On Weak and Strong Forms of Fuzzy Soft Open SetsAcademic research paper on "Mathematics"

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{"Fuzzy soft sets" / "Fuzzy soft topology" / "Fuzzy soft α -open (closed)" / "Fuzzy soft pre-open(closed)" / "Fuzzy soft neighborhood" / "Fuzzy soft point" / "Fuzzy soft regular-open (closed)"}

Abstract of research paper on Mathematics, author of scientific article — Sabir Hussain

Abstract We introduce and examined some basic properties of fuzzy soft α -open (closed) sets in fuzzy soft topological spaces. Fuzzy soft pre-open (closed) sets are also defined and discussed. We also initiate and explore fuzzy soft neighborhood at fuzzy soft point. Some properties of fuzzy soft neighborhood system, fuzzy soft basic neighborhood and fuzzy soft neighborhood (nbd) base at fuzzy soft point are studied. Moreover, we define and study fuzzy soft regular open (closed) sets. We analyze the relationship between these notions by providing examples and counter examples.

Academic research paper on topic "On Weak and Strong Forms of Fuzzy Soft Open Sets"

﻿Fuzzy Inf. Eng. (2016 ) 8: 451^63

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Fuzzy Information and Engineering

ORIGINAL ARTICLE

On Weak and Strong Forms of Fuzzy Soft Open Sets

Sabir Hussain

CrossMark

Received: 13 December 2015 / Revised: 16 June 2016/ Accepted: 30 June 2016/

Abstract We introduce and examined some basic properties of fuzzy soft or-open (closed) sets in fuzzy soft topological spaces. Fuzzy soft pre-open (closed) sets are also defined and discussed. We also initiate and explore fuzzy soft neighborhood at fuzzy soft point. Some properties of fuzzy soft neighborhood system, fuzzy soft basic neighborhood and fuzzy soft neighborhood (nbd) base at fuzzy soft point are studied. Moreover, we define and study fuzzy soft regular open (closed) sets. We analyze the relationship between these notions by providing examples and counter examples.

Keywords Fuzzy soft sets • Fuzzy soft topology • Fuzzy soft ot-open (closed) • Fuzzy soft pre-open(closed) • Fuzzy soft neighborhood • Fuzzy soft point • Fuzzy soft regular-open (closed)

© 2016 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

1. Introduction

Fuzzy topology which generalizes the basic notion of classical topology is very helpful to solve the problems in practical life. In 1965, L.A. Zadeh [36] introduced fuzzy sets and provided a natural base for dealing mathematically with fuzzy processes already found in our real world. In 1968, C.L. Chang [6] initiated the new and modern

Sabir Hussain (E3)

Department of Mathematics, College of Science, Qassim University, R O. Box 6644, Buraydah 51482, Saudi Arabia

email: sabiriub@yahoo.com sh.hussain@qu.edu. sa

Peer review under responsibility of Fuzzy Information and Engineering Branch of the Operations Research Society of China.

© 2016 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

This is an open access article under the CC BY-NC-ND license

theory of fuzzy topology which generalizes the notion of general topology. In [27, 28], D. Molodtsov established soft sets to deal with uncertainty and vague objects. In [28], D. Molodtsov et al. applied soft sets to may field like engineering, computer, medical and social sciences. Many researchers [13, 14, 22-25, 29, 31, 35] studied on the algebraic structures of soft sets theory and presented the applications of soft sets in problems including decision making, information systems, measurement of sound quality and classification of natural textures. In 2011, M. Shabir and M. Naz [32] introduced soft topological spaces. Researchers [1, 8,17,18, 37] discussed this defined notions and studied its structures. Soft semi-open(closed) sets in soft topology were defined and discussed in [7, 8, 19]. The characterizations and properties were also explored as well. In 2001, P.K. Maji et al. [26] initiated the concept of fuzzy soft sets. The new methods which involved construction of a comparison table from a fuzzy soft set in a parametric sense for decision making from an imprecise multiobserver data were developed in the literature. Using fuzzy soft sets, the topological relations in geographic information systems (GIS) were discussed in [9-11]. In 2009, B. Ahmad and A. Kharal [2] discussed fuzzy soft sets and studied the mappings on fuzzy soft sets. The work of Maji et al. was improved in [3, 4, 12]. Fuzzy soft topology as well as the structural properties of fuzzy soft topological spaces have been introduced and discussed in [15, 16, 21, 30, 33-34]. Recently in 2015, S. Hussain [20] defined and established the topological structures of fuzzy soft semi-open (closed) sets in fuzzy soft topological spaces and discussed the relationship between fuzzy soft semi-open (closed), fuzzy soft semi-interior (closure) and fuzzy soft open (closed) sets, fuzzy soft interior (closure). Moreover, S. Hussain also defined fuzzy soft semi-boundary, fuzzy soft semi-continuous and fuzzy soft semi-open mappings in fuzzy soft topological spaces and explored their characterizations and properties as well.

2. Preliminaries

First we recall some definitions and results which will use in the sequel.

Definition 2.1 [36] A fuzzy set f on X is a mapping f : X —» / = [0,1]. The value f(x) represents the degree of membership ofx e X.

Definition 2.2 [27] Let X be an initial universe and E be a set of parameters. Let P(X) denotes the power set of X and A be a non-empty subset of E. A pair (F,A) is called a soft set over X, where F is a mapping given by F : A -* P(X). In other words, a soft set over X is a parameterized family of subsets of the universe X. For e e A, F(e) may be considered as the set of e-approximate elements of the soft set (F, A). Clearly, a soft set is not a set.

Definition 2.3 [26] Let Ix denotes the set of all fuzzy sets on X and A QX. A pair (J, A) is called a fuzzy soft set over X, where f : X —» Ix is a function. That is, for each a £ A, f{a) = fa: X —»lis a fuzzy set on X.

Definition 2.4 [26] For two fuzzy soft sets (/, A) and (g, E) over a common universe X, we say that (J, A) is a fuzzy soft subset of(g, B) if

(1) ACS,

Fuzzy Inf. Eng. (2016) 8: 451-463_453

(2) for all ae A, fa < ga; implies fa is a fuzzy subset of ga.

We denote it by (J,A)<{g,B). (J, A) is said to be a fuzzy soft super set of(g,B), if (g, B) is a fuzzy soft subset of (J,A). We denote it by (f,A)>(g, B).

Definition 2.5 [26] Two fuzzy soft sets (f,A) and (g, B) over a common universe X are said to be fuzzy soft equal, if (J, A) is a fuzzy soft subset of (g, B) and (g, B) is a fuzzy soft subset of (J, A).

Definition 2.6 [26] The union of two fuzzy soft sets of (/, A) and (g, B) over the common universe X is the fuzzy soft set (h, C), where C = A U B and for all c e C,

We write (/, A)V(g, B) = (h, C).

fc, ifceA- B, gc, ifceB- A,

fc v 8c-, ifceACi B.

Definition 2.7 [26] The intersection (h, C) of two fuzzy soft sets (J, A) and (g, B) over a common universe X, denoted (/, A) A(g,B), isdefinedasC - AflB, andhc = fcAgc for all ceC.

Definition 2.8 [2] The relative complement of a fuzzy soft set (/, A) is the fuzzy soft set (JC,A), which is denoted by (J,A)C, where f° : A -» Iv is a fuzzy set-valued function. That is, for each a e A, fa is a fuzzy set over U, whose membership function is fa(x) = 1 -fa(x)for all xe.U. Here fl is the membership function of fc{a).

Definition 2.9 [16] The difference (h, C) of two fuzzy soft sets (J, A) and (g, B) over X, denoted by (f, A)\(g, A), is defined as (J, A)\(g, B) = (/, A)A(/, B f.

For convenience, we will use the notation f^ for fuzzy soft set instead of (/, A).

Definition 2.10 [33] Let r be the collection of fuzzy soft sets over X. Then r is said to be a fuzzy soft topology on X if

(1) 0a, 1 a belong to t.

(2) If (JA)i e rfor all i e I, then yi£l(fA)i e r.

(3) /a, 8b 6 t implies that fA/\gb€t.

The triplet (X, t, A) is called a fuzzy soft topological space over X. Every member of r is called fuzzy soft open set. A fuzzy soft set is called fuzzy soft closed if and only if its complement is fuzzy soft open.

Definition 2.11 [34] Let (X, r, A) be a fuzzy soft topological space over X and /a be a fuzzy soft set over X. Then

(1) fuzzy soft interior of fuzzy soft set fA over X is denoted by (Ja)° and is defined as the union of all fuzzy soft open sets contained in fA. Thus {Ja)° is the largest fuzzy soft open set contained in fA.

(2) fuzzy soft closure of fa, denoted by fx is the intersection of all fuzzy soft closed super sets of fa. Clearly, is the smallest fuzzy soft closed set over X which contains fA.

Definition 2.12 [20] Let (X,t,A) be a fuzzy soft topological space over X. A fuzzy soft set fx is called fuzzy soft semi-open, if there exists a fuzzy soft open set gA such that gA < fA ^ 8a- The class of all fuzzy soft semi-open sets in X is denoted by FSSO(X). Note that every fuzzy soft open set is fuzzy soft semi-open but the converse is not true in general.

3. Properties of Fuzzy Soft a-open Sets

In this section, we introduce and establish the topological structures of fuzzy soft a-open sets and fuzzy soft a-closed sets. We also define and discuss fuzzy soft pre-open sets and fuzzy soft pre-closed sets.

Now we define:

Definition 3.1 Let (X, t, A) be a fuzzy soft topological space over X, where X is a nonempty set and t is a family of fuzzy soft sets. Then a fuzzy soft set fA is said to be a fuzzy soft a-open iffA<((fA)°T-

Definition 3.2 Let (X, t, A) be a fuzzy soft topological space over X and fa be a fuzzy soft set over X. Then fA is said to be a fuzzy soft a-closed if ((Ja )°)<fA-

Theorem 3.1 Let (X, r, A) be a fuzzy soft topological space over X and fA be a fuzzy soft set over X. Then fA is fuzzy soft a-open <=> f^ is fuzzy soft a-closed.

Proof The proof follows directly using Definitions 3.1, 3.2 and property of the complement.

Definition 3.3 Let (X,t, A) be a fuzzy soft topological space over X. Then a fuzzy soft set fA over X is said to be a fuzzy soft pre-open ¡//a ¿(/a )"•

Definition 3.4 Let (X,t, A) be a fuzzy soft topological space over X. Then a fuzzy soft set fx over X is said to be a fuzzy soft pre-closed i/(/a)d</a-

The following theorem directly follows from Definitions 3.3 and 3.4.

Theorem 3.2 Let (X, r, A) be a fuzzy soft topological space over X and fA be a fuzzy soft set over X. Then fA is fuzzy soft pre-open <=> fcA is fuzzy soft pre-closed

Remark 3.1 It is clear from the definitions that

(1) Fuzzy soft open (closed) => fuzzy soft or-open (closed) => fuzzy soft semi-open (closed).

(2) Fuzzy soft open (closed) => fuzzy soft or-open(closed) => fuzzy soft pre-open (closed).

The following example shows that the converse of above remark is not true in general. For this, we consider Example 3.3 [20] as:

Fuzzy Inf. Eng. (2016) 8: 451-463_455

Example 3.1 LetX = {huh2,h3}, A = (e,, e2\ and r = {0, T, (fAh, (Ja)i, (Ja)3, (Ja)*\, where (Ja)i, (Ja)i< (fah, (Ja)a are fuzzy soft sets over X. Defined as follows

/i(«iX*i) = 0.5, /i(ei)(/i2) = 0.3, Mei)(h3) = 0.2, f(e2)(hx) = 0.3, Me2)(h2) = 0.5, f(e2)(h3) = 0.2, /2(61)^1) = hfiieOih) = = 0.5,

MeiXh) = 0.5, f2(e2)(h2) = 0.3, f2(e2)(h3) = 1, MeOih) = 0.5, f3(ei)(h2) - 0, Mei)(h) = 0.2, Me2№) = 0.3, Me2)(h2) = 0.3, f3(e2)(h3) = 0.2, MeOCAi) = 1, /4(61)^2) = 0.3, MeiKh) = 0.5, /4fe)№i) = 0.5, Me2)(h2) = 0.5, Me2)(h3) = 1. Then t is a fuzzy soft topology on X and hence (X, t, A) is a fuzzy soft topological space over X.

Note that the fuzzy soft closed sets are

fa = {ex={hl,h\,hl5},{e2 = hl5,hlvhl}}, ¡e1=(ht)5,hlhlg}Ae2 = hl1XM}, {ei = {hl,h201,hl5},{e2 = hla5,h205,hl}}, 1 andO. Let us take fuzzy soft set gA over X defined by

gfaX^) = 0.5, g(ei)(h2) = 0, g(ei)(h3) = 0.3, g(e2)(h j) = 0.4, g(e2)(h2) = 0.4, g(e2)(h3) = 0.2.

That is, gA = [e\ = {hQ5,hg,f^3},e2 = \hlQA,h^A,h?Q2)}. Then there exists fuzzy soft open set (Ja)3 such that (fA)3<gAi(fA)3- Hence gA is fuzzy soft semi-open set, but gA is not fuzzy soft open set. Simple calculations shows that gAi((gA)°)° = {{/ij 5, Zip 3, hg 2}, [hg3,hg5, hg 2}}. Which implies that gA is not fuzzy soft a-open. Thus it is clear that gA is fuzzy soft semi-open but not fuzzy soft open as well as fuzzy soft a-open.

Lemma 3.1 Let (Ja)i be a collection of fuzzy soft sets in fuzzy soft topological space (X, r, A) over X. Then \J¡¡=/C//t)i<V¡¡e/C/aX- Moreover, if (/a), is finite, then ViJfr)i=\/izl(fA)i and Vte/((/AW°=(Vte7(/A)i)°-

Proof The proof follows using Theorems 3.9 and 3.11 [34]. Theorem 3.3 Let (X,t, A) be a fuzzy soft topological space over X. Then

(1) Arbitrary union of fuzzy soft a-open (soft pre-open) sets is a fuzzy soft a-open (soft pre-open) set.

(2) Any intersection of fuzzy soft a-open (soft pre-open) sets is a fuzzy soft a-open (soft pre-open) set.

Proof (1) Let (/a), be an arbitrary collection of fuzzy soft a-open set in a fuzzy soft topological space over X. Then by definition, for each i, C/A)i<(((/x)i)°)°- This implies that v,£7(/a),<v,e/(((/a)<)0)0<v,e/((v,e/(/a),)0)0- This follows that \fieI(fA)i is a fuzzy soft a-open set.

Again let (Ja)í be an arbitrary collection of fuzzy soft pre-open set in a fuzzy soft topological space over X. Then by definition, for each i, (Ja)¡<((Ja)¡ This implies that V¡€7(/a)¡<V¡e7(C/^ )°<(V¡e/(/a)¡ )"• This follows that V¡e/(/a)¡ is a fuzzy soft pre-open set.

(2) The proof is similar to (1). Hence the proof.

Theorem 3.4 Let (X, r, A) be a fuzzy soft topological space over X and fA be a fuzzy soft set over X. Then fA is fuzzy soft a-open if and only if it is a fuzzy soft semi-open and fuzzy soft pre-open set.

Proof The necessity follows form Remark 3.1.

For sufficiency, suppose that fA is a fuzzy soft semi-open and fuzzy soft pre-open set. Since fA is fuzzy soft semi-open set , then Proposition 3.4 [20] follows that fA<Wf- This implies that JaSJ^=WT- Therefore (A)°<(CA)°)°. Since fA is fuzzy soft pre-open, then fA<(fA)°. Which follows that fA<((JA)°)°. This completes the proof.

Theorem 3.5 Let fA and gA be fuzzy soft sets in fuzzy soft topological space (X, r, A) over X. If fA is fuzzy soft semi-open with fA<gA<(fA)°, then gA is fuzzy soft a-open.

Proof Since fA is fuzzy soft semi-open, then Proposition 3.4 [20] implies that fA<Wr- This follows that gA<(A)°<(C^)°=(C^)0<(fe^)0- Hence the proof.

Definition 3.5 Let fA be a fuzzy soft set in fuzzy soft topological space (X, t, A) over X. Then fuzzy soft a-closure is denoted by F"cla(fA) and is defined as: Fscla(JA)= Inf{gA | gA>fA, where gA is fuzzy soft a-closed set}. Clearly, Fscla(fA) is the smallest fuzzy soft a-closed set over X containing fA.

Definition 3.6 Let fA be a fuzzy soft set in fuzzy soft topological space (X, t, A) over X. Then fuzzy soft a-interior is denoted by Fsinf(fA) and is defined as:

Fsinf(fA)= Sup {gA | gA<fA, where gA is fuzzy soft a-open set}. Clearly, FHnfifA) is the largest fuzzy soft a-open set over X contained in fA.

Theorem 3.6 Let fA and gA be a fuzzy soft sets in fuzzy soft topological space (X, r, A) overX. Then

(1) (Fscla(fA))c=Fsinf(fA)c.

(2) (FHnf(JA)r=FscP(JAy.

(3) Fsint"(0)=Fscl"(0)=0 and Fsinta(í)=Fscr'(í)=í.

(4) If fA<gA, then (Fscla(jA))<{Fscla(gA)) and {Fsinf<JA))<{Fsinf(gA)).

(5) Fscla(FscP(JA))=Fscla(JA) and FHnta{.Fsinta(JA))=Fsin1a(JA).

(6) FscP(JA)(respt. Fsinf (JA)) is fuzzy soft a-closed (respt. open) in fuzzy soft topological space (X, r, A) over X.

(7) fA is fuzzy soft a-closed (respt. open) if and only if FscP(fA)=fA( respt. FHnf(JA)=fA).

Fuzzy Inf. Eng. (2016) 8: 451-463_457

(8) Fscla(fA V gA)=FscP(fA)\/Fscla(gA) and FsinP{fAygA)>FHnf{fA)\JFsinP{gA).

(9) FHne(JAKgA)=Fsinf(JA)KFHne{gA) and Fscl"(JA KgA)<Fscla(fA)KF!cl"(gA).

Proof (1) - (7) follows directly by the definitions of fuzzy soft or-closure and fuzzy soft a-interior.

(8) Since fA<fA\/gA and gA<fA\/gA, then by (4), Fscl"(fA)<Fscl"(JA\JgA) and Fscl"(gA)<FscP(JA\/gA). Fscla(fA) and FscP{gA) are fuzzy soft a-closed sets follows that Fscla(fA)\jFscla(gA) is fuzzy soft or-closed. Also fA<Fscla(JA) and gA< Fscl"{gA) implies that fA\JgA<FscP<JA)\/FscP(gA). F*cP(JA\JgA) is the smallest fuzzy soft ot-closed set such that fA\/gA<FscP(fA\/gA). Therefore, FscP(JA\JgA)< Fscl"(fA)\JFscl"(gA). Hence

FscP(fA\/gA)=FscP(fA)\/FscP(gA). Now fA<fA\/gA and gA<fA\JgA implies that

Fsinf(fA)<Fsine(JA\JgA) and FsinP(gA)<FsinP(fA\JgA). This follows that Fsinta(fA)\/Fsinf,(gA)<Fsinta(fAygA).

(9) The proof is the same as of (8). This completes the proof.

Proposition 3.1 Let fA and gA be fuzzy soft sets in fuzzy soft topological space (X, t, A) over X. Then fA is a fuzzy soft pre-open set if and only if there exists a fuzzy soft open set gA such that fA<gA<fA.

Proof Suppose that fA is fuzzy soft pre-open set. This implies that fA<(fA Take gA=(JA Then gA is fuzzy soft open and fA<gA<fA. _

Conversely, Suppose that gA is a fuzzy soft open set with fA <gA <fA. We prove that fA is a fuzzy soft pre-open set. Our supposition follows that fA<(gAy<(fA)". This implies fA is a fuzzy soft pre-open. Hence the proof.

Proposition 3.2 Let fA and gA be fuzzy soft sets in fuzzy soft topological space (X, t, A) over X. Then fA is a fuzzy soft pre-closed set if and only if there exists a fuzzy soft closed set gA such that (fA)°<gA<fA.

Using Proposition 3.1 and 3.2, we have the following theorem.

Theorem 3.7 Let fA be a fuzzy soft set in fuzzy soft topological space (X, t, A) over X. Then fA is a fuzzy soft pre-open set if and only if (fA)c is a fuzzy soft pre-closed set.

4. Fuzzy Soft Neighborhood at Fuzzy Soft Point

In this section, we study and explore fuzzy soft neighborhood at fuzzy soft point.

Definition 4.1 A fuzzy soft set fA is said to be a fuzzy soft point in (X, r, A) denoted by e(fA), if for the element e € A, f(e)+0 and f(ec)=0, for all ec6A\{e).

Definition 4 J, The fuzzy soft point e(fA) is said to be in the fuzzy soft set gA, denoted by e(fA)egA, if for the element e 6 A, /(e) < g(e). Clearly, every fuzzy soft set gA can be expressed as the union of all fuzzy soft points which belong to gA.

Definition 4.3 The fuzzy soft point (e(/x))c is called the complement of a fuzzy soft point e(/i), if for all ec 6 A - {e}, f(e')=0 and /(e)£ 0, for any element eeA.

Example 4.1 Let X = {hi,h2,h3}, A = {ej, e2] and consider the fuzzy soft set (Ja)i sets over X is defined as follows:

/i(ei)(fci) = 0.5, Me,)(h2) = 0.3, /1(^3) = 0.2, Mei№) = 0.3, Me2)(h2) = 0.5, Me2)(h3) = 0.2,

That is (/A) 1 = {ei = {hl5,h203,hl2},e2 = №¿.3,^ 5, ^ Then £((/a)i)={ei = {hl)S,hfj3,h^2}} is a fuzzy soft point. Moreover the complement w(/a)i)c of £((/a)i) is (£((/a)l)cs{ei = h2v ft^}}.

Definition 4.4 Let e(Jjd be fuzzy soft point and gA be fuzzy soft set. Then e(jA)igA called e(fA) be in gA, if f(e)<g(e) for e e A.

Remark 4.1 Note that, if the soft point e(/t) is in the soft set gA, then it is not necessary that the complement (e(/x))c is in the s°ft set (g/Cf.

The following example verify the above remark.

Example 4.2 Let X = {hi,h2,h3}, A = {ei,e2}. Then it is to be noted that the fuzzy soft point eC/k)={ei = {/¡¿2,/ijjp/ig4}} is contained in the fuzzy soft set gA = {ei = {h'05,hg3, 6}, e2 = {hg3,ft^5, h\ 2}}. Now we can see that the complement of fuzzy soft point (e(jA))c={e1 = (/¡¿8, /ip 9, h^ 6}} is not contained in the complement of fuzzy soft set (gAf = {ei = {hl5,h2vhlA},e2 = {h]01,h2Q5,hlg}}.

Definition 4.5 Let fx be fuzzy soft set in fuzzy soft topological space (X, r, A) over X and e(jA) be fuzzy soft point. If there exists a fuzzy soft open set gA with then fA is called fuzzy soft neighborhood (nbd) of a fuzzy soft point c(Ja).

The collection of all fuzzy soft nbds of fuzzy soft point e(fA) is denoted as N(e(fA)) and is known as fuzzy soft nbd system of fuzzy soft point c(Ja).

Example 4.3 Let us consider the fuzzy soft topological space as in Example 3.1 and take fuzzy soft set gA over X defined by

g(eiXfci) = 1, g(ei)(h2) = 0.4, g(ei)(h3) = 0.6, g(e2)(hl) = 0.6, g(e2)(h2) = 0.7, g(e2)(h3) = 1.

That is, gA = {«l = {hi,ho,4,ho,6},e2 = {/¡og, hoj,hi}} and consider the fuzzy soft point e(gA)={e 1 = {ho.5,ho.2,hoA}}- Then gA is fuzzy soft nbd of fuzzy soft point e(gA), because there exists fuzzy soft open set (Ja)a such that e(gA)£(fA)4<gA-

In the following theorem, we discuss some important properties of fuzzy soft nbd system:

Theorem 4.1 Let fA, gA and /¡a be fuzzy soft sets in fuzzy soft topological space (X,t,A) over X and e(/x) be fuzzy soft point. Then fuzzy soft nbd system N(e(fa)) of fuzzy soft point e(fA) has the following properties:

(1) IfgAeN(e(fA)), then e(fA)egA.

(2) IfgA,hAme(fA)), then gAAhAeNWA)).

Fuzzy Inf. Eng. (2016) 8: 451-463_459

(3) If gAeN(e(fA)) andgA<hA, then hAeN(e(fA)).

(4) If gA&N{e(fA)), then there is an hAe.N{e(fA)) such that gAe.N{e(kA)) for each e(kA)ehA.

(5) gA is fuzzy soft open if and only if it contains a fuzzy soft nbd of each of its points.

Proof (1) is obvious, since gA is a fuzzy soft open nbd of fuzzy soft point e(fA). Therefore, gA is a fuzzy soft open set with e(fA)egA.

(2) If gA,hAeN(e(fA)), then there exist fuzzy soft open sets mA and lA such that e(JA)emA<gA and e(fA)elA<hA. Therefore e(JA)€mAKlA<gAKhA and hence gAKhAe N(e(fA)).

(3) Since gA£N(e(fA)), then there exists a fuzzy soft open set mA such that e(fA)emA<gA. Therefore, e(fA)emA<gA<hA or e(JA)emA<hA. Hence, hAeN(e(fA)).

(4) Since gAeN(e(JA)), then e(fA)ehA<gA for fuzzy soft open set hA. Since e(fA)ehA<hA, then hAeN(e(fA)). If e(kA)ehA, then by (3) hA<gA implies gAeN(e(kA)) for each e(kA)ehA.

(5) (a) Suppose gA is a fuzzy soft open in (X, r, A), then e(fA))egA<gA implies gA is a fuzzy soft nbd of each e(fA)egA.

(b) If each e(fA)egA has a fuzzy soft nbd hAetM<gA, then gA={e(fA) \ e(JA)egA}< Veu^gM^sa or gA=\J eifjugMw This gives Sa. ™ fuzzy soft open in (X, t,A). This completes the proof.

Definition 4.6 Let fA, gA and hA be fuzzy soft sets in fuzzy soft topological space (X, t, A) over X and e(JA) be fuzzy soft point. A fuzzy soft nbd base at fuzzy soft point e(fA) is a subcollection fj(e(fA)) of fuzzy soft nbd N(e(fA)) having the property that each gAS.N(e(JA)) contains some hA£rj(e(fA)). That is, N(e(JA)) must be determined by fj(e(fA)) as follows:

N(e(fA)))=[gA | hA<gA, for some hA£i](e(fA))}. Each hA£fj(e(fA)) is called a basic fuzzy soft open nbd of fuzzy soft point e(fA).

The following properties for fuzzy soft basic nbd are easily verified by referring to the corresponding properties of fuzzy soft nbd in Theorem 4.1.

Proposition 4.1 Let fA, gA and hA be fuzzy soft sets in fuzzy soft topological space (X,t,A) over X and e(fA) be fuzzy soft point. Suppose fj(e(fA)) be a fuzzy soft nbd base at e(fA). Then we have

(1) IfhAerj(e(fA)), then e(fA)ehA.

(2) If cA, dA<=S)(e(fA)), then there is some eA<=S)(e(fA)) such that eA<cAkdA.

(3) IfhAefj(e(fA)), then there is some pA<=Si(e(fA)) such that if for soft point e(qA)epA, then there is some mAeij(e(qA)) with mA<hA.

(4) gA is fuzzy soft open if and only if gA contains a fuzzy soft basic nbd of each of its points.

5. Fuzzy Soft Regular Open (closed) Sets

In this section, we define and study fuzzy soft regular open sets and fuzzy soft regular closed sets in fuzzy soft topological spaces.

Definition 5.1 Let be a fuzzy soft set in fuzzy soft topological space (X, t, A) over X. Then fx is said to be fuzzy soft regular open set over X if (Ja T=fA-

Definition 5.2 Let fA be a fuzzy soft set in fuzzy soft topological space (X, t, A) over X. Then fa is said to be fuzzy soft regular closed set over X if (JaT =fA.

Lemma 5.1 Let fA, g,i and hA be fuzzy soft sets in fuzzy soft topological space (X, r, A) overX. Then

(1) (JCAT = GaY-

(2) W = «/A)T-

Proof (1) Consider I hA>fA, where hcA is fuzzy soft open })c =\J{VA I

hA>fA, where hcA is fuzzy soft open) = Vfei I gAi=C/A)c> where gA is fuzzy soft open} =(fAf by setting gA=hfA.

(2) The proof is similar to (1).

The proof of the following theorem follows directly from Definitions 5.1, 5.2 and Lemma 5.1.

Theorem 5.1 Let fA be a fuzzy soft set in fuzzy soft topological space (X,t,A) over X. Then fA is said to be fuzzy soft regular open set over X if and only if(fA)c is said to be fuzzy soft regular closed set over X.

The following example shows that every fuzzy soft regular open (closed) set is a fuzzy soft open (closed) set. But the converse is not true in general.

Example 5.1 In Example 3.1, we note that the fuzzy soft closed sets are ^={hl5,hln,hlt),e2 = K7,hl5,hl,}}, {el={h10,h2,^5},e2 = {h105,h21,h30}},

(e, ={hlh2avhl5},e2 = K5,h25,h3}}, 1 andO. _

Since (/A)i ={ei = K5, h\n, = \h]Q1, h2y h\ 8}} and ((fAh )°={ei = {h\5, h\,

hl2], e2 - K3,h25,h30S}}, then ((/A)i )°+(fA)i- This shows that (fA)i is fuzzy soft open but not fuzzy soft regular open.

The following example shows that union of two fuzzy soft regular open sets is not a fuzzy soft regular open set.

Example 5.2 In Example 3.1, simple calculations shows that union of two fuzzy soft regular open sets is not a fuzzy soft regular open set.

Remark 5.1 In view of Theorem 5.1 and Example 5.1, it is clear that intersection of two fuzzy soft regular closed sets is not a fuzzy soft regular closed set.

Theorem 5.2 Let fA and gA be fuzzy soft sets in fuzzy soft topological space (X, r, A) over X.

Fuzzy Inf. Eng. (2016) 8: 451-463_461

(1) If fA and gA be fuzzy soft regular open sets, then /aA8a is fuzzy soft regular open.

(2) If fA and gA be fuzzy soft regular closed sets, then fA\/gA is fuzzy soft regular closed.

Proof (1) Let fA and gA be fuzzy soft regular open sets in fuzzy soft topological space (X, r, A) over X. This follows that /a and gA be fuzzy soft open. Which implies that fAf\gA is also fuzzy soft open. Thus we have fAAgA^{fA/\gA) Moreover, VaAsa) °S{/a} °=/a and {/aAsa) °<{ga) °=8a- This implies that {fA/\gA} °</a A«a-Hence {/a Asa) °=/a A«a follows that fA/\gA is fuzzy soft regular open. (2) This follows directly by (1) and Theorem 5.1.

Theorem 5.3 Let fA be fuzzy soft set in fuzzy soft topological space (X, r, A) over X.

(1) If fa is a fuzzy soft open set, then fA is a fuzzy regular closed set.

(2) If fa is a fuzzy soft closed set, then (fA)° is a fuzzy regular open set.

Proof (1) Let fA be fuzzy soft open set in fuzzy soft topological space (X, r,A) over X. Then (JaT^Ja- This follows that (<Ja)°)<Ta- Since fA in fuzzy soft open, then /a<(/a)°. This implies that fA<(fA)°. Therefore fA is a fuzzy regular closed set. (2) This follows by Theorem 5.1 and (1).

Remark 5.2 It is clear from the definitions of Sections 3 and 5 that:

(1) Fuzzy soft regular open (closed) => Fuzzy soft open (closed) => fuzzy ar-open (closed) => fuzzy soft semi-open (closed).

(2) Fuzzy soft regular open (closed) => Fuzzy soft open (closed) => fuzzy or-open (closed) => fuzzy soft pre-open (closed).

6. Conclusion

Many researchers worked on the finding of structures of soft sets theory and applied to many problems having uncertainties and not clear objects. A lot of researches have been done for the fuzzification of soft set theory now a days. The new methods which involved construction of a comparison table from a fuzzy soft set in a parametric sense for decision making from an imprecise multiobserver data were developed in the literature. In the present work, we introduced and explore the topological structures of fuzzy soft or-open (closed) sets, fuzzy soft pre-open (closed) sets, fuzzy soft neighborhood at fuzzy soft point and fuzzy soft regular open (closed) sets in fuzzy soft topological spaces. Moreover, we analyzed the relationship among them. It is observed that, Fuzzy soft Regular open (closed) => Fuzzy soft open (closed) => fuzzy or-open (closed) => fuzzy soft semi-open (closed)(=> fuzzy soft pre-open (closed)). It is shown by counter examples that converse is not true in general. In particular, we proved that a fuzzy soft set in a fuzzy soft topological space is fuzzy soft or-open if and only if it is fuzzy soft pre-open and fuzzy soft semi-open. The obtained results

make a useful contribution to both fuzzy soft sets and fuzzy soft topology. The separation properties for fuzzy soft sets seems to be of special similarity to the problem of pattern discrimination. The applications of fuzzy soft sets to these properties as well as the problems of optimizations may be further explored in the future study.

Acknowledgments

Author would like to thank referees for their valuable and helpful comments. References

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