American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS)
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402
© Global Society of Scientific Research and Researchers
http://asrjetsjournal.org/
Soft Pre-Open Sets In Soft Bitopological Spaces
Ameer Mohammed Hussein Hasan*
Department of mathematics Faculty of mathematics and computer science university of kufa, Najaf, Iraq
Email: [email protected]
Abstract
In this work , I introduce the concept of soft bitopological space on a soft set and some definitions on soft preopen set on soft bitopological space . Also introduce soft pre separation axioms , Spre- T , Spre- T1 and Spre-
T2 , with study some properties in soft bitopological space.
Keywords: Soft set; Soft pre-open; Soft topology; Soft bitopological spaces.
1. Introduction
Many classical methods have been used to solve some complicated problems in engineering economics and
environment. For instance, the interval mathematics, theory of fuzzy, theory of probability, and sets which can
consider as mathematical tools for dealing with uncertainties since all these theories have their own problems
and difficulties. In [2], the author in 1999 introduced the notion of soft set, which is free of difficulties in solving
aforementioned problems, and it has been applied over many different fields. In 2011, Naim Cagman and his
colleagues introduced a new concept of soft set called soft topology define by using the soft power set of soft set
, and this first idea to soft mathematical concepts and structures that are based on the operations of theoretic soft
set [6]. In 2011[7], the authors defined the concept of"soft topology on the collection of soft sets over"with
some basic notations of soft topological spaces. In [6], the notion of soft topology was more general than that in
[7]. Therefore, algebraists continue investigating the work of Cagman [6]"and follow their notations and
mathematical formalism. In 2013 J. Subhashini and C. Sekar defined soft pre-open sets [4]"by following
Cagman’s theory of soft topology. Therefore, this paper has introduced soft bitopological space relying on [6]
and defined the soft pre-open set of soft bitopological space. Also, I have discussed soft pre separation axioms ,
Spre- T , Spre- T1 and Spre- T2 .
-----------------------------------------------------------------------* Corresponding author.
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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
2. Preliminaries
Through this section , I give and introduction some important definitions and facts about soft topology and
recall primary definition soft set that need in this work .
2.1. Definition [2]
The set of ordered pairs
U , where
represents a soft set
is mapping and
the set of all soft sets over
FA on the universe
f A is called an approximate function of the soft set FA . However ,
U is denoted by S S (U ) .
2.2. Definition [5]
Let
FA ∈ S S (U ) . If
Moreover , If
x ∈ A , then FA is called an empty set and denoted by Fφ .
for all
f A = U for all x ∈ A , then FA is called an A -universal soft set and denoted by FAˆ . But , If
A = E , then A -universal soft set is called universal soft set and denoted by FE~ .
2.3. Definition [5]
Let
FA , FB ∈ S S (U ) . If f A ( x ) ⊆ f B ( x ) for all x , then , FA is a soft subset of FB and denoted by
.
2.4. Definition [5]
Let
~ F , however , the soft intersection is denoted
FA , FB ∈ S S (U ) . Then, the soft union is denoted by FA ∪
B
by
~
~ F Also , the soft difference of F and F is denoted by F ∆
FA ∩
A FB , are defined by the approximate
B
A
B
functions
,
,
respectively, on the onther hand , the soft complement
,
~
FAc of FA is defined by the approximate function
f c ( x ) = f Ac ( x ) , where f Ac (x ) is the complement of the set f A (x ) ; that is, f Ac ( x ) = U − f A ( x ) for all
A
~ ~
~
x ∈ A . It is easy to see that ( FAc ) c = FA and Fφc = FE~ .
2.5. Proposition [5]
Let
1-
FA , FB , FC ∈ S S (U ) . Then ,
.
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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
2-
.
3-
.
4-
.
5-
.
6-
.
7-
.
~
8- FA ∪ ( FB
~ F ) = (F ∪
~ F )∩
~ (F ∪
~ F )&F ∩
~ (F ∪
~ F ) = (F ∩
~ F )∪
~ (F ∩
~ F ).
∩
C
A
B
A
C
A
B
C
A
B
A
C
2.6. Definition [6]
Let
FA ∈ S S (U ) ."The soft power set of FA is defined by".
cardinality is defined by
, where
and its
| f A ( x ) | is the cardinality of f A (x ) .
2.7. Definition [6]
Let
FA ∈ S S (U ) . A soft topology on FA , denoted by τ~ , is a collection of soft subsets of FA having the
following properties:
•
Fφ , FA ∈τ~ .
•
.
•
.
2.8. Definition [6]
Let
( FA ,τ~ ) be a soft space on FA . Every element of τ~ is called a soft open sets .
2.9. Definition [6]
Let
( FA ,τ~ )
be
a
soft
space
on
FA and
44
~F
FB ⊆
A
.
Then
,
the
collection
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
~ F : F ∈τ~, i ∈ I ⊆ N }
τ~FB = {FAi ∩
B
Ai
called a soft topological subspace of
is called a soft subspace topology on
FB . Hence, ( FB ,τ~F ) is
B
( FA ,τ~ ) .
2.10. Definition [6]
Let
~ F . The soft interior of F , denoted F  , is defined as the
( FA ,τ~ ) be a soft space on FA and FB ⊆
A
B
B
union of all soft open subsets of
FB . Note that FB is the biggest soft open set that is contained by FB .
2.11. Theorem [6]
Let
~ F . Then ,
( FA ,τ~ ) be a soft space on FA and FB , FC ⊆
A
 
1- ( FB )
2- FB
= FB
~ F
~ F . Then , F  ⊆
⊆
B
C
C
3- FB

~ F  = (F ∩
~ F )
∩
C
B
C

~ F ⊆
~ F )
~ (F ∪
∪
C
B
C
4- FB
2.12. Definition [6]
Let
~ F . Then, the soft closure of F , denoted F B is defined as the
( FA ,τ~ ) be a soft space on FA and FB ⊆
A
B
soft intersection of all soft closed superset of
FB . Note that F B is the smallest soft closed set that containing
FB .
2.13. Theorem [6]
Let
~ F . Then ,
( FA ,τ~ ) be a soft space on FA and FB , FC ⊆
A
1- ( F B ) =
2- ( F B )
3- FB
c~
FB .
~
= ( FBc ) .
~ F . Then ,
⊆
C
.
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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
4-
.
5-
.
2.14. Theorem [6]
Let
~F .
~F ⊆
~ F Then F  ⊆
( FA ,τ~ ) be a soft space on FA and FB ⊆
B
B
B
A
2.15. Definition
Let
( FA ,τ~1 ) and ( FA ,τ~2 ) be the two different soft topologies on FA . Then ( FA ,τ~1 ,τ~2 ) is called a soft
bitopological space .
2.16. Example
Let
U = {u1 , u2 , u3 , u4 }
,
E = {w1 , w2 , w3}
,
then
FA = {( w1 , {u1})}
1
FA = {( w1 , {u2 })}
2
FA = {( w2 , {u3})}
4
FA = {( w2 , {u4 })}
5
FA = {( w2 , {u3 , u4 })}
6
FA = {( w1 , {u1}), ( w2 , {u3})}
7
FA = {( w1 , {u1}), ( w2 , {u4 })}
8
FA = {( w1 , {u2 }), (( w2 , {u3})}
10
46
A = {w1 , w2 }
such
that
~E
A⊆
and
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
FA = {( w1 , {u2 }), ( w2 , {u4 })}
11
FA = {( w1 , {u2 }), ( w2 , {u3 , u4 })}
12
FA = {( w1 , {u1 , u2 }), ( w2 , {u4 })}
14
FA = FA
15
FA = Fφ
16
Then
τ~1 = {Fφ , FA}
and
τ~2 = {Fφ , FA , FA2 , FA11 }
are a soft topology of
FA then ( FA ,τ~1 ,τ~2 ) , is a soft
bitopological space .
2.17. Definition
~F
. Then
( FA ,τ~1 ,τ~2 ) be a soft bitopological space over FA and FB ⊆
A
~ F : F ∈τ~ , i ∈ I ⊆ N } are said to be the
~ F : F ∈τ~ , i ∈ I ⊆ N } and τ~ F = {F ∩
τ~1FB = {FA ∩
2 B
A
B
A
2
1
B
A
Let
i
i
i
relative topologies on
i
FB . Then ( FB ,τ~1FB ,τ~2 FB ) is called a relative soft bitopological space of ( FA ,τ~1 ,τ~2 )
2.18. Theorem
If
~ τ~ is a soft topological space over F .
( FA ,τ~1 ,τ~2 ) is a soft bitopological space then τ~1 ∩
2
A
Proof :-
~
• Fφ , FA ∈τ~1 ∩ τ~2 .
• Let
~ τ~ ⇒ F ∈τ~ and F ∈τ~ for all i ∈ I . Therefore
{FA , i ∈ I } be a family of soft sets in τ~1 ∩
2
1
A
2
A

FA ∈τ~1 and
i∈I
i
i
i

i∈I
FA ∈τ~2 . Thus
i
• Let
1
n′
i =1

i∈I
~ τ~ .
FA ∈τ~1 ∩
2
i
FA ∈τ~1 and
. Then
FA ∈τ~1 and
i
1
n′
i =1
FA ∈τ~2 . Therefore
i
i
1
n′
i =1
i
~ τ~ ,1 ≤ i ≤ n′, n′ ∈ N .
FA ∈τ~1 ∩
2
i
47
. Since
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
2.19. Remark
If
~ τ~ is not a soft topological space over F .
( FA ,τ~1 ,τ~2 ) is a soft bitopological space then τ~1 ∪
2
A
2.20. Example
Let us consider 2.16 and let
τ~1 = {Fφ , FA , FA1 }
and
τ~2 = {Fφ , FA , FA2 }
are soft topology of
FA then
~ τ~ = {F , F , F , F } . If take F , F
( FA ,τ~1 ,τ~2 ) is soft bitopological space . Now τ~1 ∪
φ
A
A
A
2
A
A
1
2
1
2
,
~ τ~ is not soft topology on F .
~ F = {( x , {u }), ( x , {u })} ∉τ~ ∪
~ τ~ . Thus τ~ ∪
FA ∪
A
1
1
1
2
1
2
1
2
A
1
2
3. Some Definition of Soft Pre-open set in soft bitopological space
In this section introduce some definitions of soft pre-open set , soft pre-closed , soft pre-neighborhood , soft preclosure and soft pre-interior on soft bitopological space .
3.1. Definition
Let
~ F , F is called soft pre-open set with respect to
( FA ,τ~1 ,τ~2 ) be soft bitopological space and let FB ⊆
A
B
the two soft topological spaces
τ~1
and
τ~2
if
~ ( F B ) .
FB ⊆
3.2. Notes
1- The set of all soft pre-open set with respect to the two soft topologies is denoted by Pre( FA ).
2- The relative soft bitopological space for
FB with respect to Soft pre-open sets is the collection Pre
~ F : F ∈ Pre( F )}
( FA ) F given by Pre ( FA ) F = {FC ∩
B
C
A
B
B
3- Any
τ~2 -open soft set is not necessarily to be soft pre-open set .
4- Any soft pre-open set is not necessarily to be of τ~1 -open ( τ~2 -open ) soft set .
3.3. Example
Let us consider example 2.20 ,
~ F then
( FA ,τ~1 ,τ~2 ) is soft bitopological space . Take FA ⊆
A
1
~ ( F A )A ⇒ F is soft per-open set with respect to the two soft topological spaces τ~ and τ~ .
FA ⊆
A
1
2
1
1
1
3.4. Remarks
1- The intersection of any soft pre-open sets is not necessary a soft pre-open set
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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
2- The union of any soft pre-open sets is soft pre-open set with respect to soft bitopological space .
The example to part (1) is simply . The following proof explain the part (2) of the remark 3.4 . Let
FB and
~ ( F B ) and
FC be a two soft pre-open sets with respect to soft bitopological space ( FA ,τ~1 ,τ~2 ) . i.e. FB ⊆
~ ( F C )
FC ⊆
with
respect
to
the
two
~F ⊆
~ ( F C ) ⊆
~ F C )
~ ( F B ) ∪
~ (F B ∪
τ~2 ⇒ FB ∪
C
.
soft
topological
spaces
τ~1
Since
and
then
~F ⊆
~ F ) with respect to the two soft topological spaces τ~ and τ~ .
~ (F ∪
FB ∪
C
B
C
1
2
3.5. Definition
Let
~ F , F is called soft pre-closed set of F if and
( FA ,τ~1 ,τ~2 ) is soft bitopological space and let FB ⊆
A
B
A
only if
~
FBc is soft pre-open set of FA .
3.6. Definition
Let
~ F is said to be soft pre-neighborhood of a
( FA ,τ~1 ,τ~2 ) is soft bitopological space and α ∈ FA , FB ⊆
A
point
α
if there is a soft pre-open set
point
α
is denoted by Spre- υ~ (α ) .
~ F . The set of all soft pre-neighborhoods of a
FC such that α ∈ FC ⊆
B
3.7. Definition
Let
~ F . A point α ∈ F is said to be soft pre-interior
( FA ,τ~1 ,τ~2 ) is soft bitopological space , and FB ⊆
A
A
FB with respect to the two soft topological spaces τ~1 and τ~2 if there is a soft pre-open FC such that
point of
~F
α ∈ FC ⊆
B
and
τ~2
. The set of all soft pre-interior points of
FB with respect to the two soft topological spaces τ~1
denoted by Spre-int( FB ).
3.8. Definition
Let
( FA ,τ~1 ,τ~2 ) is soft bitopological space . A point α is called soft pre-limit point of soft subset FB of FA
with respect to the two soft topological spaces
containing another point different from
points of
α
in
τ~1
and
τ~2
if and only if for each a soft pre-open set
FC
~ F ≠ φ . The set of all soft pre-limit
FB , that is ( FC /{α }) ∩
B
FB be denoted by Spre-lm( FB ).
3.9. Definition
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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
Let
~ F , the intersection of all soft pre-closed sets
( FA ,τ~1 ,τ~2 ) is soft bitopological space , and FB ⊆
A
containing
FB is called soft pre-closure of FB , and is denoted by Spre-cl( FB ).
In the year 2014 J. Subhashinin and Dr. C.Sekar [3] by depending on the [6] and [4] introduces soft pre
separation axioms , soft P T -space and some of its properties in the soft topological spaces . Now begin to
important section to discuss soft pre separation axioms and some result .
4. The Separation Axioms in Soft Bitopological Space
In section four , I introduce some soft pre separation axioms , Spre- T , Spre- T1 and Spre- T2 and illustrate
transmission this Properties to The relative soft bitopological space with some result of soft pre separation
axioms .
4.1. Definition
Let
( FA ,τ~1 ,τ~2 ) is soft bitopological space , then ( FA ,τ~1 ,τ~2 ) is called Spre- T space if and only if for all
pair of soft point
α2
α1 , α 2 ∈ FA
or soft pre-open set
such that
α1 ≠ α 2
, there exists soft pre-open set
FB containing α1 but not
FC containing α 2 but not α1 .
4.2. Theorem
A soft bitopological space
( FA ,τ~1 ,τ~2 ) is Spre- T space if and only if for each distinct soft points α1 , α 2 in
FA , Spre-cl( {α1} ) ≠ Spre-cl( {α 2 } ) .
Proof :-
Let
α1 , α 2 ∈ FA
such that
α1 ≠ α 2
and Spre-cl( {α1} ) ≠ Spre-cl( {α 2 } ) .
Then there exists at least one soft point
Suppose
α3
in
FA such that , α 3 ∈ Spre-cl( {α1} ) but α 3 ∉ Spre-cl( {α 2 } ) .
~ Spreα 3 ∈ Spre-cl( {α1} ) , to show that α1 ∉ Spre-cl( {α 2 } ) . If α1 ∈ Spre-cl( {α 2 } ) , then {α1} ⊂
~ Spre-cl(Spre-cl( {α } ))=Spre-cl( {α } ) , hence
cl( {α 2 } ) . So Spre-cl( {α1} ) ⊂
2
2
α 3 ∈ Spre-cl( {α 2 } )
which is contradiction . Hence
α1 ∉ Spre-cl( {α 2 } )
α 3 ∈ Spre-cl( {α1} )
, consequently
, then
α1 ∈ FA -Spre-cl(
{α 2 } ) but Spre-cl( {α 2 } ) is soft pre-closed , so FA -Spre-cl( {α 2 } ) is soft pre-open which contains α1 but not
α2
. It follows that
Conversely , since
( FA ,τ~1 ,τ~2 ) is Spre- T space .
( FA ,τ~1 ,τ~2 ) is Spre- T space, then for each tow distinct soft points α1 , α 2 ∈ FA there
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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
exists soft pre-open set
α1
but contains
contain
α2
FB such that α1 ∈ FB , α 2 ∉ FB . FA − FB is soft closed set which does not contain
, by definition (3.9) Spre-cl( {α 2 } ) is the"soft intersection of all soft pre-closed"which
~ F − F then α ∉ F − F .This implies that α ∉ Spre-cl( {α } ) .
{α 2 } . Thus , Spre-cl( {α 2 } ) ⊂
2
A
1
B
1
A
B
So we have
α1 ∈ Spre-cl( {α1} ) , α1 ∉ Spre-cl( {α 2 } ) . Therefore Spre-cl( {α1} ) ≠
Spre-cl( {α 2 } )
4.3. Theorem
Every soft subspace of Spre- T space is Spre- T space .
Proof :-
Let
( FB ,τ~1FB ,τ~2 FB ) be a soft sub space of Spre- T space ( FA ,τ~1 ,τ~2 ) . To prove that the soft sub space is
β1 ≠ β 2
~ F then β ≠ β ∈ F and ( F ,τ~ ,τ~ ) is
FB ⊆
A
1
2
A
A 1 2
~
Spre- T space , then there is a soft pre-open set FC in FA , such that β1 ∈ FC , β 2 ∉ FC . So FC ∩ FB is soft
Spre- T space , let
β1 , β 2 ∈ FB
such that
. Since
~ F and β ∉ F ∩
~ F . Hence ( F ,τ~ F ,τ~ F ) is Spre- T space .
FB and β1 ∈ FC ∩
B
2
C
B

B 1 B
2 B
pre-open set in
4.4. Definition
Let
( FA ,τ~1 ,τ~2 ) is soft bitopological space , then ( FA ,τ~1 ,τ~2 ) is called Spre- T1 space if and only if for all pair
of soft point
α1 , α 2 ∈ FA
, there are two soft pre-open sets
FB , FC such that FB contains α1 but not α 2 and
FC contains α 2 but not α1 .
4.5. Theorem
Every soft subspace of Spre- T1 space is Spre- T1 space .
Proof :-
Let
( FB ,τ~1FB ,τ~2 FB ) be a soft sub space of Spre- T1 space ( FA ,τ~1 ,τ~2 ) . To prove that the soft sub space is
Spre- T1 space , let
β1 , β 2 ∈ FB
such that
β1 ≠ β 2
. Since
Spre- T space , then there exists two soft pre-open set
β 2 ∈ FD
but
β1 ∉ FD
but
FC , FD in FA , such that β1 ∈ FC but β 2 ∉ FC and
.
Then we obtain two soft set
β1 ∈ FC1
~ F then β ≠ β ∈ F and ( F ,τ~ ,τ~ ) is
FB ⊆
A 1 2
A
A
1
2
~ F ,F = F ∩
~ F are soft pre-open sets in F , we have
FC = FC ∩
B
D
D
B
B
β 2 ∉ FC1 ; β 2 ∈ FD1
1
1
, but
β1 ∈ FD1
. Hence
51
( FB ,τ~1FB ,τ~2 FB ) is Spre- T1 space .
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
4.6. Theorem
If Every singleton soft subset of soft bitopological space
( FA ,τ~1 ,τ~2 ) is soft pre-closed , then ( FA ,τ~1 ,τ~2 ) is
Spre- T1 space .
Proof :-This is clearly seen .
4.7. Theorem
A soft bitopological space
( FA ,τ~1 ,τ~2 ) is a Spre- T1 space if and only if Spre-cl ({α }) = φ , for each α ∈ FA
.
Proof :-This is clearly by using prove contradiction .
4.8. Definition
Let
( FA ,τ~1 ,τ~2 ) is soft bitopological space , then ( FA ,τ~1 ,τ~2 ) is called Spre- T2 space (Spre-Hausdorf) if and
only if for each pair of distinct soft point
that
α1 ∈ FB , α 2 ∈ FC
α1 , α 2 ∈ FA
, there exists two soft pre-open sets
FB , FC in FA such
~ F =φ .
FB ∩
C
and
4.9. Theorem
Each soft subspace of Spre- T2 space is Spre- T2 space .
Proof :-
Let
( FB ,τ~1FB ,τ~2 FB ) be a soft sub space of Spre- T2 space ( FA ,τ~1 ,τ~2 ) and let FB ≠ φ be a soft subset of
FA , and α1 ≠ α 2 ∈ FB then α1 , α 2 ∈ FA , since ( FA ,τ~1 ,τ~2 ) is Spre- T2 space ,there exists two soft pre~
~
~
open sets FD , FC in FA such that α1 ∈ FD , α 2 ∈ FC and FD ∩ FC = φ . So FD ∩ FB , FC ∩ FB are soft
pre-open
sets
in
FB
~ F ,α ∈ F ∩
~F
α1 ∈ FD ∩
2
B
C
B
and
;
~ F )∩
~ (F ∩
~ F ) = (F ∩
~ F )∩
~ F = φ . Hence ( F ,τ~ F ,τ~ F ) is a Spre- T space .
( FD ∩
2
2 B
B 1 B
B
C
B
D
C
B
4.10. Theorem
Each singleton soft subset of Spre- T2 space is a soft pre-closed .
Proof :-This is clearly seen .
52
and
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53
5. Conclusion
In the conclusion of a work paper , many of the basic concepts on soft bitopological space , introduced soft
bitopology . Furthermore , introduced relative soft bitopological space , soft pre-open set and some definitions
on bitopology by"soft pre-open set as (soft pre-closed , soft pre-neighborhood , soft pre-interior ,soft pre-limit
point and soft pre-closure) these"definitions using in other sections from the work and introduce some soft pre
separation axioms and studied Properties on soft bitopological space with some important results, one could
study the soft ideal bitopology and get some important results too .
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