Intern at ional Journal of Applied Research 20 15; 1(11): 615-623
ISSN Print: 2394-7500
ISSN Online: 2394-5869
Impact Factor: 5.2
IJAR 2015; 1(11): 615-623
www.allresearchjournal.com
Received: 14-08-2015
Accepted: 17-09-2015
On Soft B- Open Sets In Soft Bitopological Space
Revathi N, Bageerathi K
Revathi N
Research scholar & Lecturer,
Department of Mathematics,
Rani Anna Government
College for Women,
Tirunelveli, Tamilnadu, India.
Abstract
In this paper, we introduce(1,2)*- softb- open sets and (1,2)*- soft b –closed sets in soft bitopological
space and exhibit the properties of(1,2)*-softb- open sets. Then we discuss the relation with (1,2)*-soft
regular- open, (1,2)*- soft preopen, (1,2)*-soft semi open, (1,2)*- soft  - open, (1,2)*-soft - open in
soft bitopological space. Also we study about(1,2)*- soft b-interior, (1,2)*- soft b- closurein soft
bitopological space and analyze the relation between other functions.
Dr. Bageerathi K
Assistant Professor,
Department of Mathematics,
Aditanar College of Arts and
Science, Tiruchendur,
Tamilnadu, India.
Keywords: (1,2)*- softb- open set, (1,2)*- soft b –closed set, (1,2)*-soft regular- open, (1,2)*- soft pre
open, (1,2)*-soft semi open, (1,2)*- soft  - open, (1,2)*-soft -open, (1,2)*- soft b-interior, (1,2)*- soft
b- closure.
Mathematics Subject Classification: 54E55, Primary 54C08. Secondary 54C19.
Introduction
In the year 1999, Russian researcher Molodtsov [7], initiated the concept of soft sets as a new
mathematical tool to deal with uncertainties while modelling problems in computer science,
engineering physics, economics, social sciences and medical sciences. In 2003, Maji, Biswas
and Roy [8], studied the theory of soft sets. They discussed the basic soft sets definition with
examples. In 2011, Muhammad Shabir and Munazza Naz [12] defined the theory of soft
topological over an initial universe with a fixed set of parameters. N. Cagmen and S. Karatas
[5]
introduced a topology on a soft set called “soft topology” and presented the foundations of
the theory of soft topological spaces. In 1963, J.C. Kelly [6], first intiated the concept of
bitopological spaces. He defined a bitopological space (X, 1,2) to be a set X with two
topologies 1 and 2 on X and intiated the systematic study of bitopological spaces. In 2014,
Basavaraj M. Ittanagi [2] introduced the concept of Soft bitopological spaces. D. Andrijevic [1]
introduced b- open sets which are some of the weak forms of open sets.
2. Preliminaries
In this section, we have presented some of the basic definitions and results of soft theory, soft
topological space, bitopological spaces and soft bitopological space to use in the sequel.
Throughout this paper, X is an initial universe, E is the set of parameters, P(X) is the power
set of X, and A X
Definition 2.1 : [4] A soft set FA on the universe X is defined by the set of ordered pairs FA =
{ ( x, fA(x)) : xE}, where fA : E  P(X) such that fA(x) =  if x  A. Here fA is called
approximate function of the soft set FA . The value of fA (x) may arbitrary, some of them
maybe empty, some may have non empty intersection. The set of all soft sets over X will be
denoted by S(X).
Correspondence
Revathi N
Research scholar & Lecturer,
Department of Mathematics,
Rani Anna Government
College for Women,
Tirunelveli, Tamilnadu, India.
Example 2.2:[4]Suppose that there are six houses in the universe X ={h1, h2,h3,h4,h5,h6}under
consideration, and that E = { e1,e2,e3,e4,e5} is a set of decision parameters. The ei ( i = 1, 2, 3,
4,5) stand for the parameters “expensive”, “beautiful”, “wooden”, “cheap”, and in “green
surroundings” respectively. Consider the mapping fA given by “houses(.)”, where (.) is to be
filled by one of the parameters ei E. For instance, fA(e1) means “ houses (expensive)”,and
its functional value is the set { h X: h is an expensive house}.
~ 615 ~ International Journal of Applied Research
Suppose that A ={ e1,e2,e4}E and fA( e1) = { h2,h4},fA( e2) = X, fA( e4) = { h1,h3,h5}.
Then the soft set FA as consisting of the following collection of approximations, FA = { (e1, { h2,h4} ),(e2,X),(e4, { h1,h3,h5})}.
~ S(X). If fA(x) =  for all x E, then F is called an empty set, denoted by F.fA(x) =  means that
Definition 2.3: [4] Let FA 
A
there is no element in X related to the parameter x E. Therefore, we do not display such elements in the soft sets, as it is
meaningless to consider such parameters.
~ S(X). If fA(x) = X for all x A, then F is called an A- universal soft set, denoted by FÃ. If A = E,
Definition 2.4: [4] Let FA 
A
~
then the A- universal soft set is called a universal soft set, denoted by X ..
Definition 2.5:
FA and FB
[4]
~ F if fA(x)fB(x)for all x E. Let
FB , denoted by FA 
B
~ S(X).Then F is a soft subset of
Let FA , FB 
A
are soft equal denoted by FA = FB if fA(x) = fB(x) for all x E.
Definition 2.6: [8] Let
~
FA , FB 
S(X). Then soft union of FA and
~ F , defined by F ~ = F where
FB , denoted by FA 
B
AB
C,
C = AB, and for alleC.
 F (e)

H (e)  G (e)
 F ( e )  G ( e)

Definition

2.7:
[5]
if e  A \ B
if e  B \ A
if e  A  B
~
Let FA 
S(X).
The
soft
power
set
of
~
~ F i  I  and its cardinality is defined by
P FA   FAi 
A
~
f ( x)

xE A
| P F A  | = 2
, where |fA(x)| is cardinality off A(x).
FA is
defined
by P FA 
~
~
is
defined
by
~
Example 2.8: [5] Let X= { x1,x2},E = { e1, e2 }and X = { ( e1, { x1,x2}), (e2, { x1, x2})}.Then the soft subsetsover X are
FE1 = { ( e1,{ x1 })}
FE9 = { ( e1, { x1}), (e2, { x1, x2})}.
FE2 ={ ( e1,{ x2 }) }.
FE10 = { ( e1, { x2}), (e2, { x1})}.
FE3 = { ( e1,{ x1, x2})}.
FE11 ={ ( e1, { x2}), (e2, { x2})}.
FE4 = { (e2,{x1})}.
FE12 = { ( e1, { x2}), (e2, { x1, x2})}.
FE5 ={ (e2,{ x2})}.
FE13 = { ( e1, { x1, x2}), (e2, { x1})}.
FE6 = { (e2,{ x1,x2})}
FE14 = { ( e1, { x1, x2}), (e2, { x2})}.
~
FE15 = X
FE7 ={ ( e1, { x1}), (e2, { x1 })}
FE8 = { ( e1, { x1}), (e2, {x2})}.
~
~
Are all subsets of X .So | P  FE
FE16 = F
 | = 24 = 16.
~
Definition 2.9: [12] Let ~ be the collection of soft sets over X ,then
following axioms.
(i)
~
is said to be a soft topology on X if statisfies the
~
F , X belong to ~ .
(ii)the union of any member of soft sets in ~ belongs to ~ .
(iii) the intersection of any two soft sets in ~ belongs to
members of ~ are said to be soft open set.
~ .The triplet ( X~ , ~ ,E) is called soft topological space over X. The
~
~
Example 2.10: Let us consider the soft subsets of X that are given in Example 2.8. Then ~ 1 ={ X , F , FE4 , FE10 },
~
~
~
X , F , FE1 , FE7 , FE13 },  3 ={ P  FE
 } are soft topologies on
~
X
.
Definition 2.11: [6] Let X , 1 and 2 are two different topologies on X. Then (X, 1,2)is called a Bitopological space.
~ 616 ~ ~ 2 = {
International Journal of Applied Research
Definition 2.12: [6] A subset S of X is called 1,2-open if S =HK such that H1 and K2 and the complement of1,2-open set
is1,2-closed set.
Example 2.13: [6] Let X = {a, b,c}, 1= {, X,{a}} and 2= {,X, {b}}.The sets in{ { , X, {a},{b},{a,b}}are called 1,2-open
set and sets in{ , X, {b,c},{a,c},{c}}are called1,2-closed set.
Definition 2.14: [6] Let S be subset of X. Then,(i) The 1,2-closure of S, denoted by 1,2-cl(S), isdefined by
is a1,2-closed set}(ii) The 1,2-interior of S, denoted by 1,2-int(S), is defined by
Definition 2.15:
~
~
 A : A  S , A is a 
Let X be a nonempty soft set on the universe X, ~1 and are
[13]

 F : S  F , F
1,2-open
~2
set}
~
two different soft topologies on X .
Then X ,~1 ,~2 is called a soft Bitopological space.
Definition 2.16:
[13]
~

Let X ,~1 ,~2 is be a soft bitopological space and
~ X~ .Then F is called ~ - soft open if F =
FA 
A
A
1, 2
~ F where F 
~ ~ and F 
~ ~ . The complement of ~ – soft open set is called ~ - soft closed
FB 
B
1
2
C,
C
1, 2
1, 2
~
~
~
~
F , FE4 , FE10 }, ~2 = { X ,
~
} and ~ - soft closed set are{ X ,
Example 2.17: Let us considerthe soft subsets X ofthat are given in Example 2.8 and1 ={ X ,
F , FE1 , FE7 , FE13 }. Then ~1, 2 - soft open set are { X~ , F , FE1 , FE4 , FE7 , FE10 , FE13
1, 2
F , FE12 , FE14 , FE11 , FE8 , FE5 }
FA be a soft subset of X~ , Then
i) The ~1, 2 - soft closure of FA , denoted by ~1, 2 -cl ( FA ), is defined by 
Definition 2.18: [12] Let

~ F , F isa ~ - soft closed set}
FK : FA 
K
K
1, 2
(ii) The ~1, 2 - soft interior of
~ F , F is a ~ - soft open set}
FA , denoted by ~1, 2 -int( FA ), is defined by {FC : FC 
A
C
1, 2
Note that ~1, 2 -int( FA ) is the biggest ~1, 2 - soft open set that contained by FA and ~1, 2 -cl ( FA ) is the smallest ~1, 2 - soft
closed set that containing FA .
Definition 2.19: [1] A subset A of the topological space( X,) is calledb- open set if Acl(int(A)) int(cl(A)) andb- closed if cl
(int(A)) int (cl(A))  A.
The following concepts are used in the sequel.
FA in a soft bitopological space X~ is called
(i) (1, 2)*- soft regular open set if FA = ~1, 2 -int( ~1, 2 –cl( FA ))and (1, 2)*- soft regular closed setif FA = ~1, 2 -cl( ~1, 2 –int(
Definition 2.20: A softset
FA ))
~ ~ -int( ~ -cl( ~ –int( F ))) and(1, 2)-*soft -closed setif ~ -cl( ~ -int( ~ –cl( F
(ii)(1, 2)*-soft -open set if FA 
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
~
))) 
FA
~ ~ -int( ~ –cl( F )) and (1,2)*- soft pre closed if ~ -cl( ~ –int( F )) 
~
(iii) (1,2)*- soft pre open if FA 
A
A
1, 2
1, 2
1, 2
1, 2
FA .
~ ~ -cl( ~ –int( F )) and(1,2)*- soft semi closed if ~ -int( ~ –cl( F )) 
~ F
(iv) (1,2)*- soft semi open if FA 
A
A
A
1, 2
1, 2
1, 2
1, 2
.
~
~
~
~
~
~
~
(v)(1,2)*- soft - open if FA   1, 2 -cl(  1, 2 -int(  1, 2 –cl( FA ))) and (1,2)*-soft - closed if  1, 2 -int(  1, 2 -cl(  1, 2 –int( FA )))
~ F

A
The family of all (1, 2) *- soft regular open ( resp. (1, 2)*-soft  open, (1, 2)*- soft preopen,(1, 2)*-soft semi open, (1, 2)*-soft
 open) sets in may be denoted by (1, 2)*-sr open ( resp. (1, 2)*- s open, (1, 2)*-sp open, (1, 2)*-ss open, (1, 2)*-s open)
sets.
~

Lemma 2.21: In X , ~1 ,~2 be a soft bitopological space. we have the following results.
(i) Every (1, 2)*-soft regular open set is (1, 2)*-soft open
(ii)Every (1, 2)*-softopen set is (1, 2)*-soft - open.
(iii) Every (1, 2)*-soft  open set is (1, 2)*-soft semi open.
~ 617 ~ International Journal of Applied Research
(iv) Every (1, 2)*-soft preopen set is (1, 2)*-soft - open.
(v) Every (1, 2)*-soft semi open set is (1, 2)*-soft - open.
(vi) Every (1, 2)*-soft  open set is (1, 2)*-soft preopen
~

Proof: Let X ,~1 ,~2 be a soft bitopological space and
~1, 2
~ X~ Suppose F be a ( 1,2)* -soft regular open set. Then F =
FA 
A
A
.
-int ( ~1, 2 –cl( FA )) since ~1, 2 –cl( FA ) isa closed set in soft bitopological space and interior of any set is open. Therefore,
(i) proved.
~ ~ –cl( F ) = ~ –cl( ~ -int(
Let FA be a (1, 2)* -soft open set. This implies FA = ~1, 2 -int( ~1, 2 –cl( FA ))since FA 
A
1, 2
1, 2
1, 2
~1, 2
–cl( FA ))). Thus,
(ii) proved.
~ ~ -int( ~ -cl( ~ –int( F ))) 
~ ~ -cl( ~ –int( F ))Thus, (iii)
Let FA be a (1,2)* -soft  open set. This implies FA 
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
proved.
~ ~ -int( ~ –cl( F )) 
~ ~ -cl( ~ -int( ~ –cl( F ))). Thus, (iv)
Let FA be a (1, 2)* -soft preopen set. This implies FA 
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
proved.
~ ~ -cl( ~ –int( F )) 
~ ~ -cl( ~ -int( ~ –cl( F ))). Thus,
Let FA be a (1, 2)* -soft semi open set. This implies FA 
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
(v) proved.
Let FA be a (1, 2)* -soft - open set. This implies FA
~ ~ -int( ~ -cl( ~ –int( F ))) 
~ ~ -int( ~ -clt( F )). Thus, (vi)

A
A
1, 2
1, 2
1, 2
1, 2
1, 2
proved.
Remark 2.22: The converse of the above lemma is need not be true as seen in the following examples.
~

Example2.23: Let X ,~1 ,~2 be a soft bitopological space, where
~1 ={ X~ , F , FE4 , FE10 }, ~2 = { X~ , F , FE1 , FE7 , FE13 }. Then ~1, 2 - soft open set are { X~ , F , FE1 , FE4 , FE7 , FE10 ,
~
F } and ~ - soft closed set are{ X , F , F , F , F , F , F }

1, 2
E13
E12
E14
E11
E8
E5
(i) FE7 is a (1, 2)*-soft open but not (1, 2)*-soft regular open.
(ii) FE9 is a (1, 2)*-soft  open but not (1, 2)*-soft open.
(iii) FE6 is a (1, 2)*-soft semi open but not (1, 2)*-soft  open.
(iv) FE6 is a (1, 2)*-soft open set but not (1, 2)*-soft preopen.
~
~
Example2.24: Let us consider the soft subsets of X that are given in Example 2.8.Let X ,~1 ,~2
 be a soft bitopological
~
~
~
~
~
space, where 1 = { X , F , FE1 }, 2 ={ X , F FE2 }. Then ~1, 2 - soft open set are { X , F , FE1 , FE2 FE3 }, ~1, 2 - soft
~
closed set are{ X , F , FE12 , FE9 , FE6 }.
(v)The soft subset FE4 is (1, 2)*-soft  -open set but not (1, 2)*-soft semi open.
~
~
~
~
~
x3,x4})}.Then 1 = { X , F , FE1 , FE2 FE3 , FE4 , FE5 , FE6 , FE7 , FE8 , FE9 , FE10 , FE11 , FE12 , FE1 , FE14 , FE15 }, 2 ={ X ,
3
Example 2.25: Let X= { x1, x2, x3}, E = { e1, e2, e3 }and X = { ( e1, { x1, x2, x3,x4}), (e2, { x1, x2, x3,x4}), ( e3, { x1, x2,
F }. Where FE1 = { ( e1, { x1 }), (e2, { x2, x3}), ( e3, { x1,x4})}. FE2 ={ ( e1, {x2,x4}), (e2, { x1, x3,x4}), ( e3, { x1, x2,x4})}. FE3
={(e2, {x3}), (e3, {x1})}. FE4 = {(e1, {x1, x2, x4}), (e2, X), (e3, X}. FE5 ={ ( e1, { x1, x3}), (e2, { x2,,x4}), ( e3, X)}. FE6 ={(e1,
{x1}), (e2, {x2}}. FE7 ={ ( e1, {x1, x3}), (e2, {x2, x3,x4}), ( e3, { x1, x2,x4})}. FE8 ={(e2, {x4}), (e3, {x2})}. FE9 ={(e1, X), (e2, X),
(e3, {x1, x2, x3})}. FE10 = { ( e1, { x1, x3}), (e2, {x2, x3,x4}), ( e3, { x1, x2})}. FE11 ={ ( e1, {x2, x3,x4}), (e2, X), ( e3, { x1, x2, x3})}.
FE12 ={ ( e1, { x1}), (e2, {x2, x3,x4}), ( e3, {x1, x2,x4})}. FE13 = {(e1, {x1}), (e2, {x2,x4}), ( e3, {x2})}. FE14 = {(e1, {x3,x4}), (e2, {
~
x1, x2})}. F ={( e1, {x1 }), (e2, {x2, x3}), ( e3, { x1})}. Then X ,~ ,~ is a soft bitopological space.(vi)Consider F , the soft

E15
1
2

E
~
~
~ X~
~
subset of X .Where FE = { ( e1, { x4}), (e2, { x1, x2, x3,}), ( e3, {x2, x4})}. ~1, 2 -int( ~1, 2 -cl( FE )) = X and FE 
. But  1, 2 int( ~ -cl( ~ –int( F ))) = F .Hence F is (1, 2)*-soft preopen set but not (1, 2)*-soft  open.
1, 2
1, 2
A

E
~ 618 ~ International Journal of Applied Research
3. (1,2)*- soft b- open sets in this section we introduce (1, 2)*- soft b-open sets in soft bitopological spaces and study some of
their properties.
~

~ X~ Then F is called( 1, 2)*- soft b-open set(briefly
FA 
A
Definition 3.1: Let X ,~1 ,~2 be a soft bitopological space and
( 1, 2)* -sb-open)if
~ ~ -cl ( ~ -int( F )).
~ ~ -int( ~ -cl ( F )) 
FA 
A
A
1, 2
1, 2
1, 2
1, 2
~
Example 3.2: In Example 2.17, the( 1, 2)*- soft b-open sets are { X , F , FE1 , FE4 , FE6 , FE7 , FE8 , FE9 , FE10 , FE12 , FE1 }.
3
~
Theorem 3.3: Let X ,~1 ,~2
 be a soft bitopological space. Then(i) Every (1, 2)* -soft preopen set is (1 2)* -soft b-open
set.(ii) Every (1, 2)* -soft b- open set is (1, 2)* -soft -open set.(iii) Every (1, 2)* -soft semi open set is (1, 2)* -soft b-open set.
Proof: Let
~1, 2
X~ ,~ ,~  be a soft bitopological space and F  X~ Let F is a (1, 2)* -soft preopen set. Then F
1
A
2
.
A
A
~ ~ -int(

1, 2
~ ~ -int( F ) 
~ ~ -cl( ~ –int( F )). Thus (i)
~ ~ -int( ~ –cl( F )) 
~ ~ -int( ~ –cl( F )) 
–cl( FA )) 
A
A
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
proved
~ ~ -int ( ~ –cl( F )) 
~ ~ -cl( ~ –int( F )) 
~ ~ -cl( ~ -int( ~ –cl(
Let FA be a (1,2)* -soft b-open set. Then FA 
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
~ ~ -int( ~ -cl ( F )) 
~ ~ -cl( ~ -int( ~ –cl( F ))). Thus
FA ))) 
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
(ii) proved.
~ ~ -int( F ) 
~ ~ -cl( ~ –int( F )) 
~ ~ -cl( ~ –int( F )) 
~
Let FA is a ( 1,2)* -soft semi open set.This implies FA 
A
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
~1, 2 -cl( ~1, 2
~
–int( FA ))  ~1, 2 -int( ~1, 2 –cl( FA )). Thus
(iii) proved.
Remark 3.4: The converse of the above lemma is need not be true as seen in the following example.
~
Example3.5: Let us consider the soft subsets of X that are given in Example 2.8.Let
space, where
~1 = { X~ , F , FE },~2 ={ X~ , F FE
1
X~ ,~ ,~  be a soft bitopological
1
2
~
}. Then ~1, 2 - soft open set are{ X , F , FE1 , FE2 FE3 }, ~1, 2 - soft
2
~
~
closed set are{ X , F , FE12 , FE9 , FE6 }.(i) The soft set FE7 in X is (1, 2)* -soft b-open set but not (1, 2)* -soft preopen
~
~
set.(ii)The soft set FE5 in X is (1, 2)* -soft -open set but not (1, 2)* -soft b- open set.(iii) The soft set FE4 in X is (1, 2)*
-soft b-open set but not (1, 2)* -soft semi open set.
Remark 3.6: The above discussions are summarized in the following diagrams:
(1, 2)*-soft regular open
(1, 2)*-soft - open
(1, 2)*-soft open
Theorem 3.7: An arbitrary union of (1, 2)*-soft b-open sets are (1, 2)*- soft b-open set.
~
~ ~ -int( ~ -cl ( F ))  ~ -cl (
Proof: Let {( FA )} be a collection of (1, 2)* soft b-open sets. Then for each,( FA ) 
A
1, 2
1, 2
1, 2
~1, 2 -int( FA )). 

~
( ( FA )) 

~
~ [
{ ~1, 2 -int( ~1, 2 -cl ( FA ))  ~1, 2 -cl ( ~1, 2 -int( FA ))} 
~ 619 ~ 
{ ~1, 2 -int( ~1, 2 -cl ( FA
International Journal of Applied Research
~
))}]  [
{ ~1, 2 -cl ( ~1, 2 -int( FA ))}]

~1, 2 -int {( ~1, 2 -cl{ 

~ [{ ~ -int {

1, 2
~
( FA )})}]  [ ~1, 2 -cl{( ~1, 2 -int {
~
( ~1, 2 -cl ( FA ))}]  [ ~1, 2 -cl{


~[
( ~1, 2 -int( FA ))}] 

( FA )})}]. Hencethe theorem.
Result 3.8: The intersection of two ( 1,2)* -soft b-open sets need not be( 1,2)* -soft b-open set. In example 3.2, FE6
~ F =

E8
FE5 , which is not ( 1,2)* -soft b-open set.
~

Theorem 3.9: Let X ,~1 ,~2 be a soft bitopological space.(i) The intersection of (1,2)*-soft open set and (1,2)*-sb-open set
is (1,2)*-sb-open set.(ii) The intersection of (1,2)*-s- open set and (1,2)*-sb-open set is (1,2)*-sb-open set.


~
~F 
~[
~ F 
FA , FB  X ,~1 ,~2 bea(1, 2)*-soft open set and (1,2)*-sb-open set respectively then FA 
A
B
~ ~ -int( ~ –cl( F ))]=[ F 
~ ~ -cl( ~ –int( F )] 
~ [F 
~ ~ -int( ~ –cl( F )]=[ F 
~
~1, 2 -cl( ~1, 2 –int( FB )) 
B
A
B
A
B
A
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
~ [F 
~ ~ -int( ~ –cl( F )] 
~ cl( ~ –int( F )] 
~ [ ~ -int- ( ~ –
~ [ ~ -cl( ~ –int( F ) 
~ -cl( ~ –int( F )] 
Proof: Let
1, 2
B
1, 2
A
1, 2
B
1, 2
1, 2
A
1, 2
B
1, 2
1, 2
1, 2
~
~ F )] 
~ [ ~ -int -( ~ –cl( F 
~ F )].Thus (i)proved.
~ [ ~ -cl( ~ –int( F 
cl( FA )  ~1, 2 -int( ~1, 2 –cl( FB )] 
A
B
A
B
1, 2
1, 2
1, 2
1, 2
~ ~ ~
~
~ [ ~ -int -( ~
Let F , F  X , , bea(1, 2)*-soft - open setand (1, 2)*-sb-open set respectively, then F  F 
A

B
1
2

A
B
1, 2
1, 2
~ ~
~ ~
~ ~
~
~
~
~
~
~
1, 2 –int( FA )]  [  1, 2 -cl(  1, 2 –int( FB ))   1, 2 -int(  1, 2 –cl( FB ))]=[  1, 2 -int(  1, 2 –cl(  1, 2 –int( FA )   1, 2 -cl(  1, 2 –
~
~
~ ~ -cl( ~ –int(
~ [ ~ –cl( ~ –int( F ) ) 
int( FB ))]  [ ~1, 2 -int( ~1, 2 –cl( ~1, 2 –int( FA )  ~1, 2 -int( ~1, 2 –cl( FB ))] 
A
1, 2
1, 2
1, 2
1, 2
~
~
~
~
~
~
~
~
~
~
~
~
~
~
F ))]  [  -int(  –cl( F ))   -int(  –cl( F ))]  [  –cl(  –int( F  F ))]  [  -int(  –cl( F 
–cl( ~
B
1, 2
A
1, 2
1, 2
B
1, 2
1, 2
A
1, 2
B
1, 2
A
1, 2
FB ))]. Thus
(ii)proved.
~ ~
,  1, 2 -cl( FA )is a(1, 2)*-sr-closed set.
Theorem 3.10: Any (1, 2)*-sb-open set in soft topological space X
~ ~ -int( ~ –cl( F )). Also, ~ –
~ ~ -cl( ~ –int( F )) 

A
A
1, 2
1, 2
1, 2
1, 2
1, 2
~
~
~
~
~
~
~
~
~
~
~
~
~
cl( FA )   1, 2 –cl(  1, 2 -cl(  1, 2 –int( FA ))   1, 2 –cl(  1, 2 -int(  1, 2 –cl( FA )))   1, 2 –cl(  1, 2 -int(  1, 2 –cl( FA ))) and  1, 2 –
~ ~ –cl( F ). Therefore, ~ –cl( F )= ~ –cl( ~ -int( ~ –cl( F ))). So, ~ -cl( F )is
cl( ~1, 2 -int( ~1, 2 –cl( FA ))) 
A
A
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
Proof: Let FA be a (1,2)* -soft b-open set. This implies FA
a(1,2)*-sr-closed set.
Theorem 3.11: Let
set.(ii)If
FA be a (1, 2)*-sb-open set in soft topological space X~ ,(i) If ~1, 2 -cl( FA )=then FA is a(1, 2)*-ss-open
~1, 2 -int( FA
)=then FA is a(1, 2)*-sp-open set.
~ ~ -int( ~ –cl( F )).Since ~ -cl(
~ ~ -cl( ~ –int( F )) 

A
A
1, 2
1, 2
1, 2
1, 2
1, 2
~
~
~
~
–int( FA )). Thus (i) proved. Since  1, 2 -int( FA )=, FA   1, 2 -int(  1, 2 –cl( FA )). Thus (ii)
Proof: Let FA be a (1, 2)* -soft b-open set. This implies FA
~ ~ -cl( ~
FA )=, FA 
1, 2
1, 2
proved.
FA be a (1, 2)*-sb-open set in soft bitopologicalspace X~ ,(i) If FA is a(1, 2)*-sr-closed set then FA is
a(1,2)*-sp-open set.(ii)If FA is a(1, 2)*-sp-open set then FA is a(1,2)*-ss-open set.
~ ~ -int( ~ –cl( F )).Let F be a ( 1, 2)* -sr~ ~ -cl( ~ –int( F )) 
Proof: Since FA be a ( 1, 2)* -soft b-open set, FA 
A
A
A
1, 2
1, 2
1, 2
1, 2
~ ~ -int( ~ –cl( F )).which implies, F 
~ F 
~ ~ -int( ~ –cl(
closed set, F = ~ -cl( ~ –int( F )).Then F 
Theorem 3.12: Let
A
1, 2
A
1, 2
A
A
1, 2
1, 2
A
A
1, 2
1, 2
~ ~ -cl( ~ –int( F
~F 
FA )).Thus (i) proved. Let FA be a (1,2)* -soft sr-open set, FA = ~1, 2 -int( ~1, 2 –cl( FA )). FA 
A
A
1, 2
1, 2
~
~
~
))which implies, F   -cl(  –int( F )).Thus (ii) proved.
A
1, 2
1, 2
A
~
Theorem 3.13: FA be a (1, 2)*-sb-open set in X if and only if
FA is the union of(1, 2)*-ss-open set and (1, 2)*-sp-open set.
~
Proof: Follows from the definition of (1, 2)*-sb-open set in X
.
~ 620 ~ International Journal of Applied Research
4. (1,2)*-soft b- closed sets
In this section we introduce (1, 2)*- soft b-closed sets in soft bitopological spaces and study some of their properties.
~

~ X~ . Then F is called(1,2)*- soft b-closed set (briefly

A
~
~
~
~
~
~
(1, 2)* -sb-closed) if  1, 2 -int(  1, 2 -cl ( FA ))   1, 2 -cl (  1, 2 -int( FA ))  FA . The complement of (1, 2)*- soft b-closed set is
Definition 4.1: Let X ,~1 ,~2 be a soft bitopological space and FA
(1,2)*- soft b-open set.
~
Example 4.2: In Example 2.17, the( 1, 2)*- soft b-closed sets are{ X , F , FE12 , FE14 , FE8 , FE11 , FE10 , FE2 , FE8 , FE1 , FE5 }
Theorem 4.3: An arbitrary intersection of (1,2)*-soft b-closed sets is( 1,2)*-soft b-closed set.
~
~
Proof: Let {( FB )} be a collection of( 1,2)* soft b-closed sets in X . Then for each , ~1, 2 -int( ~1, 2 -cl ( FB ))  ~1, 2 -cl (
~ ( F ).
~1, 2 -int( FB )) 
B
 F 
C
B
~
Since{( FB )C} is an arbitrary family of(1, 2)*-sb-open set in X . Hence, by theorem3.7,
is a(1,2)*-sb-open set. But
 F 
C
B
C


=  FB   . Therefore


 F  is a (1,2)*-sb-closed set.
B
Result 4.4: The union of two ( 1, 2)*-soft b-closed sets need not be ( 1, 2)*-soft b-closed set. In example 3.5, FE1 and FE2 are (
~
1, 2)*-soft b-closed sets. But the union FE1  FE2 = FE3 ,which is not ( 1, 2)*-soft b-closed set.
FA be a (1, 2)*-sb-closed set in soft bitopologicalspace X~ ,(i) If FA is a(1, 2)*-sr-closed set then FA is a(1,
2)*-ss-closed set.(ii) If FA is a(1, 2)*-sr-open set then FA is a(1, 2)*-sp-closed set.
~
~ F Since F is a(1, 2)*-srProof: Since FA be a( 1, 2)*-sb-closed set, ~1, 2 -cl ( ~1, 2 –int( FA ))  ~1, 2 -int( ~1, 2 –cl( FA )) 
A.
A
~
~
~
~
~
~
~
~
~ F .
closed set, F =  -cl (  -int( F )). Therefore, F   -int( 
–cl( F ))  F . Thus,  -int( 
–cl( F )) 
Theorem 4.5: Let
A
1, 2
A
1, 2
A
1, 2
A
1, 2
Thus (i) proved.
Since FA is a(1,2)*-sr-open set, FA = ~1, 2 -int ( ~1, 2 -cl( FA )). Therefore, FA
~1, 2 -cl( ~1, 2
~
–int( FA )) 
A
1, 2
A
1, 2
A
~ ~ -cl( ~ –int( F )) 
~ F . This implies,

A
A
1, 2
1, 2
FA .Thus (ii) proved.
~ ~
,  1, 2 -int( FA )is a(1,2)*-sr-open set.
~
~
~
~ F . Also, ~ –
Proof: Let FA be a ( 1, 2)* -sb-closed set. This implies  1, 2 -cl(  1, 2 –int( FA ))  ~1, 2 -int( ~1, 2 –cl( FA )) 
A
1, 2
~
~
~
~
~
~
~
~
~
~
~
~
int( F )  
–int(  -cl( 
–int( F ))  
–int(  -int( 
–cl( F )))  
–int( F ).Thus 
–int(  -cl(
Theorem 4.6: Any (1,2)*-sb-closed set in soft bitopological space X
A
1, 2
1, 2
1, 2
A
1, 2
1, 2
A
1, 2
1, 2
A
1, 2
1, 2
~
~ ~
~
~ ~
~
~
~
~
1, 2 –int( FA )))   1, 2 –int( FA ).Also  1, 2 –int( FA )   1, 2 –int(  1, 2 -cl(  1, 2 –int( FA )))). Therefore,  1, 2 –int( FA )=  1, 2 –
int( ~1, 2 -cl ( ~1, 2 –int( FA ))).So, ~1, 2 -int( FA )is a(1, 2)*-sr-open set.
~
Theorem 4.7: FA be a (1, 2)*-sb-closed set in X if and only if
closed set.
FA is the intersection of(1, 2)*-ss-closed setand (1, 2)*-sp-
Proof: Follows from the definition 4.1.
5. (1,2)* - soft b- interior and(1,2)* - soft b- closure
~

FA be a soft set over X~ (i) (1, 2)*-soft b-closure (briefly ( 1,
~
~
~ F : F is a ( 1, 2)*- soft b-closed set in X~
2)* -sbcl( FA )) of a set FA in X defined by(1, 2)*- sbcl( FA ) =  { FE 
A
E
~
~
}(ii) (1, 2)*-soft b-interior (briefly ( 1, 2)* -sbint( F )) of a set F in defined by(1, 2)*- sbint( F ) =  {
Definition 5.1: Let X ,~1 ,~2 be a soft bitopological space and
~ F : F is a ( 1, 2)*- soft b-open set in X~ }
FB 
A
B
A
A
~ 621 ~ X
A
International Journal of Applied Research
~
(1, 2)*- sbcl( FA ) is the smallest( 1, 2)*- soft b-closed set in X which contains
FA and(1, 2)*- sbint( FA )is the largest( 1,
~
2)*- soft b-open set in X which is contained in FA .
The following lemma is used in the sequal.
~ ~ -int( ~ –cl( F
FA be a soft set in a soft bitopologicalspace X~ . Then(i) (1, 2)*-sscl( FA )= FA 
A
1, 2
1, 2
~
~
~
~
~
~
))(ii) (1, 2)*-ssint( F )= F   -cl (  -int( F ))(iii) (1, 2)*-spcl( F )= F   -cl (  -int( F ))(iv) (1, 2)*-spint(
Lemma 5.2: Let
A
A
1, 2
A
1, 2
A
A
1, 2
A
1, 2
~ ~ -int( ~ –cl( F ))
FA )= FA 
A
1, 2
1, 2
~ [ ~ -int( ~ –cl( F ))
FA be a soft set in a soft bitopologicalspace X~ . Then(i) (1,2)*- sbcl( FA )= FA 
A
1, 2
1, 2
~ ~ -cl ( ~ -int( F ))].(ii) (1, 2)*- sbint( F )= F 
~ [ ~ -int( ~ –cl( F )) 
~ ~ -cl ( ~ -int( F ))].

A
A
A
A
A
1, 2
1, 2
1, 2
1, 2
1, 2
1, 2
Therem5.3: Let
Proof: Proof is obvious.
~

~ F )
~ (1,2)*- sbcl(

B
~ (1,2)*- sbcl( F ).(ii)(1,2)*- sbcl( F 
~ F )
~ (1,2)*- sbcl( F ).(iii)(1,2)*- sbint( F 
~ F )
~ (1,2)*- sbcl( F ) 
~
FA ) 
B
A
B
A
B
A
B
~
~
~ (1,2)*- sbint( F ).
~ (1,2)*- sbintl( F ) 
(1,2)*- sbint( FA )  (1,2)*- sbint( FB ).( iv)(1,2)*- sbint( FA  FB ) 
A
B
Therem5.4: In X ,~1 ,~2 be a soft bitopological space have the following results:(i)(1,2)*- sbcl( FA
~ F or F 
~ F This implies,(1,2)*- sbcl( F ) 
~ F ) or(1,2)*- sbcl(
~ F 
~ F 
~ (1,2)*- sbcl( F 
FA 
A
B
B
A
B
A
A
B
~
~
~
~
~
FB )  (1,2)*- sbcl( FA  FB ).Thus, (1,2)*- sbcl( FA  FB )  (1,2)*- sbcl( FA )  (1,2)*- sbcl( FB ).
~ F or F 
~ F This implies,(1,2)*- sbint( F ) 
~ F
~ F 
~ F 
~ (1,2)*- sbint( F 
(ii) Similar to that of (i).Since FA 
A
B
B
A
B
A
A
B
~ F ).Thus,(1,2)*- sbint( F 
~ F )
~ (1,2)*- sbint( F
~ (1,2)*- sbint( F 
~ (1,2)*- sbint( F ) 
) or(1,2)*- sbint( FB ) 
A
B
A
B
A
B
Proof: Since
).(iv) Similar to that of (iii).
~ X~ ( i) [(1,2)*- sbcl( F )]C = X~ \(1, 2)*-sbint( F )(
FA be a soft set in a soft bitopologicalspace X~ . FA 
A
A
.
~
C
C
F
F
ii) [(1,2)*- sbint( A )] = X \(1, 2)*-sbcl( A )
Therem5.5: Let
~ {F 
~ F : F is a ( 1, 2)*- soft b-closed set in X~ }[(1,2)*- sbcl( F )]C = [
~ X~ (1,2)*- sbcl( F )= 
FA 
A
E
A
E
A
.
~ C
~
~
 { FE  FA : FE is a ( 1, 2)*- soft b-closed set in X }]
C ~
C
C
~
F : F is a ( 1, 2)*- soft b-open set in X~ }
= { F 
Proof:(i) Let
E
A
E
~
= (1, 2)*-sbint( FA ) = X \(1, 2)*-sbint( FA )(ii)Apply soft interior, we get the result.
C
~ (1, 2)*-spcl( F
~ (1, 2)*-sscl( F ) 
FA be a soft set in a soft bitopologicalspace X~ .(i) (1, 2)*- sbcl( FA ) 
A
A
~
~
)(ii) (1, 2)*- sbint( FA )  (1,2)*-ssint( FA )  (1, 2)*-spint( FA )
Therem5.6: Let
~
~ ~ -int( ~ –cl( F )) 
~ F 
~ ~ -cl ( ~ -int( F ))] = F 
~[
~ [F 

A
A
A
A
A
1, 2
1, 2
1, 2
1, 2
~ ~ -cl ( ~ -int( F ))] =(1,2)*- sbcl( F ). (ii) (1,2)*-ssint( F ) 
~ (1,2)*-spint( F ) 
~
~ [( F 
~1, 2 -int( ~1, 2 –cl( FA )) 
A
A
A
A
A
1, 2
1, 2
~ (F 
~ ~ -int( ~ –cl( F )))]= F 
~ [ ~ -int( ~ –cl( F )) 
~ ~ -cl ( ~ -int( F ))] =
~ -cl ( ~ -int( F ))) 
Proof: (i)(1, 2)*-sscl( FA )  (1,2)*-spcl( FA )
1, 2
1, 2
A
A
1, 2
A
1, 2
A
1, 2
1, 2
A
1, 2
1, 2
A
(1,2)*- sbint( FA ).
~
Therem5.7: In asoft set in a soft bitopologicalspace X . (i) FA is (1, 2)*-sb-closed if and only if
is (1, 2)*-sb-open if and only if FA =(1, 2)*- sbint( FA )
FA =(1, 2)*- sbcl( FA )(ii) FA
~ F 
~ F : F is a ( 1, 2)*- sb-closed set in X~ }. Therefore, F is (1, 2)*FA = (1, 2)*-sbcl( FA ) = 
A
E
A
E
~
~ F and F is (1, 2)*-sb-closed. Therefore, F 
sb-closed. Conversely, suppose FA is (1, 2)*-sb-closed in X . If we take FA 
A
A
A
~ F ~ F
~
~
~
~
 { FE  FA : FE is a ( 1, 2)*- soft b-closed set in X }. A  E implies, FA =  { FE  FA : FE is a ( 1, 2)*- soft b~
closed set in X } =(1, 2)*- sbcl( FA ).
Proof: (i) Suppose
~ 622 ~ International Journal of Applied Research
Conclusion
In this paper, we introduce the concept of (1, 2)*- soft b-open sets and (1, 2)*-soft b-closed sets insoft bitopological spaces.
Also westudy the properties of (1,2)*-soft b-interior and (1,2)*-soft b-closure. In future, using this concept various type of
open sets on soft bitopological spaces shall be studied and developed.
References
1. D Andrijievic. On b-open sets Mat Vesnik. 1996; 48:59-64.
2. Basavaraj M Ittanagi. Soft Bitopological Spaces, Comp. and Math. with App. 2014, 107(7)
3. Caldas M, JafariS. On some applications of b-open setsin topological spaces, Kochi, Jr. Math., Math., 2007; 2:11-19.
4. Çagman N, Enginoglu S. Soft set theory and uniting decision making, European Journal of Operational Research.
10.16/j.ejor.2010.05.004,2010.
5. Çagman N, Karatas S, Enginoglu S. Soft topology, Comp and Math with App. 2011; 62(1):351-358.
6. KellyJC. Bitopological Spaces, proc. London Math. Soc (1963); 13(3):71-83.
7. Molodtsov DA. Soft set theory- first results Comp and Math with App. 1999; 37(4-5):19-31.
8. Maji PK, Biswas R, Roy AR. Soft set theory Comp and Math. with App. 2003; 45:555-562
9. Min WK. Note on Soft Topological spaces. Comp and with App. 2011; 62:3524-3528.
10. Metin Akdag, Alkan Ozkan. Soft b-open sets and soft b-continuous functions. Math Sci, 2014.
11. Patty CW. Bitopological spaces. Duke Math J. 1967; 34:387-392.
12. ShabirM, Naz M. On Soft topological spaces. Comp and Math with App. 2011; 61:1786-1799.
13. šenel G, Çagman N. Soft bitopological spaces, submitted.
14. šenel G, Çagman N. Soft Closed sets on Soft bitopological Spaces. Journal of New Results in Science, 2014, 57-66.
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