological space is exanimated. Our main objective is

to establish new results about some nano open sets and the relationships between them. Many concepts available

in the classical theorem maybe discussed using the theory of nano ideal topological space.

Keywords: Nano ideals, nano local functions, topologica ideals, nano topological ideals, Approxi-

mation space, Nano I-open set, Nano I-closed set.

2020 AMS Classification Codes: 54A05, 54A10, 54B05.

Received: November 16, 2023. Revised: March 12, 2024. Accepted: April 2, 2024. Published: May 7, 2024.

1 Introduction

Some early applications of ideal topological struc-

ture can be found in various branches of mathemat-

ical modellings[1]. Many perspectives have been

used to investigate approximate topological space [2].

Since 1930, [3] has explored ideals in topological

spaces.The study written by Vaidyanathaswamy[4] in

1945 contributed to establishing the topic’s signifi-

cance.An ideal or dual filter on X is a nonempty set

of finitely many subsets of X with hereditary condi-

tions on additivity. Specifically a nonempty family I

⊆P(X)(where P(x) is the set of all subsets of X is re-

ferred to an ideal if and only if (i) A∈Igives P(A) ⊆

Iand(ii)A, B ∈Igives A∪B∈I. Given a topological

space (X , τ)with an ideal Ion X, a set operator()*:

P(X)→P(X), is call a local function [4] of A with re-

spect to τand Iis defined as follows: for A⊆X ,

A*(I, τ) = {x∈X:G ∩A/∈I, for every G∈τ(x)}

where , τ(x)= {G∈τ: x∈G},A kuratowski closure

operator C1*() for a topology τ∗(I, τ)is called *-

topology, finer than τis defined by C1* (A)= A ∪A*

(I, τ).When there is no possibility of misunderstand-

ing , we will simply write A* for A*(I, τ)and τ∗for

τ∗(I,τ). The space (X, τ, I) is referred to as an ideal

topological space if Iis an ideal on X.

In [5] the idea of a nano-topology was first pro-

posed, which they characterized in terms of approxi-

mations and the boundary area of a subset of the uni-

verse using an equivalence relation. They also intro-

duced the concepts of nano closed sets, nano-interior,

and nano-closure.In [6] the idea of topological nano-

spaces was introduced and its properties were inves-

tigated. The links between some weak forms of nano

open sets in nano topological spaces and some weak

forms of nano open sets in nano ideal topological

spaces are examined in this research. We addition-

ally draw attention to some findings in ([6][7]) that

are not correct.

2 Preliminaries

We recall the following terms, which are vital in the

sequel, before commencing our task.

Definition 2.1. [8] Let R be an equivalence relation

on Uknown as the indiscernibility relation, and let

Ube a nanempty finite set of objects, Then differ-

ent equivalence classes for Uare created. It is argued

that elements in some equivalence classes are indis-

cernible from one another. The approximation space

is referred to as the pair (U, R).

Definition 2.2. [8] Check Figure 1. LetX⊆Uand

(U, R) be an approximation space.

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Figure 1: A rough set in a rough approximation

space.

(i) The lower approximation of Xwith respect toRis

the set of all objects, which can be for certain cate-

gorized as Xwith regard to Rand it is indicated by

LR(X), that is LR(X) =∪x∈∪ {R(χ): R(x) ⊆X}

where R(x) denotes the equivalence class specified by

x∈ ∪.

(ii) The set of all objects that mightbe categorized as

Xwith respect to Ris the upper approximation of X

with respect to R, and it is indicated by HR(X), that is

HR(X) ∪x∈∪ {R(x): R(x) ∩X=ϕ}.

(iii) The collection of all objects that connot be clas-

sified as either Xor -X with respect to Ris known

as the boundary region of Xwith respect to Rand is

indicated by BR(X), where BR(X) = HR(X) - LR

(X).

Pawlak’s definition states that Xis a rough set if

HR(X) =LR(X).

Proposition 2.3. [8] If (U, R) is an approximation

space and X, y ⊆U, which possess the qualities of

pawlak’s rough sets.

(i) LR(X)⊆X⊆HR(X)(Contraction and Exten-

sion).

(ii) LR(ϕ)=HR(ϕ)=ϕ(Normality) and LRU = HR

U = U (Co-normality).

(iii) HR(X∪Y)=HR(X)∪HR(Y)(Addition).

(iv) HR(X∩Y)⊆HR(X)∩HR(Y).

(v) LR(X∪Y)⊇LR(X)∪LR(Y).

(vi) LR(X∩Y)=LR(X)∩LR(Y)(Multiplication).

(vii) LR(X)⊆LR(Y)and HR(X)⊆HR(Y)when-

ever X⊆Y(Monotone).

(viii) HR(Xc)=[HR(X)]cand LR(Xc)=[HR(X)]c

where Xcdenotes the complement of Xin U(Dual-

ity).

(ix) HR(HR(X)) =LR(HR(X)) =HR(X)(Idem-

potency).

(x) LR(LR(X)) =HR(LR(X)) =LR(X)(Idempo-

tency).

Definition 2.4. [5] Let Ube the universe, Rbe an

equivalence relation on U, then for X⊆U,τR(X)=

{U, ϕ,LR(X),HR(X),BR(X)}is referred to the

nano topology on Uwhich complies the following ax-

ioms:

(i) U and ϕ∈τR(X).

(ii) The combination of any subcollection’s parts of

τR(X)is in τR(X);

(iii) Any finite subcollection’s intersection of compo-

nents of τR(X)is in τR(X).

In other words, the pair (U, R(X)) is referred to as a

nano topological space, and R(X)is a topology on U

that is known as the nano topology on U with regard to

X. In U, the components of R(X)are known as nano

open sets, and a nano open set’s complement is known

as a nano closed set. The components of [R(X)]care

referred to as R′sdual nano topology (X).

Definition 2.5.[5] If τR(X)is nano topology on U

with respect to X, then the family β={U , LR(X),

BR(X)}is the basic for τR(X).

Remark 2.6. With respect to X, let (U,τR(X)) be

anano topological space, and let X,...U, and Reach

represent an equivalence relation on U. The family of

equivalence classes of Uby Ris thus denoted by U/R.

Definition 2.7. [5] If (U,τR(X)) is anano topolog-

ical space regarding X, where X⊆Uand if A⊆U,

then:

(i) The union of all the set Ais nano open subsets is

the definitionof the set’s nano interior, which is rep-

resented by the symbol nInt(A). This means that the

greatestnano open subset of Ais nInt(A).

(ii) The intersection of all nano closed sets contain-

ing Ais known as the nano closure of the set A, and it

is represented by the symbol nCI(A). This means that

the smallest nano closed set that contains Ais nCI(A).

Definition 2.8. ([5], [9], [10])Let (U,τR(X))be a

nano topological space and A⊆U. Then Ais said to

be :

(i)Nano regular open if A=nInt(nCI(A)),

(ii) Nano α-open if A⊆nInt(nCI(nInt(A))),

(iii) Nano semi-open if A⊆nCI(nInt(A)),

(iv) Nano preopen if A⊆nInt(nCI(A)),

(v) Nano γ-open (or nano b-open ) if A⊆nCI(nInt(A))

∪nInt(nCI(A))

(vi) Nano β-open if A⊆nCI(nInt(nInt(A)))

Nano regular closed (resp. nano-closed,nano semi-

closed, nano preclosed, nano-close, nano-close)sets

are the complement of a nano regular open (resp.

nano-open, nano semi-open, nano preopen, nano-

open, nano-open) set. NSO stands for the family of

all nano semi-open sets in a nano topological space

(U,τR(X))(U, X).

3 Nano ideal topological spaces

The nano lacal function in a nano ideal topological

space is examined in this section.

Definition 3.1. [6] Let (U,τR(X), I ) be anano ideal

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topological space. Aset operatar ()∗

n:P(U)→P(U) is

called the nano local function. And for asubset A⊆

U,A∗

n(I, τR(X))= {x∈U : Gχ∩A/∈I, for every Gx

∈τR(X)}is called the nano local function of Awith

respect to Iand τR(X), we shall merely write A∗

nfor

A∗

n(I, τR(X)).

Example 3.2. Let (U,τR(X))be anano topological

space with an ideal IonUand for every A⊆U:

(i) If I = {ϕ}, then A∗

n=nCI(A),

(ii)If I=P(U), then A∗

n=ϕ.

The following theorem contains many basic and

useful facts concerning the nano local function.

Theorem 3.3. [7] Let (U,τR(X)) be anano topolog-

ical space with an ideal I,Jon Uand A, B be subsets

of U. Then the subsequent statements are true:

(i) If A⊆B, then A∗

n⊆B∗

(ii) If I⊆J, then A∗

n(J)⊆A∗

n(I),

(iii) A∗

n=nCI(A∗

n)⊆nCI(A), this means that A∗

nisa

nano closed subset of nCI( A ),

(iv) (A∗

n)∗

n⊆A∗

(v) (A∪B)∗

n=A∗

n∪B∗

(vi) (A∩B)∗

n⊆A∗

n∩B∗

(vii) A∗

n−B∗

n=(A−B)∗

n-B∗

n⊆(A−B)∗

(viii) If V∈τR(X), thenV∩A∗

n= V ∩(V∩A)∗

⊆(V∩A)∗

(ix) If E⊆I , then (A∪E)∗

n=A∗

n=(A−E)∗

Proof .

(i) Let A⊆Band x /∈A∗

n, Then G∈τR(X)contains

x such that G∩B∈I. Since A⊆B, G ∩A∈Iand

hence x/∈A∗

n, Thus A∗

n⊆B∗

(ii) Let x ∈A∗

n(J), then for every G∈τR(X),G∩A

/∈J, this implies that G∩A/∈I, so x∈A∗

n(I). Hence

A∗

n(J)⊆A∗

n(I).

(iii) In general A∗

n⊆nCI( A∗

n). Let x∈nCI( A∗

n).

Then A∗

n∩G=ϕfor every Gx∈τR(X). Therefore,

there exist some y∈A∗

n∩Gand G∈τR(X)con-

taining x. Since y∈A∗

n,A∩G/∈Iand hence nCI(

A∗

n)⊆A∗

n,A∗

n=nCI(A∗

n).

(iv) Let x ∈(A∗

n)∗

n,then for every G∈τR(X)con-

taining x , G ∩A∗

n/∈Iand hence G∩A∗

n=ϕ. Let

y∈G∩A∗

n. Then G∈τR(X)containing yand

y∈A∗

n. Hence G∩A∗

n/∈Iand x ∈A∗

n. This im-

plies that (A∗

n)∗

n⊆A∗

(v) Since A⊆A∪Band B⊆A∪B, then from

(i), A∗

n⊆(A∪B)∗

nand B∗

n⊆(A∪B)∗

n.Hence

A∗

n∪B∗

n⊆(A∪B)∗

n. Conversely, let x ∈(A∪

B)∗

n. Then for every nano open set Gof x such that

(G∩A)∪(G∩B) = G∩(A∪B) /∈I. There-

fore, G∩A/∈Ior G∩B/∈I. This implies that

x∈A∗

nor x ∈B∗

n. That is x ∈A∗

n∪B∗

n. There-

fore, we have (A∪B)∗

n⊆A∗

n∪B∗

n. Thus we get

(A∪B)∗

n⊆Ankm∗∪B∗

(vi) Since A∩B⊆Aand A∩B⊆B, then by

(i), (A∩B)∗

n⊆A∗

nand (A∩B)∗

n⊆B∗

n. Hence

(A∩B)∗

n⊆A∗

n∩B∗

(vii) Since A∪B= (A−B)∪B, by (i), (A∪B)∗

[(A−B)∪B]∗

n, by(v), A∗

n∪B∗

n= (A−B)∗

n∪B∗

nand

hence A∗

n⊆(A−B)∗

n∪B∗

n, therefore A∗

n−B∗

n⊆

(A−B)∗

(viii) If V∈τR(X)and x ∈V∩A∗

n. Then x ∈V

and x ∈A∗

n. Let Gbe any nano open set containing

x. Then G∩V∈τR(X)and G∩(V∩A)=(G∩

V)∩A/∈I. This shows that x ∈(V∩A)∗

nand hence

V∩A∗

n⊆(V∩A)∗

n. Hence V∩A∗

n⊆V∩(V∩A)∗

By (i), (V∩A)∗

n⊆A∗

nand V∩A∗

n⊇V∩(V∩A)∗

Therefore V∩A∗

n=V∩(V∩A)∗

n⊆(V∩A)∗

(ix)Since A−E⊆A, by (i), (A−E)∗

n⊆A∗

n. Also,

by (v), (A∪E)∗

n=A∗

n∪E∗

n=A∗

n∪ϕ=A∗

n. Hence

(A−E)∗

n⊆A∗

n= (A∪E)∗

n, since E∗

n=ϕ.

The instance that follows demonstrates that the con-

verse implications of the preceding theorem’s (i), (ii)

and (iii) do not generally hold.

Example 3.4. Let U={a, b, c, d}be the universe.

(i) If X={a, b};U/R={{a},{c},{b, d}} are

the family of equivalence classes of Uby the equiv-

alence relation R. One can deduce that LR(X) =

{a}, HR(X) = {a, b, d}and BR(X) = {b, d}, then

τR(X) = {U, ϕ, {a},{b, d},{a, b, d}}. Let I=

{ϕ, {a}}. For A={A, C}and B={a, d}, we

have A∗

n={c}, B∗

n={b, c, d}, that is A∗

n⊆B∗

n, but

A⊈B. Also, let I={ϕ, {a}} and J={ϕ, {b}}.For

A, it is obvious that, for A={a, c, d}.A∗

n(I) =

{b, c, d}, A∗

n(J) = {a, b, c, d}=U, that is A∗

n(I)⊆

A∗

n(J)while I⊈J.

(ii) Let X={a, d};U/R={{b},{d},{a, c}}.

One can draw the conclusion that LR(X) =

{d}, HR(X) = {a, c, d}and BR(X) = {a, c}, then

τR(X) = {U, ϕ, {d},{a, c},{a, c, d}}. Let I=

{ϕ, {{d}}. For A={b, d}, we have nCl(A) =

nCl({b, d}) = {b, d}, A∗

n={b, d}∗

n={b}and

nCl(A∗

n) = nCl({b}) = {b}. Therefore, nCl(A)⊈

A∗

n=nCl(A∗

n).

Theorem 3.5. Let (U, τR(X), I)be anano ideal

topological space and A, B be subsets of U. Then:

(i) A∗

nis anano closed.

(ii) ϕ∗

n=ϕ,

(iii) (U−E)∗

n=U∗

nif E∈I,

(iv) [U−(A−E)]∗

n= [(U−A)∪E]∗

n, if E∈I.

Proof.

(i) It is clear.

(ii) Obvious, since ϕalways belongs to I.

(iii) Let x ∈(U−E)∗

n, then for every nano open

neighbourhood Gcontaining Gx∩(U−E) /∈I, im-

plies, (Gx∩U)−(Gx∩E) /∈I, implies, Gx∩U/∈I,

so x∈U∗

n. Thus (U−E)∗

n⊆U∗

n. Also, let x∈U∗

implies, Gx∩U/∈I, for every nano open neighbour-

hood Gcontaining x, implies, Gx∩(U−E) /∈I, so

x∈(U−E)∗

nand thus concludes the proof.

(iv) Follow by using Theorem 3.3(ix).

Definition 3.6. Let (U, τR(X)) be anano topolog-

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ical space with an ideal Ion U, the set operator

nCl∗is called anano ∗-closure and is defined as :

nCl∗(A) = A∪A∗

n, for A⊆X.

Theorem 3.7. [11] The set operator nCl∗meets the

requirement listed below:

(i) A⊆nCl∗(A),

(ii) nCl∗(ϕ) = ϕand nCl∗(U) = U,

(iii) If A⊆B, then nCl∗(A)⊆nCl∗(B),

(iv) nCl∗(A)∪nCl∗(B) = nCl∗(A∪B),

(v) nCl∗(nCl∗(A)) = nCl∗(A).

Definition 3.8. ([5], [7]) Let (U, τR(X), I)be anano

ideal topological space, then A⊆Uis said to be :

(i) Nano regular I-open (nano RI-open ) if A=

nInt(nCl∗(A)),

(ii) Nano regular I-closed(nano RI-closed) if its com-

plement is nano RI-open

(iii) Nano semi I-open if A=nCl(nInt∗(A)),

(iv) Nano semi I-closed if its complement is nano

semi I-open ,

(v) Nano αI-open if A⊆nInt(nCl∗(nInt(A))),

(vi) Nano αI-closed if its complement is nano αI-

open ,

(vii) Nano pre-I-open if A⊆nInt(nCl∗(A)),

(viii) Nano pre-I-closed if its complement is nano

pre-I-open ,

(ix) Nano βI-open if A⊆nCl(nInt(nCl∗(A))),

(x) Nano βI-closed if its complement is nano βI-open

Definition 3.9. [6] An ideal Iin anano ideal topo-

logical space (U, τR(X), I)is called τR(X)- codense

ideal if τR(X)∩I={ϕ}.

Definition 3.10. [6] Asubset Ain anano ideal topo-

logical space (U, τR(x), I)is said to be :

(i) Nano ⋆-dense-in-itself if A⊆A∗

(ii) Nano ⋆-closed if A∗

n⊆A,

(iii) Nano ⋆-perfect if A=A∗

(iv) Nano I-dense if A∗

n=U.

The connections between the aforementioned nano

sets are depicted in the diagram below.

Nano ⋆-dense-in-itself ⇐Nano ⋆-perfect ⇒

Nano ⋆-closed

The examples that follow demonstrate that the dia-

gram’s converse implications cannot be satisfied.

Example 3.11. Let U={a, b, c, d}be the uni-

verse, U/R={{a},{c},{b, d}} be the family of

equivalence classes of Uby the equivalence rela-

tion Rand X={a, b}. Then one can deduce that

LR(X) = {a}, HR(X) = {a, b, d},then the nano

topology τR(X) = {U, ϕ, {a},{b, d},{a, b, d}}. For

I ={ϕ, {a}}, we have :

(i) If A={a, c}, then parimala et al. [6] observed

that A∗

n={c}. Here Ais anano ⋆closed, but not

nano ⋆-perfect .

(ii) If B={c, d}, then we have B∗

n={b, c, d}. So

Bis anano ⋆-dense-in-itself but not nano ⋆-perfect.

4 More on nano I-open sets

Definition 4.1. [11] Asubset Ain anano ideal topo-

logical space (U, τR(x), I)is said to be nano I-open

if A⊆nINT (A∗

n).Asubset F⊆(U, τR(X), I)is

said to nano I-closed if its complement U\Fis nano

I-open .

We connote by NIO(U, τR(X), I) = {A⊆U:

A⊆nInt(A∗

n)}or simply write NIO(U, τR(X))

or NIO(U)for N IO(U, τR(X), I)when there is no

possibility of mistake with the ideal.

The largest nano I-open set contained in Ais

known as the nano I-interior of A, denoted by nI-

Int(A).

Remark 4.2. The following example demonstrates

how nano I-openness and nano openness are concep-

tually distinct.

Example 4.3. Let U={a, b, c, d}be the

universe, X={a, b} ⊂ Uand U/R=

{{a},{c},{b, d}}. This can be extrapolated as:

LR(X) = {a},HR(X) = {a, b, d}. Then τR(X) =

{U, ϕ, {a},{b, d},{a, b, d}}. For I={ϕ, {a}}.

(i) Set A={a, b, d}, we have A∗

n={b, c, d}and

nInt(A∗

n) = {b, d}, that is A⊈nInt(A∗

n). As can

be seen, Ais nano open but nano I-open.

(ii) Set B={d}, we have B∗

n={b, c, d}and

nInt(B∗

n) = {b, c, d}, that is B⊆nInt(B∗

n).Ais

nano I-open but not nano open, as evidenced by this.

Remark 4.4. The following example demonstrates

that every nano I-open set is anano preopen set and

that, generally speaking, the opposite is not true.

Example 4.5. Let U={a, b, c, d}be the uni-

verse, and let X={a, b} ⊂ Uand U/R=

{{a},{c},{b, d}}. This can be presumed as:

LR(X) = {a}, HR(X) = {a, b, d}. Then τR(X) =

{U, ϕ, {a},{b, d},{a, b, d}}. For I={ϕ, {a}}

and A={a, b, c}, we have A∗

n={b, c, d}and

nInt(A∗

n) = {b, d}, that is A⊈nInt(A∗

n), but

nCl(A) = Uand nInt(nCl(A)) = U.Ais anano

preopen, but not nano I-open, as demonstratedby this.

Proposition 4.6. Nano I-open also refers to the arbi-

trary union of nano I-open sets.

Proof. Let (U, τR(X), I)be a nano ideal topologi-

cal space and Wi∈NIO(U, τR(X), I)for i∈ ▽;

this implies that for every i∈ ▽, W, ∈nInt((Wi)∗

and so ∪i{Wi:i∈ ▽} ⊆ ∪i(nInt((Wi)∗

n)⊆

nIn(∪i((Wi)∗

n)⊆nIn(∪i((Wi))∗

nfor every i∈ ▽.

Hence ∪i{Wi:i∈ ▽} ∈ NIO(U, τR(X), I)

Theorem 4.7. [11] Let (U, τR(X), I)be anano ideal

topological space and A, B ⊆U.Then:

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(i) If A∈NIO(U, τR(X), I)and B∈τR(X), then

A∩B∈NIO(A),

(ii) If A∈NIO(U, τR(X), I)and B∈

NSO(U, τR(X)), then A∩B∈NSO(A),

(iii) If A∈NIO(U, τR(X), I)and B∈τR(X), then

A∩B⊆nInt(B∩(B∩A)∗

n),

Proposition 4.8.For anano ideal topological space

(U, τR(X), I)and A⊆U, we have:

(i) If I=ϕ, then A∗

n=nCl(A).and hence each of

nano I-open set and nano preopen sets coincide.

(ii) If I=P(U), then A∗

n=ϕ. and hence Ais anano

I-open if and only if A=ϕ.

Proposition 4.9.For any nano I-open set Aof anano

ideal topological space (U, τR(X), I), we have A∗

nInt(A∗

n)∗

Proposition 4.10. If A⊆(U, τR(X), I)is anano I-

closed, then A⊇(nInt(A))∗

Proof. It results from Theorem 3.3 and the concept of

namo I-closed sets (iii).

Remark 4.11. The idea of nano I-closeness creates

asignificant departure from the notion of nano topol-

ogy in general.

Proposition 4.12. If A⊆(U, τR(X), I), we have

((nInt(A))∗

n)c=nInt((Ac)∗

n)in general (Example

4.13), where Acdenotes the complement of A.

Example 4.13. Let U={a, b, c, d}be the universe,

X={a, c} ⊂ U, let U/R={{a},{d},{b, c}}.

The following may be inferred as: LR(X) =

{a}, HR(X) = {a, b, c}and BR(X) = {b, c}. Then

τR(X) = {U, ϕ, {a},{b, c},{a, b, c}}. For I=

{ϕ, {c},{d},{c, d}}. Set A={a, d}, then it is

clear that nInt(A) = {a}and ({a})∗

n={a, d},

so ((nInt(A))∗

n)c={b, c}. But nInt(Ac) =

nInt({b, c}) = {b, c}, so (nInt(Ac))∗

n) =

({b, c})∗

n={b, c, d}. This also complies with propo-

sition 4.12.

Proposition 4.14. If A⊆(U, τR(X), I)is anano I-

open and nano semi- closed, then A=nInt(A∗

n).

Proof. Theorem 3.3 (iii) dictates this.

Proposition 4.15. Every nano I-open set is nano pre-

I-open.

Proof. Let (U, τR(X), I)be anano ideal topological

space and let A⊆Ube nano.

Proposition 4.16. We have the following for asubset

Aof anano ideal topological space (U;τR(X); I):

(i) If Ais nano ⋆-closed and A∈NIO(U), then

nLnt(A) = nInt(A∗

n).

(ii) Ais nano ⋆-closed if and only if Ais nano open

and nano I-closed .

(iii) If Ais nano ⋆-perfect , then A=nInt(A∗

n(In)),

for every A∈NIO(U, τR(X)).

(iv) If Ais nano regular closed and nano I-open,

then A∗

n(In) = nInt(A∗

n(In)), where Inis the ideal

of nano nowhere dense sets (In={A⊆U:

nlnt(nCl(A))}=ϕ).

Proof. (i), (ii), and (iii) are obvious.

(vi) Is implied by the description of nano I-open and

the assumption that Ais nano regular closed if and

only if A=A∗

n(In).

Proposition 4.17. The implications of various weak

nano open set types as stated above in the nano ideal

topological space (U;τR(X); I)are depicted in Fig-

ure 2.

Nonetheless, Examples 2.1 and 2.2[9] and the sub-

sequent example show that the converses of the pre-

ceding diagram’s assumptions are not generally true.

Example 4.18. Let U={a, b, c, d}be the uni-

verse and U/R={{b},{d},{a, c}} and X=

{a, d} ⊂ U. Hence, it entails that: τR(X) =

{U, ϕ, {d},{a, c},{a, c, d}}.

(a) If I={ϕ, {d}}. Then:

(i) A={a, d}is nano pre-I-open but not nano αI-

open.

(ii) B={a, b, c}is nano semi-I-open but not nano

αI-open.

(iii) Also B={a, b, c}is nano βI-open but not nano

pre-I-open.

(iv) C={b, d}is nano semi-open but not nano semi-

I-open.

Figure 2: Weak nano open sets

(b) If I={ϕ, {a}}. Then A={a, d}is nano pre-

open but not nano pre-I-open.

is nano β-open but not nano βI-open.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.31

Arafa A. Nasef, Radwan Abu-Gdairi

E-ISSN: 2224-2880

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5 Conclusions

We anticipate that this research is merely the start of

a new framework. Many people will be motivated to

contribute to the development of nano ideal topology

in the area of mathematical nanostructures.

Acknowledgments

The authors express their gratitude to Zarqa

University-Jordan, as this research is funded by

them. The authors also extend their thanks to

the Editor-in-Chief, Area Editor, and referees for

their valuable comments and suggestions, which

significantly enhanced the quality of this work.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

The authors express their gratitude to Zarqa

University-Jordan, as this research is funded by them.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.31

Arafa A. Nasef, Radwan Abu-Gdairi

E-ISSN: 2224-2880

287

Volume 23, 2024