On fuzzy metric spaces

sets and systems ELSEVIER

Fuzzy Sets and Systems 99 (1998) 111-114

On fuzzy metric spaces Kankana Chakrabarty, Ranjit Biswas*, Sudarsan Nanda Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, West Bengal, India Received July 1996; revised January 1996

Abstract

In the present paper, the authors define F-open sets, F-closed sets, F-adherent points, F-limit points, F-isolated points, F-isolated sets, F-derived sets, F-closures, F-interior points, F-interior, F-exterior points, F-exterior, F-everywhere dense sets, F-nowhere dense sets and make some characterizations of fuzzy metric spaces. (~) 1998 Elsevier Science B.V. All rights reserved Keywords: Fuzzy set; Fuzzy metric; Fuzzy metric space; F-convergence; F-Cauchy's sequence; F-complete fuzzy metric space; F-continuity; Fixed fuzzy set; F-open sphere; F-closed sphere; F-spherical neighbourhood; F-open set; F-closed set; F-adherent point; F-limit point; F-isolated point; F-isolated set; F-derived set; F-closure; F-interior point; F-interior; F-exterior point; Fexterior; F-everywhere dense set; F-nowhere dense set

1. Introduction Fuzzy metric spaces were introduced by Kaleva and Seikkala [6] and by Abu Osman [1] independently with two different approaches. In [1], Abu Osman also fuzzified the classical Banach's fixed point theorem. A number of fixed fuzzy set theorems have been studied by many authors (e.g. [2, 3, 5]). In [2] Biswas defined F-open sphere and F-closed sphere in a fuzzy metric space with centre at some fuzzy set and with radius r > 0. He also defined F-neighbourhood of a fuzzy set. In the present paper, we define F-open sets, F-closed sets, F-adherent point, Flimit point, F-isolated point, F-isolated set, Fderived set, F-closure, F-interior point, F-interior, F-exterior point, F-exterior, F-everywhere dense set, F-nowhere dense set and make some characterizations of fuzzy metric spaces. We furnish * Correspondingauthor. E-mail: [email protected].

below some basic preliminaries required for this paper.

2. Preliminaries

In the context of the present paper, we consider the metric spaces defined by Abu Osman [1]. Some definitions and propositions from [1, 2, 5] are given below. Definition 2.1. I f X is a set, a mapping # : X ~ [ 0 , 1] is called a fuzzy subset of X. The function value #(x) is called the grade of membership o f x in #. The collection of all fuzzy sets of X is denoted by Fx. Definition 2.2. Consider the metric # x on Fx defined by # x : F x × F x ~ [ 0 , 1 ] and # x ( A , B ) = SuPx~xlA(x) - B(x)l. Then we call ( F x , # x ) a fuzzy metric space of X and # x a fuzzy metric on X.

0165-0114/98/$19.00 (~) 1998 Elsevier ScienceB.V. All rights reserved PII S0165-0114(97)00037-7

K. Chakrabarty et al./Fuzzy Sets and Systems 99 (1998) 111-114

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Definition 2.3. A sequence {Ai} where Ai E Fx, is said to be F-convergent to a fuzzy set A E Fx if Ve > 0, 3 a positive integer N such that lax(A,,A) < Vn >N. Definition 2.4. A sequence {Ai} in Fx is a FCauchy's sequence if Ve > 0, 3 a positive integer N such that lax(Am,A,) < ~ Vm, n > N. Definition 2.5. A fuzzy metric space of X is Fcomplete if every F-Cauchy's sequence in X, Fconverges to a fuzzy set in X. Definition 2.6. Let f : X ~ Y be a mapping from a set X into a set Y. Let B be a fuzzy set in Y with membership function/as. Then f - l [ B ] is a fuzzy set in X whose membership function is defined by

laf-~[B](x)=las(f(x))

VxEX.

Conversely, let A be a fuzzy set in X with membership function laa. The image of A written as f[A], is a fuzzy set in Y whose membership function is given by f

Sup

{laA(z)}

] zEf-l[Y] laf[A](Y) = ]

i f f - l [ y ] is n o n - e m p t y ,

F-neighbourhood of A with radius e > 0, we mean S(A, e). Clearly, if 0 < rl < ?'2 then

S(A, rl ) C S(A, r2) C S(A, r2). 3. Some characterizations of fuzzy metric spaces In [2], the concept of F-open sphere S(A, r) and F-closed sphere ,q(A, r) are introduced. Here first of all we define F-open set. Definition 3.1. Suppose G c_ Fx. G is said to be Fopen set if Vii E G, 3 r > 0 such that the F-open sphere S(A, r) C_G. Clearly, ~b and Fx are F-open sets.

Theorem 3.1. Let (Fx, lax) be a fuzzy metric space. Then each F-open sphere in it is an F-open set. Proof. Let S(A,r) be an F-open sphere in (Fx,#x). Let B E S( A, r ). It implies that # x( B, A ) < r ~ S( B, rl ) C_S( A, r ), where rl = r - p x( B,A ); hence the theorem is proved. []

Theorem 3.2. Let (Fx, #x ) be a fuzzy metric space. A subset G of Fx is F-open iffit is a union ofF-open spheres.

1, 0 otherwise for all y E Y, where f - l [ y ] = {x : f ( x ) = y}. We say that f is F-continuous on a fuzzy set A in X if Ve > 0, 36 > 0 such that lax(f[A],f[B]) < e whenever lax(A,B) < 3, where B E Fx. Definition 2.7. Let f be a mapping of X into itself. We say that f has a fixed fuzzy set A iff f[A] =A. Definition 2.8. Let (Fx, lax) be a fuzzy metric space inX, r E (0, I) andA E Fx. Consider the subsetS(A,r) of Fx defined by

Proof. The first part is straightforward. We prove the converse only. Suppose G is the union of a collection F of F-open spheres. If F = 0, the proof is complete. If F ¢ 0, G ¢ 0. Let B E G. Since G is the union of F-open spheres, 3 an F-open sphere S(A, r) E F such that B E S ( A , r ) . Since each F-open sphere is an Fopen set, hence 3rl > 0 such that

S(B, rl ) C S(A, r) =~ S(B, rl ) C_G =~ GisF-open.

[]

S(A,r) = {B :B EFx, lax(B,A)<~, r}.

Theorem 3.3. On a fuzzy metric space ( F x , # x ) the followin9 holds: (i) The union of any number ofF-open sets is Fopen. (ii) The intersection of a finite number ofF-open sets is F-open.

The sets S(A, r) and ~q(A,r) are, respectively, called F-open sphere and F-closed sphere with centre A and radius r. By an F-spherical neighbourhood or

Proof. (i) Straightforward. (ii) Let Gi----1,2. . . . . n be a finite number of Fn open sets in Fx. Let G--- Ni=l Gi. If Gi=O Vi, then

S(A,r) = {B:B EFx, lax(B,A) < r} and the subset S(A,r) of Fx defined by

K Chakrabartyet al./Fuzzy Sets and Systems 99 (1998) 111-114 the proof is complete. If G # 0, then take A E G. Therefore A E Gi Vi. But each Gi is F-open which implies Vi, 3 r; > 0 such that S(A, ri) C_ Gi. If r = M.in{ri},then S(A,r) C_Gi Vi

S(A,r)C_G ~

G isF-open.

[]

Corollary 3.1. Every F-open sphere (or F-closed sphere) in a fuzzy metric space ( F x , # x ) is nonempty. Definition 3.2. Let (Fx, # x ) be a fuzzy metric space. Then an element A E Fx is said to be an F-adherent point of a subset G of Fx if every F-open sphere with centre A contains atleast one element of G. F-adherent points are of two types (i) F-limit point or F-accumulation point. (ii) F-isolated point. Definition 3.3. Let (Fx, # x ) be a fuzzy metric space. Then an element A E Fx is said to be an F-limit point or F-accumulation point of a subset G of Fx if every F-open sphere with centre A contains atleast one element of G other than A. An F-limit point of G is not necessarily a point of G.

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or an F-limit point of G. In either case d(A, G) = 0; hence the theorem is proved. [] Definition 3.5. Let (Fx, # x ) be a fuzzy metric space and let G be any subset of Fx. Then the set of all Flimit points of G in Fx is said to be the F-derived set of G in Fx. It is denoted by G'. Definition 3.6. Let (Fx, # x ) be a fuzzy metric space and let G c_Fx. Then the set of all F-adherent points of G in Fx is said to be the F-closure G in Fx and it is denoted by G. Thus, if A belongs to G, then every F-open sphere with centre A contains a point of G and conversely. Definition 3.7. Let (Fx, # x ) be a fuzzy metric space. Then an element A E Fx is said to be an F-interior point of a subset G of Fx if there exists an F-open sphere S(A, r) contained in G. The set of all F-interior points of G is said to be the F-interior of G and is denoted by Fint(G). Also Fint(G) is an F-open set. Definition 3.8. Let (Ix, # x ) be a fuzzy metric space. Then an element A E Fx is called to be the F-exterior point of a subset G ofFx ifA is an F-interior point of the complement of G. The set of all F-exterior points of G is said to be the F-exterior of G and it is denoted by Fext(G).

Definition 3.4. An F-adherent point A of a subset G of Fx is called an F-isolated point if there exists atleast one F-open sphere with centre A which contains no point of G other than A itself. If every element of G is an F-isolated point, then G is called to be an F-isolated set. Let ( F x , # x ) be a fuzzy metric space and let G C_Fx. Then the distance between A and G where A E Fx is defined as d(A, G) = inf{px(A, y) : y E G}.

Definition 3.9. Let (Fx, # x ) be a fuzzy metric space. Then a subset G of Fx is said to be F-everywhere dense in Fx if Fx is the F-closure of G. A subset G of the fuzzy metric space (Fx, # x ) is called to be Fnowhere dense in Fx iff the F-closure of G has no F-interior points.

Theorem 3.4. The element A of a fuzzy metric space ( F x , # x ) is an F-adherent point of the subset G of Fx iff d(A, G) = O.

Fx.

Proof. We have d(A, G) = inf{#x(A, y): y E G}. Therefore, d(A, G) = 0 =~ Every F-open sphere S(A, r) contains an element of G, which implies A is an Fadherent point of G. Conversely, ifA is an F-adherent point of G, then either A is an F-isolated point of G

Definition3.10. Let ( F x , # x ) be a fuzzy metric space. Then a subset M of Fx is said to be an F-closed set in Fx if the complement of M in Fx is F-open in

Corollary 3.2. (i) The complement of an F-closed set is F-open and the complement of an F-open set is F-closed. (ii) q9 and Fx are both F-open and F-closed. Theorem 3.5. Every F-closed sphere is a F-closed set in a fuzzy metric space.

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Proof. Let (Fx, #x) be a fuzzy metric space and let S(A,r)={B:BCFx,#x(B,A)<~ r} be any F-closed sphere in it. We prove that {S(A, r)} c is an F-open set which implies that ~q(A,r) is F-closed. Let B E {S(A, r)} ~ be arbitrary. Then px(B,A) ~ r i.e. #x(B,A) > r. Let #x(B,A) = r + 2~, where e > 0 is arbitrary. Now we consider the F-open sphere S(B, ~) which is contained in {~q(A,r)} c since every element ofS(B, ~) is at a distance greater than r + e from A. Therefore, {S(A,r)} c is an F-open set and consequently S(A, r) is F-closed. Hence, the theorem is proved. []

Theorem 3.6. In a fuzzy metric space the intersection of an arbitrary family of F-closed sets is Fclosed Proof. Let {M~} be an arbitrary family of F-closed sets in a fuzzy metric space (Fx, Px). Let M = n ~ M~. Clearly, M c = U~ M~c which is F-open since M~ is F-open for each ct and arbitrary union ofF-open sets is F-open. Hence M c is F-open. Therefore M is F-closed. Hence the theorem is proved. []

Theorem 3.7. In a fuzzy metric space the intersection of an arbitrary family of F-closed sets is Fclosed

ProoL Let {G1, G2 . . . . . Gn} be a finite family of Fclosed sets in a fuzzy metric space (Fx, I~x). Let G = U]=l G~. Then G c = n~=l G~ which is F-open since every G~c is F-open and the intersection of a finite number of F-open sets in Fx is F-open. Therefore, G is F-closed; hence the theorem is proved. []

Acknowledgements We are thankful to the referees for their valuable comments in modifying the first version of this paper.

Reference [1] M.T. Abu Osman, Fuzzy metric spaces and fixed fuzzy set theorem, Bull. Malaysian Math. Soc. 6(1) (1983) 1-4. [2] Biswas Ranjit, On fixed fitzzy set theorem, J. Fuzzy Math. 3 (1995) 611-615. [3] Biswas Ranjit, On fixed fuzzy set theorems in fuzzy metric spaces, 3. Fuzzy Math., to appear. [4] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. [5] S. Heilpem, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl. 83 (1981) 566-569. [6] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984) 215-229. [7] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.