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

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 set f on the universe U is defined by the set of ordered pairs

f = {(e, f(e)): e [member of] E}, (1)

where f : E [right arrow] P(U) such that f(e) = 0; if e [member of] E \ A, then f = [f.sub.A].

Note that the set of all soft sets over U will be denoted by S.

Definition 3 (see [10]). Let f [member of] S. Then,

if f(e) = 0 for all e [member of] E, then f is called an empty set, denoted by [PHI];

if f(e) = U for all e [member of] E, then f is called universal soft set, denoted by [??].

Definition 4 (see [10]). Let f, g [member of] S. Then,

f is a soft subset of g, denoted by f [??] g, if f [[subset].bar] g for all e [member of] E;

f and g are soft equal, denoted by f = g, if and only if f(e) = g(e) for all e [member of] E.

Definition 5 (see [10]). Let f g [member of] S. Then, soft union and soft intersection of f and g are defined by the soft sets, respectively:

f [??] g = {f (e) [union] g (e) : e [member of] E},

f [??] g = {f (e) [intersection] g (e) : e [member of] E}, (2)

and the soft complement of f is defined by

[f.sup.[??]] = {f [(e).sup.c] : e [member of] E}, (3)

where [f.sup.[??]] is the complement of the set f(e); that is, f [(e).sup.c] = U \ [f.sub.A] (e) for all e [member of] E.

It is easy to see that [mathematical expression not reproducible].

Proposition 6 (see [10]). Let f [member of] S. Then,

(i) f [??] f = f, f [??] f = f;

(ii) f [??] [PHI] = f, f [??] [PHI] = [PHI];

(iii) [mathematical expression not reproducible];

(iv) [mathematical expression not reproducible].

Proposition 7 (see [10]). Let f, g, h [member of] S. Then,

(i) [mathematical expression not reproducible];

(ii) [mathematical expression not reproducible];

(iii) [mathematical expression not reproducible];

(iv) [mathematical expression not reproducible].

Definition 8 (see [8]). Let f e S. The power soft set of f is defined by

P (f) = {[f.sub.i] [??] f : i [member of] I} (4)

and its cardinality is defined by

[mathematical expression not reproducible], (5)

where [absolute value of f(e)] is the cardinality of f(e).

Example 9. Let U = {[u.sub.1], [u.sub.2], [u.sub.3]] and E = {[e.sub.1], [e.sub.2]}. f [member of] S and f = {{[e.sub.1], {[u.sub.1], [u.sub.2]}),{[e.sub.2], {[u.sub.2], [u.sub.3]})}. (6)

Then,

[mathematical expression not reproducible] (7)

are all soft subsets of f. So [absolute value of [??](f)] = 24 = 16.

Definition 10 (see [23]). The soft set f [member of] S is called a soft point in [??], denoted by [e.sub.f], if there exists an element e [member of] E such that f(e) [not equal to] 0 and f(e') = 0, for all e [member of] E \ {e}.

Definition 11 (see [23]). The soft point [e.sub.f] is said to belong to a soft set g [member of] S, denoted by [e.sub.f] [??] g, if e [member of] E and f(e) [[subset].bar] g(e).

Theorem 12 (see [23]). Let E and U be finite sets. The number of all soft points in f [member of] S is equal to

[mathematical expression not reproducible]. (8)

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

Theorem 14 (see [23]). Let f, g [member of] S. Then,

[mathematical expression not reproducible] (9)

for all [mathematical expression not reproducible].

Definition 15 (see [37]). (i) Let X [[subset].bar] E and f [member of] [S.sub.X](U) be a soft set in S. The image of f under [[phi].sub.[psi]] is a soft set in [S.sub.K](V) such that

[mathematical expression not reproducible] (10)

for all [k.sub.j] [member of] K.

(ii) Let Y [[subset].bar] K and g [member of] [S.sub.Y](V). Then, the inverse image of g under [[phi].sub.[psi]] is a soft set in [S.sub.E](U) such that

[mathematical expression not reproducible] (11)

for all [e.sub.i] [member of] E.

Definition 16 (see [38]). Let [PHI] [not equal to] X [[subset].bar] E and f [member of] S. Let [??] = {[g.sub.i]}.sub.i [member of] I] be the collection of soft sets over f. Then, [??] is called a soft topology on f if [??] satisfies the following axioms:

(i) [PHI], f [member of] [??],

(ii) {[g.sub.i]}.sub.i [member of] I] [[subset].bar] [??] [right arrow] [[??].sub.i [member of] I] [g.sub.i] [member of] [??],

(iii) [{[g.sub.i]}.sup.n.sub.i=1] [[subset].bar] [right arrow] [[??].sup.n.sub.i=1] [g.sub.i] [member of] [??].

The pair (f, [??]) is called a soft topological space over f and the members of [??] are said to be soft open in f.

Example 17. Let us consider the soft subsets of f that are given in Example 9. Then, [mathematical expression not reproducible], and [??] = {[PHI], f, [f.sub.2], [f.sub.11], [f.sub.13]} are some soft topologies on f.

Definition 18 (see [38]). Let (f, [??]) and g [member of] S. Then, g is soft closed in [mathematical expression not reproducible].

Definition 19 (see [24]). Let X [not equal to] 0, and let [[tau].sub.1] and [[tau].sub.2] be two different topologies on X. Then, (X, [[tau].sub.1], [[tau].sub.2]) is called a bitopological space. Throughout this paper, (X, [[tau].sub.1], [[tau].sub.2]) [or simply X] denote bitopological space on which no seperation axioms are assumed unless explicitly stated.

Definition 20 (see [24]). A subset S of X is called [[tau].sub.1] [[tau].sub.2]- open if S [member of] [[tau].sub.1] [union] [[tau].sub.2] and the complement of [[tau].sub.1] [[tau].sub.2]-open is [[tau].sub.1] [[tau].sub.2]-closed.

Example 21. Let X = {a, b, c}, [[tau].sub.1] = {0, X, {a}}, and [[tau].sub.2] = {0, X, {b}}. The sets in {0, X, {a}, {b},{a, b}} are called [[tau].sub.1] [[tau].sub.2]-open and the sets in {0, X, {b, c}, {a, c}, {c}} are called [[tau].sub.1] [[tau].sub.2]-closed.

Definition 22 (see [24]). Let S be a subset of X. Then,

(i) the [[tau].sub.1] [[tau].sub.2]-interior of S, denoted by [[tau].sub.1] [[tau].sub.2] int(S), is defined by

U [F : S [subset] F, F is a [[tau].sub.1] [[tau].sub.2]-open}; (12)

(ii) the [[tau].sub.1] [[tau].sub.2]-closure of S, denoted by [[tau].sub.1] [[tau].sub.2] cl(S), is defined by

[intersection] [F : S [subset] F, F is a [[tau].sub.1] [[tau].sub.2]-closed}. (13)

Definition 23 (see [39]). Let f be a nonempty soft set on the universe U, and let [??].sub.1] and [[??].sub.2] be two different soft topologies on f. Then, (f, [??].sub.1], [[??].sub.2]) is called a soft bitopological space which is abbreviated as SBT space.

Definition 24 (see [39]). Let (f, [??].sub.1], [[??].sub.2]) be a SBT space and g [??] f. Then, g is called [??].sub.1] [[??].sub.2])-soft open if g = h [??] k, where h [member of] [??].sub.1] and k [member of] [[??].sub.2]).

The soft complement of [??].sub.1] [[??].sub.2])-soft open set is called [??].sub.1] [[??].sub.2])-soft closed.

Definition 25 (see [39]). Let g be a soft subset f. Then, [[??].sub.1] [[??].sub.2]-interior of g, denoted by [mathematical expression not reproducible], is defined by the following:

[mathematical expression not reproducible]. (14)

The [[??].sub.1] [[??].sub.2]-closure of g, denoted by ([mathematical expression not reproducible], is defined by the following:

[mathematical expression not reproducible]. (15)

Note that [mathematical expression not reproducible] the biggest [[??].sub.1] [[??].sub.2]-soft open set contained in g and [mathematical expression not reproducible] is the smallest [[??].sub.1] [[??].sub.2]-soft closed set contained in g.

Example 26 (see [39]). Considering Example 9, [[??].sub.1] = {[PHI] f, [f.sub.2]} and [[??].sub.2] = {[PHI], f [f.sub.1], [f.sub.4]}. Then, {[PHI], f [f.sub.1], [f.sub.2], [f.sub.3], [f.sub.4]} are [[??].sub.1] [[??].sub.2]-soft open sets and {[PHI], f [f.sub.1], [f.sub.2], [f.sub.5]} are [[??].sub.1] [[??].sub.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, [[??].sub.1], [[??].sub.2]) be a SBT space and g [??] f. Then, g is called [[??].sub.1] [[??].sub.2]-soft point if g is a soft point in S and is denoted by [e.sub.g] [??] f.

Definition 28. Let (f, [[??].sub.1], [[??].sub.2]) be a SBT space and let g be a soft set over U. The soft point [e.sub.f] [??] S is called a (f, [[??].sub.1] [[??].sub.2])-interior point of a soft set g if there exists a soft open set h such that [mathematical expression not reproducible].

Definition 29. Let (f, (f, [[??].sub.1], [[??].sub.2])) and (g, (f, [[??].sub.1], [[??].sub.2])) be two SBT spaces and ([[phi].sub.[psi]] : SE(U) [right arrow] [S.sub.K](V) be a soft function. If [mathematical expression not reproducible] for all [mathematical expression not reproducible].

Definition 30. Let (f, [[??].sub.1], [[??].sub.2])) and (g, (f, [[??]..sup.*.sub.1], [[??].sup.*.sub.2])) be two SBT spaces and [[phi].sub.[psi]] : [S.sub.E](U) [right arrow] [S.sub.K](V) be a soft function and [e.sub.f] [??] f.

(i) [[phi].sub.[psi]] soft function is [[??].sub.1], [[??].sub.2] continuous function at [e.sub.f] [??] f if, for each [mathematical expression not reproducible], there exists h [??] t, [e.sub.f] [mathematical expression not reproducible], such that [[phi].sub.[psi]] ([e.sub.f]) [??] g.

(ii) [[phi].sub.[psi]] is [[??].sub.1] [[??].sub.2] continuous on f if [[phi].sub.[psi]] is soft continuous at each soft point in f.

Definition 31. A soft function [[phi].sub.[psi]] : [S.sub.E](U) [right arrow] [S.sub.K](V) between two SBT spaces (f, [[??].sub.1] [[??].sub.2]) and (g, [[??]..sup.*.sub.1], [[??].sup.*.sub.2]) is called a SBT homeomorphism if it has the following properties:

(i) [[phi].sub.[psi]] is a soft bijection (soft surjective and soft injective).

(ii) [[phi].sub.[psi]] is [[??].sub.1], [[??].sub.2] continuous.

(iii) [[phi].sup.-1.sub.[psi]] is [[??].sub.1] [[??].sub.2] continuous.

A soft function with these three properties is called [[??].sub.1] [[??].sub.2] homeomorphism. If such a soft function exists, we say (f, [[??].sub.1], [[??].sub.2]) and (g, [[??]..sup.*.sub.1], [[??].sup.*.sub.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, [[??].sub.1], [[??].sub.2]) be a SBT space. If for each pair of distinct soft points [mathematical expression not reproducible] there exist a [[??].sub.1] open set g and [[??].sub.2] open set h such that [mathematical expression not reproducible], then (f, [[??].sub.1], [[??].sub.2]) is called a SBT Hausdorff space.

Example 34. Let f = {([e.sub.1], {[u.sub.1], [u.sub.2]}), ([e.sub.2], {[u.sub.2], [u.sub.3]})}, [[??].sub.1] = {[PHI], f, ([e.sub.1], {[u.sub.1]}), ([e.sub.1],{[u.sub.2]})}, and [[??].sub.2] = {[PHI], f, ([e.sub.2], {[u.sub.2]})}. Then, [??].sub.1], [[??].sub.2]-soft open sets are

{[PHI], f, ([e.sub.1], {[u.sub.1]}), ([e.sub.1], {[u.sub.2]}), ([e.sub.2], {[u.sub.2]}), ([e.sub.1], {[u.sub.1], [u.sub.2]})}. (16)

Let [mathematical expression not reproducible].

Hence, (f, [[??].sub.1], [[??].sub.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 of (f, [[??].sub.1] [[??].sub.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, [[??].sub.1], [[??].sub.2]) be a SBT space and [mathematical expression not reproducible]. If every element of [[??].sub.1] [??] [[??].sub.2] can be written as the union of elements of [mathematical expression not reproducible] is called [[??].sub.1] [[??].sub.2]-soft basis for (f, [[??].sub.1], [[??].sub.2]).

Each element of [mathematical expression not reproducible] is called soft bitopological basis element.

Theorem 36. Let (f, [[??].sub.1], [[??].sub.2]) be a SBT space and [mathematical expression not reproducible] be a soft basisfor (f, [[??].sub.1], [[??].sub.2]). Then, [[??].sub.1] [union] [[??].sub.2] equals the collections of all soft unions of elements [mathematical expression not reproducible].

Proof. It is clearly seen from Definition 35.

Theorem 37. Every finite point [[??].sub.1] [[??].sub.2]-soft set in a SBT Hausdorff space is [[??].sub.1] [[??].sub.2]-soft closed set.

Proof. Let (f, [[??].sub.1], [[??].sub.2]) be a SBT Hausdorff space. It suffices to show that every soft point {[e.sub.f]} is [[??].sub.1] [[??].sub.2]-soft closed. If [e.sub.g] is a soft point of f different from [e.sub.f] then [e.sub.f] and [e.sub.g] have disjoint [[??].sub.1] [[??].sub.2]-soft neighborhoods [g.sub.1] and [g.sub.2], respectively. Since [g.sub.1] does not soft-intersect {[e.sub.g]}, the soft point [e.sub.f] cannot belong to the [[??].sub.1] [[??].sub.2]-soft closure of the set {[e.sub.g]}. As a result, the [[??].sub.1] [[??].sub.2]-soft closure of the set {[e.sub.f]} is {[e.sub.f]} itself, so that it is [[??].sub.1] [[??].sub.2]-soft closed.

In order to show Theorem 37, we have the following example.

Example 38. Consider the SBT Hausdorff space in Example 34. Define finite soft point [[??].sub.1] [[??].sub.2]-soft sets [f.sub.1] = {([e.sub.1], {[u.sub.1]})} and [f.sub.2] = {([e.sub.1], {[u.sub.2]})} such that soft points are [mathematical expression not reproducible]. By taking account of the [mathematical expression not reproducible] notion that [mathematical expression not reproducible] is a soft point of f different from [mathematical expression not reproducible] have disjoint [[??].sub.1] [[??].sub.2]-soft neighborhoods [g.sub.1] and [g.sub.2] such that

[g.sub.1] = {([e.sub.1], {[u.sub.1]})},

[f.sub.2] = {([e.sub.1], {[u.sub.2]})}. (17)

Since {([e.sub.1], {[u.sub.1]})} [??] {([e.sub.1], {[u.sub.2]})} = [PHI], [[??].sub.1] [[??].sub.2]-soft closure of the set [mathematical expression not reproducible] is itself, so that it is [[??].sub.1] [[??].sub.2]-soft closed.

Theorem 39. If (f, [[??].sub.1], [[??].sub.2]) is a SBT Hausdorff space and [[phi].sub.[psi]] : [S.sub.E](U) [right arrow] [S.sub.K](V) between two SBT spaces (f, [[??].sub.1], [[??].sub.2]) and (g, [[??].sup.*.sub.1], [[??].sup.*.sub.1]) is a SBT homeomorphism, then (g, [[??].sup.*.sub.1], [[??].sup.*.sub.1]) is a SBT Hausdorff space.

Proof. Let [mathematical expression not reproducible]. Since [[phi].sub.[psi]] is soft surjective, there exist [mathematical expression not reproducible]. From the hypothesis, ([[??].sub.1], [[??].sub.2]) is a SBT Hausdorff space, so there exist [mathematical expression not reproducible]. For each [mathematical expression not reproducible] and h(e) [intersection] k(e) = 0. So, [mathematical expression not reproducible]. Hence, [mathematical expression not reproducible]. Since [[phi].sub.[psi]] is soft open, then [mathematical expression not reproducible] and since [[phi].sub.[psi]] is soft injective, [[phi].sub.[psi]] (h) [??][[phi].sub.[psi]](k) = [[phi].sub.[psi]] (h [??] k) = [PHI]. Thus,(g, [[??].sup.*.sub.1], [[??].sup.*.sub.1]) is a SBT Hausdorff space.

From Definition 32 and Theorem 39, we have the following.

Remark 40. The property of being SBT Hausdorff space is a SBT property.

Theorem 41. Let (f, [??].sub.1], [[??].sub.2]) be a SBT space and g [??] f. Then, collections

[mathematical expression not reproducible] (18)

are soft bitopologies on g.

Proof. Indeed, the union of the soft topologies contains [PHI] and g because [PHI] [??] g = [PHI] and f [??] g = g, where [mathematical expression not reproducible]; it is closed under finite soft intersections and arbitrary soft unions:

[mathematical expression not reproducible]. (19)

In order to show Theorem 41, we have the following example.

Example 42. Let us consider the soft subsets of f that are given in Example 9. Then, [[??].sup.1] = [??](f), [[??].sup.0] = {[PHI], f}, and [??] = {[PHI], f, [f.sub.2], [f.sub.11], [f.sub.13]} are some soft topologies on f.

By taking account of g = [f.sub.9], then [[??].sub.g] = {[PHI], [f.sub.5], [f.sub.7], [f.sub.9]}, and so (g, [[??].sub.g]) is a soft topological subspace of (f, [??]). Hence, we get that ([mathematical expression not reproducible]) is a soft bitopological space on g.

Definition 43. Let (f, T1, f2) be a SBT space and g [??] f. If collections [mathematical expression not reproducible] are two soft topologies on g, then a SBT space ([mathematical expression not reproducible]) is called a SBT subspace of (f, [[??].sub.1], [[??].sub.2]).

In order to show Definition 43, we have the following example.

Example 44. By taking account of Example 42 and considering that (f, [[??].sub.1], [[??].sub.2]) is a SBT Hausdorff space ordered by inclusion, we have that (g, [[??].sub.1], [[??].sub.2]) is called a SBT Hausdorff space of (f, [[??].sub.1], [[??].sub.2]).

Theorem 45. Every SBT open set in (f, [[??].sub.1], [[??].sub.2]) is SBT open in SBTsubspace of (f, [[??].sub.1], [[??].sub.2]).

Proof. It is clearly seen from Definition 43. ?

Theorem 46. Let (f, [[??].sub.1], [[??].sub.2]) be a SBT Hausdorff space and g [??] f. Then, ([mathematical expression not reproducible]) is a SBT Hausdorff space.

Proof. Let[mathematical expression not reproducible]. From the hypothesis, [mathematical expression not reproducible]. Since (f, [[??].sub.1], [[??].sub.2]) is a SBT Hausdorff space, there exist [mathematical expression not reproducible]. So [mathematical expression not reproducible].

[mathematical expression not reproducible]. (20)

Thus, ([mathematical expression not reproducible]) is SBT Hausdorff space.

From Definition 43 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 of [??].sub.1] [[??].sub.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.

http://dx.doi.org/10.1155/2016/2196743

Competing Interests

The author declares that there are no competing interests.

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Guzide Senel

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

Correspondence should be addressed to Guzide Senel; g.senel@amasya.edu.tr

Received 1 April 2016; Accepted 8 August 2016