PROPERTIES OF FUZZY DITANCE ON FUZZY SET

In this paper we introduce the definition of fuzzy distance space on fuzzy set then we study and discuss several properties of this space after some illustrative examples are given . Furthermore we introduce the definition of fuzzy convergence, fuzzy Cauchy sequence of fuzzy point and fuzzy bounded fuzzy distance space .


INTRODUCTION
In 1965 fuzzy sets was introduced by Zadeh [20] many authors used this concept to formulate the notion of fuzzy metric on ordinary set [1,2,3,7,8,9,10,13,15,16]. In 1975 Kramosil and Michalek [14] introduce the definition of fuzzy metric on ordinary set which was called later the KM-fuzzy metric space. In 1994 George and Veermani [5] modified the KM-fuzzy metric space by taking instead of minimum function the binary operation t-norm which was called later GV-fuzzy metric space on ordinary set. In this paper we modified the definition of GV-fuzzy metric to a fuzzy space on fuzzy set This paper consist of three sections. In section two we introduce basic properties of fuzzy set and explain the difference between continuous and discrete fuzzy sets by example. After that we introduce the definition of fuzzy distance on fuzzy set then we give some examples to illustrate this notion. In section three we define fuzzy open fuzzy ball, fuzzy convergence of sequence of fuzzy points, fuzzy closed fuzzy set, fuzzy bounded fuzzy set, fuzzy dense fuzzy set and fuzzy Cauchy fuzzy sequence. It was proved that, let(Ã, , ) be a fuzzy distance space on the fuzzy set Ã and let be a subset of Ã. then is a fuzzy dense in Ã if and only if for every Ã there is  such that ( , )  (1- ) for some 0    1. (see Theorem 2.12).

FUZZY DISTANCE SPACE ON A FUZZY SET Definition 1.1:[20]
Let X be a nonempty set of elements, a fuzzy set in X is characterized by a membership function, ( We now recall an example of a continuous fuzzy set.

Example 1.2:[17]
Let X = and let be a fuzzy set in with membership function by:

Definition 1.3:[4]
Let Ã and be two fuzzy sets in X. then

Definition 1.6:[20]
Let be a fuzzy point and Ã be a fuzzy set in X. then is said to be in Ã or belongs to Ã which is denoted by  Ã if and only if (x) α.

Definition 1.7:[11]
Let f be a function from a nonempty set X into a nonempty set Y. If is a fuzzy set in Y then is a fuzzy set in X defined by:  Hence is a continuous fuzzy set Remark 1.20: 1-In example 1.19 we can replace by any nonempty set X and the usual metric on by any metric d.

Example 1.22:
Let X= and let Ã=[2, ] be a fuzzy set in X, consider the mapping Proof:

Fuzzy Convergence, Fuzzy Cauchy Sequences, Fuzzy bounded, Fuzzy open and Fuzzy closed Fuzzy sets
In this section Ã will be a fuzzy set in the nonempty set X. Next, we assume that ≠ , we may assume without loss of generality that this implies that (

Definition 2.4:
A sequence )}of fuzzy points in a fuzzy distance space (Ã, , ) is said to be fuzzy converges to a fuzzy point Ã

Definition 2.9:
Let (Ã, , ) be a fuzzy distance space on a fuzzy set Ã and let Ã then the fuzzy closure of is denoted by or FCL( ) and is defined to be the smallest fuzzy closed fuzzy set contains .

Definition 2.10:
A fuzzy subset of a fuzzy distance space (Ã, , ) on a fuzzy set Ã is said to be fuzzy dense in Ã if = Ã.

Proof:
Let  , if  then we take sequence of that type is ( , , ,…, ,…). If  , it is a limit fuzzy point of .

Definition 2.13:
A sequence )} of fuzzy points in a fuzzy distance space (Ã, , ) is said to be fuzzy Cauchy if for each 0 1 there is a positive number N such that ( ) , ) ) (1-for all n, m .

Theorem 2.14:
In a fuzzy distance space every fuzzy convergent sequence of fuzzy points is fuzzy Cauchy.

Proof:
Let )} be a sequence of fuzzy points in a fuzzy distance space (Ã, , ) that is fuzzy converges to  Ã , then for

Proof:
Suppose that )} is a Cauchy sequence in (Ã, d), then there is a positive number N such that d( ) , ) ) for given and for all m, n N. Now t+ d( ) , ) ) , implies Put = (1-r) for some 0 r 1.
It follows that (1-r) for all m, n N. Hence )} is a fuzzy Cauchy sequence in (Ã, , ).
Conversely, assume that )} is a fuzzy Cauchy sequence in (Ã, , ) then given 0 1, there is a positive number N such that (1-. Put (1-= r then r for all n, m  N. This implies , it follows that t) for all n, m N, putt = k. Then k for all n, m N. Hence )} is Cauchy sequence in (Ã, d)

Definition 2.16:
Let )} be a given sequence of fuzzy points in a fuzzy distance space (Ã, , ) and let ( ) be a sequence of positive integer such that …Then the sequence {( )} is called a subsequence of )}. If ( ) fuzzy converges, its limit is called a sub sequential limit of )}.It is clear that a sequence )} in Ã fuzzy converges to if and only if every subsequence of it fuzzy converges to .

Proposition 2.17:
If a fuzzy Cauchy sequence of fuzzy points in a fuzzy distance space (Ã, , ) contains a fuzzy convergent subsequence, then the sequence fuzzy converges to the same fuzzy limit as thesubsequence. O c t o b e r 2 1 , 2 0 1 5

Lemma 2.20:
A fuzzy convergent sequence of fuzzy points in a fuzzy distance space (Ã, , ) is fuzzy bounded and its fuzzy limit is unique.

Proof:
Suppose that )} fuzzy converges to then given 0 1 we can find a positive number N such that   Proof: The prove follows from the fact (( ,( )) = 0  ( , )) = 1

Theorem 2.26:
Every fuzzy distance space on a fuzzy set is a fuzzy Hausdorff space.