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Cite the article: Arif Mehmood Khattak, Zia Ullah, Fazli Amin, Nisar Ahmad Khattak, Shamona Jbeen (2018). Binary Soft Pre-Separation Axioms In Binary Soft Topological Spaces. Matrix Science Mathematic, 2(2) : 18-24.

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Article History:

Received 12 November 2017 Accepted 12 December 2017 Available online 1 January 2018

ABSTRACT

In this article, we introduce binary soft pre-separation axioms in binary soft Topological space along with several properties of binary soft pre τ△i , i = 0; 1; 2, binary soft pre-regular, binary soft pre τ△3 , binary soft pre-normal and binary soft τ△4 axioms using binary soft points. We also mention some binary soft invariance properties namely binary soft topological property and binary soft hereditary property. We hope that these results will be useful for the future study on binary soft topology to carry out general background for the practical applications and to solve the thorny problems containing doubts in different grounds.

KEYWORDS

Binary soft topology, Binary soft pre-open and binary soft pre-closed sets Binary soft pre-separation axioms.

1. INTRODUCTION

Based on a study, the concept of soft sets was first introduced in 1999 as a

general mathematical technique for dealing with uncertain substances [1].

A researcher magnificently applied the soft theory in numerous ways, such

as smoothness of functions, game theory, operations research, Riemann

integration, Perron integration, probability, theory of measurement, and

so on [2]. Point soft set topology deals with a non-empty setX to gether

with a collection τ of sub set X under some set of parameters satisfying

certain conditions. Such a collection τ is called a soft topological structure

onX. General soft topology studied the characteristics of sub set of X by

using the members of. Therefore, the study of point soft topology can be

thought of the study of information. But in the real-world situation there

may be two or more universal sets. Our attempt is to introduce a single

structure which carries the sub sets of X and Y for studying the

information about orderd pair of sub sets of X and Y. Such a structure is

called a binary soft structure from X toY.

In 2016, a group of researchers introduced the notion of binary soft set

Theory on two master sets and studied some basic characteristics [3]. In

prolongation, another researcher planned the idea of binary soft topology

and linked fundamental properties which are defined over two master sets

with appropriate parameters [4]. In continuation, in the present paper we

have defined and explored several properties of binary soft pre τ△i , i =

0 ; 1; 2 binary soft pre regular, binary soft preτ△3, binary soft pre normal

and binary soft pre τ△4 axioms using binary soft points. Also, we have

talked over some binary soft invariance properties i.e. binary soft

topological property and binary soft hereditary property in binary soft

topological spaces.

The arrangement of this paper is as follows: Section 2 briefly reviews some

basic concepts about soft sets, binary soft sets and their related properties;

Section 3 some hereditary properties are discussed in a beautiful way.

Section 4 is devoted to Binary Soft Pre-Separation Axioms. Section 5 is

devoted to Binary Soft Pre-Regular, Binary Soft pre-Normal and Binary

pre-Soft 𝛕∆𝐢 (i=4, 3) Spaces

2. PRELIMINARIES

Definition 1: Let X be an initial universe and let E be a set of parameters

[5]. Let P(X) denote the power set of X and let A be a nonempty subset ofE.

A pair (F, A) iscalled a soft set overX, where F is a mapping given by ∶ A →

P(X) . In other words, a soft set over X is a parameterized family of subsets

of the universe X. For ε ∈ A, F (ε ) may be considered as the set of ε -

approximate elements of the soft set (F, A). Clearly, a soft set is not a set.

Let U1, U2 be two initial universe sets and E be a set of parameters.

LetP(U1), P(U2) denote the power set of U1, U2 respectively. Also,

letA, B, C ⊆ E.

Definition 2: A pair (F, A) is said to be a binary soft set over U1, U2 where

F is defined as below:

F: A → P(U1) × P(U2), F(e) = (X, Y) for each e ∈ A such that ⊆ U1 , Y ⊆ U2

.

Definition 2.1: A binary soft set (G, A) over U1, U2 is called a binary

absolute soft set, denoted byA A⏞ if F(e)= U1, U2 for each e ∈ A.

CODEN: MSMADH

BINARY SOFT PRE-SEPARATION AXIOMS IN BINARY SOFT TOPOLOGICAL SPACESArif Mehmood Khattak1, Zia Ullah2, Fazli Amin2, Nisar Ahmad Khattak3, Shamona Jbeen4 1Department of Mathematics and Statistics, Riphah International University Islamabad, Pakistan 2Department of Mathematics, Hazara University Mansehra, Pakista3Department of Mathematics, Kohat University of Science and Technology, Pakistan4School of Mathematics and System Sciences Beihang University Beijing (China)*Corresponding Author’s E-mail: [email protected]

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Matriks Sains Matematik (MSMK)

DOI : http://doi.org/10.26480/msmk.02.2018.18.24

Matrix Science Mathematic(MSMK) 2(2) (2018) 18-24

mailto:[email protected]


Definition 3: The intersection of two binary soft sets of (F, A) and (G, B)

over the common U1, U2 is the binary soft set (H, C), where C = A ∩ B and

for all e ∈ C

H(e) = {

(X1, Y1) if e ∈ A − B (X2, Y2) if e ∈ B − A

(X1 ∪ X2, Y1 ∪ Y2) if e ∈ A ∩ B

Such that F(e) = (X1, Y1) for each e ∈ A and G(e) = (X2, Y2) for eache ∈ B.

We denote it (F, A) ∪ (G, A) = (H, C)

Definition 5: The intersection of two binary soft sets (F, A) and (G, B) over

a common U1, U2 is the binary soft set (H, C), whereC = A ∩ B, and

H(e) = (X1 ∩ X2, Y1 ∩ Y2) ) for each e ∈ C such that F(e ) = (X1, Y1) for

each e ∈ A and G(e) = (X2, Y2)for eache ∈ B. We denote it as

(F, A) ∩ (G, B) = (H, C)

Definition 6: Let (F, A) and (G, B) be two binary soft sets over a common

U1, U2 . (F, A) is called a binary soft subset of (G, B) if

(i) A ⊆ B ,

(ii) X1 ⊆ X2 and Y1 ⊆ Y2 Such that F(e) = (X1, Y1) , G(e) = X2, Y2 for

eache ∈ A. We denote it as (F, A) ⊆ (G, B).

Definition 7: A binary soft set (F, A) over U1, U2 is called a binary null soft

set, denoted by if F(e) = (φ, φ ) for each e ∈ A.

Definition 8: The difference of two binary soft sets (F, A) and (G, A) over

the

Common U1, U2 is the binary soft set (H, A), where H(e) (X1 − X2, Y1 − Y2)

for each e ∈ A such that(F, A) = (X1, Y1) and (G, A) = (X2, Y2).

Definition 9: Let τ△ be the collection of binary soft sets over U1, U2 then

τ△ is said to be a binary soft topology on U1, U2 if

(i) φ, X ∈ τ△

(ii) The union of any member of binary soft sets in τ△ belongs to τ△ .

(iii) The intersection of any two binary soft sets in τ△ belongs to τ△ .

Then (U1, U2, τ△, E) is called a binary soft topological space over U1, U2 .

Definition 10: Let (U1, U2, τ△, E) be a binary soft topological space on X

over U1 × U2 and Y be non-empty binary soft subset of X.

Then τ△Y= {Y (F, E)/(F, E) ∈ ∆ is said to be the binary soft relative

topology on Yand (Y, τ△Y, E) is called a binary soft subspace of(

(U1, U2, τ∆, E) . We can easily verify that τ△Y is a binary soft topology on Y.

Example1: Any binary soft subspace of a binary soft indiscrete topological

space is binary soft indiscrete topological space.

3. SOME HEREDITARY PROPERTIES

Definition 11: Let (F, A) be any binary soft sub set of a binary soft topological space (U1, U2, τ△, E) then (F, A) is called

1) (F, A) binary soft pre-open set of (U1, U2, τ△, E) if (F, A) ⊆int(cl((F, A) ) and

2) (F, A) binary soft pre-closed set of (U1, U2, τ△, E) if (F, A) ⊇cl(int(F, A)) )

The set of all binary pre-open soft sets is denoted by bspo (U)

Proposition 1. Let (U1, U2, τ∆, E) be a binary soft topological space on X

over U1 × U2 and Y be a non-empty binary soft subset ofX. Then

(U1, U2, τ△Y, α)is subspace of (U1, U2, τ△Y

, E) for eachα ∈ E.

Proof. Let (U1, U2, τ△Y, α) is a binary soft topological space for eachα ∈ E.

Now by definition for any α ∈ E

τ△Y= {YF(α)/(F, E) is binary soft pre − open set}

= {Y ∩ F(α)/(F, E) is binary soft pre − open set }

= {Y ∩ F(α)/(F, E)is binary soft pre − open set}

= {Y ∩ F(α)/F(α) ∈ τ△α}

Thus (U1, U2, τ△Y, α) is a subspace of(U1, U2, τ∆, α).

Proposition 2. Let (U1, U2, τ△Y, E) be a binary soft subspace of a binary soft

Topological space (U1, U2, τ∆, E)and (G, E) be a binary soft pre-open inY. If

Y ∈ τ∆

Then (G, E) ∈ τ∆ .

Proof. Let (G, E) be a binary soft pre-open set in Y, then there exists a

binary soft pre-open set (H, E) in X over U1 × U2 such that (G,E) = Y ∩

(H, E). Now, if Y ∈ τ∆

Then Y ∩ (H, E) ∈ τ∆ by the third axiom of the definition of binary soft topological space and hence (G, E) ∈ τ∆.

Proposition 3. Let (U1, U2, τ△Y, E) be a binary soft subspace of a binary soft

Topological space (U1, U2, τ∆, E) and (G, E) be a binary soft pre-open set of

XoverU1 × U2, then

(i) (G, E) is binary soft pre-open in Yif and only if (G, E) = Y ∩ (H, E) for some(H, E) ∈ τ∆.

(ii) (G, E) is binary soft pre-closed in Y if and only if (G, E) = Y ∩ (H, E) for

some binary soft pre-closed set in (H, E) ∈ X overU1 × U2.

Proof. (i) Follows from the definition of binary soft subspace.

(ii) If (G, E) is binary soft pre-closed in Y then we have (G, E) = Y then we


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have(G, E) = Y − (H, E), for some binary soft pre-open(H, E) ∈ τ△Y, now

(H, E) = Y ∩ (H, E)

For some binary soft pre-open (K, E) ∈ τ∆ for anyβ ∈ E, G(β) = Y(β) −H(β)

= Y − H(β)

= Y − [ Y(β) ∩ K(β)]

= Y − [ Y ∩ K(β)]

= Y − K(β)

= Y ∩ (X − K(β))]

= Y ∩ [K(β)]

= Y(β) ∩ [K(β)]C

Thus (G, E) = Y(β) ∩ [K(β)]CWhere (K, E)Cis binary soft pre-closed set in

X over U1 × U2 as (K, E) ∈ τ∆.

Conversely, assume that (G, E) = Y ∩ (H, E) for some binary soft pre-

closed set (H, E) in X over U1 × U2 which means that(H, E) ∈ τ∆. Now if

(H, E) = X − (K, E) where (K, E) ∈ τ∆ then for anyβ ∈ E,

G(β) = Y(β) ∩ H(β)

= Y ∩ H(β)

= Y ∩ [X − K(β)]

= Y − [Y ∩ K(β)]

= Y(β) − [Y(β) ∩ K(β)]

Thus= Y − [Y ∩ (K, E)]. Since(K, E) ∈ τ∆ , so [Y ∩ (K, E)] ∈ τ∆Y and hence

(G,E) is binary soft closed pre-open set in Y.This finishes the proof.

Let (U1, U2, τ∆, E) be a binary soft topological space. Let (U1, U2, τ∆Y, E)

be a binary soft subspace of (U1, U2, τ∆, E). Let (F, E) ⊆ Y be a binary soft

subset ofY. Then we can find the binary soft pre-closure of (F, E) in the space (U1, U2, τ∆Y

, E) . The binary soft pre-closure of (F, E) in

(U1, U2, τ∆Y, E)

is denoted by (F, E)y

.

Proposition 4. Let (U1, U2, τ∆Y, E) be a binary soft subspace of binary soft

Topological space (U1, U2, τ∆, E) . Let (F, E) ⊆ Y be a binary soft subset ofY. Then we have the following results as follows.

(i) (F, E) y

= Y ∩ (F, E).

(ii) (F, E)∗y = Y ∩ (F, E)∗

(iii) (F, E) y ⊆ Y ∩ (F, E)

Proof. (i) To prove, let (F, E) y

= Y ∩ (F, E). we have (F, E) y

= the binary soft intersection of all the binary soft Pre-closed sets containing (F, E) =

∩{(G, E) y: (G, E) y is τ∆Y-binary soft pre-closed set and (G, E) y⊃ (F, E)} =

∩{Y ∩ (G, E): (G, E) is -binary soft pre-closed set and Y ∩ (G, E) ⊃ (F, E)} =

∩{Y ∩ (G, E): (G, E) is τ∆Y-binary soft pre-closed set and (G, E) ⊃ (F, E)}

=Y ∩{∩ (G,E) : (G,E) is τ∆ -binary soft pre-closed set and (G,E) ⊃ (F,E)}

=Y ∩ (F, E). Thus (F, E) y

= Y ∩ (F, E) .

(ii) To prove that (F, E) y = Y ∩ (F, E)∗ , we know that, (F,E) ey = The binary soft union of all the τ∆Y

-binary soft pre- open Sets contained in (F,E).

= ∪ {(H, E): (H, E) is τ∆Y -binary soft pre-open and (H, E) ⊃ (F, E)}

= ∪ {(H, E)= Y ∩ (K, E): (K,E) is τ∆ -binary soft pre-open set and

Y ∩ (K, E) ⊃ (F,E)}.

Also we know that (F, E) e = Y ∩ [∪(L, E) γ ]: (L, E) is τ∆-binary soft pre-

open set and (L, E)γ ⊆ (F,E)}

Now let (M, E) ∈ Y ∩ (F, E)∗ which implies (M, E) ∈ Y and (M, E) ∈ (F, E)∗

(M, E) ∈ Y and (M, E) ∈∪(L,E) γ ∶ (L, E) γ is τ∆ -binary soft pre-open set and

(L, E) γ ⊆ (F, E)

⟹ (M, E) ∈ Yand (M, E) ∈ (L, E) γ, where (L, E)γi , is τ∆ -pre-open and

(L, E)γi⊆ (F, E)

⟹ (M, E) ∈ Y ∩ (L, E)γi Where (L, E)γi

is τ∆-pre-open and (L, E)γi⊆ (F, E)

that is Y ∩ (L, E)γi⊆ Y ∩ (F, E)

⟹ (M, E) ∈∪ {Y ∩ (K, E): (K, E) is τ∆- pre-open Y ∩ (K, E) ⊆ (F, E)}

⟹ (M, E) ∈ (F, E)∗y. Thus (M, E) ∈ Y ∩ (F, E)∗ which implies

(M, E) ∈ (F, E)∗.

Therefore Y ∩ (F, E)∗ ⊆ (F, E)∗.


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(iii) To prove, (F, E) y ⊆ Y ∩ (F, E)

Now consider (F, E) y = (F, E) y

∩ Y − (F, E)

y

⟹ [Y ∩ (F, E)] ∩ Y ∩ [Y − (F, E)].

Since using the result (i) [Y ∩ (F, E)] ∩ Y ∩ [Y − (F, E)].(since Y ⊆ X).

⊆ Y ∩ (F, E)] ∩ [X − (F, E)]

=Y ∩ (F, E)

Thus (F, E) y ⊆ Y ∩ (F, E)

This finishes the proof.

4. BINARY SOFT PRE-SEPARATION AXIOMS

In this section binary soft pre-separation axioms in Binary Soft

Topological Spaces are discussed.

Definition 12: Let (U1, U2, τ∆, A) be a binary soft topological space of Xover

(U1 × U2) and Fe, Ge ∈ XA such that Fe ≠ Ge. Then the binary soft

topological space is said to be a binary soft pre- τo space denoted as pre-

T∆o.If there exists at least one binary soft pre-open set (F1, A) or (F2, A)

such that Fe ∈ (F1, A) , Ge ∈ (F1, A) or Fe ∈ (F2, A) , Ge ∈ (F2, A).

Definition 13: Let (U1, U2, τ∆, A) be a binary soft topological space of Xover

(U1 × U2) and Fe, Ge ∈ XA such that Fe ≠ Ge. Then the binary soft

topological space is said to be a binary soft pre- τ1 space denoted as pre-

T∆1. If there exists at least one binary soft pre-open set (F1, A) or (F2, A)

such that Fe ∈ (F1, A) , Ge ∉ (F1, A) or Fe ∈ (F2, A) , Ge ∉ (F2, A).

Definition 14: Let (U1, U2, τ∆, A) be a binary soft topological space of

Xover (U1 × U2) and Fe, Ge ∈ XA such that Fe ≠ Ge. Then the binary soft

topological space is said to be a binary soft pre- τ2 space denoted as pre-

T∆2. If there exists at least one binary soft pre-open set (F1, A) or (F2, A)

such that Fe ∈ (F1, A) , He ∈ (F2, A) and (F1, E) ∩ (F2, E) = φA.

Proposition 5. (i) Every pre- T∆1-space is pre- T∆o

space.

(ii)Every T∆2-space is a T∆1

-space.

Proof. (i) is obvious. (ii) If (U1, U2, τ∆, A) is a T∆2-space then by definition,

for Fe, Ge ∈ XA , Fe ≠ Ge there exists at least one binary soft pre-open set

(F1, A) and (F2, A) such that Fe ∈ (F1, A) , He ∈ (F2, A) and

(F1, E) ∩ (F2, E) = φA. Since (F1, E) ∩ (F2, E) = φA ; Fe ∉ (F2, A) and

Ge ∉ (F1, A). Thus it follows that (U1, U2, τ∆, A) is pre- T∆1space.

Note that every pre- T∆1space is pre- T∆o

space. Every pre- T∆2space is pre-

T∆1space.

Proposition 6. Let (U1, U2, τ∆, E) be a binary soft topological space of

Xover (U1 × U2) and Y be a non-empty subset of X. If (U1, U2, τ∆, E) is a

binary soft pre-

T∆oSpace then (U1, U2, τ∆y

, E) is a binary soft pre- T∆ospace.

Proof. Let (U1, U2, τ∆, E) be a binary soft topological space of Xover

(U1 × U2) Now let Fe, Ge ∈ Y such that Fe ≠ Ge. If there exist a binary soft

pre-open set (F1, E) in X such that Fe ∈ (F1, E) andGe ∉ (F1, E). Now if

Fe ∈ Y implies thatFe ∈ Y. So Fe ∈ Y andFe ∈ (F1, E).

HenceFe ∈ Y ∩ (F1, E) = [Y(F1, E)], where, (F1, E) is binary soft pre-open

set. That is(F1, E) ∈ τ∆. let Ge ∉ (F1, E) , this means that Ge ∉ F(β) for

some β ∈ E . Ge ∉ Y ∩ (F1, E)α = Yβ(F1, E)α . Therefore, Ge ∉ Y ∩ (F1, E) =

[Y(F1, E)]. Similarly, it can prove that if Ge ∈ (F2, E) and Fe ∉ (F2, E) then

Ge ∈ [Y(F2, E)] andFe ∈ [Y(F2, E)]. Thus (U1, U2, τ∆y, E) is a binary soft pre-

T∆ospace.

Example2: Let U1 = {c1, c2, c3}, U2 = {m1, m2} E = {e1, e2} and

τ∆ = {X, φ, {(e1({c2}{m2})) , (e2({c1}{m1}))},

{(e1({c1}{m1})) , (e2({c2}{m2}))} , {(e1({c1}{m1}))}, {(e1 ({X} {X})) , (e2({c1}{m1}))}}

where (F1, E) = {(e1({c2}{m2}) , (e2({c1}{m1}))} ,

(F2, E) = {(e1({c1}{m1}) , (e2({c2}{m1}))} ,

(F3, E) = {(e1({c1}{m1}))}

(F4, E) = {(e1 ({X} {X})) , (e2({c1}{m1}))}

Clearly (U1, U2, τ∆, E) is binary sot topological space of X over (U1 × U2).

Note that τ∆1= {X, φ, {(e1({c1}{m1}))}, (e1({c2}{m2}))}

T∆2= {X, φ, {(e2({c1}{m1}))}, (e2({c2}{m2}))} are binary soft topological

spaces on X over (U1 × U2). There are two pairs of distinct binary soft

points namely

Fe1= {(e1({c2}{m2})}, Ge1

= {(e1({c1}{m1})} and Fe2= {(e2({c1}{m1})},

Ge2= {(e2({c2}{m2})}. Then for binary soft pair Fe1

≠ Ge1 of points there

are binary soft open sets (F`1, E) and (F2, E) such that Fe1∈ (F1, E),

Ge1∉ (F1, E) and Ge1

∈ (F1, E), Fe1∉ (F2, E). Similarly for the pair Fe2

≠

Ge2 , there are binary soft pre-open sets (F`1, E) and (F2, E) such that

Fe2∉ (F2, E) , Ge2

∈ (F2, E) and Ge2∉ (F1, E) , Fe2

∈ (F1, E). This shows

that (U1, U2, τ∆, E) is binary soft space pre- T∆1-space and hence a binary

soft pre- T∆ospace. Note that (U1, U2, τ∆, E) is binary soft pre- T∆2

space.

Proposition 7. Let (U1, U2, τ∆, E) is binary soft topological space on X over

(U1 × U2). Then each binary soft point is binary soft pre-closed if and only

if (U1, U2, τ∆, E) is binary soft pre- T∆1space.

Proof. Let (U1, U2, τ∆, E) is binary soft topological space on X over

(U1 × U2). Now to prove let (U1, U2, τ∆, E) is binary soft pre- T∆1space,

suppose binary soft points Fe1≡ (F, E), Ge1

≡ (G, E) are binary soft pre-

closed andFe1≠ Ge1

. Then (F, E)c and (G, E)c are binary soft pre-open

in(U1, U2, τ∆, E). Then by definition (F, E)c = (Fc, E) where Fc(e1) = X −

F(e1) and (G, E)c = (Gc, E) , where Gce1

= X − G(e1). Since F(e1) ∩ G(e1) =

φ. This implies

F(e1) = X − G(e1) = Gce1

∀ e. This impliesF(e1) = (F, E)∈ (G, E)c.

SimilarlyG(e1) = (G, E)∈ (F, E)c. Thus we have (e1) ∈ (G, E)c ,

G(e1) ∉ (G, E)c andF(e1) ∉ (F, E)c, G(e1) ∈ (F, E)c. this prove that

(U1, U2, τ∆, E) is binary soft pre- T1space.

Conversely, let (U1, U2, τ∆, E) is binary soft pre- T∆1space, to prove

thatF(e1) = (F, E)∈ X is binary soft pre-closed, we show that (F, E)c is

binary soft pre-open in(U1, U2, τ∆, E). LetGe1= (G, E)∈ (F, E)c is binary soft

pre-closed. Then Fe1≠ Ge1

, since (U1, U2, τ∆, E) is binary soft pre-

T∆1space, there exists binary soft pre-open set

(L, E) such that G(e1) ∈ (L, E) ⊆ (F, E)c and hence ∪G e1{(L, E),

Ge1∈ (F, E)c }. This proves that (F, E)c is binary soft pre-open in

(U1, U2, τ∆, E) that is Fe1= (F, E) is binary soft pre-closed in(U1, U2, τ∆, E).

Which completes the proof.

Proposition 8. Let (U1, U2, τ∆, E) be a binary soft topological space of X

over (U1 × U2) and Fe, Ge ∈ X such that Fe ≠ Ge . If there exist binary soft

pre-open sets (F1, E), (F2, E) such that Fe ∈ (F1, E) and Ge ∈ (F2, E)c, then

(U1, U2, τ∆, E) is a binary soft pre- T∆ospace and (U1, U2, τ∆, E) is a binary

soft pre- T∆ospace for each e ∈ E.


21


Proof. Clearly Ge1∈ (F1, E)c = (F1

c, E) implies Ge1∉ (F2, E) similarly

Fe ∈ (F2, E)c = (F2c, E) implies Fe ∉ (F2, E) . Thus we have Fe ∈ (F1, E) ,

Ge ∉ (F1, E) orGe ∈ (F2, E) , Fe ∉ (F2, E). This proves (U1, U2, τ∆, E) is a

binary soft pre- T∆ospace. Now for any ∈ E , (U1, U2, τ∆, E) is a binary soft

topological space and Fe ∈ (F1, E) and Ge ∈ (F1, E)cor Ge ∈ (F2, E)

andFe ∉ (F2, E)c. So that Fe ∈ F1(e) , Ge ∉ F1(e) , Ge ∈ F2(e) , Ge ∉ F2(e).

Thus (U1, U2, τ∆, E) is a binary soft pre- T∆ospace.



pre-open sets (F1, E), (F2, E) such that Fe ∈ (F1, E) and Ge ∈ (F1, E)cor

Fe ∈ (F2, E) and Ge ∈ (F2, E) , then (U1, U2, τ∆, E) is a binary soft pre-

T∆ospace and (U1, U2, τ∆, E) is a binary soft pre- T∆o

space for each e ∈ E.

Proof. Clearly Ge ∈ (F1, E)c = (F1c, E) implies Ge ∉ (F2, E) similarly

Fe ∈ (F2, E)c = (F2c, E) implies Fe ∉ (F2, E) . Thus we have Fe ∈ (F1, E) ,

Ge ∉ (F1, E) orGe ∈ (F2, E) , Fe ∉ (F2, E). This proves (U1, U2, τ∆, E) is a

binary soft pre- T∆ospace. Now for any ∈ E , (U1, U2, τ∆, E) is a binary soft

topological space and Fe ∈ (F1, E) and Ge ∈ (F1, E)cor Ge ∈ (F2, E) and

Fe ∉ (F2, E)c. So that Fe ∈ F1(e) , Ge ∉ F1(e) or Ge ∈ F2(e) , Fe ∉ F1(e).

Thus (U1, U2, τ∆, E) is a binary soft pre- T∆ospace.



pre-open sets (F1, E), (F2, E) such that Fe ∈ (F1, E) and Ge ∈ (F1, E)cor

Ge ∈ (F2, E) and Fe ∈ (F2, E)c , then (U1, U2, τ∆, E) is a binary soft pre-

T∆ospace and (U1, U2, τ∆, E) is a binary soft pre- T∆1

space and (U1, U2, τ∆, E)

is a binary soft pre- T∆1space for each e ∈ E.

Proof. The proof is similar to the proof 9.

Now we shill discuss some of the binary soft hereditary properties of

T∆i (i = 0, 1) spaces.


over (U1 × U2) and Y ⊆ X. Then if (U1, U2, τ∆, E) is a binary soft pre-

T∆ospace then (U1, U2, τ∆y

, E) is binary soft pre- T∆ospace.

Proof. Fe, Ge ∈ Ysuch that Fe ≠ Ge . Then Fe, Ge ∈ X. Since (U1, U2, τ∆, E) is

a binary soft pre- T∆ospace, thus there exists binary soft pre-open sets

(F, E) and (G, E) in (U1, U2, τ∆, E) such that Fe ∈ (F, E) and Ge ∉ (F, E) or

Ge ∈ (G, E) and Fe ∉ (G, E). Therefore Fe ∈ Y ∩ (F, E) =Y (F, E). Similarly I

can be shown that if Ge ∈ (G, E) andFe ∉ (G, E), then Ge ∈Y (G, E) and

Fe ∈Y (G, E) and Fe ∉Y (G, E). Thus (U1, U2, τ∆y, E) is binary soft pre-

T∆ospace.


over (U1 × U2) andY ⊆ X. Then if (U1, U2, τ∆, E) is a binary soft pre-

T∆1space then (U1, U2, τ∆y

, E) is binary soft pre- T∆1space.

Proof. The proof is similar to the proof 11.


over (U1 × U2). If (U1, U2, τ∆, E) is a binary soft pre- τ∆2space on Xover

(U1 × U2) then (U1, U2, τ∆e, E) is binary soft pre- T∆2


Proof. Let (U1, U2, τ∆, E) be a binary soft topological space on X over

(U1 × U2). For anye ∈ E, τ∆e= {F(e): (F, E) ∈ τ∆} is a binary soft topology

on X over (U1 × U2). Let x, y ∈ X such that x ≠ y, since (U1, U2, τ∆, E) is a

binary soft pre- T∆2space, therefore binary soft points Fe, Ge ∈ X such that

Fe ≠ Ge and x ∈ F(e) , y ∈ G(e) , there exists binary soft pre-open sets

(F1, E) , (F2, E) such that Fe ∈ (F1, E) , Ge ∈ (F2, E) and (F1, E) ∩ (F2, E) = φ.

Which implies that ∈ F(e) ⊆ F1(e) , y ∈ G(e) ⊆ F2(e) and F1(e) ∩ F2(e) =

φ. This proves that (U1, U2, τ∆e, E) is binary soft pre- T∆2

space.


over (U1 × U2) and Y ⊆ X. Then if (U1, U2, τ∆, E) is a binary soft pre-

τ∆2space then (U1, U2, τ∆y

, E) is binary soft pre- T∆2space and

(U1, U2, τ∆e, E) is binary soft pre- T∆2


Proof. Let Fe, Ge ∈ Ysuch that Fe ≠ Ge . ThenFe, Ge ∈ X. Since (U1, U2, τ∆, E)

is a binary soft pre- T∆2space, thus there exists binary soft pre-open sets

(F1, E) and (F2, E) such that Fe ∈ (F1, E) and Ge ∈ (, E)

and(F1, E) ∩ (F2, E) = φ. Thus Therefore Fe ∈ Y ∩ (F2, E) =Y (F2, E) andY(F2, E) ∩Y(F2, E) = φ . Thus it proves that (U1, U2, τ∆y

, E) is binary soft

pre- T∆2space.


over (U1 × U2). If (U1, U2, τ∆, E) is a binary soft pre- T∆2space and for any

two binary soft points Fe, Ge ∈ X such that Fe ≠ Ge . Then there exist

binary soft pre-closed sets (F1, E) and (F2, E) such that Fe ∈ (F1, E) and

Ge ∉ (F1, E)or Ge ∈ (F2, E) and

(F1, E) ∪ (F2, E) = X.

Proof. Let (U1, U2, τ∆, E) be a binary soft topological space of X over

(U1 × U2).Since (U1, U2, τ∆, E) is a binary soft pre- τ∆2space and

Fe, Ge ∈ Xsuch that Fe ≠ Ge there exists binary soft pre-open sets (H, E)

and (L, E) such that Fe ∈ (H, E) and Ge ∈ (L, E) and (H, E) ∩ (L, E) =

φ.Clearly (H, E) ⊆ (L, E)cand (L, E) ⊆ (H, E)c. Hence Fe ∈ (L, E)c , put

(L, E)c = (F1, E) which gives Fe ∈ (F1, E) and Ge ∉ (F1, E). Also

Ge ∈ (F1, E)c, then put (H, E)c = (F2, E). Therefore Fe ∈ (F1, E) and

Ge ∈ (F2, E) . Moreover, (F1, E) ∪ (F2, E) = (L, E)c ∪ (H, E)c = X. Which

completes the proof.

5. BINARY SOFT PRE- 𝐓∆𝐢 (I=4, 3) SPACES

In this section binary soft pre-separation axioms in Binary Soft Topological

Spaces are discussed.

In this section, we define binary soft pre-regular and binary soft pre- T∆i-

spaces using binary soft points. We also characterize binary soft pre-

regular and binary soft pre- normal spaces. Moreover, we prove that

binary soft pre-regular and binary soft pre- T∆3 properties are binary soft

hereditary, whereas binary soft pre- normal and binary soft pre- T∆4 are

binary soft pre-closed hereditary properties.

Now we define binary soft pre-regular space as follows:

Definition 15: Let (U1, U2, τ∆, E) be a binary soft topological space of X

over

(U1 × U2). Let (F, E) be a binary soft pre-closed set in (U1, U2, τ∆, E)

andFe ∉ (F, E). If there exists binary soft pre-open sets (G, E) and (H, E)

such that Fe ∈ (G, E) , (F, E) ⊆ (H, E) and (F, E) ∩ (H, E) = φ , then

(U1, U2, τ∆, E) is called a binary soft pre- regular space.


over

(U1 × U2). Then the following statements are equivalent:

(i) (U1, U2, τ∆, E) is binary soft pre-regular.

(ii) For any binary soft pre- open set (F, E) in (U1, U2, τ∆, E)

and Ge ∈ (F, E), there is binary soft pre-open set (G, E)

containing Ge such that Ge ∈ (G, E) ⊆ (F, E).

(iii) Each binary soft point in (U1, U2, τ∆, E) has a binary soft

neighborhood base consisting of binary soft pre-closed

sets.

Proof. (i)⇒ (ii)

Let (F, E) be a binary soft pre-open set in (U1, U2, τ∆, E) andGe ∈ (F, E).

Then

(F, E)cis binary soft pre-closed set such that Ge ∉ (F, E)c. By he binary soft

regularity of (U1, U2, τ∆, E) there are binary soft pre-open sets

(F1, E), (F2, E) such thatGe ∉ (F1, E) , (F, E)c ⊆ (F2, E)

Cite the article: Arif Mehmood Khattak, Asad Zaighum, Zia Ullah, Shamoona Jbeen (2018). Binary Soft Pre-Separation Axioms In Binary Soft Topological Spaces. Matrix Science Mathematic, 2(2) : 18-24.


22


and(F1, E) ∩ (F2, E) = φ. Clearly (F2, E)c is a binary soft set contained

in(F, E). Thus, (F1, E) ⊆ (F2, E)c ⊆ (F, E) . This

gives(F1, E) ⊆ (F2, E)c ⊆ (F, E), put(F1, E) = (G, E). Consequently

Ge ∈ (G, E) and(G, E) ⊆ (F, E).

This proves (ii)

(ii) ⇒ (iii)

Let Ge ∈ X, for binary soft pre-open set (F, E) in (U1, U2, τ∆, E) there is a

binary soft pre- open set (G, E) containing Ge such that Ge ∈ (G, E) ,

(G, E) ⊆ (F, E). Thus for each Ge ∈ X , the sets (G, E) from a binary soft

neighborhood base consisting of binary soft pre-closed sets of

(U1, U2, τ∆, E) which proves (iii).

(iii) ⇒ (i)

Let (F, E) be a binary soft pre-closed set such thatGe ∉ (F, E). Then (F, E)cis

a binary soft open neighbourhood ofGe. By (iii) there is a binary soft pre-

closed set (F1, E) Which contains Ge and is a binary soft neighborhood of

Ge with (F1, E) ⊆ (F1, E)c . ThenGe ∉ (F, E)c, (F, E) ⊆ (F1, E)c = (F2, E) and

(F1, E) ∩ (F2, E) = φ . Therefore (U1, U2, τ∆, E) is binary soft pre-regular.

Proposition 17. Let (U1, U2, τ∆, E) be a binary soft pre-regular space on X

over

(U1 × U2). Then every binary soft subspace of (U1, U2, τ∆, E) is binary soft

pre- regular.

Proof. Let (U1, U2, τ∆, E) be a binary soft subspace of a binary soft pre-

regular space(U1, U2, τ∆, E). Suppose (F, E) is a binary soft pre-closed set in

(U1, U2, τ∆y, E) and Fe ∈ Ysuch that Fe ∉ (F, E). Then(F, E) = (G, E) ∩ Y;

Where (G, E) is binary soft pre-closed set in(U1, U2, τ∆, E). ThenFe ∉ (F, E),

since (U1, U2, τ∆, E) be a binary soft subspace of a binary soft pre-regular,

there exists soft disjoint binary pre-open sets(F1, E) , (F2, E) in

(U1, U2, τ∆, E). Then Fe ∉ (G, E), Since (U1, U2, τ∆, E) is binary soft pre-

regular , there exist binary soft disjoint binary pre-open sets (F1, E) , (F2, E)

in (U1, U2, τ∆, E) such that Fe ∈ (F1, E) , (G, E) ∈ (F2, E). Clearly Fe

∈ (F1, E) ∩ Y =Y (F2, E) and (F, E) ⊆ (F2, E) ∩ Y =Y (F2, E) such thatY(F1, E) ∩Y(F2, E) = φ. Therefore, it proves that (U1, U2, τ∆y

, E) is a binary

soft pre- regular subspace of (U1, U2, τ∆, E).

Proposition 18. Let (U1, U2, τ∆, E) be a binary soft regular space on X over

(U1 × U2). A binary space (U1, U2, τ∆, E) is binary soft pre-regular if and

only if for each Fe ∈ X and a binary soft pre-closed set (F, E) in

(U1, U2, τ∆, E) such that Fe ∉ (F, E) there exist binary soft pre- open sets

(F1, E) , (F2, E) in (U1, U2, τ∆, E) such that Fe ∈ (F1, E) , (F1, E) ⊆ (F2, E) and

(F1, E) ∩ (F2, E) = φ.

Proof. For each Fe ∈ X and a binary soft pre-closed set (G, E) such that

Fe ∉ (F, E) by theorem 16 there is a binary soft pre-open set (G, E) such

that Fe ∈ (G, E) , (G, E) ⊆ (F1, E)c. Again by theorem16 there is a binary

soft pre- open (F1, E) containing Fe such that(F1, E) ⊆ (G, E). Let(F2, E) =

((G, E))c, then

(F1, E) ⊆ (G, E) ⊆ (G, E) ⊆ (F, E)c Implies (F1, E) ⊆ ((G, E))c = (F2, E) or

(F, E) ⊆ (F2, E) .

Also

(F1, E) ∩ (F2, E)

= (F1, E) ∩ ((G, E))c

⊆ (G, E) ∩ ((G, E))c

⊆ (G, E) ∩ ((G, E))c

= φ = φ .

Thus (F1, E) , (F2, E) are the required binary soft pre-open sets

in(U1, U2, τ∆, E). This proves the necessity. The sufficiency is immediate.

Definition 16: Let (U1, U2, τ∆, E) be a binary soft regular space on X over

(U1 × U2). (F, E) , (G, E) are binary soft pre-closed sets over (U1 × U2) such

that (F, E) ∩ (G, E) = φ . If there exist binary soft pre-open sets

(F1, E) , and (F2, E) such that (F, E) ⊆ (F1, E) , (G, E) ⊆ (F2, E) and

(F1, E) ∩ (F2, E) = φ , then (U1, U2, τ∆, E) is called a binary soft pre-normal

space.

Definition 17: Let (U1, U2, τ∆, E) be a binary soft regular space on X over

(U1 × U2). Then (U1, U2, τ∆, E) is said to be a binary soft pre- τ∆3space if it

is binary soft pre-regular and a binary soft pre- τ∆1space.

Proposition 19. Let (U1, U2, τ∆, E) be a binary soft regular space on X

over(U1 × U2) and Y ⊆ X. If (U1, U2, τ∆, E) is a binary soft pre- τ∆3space

then (U1, U2, τ∆y, E) is a binary soft pre- τ∆3

space.

Definition 18: A binary soft topological space (U1, U2, τ∆, E) on X over

(U1 × U2) is said to be a binary soft pre- τ∆4space if it is binary soft pre-

normal and binary soft pre- τ∆1space.

τ∆3-space.

Proposition 20. A binary soft topological space (U1, U2, τ∆, E) is binary

soft pre-normal if and only if for soft pre-closed set (F, E) and a binary soft

pre-open set (G, E), such that (F, E) ⊆ (G, E) these exist at least one binary

soft pre-open set (H, E) containing (F, E) such that

(F, E) ⊆ (H, E) ⊆ (H, E) ⊆ (G, E).

Proof. Let us suppose that (U1, U2, τ∆, E) is a binary soft normal space and (F, E) is any binary soft pre- closed subset of (U1, U2, τ∆, E) and (G, E) is a

binary soft pre-open set such that (F, E) ⊆ (G, E). Then (G, E)c is binary

soft pre- closed and (F, E) ∩ (G, E)c = φ. So by supposition, there are

binary soft pre-open sets (H, E) and (K, E) such that (F, E) ⊆ (H, E) ,

(G, E)c ⊆ (K, E) and ∩ (K, E) = φ. Since (H, E) ∩ (K, E) = φ ,

(H, E) ⊆ (K, E)c. But (K, E)cis binary soft pre-closed, so that

(F, E) ⊆ (H, E) ⊆ (H, E) ⊆ (K, E)c ⊆ (G, E) .

Hence (F, E) ⊆ (H, E) ⊆ (H, E) ⊆ (K, E)c ⊆ (G, E) .

Conversely, suppose that for every binary soft pre-closed set (F, E) and a

binary soft pre-open set (G, E) such that (F, E) ⊆ (H, E) , there is a binary

soft pre- open set (H, E) such that (F, E) ⊆ (H, E) ⊆ (H, E) ⊆ (G, E) . Let

(F1, E) , (F2, E) be any two-soft disjoint pre-closed sets, then

(F1, E) ⊆ (F2, E)c where (F2, E) cbinary soft pre-open. Hence there is a

binary soft pre-open set (H, E) such that

(F, E) ⊆ (H, E) ⊆ (H, E) ⊆ (F2, E)c. But then (F2, E) ⊆ ((H, E))c and

(H, E) ∩ ((H, E))c ≠ φ .

Hence (F1, E) ⊆ (H, E) and (F2, E) ⊆ ((H, E))c with (H, E) ∩ ((H, E))c = φ .

Hence (U1, U2, τ∆, E) is binary soft pre- normal space.

6. CONCLUSION

A soft topology between two sets other than the product soft topology has

been touched through proper channel. A soft set with single specific

topological structure is unable to shoulder up the responsibility to build

the whole theory. So, to make the theory strong, some additional

structures on soft set has to be introduced. It makes it bouncier to grow

the soft topological spaces with its infinite applications. In this regards we

familiarized soft topological structure known as binary soft pre-

separation axioms in binary soft topological structure with respect to soft

pre-open sets.

Topology is the most important branch of pure mathematics which deals

with mathematical structures by one way or the others. Recently, many

scholars have studied the soft set theory which is carefully applied to many

difficulties which contain uncertainties in our social life. A few researchers

familiarized and profoundly studied the foundation of soft topological

spaces. They also studied topological structures and displayed their


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several properties with respect to ordinary points.

In the present work, we constantly study the behavior of binary soft pre-

separation axioms in binary soft topological spaces with respect to soft

points as well as ordinary points. We introduce (pre- τ∆o, pre- τ∆1

, pre-

τ∆2 , pre- τ∆3

and pre- τ∆4 )structures with respect to soft points. In future

we will plant these structures in different results. We also planted these

axioms to different results. These binary soft pre-separation structure

would be valuable for the development of the theory of soft in binary soft

topology to solve complicated problems, comprising doubts in economics,

engineering, medical etc. We also attractively discussed some soft

transmissible properties with respect to ordinary as well as soft points. I

have fastidiously studied numerous homes on the behalf of Soft Topology.

And lastly, I determined that soft Topology is totally linked or in other

sense we can correctly say that Soft Topology (Separation Axioms) are

connected with structure. Provided if it is related with structures then it

gives the idea of non-linearity beautifully. In other ways we can rightly say

Soft Topology is somewhat directly proportional to non-linearity.

Although we use non-linearity in Applied Math. So, it is not wrong to say

that Soft Topology is applied Math in itself. It means that Soft Topology has

the taste of both of pure and applied math. In future I will discuss

Separation Axioms in Soft Topology With respect to soft points. We expect

that these results in this article will do help the researchers for

strengthening the toolbox of soft topological structures. Soft topology

provides less informations on the behalf of a few choices. The reason for

this is that we use a single set in soft topology and in binary soft topology

we use double sets. It means that binary soft topology exceeds soft

topology in all respect. In the light of above mentioned discussion I can

literary say that number of sets is directly proportional to choices.

Therefore, all mathematicians are kindly informed to emphasize upon it.

REFERENCES

[1] Molodtsov, D. 1999. Soft Set Theory First Results. Computers &

Mathematics with Applications, 37, 19-31.

[2] Molodtsov, D., Leonov, V.Y., Kovkov, D.V. 2006. Soft sets technique

and its Application, Nechetkie Sistemy i Myagkie Vychisleniya, 1,

(1): 8-39.

[3] Acikg oz, A., Tas N. 2016. Binary Soft Set Theory, European Journal

of Pure and Applied Mathematics, 9, (4): 452- 463.

[4] Benchalli, S.S., Patil, G., Abeda, S.D., Pradeepkumar, J. 2017. On

Binary Soft Separation Axioms in Binary Soft Topological Spaces,

Global Journal and Applied Mathematics, 13, (9): 5393-5412.

[5] Shabir, M.N. 2011. On Some New Operations in Soft Set Theory, Computers & Mathematics with Applications, 57, 1786 - 1799.


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