# A New Approach to Hausdorff Space Theory via the Soft Sets

Authors: Güzide Şenel
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

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


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## Body

## 1. Introduction

In applied and theoretical areas of mathematics, we often deal with sets evolved with various structures. However, it may happen that the consideration of a set with classical mathematical approaches is not useful to characterize uncertainty. To overcome these difficulties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool. Later, he developed and applied this theory to several directions [2–4]. New soft set definitions are made, and new classes of soft sets and mappings between different classes of soft sets are studied by many researchers [5–16]. Topology depends strongly on the ideas of set theory. The theory of soft topological spaces is investigated by defining a new soft set theory which can lead to the development of new mathematical models. The topological structure of soft sets also was studied by many authors [7, 11, 17–23] which are defined over an initial universe with a fixed set of parameters.In 1963, Kelly [24] defined the bitopological space as an original and fundamental work by using two different topologies. It is an extension of general topology. Before Kelly, bitopological space appeared in a narrow sense in [25] as a supplementary work to characterize Baire spaces. In 1990, Ivanov [26] presented a new viewpoint for the theory of bitopological spaces by using a topologic structure on the cartesian product of two sets. There are several works on theory (e.g., [26–31]) and application (e.g., [32–36]) of bitopological spaces.A soft set with one specific topological structure is not sufficient to develop the theory. In that case, it becomes necessary to introduce an additional structure on the soft set. To confirm this idea, soft bitopological space (SBT) by soft bitopological theory was introduced. In this theory, a soft set was equipped with arbitrary soft topologies.In this paper, I present the definition of soft bitopological Hausdorff space and construct some basic properties. I introduce the notions of SBT point, SBT continuous function, and SBT homeomorphism. Moreover, I define SBT property and hereditary SBT by SBT homeomorphism and investigate the relations between these concepts.


## 2. Preliminaries

In this section, I will recall the notions of soft sets [1], soft point [23], soft function [37], soft topology [38], bitopological space [24], and soft bitopological space [39]. Then, I will give some properties of these notions.Throughout this work,U refers to an initial universe, E is a set of parameters, and P(U) is the power set of U.Definition 1 (see [1]).
A pair(f,E) is called a soft set (over U) if and only if f is a mapping or E is the set of all subsets of the set U.From now on, I will use definitions and operations of soft sets which are more suitable for pure mathematics based on the study of [10].Definition 2 (see [10]).
A soft setf on the universe U is defined by the set of ordered pairs(1)f=e,fe:e∈E,where f:E→P(U) such that f(e)=∅; if e∈E∖A, then f=fA.Note that the set of all soft sets overU will be denoted by S.Definition 3 (see [10]).
Letf∈S. Then,iff(e)=∅ for all e∈E, then f is called an empty set, denoted by Φ;iff(e)=U for all e∈E, then f is called universal soft set, denoted by E~.Definition 4 (see [10]).
Letf,g∈S. Then,f is a soft subset of g, denoted by f⊆~g, if f⊆g for all e∈E;f and g are soft equal, denoted by f=g, if and only if f(e)=g(e) for all e∈E.Definition 5 (see [10]).
Letf,g∈S. Then, soft union and soft intersection of f and g are defined by the soft sets, respectively: (2)f∪~g=fe∪ge:e∈E,f∩~g=fe∩ge:e∈E,and the soft complement of f is defined by (3)fc~=fec:e∈E,where fc~ is the complement of the set f(e); that is, f(e)c=U∖fA(e) for all e∈E.It is easy to see that(fc~)c~=f and Φc~=E~.Proposition 6 (see [10]).
Letf∈S. Then,(i)
f∪~f=f, f∩~f=f;(ii)
f∪~Φ=f, f∩~Φ=Φ;(iii)
f∪~E~=E~, f∩~E~=f;(iv)
f∪~fc~=E~, f∩~fc~=Φ.Proposition 7 (see [10]).
Letf,g,h∈S. Then,(i)
f∪~g=g∪~f, f∩~g=g∩~f;(ii)
(f∩~g)c~=gc~∪~fc~, (f∪~g)c~=gc~∩~fc~;(iii)
(f∪~g)∪~h=f∪~(g∪~h), (f∩~g)∩~h=f∩~(g∩~h);(iv)
f∪~g∩~h=f∪~g∩~f∪~h;f∩~(g∪~h)=(f∩~g)∪~(f∩~h).Definition 8 (see [8]).
Letf∈S. The power soft set of f is defined by (4)Pf=fi⊆~f:i∈Iand its cardinality is defined by (5)Pf=2∑e∈Efe,where fe is the cardinality of f(e).Example 9.
LetU={u1,u2,u3} and E={e1,e2}. f∈S and (6)f=e1,u1,u2,e2,u2,u3.Then, (7)f1=e1,u1,f2=e1,u2,f3=e1,u1,u2,f4=e2,u2,f5=e2,u3,f6=e2,u2,u3,f7=e1,u1,e2,u2,f8=e1,u1,e2,u3,f9=e1,u1,e2,u2,u3,f10=e1,u2,e2,u2,f11=e1,u2,e2,u3,f12=e1,u2,e2,u2,u3,f13=e1,u1,u2,e2,u2,f14=e1,u1,u2,e2,u3,f15=f,f16=Φare all soft subsets of f. So P~f=24=16.Definition 10 (see [23]).
The soft setf∈S is called a soft point in E~, denoted by ef, if there exists an element e∈E such that f(e)≠∅ and f(e′)=∅, for all e′∈E∖{e}.Definition 11 (see [23]).
The soft pointef is said to belong to a soft set g∈S, denoted by ef∈~g, if e∈E and f(e)⊆g(e).Theorem 12 (see [23]).
LetE and U be finite sets. The number of all soft points in f∈S is equal to (8)∑e∈E2fe-1.Theorem 13 (see [23]).
A soft set can be written as the soft union of all its soft points.Theorem 14 (see [23]).
Letf,g∈S. Then, (9)eif∈~g⟹f⊆~gfor all eif∈~f.Definition 15 (see [37]).
(i) LetX⊆E and f∈SX(U) be a soft set in S. The image of f under φψ is a soft set in SK(V) such that (10)φψfkj=⋃ei∈ψ-1kj∩Xφfei,ψ-1kj∩X≠∅∅,ψ-1kj∩X=∅for all kj∈K.
(ii) LetY⊆K and g∈SY(V). Then, the inverse image of g under φψ is a soft set in SE(U) such that (11)φψ-1gei=φ-1gψei,ψei∈Y∅,ψei∉Yfor all ei∈E.Definition 16 (see [38]).
LetΦ≠X⊆E and f∈S. Let τ~={gi}i∈I be the collection of soft sets over f. Then, τ~ is called a soft topology on f if τ~ satisfies the following axioms: (i)
Φ,f∈τ~,(ii)
{gi}i∈I⊆τ~⇒⋃~i∈Igi∈τ~,(iii)
{gi}i=1n⊆τ~⇒⋂~i=1ngi∈τ~.The pair(f,τ~) is called a soft topological space over f and the members of τ~ are said to be soft open in f.Example 17.
Let us consider the soft subsets off that are given in Example 9. Then, τ~1=P~(f), τ~0={Φ,f}, and τ~={Φ,f,f2,f11,f13} are some soft topologies on f.Definition 18 (see [38]).
Let(f,τ~) and g∈S. Then, g is soft closed in τ~ if gc~∈τ~.Definition 19 (see [24]).
LetX≠∅, and let τ1 and τ2 be two different topologies on X. Then, (X,τ1,τ2) is called a bitopological space. Throughout this paper, (X,τ1,τ2) [or simply X] denote bitopological space on which no seperation axioms are assumed unless explicitly stated.Definition 20 (see [24]).
A subsetS of X is called τ1τ2-open if S∈τ1∪τ2 and the complement of τ1τ2-open is τ1τ2-closed.Example 21.
LetX={a,b,c}, τ1={∅,X,{a}}, and τ2={∅,X,{b}}. The sets in {∅,X,{a},{b},{a,b}} are called τ1τ2-open and the sets in {∅,X,{b,c},{a,c},{c}} are called τ1τ2-closed.Definition 22 (see [24]).
LetS be a subset of X. Then, (i)
theτ1τ2-interior of S, denoted by τ1τ2int⁡(S), is defined by (12)⋃F:S⊂F,Fis  aτ1τ2-open;(ii)
theτ1τ2-closure of S, denoted by τ1τ2cl(S), is defined by (13)⋂F:S⊂F,Fis  aτ1τ2-closed.Definition 23 (see [39]).
Letf be a nonempty soft set on the universe U, and let τ~1 and τ~2 be two different soft topologies on f. Then, (f,τ~1,τ~2) is called a soft bitopological space which is abbreviated as SBT space.Definition 24 (see [39]).
Let(f,τ~1,τ~2) be a SBT space and g⊂~f. Then, g is called τ~1τ~2-soft open if g=h∪~k, where h∈τ~1 and k∈τ~2.
The soft complement ofτ~1τ~2-soft open set is called τ~1τ~2-soft closed.Definition 25 (see [39]).
Letg be a soft subset f. Then, τ~1τ~2-interior of g, denoted by (g)τ~1τ~2∘, is defined by the following: (14)gτ~1τ~2∘=⋃⁡~g:h⊂~g,hisτ~1τ~2-soft  open.The τ~1τ~2-closure of g, denoted by (g¯)τ~1τ~2, is defined by the following: (15)g¯τ~1τ~2=⋂⁡~h:g⊂~h,hisτ~1τ~2-soft  closed.Note that(g)τ~1τ~2∘ is the biggest τ~1τ~2-soft open set contained in g and (g¯)τ~1τ~2 is the smallest τ~1τ~2-soft closed set contained in g.Example 26 (see [39]).
Considering Example9, τ~1={Φ,f,f2} and τ~2={Φ,f,f1,f4}. Then, {Φ,f,f1,f2,f3,f4} are τ~1τ~2-soft open sets and {Φ,f,f1,f2,f5} are τ~1τ~2-soft closed sets.


## 3. SBT Hausdorff Space

In this section, I present the definition of soft bitopological Hausdorff space and construct some basic properties. I introduce the notions of SBT point, SBT continuous function, and SBT homeomorphism. I analyse whether a SBT space is Hausdorff or not by SBT homeomorphism defined from a SBT Hausdorff space to researched SBT space. Moreover, I define SBT property and hereditary SBT by SBT homeomorphism and investigate the relations between these concepts.Definition 27.
Let(f,τ~1,τ~2) be a SBT space and g⊂~f. Then, g is called τ~1τ~2-soft point if g is a soft point in S and is denoted by eg∈~f.Definition 28.
Let(f,τ~1,τ~2) be a SBT space and let g be a soft set over U. The soft point ef∈~S is called a τ~1τ~2-interior point of a soft set g if there exists a soft open set h such that ef∈~h∈~g.Definition 29.
Let(f,τ~1,τ~2) and (g,τ~1∗,τ~2∗) be two SBT spaces and φψ:SE(U)→SK(V) be a soft function. If φψ-1(h)∈~τ~1 for all h∈~τ~1∗ and φψ-1(k)∈~τ~2 for all k∈~τ~2∗, then φψ soft function is called τ~1τ~2 continuous function.Definition 30.
Let(f,τ~1,τ~2) and (g,τ~1∗,τ~2∗) be two SBT spaces and φψ:SE(U)→SK(V) be a soft function and ef∈~f.(i)
φψ soft function is τ~1τ~2 continuous function at ef∈~f if, for each g∈~k, φψ(ef)∈~k∈~τ~1∗∪~τ~2∗, there exists h∈~t, ef∈~t∈~τ~1∪~τ~2, such that φψ(ef)⊆~g.(ii)
φψ is τ~1τ~2 continuous on f if φψ is soft continuous at each soft point in f.Definition 31.
A soft functionφψ:SE(U)→SK(V) between two SBT spaces (f,τ~1,τ~2) and (g,τ~1∗,τ~2∗) is called a SBT homeomorphism if it has the following properties: (i)
φψ is a soft bijection (soft surjective and soft injective).(ii)
φψ is τ~1τ~2 continuous.(iii)
φψ-1 is τ~1τ~2 continuous.A soft function with these three properties is called τ~1τ~2 homeomorphism. If such a soft function exists, we say (f,τ~1,τ~2) and (g,τ~1∗,τ~2∗) are SBT homeomorphic.Definition 32.
SBT property is a property of a SBT space which is invariant under SBT homeomorphisms.That is, a property of SBT spaces is a SBT property if whenever a SBT space possesses that property every space SBT homeomorphic to this space possesses that property.Definition 33.
Let(f,τ~1,τ~2) be a SBT space. If for each pair of distinct soft points eifi, ejfj∈~f there exist a τ~1 open set g and τ~2 open set h such that eifi∈~g,ejfj∈~h, and g∩~h=Φ, then (f,τ~1,τ~2) is called a SBT Hausdorff space.Example 34.
Letf={(e1,{u1,u2}),(e2,{u2,u3})}, τ~1={Φ,f,(e1,{u1}),(e1,{u2})}, and τ~2=Φ,f,e2,u2.
Then,τ~1τ~2-soft open sets are (16)Φ,f,e1,u1,e1,u2,e2,u2,e1,u1,u2.Let e1f1={(e1,{u1})}, e1f2={(e1,{u2})}, and e1f1≠e1f2.
g 1 = { ( e 1 , { u 1 } ) }, g2={(e1,{u2})}, e1f1∈~g1, e1f2∈~g2, and g1∩~g2=Φ.
Hence,(f,τ~1,τ~2) is a SBT Hausdorff space.


## 4. More on SBT Hausdorff Space

We continue the study of the theory of SBT Hausdorff spaces. In order to investigate all the soft bitopological modifications of SBT Hausdorff spaces, I present new definitions ofτ~1τ~2-soft closure, SBT homeomorphism, SBT property, and hereditary SBT. I have explored relations between SBT space and SBT subspace by hereditary SBT.Definition 35.
Let(f,τ~1,τ~2) be a SBT space and B~τ~1τ~2⊆~τ~1∪~τ~2. If every element of τ~1∪~τ~2 can be written as the union of elements of B~τ~1τ~2, then B~τ~1τ~2 is called τ~1τ~2-soft basis for (f,τ~1,τ~2).
Each element ofB~τ~1τ~2 is called soft bitopological basis element.Theorem 36.
Let(f,τ~1,τ~2) be a SBT space and B~τ~1τ~2 be a soft basis for (f,τ~1,τ~2). Then, τ~1∪τ~2 equals the collections of all soft unions of elements B~τ~1τ~2.Proof.
It is clearly seen from Definition35.Theorem 37.
Every finite pointτ~1τ~2-soft set in a SBT Hausdorff space is τ~1τ~2-soft closed set.Proof.
Let(f,τ~1,τ~2) be a SBT Hausdorff space. It suffices to show that every soft point {ef} is τ~1τ~2-soft closed. If eg is a soft point of f different from ef, then ef and eg have disjoint τ~1τ~2-soft neighborhoods g1 and g2, respectively. Since g1 does not soft-intersect {eg}, the soft point ef cannot belong to the τ~1τ~2-soft closure of the set {eg}. As a result, the τ~1τ~2-soft closure of the set {ef} is {ef} itself, so that it is τ~1τ~2-soft closed.In order to show Theorem37, we have the following example.Example 38.
Consider the SBT Hausdorff space in Example34. Define finite soft point τ~1τ~2-soft sets f1={(e1,{u1})} and f2={(e1,{u2})} such that soft points are e1f1={(e1,{u1})} and e1f2={(e1,{u2})}. By taking account of the notion that e1f2 is a soft point of f different from e1f1, then e1f1 and e1f2 have disjoint τ~1τ~2-soft neighborhoods g1 and g2 such that (17)g1=e1,u1,f2=e1,u2.Since {(e1,{u1})}∩~{(e1,{u2})}=Φ, τ~1τ~2-soft closure of the set e1f1 is itself, so that it is τ~1τ~2-soft closed.Theorem 39.
If(f,τ~1,τ~2) is a SBT Hausdorff space and φψ:SE(U)→SK(V) between two SBT spaces (f,τ~1,τ~2) and (g,τ~1∗,τ~2∗) is a SBT homeomorphism, then (g,τ~1∗,τ~2∗) is a SBT Hausdorff space.Proof.
Lete1g1,e2g2∈~g such that e1g1≠e2g2. Since φψ is soft surjective, there exist e1f1,e2f2∈~f such that φψ(e1f1)=e1g1, φψ(e2f2)=e2g2, and e1f1≠e2f2. From the hypothesis, (f,τ~1,τ~2) is a SBT Hausdorff space, so there exist h,k∈~τ~1∪~τ~2 such that e1f1∈~h, e2f2∈~k, and h∩~k=Φ. For each e∈~E, e1f1∈~h(e), e2f2∈~k(e), and h(e)∩k(e)=∅. So, φψ(e1f1)=e1g1∈~φψ(h(e)) and φψ(e2f2)=e2g2∈~φψ(k(e)). Hence, e1g1∈~φψ(h), e2g2∈~φψ(k). Since φψ is soft open, then φψ(h),φψ(k)∈~τ~1∗∪~τ~2∗ and since φψ is soft injective, φψ(h)∩~φψ(k)=φψ(h∩~k)=Φ. Thus, (g,τ~1∗,τ~2∗) is a SBT Hausdorff space.From Definition32 and Theorem 39, we have the following.Remark 40.
The property of being SBT Hausdorff space is a SBT property.Theorem 41.
Let(f,τ~1,τ~2) be a SBT space and g⊆~f. Then, collections(18)τ~1g=gi∩~g:gi∈τ~1,i∈I⊆~N,τ~2g=hi∩~g:hi∈τ~2,i∈I⊆~Nare soft bitopologies on g.Proof.
Indeed, the union of the soft topologies containsΦ and g because Φ∩~g=Φ and f∩~g=g, where τ~1∪~τ~2={gi∪~hi:gi∪~hi⊆~f,i∈I}, τ~1∪~τ~2={gi∪~hi:gi∪~hi⊆~f,i∈I}; it is closed under finite soft intersections and arbitrary soft unions: (19)
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.In order to show Theorem41, we have the following example.Example 42.
Let us consider the soft subsets off that are given in Example 9. Then, τ~1=P~(f), τ~0={Φ,f}, and τ~={Φ,f,f2,f11,f13} are some soft topologies on f.
By taking account ofg=f9, then τ~g={Φ,f5,f7,f9}, and so (g,τ~g) is a soft topological subspace of (f,τ~). Hence, we get that (g,τ~1g,τ~2g) is a soft bitopological space on g.Definition 43.
Let(f,τ~1,τ~2) be a SBT space and g⊂~f. If collections τ~1g={gi∩~g:gi∈τ~1,i∈I⊆~N} and τ~2g={hi∩~g:hi∈τ~2,i∈I⊆~N} are two soft topologies on g, then a SBT space (g,τ~1g,τ~2g) is called a SBT subspace of (f,τ~1,τ~2).In order to show Definition43, we have the following example.Example 44.
By taking account of Example42 and considering that (f,τ~1,τ~2) is a SBT Hausdorff space ordered by inclusion, we have that (g,τ~1g,τ~2g) is called a SBT Hausdorff space of (f,τ~1,τ~2).Theorem 45.
Every SBT open set in(f,τ~1,τ~2) is SBT open in SBT subspace of (f,τ~1,τ~2).Proof.
It is clearly seen from Definition43.Theorem 46.
Let(f,τ~1,τ~2) be a SBT Hausdorff space and g⊂~f. Then, (g,τ~1g,τ~2g) is a SBT Hausdorff space.Proof.
Lete1g1,e2g2∈~g such that e1g1≠e2g2. From the hypothesis, g⊂~f, so e1g1,e2g2∈~f. Since (f,τ~1,τ~2) is a SBT Hausdorff space, there exist h,k∈~τ~1∪~τ~2 such that e1f1∈~h, e2f2∈~k and h∩~k=Φ. So e1g1∈~h∩~g and e2g2∈~k∩~g. (20)h∩~g∩~k∩~g=h∩~k∩~g=Φ.Thus, (g,τ~1g,τ~2g) is SBT Hausdorff space.From Definition43 and Theorem 46, we have the following.Remark 47.
The property of being a soft SBT Hausdorff space is hereditary.


## 5. Conclusion

A soft set with one specific topological structure is not sufficient to develop the theory. In that case, it becomes necessary to introduce an additional structure on the soft set. To confirm this idea, soft bitopological space (SBT) by soft bitopological theory was introduced. It makes it more flexible to develop the theory of soft topological spaces with its applications. Thus, in this paper, I make a new approach to the SBT space theory.In the present work, I introduce the concept of soft bitopological Hausdorff space (SBT Hausdorff space) as an original study. Firstly, I introduce some new concepts in soft bitopological space such as SBT point, SBT continuous function, and SBT homeomorphism. Secondly, I define SBT Hausdorff space. I analyse whether a SBT space is Hausdorff or not by SBT homeomorphism defined from a SBT Hausdorff space to researched SBT space. In order to investigate all the soft bitopological modifications of SBT Hausdorff spaces, I present new definitions ofτ~1τ~2-soft closure, SBT homeomorphism, SBT property, and hereditary SBT. I have explored relations between SBT space and SBT subspace by hereditary SBT.I hope that findings in this paper will be useful to characterize the SBT Hausdorff spaces; some further works can be done on the properties of hereditary SBT and SBT property to carry out a general framework for applications of SBT spaces.



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*Source: 2196743-2016-09-07.xml*