On Some Characteristics of Generalized γ-Closure Spaces

Abstract: In this paper, we study that a pointwise symmetric γ-isotonic (γIso), γ-closure (γCl) mapping is

uniquely specified by the pairs of sets it separates. Then, we demonstrate that when the γCl mapping of the

domain is -γIso and the γCl mapping of the co-domain is γIso and pointwise γ-symmetric (γsym), mappings

that only separate already separated pairs of sets are γ-continuous.

Key-words: - γCl separated; γCl mapping; γ-continuous mappings.

Received: April 9, 2024. Revised: September 4, 2024. Accepted: September 24, 2024. Published: October 22, 2024.

2020 AMS Classification Codes: 54A05, 54A10, 54B05,54C10, 54D10.

1 Introduction

The Nobel Prize 2016 in Physics was jointly

bestowed upon three researchers in recognition of

their work on phase transitions and topological phases

of matter. As this occurrence, people are more

aware of the necessity of gaining further topological

understanding.

General topological spaces are becoming more and

more important in several fields of use, for example

data mining, [1]. Information systems are the

essential tools used in any real-world field to learn

from data. Mathematical models can be made for both

quantitative and qualitative data based on topological

structures that describe how data is collected. So far,

there have been a lot of attempts by topologists to

use the idea of closure spaces to investigate different

topological problems, [2, 3, 4]. Within this context,

the symbols (U1;τ)and (U2;σ)or just U1andU2

stand for topological spaces. Let K1⊆U1is called

a (γ−open, [5]) or (b−open, [6]) or (sp-open,

[7]), if K1⊆(Cl(Int(K1))) ∪(Int(Cl(K1))). The

complement of a γ−open set is called γ−closed.

The intersection of all γ−closed sets containing a set

K1is called the γCl(K1).

We may not have at the moment an application of

theoretical mathematics that is being formulated, but

someone will find an application for it and who will

be fluent in using it, and that is why theoretical

mathematics is very important. Recently, a lot

of research has been published in Gamma, Nano

Topology and Soft, [8]. Previously, we have used

topology to study the similarity of DNA sequences

and to identify mutations in genes, chromosomes,

[9]. Many topologists studied topological models

in medicine, [10, 11]. We also used topology to

study the recombination of DNA and to form a

mathematical model for the recombination process.

El-Sharkasy[12] used τto study the recombination of

DNA and it was better as it gives more than one space.

In biomathematics, topological concepts can be used

to build flexible mathematical models.

2 Generalized γCl structures

Throughout this section, we will proceed under the

assumption that set W is a nonempty finite universal

set and that set 2Wis its power set.

Definition 2.1. A closure structure ([13], [14], [15])

is a pair (W, Cl)and Cl : 2W→2Wis a mapping

associating with ∀U1⊆Wand Cl(U1)⊆W, called

the closure of U1, where:

(1) Cl(ϕ) = ϕ,

(2) U1⊆Cl(U1),

(3) Cl(U1∪U2) = Cl(U1)∪Cl(U2),

(4) The interior Int(U1) of U1is (Cl(Uc

1))c,

(5) The set U1is a neighborhood of an element u1∈

Wif u1∈Int(U1),

(6) U1is closed if U1=Cl(U1),

(7) U1is open if U1=Int(U1).

Because the subsequent lemmas’ proofs are clear,

we omit them.

Lemma 2.2. In a closure structure (W, Cl), the

following hold:

M. BADR1, RADWAN ABU-GDAIRI2

1Department of Mathematics, Faculty of Science,New Valley University,

EGYPT

2Department of Mathematics,Faculty of Science,

Zarqa university, zarqa 13132,

JORDAN

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(1) U1is open if and only if Uc

1is closed, for any

U1⊆W.

(2) If U1⊆U2, then Cl(U1)⊆Cl(U2)and

consequently Int(U1)⊆Int(U2).

(3) Cl(∩

i∈I

U1i)⊆∩

i∈I

Cl(U1i).

Lemma 2.3. The open sets of a closure structure

(W, Cl), satisfy

(1) ϕand Ware open,

(2) If U1,U2is open, therefore (U1∩U2)is open,

(3) If U1iis open, ∀i∈I, then ∪

i∈I

U1iis open.

Definition 2.4. (1) The generalized γCl structure is

a pair (U1, γCl)consisting of a set U1, a γCl

mapping (γCl), a mapping from the power set

of U1to itself.

(2) γCl of K1⊆U1, denoted γCl(K1)is the image

of K1under γCl.

(3) The γ-Exterior (γExt)of K1is γExt(K1) =

U1\γCl(K1)and γ-Interior (γInt)of K1is

γInt(K1) = U1\{γCl(U1\K1)}.

(4) K1is γ-closed when K1=γCl(K1),K1is

γ-open when K1=γInt(K1)and Mis a

γ-neighborhood (γnbd)of u1if u1∈γInt(M).

Definition 2.5. AγCl mapping defined on U1is

called:

(1) γ-grounded if γCl(ϕ) = ϕ.

(2) γIso if γCl(K1)⊆γCl(K2)for K1⊆K2.

(3) γ-enlarging if K1⊆γCl(K1),∀K1⊆U1.

(4) γ-idempotent if γCl(K1) = γCl(γCl(K1)),

∀K1⊆U1.

(5) γ-sublinear if γCl(K1∪K2)⊆γCl(K1)∪

γCl(K2),∀K1, K2⊆U1.

(6) γ-additive if ∪i∈U1γ(K1i) = γCl(∪i∈IK1i)

for K1i⊆U1.

Definition 2.6. (1) A, B ⊆U1are called γCl-

separated(γCl- sep) in a generalized γCl

structure (U1, γCl) if K1∩γCl(K2) = ϕand

γCl(K1)∩K2=ϕ, or a similar expression, if

K1⊆γExt(K2)and K2⊆γExt(K1).

(2) γExt points are called γCl- sep in a generalized

γCl structure (U1, γCl)if ∀K1⊆U1and ∀u1∈

γExt(K1),K1and {u1}are γCl −sep.

3 Some Basic Properties

Theorem 3.1. Let (U1, γCl)be a generalized

γ-closure structure in which γExt points are γCl-sep

and let K1be the pairs of γCl-sep sets in U1.

Therefore, ∀K1⊆U1, the γCl of K1is γCl(K1) =

{u1∈U1: ({u1}, K1) /∈K1}.

Proof. . For any generalized γCl structure

γCl(K1)⊆ {u1∈U1: ({u1}, K1) /∈K}.

Indeed, Suppose that u2/∈ {u1∈U1:

({u1}, K1) /∈K}which is, ({u2}, K1)∈K,

therefore {u2}∩γCl(K1) = ϕ, then u2/∈γCl(K1).

Let u2/∈γCl(K1). According to the hypothesis,

({u2}, K1)∈K. Hence u2/∈ {u1∈U1:

({u1}, K1) /∈K}.

Definition 3.2. (1) A γCl mapping defined on a

set U1is called pointwise γsym when, for any

u1,u2∈U1, if u1∈γCl({u2}), then u2∈

γCl({u1}).

(2) A generalized γCl structure (U1, γCl)is said to

be R0γwhen, for any u1, u2∈U1, if u1is in

every γnbd of u2, then u2is in every γnbd of u1.

Corollary 3.3. Let (U1, γCl)be a generalized γCl

structure in which γExt points are γCl-sep. Then

γCl is pointwise γsym and (U1, γCl)is R0γ.

Proof. . Assume that γExt points be γCl-sep in

(U1, γCl). If u1∈γCl({u2}), then {u1}and

{u2}are not γCl-separated. This means that u2∈

γCl({u1}). Then, γCl is pointwise γsym. Assume

that u1belongs to every γnbd of u2, that is, U1∈M

whenever y∈γInt(M). Letting K1={U1\M}

and rewriting in the other direction, u2∈γCl(K1)

whenever u1∈K1. Suppose that u1∈γInt(M),

u1/∈γCl(U1\M). So u1is γCl-sep from {U1\M},

and hence, γCl({u1})⊆M,u1∈({u1}), so u2∈

γCl({u1})⊆M. Hence (U1, γCl)is R0γ.

Note that these three axioms are not equal with one

another in general; nevertheless, they are equivalent

with one another when γCl mapping is γIso

Theorem 3.4. If (U1, γCl)is a generalized γCl

structure with γCl γIso, then the next statement are

equivalent:

(1) γExt points are γCl-sep.

(2) (γCl)is pointwise γsym.

(3) (U1, γCl)is R0γ.

Proof. . Let (2) be true. Assume that K1⊆U1, and

u1∈γExt(K1). Then, as γCl is γIso,∀u2∈K1,

u1/∈γCl({u2}), and hence, u2/∈γCl({u1}). Thus

K1∩γCl{u1}=ϕ. Then (2) →(1), and by

Corollary 3.1,(1) →(2). Let (2) be true and suppose

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that u1, u2∈U1since u1is in each γnbd of u2,

i. e., u1∈Mwhen u2∈γInt(M). Therefore

u2∈γCl(K1)when u1∈K1, and especially since

u1∈ {u1},u2∈γCl({u1}).As a consequence,

u1∈γCl({u2}). Hence if u2∈K2, therefore

u1∈γCl({u2})⊆γCl(K2), as γCl is γIso. Then,

if u1∈γCl(K3), then u2∈K3, that is, u2is in each

γnbdofu1. Then, (2) implies (3).

Now, assume that (U1, γCl)is R0γand u1∈

γCl({u2})such that γCl is γIso,u1∈γCl(K2)

whenever u2∈K2or, u2is in each γnbd of u1

where (U1, γCl)is R0γ,u1∈γInt(M). Then,

u2∈γCl(K1)whenever u1∈K1, and especially

since u1∈ {u1},u2∈γCl({u1}). Hence, (3) =⇒

(2).

Theorem 3.5. Let Wbe a set of unordered pairs of

subsets of a set of U1. Then:

(1) If K1⊆K2and (K2, K3)∈W, then

(K1, K3)∈W,∀K1, K2, K3⊆U1;

(2) If ({u1}, K2)∈W,∀u1∈K1and

({u2}, K1)∈W, ∀u2∈K2, then (K1, K2)∈

W,∀K1, K2, K3⊆U1. Then there is a unique

pointwise γsym,γIso and γCl mappings γCl

on U1which γCl-sep the elements of W.

Proof. . Define γCl by γCl(K1) = {u1∈U1:

({u2}, K1) /∈W}for each K1⊆U1. If K1⊆

K2⊆U1and u1∈γCl(K1), then ({u2}, K1) /∈

W, thus ({u1}, K2) /∈W, meaning that, u1∈

γCl(K2). Then, γCl({u1})is γIso, Moreover,

u1∈γCl({u2})iff ({u1},{u2}) /∈Wiff u2∈

γCl({u1}). Hence γCl is pointwise γsym. Let

(K1, K2)∈W. Therefore K1∩{γCl(K2)}=

K1∩{u1∈U1: ({u1}, K2) /∈W}={u1∈K1:

({u1}, K1) /∈W}=ϕ. Similarly, γCl(K1)∩K2=

ϕ. Then, if (K1, K2)∈W, then K1, and K2are

γCl-sep.

Now, Let K1and K2be γCl-sep. Then {u1∈K1:

({u1}, K2) /∈W}=K1∩γCl(K2) = ϕand {u1∈

K2: ({u2}, K1) /∈W}=γCl{K1∩K2=ϕ.

Thus, ({u1}, K2)∈W, ∀u1∈K1and ({u2}, K1)∈

W, ∀u2∈K2. Then, (K1, K2)∈W.

Many features of γCl mappings can be stated wise

the sets they separate, as shown below:

Theorem 3.6. If Wis the pairs of γCl-sep sets of a

generalized γCl structure (U1, γCl)in which γExt

points are γCl-sep, then γCl is

(1) γ-grounded iff ∀u1∈U1,({u1}, ϕ)∈W.

(2) γ-enlarging iff ∀(K1, K2)∈W,K1∩K2=ϕ.

(3) γ-sub linear iff (K1, K2∪K3)∈Wwhenever

(K1, K2)∈Wand (K1, K3)∈W.

In addition, if γCl is γ-enlarging and for

K1, K2⊆U1.({u1}, K1) /∈Wwhenever

({u1}, K2) /∈W}and ({u2}, K1) /∈W, ∀u2∈

K2, then γCl is γ-idempotent. Also,if γCl-Iso

and γ-idepotent, then ({u1}, K1) /∈W

whenever ({u1}, K1) /∈Wand ({u2}, K1) /∈

W, ∀u2∈K2.

Proof. . By Theorem 3.1, γCl(K1) = {u1∈

U1: ({u1}, K1) /∈W}for each K1⊆U1.

Let ∀u1∈U1,({u1}, ϕ)∈W. Therefore

γCl(ϕ) = {u1∈U1,({u1}, ϕ) /∈W}=ϕ.

Then γCl is γ-grounded. Conversely, if ϕ=

γCl(ϕ) = {u1∈U1,({u1}, ϕ) /∈W}, and hence

({u1}, ϕ)∈W, for every u1∈U1. Let for each

(K1, K2)∈W,K1∩K2=ϕ. Since ({a}, K1) /∈W

if a∈K1,K1⊆γCl(K1),∀K1⊆U1. Then

γCl is γ-enlarging. Conversely, suppose that γCl

is γ-enlarging and (K1, K2)∈W. Therefore

K1∩K2⊆γCl(K1)∩K2=ϕ.

Let (K1, K2∪K3)∈Wwhenever (K1, K2)∈W

and (K1, K3)∈W, and Let u1∈U1and

K2, K3⊆U1where ({u1}, K2∪K3) /∈W.

Therefore ({u1}, K2) /∈Wor ({u1}, K3) /∈W.

Thus γCl(K2∪K3)⊆γCl(K2)∪γCl(K3).

Hence, γCl is γ-sublinear.

Conversely, let γCl be γ-sublinear,

and (K1, K2),(K1, K3)∈W.

Therefore γCl(K2∪K3)∩K1⊆

(γCl(K2)∪{γCl(K3)) ∩K1=

(γCl(K2)∩K1)∪(γCl(K3)∩K1) =

ϕand (K2∪K3)∩γCl(K1) =

(K2∩γCl(K1)) ∪(K3∩γCl(K1)) = ϕ. Assume

that γCl is γ-enlarging and let ({u1}, K1) /∈W

whenever ({u2}, K2) /∈Wand ({u2}, K1) /∈

W, ∀u2∈K2, then γCl(γCl(K1)) ⊆γCl(K1). If

u1∈ {γCl{γCl(K1)}}, hence ({u1}, γCl(K1)) /∈

W,({u2}, K1) /∈W, ∀u2∈γCl(K1), and hence

({u1}, K1) /∈W. Where γCl is γ-enlarging,

therefore γCl(K1)⊆γCl(γCl(K1)). Then,

γCl(γCl(K1)) = γCl(K1),∀K1⊆U1. Lastly,

let’s say that γCl be γIso and γ-idempotent.

Suppose that u1∈U1and K1, K2⊆U1

since ({u1}, K2) /∈Wand ∀u2∈K2,

({u2}, K1) /∈W, then u1∈γCl(K2)and ∀u2∈K2,

u2∈γCl(K1), (i. e., K2⊆γCl(K1). Then,

u1∈γCl(K2)⊆γCl(γCl(K1)) = γCl(K1).

Definition 3.7. If (U1,(γCl)U1)and (U2,(γCl)U2)

are generalized γCl structures, then a mapping T:

U1→U2is called:

(1) γCl preserving if T((γCl)U1(K1)) ⊆

(γCl)U2(T(K1)),∀K1⊆U1.

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(2) γ-continuous if (γCl)U1(T−1(K2)) ⊆

T−1((γCl)U2(K2)),∀K2⊆U2.

Theorem 3.8. Let (U1,(γCl)U1)and (U2,(γCl)U2)

be generalized γCl structures and T:U1→U2be a

mapping:

(1) If Tis γCl preserving and (γCl)U2is γIso, then

Tis γ−continuous.

(2) If Tis γ-continuous and (γCl)U1is γIso, then

Tis γClpreserving.

Proof. . Assume that Tis γCl preserving

and, (γCl)U2is γIso. Let K2⊆U2,

therefore T(γCl)U1((T−1(K2))) ⊆

(γCl)U2((T(T−1(K2))) ⊆(γCl)U2(K2)

hence, (γCl)U1((T−1T(K2))) ⊆

T−1((T(γCl)U1T−1(K2))) ⊆T−1((γCl)U2(K2)).

Let Tbe γ-continuous and (γCl)U1is

γIso. Suppose that K1⊆U1. Then

(γCl)U1(K1)⊆(γCl)U1(T−1(T(K1))) ⊆

T−1((γCl)U2(T−1(K1))). Then, T(γCl)U1(K1)⊆

T(T−1(γCl)U2T(K1)) ⊆(γCl)U2(T(K1).

Definition 3.9. Let (U1,(γCl)U1)and (U2,(γCl)U2)

be generalized γCl structures and T:U1→U2be

a mapping, if ∀K1, K2⊆U1,T(K1)and T(K2)

are not (γCl)U2-sep whenever K1and k2are not

(γCl)U1-sep, Then, we say, Tis non γ-sep.

Notice that Tis non γ-sep iff K1and K2are

(γCl)U1-sep, whenever T(K1)and T(K2)are

(γCl)U2-sep.

Theorem 3.10. Suppose that (U1,(γCl)U1)and

(U2,(γCl)U2)are a generalized γCl structures and

T:U1→U2is a mapping:

(1) If (γCl)U2is γIso and Tis non γ-sep, then

T−1(C)and f−1(D)are ((γCl)U1)-sep for

every Cand Dare ((γCl)U2)-sep.

(2) If ((γCl)U1)is γIso and T−1(C)and T−1(D)

are (γCl)U1-sep for every C,Dis (γCl)U2-sep,

then Tis non γ−sep.

Proof. . Let Cand Dbe (γCl)U2-sep subsets, such

that (γCl)U2is γIso. Suppose that K1=T−1(C)

,K2=T−1(D)then T(K1)⊆C,T(K2)⊆D

and (γCl)U2is γIso,T(K1)and T(K2)are also

(γCl)U2-sep, as a result of this, K1and K2are

(γCl)U2-sep in U1. Assume that (γCl)U1is γIso and

let K1, K2⊆U1where C=T(K1)and D=T(K2)

are (γCl)U1-sep, therefore T−1(C)and T−1(D)

are (γCl)U1-sep and since (γCl)U1γIso,K1⊆

T−1(T(K1)) = T−1(C)and K2⊆T−1(T(K2)) =

T−1(D)are (γCl)U1-sep as well.

Theorem 3.11. Let (U1,(γCl)U1)and ((U2, γCl)U2)

be generalized γCl structures and Assume that T:

U1→U2be a mapping. If Tis γCl preserving, then

Tis non γ-sep.

Proof. . Let Tbe γCl preserving and

K1, K2⊆U1be not (γCl)U1-sep. Assume

that (γCl)U1(K1)∩K2=ϕ. Therefore

ϕ=T((γCl)U1(K1)) ∩K2)) ⊆T((γCl)U1(K1))

∩T(K2)⊆(γCl)U2(T(K1)) ∩T(K2).

Similarly K1∩(γCl)U1(K2)=ϕ, hence

T(K1)∩((γCl)U2(T(K2))) =ϕ. Then T(K1)

and T(K2)are not (γCl)U2-sep.

Theorem 3.12. Let (U1,(γCl)U1)and (U2,(γCl)U2)

be generalized γCl structures which γExt points

(γCl)U2-sep in U2and let T:U1→U2be a

mapping. Then Tis γCl preserving iff Tis non γ-sep.

Proof. . If Tis γCl preserving, then Tis non

γ-sep. Let Tbe non γ-sep and K1⊆U1. If

(γCl)U1(K1) = ϕ, therefore T(γCl)U1(K1)) =

ϕ⊆(γCl)U2(T(K1)). Assume (γCl)U1(K1)=

ϕ. Let WU1and WU2be denote pairs of (γCl)U1−

sep ⊆U1and the pairs of (γCl)U2-sep subsets of

U2, respectively. Suppose u2∈T((γCl)U1(K1))

,u1∈(γCl)U1(K1)∩T−1(u2). Where u1∈

{(γCl)U1(K1),({u2}, K1)}/∈WU1and since Tis

non γ-sep, {{u2}, T (K1)}/∈WU2. Where γExt

points are (γCl)U2-sep, u2∈(γCl)U2(T(K2)).

Thus T(γCl)U1(K1)) ⊆(γCl)U2(T(K1)),∀K1⊆

U1.

Corollary 3.13. Let (U1,(γCl)U1)) and

(U2,(γCl)U2)) be generalized γCl structures with

(γCl)U1gamma-Iso and assume that T:U1→U2

be a mapping, if Tis γ-continuous. Then Tis non

γ-sep.

Proof. . If Tis γ-continuous and (γCl)U1γIso,

then by (Theorem 3.4) if Tis γCl preserving. by

(Theorem 3.7), Tis non γ-sep.

Corollary 3.14. Let (U1,(γCl)U1)and

(U2,(γCl)U2)be a generalized γCl structures with

γIso closure mappings and with (γCl)U2-pointwise

Syγand let f:U1→U2be a mapping. Then fis

γ-continuous iff fnon-γ-sep.

Proof. . Since (γCl)U2is γIso and pointwise Syγ,

Ext γpoints are γCl sep in (U2,(γCl)U2)(Theorem

3.1). Since both γCl mappings are γIso,fis γCl

preserving (Theorem 3.4) iff fis γ-continuous. Thus,

we can apply the (Theorem 3.7).

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4γ-Connected Generalized γCl

structures

Definition 4.1. Let (U1, γCl)be a generalized γCl

structure. U1is called γ-connected if U1is not a union

of disjoint nontrivial γCl-sep pair of sets.

Theorem 4.2. If (U1, γCl)be a generalized γCl

structure with γ-grounded γIso and γ-enlarging γCl,

then the next are equivalent:

(1) (U1, γCl)is γ-connected,

(2) U1can not be a union of nonempty disjoint

γ-open sets.

Proof. .(1)⇒(2): Assume that K1, K2beγ-open sets.

Then U1=K2∪K2and K1∩K2=ϕ. This implies

K2=U1K1and K1is a γ−open set. Thus, K2is

γ−closed, hence K1∩γCl(K2) = K1∩K2=ϕ.

By using the same method, we get γCl(K1)∩K2=

ϕ. Hence, K1and K2are γCl- sep and hence U1is

not γ-connected. A contradiction.

(2)⇒(1): Assume that U1is not γ-connected. Then

U1=K1∪K2, since K1∩K2=ϕ,γCl- sep sets,

i.e, K1∩γCl(K2) = γCl(K1)∩K2=ϕ. We

have γCl(K2)⊂U1−K1⊂K2such that γCl

is γ-enlarging, we get γCl(K2) = K2, then, K2is

γ-closed. By using γCl(K1)∩K2=ϕ. Similarly, it

is clear that K1is γ-closed. Inconsistency.

Definition 4.3. Let (U1, γCl)be a generalized γCl

structure with γ-grounded γIsoγCl. Then, (U1, γCl)

is called a T1−γ-grounded γIso structure if

γCl(u1)⊂ {u1},∀u1∈U1.

Theorem 4.4. If (U1, γCl)is a generalized γCl

structure with γ-grounded γIso γCl,then the next

statement are equivalent:

(1) (U1, γCl)is γ-connected,

(2) Every γ-continuous mapping T:U1→U2is

constant for all T1−γ-grounded γIso structure

U2={0,1}.

Proof. .(1)⇒(2): Assume that U1is γ-connected

and T:U1→U2is γ-continuous and it isn’t

constant. Therefore there is a set U11 ⊆U1where

U11 ={T−1{0}} and U1\U11 =T−1({1}). such

that Tis γ-continuous and U2is T1−γ-grounded

γIso structure, then γCl(U11) = γCl(T−1{0})⊂

T−1(γCl({0})) ⊂T−1({0}) = U11 and hence

γCl(U11)∩{U1\U11}=ϕ. Similarly, we have

U11 ∩γCl(U1\U11) = ϕ. Contradiction, then, Tis

constant.

(2)⇒(1): Let U1is not γ-connected. Therefore

there exists γCl sep sets U11 and U12 where

U11 ∪U12 =U1. We have γCl(U11)⊂U11

and γCl(U12)⊂U12 and U1U11 ⊂U12. Such

that γCl is γIso and U11, U12 are γCl- sep, hence

{γCl(U1\U11)⊂γCl(U12)⊂ {U1\U11}. Let

the structure (U2, γCl)by U2={0,1},γCl(ϕ) =

ϕ,γCl({0}) = {0},γCl({1}) = {1}, and

γCl(U2) = U2, therefore the structure (U2, γCl)

is a T1−γ-grounded γ-Iso structure, we define the

mapping T:U1→U2as T(U11) = {0}and

T(U1\U11) = {1}. Assume that K1=ϕand

K1⊂U2. If K1=U2, and hence T−1(K1) = U1,

γCl({U1}) = γCl(T−1(K1)) ⊂U1=T−1(K1) =

T−1(γCl(K1)). If K1={0}, therefore T−1(K1) =

U11, then γCl(U11) = γCl(T−1(K1)) ⊂U11 =

T−1(K1) = T−1(γCl(K1)). If K1={1}, then

T−1(K1) = {U1\U11}, hence γCl(U1\U11) =

γCl(T−1(K1)) ⊂U1\U11 =T−1(K1) =

T−1(γCl(K1)). Thus, Tis γ-continuous such that

Tis not constant. Inconsistency.

Theorem 4.5. If T: (U1, γCl)→(U2, γCl)and

g: (U2, γCl)→(U3, γCl)are γ-continuous maps,

then, g◦T:U1→U3is γ-continuous.

Proof. . Let Tand gbe γ-continuous. For

every K1⊂U3we get γCl((g◦T)−1(K1)) =

γCl(T−1(g−1(K1))) ⊂T−1(γCl(g−1(K1))) ⊂

T−1(g−1(γCl(K1))) = (g◦T)−1(γCl(K1)). Then,

g◦T:U1→U3is γ-continuous.

Theorem 4.6. Let (U1, γCl),(U2, γCl)be

generalized γCl structures with γ-grounded

γIsoγCl and T: (U1, γCl)→(U2, γCl)

be a γ-continuous mapping from U1onto U2.

If U1is γ-connected, then U2is γ-connected,

g◦T:U1→ {0,1}is constant and then ”g”

is constant mapping. By (Theorem 4.2), U2is

γ-connected.

Proof. . Let {0,1}be a generalized γCl structures

with γ-grounded γIso γCl and g:U2→ {0,1}is

aγ-continuous mapping. Since Tis γcontinuous, by

(Theorem 3.3), g◦T:U1→ {0,1}is γ-continuous.

where U1is γ-connected, g◦Tis constant, then gis

constant. Consequently, U2is γ-connected.

Definition 4.7. Let (U2, γCl)be a generalized γCl

structure with γ-grounded γIso γCl, and more than

one element. A generalized γCl structure (U1, γCl)

with γ-grounded γIso γCl is called U2-γ-connected

if any γ-continuous mapping T:U1→U2is

constant.

Theorem 4.8. Let (U2, γCl)be a generalized

γCl structure with γ-grounded γIso,γ-enlarging

γCl, and more than one element. Then every

U2-γ-connected generalized γCl structure with

γ-grounded γIso is γ-connected.

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Proof. . Assume that (U1, γCl)be a U2-γ-connected

generalized γCl structure with γ-grounded γIso γCl.

Let T:U1→ {0,1}is a γ-continuous mapping, since

{0,1}is a T1γ-grounded γIso structure. Where U2

is a generalized γCl structure with γ-grounded γIso,

γ-enlarging γCl and more than one element, therefore

there is a γ-continuous injection g:{0,1} → U2. By

(Theorem 4.3), g◦T:U1→U2is γ-continuous.

Such that U1is U2-γ-connected, then g◦Tis constant.

Then, Tis constant hence, by (Theorem 4.2), U1is

γ-connected.

Theorem 4.9. Let (U1, γCl)and (U2, γCl)be

a generalized γCl structures with γ-grounded

γIso γCl and T: (U1, γCl)→(U2, γCl)

be a γ-continuous mapping onto U2. If U1is

Z-γ-connected, then U2is Z-γ−connected.

Proof. . Let g:U2→Zis a γ-continuous mapping.

Then g◦T:U1→Zis γ-continuous. Since U1is

Z-γ-connected, therefore g◦Tis constant. Therefor

”g’ is constant. Then U2is Z-γ-connected.

Definition 4.10. the generalized γCl structure

(U1, γCl)is strongly γ-connected if there is no a

countable collection of pairwise γCl- sep sets {Kn}

where U1=∪{Kn}.

Theorem 4.11. Every strongly γ-connected

generalized γCl structure with γ-grounded γIso

γCl is γ-connected.

Theorem 4.12. Let (U1, γCl)and (U2, γCl)be a

generalized γCl structure with γ-grounded γIso γCl

and T: (U1, γCl)→(U2, γCl)be a γ-continuous

mapping onto U2. If U1is strongly γ-connected, then

U2is strongly γ-connected.

Proof. . let U2is not strongly γ-connected. Then,

there exists a countable collection of pairwise γCl

sep sets {Kn}such that U2=∪{Kn}.

Since T−1(Kn)∩γCl(T−1(Km)) ⊂

(T−1(Kn)) ∩(T−1(γCl(Km))) = ϕfor all n=m,

then the collection {T−1(Kn)}is pairwise γCl-

sep. This is a contradiction. Thus, U2is strongly

γ-connected.

Theorem 4.13. Let (U1, γCl)U1,(U2, γCl)U2be a

generalized γCl structures. Then the subsequent are

equivalent for a mapping T:U1→U2

(1) Tis γ-continuous,

(2) T−1(γInt(K2)) ⊆γInt(T−1(K2)),∀K2⊆

U2.

Theorem 4.14. If (U1, γCl)is a generalized

γCl structure with γ-grounded γIso γCl, then

(U1, γCl)is strongly γ-connected iff (U1, γCl)is

U2-γ-connected for any countable T1−γ-grounded

γIso structure (U2, γCl).

Proof. . (Necessity): Assume that (U1, γCl)is

strongly connected and (U1, γCl) is not U2−

γ-connected for some countable T1−γ-grounded.

There is a γ-continuous mapping T:U1→U2

this isn’t constant and as a result H=T(U1)is a

type of countable set that contains several elements.

for every an∈H, there exists Kn⊂U1since

Kn=T−1(an)hence U2=∪Kn. Such that Tis

γ-continuous and U2is γ-grounded, then ∀n=m,

{Kn}∩γClKm)}=T−1(an)∩γCl(T−1(am)) ⊂

T−1{an}∩T−1(γCl{am})⊂

T−1{an}∩T−1{am}=ϕ. Contradicts, for

strong γ-connectedness of U1. Hence, U1is

U2-γ-connected.

(Sufficiency): Assume that U1is U2−γ-connected

for any countable T1−γ-grounded γIso structure

(U2, γCl). Let U1b not strongly γ-connected.

There is a countable collection of pairwise

γCl sep sets {Kn}where U1=∪Kn. We

take the structure (Z, γCl)such that Zis the

set of integers and γCl :P(Z)→P(Z)is

defined as γCl(H) = H, ∀H⊂Z. Obviously

(Z, γCl)is a countable T1-γ-grounded γIso

structure. Let Kk∈ {Kn}. We define a

mapping T:U1→Zby T(Kk) = {u1}and

T(U1\Kk) = {u2}since u1, u2∈Zand u1=u2.

Where γCl(Kk)∩{Kn}=ϕ, for each n=k,

therefore γCl(Kk)∩(∪{Kn}) = ϕ,n=k

hence, γCl(Kk)⊂ {Kk}. Put ϕ=H⊂Z. If

u1, u2∈Hthen T−1(H) = U1and γCl(T−1(H)) =

γCl(U1)⊂U1=T−1(H) = T−1(γCl(H)). If

u1∈Hand u2/∈H, then T−1(H) = Kkand

γCl(T−1(H)) = γCl(Kk)⊂Kk=T−1(H) =

T−1(γCl(H)). If u2∈Hand u1/∈Hthen

T−1(H) = {U1\Kk}. Since γCl(H) = H, ∀H⊂

Z, then γInt(H) = H,∀H⊂Z. Also,

(U1\Kk)⊂Kn=k{Uk} ⊂ {U1\γCl(Kk)}=

γInt(U1\Kk). Then, T−1(γInt(H)) = U1\Kk=

T−1(H)⊂γInt(U1\Kk) = γInt(T−1(H)).

Alternatively, ∀n=k,Kk∩γCl(kn) = ∅, hence

Kk∩∪{γCl(Kn), n =k}=∅. As a result, it can

be concluded that Kk∩γCl(Kn), n =k) = ∅.

Thus, γCl(U1\Kk)⊂U1\Kk. Such that

γCl(H) = H, ∀H=Z, we get γCl(T−1(H)) =

γCl(U1\Kk)⊂U1\Kk. Hence, Tis γ-continuous.

Since Tis not constant, this goes against what was

stated in Z-γ-connectedness of U1. Hence, U1is

strongy γ-connected.

5 Conclusion

This research explores closure structures in point-set

topology, with emphasis on γ-closure operators.

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DOI: 10.37394/23206.2024.23.70

M. Badr, Radwan Abu-Gdairi

E-ISSN: 2224-2880

680

Volume 23, 2024

We investigate the characteristics of γ-isotonic and

γ-closure mappings and provide useful definitions,

lemma, and propositions related to the γ-closure

structure.

Closure structure in point-set topology gives novel

topological qualities (such as separation axioms,

connectedness, and continuity) that are useful in

studying digital topology, [16]. Thus, we may

emphasize γclosure operators as a branch of them

and their application in quantum physics, [17], and

computer graphics, [18]. In the future, we can use

these results to study the processes of nucleic acid

”mutation, recombination, and crossover.”

This research is funded by Zarqa University, Jordan.

The authors declare no conflict of interest.

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problem to the final findings and solution.

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.70

M. Badr, Radwan Abu-Gdairi

E-ISSN: 2224-2880

681

Volume 23, 2024