research article a new approach to hausdorff space theory...

Research ArticleA New Approach to Hausdorff Space Theory via the Soft Sets

Guumlzide Fenel

Department of Mathematics Faculty of Arts and Science Amasya University 05100 Amasya Turkey

Correspondence should be addressed to Guzide Senel gsenelamasyaedutr

Received 1 April 2016 Accepted 8 August 2016

Academic Editor Anna M Gil-Lafuente

Copyright copy 2016 Guzide SenelThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of this paper is to present the concept of soft bitopological Hausdorff space (SBT Hausdorff space) as an originalstudy Firstly I introduce some new concepts in soft bitopological space such as SBT point SBT continuous function andSBT homeomorphism Secondly I define SBT Hausdorff space I analyse whether a SBT space is Hausdorff or not by SBThomeomorphism defined from a SBT Hausdorff space to researched SBT space I end my study by defining SBT property andhereditary SBT by SBT homeomorphism and investigate the relations between SBT space and SBT subspace

1 Introduction

In applied and theoretical areas ofmathematics we often dealwith sets evolved with various structures However it mayhappen that the consideration of a set with classical mathe-matical approaches is not useful to characterize uncertaintyTo overcome these difficulties Molodtsov [1] introduced theconcept of soft set as a new mathematical tool Later hedeveloped and applied this theory to several directions [2ndash4]New soft set definitions are made and new classes of soft setsandmappings between different classes of soft sets are studiedby many researchers [5ndash16] Topology depends strongly onthe ideas of set theory The theory of soft topological spacesis investigated by defining a new soft set theory whichcan lead to the development of new mathematical modelsThe topological structure of soft sets also was studied bymany authors [7 11 17ndash23] which are defined over an initialuniverse with a fixed set of parameters

In 1963 Kelly [24] defined the bitopological space asan original and fundamental work by using two differenttopologies It is an extension of general topology BeforeKelly bitopological space appeared in a narrow sense in [25]as a supplementary work to characterize Baire spaces In1990 Ivanov [26] presented a new viewpoint for the theoryof bitopological spaces by using a topologic structure onthe cartesian product of two sets There are several works

on theory (eg [26ndash31]) and application (eg [32ndash36]) ofbitopological spaces

A soft set with one specific topological structure is notsufficient to develop the theory In that case it becomesnecessary to introduce an additional structure on the soft setTo confirm this idea soft bitopological space (SBT) by softbitopological theory was introduced In this theory a soft setwas equipped with arbitrary soft topologies

In this paper I present the definition of soft bitopologicalHausdorff space and construct some basic properties Iintroduce the notions of SBT point SBT continuous functionand SBT homeomorphism Moreover I define SBT propertyand hereditary SBT by SBT homeomorphism and investigatethe relations between these concepts

2 Preliminaries

In this section I will recall the notions of soft sets [1] softpoint [23] soft function [37] soft topology [38] bitopologicalspace [24] and soft bitopological space [39] Then I will givesome properties of these notions

Throughout this work119880 refers to an initial universe 119864 isa set of parameters and 119875(119880) is the power set of 119880

Definition 1 (see [1]) A pair (119891 119864) is called a soft set (over119880)if and only if 119891 is a mapping or 119864 is the set of all subsets of theset 119880

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2196743 6 pageshttpdxdoiorg10115520162196743

2 Mathematical Problems in Engineering

From now on I will use definitions and operations of softsets which are more suitable for pure mathematics based onthe study of [10]

Definition 2 (see [10]) A soft set 119891 on the universe 119880 isdefined by the set of ordered pairs

119891 = (119890 119891 (119890)) 119890 isin 119864 (1)

where 119891 119864 rarr P(119880) such that 119891(119890) = 0 if 119890 isin 119864 119860 then119891 = 119891

119860

Note that the set of all soft sets over 119880 will be denoted byS

Definition 3 (see [10]) Let 119891 isin S Then

if 119891(119890) = 0 for all 119890 isin 119864 then 119891 is called an empty setdenoted by Φif 119891(119890) = 119880 for all 119890 isin 119864 then 119891 is called universal softset denoted by

Definition 4 (see [10]) Let 119891 119892 isin S Then

119891 is a soft subset of 119892 denoted by 119891 sube 119892 if 119891 sube 119892 forall 119890 isin 119864119891 and 119892 are soft equal denoted by 119891 = 119892 if and onlyif 119891(119890) = 119892(119890) for all 119890 isin 119864

Definition 5 (see [10]) Let 119891 119892 isin S Then soft union andsoft intersection of 119891 and 119892 are defined by the soft setsrespectively

119891 cup 119892 = 119891 (119890) cup 119892 (119890) 119890 isin 119864

119891 cap 119892 = 119891 (119890) cap 119892 (119890) 119890 isin 119864

(2)

and the soft complement of 119891 is defined by

119891= 119891 (119890)

119888 119890 isin 119864 (3)

where 119891 is the complement of the set 119891(119890) that is 119891(119890)119888 =119880 119891119860(119890) for all 119890 isin 119864

It is easy to see that (119891) = 119891 and Φ =

Proposition 6 (see [10]) Let 119891 isin S Then

(i) 119891 cup 119891 = 119891 119891 cap119891 = 119891(ii) 119891 cupΦ = 119891 119891 capΦ = Φ

(iii) 119891 cup = 119891 cap = 119891

(iv) 119891 cup 119891 = 119891 cap119891 = Φ

Proposition 7 (see [10]) Let 119891 119892 ℎ isin S Then

(i) 119891 cup 119892 = 119892 cup 119891 119891 cap 119892 = 119892 cap 119891

(ii) (119891 cap 119892) = 119892 cup 119891 (119891 cup 119892) = 119892 cap 119891

(iii) (119891 cup 119892) cup ℎ = 119891 cup (119892 cup ℎ) (119891 cap 119892) cap ℎ = 119891 cap (119892 cap ℎ)(iv) 119891 cup (119892 cap ℎ) = (119891 cup 119892) cap (119891 cup ℎ)

119891 cap (119892 cup ℎ) = (119891 cap 119892) cup (119891 cap ℎ)

Definition 8 (see [8]) Let 119891 isin S The power soft set of 119891 isdefined by

P (119891) = 119891119894sube 119891 119894 isin 119868 (4)

and its cardinality is defined by

1003816100381610038161003816P (119891)1003816100381610038161003816 = 2sum119890isin119864|119891(119890)|

(5)

where |119891(119890)| is the cardinality of 119891(119890)

Example 9 Let 119880 = 1199061 1199062 1199063 and 119864 = 119890

1 1198902 119891 isin S and

119891 = (1198901 1199061 1199062) (1198902 1199062 1199063) (6)

Then

1198911= (1198901 1199061)

1198912= (1198901 1199062)

1198913= (1198901 1199061 1199062)

1198914= (1198902 1199062)

1198915= (1198902 1199063)

1198916= (1198902 1199062 1199063)

1198917= (1198901 1199061) (1198902 1199062)

1198918= (1198901 1199061) (1198902 1199063)

1198919= (1198901 1199061) (1198902 1199062 1199063)

11989110= (1198901 1199062) (1198902 1199062)

11989111= (1198901 1199062) (1198902 1199063)

11989112= (1198901 1199062) (1198902 1199062 1199063)

11989113= (1198901 1199061 1199062) (1198902 1199062)

11989114= (1198901 1199061 1199062) (1198902 1199063)

11989115= 119891

11989116= Φ

(7)

are all soft subsets of 119891 So |(119891)| = 24 = 16

Definition 10 (see [23]) The soft set119891 isin S is called a softpointin denoted by 119890

119891 if there exists an element 119890 isin 119864 such that

119891(119890) = 0 and 119891(1198901015840) = 0 for all 1198901015840 isin 119864 119890

Definition 11 (see [23]) The soft point 119890119891is said to belong to

a soft set 119892 isin S denoted by 119890119891isin 119892 if 119890 isin 119864 and 119891(119890) sube 119892(119890)

Mathematical Problems in Engineering 3

Theorem 12 (see [23]) Let 119864 and119880 be finite sets The numberof all soft points in 119891 isin S is equal to

sum

119890isin119864

(2|119891(119890)|

minus 1) (8)

Theorem 13 (see [23]) A soft set can be written as the softunion of all its soft points

Theorem 14 (see [23]) Let 119891 119892 isin S Then

119890119894119891

isin 119892 997904rArr 119891 sube 119892 (9)

for all 119890119894119891

isin 119891

Definition 15 (see [37]) (i) Let119883 sube 119864 and119891 isin 119878119883(119880) be a soft

set in S The image of 119891 under 120593120595is a soft set in 119878

119870(119881) such

that

120593120595(119891) (119896

119895)

=

⋃

119890119894isin120595minus1(119896119895)cap119883

120593 (119891 (119890119894)) 120595

minus1(119896119895) cap 119883 = 0

0 120595minus1(119896119895) cap 119883 = 0

(10)

for all 119896119895isin 119870

(ii) Let 119884 sube 119870 and 119892 isin 119878119884(119881) Then the inverse image of

119892 under 120593120595is a soft set in 119878

119864(119880) such that

120593minus1

120595(119892) (119890

119894) =

120593minus1(119892 (120595 (119890

119894))) 120595 (119890

119894) isin 119884

0 120595 (119890119894) notin 119884

(11)

for all 119890119894isin 119864

Definition 16 (see [38]) Let Φ = 119883 sube 119864 and 119891 isin S Let = 119892

119894119894isin119868

be the collection of soft sets over 119891 Then iscalled a soft topology on 119891 if satisfies the following axioms

(i) Φ119891 isin

(ii) 119892119894119894isin119868sube rArr ⋃

119894isin119868119892119894isin

(iii) 119892119894119899

119894=1sube rArr ⋂

119899

119894=1119892119894isin

The pair (119891 ) is called a soft topological space over119891 andthe members of are said to be soft open in 119891

Example 17 Let us consider the soft subsets of 119891 that aregiven in Example 9 Then 1 = P(119891) 0 = Φ 119891 and = Φ 119891 119891

2 11989111 11989113 are some soft topologies on 119891

Definition 18 (see [38]) Let (119891 ) and 119892 isin S Then 119892 is softclosed in if 119892 isin

Definition 19 (see [24]) Let 119883 = 0 and let 1205911and 120591

2be

two different topologies on 119883 Then (119883 1205911 1205912) is called a

bitopological space Throughout this paper (119883 1205911 1205912) [or

simply119883] denote bitopological space on which no seperationaxioms are assumed unless explicitly stated

Definition 20 (see [24]) A subset 119878 of119883 is called 12059111205912-open if

119878 isin 1205911cup 1205912and the complement of 120591

11205912-open is 120591

11205912-closed

Example 21 Let 119883 = 119886 119887 119888 1205911= 0 119883 119886 and 120591

2=

0 119883 119887 The sets in 0 119883 119886 119887 119886 119887 are called 12059111205912-

open and the sets in 0 119883 119887 119888 119886 119888 119888 are called 12059111205912-

closed

Definition 22 (see [24]) Let 119878 be a subset of119883 Then

(i) the 12059111205912-interior of 119878 denoted by 120591

11205912int(119878) is defined

by

⋃119865 119878 sub 119865 119865 is a 12059111205912-open (12)

(ii) the 12059111205912-closure of 119878 denoted by 120591

11205912cl(119878) is defined

by

⋂119865 119878 sub 119865 119865 is a 12059111205912-closed (13)

Definition 23 (see [39]) Let 119891 be a nonempty soft set on theuniverse 119880 and let

1and 2be two different soft topologies

on 119891 Then (119891 1 2) is called a soft bitopological space

which is abbreviated as SBT space

Definition 24 (see [39]) Let (119891 1 2) be a SBT space and

119892 sub 119891 Then 119892 is called 12-soft open if 119892 = ℎ cup 119896 where

ℎ isin 1and 119896 isin

2

The soft complement of 12-soft open set is called

12-

soft closed

Definition 25 (see [39]) Let 119892 be a soft subset 119891 Then 12-

interior of 119892 denoted by (119892)∘12

is defined by the following

(119892)∘

12

=⋃119892 ℎ sub 119892 ℎ is

12-soft open (14)

The 12-closure of 119892 denoted by (119892)

12

is defined by thefollowing

(119892)12

=⋂ℎ 119892 sub ℎ ℎ is

12-soft closed (15)

Note that (119892)∘12

is the biggest 12-soft open set con-

tained in 119892 and (119892)12

is the smallest 12-soft closed set

contained in 119892

Example 26 (see [39]) Considering Example 9 1

=

Φ 119891 1198912 and

2= Φ 119891 119891

1 1198914Then Φ 119891 119891

1 1198912 1198913 1198914 are

12-soft open sets and Φ 119891 119891

1 1198912 1198915 are

12-soft closed

sets

3 SBT Hausdorff Space

In this section I present the definition of soft bitopologicalHausdorff space and construct some basic properties Iintroduce the notions of SBT point SBT continuous functionand SBT homeomorphism I analyse whether a SBT space isHausdorff or not by SBT homeomorphism defined from aSBT Hausdorff space to researched SBT space Moreover Idefine SBT property and hereditary SBT by SBT homeomor-phism and investigate the relations between these concepts


Definition 27 Let (119891 1 2) be a SBT space and119892 sub 119891Then119892

is called 12-soft point if 119892 is a soft point in S and is denoted

by 119890119892isin 119891

Definition 28 Let (119891 1 2) be a SBT space and let 119892 be a

soft set over 119880 The soft point 119890119891isinS is called a

12-interior

point of a soft set 119892 if there exists a soft open set ℎ such that119890119891isin ℎ isin 119892

Definition 29 Let (119891 1 2) and (119892 lowast

1 lowast

2) be two SBT spaces

and 120593120595 119878119864(119880) rarr 119878

119870(119881) be a soft function If 120593minus1

120595(ℎ) isin

1for

all ℎ isin lowast1and 120593minus1

120595(119896) isin

2for all 119896 isin lowast

2 then 120593

120595soft function

is called 12continuous function


1 lowast


and 120593120595 119878119864(119880) rarr 119878

119870(119881) be a soft function and 119890

119891isin 119891

(i) 120593120595soft function is

12continuous function at 119890

119891isin 119891

if for each 119892isin119896 120593120595(119890119891) isin 119896 isin

lowast

1cup lowast

2 there exists ℎ isin 119905

119890119891isin 119905 isin

1cup 2 such that 120593

120595(119890119891) sube 119892

(ii) 120593120595is 12continuous on 119891 if 120593

120595is soft continuous at

each soft point in 119891

Definition 31 A soft function 120593120595 119878119864(119880) rarr 119878

119870(119881) between

two SBT spaces (119891 1 2) and (119892 lowast

1 lowast

2) is called a SBT

homeomorphism if it has the following properties

(i) 120593120595is a soft bijection (soft surjective and soft injective)

(ii) 120593120595is 12continuous

(iii) 120593minus1120595

is 12continuous

A soft function with these three properties is called 12

homeomorphism If such a soft function exists we say(119891 1 2) and (119892 lowast

1 lowast

2) are SBT homeomorphic

Definition 32 SBT property is a property of a SBT spacewhich is invariant under SBT homeomorphisms

That is a property of SBT spaces is a SBT property ifwhenever a SBT space possesses that property every spaceSBT homeomorphic to this space possesses that property

Definition 33 Let (119891 1 2) be a SBT space If for each pair of

distinct soft points 119890119894119891119894

119890119895119891119895

isin 119891 there exist a 1open set 119892 and

2open set ℎ such that 119890

119894119891119894

isin 119892 119890119895119891119895

isin ℎ and 119892 cap ℎ = Φ then(119891 1 2) is called a SBT Hausdorff space

Example 34 Let 119891 = (1198901 1199061 1199062) (1198902 1199062 1199063) 1=

Φ 119891 (1198901 1199061) (1198901 1199062) and

2= Φ 119891 (119890

2 1199062)

Then 12-soft open sets are

Φ 119891 (1198901 1199061) (1198901 1199062) (1198902 1199062) (1198901 1199061 1199062) (16)

Let 11989011198911

= (1198901 1199061) 11989011198912

= (1198901 1199062) and 119890

11198911

= 11989011198912

1198921= (1198901 1199061) 1198922= (1198901 1199062) 11989011198911

isin 1198921 11989011198912

isin 1198922 and

1198921cap 1198922= Φ

Hence (119891 1 2) is a SBT Hausdorff space

4 More on SBT Hausdorff Space

We continue the study of the theory of SBTHausdorff spacesIn order to investigate all the soft bitopological modificationsof SBT Hausdorff spaces I present new definitions of

12-

soft closure SBT homeomorphism SBT property and hered-itary SBT I have explored relations between SBT space andSBT subspace by hereditary SBT

Definition 35 Let (119891 1 2) be a SBT space and

B12

sube 1cup 2 If every element of

1cup 2can be written as

the union of elements of B12

then B12

is called 12-soft

basis for (119891 1 2)

Each element of B12

is called soft bitopological basiselement

Theorem 36 Let (119891 1 2) be a SBT space and B

12

be a softbasis for (119891

1 2)Then

1cup2equals the collections of all soft

unions of elements B12

Proof It is clearly seen from Definition 35

Theorem 37 Every finite point 12-soft set in a SBT Haus-

dorff space is 12-soft closed set

Proof Let (119891 1 2) be a SBT Hausdorff space It suffices to

show that every soft point 119890119891 is 12-soft closed If 119890

119892is a soft

point of119891 different from 119890119891 then 119890

119891and 119890119892have disjoint

12-

soft neighborhoods 1198921and 119892

2 respectively Since 119892

1does not

soft-intersect 119890119892 the soft point 119890

119891cannot belong to the

12-

soft closure of the set 119890119892 As a result the

12-soft closure of

the set 119890119891 is 119890119891 itself so that it is

12-soft closed

In order to show Theorem 37 we have the followingexample

Example 38 Consider the SBT Hausdorff space in Exam-ple 34 Define finite soft point

12-soft sets 119891

1= (1198901 1199061)

and 1198912= (119890

1 1199062) such that soft points are 119890

11198911

=

(1198901 1199061) and 119890

11198912

= (1198901 1199062) By taking account of the

notion that 11989011198912

is a soft point of 119891 different from 11989011198911

then11989011198911

and 11989011198912

have disjoint 12-soft neighborhoods 119892

1and 119892

2

such that

1198921= (1198901 1199061)

1198912= (1198901 1199062)

(17)

Since (1198901 1199061) cap (119890

1 1199062) = Φ

12-soft closure of the

set 11989011198911

is itself so that it is 12-soft closed

Theorem 39 If (119891 1 2) is a SBT Hausdorff space and 120593

120595

119878119864(119880) rarr 119878

119870(119881) between two SBT spaces (119891

1 2) and

(119892 lowast

1 lowast

2) is a SBT homeomorphism then (119892 lowast

1 lowast

2) is a SBT

Hausdorff space

Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922

Since 120593120595is

soft surjective there exist 11989011198911

11989021198912

isin 119891 such that 120593120595(11989011198911

) =


11989011198921

120593120595(11989021198912

) = 11989021198922

and 11989011198911

= 11989021198912

From the hypoth-esis (119891

1 2) is a SBT Hausdorff space so there exist

ℎ 119896 isin 1cup 2such that 119890

11198911

isin ℎ 11989021198912

isin 119896 and ℎ cap 119896 = Φ Foreach 119890 isin 119864 119890

11198911

isin ℎ(119890) 11989021198912

isin 119896(119890) and ℎ(119890) cap 119896(119890) = 0 So120593120595(11989011198911

) = 11989011198921

isin 120593120595(ℎ(119890)) and 120593

120595(11989021198912

) = 11989021198922

isin 120593120595(119896(119890))

Hence 11989011198921

isin 120593120595(ℎ) 119890

21198922

isin 120593120595(119896) Since 120593

120595is soft open

then 120593120595(ℎ) 120593120595(119896) isin

lowast

1cup lowast

2and since 120593

120595is soft injective

120593120595(ℎ) cap 120593

120595(119896) = 120593

120595(ℎ cap 119896) = Φ Thus (119892 lowast

1 lowast

2) is a SBT

Hausdorff space

From Definition 32 andTheorem 39 we have the follow-ing

Remark 40 The property of being SBT Hausdorff space is aSBT property

Theorem 41 Let (119891 1 2) be a SBT space and 119892 sube 119891 Then

collections

1119892

= 119892119894cap 119892 119892

119894isin 1 119894 isin 119868 subeN

2119892

= ℎ119894cap 119892 ℎ

119894isin 2 119894 isin 119868 subeN

(18)

are soft bitopologies on 119892

Proof Indeed the union of the soft topologies containsΦ and119892 becauseΦ cap 119892 = Φ and119891 cap 119892 = 119892 where

1cup 2= 119892119894cup ℎ119894

119892119894cup ℎ119894sube 119891 119894 isin 119868

1cup 2= 119892119894cup ℎ119894 119892119894cup ℎ119894sube 119891 119894 isin 119868

it is closed under finite soft intersections and arbitrary softunions

119899

⋂

119894=1

(119892119894cap 119892) = (

119899

⋂

119894=1

119892119894) cap 119892

⋃

119894isin119868

(119892119894cap 119892) = (

⋃

119894isin119868

119892119894) cap 119892

(19)




By taking account of 119892 = 1198919 then

119892= Φ 119891

5 1198917 1198919 and

so (119892 119892) is a soft topological subspace of (119891 ) Hence we get

that (119892 1119892

2119892

) is a soft bitopological space on 119892

Definition 43 Let (119891 1 2) be a SBT space and 119892 sub 119891 If

collections 1119892

= 119892119894cap 119892 119892

119894isin 1 119894 isin 119868 subeN and

2119892

=

ℎ119894cap 119892 ℎ

119894isin 2 119894 isin 119868 subeN are two soft topologies on 119892 then

a SBT space (119892 1119892

2119892

) is called a SBT subspace of (119891 1 2)

In order to show Definition 43 we have the followingexample

Example 44 By taking account of Example 42 and consid-ering that (119891

1 2) is a SBT Hausdorff space ordered by

inclusion we have that (119892 1119892

2119892

) is called a SBT Hausdorffspace of (119891

1 2)

Theorem 45 Every SBT open set in (119891 1 2) is SBT open in

SBT subspace of (119891 1 2)


Theorem 46 Let (119891 1 2) be a SBT Hausdorff space and

119892sub119891 Then (119892 1119892

2119892

) is a SBT Hausdorff space

Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922

From thehypothesis 119892 sub 119891 so 119890

11198921

11989021198922

isin 119891 Since (119891 1 2) is a SBT

Hausdorff space there exist ℎ 119896 isin 1cup 2such that 119890

11198911

isin ℎ11989021198912

isin 119896 and ℎ cap 119896 = Φ So 11989011198921

isin ℎ cap 119892 and 11989021198922

isin 119896 cap 119892

(ℎ cap 119892) cap (119896 cap 119892) = (ℎ cap 119896) cap 119892 = Φ (20)

Thus (119892 1119892

2119892

) is SBT Hausdorff space


Remark 47 The property of being a soft SBTHausdorff spaceis hereditary

5 Conclusion

A soft set with one specific topological structure is notsufficient to develop the theory In that case it becomesnecessary to introduce an additional structure on the softset To confirm this idea soft bitopological space (SBT) bysoft bitopological theory was introduced It makes it moreflexible to develop the theory of soft topological spaces withits applicationsThus in this paper I make a new approach tothe SBT space theory

In the present work I introduce the concept of softbitopological Hausdorff space (SBT Hausdorff space) as anoriginal study Firstly I introduce some new concepts insoft bitopological space such as SBT point SBT continuousfunction and SBT homeomorphism Secondly I define SBTHausdorff space I analyse whether a SBT space is Hausdorffor not by SBT homeomorphism defined from a SBT Haus-dorff space to researched SBT space In order to investigateall the soft bitopological modifications of SBT Hausdorffspaces I present new definitions of

12-soft closure SBT

homeomorphism SBT property and hereditary SBT I haveexplored relations between SBT space and SBT subspace byhereditary SBT

I hope that findings in this paper will be useful tocharacterize the SBT Hausdorff spaces some further workscan be done on the properties of hereditary SBT and SBTproperty to carry out a general framework for applicationsof SBT spaces


Competing Interests

The author declares that there are no competing interests

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] D A Molodtsov ldquoThe description of a dependence with thehelp of soft setsrdquo Journal of Computer and Systems SciencesInternational vol 40 no 6 pp 977ndash984 2001

[3] D Molodtsov The Theory of Soft Sets URSS PublishersMoscow Russia 2004 (Russian)

[4] D A Molodtsov V Y Leonov and D V Kovkov ldquoSoft setstechnique and its applicationrdquo Nechetkie Sistemy i MyagkieVychisleniya vol 1 no 1 pp 8ndash39 2006

[5] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[6] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[7] A Aygunoglu and H Aygun ldquoSome notes on soft topologicalspacesrdquoNeural Computing and Applications vol 21 supplement1 pp 113ndash119 2011

[8] N Cagman and S Enginoglu ldquoSoft matrix theory and itsdecision makingrdquo Computers ampMathematics with Applicationsvol 59 no 10 pp 3308ndash3314 2010

[9] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 pp 848ndash855 2010

[10] N Cagman ldquoContributions to the theory of soft setsrdquo Journal ofNew Result in Science vol 4 pp 33ndash41 2014

[11] D N Georgiou and A C Megaritis ldquoSoft set theory andtopologyrdquo Applied General Topology vol 15 no 1 pp 93ndash1092014

[12] O Kazanci S Yilmaz and S Yamak ldquoSoft sets and soft BCH-algebrasrdquo Hacettepe Journal of Mathematics and Statistics vol39 no 2 pp 205ndash217 2010

[13] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[14] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[15] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[16] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[17] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

[18] E F Lashin A M Kozae A A Abo Khadra and T MedhatldquoRough set theory for topological spacesrdquo International Journalof Approximate Reasoning vol 40 no 1-2 pp 35ndash43 2005

[19] W K Min ldquoA note on soft topological spacesrdquo Computers ampMathematics with Applications vol 62 no 9 pp 3524ndash35282011

[20] E Peyghan B Samadi and A Tayebi ldquoAbout soft topologicalspacesrdquo Journal of New Results in Science vol 2 pp 60ndash75 2013

[21] G Senel Soft metric spaces gaziosmanpas [PhD thesis]University Graduate School of Natural and Applied SciencesDepartment of Mathematics 2013

[22] M Shabir and M Naz ldquoOn soft topological spacesrdquo Computersamp Mathematics with Applications vol 61 no 7 pp 1786ndash17992011

[23] I Zorlutuna M Akdag W K Min and S Atmaca ldquoRemarkson soft topological spacesrdquo Annals of Fuzzy Mathematics andInformatics vol 3 no 2 pp 171ndash185 2012

[24] J C Kelly ldquoBitopological spacesrdquo Proceedings of the LondonMathematical Society vol 13 no 3 pp 71ndash89 1963

[25] L Motchane ldquoSur La Notion Diespace Bitopologique et Sur LesEspaces de Bairerdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 224 pp 3121ndash3124 1957

[26] A A Ivanov ldquoProblems of the theory of bitoplogical spacesrdquoZap Nauchn Sem Leningrad Otdel Mat Inst Steklov (LOMI)vol 167 no 6 pp 5ndash62 1988 (Russian) English TranslationJournal of SovietMathematics vol 52 no 1 pp 2759ndash2790 1990

[27] G C L Brummer ldquoTwo procedures in bitopologyrdquo in Categori-cal Topology Proceedings of the International Conference BerlinAugust 27th to September 2nd 1978 vol 719 of Lecture Notes inMathematics pp 35ndash43 Springer Berlin Germany 1979

[28] M C Datta ldquoProjective bitopological spacesrdquoAustralianMath-ematical Society Journal Series A Pure Mathematics and Statis-tics vol 13 pp 327ndash334 1972

[29] M C Datta ldquoProjective bitopological spaces IIrdquo Journal of theAustralianMathematical Society vol 14 no 1 pp 119ndash128 1972

[30] B P Dvalishvili Bitoplogical Spaces Theory Relations withGeneralized Algebraic Structures and Applications vol 199 ofNorth-Holland Mathematical Studies Elsevier Science 2005

[31] C W Patty ldquoBitopological spacesrdquo Duke Mathematical Journalvol 34 pp 387ndash391 1967

[32] D Adnadjevic ldquoOrdered spaces and bitopologyrdquo GlasnikMatematicki Serija III vol 10 no 30 pp 337ndash340 1975

[33] B Banaschewski and G C Brummer ldquoStably continuousframesrdquoMathematical Proceedings of the Cambridge Philosoph-ical Society vol 104 no 1 pp 7ndash19 1988

[34] H A Priestley ldquoOrdered topological spaces and the represen-tation of distributive latticesrdquo Proceedings LondonMathematicalSociety vol 24 no 3 pp 507ndash530 1972

[35] O Ravi and M L Thivagar ldquoA bitopological (1 2)lowast semi-generalised continuous mapsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 29 no 1 pp 79ndash88 2006

[36] R E Smithson ldquoMultifunctions and bitopological spacesrdquoJournal of Natural Sciences andMathematics vol 11 pp 191ndash1981971

[37] S Hussain and B Ahmad ldquoSome properties of soft topologicalspacesrdquoComputersampMathematics withApplications vol 62 no11 pp 4058ndash4067 2011

[38] N Cagman S Karatas and S Enginoglu ldquoSoft topologyrdquoComputers amp Mathematics with Applications vol 62 no 1 pp351ndash358 2011

[39] G Senel andN Cagman ldquoSoft topological subspacesrdquoAnnals ofFuzzy Mathematics and Informatics vol 10 no 4 pp 525ndash5352015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


From now on I will use definitions and operations of softsets which are more suitable for pure mathematics based onthe study of [10]

Definition 2 (see [10]) A soft set 119891 on the universe 119880 isdefined by the set of ordered pairs

119891 = (119890 119891 (119890)) 119890 isin 119864 (1)

where 119891 119864 rarr P(119880) such that 119891(119890) = 0 if 119890 isin 119864 119860 then119891 = 119891

119860

Note that the set of all soft sets over 119880 will be denoted byS

Definition 3 (see [10]) Let 119891 isin S Then

if 119891(119890) = 0 for all 119890 isin 119864 then 119891 is called an empty setdenoted by Φif 119891(119890) = 119880 for all 119890 isin 119864 then 119891 is called universal softset denoted by

Definition 4 (see [10]) Let 119891 119892 isin S Then

119891 is a soft subset of 119892 denoted by 119891 sube 119892 if 119891 sube 119892 forall 119890 isin 119864119891 and 119892 are soft equal denoted by 119891 = 119892 if and onlyif 119891(119890) = 119892(119890) for all 119890 isin 119864

Definition 5 (see [10]) Let 119891 119892 isin S Then soft union andsoft intersection of 119891 and 119892 are defined by the soft setsrespectively

119891 cup 119892 = 119891 (119890) cup 119892 (119890) 119890 isin 119864

119891 cap 119892 = 119891 (119890) cap 119892 (119890) 119890 isin 119864

(2)

and the soft complement of 119891 is defined by

119891= 119891 (119890)

119888 119890 isin 119864 (3)

where 119891 is the complement of the set 119891(119890) that is 119891(119890)119888 =119880 119891119860(119890) for all 119890 isin 119864

It is easy to see that (119891) = 119891 and Φ =

Proposition 6 (see [10]) Let 119891 isin S Then

(i) 119891 cup 119891 = 119891 119891 cap119891 = 119891(ii) 119891 cupΦ = 119891 119891 capΦ = Φ

(iii) 119891 cup = 119891 cap = 119891

(iv) 119891 cup 119891 = 119891 cap119891 = Φ

Proposition 7 (see [10]) Let 119891 119892 ℎ isin S Then

(i) 119891 cup 119892 = 119892 cup 119891 119891 cap 119892 = 119892 cap 119891

(ii) (119891 cap 119892) = 119892 cup 119891 (119891 cup 119892) = 119892 cap 119891

(iii) (119891 cup 119892) cup ℎ = 119891 cup (119892 cup ℎ) (119891 cap 119892) cap ℎ = 119891 cap (119892 cap ℎ)(iv) 119891 cup (119892 cap ℎ) = (119891 cup 119892) cap (119891 cup ℎ)

119891 cap (119892 cup ℎ) = (119891 cap 119892) cup (119891 cap ℎ)

Definition 8 (see [8]) Let 119891 isin S The power soft set of 119891 isdefined by

P (119891) = 119891119894sube 119891 119894 isin 119868 (4)

and its cardinality is defined by

1003816100381610038161003816P (119891)1003816100381610038161003816 = 2sum119890isin119864|119891(119890)|

(5)

where |119891(119890)| is the cardinality of 119891(119890)

Example 9 Let 119880 = 1199061 1199062 1199063 and 119864 = 119890

1 1198902 119891 isin S and

119891 = (1198901 1199061 1199062) (1198902 1199062 1199063) (6)

Then

1198911= (1198901 1199061)

1198912= (1198901 1199062)

1198913= (1198901 1199061 1199062)

1198914= (1198902 1199062)

1198915= (1198902 1199063)

1198916= (1198902 1199062 1199063)

1198917= (1198901 1199061) (1198902 1199062)

1198918= (1198901 1199061) (1198902 1199063)

1198919= (1198901 1199061) (1198902 1199062 1199063)

11989110= (1198901 1199062) (1198902 1199062)

11989111= (1198901 1199062) (1198902 1199063)

11989112= (1198901 1199062) (1198902 1199062 1199063)

11989113= (1198901 1199061 1199062) (1198902 1199062)

11989114= (1198901 1199061 1199062) (1198902 1199063)

11989115= 119891

11989116= Φ

(7)

are all soft subsets of 119891 So |(119891)| = 24 = 16

Definition 10 (see [23]) The soft set119891 isin S is called a softpointin denoted by 119890

119891 if there exists an element 119890 isin 119864 such that

119891(119890) = 0 and 119891(1198901015840) = 0 for all 1198901015840 isin 119864 119890

Definition 11 (see [23]) The soft point 119890119891is said to belong to

a soft set 119892 isin S denoted by 119890119891isin 119892 if 119890 isin 119864 and 119891(119890) sube 119892(119890)



sum

119890isin119864

(2|119891(119890)|

minus 1) (8)



119890119894119891

isin 119892 997904rArr 119891 sube 119892 (9)

for all 119890119894119891

isin 119891



119870(119881) such

that

120593120595(119891) (119896

119895)

=

⋃

119890119894isin120595minus1(119896119895)cap119883

120593 (119891 (119890119894)) 120595

minus1(119896119895) cap 119883 = 0

0 120595minus1(119896119895) cap 119883 = 0

(10)

for all 119896119895isin 119870



119864(119880) such that

120593minus1

120595(119892) (119890

119894) =

120593minus1(119892 (120595 (119890

119894))) 120595 (119890

119894) isin 119884

0 120595 (119890119894) notin 119884

(11)

for all 119890119894isin 119864


119894119894isin119868


(i) Φ119891 isin

(ii) 119892119894119894isin119868sube rArr ⋃

119894isin119868119892119894isin

(iii) 119892119894119899


119899

119894=1119892119894isin






2be






11205912-open is 120591

11205912-closed

Example 21 Let 119883 = 119886 119887 119888 1205911= 0 119883 119886 and 120591

2=



closed




by

⋃119865 119878 sub 119865 119865 is a 12059111205912-open (12)


11205912cl(119878) is defined

by

⋂119865 119878 sub 119865 119865 is a 12059111205912-closed (13)








2


12-

soft closed




(119892)∘

12

=⋃119892 ℎ sub 119892 ℎ is

12-soft open (14)


12


(119892)12

=⋂ℎ 119892 sub ℎ ℎ is

12-soft closed (15)

Note that (119892)∘12


tained in 119892 and (119892)12


contained in 119892


=

Φ 119891 1198912 and

2= Φ 119891 119891

1 1198914Then Φ 119891 119891

1 1198912 1198913 1198914 are


1 1198912 1198915 are

12-soft closed

sets






by 119890119892isin 119891



12-interior



1 lowast


and 120593120595 119878119864(119880) rarr 119878


120595(ℎ) isin

1for


120595(119896) isin


2 then 120593

120595soft function



1 lowast


and 120593120595 119878119864(119880) rarr 119878


119891isin 119891



119891isin 119891


lowast

1cup lowast


119890119891isin 119905 isin


120595(119890119891) sube 119892





119870(119881) between


1 lowast

2) is called a SBT




(iii) 120593minus1120595

is 12continuous



1 lowast






119890119895119891119895



119894119891119894

isin 119892 119890119895119891119895


Example 34 Let 119891 = (1198901 1199061 1199062) (1198902 1199062 1199063) 1=

Φ 119891 (1198901 1199061) (1198901 1199062) and

2= Φ 119891 (119890

2 1199062)


Φ 119891 (1198901 1199061) (1198901 1199062) (1198902 1199062) (1198901 1199061 1199062) (16)

Let 11989011198911

= (1198901 1199061) 11989011198912

= (1198901 1199062) and 119890

11198911

= 11989011198912

1198921= (1198901 1199061) 1198922= (1198901 1199062) 11989011198911

isin 1198921 11989011198912

isin 1198922 and

1198921cap 1198922= Φ




12-



B12




then B12

is called 12-soft

basis for (119891 1 2)

Each element of B12



12


1 2)Then








119892is a soft



12-



1does not



12-


12-soft closure of


12-soft closed



12-soft sets 119891

1= (1198901 1199061)

and 1198912= (119890


11198911

=

(1198901 1199061) and 119890

11198912


notion that 11989011198912


then11989011198911

and 11989011198912


1and 119892

2

such that

1198921= (1198901 1199061)

1198912= (1198901 1199062)

(17)

Since (1198901 1199061) cap (119890

1 1199062) = Φ


set 11989011198911



120595

119878119864(119880) rarr 119878


1 2) and

(119892 lowast

1 lowast


1 lowast

2) is a SBT

Hausdorff space

Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922

Since 120593120595is


11989021198912

isin 119891 such that 120593120595(11989011198911

) =


11989011198921

120593120595(11989021198912

) = 11989021198922

and 11989011198911

= 11989021198912




11198911

isin ℎ 11989021198912


11198911

isin ℎ(119890) 11989021198912

isin 119896(119890) and ℎ(119890) cap 119896(119890) = 0 So120593120595(11989011198911

) = 11989011198921

isin 120593120595(ℎ(119890)) and 120593

120595(11989021198912

) = 11989021198922

isin 120593120595(119896(119890))

Hence 11989011198921

isin 120593120595(ℎ) 119890

21198922

isin 120593120595(119896) Since 120593

120595is soft open

then 120593120595(ℎ) 120593120595(119896) isin

lowast

1cup lowast

2and since 120593


120593120595(ℎ) cap 120593

120595(119896) = 120593


1 lowast

2) is a SBT

Hausdorff space




collections

1119892

= 119892119894cap 119892 119892

119894isin 1 119894 isin 119868 subeN

2119892

= ℎ119894cap 119892 ℎ

119894isin 2 119894 isin 119868 subeN

(18)



1cup 2= 119892119894cup ℎ119894

119892119894cup ℎ119894sube 119891 119894 isin 119868



119899

⋂

119894=1

(119892119894cap 119892) = (

119899

⋂

119894=1

119892119894) cap 119892

⋃

119894isin119868

(119892119894cap 119892) = (

⋃

119894isin119868

119892119894) cap 119892

(19)





119892= Φ 119891

5 1198917 1198919 and


that (119892 1119892

2119892



collections 1119892

= 119892119894cap 119892 119892


2119892

=

ℎ119894cap 119892 ℎ


a SBT space (119892 1119892

2119892






2119892


1 2)





119892sub119891 Then (119892 1119892

2119892


Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922


11198921

11989021198922



11198911

isin ℎ11989021198912

isin 119896 and ℎ cap 119896 = Φ So 11989011198921

isin ℎ cap 119892 and 11989021198922

isin 119896 cap 119892


Thus (119892 1119892

2119892




5 Conclusion



12-soft closure SBT




Competing Interests


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

119890isin119864

(2|119891(119890)|

minus 1) (8)



119890119894119891

isin 119892 997904rArr 119891 sube 119892 (9)

for all 119890119894119891

isin 119891



119870(119881) such

that

120593120595(119891) (119896

119895)

=

⋃

119890119894isin120595minus1(119896119895)cap119883

120593 (119891 (119890119894)) 120595

minus1(119896119895) cap 119883 = 0

0 120595minus1(119896119895) cap 119883 = 0

(10)

for all 119896119895isin 119870



119864(119880) such that

120593minus1

120595(119892) (119890

119894) =

120593minus1(119892 (120595 (119890

119894))) 120595 (119890

119894) isin 119884

0 120595 (119890119894) notin 119884

(11)

for all 119890119894isin 119864


119894119894isin119868


(i) Φ119891 isin

(ii) 119892119894119894isin119868sube rArr ⋃

119894isin119868119892119894isin

(iii) 119892119894119899


119899

119894=1119892119894isin






2be






11205912-open is 120591

11205912-closed

Example 21 Let 119883 = 119886 119887 119888 1205911= 0 119883 119886 and 120591

2=



closed




by

⋃119865 119878 sub 119865 119865 is a 12059111205912-open (12)


11205912cl(119878) is defined

by

⋂119865 119878 sub 119865 119865 is a 12059111205912-closed (13)








2


12-

soft closed




(119892)∘

12

=⋃119892 ℎ sub 119892 ℎ is

12-soft open (14)


12


(119892)12

=⋂ℎ 119892 sub ℎ ℎ is

12-soft closed (15)

Note that (119892)∘12


tained in 119892 and (119892)12


contained in 119892


=

Φ 119891 1198912 and

2= Φ 119891 119891

1 1198914Then Φ 119891 119891

1 1198912 1198913 1198914 are


1 1198912 1198915 are

12-soft closed

sets






by 119890119892isin 119891



12-interior



1 lowast


and 120593120595 119878119864(119880) rarr 119878


120595(ℎ) isin

1for


120595(119896) isin


2 then 120593

120595soft function



1 lowast


and 120593120595 119878119864(119880) rarr 119878


119891isin 119891



119891isin 119891


lowast

1cup lowast


119890119891isin 119905 isin


120595(119890119891) sube 119892





119870(119881) between


1 lowast

2) is called a SBT




(iii) 120593minus1120595

is 12continuous



1 lowast






119890119895119891119895



119894119891119894

isin 119892 119890119895119891119895


Example 34 Let 119891 = (1198901 1199061 1199062) (1198902 1199062 1199063) 1=

Φ 119891 (1198901 1199061) (1198901 1199062) and

2= Φ 119891 (119890

2 1199062)


Φ 119891 (1198901 1199061) (1198901 1199062) (1198902 1199062) (1198901 1199061 1199062) (16)

Let 11989011198911

= (1198901 1199061) 11989011198912

= (1198901 1199062) and 119890

11198911

= 11989011198912

1198921= (1198901 1199061) 1198922= (1198901 1199062) 11989011198911

isin 1198921 11989011198912

isin 1198922 and

1198921cap 1198922= Φ




12-



B12




then B12

is called 12-soft

basis for (119891 1 2)

Each element of B12



12


1 2)Then








119892is a soft



12-



1does not



12-


12-soft closure of


12-soft closed



12-soft sets 119891

1= (1198901 1199061)

and 1198912= (119890


11198911

=

(1198901 1199061) and 119890

11198912


notion that 11989011198912


then11989011198911

and 11989011198912


1and 119892

2

such that

1198921= (1198901 1199061)

1198912= (1198901 1199062)

(17)

Since (1198901 1199061) cap (119890

1 1199062) = Φ


set 11989011198911



120595

119878119864(119880) rarr 119878


1 2) and

(119892 lowast

1 lowast


1 lowast

2) is a SBT

Hausdorff space

Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922

Since 120593120595is


11989021198912

isin 119891 such that 120593120595(11989011198911

) =


11989011198921

120593120595(11989021198912

) = 11989021198922

and 11989011198911

= 11989021198912




11198911

isin ℎ 11989021198912


11198911

isin ℎ(119890) 11989021198912

isin 119896(119890) and ℎ(119890) cap 119896(119890) = 0 So120593120595(11989011198911

) = 11989011198921

isin 120593120595(ℎ(119890)) and 120593

120595(11989021198912

) = 11989021198922

isin 120593120595(119896(119890))

Hence 11989011198921

isin 120593120595(ℎ) 119890

21198922

isin 120593120595(119896) Since 120593

120595is soft open

then 120593120595(ℎ) 120593120595(119896) isin

lowast

1cup lowast

2and since 120593


120593120595(ℎ) cap 120593

120595(119896) = 120593


1 lowast

2) is a SBT

Hausdorff space




collections

1119892

= 119892119894cap 119892 119892

119894isin 1 119894 isin 119868 subeN

2119892

= ℎ119894cap 119892 ℎ

119894isin 2 119894 isin 119868 subeN

(18)



1cup 2= 119892119894cup ℎ119894

119892119894cup ℎ119894sube 119891 119894 isin 119868



119899

⋂

119894=1

(119892119894cap 119892) = (

119899

⋂

119894=1

119892119894) cap 119892

⋃

119894isin119868

(119892119894cap 119892) = (

⋃

119894isin119868

119892119894) cap 119892

(19)





119892= Φ 119891

5 1198917 1198919 and


that (119892 1119892

2119892



collections 1119892

= 119892119894cap 119892 119892


2119892

=

ℎ119894cap 119892 ℎ


a SBT space (119892 1119892

2119892






2119892


1 2)





119892sub119891 Then (119892 1119892

2119892


Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922


11198921

11989021198922



11198911

isin ℎ11989021198912

isin 119896 and ℎ cap 119896 = Φ So 11989011198921

isin ℎ cap 119892 and 11989021198922

isin 119896 cap 119892


Thus (119892 1119892

2119892




5 Conclusion



12-soft closure SBT




Competing Interests


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












by 119890119892isin 119891



12-interior



1 lowast


and 120593120595 119878119864(119880) rarr 119878


120595(ℎ) isin

1for


120595(119896) isin


2 then 120593

120595soft function



1 lowast


and 120593120595 119878119864(119880) rarr 119878


119891isin 119891



119891isin 119891


lowast

1cup lowast


119890119891isin 119905 isin


120595(119890119891) sube 119892





119870(119881) between


1 lowast

2) is called a SBT




(iii) 120593minus1120595

is 12continuous



1 lowast






119890119895119891119895



119894119891119894

isin 119892 119890119895119891119895


Example 34 Let 119891 = (1198901 1199061 1199062) (1198902 1199062 1199063) 1=

Φ 119891 (1198901 1199061) (1198901 1199062) and

2= Φ 119891 (119890

2 1199062)


Φ 119891 (1198901 1199061) (1198901 1199062) (1198902 1199062) (1198901 1199061 1199062) (16)

Let 11989011198911

= (1198901 1199061) 11989011198912

= (1198901 1199062) and 119890

11198911

= 11989011198912

1198921= (1198901 1199061) 1198922= (1198901 1199062) 11989011198911

isin 1198921 11989011198912

isin 1198922 and

1198921cap 1198922= Φ




12-



B12




then B12

is called 12-soft

basis for (119891 1 2)

Each element of B12



12


1 2)Then








119892is a soft



12-



1does not



12-


12-soft closure of


12-soft closed



12-soft sets 119891

1= (1198901 1199061)

and 1198912= (119890


11198911

=

(1198901 1199061) and 119890

11198912


notion that 11989011198912


then11989011198911

and 11989011198912


1and 119892

2

such that

1198921= (1198901 1199061)

1198912= (1198901 1199062)

(17)

Since (1198901 1199061) cap (119890

1 1199062) = Φ


set 11989011198911



120595

119878119864(119880) rarr 119878


1 2) and

(119892 lowast

1 lowast


1 lowast

2) is a SBT

Hausdorff space

Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922

Since 120593120595is


11989021198912

isin 119891 such that 120593120595(11989011198911

) =


11989011198921

120593120595(11989021198912

) = 11989021198922

and 11989011198911

= 11989021198912




11198911

isin ℎ 11989021198912


11198911

isin ℎ(119890) 11989021198912

isin 119896(119890) and ℎ(119890) cap 119896(119890) = 0 So120593120595(11989011198911

) = 11989011198921

isin 120593120595(ℎ(119890)) and 120593

120595(11989021198912

) = 11989021198922

isin 120593120595(119896(119890))

Hence 11989011198921

isin 120593120595(ℎ) 119890

21198922

isin 120593120595(119896) Since 120593

120595is soft open

then 120593120595(ℎ) 120593120595(119896) isin

lowast

1cup lowast

2and since 120593


120593120595(ℎ) cap 120593

120595(119896) = 120593


1 lowast

2) is a SBT

Hausdorff space




collections

1119892

= 119892119894cap 119892 119892

119894isin 1 119894 isin 119868 subeN

2119892

= ℎ119894cap 119892 ℎ

119894isin 2 119894 isin 119868 subeN

(18)



1cup 2= 119892119894cup ℎ119894

119892119894cup ℎ119894sube 119891 119894 isin 119868



119899

⋂

119894=1

(119892119894cap 119892) = (

119899

⋂

119894=1

119892119894) cap 119892

⋃

119894isin119868

(119892119894cap 119892) = (

⋃

119894isin119868

119892119894) cap 119892

(19)





119892= Φ 119891

5 1198917 1198919 and


that (119892 1119892

2119892



collections 1119892

= 119892119894cap 119892 119892


2119892

=

ℎ119894cap 119892 ℎ


a SBT space (119892 1119892

2119892






2119892


1 2)





119892sub119891 Then (119892 1119892

2119892


Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922


11198921

11989021198922



11198911

isin ℎ11989021198912

isin 119896 and ℎ cap 119896 = Φ So 11989011198921

isin ℎ cap 119892 and 11989021198922

isin 119896 cap 119892


Thus (119892 1119892

2119892




5 Conclusion



12-soft closure SBT




Competing Interests


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










11989011198921

120593120595(11989021198912

) = 11989021198922

and 11989011198911

= 11989021198912




11198911

isin ℎ 11989021198912


11198911

isin ℎ(119890) 11989021198912

isin 119896(119890) and ℎ(119890) cap 119896(119890) = 0 So120593120595(11989011198911

) = 11989011198921

isin 120593120595(ℎ(119890)) and 120593

120595(11989021198912

) = 11989021198922

isin 120593120595(119896(119890))

Hence 11989011198921

isin 120593120595(ℎ) 119890

21198922

isin 120593120595(119896) Since 120593

120595is soft open

then 120593120595(ℎ) 120593120595(119896) isin

lowast

1cup lowast

2and since 120593


120593120595(ℎ) cap 120593

120595(119896) = 120593


1 lowast

2) is a SBT

Hausdorff space




collections

1119892

= 119892119894cap 119892 119892

119894isin 1 119894 isin 119868 subeN

2119892

= ℎ119894cap 119892 ℎ

119894isin 2 119894 isin 119868 subeN

(18)



1cup 2= 119892119894cup ℎ119894

119892119894cup ℎ119894sube 119891 119894 isin 119868



119899

⋂

119894=1

(119892119894cap 119892) = (

119899

⋂

119894=1

119892119894) cap 119892

⋃

119894isin119868

(119892119894cap 119892) = (

⋃

119894isin119868

119892119894) cap 119892

(19)





119892= Φ 119891

5 1198917 1198919 and


that (119892 1119892

2119892



collections 1119892

= 119892119894cap 119892 119892


2119892

=

ℎ119894cap 119892 ℎ


a SBT space (119892 1119892

2119892






2119892


1 2)





119892sub119891 Then (119892 1119892

2119892


Proof Let 11989011198921

11989021198922

isin 119892 such that 11989011198921

= 11989021198922


11198921

11989021198922



11198911

isin ℎ11989021198912

isin 119896 and ℎ cap 119896 = Φ So 11989011198921

isin ℎ cap 119892 and 11989021198922

isin 119896 cap 119892


Thus (119892 1119892

2119892




5 Conclusion



12-soft closure SBT




Competing Interests


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Competing Interests


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a new approach to hausdorff space theory...

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