Scholarly article on topic 'On hesitant multi-fuzzy soft topology'

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{"Hesitant multi-fuzzy soft basis" / "Hesitant multi-fuzzy soft subspace topology" / "Hesitant multi-fuzzy soft cover" / "Hesitant multi-fuzzy soft open cover" / "Hesitant multi-fuzzy soft compactness"}

Abstract of research paper on Mathematics, author of scientific article — Asit Dey, Madhumangal Pal

Abstract The aim of this paper is to introduce and study the concept of hesitant multi-fuzzy soft topological space and some of its structural properties, such as the neighbourhood of hesitant multi-fuzzy soft sets and interior hesitant multi-fuzzy soft sets, the hesitant multi-fuzzy soft basis and hesitant multi-fuzzy soft subspace topology. Additionally, we introduce and define the hesitant multi-fuzzy soft cover, hesitant multi-fuzzy soft open cover, and hesitant multi-fuzzy soft compactness, and some important results on them are presented.

Academic research paper on topic "On hesitant multi-fuzzy soft topology"

Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—7

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ELSEVIER

Pacific Science Review B: Humanities and Social Sciences

journal homepage: www.journals.elsevier.com/pacific-science-review-b-humanities-and-social-sciences/

On hesitant multi-fuzzy soft topology

Asit Dey*, Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India

ARTICLE INFO

Article history: Received 30 May 2016 Accepted 17 July 2016 Available online xxx

AMS subject classification:

Keywords:

Hesitant multi-fuzzy soft set Interior hesitant multi-fuzzy soft set Hesitant multi-fuzzy soft basis Hesitant multi-fuzzy soft subspace topology Hesitant multi-fuzzy soft cover Hesitant multi-fuzzy soft open cover Hesitant multi-fuzzy soft compactness

ABSTRACT

The aim of this paper is to introduce and study the concept of hesitant multi-fuzzy soft topological space and some of its structural properties, such as the neighbourhood of hesitant multi-fuzzy soft sets and interior hesitant multi-fuzzy soft sets, the hesitant multi-fuzzy soft basis and hesitant multi-fuzzy soft subspace topology. Additionally, we introduce and define the hesitant multi-fuzzy soft cover, hesitant multi-fuzzy soft open cover, and hesitant multi-fuzzy soft compactness, and some important results on them are presented.

Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

In real life situations, problems in economics, engineering, social sciences, medical science, etc. do not always involve crisp data. Therefore, we cannot successfully use traditional methods because of various types of uncertainties presented in these problems. To manage these uncertainties, theories were developed, such as the theory of fuzzy sets, intuitionistic fuzzy sets, rough sets, and bipolar fuzzy sets, which we can use as mathematical tools for dealing with uncertainties. However, all these theories have inherent difficulties. Because of these difficulties, Molodtsov initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties.

Maji et al. (Maji et al., 2002) gave the first practical application of soft sets in decision making problems. They also introduced the concept of the fuzzy soft set, a more generalized concept, which is a

* Corresponding author. E-mail addresses: asitvu@gmail.com (A. Dey), mmpalvu@gmail.com (M. Pal). Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University.

combination of the fuzzy set and soft set, and they also studied some of its properties.

A new type of fuzzy set (multi-fuzzy set) was introduced in a paper by Sebastian and Ramakrishnan (Sebastian and Ramakrishnan, 20Ha) using the ordered sequences of membership function. The notion of multi-fuzzy sets provides a new method to represent some problems that are difficult to explain using other extensions of fuzzy set theory, such as the colour of pixels. Sebastian and Ramakrishnan (Sebastian and Ramakrishnan, 2010, 2011b, 2011c) discussed multi-fuzzy extensions of functions and multi-fuzzy subgroups.

The notion of multi-fuzzy complex numbers and multi-fuzzy complex sets are introduced for the first time by Dey and Pal (Dey and Pal, 2014b). Using these concepts, Dey and Pal (Dey and Pal, 2014c) created multi-fuzzy complex nilpotent matrices over a distributive lattice. Dey and Pal (Dey and Pal, 2014d) introduced multi-fuzzy vector space and multi-fuzzy linear transformation over a finite dimensional multi-fuzzy set. Additionally, the concept of multi-fuzzy soft topological spaces is discussed by Dey and Pal (Dey and Pal, 2015a). Dey and Pal (Dey and Pal, 2015b) introduced generalized multi-fuzzy soft sets and presented an application in decision making. Recently, Yang et al. (Yang et al.,

http://dx.doi.org/10.1016/j.psrb.2016.07.001

2405-8831/Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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2013) proposed the concept of the multi-fuzzy soft set, which is a more general fuzzy soft set, and also presented its application in decision making.

Torra (Torra, 2010) first introduced the concept of hesitant fuzzy sets, which permits the membership to have a set of possible values and defines some of its basic operations in expressing uncertainty and vagueness.

The purpose of our paper is to introduce the concepts of the hesitant multi-fuzzy soft set and hesitant multi-fuzzy soft topology, from which we can obtain a new notion for modern analysis. Additionally, the hesitant multi-fuzzy soft set can more precisely reflect uncertainty and vagueness. To facilitate our discussion, we first review some background on soft sets, fuzzy soft sets, multi-fuzzy sets, multi-fuzzy soft sets and hesitant fuzzy sets in Section 2. In Section 3, the concept of the hesitant multi-fuzzy soft set is introduced. In Section 4, the notion of hesitant multi-fuzzy soft topology and some of its structural properties, such as neighbourhood of a hesitant multi-fuzzy soft set, interior hesitant multi-fuzzy soft sets, hesitant multi-fuzzy soft basis and hesitant multi-fuzzy soft subspace topology, are investigated. In Section 5, the concepts of the hesitant multi-fuzzy soft cover, hesitant multi-fuzzy soft open cover, and hesitant multi-fuzzy soft compactness are introduced. In Section 6, some conclusions are noted.

2. Preliminaries

For easy reference, we present the following definitions for the development of the hesitant multi-fuzzy soft set and hesitant multi-fuzzy soft topology. Throughout this paper, U refers to an initial universal set, E is a set of parameters, P(U) is the power set of U andA4E.

Definition 1. (Molodtsov, 1999) (Soft sets) A pair (FA) is called a soft set over U, where F is a mapping given by F:A/P(U).

In other words, a soft set over U is a mapping from parameters toP(U), and it is not a set but a parameterized family of subsets of U.

Now, we elaborate the definition of soft set by the following example.

Example 1. Let U = {b1,b2,b3,b4,b5} be a set of bikes under consideration. Let A = {e1,e2,e3} be a set of parameters, where e1 = expensive, e2 = beautiful and e3 = good mileage. Suppose that F(e1) = {b2,b4}, F(e2) = {b1,b4,b5}, F(e3) = {b1,b3}. The soft set (FA) describes the "attractiveness of the bikes". F(e1) means "bikes (expensive)" whose function value is the set {b2,b4}, F(e2) means "bikes (beautiful)" whose function value is the set {b1,b4,b5} and F(e3) means "bikes (good mileage)" whose function value is the set {b1,b3}.

Definition 2. (Maji et al., 2001) (Fuzzy soft sets) Let P(U) be all fuzzy subsets ofU. A pair (F, A) is called fuzzy soft set over U, where F is a mapping given by F : A/P(U).

Example 2. Let us consider Example 1.

The fuzzy soft set (F, A) can describe the "attractiveness of the bikes" under fuzzy circumstances. p(e1) = {b1 /0.3, b2/0.8, b3/ 0.4, b4/0.7, b5/0.5}, ~F(e2) = {bi /0.7,62/0.2, b3/0.4, b4/0.8, b5= 0.9}, ~F(e3) = {bi/0.6, b2/0.4, b3/0.7, b4/0.2, b5/0.1}.

Now, we introduce the concept of a multi-fuzzy set.

Definition 3. (Sebastian and Ramakrishnan, 2011a) (Multi-fuzzy

sets) Let k be a positive integer. A multi-fuzzy set AF in U is a set of ordered sequences A ={u/(m1(u), m2(u),-■-, M-k(u)) : ueU}, where mjeP(U), i = 1,2,...,k.

The function mA = (m1(u),m2(u),-■-,mk(u)) is called the multi-membership function of multi-fuzzy set A, and k is called dimension

of A. The set of all multi-fuzzy sets of dimension k in U is denoted by MkFS(U).

Note 1A multi-fuzzy set of dimension 1 is a Zadeh's fuzzy set, and a multi-fuzzy set of dimension 2 with m1(u)+m2(u) < 1 is the Atanassov's intuitionistic fuzzy set.

Note 2 If Plk=1mi(u) < 1,for all ueU, then the multi-fuzzy set of dimension k is called a normalized multi-fuzzy set. IfYl>k=1 mi(u) = l > 1 for some ueU, we redefine the multi-membership degree (mi(u),m2(u),...,mk(u)) as }(mi(u),mi(u),-■-,mk(u)), then the non-normalized multi-fuzzy set can be changed into a normalized multi-fuzzy set.

Definition 4. (Sebastian and Ramakrishnan, 2011a) Let AeMkFS(U). If A = {u/(0,0,..., 0) : ueU}, then A is called the null multi-fuzzy set of dimension k, denoted by Fk. If A = {u/(1,1,..., 1) : ueU}, then A is called the universal multi-fuzzy set of dimension k, denoted by ~k.

An example can be used to illustrate the concept of multi-fuzzy sets.

Example 3. Suppose a colour image is approximated by an m x n matrix of pixels. Let U be the set of all pixels of the colour image. For any pixel u in U, the membership values mr(u),mg(u),mb(u) being the normalized red value, green value and blue value of the pixel u, respectively. Therefore, the colour image can be approximated by the collection of pixels with the multi-membership function (mr(u),mg(u),mb(u)), and it can be represented as a multi-fuzzy set A = {u/(mr(u),mg(u),mb(u)) : ueU}. In a two-dimensional image, the colour of the pixels cannot be characterized by a membership function of an ordinary fuzzy set, but it can be characterized by a three-dimensional membership function (mr(u),mg(u),mb(u)). In fact, a multi-fuzzy set can be understood to be a more general fuzzy set using ordinary fuzzy sets as its building blocks.

Definition 5. (Sebastian and Ramakrishnan, 2011a) Let

A = {u/(m1(u), m2(u), -■-, mk(u)) : u2U} and B = {u/(v1(u), V2(u),..., Vk(u)) : ueU} be two multi-fuzzy sets of dimension k in U. We define the following relations and operations:

(1) Â8B if and only if mi(u) < n¡(u),VueU and I < i < k.

(2) A = B if and only if mi(u) = vi(u),VueU and I < i < k.

(3) AuB ={u/(Mi(u)vni(u),M2(u)vn2(u),...,mk(u)Wk(u)) : ueU}.

(4) AnB = {u/(mi(u)AVi (u), m>2 (u) AV2 (u),..., mk(u)AVk(u)) : ueU}.

(5) Ai ={u/(m1(u),m2(u),...,mk(u)) : ueU}.

The relationships between multi-fuzzy set and soft set can be further discussed as follows:

Definition 6. [?](Multi-fuzzy soft sets) A pair (F, A) is called a multi-fuzzy soft set of dimension k over U, where F is a mapping given by F: A/MkFS(U). Q7

A multi-fuzzy soft set is a mapping from parameters to MkFS(U). It is a parameterized family ofmti-fuzzy subsets ofU. ForeeA, p(e) may be considered as the set of e-approximate elements of the multi-fuzzy soft set (F, A).

We illustrate this definition by an example:

Example 4. Suppose that U = {c1,c2,c3,c4,c5} is the set of cell phones under consideration, and A = {e1,e2,e3} is the set of parameters, where e1 stands for the parameter colour which consists of red, green and blue, e2 stands for the parameter ingredient which is made from plastic, liquid crystal and metal, and e3 stands for the parameter price which can be various: high, medium or low. We define a multi-fuzzy soft set of dimension 3 as follows:

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îr(e1) = {c1/(0.4, 0.2, 0.3), c2/(0.2, 0.1, 0.6), c3/(0.1, 0.3, 0.4), c4/(0.3, 0.1, 0.3), c5/(0.7, 0.1, 0.2)}. ~(e2) = {c1 /(0.1, 0.2, 0.6), c2/(0.3, 0.2, 0.4), c3/(0.5, 0.3, 0.1), c4/(0.6, 0.1, 0.3), c5/(0.6, 0.2, 0.1)}, ~(e3) = {c1/(0.3, 0.4, 0.1), c2/(0.4, 0.1, 0.2), c3/(0.2, 0.2, 0.5), c4/(0.7, 0.1, 0.2), c5/(0.5, 0.2, 0.3)}.

Hesitant fuzzy sets were defined to permit several membership values for a single element u in the reference set U. This is formally defined as follows:

Definition 7. (Torra, 2010) (Hesitant fuzzy sets) A hesitant fuzzy set (HFS) on U is denoted by A = ((u,hA(u)):ueU) and is defined in terms of hA(u) when applied to U, and hA(u) is a set of some different values in [0,1], indicating the possible membership degrees of the elements usU to the set A.

For convenience, we call hA(u) a hesitant fuzzy element (HFE).

Example 5. Let U = {x1,x2,x3,x4} and hA(x1) = {0.1,0.3,0.7}, hA(x2) = {0.6,0.8}, hA(x3) = {0.2,0.4,0.6,0.9} and hA(x4) = {0.5,0.7,0.8} be the membership degree sets ofx1,x2,x3 and x4 respectively. Then, the hesitant fuzzy set is A = {{x1, (0.1,0.3,0.7}}, {x2, (0.6,0.8}}, <x3, (0.2,0.4,0.6,0.9}}, (x4, (0.5,0.7,0.8}}}.

Definition 8. (Dey and Pal, 2016b) (Hesitant multi-fuzzy set) Let k

be a positive integer. A hesitant multi-fuzzy set A of dimension k in U is a set

A = {{u, ~a(u)} : ueU}, where (u) = (hj(u), hA(u), ..., hA(u)) and hjj (u) is a hesitant fuzzy element fori = 1,2,...,k.

The function ~a(u) = (hA(u), hA(u), ..., hA(u)) is denoting the possible multi-membership degrees of the element ueU to the set A For convenience, we call ~a (u) a hesitant multi-fuzzy element.

Here, k is called the dimension of the hesitant multi-fuzzy set A. The set of all hesitant multi-fuzzy sets of dimension k in U is denoted by HMkFS(U).

Note 3 A hesitant multi-fuzzy set of dimension l is a Torra's hesitant fuzzy set.

We now give the concept of hesitant multi-fuzzy soft set.

Definition 9. (Dey and Pal, 2016b) (Hesitant multi-fuzzy soft set)

Let U = (x1,x2,.,xn} be an initial universe, E = (e1,e2,.,em} be the universal set of parameters and A4E. Additionally, let HMkFS(U) be the set of all hesitant multi-fuzzy sets of dimension k in U.

A pair (~, A) is called a hesitant multi-fuzzy soft set of dimension k over U, where ~ is a mapping given by ~ : A / HMkFS(U).

A hesitant multi-fuzzy soft set is a mapping from parameters to HMkFS(U), and it is a parameterized family of hesitant multi-fuzzy subsets of U. We can consider, ~(e) as the set of e-approximate element of (~, A).

Definition 10. (Dey and Pal, 2016b) LetA,B4Eand (~, A), (G, B) be two hesitant multi-fuzzy soft sets of dimension k in U. Then, (G, B) is said to be a hesitant multi-fuzzy soft subset of (~, A) if

(1) B4A, and

(2) G(e)8~(e) forallesB.

Here, we write (G, B) 8 (~, A).

Note 4 Two hesitant multi-fuzzy soft sets are equal if (G, B) 8 (~, A) and (~, A) 8 (G, B), and it is denoted by (~, A) = (G, B).

Definition 11. (Dey and Pal, 2016b) (Null hesitant multi-fuzzy soft sets) A hesitant multi-fuzzy soft set (F, A) of dimension k over U is called the null hesitant multi-fuzzy soft set if F(e) = Fk for all esA, and it is denoted by l~A .

Definition 12. (Dey and Pal, 2016b) (Absolute hesitant multi-fuzzy soft sets) A hesitant multi-fuzzy soft set (F, A) of dimension k over U is called the absolute hesitant multi-fuzzy soft set if F(e) = Ik for all es A, and it is denoted by UA .

We now give some operations of the hesitant multi-fuzzy soft sets.

Definition 13. (Dey and Pal, 2016b) Let (F, A) be a hesitant multi-fuzzy soft set of dimension k over U. Then, the complement of (~, A) is denoted by (~, A)c and is defined by (~, A)c = (F , A) where ~c : A/HMkFS(U) is a mapping given by F(e) = (~(e))c for all esA.

~ k ~ k ~ k ~ k Clearly, ((F,A)c)c = (F,A) and ($A)c = UA, (UA)c = $A.

Definition 14. (Dey and Pal, 2016b) Let (~, A) and (G, B) be two hesitant multi-fuzzy soft sets of dimension over U. Then, (~, A)AND(G, B), denoted by (~, A)a(G, B) is defined by (~,A)a(G,B) = (H,A x B), where H(a, b) = F(a)nG(b), for all (a,b)e A x B.

Definition 15. (Dey and Pal, 2016b) If (~,A) and (G, B) be two hesitant multi-fuzzy soft sets of dimension k over U, then (~, A) OR (G, B), denoted by (~,A)v(G, B) is defined by (~, A)v(G, B) = (O,A x B), where O(a, b) = ~(a)uG(b), for all (a,b)sA x B.

Theorem 1. (Dey and Pal, 2016b) Let (~, A) and (G, B) be two hesitant multi-fuzzy soft sets of dimension k over U. Then

(1) ((~,A)a(G,B))c = ((^,A))cv((G,B))c.

(2) ((~,A)v(U,B))c = ((^,A))ca((G,B))c.

Definition 16. (Dey and Pal, 2016b) The union of two hesitant multi-fuzzy soft sets (~, A) and (G, B) of dimension k over U is the hesitant multi-fuzzy soft set (H, C), where C = AuB, and CesC,

( ~(e) if esA - B, H(e) = < G(e) if eeB - A, I F(e)uG(e) if eeAnB.

In this case, we write (~, A)U(G, B) = (H, C).

Definition 17. (Dey and Pal, 2016b) The intersection of two hesitant multi-fuzzy soft sets (~, A) and (G, B) of dimension k over U with AnBsf is the multi-fuzzy soft set (H, C), where C = AnB, and CesC, H (e) = ~(e)nG(e).

We write (~, A)U(G, B) = (H, C).

Now, we discuss some trivial results related to union and intersection between hesitant multi-fuzzy soft sets.

Theorem 2. Dey and Pal, 2016b Let (~, A) and (G, B) are two hesitant multi-fuzzy soft sets of dimension k over U. Then

(1) (UA)U(F,A) = (F,A),

(2) (F,A)F(F,A) = A),

(3) (F,A)U$A = (F,A),

(4) (F,A)f$a = Fa,

(5) (F,A)UUA = UA,

(6) (F,A)fUA = (FF,A),

(7) (F,A)U(G, B) = (G, B)U(F,A),

(8) (F,A)f(G, B) = (G, B)F(F,A).

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3. Hesitant multi-fuzzy soft topology

Now we are ready to define the concept of hesitant multi-fuzzy soft topology. Let U be an initial universe, E be the set of parameters,

multi-fuzzy soft topology and discrete hesitant multi-fuzzy soft topology, respectively, as called in point set topology.

Example 6. Let (p, X) be as (p, A) in Example 6. Then, the subfamily

t = X),

{e1 = c1/({0.3, 0.4}, {0.6}, {0.2, 0.3, 0.6}), c2/({0.4, 0.6}, {0.2,0.4}, {0.5}), c3/({0.2, 0.4, 0.6}, {0.3, 0.7}, {0.8}), c4/({0.3, 0.5, 0.6}, {0.4, 0.6}, {0.2, 0.4, 0.6}), c5/({0.1, 0.4, 0.7, 0.8}, {0.5, 0.6, 0.9}, {0.2}),

e2 = c-[/({0.4, 0.8, 0.9}, {0.1, 0.4, 0.8}, {0.4, 0.5, 0.6}), c2/({0.2, 0.7}, {0.2, 0.4, 0.6}, {0.5, 0.7}), c3/({0.1, 0.3, 0.4, 0.7}, {0.3, 0.4, 0.7}, {0.3, 0.6}), c4/({0.1, 0.6}, {0.5}, {0.3, 0.5, 0.6}), c5/({0.5, 0.7, 0.8}, {0.1, 0.4, 0.6, 0.8, 0.9}, {0.3, 0.5,0.6}),

e3 = {ct/({0}, {0}, {0}), c2/({0}, {0}, {0}), c3/({0}, {0}, {0}), c4/({0}, {0}, {0}), c5/({0}, {0}, {0})}}, {e3 = c1 /({0.1}, {0.3, 0.4}, {0.2, 0.5, 0.7, 0.8}), c2/({0.2, 0.5, 0.7}, {0.4}, {0.1, 0.3, 0.5, 0.7, 0.9}), c3/({0.1, 0.3, 0.4}, {0.7, 0.9}, {0.6}), c4/({0.1, 0.2, 0.4}, {0.5, 0.6, 0.7, 0.8}, {0.3, 0.4, 0.6,0.7}), c5/({0.5, 0.7}, {0.1}, {0.2, 0.3, 0.4, 0.6})},

{e1 = {c1/({0}, {0}, {0}), c2/({0}, {0}, {0}), c3/({0}, {0}, {0}), c4/({0}, {0}, {0}), c5/({0}, {0}, {0})}, e2 = {c1/({0}, {0}, {0}), c2/({0}, {0}, {0}), c3/({0}, {0}, {0}), c4/({0}, {0}, {0}), c5/({0}, {0}, {0})}}, {e1 = {c1/({0}, {0}, {0}), c2/({0}, {0}, {0}), c3/({0}, {0}, {0}), c4/({0}, {0}, {0}), c5/({0}, {0}, {0})},

e2 = {c1/({0}, {0}, {0}), c2/({0}, {0}, {0}), c3/({0}, {0}, {0}), c4/({0}, {0}, {0}), c5/({0}, {0}, {0})}

e3 = c1/({0.1}, {0.3, 0.4}, {0.2, 0.5, 0.7, 0.8}), c2/({0.2, 0.5, 0.7}, {0.4}, {0.1, 0.3, 0.5, 0.7, 0.9}), c3/({0.1, 0.3, 0.4}, {0.7, 0.9}, {0.6}), c4/({0.1, 0.2, 0.4}, {0.5, 0.6, 0.7, 0.8}, {0.3, 0.4, 0.6,0.7}), c5/({0.5, 0.7}, {0.1}, {0.2, 0.3, 0.4, 0.6})}

{e1 = c1/({0.3, 0.4}, {0.6}, {0.2, 0.3, 0.6}), c2/({0.4, 0.6}, {0.2,0.4}, {0.5}), c3/({0.2, 0.4, 0.6}, {0.3, 0.7}, {0.8}), c4/({0.3, 0.5, 0.6}, {0.4, 0.6}, {0.2, 0.4, 0.6}), c5/({0.1, 0.4, 0.7, 0.8}, {0.5, 0.6, 0.9}, {0.2}),

e2 = c1/({0.4, 0.8, 0.9}, {0.1, 0.4, 0.8}, {0.4, 0.5, 0.6}), c2/({0.2, 0.7}, {0.2, 0.4, 0.6}, {0.5, 0.7}), c3/({0.1, 0.3, 0.4, 0.7}, {0.3, 0.4, 0.7}, {0.3, 0.6}), c4/({0.1, 0.6}, {0.5}, {0.3, 0.5, 0.6}), c5/({0.5, 0.7, 0.8}, {0.1, 0.4, 0.6, 0.8, 0.9}, {0.3, 0.5,0.6}),

e3 = c1/({0.1}, {0.3, 0.4}, {0.2, 0.5, 0.7, 0.8}), c2/({0.2, 0.5, 0.7}, {0.4}, {0.1, 0.3, 0.5, 0.7, 0.9}), c3/({0.1, 0.3, 0.4}, {0.7, 0.9}, {0.6}), c4/({0.1, 0.2, 0.4}, {0.5, 0.6, 0.7, 0.8}, {0.3, 0.4, 0.6,0.7}), c5/({0.5, 0.7}, {0.1}, {0.2, 0.3, 0.4, 0.6})}

{e1 = c1/({0.2, 0.3}, {0.6}, {0.2, 0.3}), c2/({0.4, 0.5}, {0.2, 0.4}, {0.5}), c3/({0.2, 0.3, 0.5}, {0.3, 0.4}, {0.4}), c4/({0.3, 0.4, 0.6}, {0.4, 0.5}, {0.2, 0.4}), c5/({0.1, 0.4, 0.6, 0.7}, {0.4, 0.6, 0.7}, {0.2}),

e2 = c1/({0.4, 0.5, 0.8}, {0.1, 0.2, 0.3}, {0.4, 0.5, 0.6}), c2/({0.2, 0.3}, {0.2, 0.3, 0.6}, {0.5, 0.6}), c3/({0.1, 0.3, 0.4, 0.5}, {0.2, 0.4, 0.5}, {0.3}), c4/({0.1, 0.3}, {0.1}, {0.3, 0.5}), c5/({0.5, 0.6, 0.7}, {0.1, 0.4, 0.6, 0.8}, {0.3, 0.5}),

e3 = {c1/({0}, {0}, {0}), c2/({0}, {0}, {0}), c3/({0}, {0}, {0}), c4/({0}, {0}, {0}), c5/({0}, {0}, {0})}}}

P(U) be the set of all subsets of U and HFP/ (U; E) be the family of all hesitant multi-fuzzy soft sets over U via parameters in E.

Definition 18. Let (F,X) be an element of H3F,- (U; E), P(F,X) be the set of all hesitant multi-fuzzy soft subsets of (p, X) and t be a subfamily of P(p, X). Then, t is called hesitant multi-fuzzy soft topology on (p, X) if the following conditions are satisfied:

(1) Fa(f,X)2t, _ _

(2) (G^A), (G2,B)et0(G1,A)n(G2,B)Et,

(3) {(Gk,Ak) : k2K}4t0 u (Gk,Ak)2t.

The pair (XF~, t) is called a hesitant multi-fuzzy soft topological space. Each member oft is called t-open hesitant multi-fuzzy soft set. A hesitant multi-fuzzy soft set is called t-closed if its complement is t-open. We shall call a t-open (t-closed) hesitant multi-fuzzy soft set simply openk(closed) set.

Here, {Fa , (p, X)} and P(p, X) are two examples for the hesitant multi-fuzzy soft topology on (F~, X) and are called indiscrete hesitant

of P(p, X) is a hesitant multi-fuzzy soft topology on (p, X) and (Xp, t) is a hesitant multi-fuzzy soft topological spaces.

Definition 19. Let t be a hesitant multi-fuzzy soft topology on (p,X)2HSP,- (U;E) and (G2,B) be a hesitant multi-fuzzy soft set in P(p,X). Then, a hesitant multi-fuzzy soft set (G1,A) eP(p,X) is a neighbourhood of (G^B) if and only if there exists^an open_hesitant multi-fuzzy soft set (G3, C)Et such that (G2, B) 8F (G3, C) 8F (G1, A).

Example 7. In Example 14, let

(G1,A) = {e3 = c1/({0.1}, {0.3,0.4}, {0.2,0.5,0.7,0.8}), c2/({0.2, 0.5,0.7},{0.4},{0.1,0.3,0.5,0.7,0.9}),c3/({0.1,0.3,0.4}, {0.7,0.9}, {0.6}), c4/({0.1,0.2,0.4}, {0.5,0.6,0.7,0.8}{0.3,0.4, 0.6,0.7}), c5/({0.5,0.7}, {0.1}, {0.2,0.3,0.4,0.6})}, (G2, B) = {e3 = ^/({0.1}, {0.1,0.2},{0.2,0.3}), c2/({0.2,0.4},{0.1},{0.1,0.3}) , c3/({0.1}, {0.4,0.5}, {0.2}), c4/({0.1,0.2}, {0.5,0.6}, {0.1,0.2}), c5/({0.5,0.6}, {0.1}, {0.2,0.3,0.4})} and (G3, C)= {e3 = c1/ ({0.1}, {0.2,0.3}, {0.2,0.5}), c2/({0.2,0.5}, {0.3}, {0.1,0.3,0.5}), c3/({0.1}, {0.7}, {0.4}), c4/({0.1,0.2,0.3}, {0.5,0.6,0.7}, {0.3,0.4,

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0.6})c5/({0.5,0.7}, {0.1}, (0.2,0.3,0.4,0.6})}.Then,JC2, B)8 (G3, C) 8 (G1,A). Therefore, (G1 ,A) is a neighbourhood of (G2,B).

Definition 20. Let t be a hesitant multi-fuzzy soft topology on (F, X) and (G, B)sP (F, X). The collection of all the neighbourhood of (G, B) is denoted by Nj B) and it is called the neighbourhood system of (G, B).

called hesitant multi soft finer than tn , or equivalently tn is hesitant multi-fuzzy soft coarser than t2.

Example 8.

following.

Let r-[ be as F in Example 14, and t2t2 be as the

fe3 = {ci/({0}, {0}, {0}), C2/({0}, {0}, {0}), C3/({0}, {0}, {0}), C4/({0}, {0}, {0}), C5/({0}, {0}, {0})},

{e3 = c1/({0.1}, {0.3, 0.4}, {0.2, 0.5, 0.7, 0.8}), c2/({0.2, 0.5, 0.7}, {0.4}, {0.1, 0.3, 0.5, 0.7, 0.9}), c3/({0.1, 0.3, 0.4}, {0.7, 0.9}, {0.6}), c4/({0.1, 0.2, 0.4}, {0.5, 0.6, 0.7, 0.8}, {0.3, 0.4, 0.6, 0.7}), c5/({0.5, 0.7}, {0.1}, {0.2, 0.3, 0.4, 0.6})},

{e1 = {c1/({0}, {0}, {0}), C2/({0}, {0}, {0}), C3/({0}, {0}, {0}), C4/({0}, {0}, {0}), C5/({0}, {0}, {0})},

e2 = {C1 /({0}, {0}, {0}), C2/({0}, {0}, {0}), C3/({0}, {0}, {0}), C4/({0}, {0}, {0}),

C5/({0}, {0}, {0})},

e3 = c1/({0.1}, {0.3, 0.4}, {0.2, 0.5, 0.7, 0.8}), c2/({0.2, 0.5, 0.7}, {0.4}, {0.1, 0.3, 0.5, 0.7, 0.9}), c3/({0.1, 0.3, 0.4}, {0.7, 0.9}, {0.6}), c4/({0.1, 0.2, 0.4}, {0.5, 0.6, 0.7, 0.8}, {0.3, 0.4, 0.6, 0.7}), c5/({0.5, 0.7}, {0.1}, {0.2, 0.3, 0.4, 0.6})}}.

Theorem 3. A hesitant multi-fuzzy soft set (G, B) in P(F,X) is an open hesitant multi-fuzzy soft set if and only if (G, B) is a neighbourhood of each hesitant multi-fuzzy soft set (H, C) contained in (G, B). The proof is trivial and it is omitted.

Definition 21. LetJX~, t) be a hesitant multi-fuzzy softjopological space and (G1,B), (G2,B)eP(F,X) such that (G2,B)8(G1,AJ^Then, (G2, B) is called an interior hesitant multi-fuzzy soft set of (G1, A) if (G1,A) is a neighbourhood of (G2,B).

The union of all interior hesitant multi-fuzzy softsets of (G1, A) is called the interior of (G1, A), and it is denoted by (G1, A)°. We record the following observation as well:

Theorem 4. Let F be a hesitant multi-fuzzy soft topology on (F, X) and (G1,A)sP(F,X). Then,

(i) (G1,A)" is the largest open hesitant multi-fuzzy soft set contained in (G1, A).

(ii) The hesitantmulti-fuzzy soft set (G1, A) is open if and only if

(f, A) = (f, A)o.

Proof.

(i) (G1,A)o = U((G2, B) : (G1,A) is a neighbourhood of (G2,B)}.

_ Thus, (G1, A)'' is itself an interior hesitant multi-fuzzy soft set of (G1,A). Therefore, there existjm open hesitant multi-fuzzy soft set (g3, C) such that (G1,A)o 8 (G3, C) 8 (ji,A). However, (G3, C) is an interior hesitant multi-fuzzy soft set of (G1, A). Thus, (G3,C)8(G1,A)^ Therefore, (G3,C) = (G1,A)o. _

This implies (G1,A)o is open and (G1, Af is the largest open hesitant multi-fuzzy soft set contained in (G1,A).

(ii) The proof is obvious.

Definition 22. Let (Xf , t1), (Xf, t2) be two hesitant multi-fuzzy soft topological spaces. If each (G,A)ei~ implies (G,A)^-^, then t2 is

Then, t-[ is hesitant multi-fuzzy soft finer than t2.

Definition 23. Let t be a hesitant multi-fuzzy soft topology on (~ , X) and be a subfamily of t. If every element of U be written as arbitrary hesitant multi-fuzzy soft union of some elements of then is called a hesitant multi-fuzzy soft basis for the hesitant multi-fuzzy soft topology tF.

Example 9. Let us consider the hesitant multi-fuzzy soft topology t2 as in Example l6. Thenk, the subfamily

~ = (Fa, (F,X) , (e3 = (c1/((0} , (0} ,(0}),

c2/((0} ,(0} ,(0}) , c3/((0} ,(0} ,(0}) ,

c4/((0} , (0} , (0}) , c5/((0} , (0} , (0})} , (e3 = ^/((0.1}, (0.3 , 0.4} , (0.2 , 0.5 , 0.7 , 0.8}) , c2/((0.2 , 0.5 , 0.7} , (0.4} , (0.1, 0.3 ,0.5 , 0.7 , 0.9}) ,c3/((0.1,0.3 ,0.4} ,(0.7,0.9} , (0.6}) , c4/((0.1, 0.2 , 0.4} , (0.5 , 0.6 , 0.7 , 0.8} , (0.3 ,0.4 , 0.6 , 0.7}) , c5/((0.5, 0.7} , (0.1} , (0.2 , 0.3 ,0.4 , 0.6})}} is a basis of t2.

The next two results can be proved using the previous results.

Proposition 1. Let 7~, t2 be two hesitant multi-fuzzy soft topologies on (F , X) and B1, B2.be hesitant multi-fuzzy soft bases for f\, t2 , respectively. If B1 8 B2 , then t2 is hesitant multi-fuzzy soft finer than

Proposition 2. Let t be a hesitant multi-fuzzy soft topology on (~ , X) and BF be a hesitant multi-fuzzy soft basis for tF. Then, Ft equals to the collection of hesitant multi-fuzzy soft unions of the elements of BF . We are now ready to show:

Theorem 5. Let (X~,t) be a hesitant multi-fuzzy soft topological

space and (G1 ,A)eP(F,X). Then, the collection U ~ = ((G1 ,A)

(G1,A)

f(G2,B) : (G2,B)st} is a hesitant multi-fuzzy soft topology on the hesitantmulti-fuzzy soft subsets (G1, A) relative to the parameter set A. Proof.

(i) Since Fa, (F,X)st.

Thus, $a = Fa f (f1,A)st(~,A) and f,A) ~ (~ ,X)e~u(~,A).

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(ii) Let (G11,A1), (G12,A2)2t ~,A).

Therefore, there exist (G21,B1), (G22,B2)eP such that (G11,A1) = (G1, A)n(G21, B1) and (G12, A2) = (f, A)n(G22, B2). ^

Therefore,JGU,A1) n(G12,A2)= [(Cf,A)n(G21,B^]n [(G1,A) n

(G22,B2)]=_(C1,A)n [(G21,B1) n (G22,B2)].

Since (G21,B1) n (G22,B2)et, therefore, (G„,A1) n (G12,A2) e t/.

(F,X)'

(iii) Let {(Gk, Bk) : IieK} be a subfamily of p ~

(G1,A)

Thus, for each IieK, there is a hesitant multi-fuzzy soft set (Hk, Ck) of t such that(Gk, Bk) = (G1,A)n(Hk, Ck). ^

Now,~U(Gk, Bk) = ~T [(G1, A)n(H|, Ck)] = (G1, A)n(1T(Hk, Ck)).

IeK IeK IeK

Since IT^ Ck)Et, therefore, IT^Bk)Et,~ A.

IeK IeK (G1,A)

Definition 24. Let (Xp, t) be a hesitant multi-fuzzy soft topological space and (G1, A)eP(F, X). Then, the hesitant multi-fuzzy soft topology t ~ given in Theorem 5 is called hesitant multi-fuzzy soft

Q8 subspace topology, and it is called a hesitant multi-fuzzy soft subspace

of (Xp, t).

The following result follows from the previous results.

Theorem 6. Let (Xp, t) be a hesitant multi-fuzzy soft topological space on (F, X), B be a hesitant multi-fuzzy soft basis for jr and (Gi,A)eP(p,X). Then, the collection p~ ={(G1,A)P(G2,B) : (G2, B) eB} is a hesitant multi-fuzzy soft bas?sfor the hesitant multi-

fuzzy soft subspace topology t ~ .

(G~,A)

4. Compactness in hesitant multi-fuzzy soft topological spaces

In this section, we introduce hesitant multi-fuzzy soft cover, hesitant multi-fuzzy soft open cover, hesitant multi-fuzzy soft compactness and the theorem of these concepts.

Let U be an initial universe, E be the set of parameters, P( U) be the set of all subsets of U and H3P,- (U; E) be the family of all hesitant multi-fuzzy soft set of dimension k over U via parameters in E.

Definition 25. Let (F,X) be an element of HFP; (U; E). A family 5. Conclusion

Proofs

Let {(Fk, Ak) : keK} be any hesitant multi-fuzzy soft open cover of (G, B).

Then, Ukx ~ Cu (Fk,Ak)} Ü (G,B)c.

Thus, {(Fk, Ak) : IceK} togetherwith hesitant multi-fuzzy soft open set (G, B)c is a hesitant multi-fuzzy soft open cover of UX.

Since, (X~, t) is compact, therefore, there exists a finite subcover. Let {(Fk, ,Ak,), (~k2,Ak2), ..., (Fkn ,Akn), (G,B)c} be the finite subcover.

Therefore, (G,B) ~{ÜL, (F^,A^)}Ü(G,B)c. This implies (G, B)G ülL,(fki,Ak.). Hence, (G, B) is compact.

Definition 27. Let (X~, t) be a hesitant multi-fuzzy soft topological space and e1re2 eXwith ejse2. If there exists two hesitant multi-fuzzy

soft open sets (F,A), (G,B) such that (F(e,)e(F,A)), (G(e2)e(G,B)) k

and (F, A)n(G, B) = <Fa, then (Xp, F) iscalleda hesitant multi-fuzzy soft Hausdroff space.

Theorem 8. Let (G, B) be a hesitant multi-fuzzy soft compact set in a hesitant multi-fuzzy soft Hausdroff space (Xf, t). Then, (G, B) is hesitant multi-fuzzy soft closed set. Proof.

Let e,eX with G(e,), but does not belong to (G,B), i.e., Q9 G (e,)e(G B)c. LetG (e2)e(G, B). Therefore, e, s e2 and e,,e2 eX.

Thus, by the Hausdroff property of (Xp, t) there exist two disjoint hesitant multi-fuzzy soft open sets (He2,A), (Ge2,A), such that H(e,)e(He2,A) andG(e2)e(Ge2,A).

Thus, the collection {(Ge2,A) : e2eX, G(e2)e(G,B)} is a hesitant multi-fuzzy soft open cover of (G, B).

Since (G,B) is a hesitant multi-fuzzy soft compact set in (Xp, t), therefore, there exists a finite subcover for (G, B).

Let {(Ger,, A), (Ger2, A),..., (Gern, A)} be the finite subcover. Thus, rin=i (Her., A) is a hesitant multi-fuzzy soft open set contained in (G, B)c. i

Therefore, (G, B)c is a multi-fuzzy soft open set. Hence, (G, B) is a multi-fuzzy soft closed set.

{(pk,Ak) : IieK}GHF (U;E) of hesitant multi-fuzzy soft sets is a

cover of (p, X) if (p, X)G IT (Fk, Ak).

If each member of the family {(F|, A|) : IieK} is a hesitant multi-fuzzy soft open set, then it is called a hesitant multi-fuzzy soft open cover of (p, X).

If a subfamily of {(F~k, A|) : IeK} is also cover of (p,X), then it is called a subcover.

Definition 26. Let (P,X) be an element of HG (U; E).Then, (P,X) is said to be a hesitant multi-fuzzy soft compact if each hesitant multi-fuzzy soft open cover of (P, X) has a finite subcover.

Additionally, a hesitant multi-fuzzy soft topological space ^, t) is called compact if each hesitant multi-fuzzy soft open cover of UX has a finite subcover.

Note 5 Let (Xp, fj), (Xp, t2) be two hesitant multi-fuzzy soft topological spaces and t2 be hesitant multi-fuzzy soft finer than t1. If (Xp, t2) is compact, then (Xp, 7~) is compact. The main result of this section is:

Theorem 7. Let (G, B) be a hesitant multi-fuzzy soft closed set in a hesitant multi-fuzzy soft compact topological space (Xp, P). Then, (G, B) is also compact.

The purpose of this paper is to discuss some important properties of hesitant multi-fuzzy soft topological spaces. We introduce the neighbourhood of a hesitant multi-fuzzy soft sets and interior hesitant multi-fuzzy soft sets. We also introduce the hesitant multi-fuzzy soft basis and hesitant multi-fuzzy soft subspace topology and have established several interesting properties. To extend this work, one could study connectedness or other interesting properties for hesitant multi-fuzzy soft to-pological spaces. We hope that this work will help enhance the study of hesitant multi-fuzzy soft topological spaces for researchers.

Uncited reference

Ali et al., 2009, Aygunoglu and Aygun, 2012a, Aygunoglu and Aygun, 2012b, Blizard, 1991, Dey and Pal, 2014a, Dey and Pal, 2016a, Jun, 2008, Katsaras and Liu, 1977, Liao and Xu, 2014, Majumdar and Samanta, 2010, Roy and Maji, 2007, Shabir and Naz, 2011, Varol et al., 2014, Verma and Sharma, 2013, Zadeh, 1965. Q10

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Acknowledgement

Q11 Financial support offered by Council of Scientific and Industrial Research, New Delhi, India (Sanction no. 09/599(0054)/2013-EMR-I) is thankfully acknowledged.

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