Soft Hausdorff spaces and their some properties

Annals of Fuzzy Mathematics and Informatics Volume 8, No. 5, (November 2014), pp. 769–783 ISSN: 2093–9310 (print version) ISSN: 2287–6235 (electronic version) http://www.afmi.or.kr

@FMI c Kyung Moon Sa Co. ⃝ http://www.kyungmoon.com

Soft Hausdorff spaces and their some properties ˙ ¨ Izzett I˙ n DemI˙ r, Oya Bedre Ozbakir Received 14 January 2014; Revised 21 April 2014; Accepted 17 May 2014 Molodtsov (1999) [24] defined the notion of a soft set for coping with uncertainties. Shabir and Naz (2011) [30] introduced the topological structures of soft sets and investigated many properties. In the present paper, we contribute to the progress of the soft topological structures. We introduce soft Hausdorff spaces using the definition of soft point in [10, 19] and investigate some of their properties. We give some new concepts such as cluster soft point, soft net and support them with examples. Then, by using these concepts, we obtain some properties with respect to soft Hausdorff spaces which are important for further research on soft topology.

Abstract.

2010 AMS Classification: 06D72, 54A40 Keywords: Soft set, Soft topology, Cluster soft point, Soft net, Soft filter, Soft Hausdorff space. ¨ Corresponding Author: Oya Bedre Ozbakir ([email protected])

1. Introduction

T

here are diverse uncertainties for most of complex problems in economics, engineering, environmental science and social science. Several set theories can be regarded as mathematical tools for dealing with these uncertainties, for example the theory of fuzzy sets [32], the theory of intuitionistic fuzzy sets [3, 4], the theory of vague sets [12], the theory of interval mathematics [4, 14] and the theory of rough sets [26]. However, these theories have their inherent difficulties because of the inadequacy of the parameterization tool of the theories as cited by Molodtsov [24]. In 1999, Molodtsov [24] initiated the concept of soft set theory as a new approach for coping with uncertainties and also presented the basic results of the new theory. This new theory does not require the specification of a parameter. We can utilize any parametrization with the aid of words, sentences, real numbers and so on. This implies that the problem of setting the membership function does not arise. Hence,

˙ Izzettin Demir et al./Ann. Fuzzy Math. Inform. 8 (2014), No. 5, 769–783

soft set theory has compelling applications in several diverse fields, most of these applications was shown by Molodtsov [24]. Maji et al. [22] gave the first practical application of soft sets in decision making problems. Chen et al. [8] presented a novel concept of parameterization reduction in soft sets. Kong et al. [18] introduced the notion of normal parameter reduction and presented a algorithm for normal parameter reduction. Then, Ma et al. [20] proposed a simpler and more easily comprehensible algorithm. Pei and Miao [27] showed that soft sets are a class of special information systems. Maji et al. [21] studied on soft set theory in detail. Ali et al. [2] presented some new algebraic operations on soft sets. Aktas and Cagman [1] introduced the soft group and also compared soft sets to fuzzy set and rough set. Jun [15] defined and studied the concept of soft BCK/BCI-algebras. Also, Feng et al. [11] worked on soft semirings, soft ideals and idealistic soft semirings. Babitha and Sunil [6] introduced the soft set relation and discussed many related concepts such as equivalent soft set relation, partition and composition. Kharal and Ahmad [17] defined the notion of a mapping on soft classes and worked some properties of images and inverse images of soft sets. Das and Samanta [9] introduced the notions of soft real sets and soft real numbers and studied their properties. Shabir and Naz [30] initiated the study of soft topological spaces. Also, they defined fundamental notions such as soft open sets, soft closed sets, soft interior, soft closure and soft separation axioms and established their several properties. Nazmul and Samanta [25] defined new concepts such as soft element, soft interior operator and soft closure operator and worked their properties. Aygunoglu and Aygun [5] introduced soft product topology, soft compactness and generalized Tychonoff theorem to the soft topological spaces. Zorlutuna et al. [33] also studied on soft topological spaces and obtained the relation between soft topology and fuzzy topology. Das and Samanta [10] introduced soft metric spaces and investigated some their fundamental properties. Also, they redefined the notion of soft point. Studies on the soft topological spaces have been accelerated [13, 28, 29, 31]. In this work, we first remind the fundamental concepts of soft set theory. Also, we recall notion of soft point which is defined in [10, 19] and some of its basic properties. We then recall some concepts of soft topological spaces and introduce some new notions such as cluster soft point and soft net. Moreover, we introduce convergence of soft net and soft filter. Finally, we define soft Hausdorff spaces by using notion of soft point in [10, 19]. We study some basic properties of soft Hausdorff spaces. Also, we show that a soft net and a soft filter converge to at most one soft point in soft Hausdorff space. These study not only can form the theoretical basis for further applications of soft topology but also lead to the advance of information systems. 2. Preliminaries In this section, we recall some basic notions regarding soft sets, most of which exist in [7, 17], and we present the concept of soft point introduced in [10, 19]. Throughout this work, let X be an initial universe, P (X) be the power set of X, E be the set of all parameters for X and A ⊆ E. Aygunoglu and Aygun [5] modified the definition of soft set defined in [7, 23, 24] as follows. 770


Definition 2.1. A soft set FA on the universe X with the set E of parameters is defined by the set of ordered pairs FA = {(e, FA (e)) : e ∈ E, FA (e) ∈ P (X)} where FA : E → P (X) such that FA (e) ̸= ∅ if e ∈ A ⊆ E and FA (e) = ∅ if e ∈ / A. Throughout this paper, the family of all soft sets over X is denoted by S(X, E). [5] Definition 2.2 ([7]). The soft set F∅ ∈ S(X, E) is called null soft set if F∅ (e) = ∅ for every e ∈ E. The null soft set is denoted by Φ. [21] Definition 2.3 ([7]). Let FA ∈ S(X, E). If FA (e) = X for all e ∈ A, then FA is called A-absolute soft set. If A = E, then the A-absolute soft set is called absolute soft set. e and E, e respectively. The A-absolute soft set and absolute soft set are denoted by A [21] Definition 2.4 ([7]). Let FA , GB ∈ S(X, E). FA is a soft subset of GB if FA (e) ⊆ GB (e) for each e ∈ E. It is denoted by FA ⊑ GB . Definition 2.5 ([7]). Let FA , GB ∈ S(X, E). FA and GB are soft equal if FA ⊑ GB and GB ⊑ FA . It is denoted by FA = GB . Definition 2.6 ([7]). Let FA ∈ S(X, E). The complement of FA is denoted by FAc , where FAc : E → P (X) is a mapping defined by FAc (e) = X − FA (e) for all e ∈ E. Definition 2.7 ([7]). Let FA , GB ∈ S(X, E). The union of FA and GB is a soft set HC , which is defined by HC (e) = FA (e) ∪ GB (e) for all e ∈ E. HC is denoted by FA ⊔ GB . Definition 2.8 ([7]). Let FA , GB ∈ S(X, E). Then, the intersection of FA and GB is a soft set HC , which is defined by HC (e) = FA (e) ∩ GB (e) for all e ∈ E. HC is denoted by FA ⊓ GB . Definition 2.9 ([33]). Let J be an arbitrary index set and let {(FA )i }i∈J be a family of soft sets over X. Then, ∪ (i) The union of these soft sets is the soft set H⊔C defined by HC (e) = i∈J (FA )i (e) for every e ∈ E and this soft set is denoted by i∈J (FA )i . ∩ (ii) The intersection of these soft sets is the soft set HC defined by HC (e) = i∈J (FA )i (e) for every e ∈ E and this soft set is denoted by ⊓i∈J (FA )i . Theorem 2.10 ([5, 7, 33]). Let J be an index set and FA , GB , (FA )i , (GB )i ∈ S(X, E), for( all ) ⊔the following statements are satisfied. ⊔ i ∈ J. Then, (1) FA ⊓ ( i∈J (GB )i ) = i∈J (FA ⊓ (GB )i ). (2) FA ⊔ ⊓i∈J (GB )i = ⊓i∈J (FA ⊔ (GB )i ). c c (3) (F ( A ) = FA .)c ⊔ (⊔ )c (4) ⊓i∈J (FA )i = i∈J (FA )ci , = ⊓i∈J (FA )ci . i∈J (FA )i c c (5) If FA ⊑ GB , then GB ⊑ FA . e FA ⊓ F c = Φ. (6) FA ⊔ FAc = E, A 771


Definition 2.11 ([10, 19]). A soft set PA over X is said to be a soft point, which is called soft element in [25] , if there is e ∈ A such that PA (e) = {x} for some x ∈ X and PA (e′ ) = ∅ for all e′ ∈ E\{e}. This soft point is denoted by Pex e FA , if The soft point Pex is said to belongs to the soft set FA , denoted by Pex ∈ x ∈ FA (e). [10, 25] From now on, the family of all soft points over X will be denoted by SP (X, E). Definition 2.12 ([10]). Two soft points Pex11 , Pex22 are said to be equal if e1 = e2 and x1 = x2 . Thus, Pex11 ̸= Pex22 ⇔ x1 ̸= x2 or e1 ̸= e2 . Proposition 2.13 ([10, 25]). Let FA , GB ∈ S(X, E) and let Pex ∈ SP (X, E). Then we have: e e FA iff Pex ∈ (1) Pex ∈ / FAc . xe e FA or Pex ∈ e GB . (2) Pe ∈ FA ⊔ GB iff Pex ∈ xe xe xe (3) Pe ∈ FA ⊓ GB iff Pe ∈ FA and Pe ∈ GB . e FA imply Pex ∈ e GB . (4) FA ⊑ GB iff Pex ∈ Definition 2.14 ([17]). Let S(X, E) and S(Y, K) be the families of all soft sets over X and Y , respectively. Let φ : X → Y and ψ : E → K be two mappings. Then, the mapping φψ is called a soft mapping from X to Y , denoted by φψ : S(X, E) → S(Y, K), for which: (1) Let FA ∈ S(X, E). Then φψ (FA ) is the soft set over Y defined as follows: { ∪ −1 (k) ∩ A ̸= ∅; e∈ψ −1 (k)∩A φ(FA (e)), if ψ φψ (FA )(k) = ∅, otherwise. for all k ∈ K. φψ (FA ) is called a soft image of a soft set FA . (2) Let GB ∈ S(Y, K). Then φ−1 ψ (GB ) is the soft set over X defined as follows: { −1 φ (GB (ψ(e))), if ψ(e) ∈ B; φ−1 (G )(e) = B ψ ∅, otherwise. for all e ∈ E. φ−1 ψ (GB ) is called a soft inverse image of a soft set GB . The soft mapping φψ is called injective, if φ and ψ are injective. The soft mapping φψ is called surjective, if φ and ψ are surjective. [5, 33] Theorem 2.15 ([17]). Let (FA )i := (Fi )Ai ∈ S(X, E) and (GB )i := (Gi )Bi ∈ S(Y, K) for all i ∈ J where J is an index set. Then, for a soft mapping φψ : S(X, E) → S(Y, K), the following conditions are satisfied. (1) If (FA )1 ⊑ (FA )2 , then φψ ((FA )1 ) ⊑ φψ ((FA )2 ). −1 (2) If (GB )1 ⊑ (GB )2 , then φ−1 ψ ((GB )1 ) ⊑ φψ( ((GB )2 ). ) (⊔ ) ⊔ (3) φψ (F ) = i∈J φψ ((FA )i ), φψ ⊓i∈J (FA )i ⊑ ⊓i∈J φψ ((FA )i ). ( ⊔i∈J A i ) ⊔ ( ) −1 −1 −1 (4) φ−1 i∈J (GB )i = i∈J φψ ((GB )i ), φψ ⊓i∈J (GB )i = ⊓i∈J φψ ((GB )i ). ψ e = E, e φ−1 (Φ) = Φ. (5) φ−1 (K) ψ

ψ

(6) φψ (Φ) = Φ. 772


Theorem 2.16 ([5, 33]). Let FA , (FA )i := (Fi )Ai ∈ S(X, E) for all i ∈ J where J is an index set and let GB ∈ S(Y, K). Then, for a soft mapping φψ : S(X, E) → S(Y, K), the following conditions are satisfied. (1) (2) (3) (4)

FA ⊑ φ−1 ψ (φψ (FA )), the equality holds if φψ is injective. φψ (φ−1 (GB )) ⊑ GB , the equality holds if φψ is surjective. ) ( ψ φψ ⊓i∈J (FA )i = ⊓i∈J φψ ((FA )i ) if φψ is injective. e =K e if φψ is surjective. φψ (E)

Definition 2.17 ([6]). Let FA ∈ S(X, E) and GB ∈ S(Y, K). The cartesian product FA × GB is defined by HA×B , where HA×B : E × K → P (X × Y ) and HA×B (e, k) = FA (e) × GB (k) for all (e, k) ∈ E × K. Definition 2.18 ([5]). Let FA ∈ S(X, E), GB ∈ S(Y, K) and let p1 : X × Y → X, q1 : E ×K → E and p2 : X ×Y → Y, q2 : E ×K → K be the projection mappings in classical meaning. The soft mappings (pq )1 := (p1 )q1 and (pq )2 := (p2 )q2 are called soft projection mappings from X × Y to X and X × Y to Y , respectively, where (pq )1 (FA × GB ) = FA and (pq )2 (FA × GB ) = GB . 3. Soft topological spaces, soft nets and convergence In this section, we recall some fundamental properties of soft topological spaces and give new definitions such as cluster soft point and soft net. In the next section, these properties and definitions will be used. Definition 3.1 ([30]). Let τ be a collection of soft sets over X, then τ is said to be a soft topology on X if e belong to τ . (st1 ) Φ, E (st2 ) the union of any number of soft sets in τ belongs to τ . (st3 ) the intersection of any two soft sets in τ belongs to τ . The pair (X, τ ) is called a soft topological space. The members of τ are called soft open sets in X. A soft set FA over X is called a soft closed in X if FAc ∈ τ . Example 3.2. Let X be a non-empty set and E be a non-empty set of parameters. Let Pex ∈ SP (X, E) be a fixed soft point. Then, ∪ e e FE , then τ = {FE : Pex ∈ / FE or if Pex ∈ X\FE (e) is f inite} ∪ {Φ} e∈E

is a soft topology on X and hence (X, τ ) is a soft topological space over X. Definition 3.3 ([5, 25]). Let (X, τ ) be a soft topological space. A subcollection B of τ is called a base for τ if every member of τ can be expressed as the union of some members of B. Proposition 3.4 ([25]). A family B of soft sets over X forms a base of a soft topology over X iff the following conditions are satisfied. (i) Φ ∈ B; e is union of the members of B; (ii) E (iii) If FA , GB ∈ B then FA ⊓ GB is union of some members of B, i.e., for e FA ⊓GB there exists a HC ∈ B such that Pex ∈ e HC ⊑ FA ⊓GB . FA , GB ∈ B and Pex ∈ 773


⊔ Remark 3.5. Let D = {(FA )i : i ∈ J}. Then, i∈∅ (FA )i = Φ. Assume that ⊔ i∈∅ (FA )i ̸= Φ. Then, for e ∈ Ai0 there exists i0 ∈ ∅ such that (FA ⊔)i0 (e) ̸= ∅, and we have a contradiction. Moreover, let C = {(FA )ci : i ∈ J}. Then, i∈∅ (FA )ci = Φ. e If we pass to the complement, then we have ⊓i∈∅ (FA )i = E. From now on, we will omit the condition (i) in Proposition 3.4. Corollary 3.6. A family of soft sets B satisfying {the } (iii) of ⊔ conditions (ii) and Proposition 3.4 generate a unique soft topology τ = F : ψ ⊆ B over X. A FA ∈ψ Example 3.7. Let R be the real numbers, E = {e} and (Fλ )E = { (e, {λ} ) }, where λ ∈ R. Consider the family B = {(Fλ )E { : λ⊔∈ R}; since B satisfy } (ii) and (iii), it is base and generate the soft topology τ = (F ) : J ⊆ R over R. λ E λ∈J Definition 3.8 ([5]). Let (X, τ1 ) and (Y, τ2 ) be two soft topological spaces. A soft mapping φψ : (X, τ1 ) → (Y, τ2 ) is called soft continuous if φ−1 ψ (GB ) ∈ τ1 for every GB ∈ τ2 . A soft mapping φψ : (X, τ1 ) → (Y, τ2 ) is called soft open if φψ (FA ) ∈ τ2 for every FA ∈ τ1 . Definition 3.9 ([31]). Let (X, τ1 ) and (Y, τ2 ) be two soft topological spaces. A soft mapping φψ : (X, τ1 ) → (Y, τ2 ) is called homeomorphism if φψ is bijective, soft continuous and soft open. Definition 3.10 ([5]). Let (X, τ ) be a soft topological space. A subcollection S of τ is called a subbase for τ if the family of all finite intersections of members of S forms a base for τ . e ∈ S. Then Theorem 3.11 ([5]). Let S be a family of soft sets over X such that Φ, ⊔ E ( S is a subbase for the soft topology τ , whose members are of the form i∈J ⊓λ∈Λi ) (FA )i,λ where J is arbitrary index set and for each i ∈ J, Λi is a finite index set, (FA )i,λ ∈ S for i ∈ J and λ ∈ Λi . Definition 3.12 ([5]). Let {(φψ )i : S(X, E) → (Yi , τi )}i∈J be a family of soft mappings and {(Yi , τi )}i∈J be a family of soft topological spaces. Then, the soft topology τ generated from the subbase S = {(φψ )−1 i (FA ) : FA ∈ τi , i ∈ J} is called the soft topology ( or initial soft topology ) induced by the family of soft mappings {(φψ )i }i∈J . Theorem 3.13 ([5]). The initial soft topology τ on X induced by the family {(φψ )i : S(X, E) → (Yi , τi )}i∈J is the coarsest soft topology making (φψ )i : (X, τ ) → (Yi , τi ) soft continuous, for all i ∈ J. Definition 3.14 ([5]). Let {(X(i , τi )} ∏i∈J be a) family of soft topological spaces. Then, the initial soft topology on X = i∈J Xi generated by the family {(pq )i }i∈J is called product soft topology on X (Here, (pq )i is the soft projection mapping from X to Xi , i ∈ J). ∏ The product soft topology is denoted by i∈J τi . Definition 3.15 ([33]). Let (X, τ ) be a soft topological space and FA ∈ S(X, E). The soft interior of FA is the soft set 774


(FA )o =

⊔

{GB : GB is a sof t open set and GB ⊑ FA }.

By property (st2 ) for soft open sets, (FA )o is soft open. It is the largest soft open set contained in FA . Definition 3.16 ([30]). Let (X, τ ) be a soft topological space and FA ∈ S(X, E). The soft closure of FA is the soft set (FA ) = ⊓{GB : GB is a sof t closed set and FA ⊑ GB }. Clearly (FA ) is the smallest soft closed set over X which contains FA . Definition 3.17 ([25]). A soft set FA in a soft topological space (X, τ ) is called a soft neighborhood of the soft point Pex if there exists a soft open set GB such that e G B ⊑ FA . Pex ∈ The soft neighborhood system of a soft point Pex , denoted by Nτ (Pex ), is the family of all its soft neighborhoods. Definition 3.18 ([25, 33]). A soft set FA in a soft topological space (X, τ ) is called a soft neighborhood of the soft set GB if there exists a soft open set HC such that GB ⊑ HC ⊑ FA . Definition 3.19 ([25]). Let (X, τ ) be a soft topological space and {Nτ (Pex ) : e be the system of soft neighborhoods. Then, e E} Pex ∈ (sn1 ) Nτ (Pex ) ̸= ∅, for all Pex ∈ SP (X, E), e FA , (sn2 ) If FA ∈ Nτ (Pex ), then Pex ∈ x (sn3 ) If FA ∈ Nτ (Pe ) and FA ⊑ GB , then GB ∈ Nτ (Pex ), (sn4 ) If FA , GB ∈ Nτ (Pex ), then FA ⊓ GB ∈ Nτ (Pex ), (sn5 ) If FA ∈ Nτ (Pex ), then there is a GB ∈ Nτ (Pex ) such that GB ⊑ FA and e GB . GB ∈ Nτ (Pαy ), for each Pαy ∈ e (FA ) Theorem 3.20 ([19]). Let (X, τ ) be a soft topological space. A soft point Pex ∈ x iff each soft neighborhood of Pe intersects FA . Definition 3.21 ([28]). Let (X, τ ) be a soft topological space and let E(Pex ) be a family of soft neighborhoods of a soft point Pex . If, for each soft neighborhood FA e GB ⊑ FA , then we say that E(Pex ) of Pex , there exists a GB ∈ E(Pex ) such that Pex ∈ x is a soft neighborhood base at Pe . Definition 3.22 ([30]). Let (X, τ ) be a soft topological space and S be a non-empty fS ⊓ FA : FA ∈ τ } is called the soft relative topology on subset of X. Then, τS = {E S and (S, τS ) is called a soft subspace of (X, τ ). fS is the soft set over X defined by E fS (e) = S for all e ∈ E. Here, E Proposition 3.23 ([30]). Let (S, τS ) be a soft subspace of (X, τ ) and FA ∈ S(X, E). fS ⊓ GB , for some GB ∈ τ . Then, FA is soft open in S if and only if FA = E Definition 3.24. Let (X, τ ) be a soft topological space, {Pexnn : n ∈ N} be a sequence of soft points in (X, τ ) and Pex ∈ SP (X, E). The sequence {Pexnn : n ∈ N} is said to converge to Pex , and we write Pexnn → Pex , if for every FA ∈ Nτ (Pex ), there exists an e FA for all n ≥ n0 . n0 ∈ N such that Pexnn ∈ 775


Example 3.25. Let (R, τ ) be a soft topological space which is defined in Example 3.7 and Pexn → Pex in (X, τ ). Then, for every FA ∈ Nτ (Pex ), there exists an n0 ∈ N e FA for all n ≥ n0 . Since Pex ∈ τ , there exists an n′ ∈ N such that such that Pexn ∈ xn e x Pe ∈ Pe for all n ≥ n′ . Hence, xn = x, for all n ≥ n′ . Definition 3.26. Let (X, τ ) be a soft topological space, {Pexnn : n ∈ N} be a sequence of soft points in (X, τ ) and Pex ∈ SP (X, E). Pex is called a cluster soft point of the sequence {Pexnn : n ∈ N} if for every FA ∈ Nτ (Pex ) and for every n0 ∈ N, there is e FA . some n ≥ n0 such that Pexnn ∈ Remark 3.27. If a sequence {Pexnn : n ∈ N} in a soft topological space (X, τ ) converges to Pex ∈ SP (X, E), then this soft point is a cluster soft point of this sequence. But the converse is not always true. For example, consider the soft (−1)n : n ∈ N} topological space (R, τ ) as defined in Example 3.7. The sequence {Pe of soft points in (R, τ ) has two cluster soft points, Pe1 and Pe−1 ; but does not converge. Definition 3.28 ([16]). A directed set is a pair (D, ≳) where D is a non-empty set and ≳ is a binary relation on D satisfying: (d1 ) For all m ∈ D, m ≳ m, (d2 ) For all m, n, p ∈ D, m ≳ n and n ≳ p imply m ≳ p, (d3 ) For all m, n ∈ D, there is some p ∈ D such that p ≳ m, p ≳ n. We also say that the relation ≳ directs the set D. Example 3.29. Let Nτ (Pex ) be a soft neighborhood system of a soft point Pex in a soft topological space (X, τ ). Then, the set Nτ (Pex ) with the relation ⊑∗ ( that is, FA ⊑∗ GB if and only if GB ⊑ FA ) forms a directed set. Definition 3.30. Let X be a set and (D, ≳) be a directed set. The function T : D → SP (X, E) is called a soft net in X. In other words, a soft net is a pair (T, ≳) such that T : D → SP (X, E) is a function and ≳ directs the domain of T . For n ∈ D, T (n) is denoted by Tn and hence a soft net is denoted by {Tn : n ∈ D}. ( ) Example 3.31. Since Nτ (Pex ), ⊑∗ is a directed set, the function T : Nτ (Pex ) → SP (X, E) is a soft net, denoted by {TFA : FA ∈ Nτ (Pex )}. Definition 3.32. Let (X, τ ) be a soft topological space, {Tn : n ∈ D} be a soft net in X and FA ∈ S(X, E). e FA , for all n ∈ D. (i) The soft net {Tn : n ∈ D} is called in FA if Tn ∈ (ii) The soft net {Tn : n ∈ D} is called eventually in FA if there exists some e FA , for all n ≳ m. m ∈ D such that Tn ∈ Definition 3.33. A soft net {Tn : n ∈ D} in a soft topological space (X, τ ) is said to converge to Pex ∈ SP (X, E), and we write Tn → Pex , if it is eventually in every soft neighborhood of Pex . Example 3.34. Let X = {a, b}, E = {e1 , e2 } and { e FE = {(e1 , X), (e2 , {b})}, GE = {(e1 , {a}), (e2 , X)}, τ = Φ, E, } FE ⊓ GE = {(e1 , {a}), (e2 , {b})} . The set Nτ (Pea1 ) with the relation ⊑∗ form a directed set. Then, {TFA : FA ∈ Nτ (Pea1 )} is a soft net in X, where TFE = Pea2 , TGE = Peb2 , TFE ⊓GE = Pea1 , TEe = Peb1 , and converges to Pea1 . 776


Definition 3.35 ([29]). A soft filter on X is a non-empty subset F ⊆ S(X, E) such that: (sf1 ) Φ ∈ / F, (sf2 ) If FA , GB ∈ F , then FA ⊓ GB ∈ F , (sf3 ) If FA ∈ F and FA ⊑ GB , then GB ∈ F . The definition implies that the intersection of a finite number of members of a filter is non-empty and the union of any number of members of a filter belongs to e ∈ F. filter. Also, from (sf3 ) it follows that E Example 3.36. Let (X, τ ) be a soft topological space. Then, the soft neighborhood system Nτ (Pex ) of a soft point Pex is a soft filter on X. It is called the soft neighborhood filter at Pex . Definition 3.37 ([29]). Let F1 and F2 be two soft filters on X. Then, F2 is finer than F1 (or F1 is coarser than F2 ) if F1 ⊂ F2 . Definition 3.38. A soft filter F on a soft topological space (X, τ ) is said to converge to Pex ∈ SP (X, E), and we write F → Pex , if Nτ (Pex ) ⊂ F. Definition 3.38 implies that F converges to Pex if F is finer than the soft neighborhood filter at Pex . 4. Soft Hausdorff spaces In this section, we introduce soft Hausdorff space and establish its some properties. Also, we show that a soft net and a soft filter converge to at most one soft point in soft Hausdorff space. Definition 4.1. A soft topological space (X, τ ) is called soft Hausdorff space or soft T2 -space if for any two distinct soft points Pex11 , Pex22 ∈ SP (X, E) there exist soft e GB and FA ⊓ GB = Φ. e FA , Pex22 ∈ open sets FA and GB such that Pex11 ∈ Example 4.2. Let (X, τ ) be a soft topological space which is defined in Example 3.2. Then, (X, τ ) is a soft Hausdorff space. Definition 4.3 ([28]). A soft topological space (X, τ ) is soft first-countable if there exists a countable soft neighborhood base at every soft point over X. Lemma 4.4 ([28]). (X, τ ) is a soft first-countable space if and only if every soft point over X has a countable soft open neighborhood base {(FA )n }n∈N such that (FA )n+1 ⊑ (FA )n for each n ∈ N. Theorem 4.5. Let (X, τ ) be a soft first-countable space. (X, τ ) is a soft Hausdorff space if and only if every sequence {Pexnn : n ∈ N} of soft points in (X, τ ) converges to at most one soft point. Proof. Let (X, τ ) be a soft Hausdorff space and let us assume that a sequence {Pexnn : n ∈ N} of soft points in (X, τ ) converges to two distinct soft points Pex and Pαy . By the soft Hausdorff property, there exist soft open sets FA and GB such e FA , Pαy ∈ e GB and FA ⊓GB = Φ. Since {Pexnn : n ∈ N} converges to Pex and that Pex ∈ y e FA and Pexnn ∈ e GB for all n ≥ p0 , Pα there exists an index p0 ∈ N such that Pexnn ∈ a contradiction. This mean that every sequence of soft points in (X, τ ) converges to 777


at most one soft point. Conversely, let (X, τ ) be a soft first-countable space and assume that every sequence of soft points in (X, τ ) converges to at most one soft point, but suppose that (X, τ ) is not a soft Hausdorff space. Then, there exist two distinct soft points Pex , Pαy ∈ SP (X, E) such that every soft neighborhood of Pex intersects every soft neighborhood of Pαy . By Lemma 4.4, Pex and Pαy have countable soft open neighborhood bases {(FA )n }n∈N and {(GB )n }n∈N , respectively such that (FA )n+1 ⊑ (FA )n and (GB )n+1 ⊑ (GB )n for each n ∈ N. Because (FA )n ⊓ (GB )n ̸= Φ, for every e (FA )n ⊓ (GB )n . Then, {Pexnn : n ∈ N} is a n ∈ N we may choose a soft point Pexnn ∈ sequence of soft points in (X, τ ) and converges to both Pex and Pαy , which contradicts our assumption. □ If (X, τ ) is a soft topological space which is not soft first-countable, then the sufficient condition of Theorem 4.5 does not hold. Example 4.6. Let R be the real numbers, E = {e} and τ = {FE : R\FE (e) is countable} ∪ {Φ}. Then, (R, τ ) is a soft topological space which is not soft first-countable. Now, let {Pexn : n ∈ N} be a sequence of soft points in (R, τ ) and let us consider this sequence converges to Pex ∈ SP (R, E). Then, this sequence does not converge to any soft x point expect for Pex since it has the form {Pex1 , Pex2 , ..., Pe n0 , Pex , Pex , ...}. Hence, every sequence of soft points in (R, τ ) converges to at most one soft point; but (R, τ ) is not soft Hausdorff space. Lemma 4.7. Let (X, τ ) be a soft topological space, where E = {e}, and let {Pexn : n ∈ N} be a sequence of soft points in (X, τ ). Then, Pex ∈ SP (X, E) is a cluster e (Fn )E , for all n ∈ N, soft point of the⊔sequence {Pexn : n ∈ N} if and only if Pex ∈ xn′ where (Fn )E = n′ ≥n Pe . Proof. The sufficiency is clear from Theorem 3.20 and the definition of cluster soft point. To prove necessity, let Pex ∈ SP (X, E) be a cluster soft point of {Pexn : n ∈ N}. Then, for every FE ∈ Nτ (Pex ) and for every n0 ∈ N, there is some n ≥ n0 such that e FE . Therefore, every soft neighborhood of Pex intersects (Fn )E for all n ∈ N. Pexn ∈ e (Fn )E for all n ∈ N. By Theorem 3.20, Pex ∈ □ Theorem 4.8. Let (X, τ ) be a soft Hausdorff space, where E = {e}, and let {Pexn : n ∈ N} be a sequence of soft points in (X, τ ). If {Pexn : n ∈ N} → Pex ∈ SP (X, E), then Pex is the unique cluster soft point of {Pexn : n ∈ N}. Proof. Firstly, we show that the soft point Pex is a cluster soft point of {Pexn : n ∈ N}. ⊔ e (Fn )E for all n ∈ N, where (Fn )E = n′ ≥n Pexn′ . By Lemma 4.7, we show that Pex ∈ Since {Pexn : n ∈ N} → Pex , for every FE ∈ Nτ (Pex ), there exists an n0 ∈ N such e FE for all n ≥ n0 . Then, FE ⊓ (Fn )E ̸= Φ for all n ≥ n0 . If n < n0 , that Pexn ∈ e (Fn )E for all n ∈ N. then (Fn0 )E ⊑ (Fn )E and so FE ⊓ (Fn )E ̸= Φ. Hence Pex ∈ x Now, we show that the soft point Pe is unique. Assume that Pey ∈ SP (X, E) (x ̸= y) is also a cluster soft point of {Pexn : n ∈ N}. Because (X, τ ) is a soft e FE , Pey ∈ e GE Hausdorff space, there exist soft open sets FE and GE such that Pex ∈ 778


and FE ⊓ GE = Φ. Since {Pexn : n ∈ N} → Pex , there exists an n0 ∈ N such that e (Fn )E e FE for all n ≥ n0 . Therefore, (Fn0 )E ⊑ FE . On the other hand, Pey ∈ Pexn ∈ e (Fn0 )E and by for all n ∈ N from Lemma 4.7. Then, for n = n0 , we have Pey ∈ Theorem 3.20, GE ⊓ (Fn0 )E ̸= Φ. This is a contradiction. □ Theorem 4.9. A soft topological space (X, τ ) is a soft Hausdorff space if and only if every soft net in (X, τ ) converges to at most one soft point. Proof. Let (X, τ ) be a soft Hausdorff space and let us assume that a soft net {Tn : n ∈ D} converges to two distinct soft points Pex11 and Pex22 . By the soft Hause FA , Pex22 ∈ e GB dorff property, there exist soft open sets FA and GB such that Pex11 ∈ x1 x2 and FA ⊓ GB = Φ. As {Tn : n ∈ D} converges to Pe1 and Pe2 there exists an index e FA and Tk ∈ e GB , a contradiction. This mean that every soft k ∈ D such that Tk ∈ net in X converges to at most one soft point. For the converse, assume that every soft net in (X, τ ) converges to at most one soft point, but suppose that (X, τ ) is not a soft Hausdorff space. This implies that there exist two distinct soft points Pex11 , Pex22 ∈ SP (X, E) such that every soft neighborhood of Pex11 intersects every soft neighborhood of Pex22 . Since Nτ (Pex11 ) and Nτ (Pex22 ) are directed sets, Nτ (Pex11 ) × Nτ (Pex22 ) with the relation ⊑∗ ( that is, (FA , GB ) ⊑∗ ((FA )1 , (GB )1 ) if and only if FA ⊒ (FA )1 and GB ⊒ (GB )1 ) form a directed set. As FA ⊓GB ̸= Φ, for every (FA , GB ) ∈ Nτ (Pex11 )×Nτ (Pex22 ) we may select e FA ⊓ GB . Then, {T(FA ,GB ) : (FA , GB ) ∈ Nτ (Pex11 ) × Nτ (Pex22 )} a soft point T(FA ,GB ) ∈ e (FA )1 ⊓ (GB )1 ⊑ FA ⊓ GB , is a soft net. Now, we observe that T((FA )1 ,(GB )1 ) ∈ x1 x2 for all ((FA )1 , (GB )1 ) ∈ Nτ (Pe1 ) × Nτ (Pe2 ) such that (FA , GB ) ⊑∗ ((FA )1 , (GB )1 ). Consequently, this soft net converges to Pex11 and Pex22 , which contradicts our assumption. □ In a soft topological space which is not soft Hausdorff space, a soft net can converge to more than one soft point. ( Example 4.10. Let R be the real numbers, E = {e} and (Fλ )E = { e, (2 − λ, 2 + ) e is a soft topological λ) }, where λ ∈ R+ . Then, τ = {(Fλ )E : λ ∈ R+ } ∪ {Φ, E} space which is not soft Hausdorff space. Since the set R\{0} with the relation ≤ 1 2− n

form a directed set, {Tn : n ∈ R\{0}} is a soft net in R, where Tn = Pe n ∈ R\{0}, and converge to more than one soft point.

for all

Theorem 4.11. A soft topological space (X, τ ) is a soft Hausdorff space if and only if every soft filter on (X, τ ) converges to at most one soft point. Proof. Let (X, τ ) be a soft Hausdorff space and F be a soft filter on (X, τ ). Suppose that F converges to two distinct soft points Pex11 and Pex22 . By the soft Hausdorff e FA , Pex22 ∈ e GB and property, there exist soft open sets FA and GB such that Pex11 ∈ x1 x2 FA ⊓ GB = Φ. Since F converges to Pe1 and Pe2 , then FA , GB ∈ F. Therefore, FA ⊓ GB = Φ ∈ F, contradicting the definition of soft filter. Conversely, assume that every soft filter on (X, τ ) converges to at most one soft point, but suppose that (X, τ ) is not a soft Hausdorff space. Then, there are two distinct soft points Pex11 , Pex22 ∈ SP (X, E) such that every pair of soft neighborhoods FA of Pex11 and GB of Pex22 intersect. Thus, F = {FA ⊓ GB : FA ∈ Nτ (Pex11 ), GB ∈ 779


Nτ (Pex22 )} is a soft filter on X. Since every soft neighborhood of Pex11 and every soft neighborhood of Pex22 belongs to F, we get F → Pex11 and F → Pex22 . This is a contradiction. □ Theorem 4.12. A soft topological space (X, τ ) is a soft Hausdorff space if and only if the intersection of all soft closed neighborhoods of a soft point equals the soft point itself. Proof. Let (X, τ ) be a soft Hausdorff space and Pex11 ∈ SP (X, E). If Pex22 ∈ SP (X, E) (Pex11 ̸= Pex22 ) then, since (X, τ ) is a soft Hausdorff space, there exist soft open sets e FA , Pex22 ∈ e GB and FA ⊓ GB = Φ. If FA ⊓ GB = Φ, FA and GB such that Pex11 ∈ e FA ⊑ GcB . Therefore, GcB is a soft closed neighborhood of Pex11 then we have Pex11 ∈ not containing Pex22 . Hence, the intersection of all soft closed neighborhoods of Pex11 does not contain any soft point except for Pex11 . Conversely, let Pex11 , Pex22 ∈ SP (X, E) be any two distinct soft points and let {(FA )i }i∈J be a family of all soft closed neighborhoods of Pex11 . Since ⊓i∈J (FA )i = Pex11 , there exists an i0 ∈ J such that x2 ∈ / (FA )i0 (e2 ). Then, there exist soft neighborhoods (FA )i0 of Pex11 and (FA )ci0 of Pex22 such that (FA )i0 ⊓ (FA )ci0 = Φ. Thus, (X, τ ) is a soft Hausdorff space. □ Proposition 4.13. Let φψ : S(X, E) → S(Y, K) be a soft mapping and Pex ∈ φ(x) SP (X, E). Then φψ (Pex ) = Pψ(e) ∈ SP (Y, K). □

Proof. It is clear from the Definition 2.14.

Proposition 4.14. Let φψ : S(X, E) → S(Y, K) be a soft mapping and Pky ∈ φ−1 (y)

y SP (Y, K). If φψ is bijective, then φ−1 ψ (Pk ) = Pψ −1 (k) ∈ SP (X, E).

Proof. Using the Definition 2.14, we can easily prove it.

□

Theorem 4.15. Let the soft mapping φψ : (X, τ1 ) → (Y, τ2 ) be injective and soft continuous. If (Y, τ2 ) is soft Hausdorff space, then (X, τ1 ) is also soft Hausdorff space. Proof. Let Pex11 , Pex22 ∈ SP (X, E) be any two distinct soft points. By Proposition φ(x ) φ(x ) 4.13, φψ (Pex11 ) = Pψ(e11) , φψ (Pex22 ) = Pψ(e22) ∈ SP (Y, K). Since φψ is injective, we φ(x )

φ(x )

have Pψ(e11) ̸= Pψ(e22) . Since (Y, τ2 ) is soft Hausdorff space, there exist soft open sets e FA , Pψ(e 2) ∈ e GB and FA ⊓ GB = Φ. Because φψ is FA and GB such that Pψ(e11) ∈ 2 −1 −1 soft continuous, the sets φψ (FA ) and φψ (GB ) are disjoint soft open sets in (X, τ1 ) containing Pex11 and Pex22 , respectively. This implies that (X, τ1 ) is a soft Hausdorff space. □ φ(x )

φ(x )

Theorem 4.16. Let the soft mapping φψ : (X, τ1 ) → (Y, τ2 ) be bijective and soft open. If (X, τ1 ) is soft Hausdorff space, then (Y, τ2 ) is also soft Hausdorff space. Proof. Let Pky11 , Pky22 ∈ SP (Y, K) be any two distinct soft points. By Proposition φ−1 (y )

φ−1 (y )

y1 y2 −1 1 2 4.14, φ−1 ψ (Pk1 ) = Pψ −1 (k1 ) , φψ (Pk2 ) = Pψ −1 (k2 ) ∈ SP (X, E). Since φψ is surjecφ−1 (y )

φ−1 (y )

tive, we have Pψ−1 (k11) ̸= Pψ−1 (k22) . Since (X, τ1 ) is soft Hausdorff space, there exist 780

˙ Izzettin Demir et al./Ann. Fuzzy Math. Inform. 8 (2014), No. 5, 769–783 −1

−1

(y2 ) φ (y ) e FA , Pψφ−1 (k e GB and FA ⊓GB = Φ. soft open sets FA and GB such that Pψ−1 (k11) ∈ ∈ 2) As φψ is soft open, the sets φψ (FA ) and φψ (GB ) are soft open sets in (Y, τ2 ) containing Pky11 and Pky22 , respectively. Also, φψ (FA ) and φψ (GB ) are disjoint because φψ is injective. Thus, (Y, τ2 ) is a soft Hausdorff space. □

Theorem 4.17. Let (X, τ ) be a soft topological space and S be a non-empty subset of X. If (X, τ ) is a soft Hausdorff space, then (S, τS ) is a soft Hausdorff space. fS be any two distinct soft points. Then, Pex1 , Pex2 ∈ e and eE eE Proof. Let Pex11 , Pex22 ∈ 1 2 x1 e x2 e so there exist soft open sets FA and GB in (X, τ ) such that Pe1 ∈ FA , Pe2 ∈ GB fS ⊓ FA and E fS ⊓ GB are soft open sets in (S, τS ) and FA ⊓ GB = Φ. Therefore, E x1 x2 fS ⊓ FA and E fS ⊓ GB are disjoint containing Pe1 and Pe2 , respectively. Also, E because FA ⊓ GB = Φ. Hence, (S, τS ) is a soft Hausdorff space. □ Lemma 4.18 ([31]). Let (X, τ1 ) and (Y, τ2 ) be two soft topological spaces. Then, X and Y are homeomorphic to a subspace of X × Y . Theorem 4.19. Let (X, τ1 ) and (Y, τ2 ) be two soft topological spaces. Then, (X, τ1 ) and (Y, τ2 ) are soft Hausdorff spaces if and only if (X × Y, τ1 × τ2 ) is soft Hausdorff space. (x ,y )

(x ,y )

Proof. Let (X, τ1 ) and (Y, τ2 ) be soft Hausdorff spaces and let P(e11,k11) , P(e22,k22) ∈ SP (X × Y, E × K) be any two distinct soft points. Then either Pex11 ̸= Pex22 or Pky11 ̸= Pky22 . Take Pex11 ̸= Pex22 . Since (X, τ1 ) is soft Hausdorff space, there exist soft e GB and FA ⊓ GB = e FA , Pex22 ∈ open sets FA and GB in (X, τ1 ) such that Pex11 ∈ e e Φ. Then, FA × K and GB × K are soft open sets in (X × Y, τ1 × τ2 ) such that (x ,y ) e and (FA × K) e ⊓ (GB × K) e = Φ. Thus, e P (x2 ,y2 ) ∈ e GB × K e FA × K, P(e11,k11) ∈ (e2 ,k2 ) (X × Y, τ1 × τ2 ) is soft Hausdorff space. For the converse, let X × Y be soft Hausdorff space. It follows directly from the Theorem 4.17 and Lemma 4.18 that X and Y are soft Hausdorff spaces. □ 5. Conclusions In the present work, we mainly introduce soft Hausdorff spaces and establish some of their properties. Also, we have shown that a soft net and a soft filter converge to at most one soft point in soft Hausdorff space. Since there are close relations between soft sets and information systems, we can utilize the results inferred from this study to improve these kinds of relations. We believe that these results will help the researchers to advance and promote the further study on soft topology to carry out a general framework for their applications in practical life. References [1] H. Aktas and N. Cagman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726–2735. [2] M.I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009) 1547–1553. [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96. [4] K. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 64 (1994) 159–174.

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˙ Izzett I˙ n DemI˙ r ([email protected]) Department of Mathematics, Faculty of Arts and Sciences, Duzce University, 81620, Duzce, Turkey 782


¨ Oya Bedre Ozbakir ([email protected]) Department of Mathematics, Faculty of Science, Ege University, 35100, Izmir, Turkey

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