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The definition of fuzzy length space on fuzzy set in this research was introduced after the studies and discussion of many properties of this space were proved, and then an example to illustrate this notion was given. Also the definitio n of fuzzy convergence, fuzzy bounded fuzzy set, and fuzzy dense fuzzy set space was introduced, and then the definition of fuzzy continuous operator was introduced.

Zadeh in 1965 [

The order pair

(i)

(ii) For all

(iii)

(iv) For all

(v)

The definition of continuous t-norm was introduced by George and Veeramani in [

The triple

(i)

(ii) For all

(iii)

(iv) For all

(v) For

(vi)

The definition of fuzzy length space is introduced in this research as a modification of the notion of fuzzy normed space due to Bag and Samanta. In Section 1, we recall basic concepts of fuzzy set and the definition of continuous t-norm. Then in Section 2 we define the fuzzy length space on fuzzy set after we give an example; then we prove that every ordinary norm induces a fuzzy length, and also the definition of fuzzy open fuzzy ball, fuzzy convergent sequence, fuzzy open fuzzy set, fuzzy Cauchy sequence, and fuzzy bounded fuzzy set is introduced. In Section 3, we prove other properties of fuzzy length space. Finally in Section 4, we define a fuzzy continuous operator between two fuzzy length spaces. Also we prove several properties for fuzzy continuous operator.

Definition 2.1:

Let X be a classical set of object, called the universal set, whose generic elements are denoted by x. The membership in a classical subject A of X is often viewed as a characteritic function

Definition 2.2:

Suppose that

(i)

(ii)

(iii)

(iv)

(v)

Definition 2.3:

Suppose that

Definition 2.4:

A fuzzy point p in Y is a fuzzy set with single element and is denoted by

Definition 2.5:

Suppose that

Definition 2.6:

Suppose that h is a function from the set

Proposition 2.7:

Suppose that

Definition 2.8:

A binary operation

(i)

(ii)

(iii)

(iv) If

Examples 2.9:

When

Remark 2.10:

Definition 2.11:

Let

(ii)

(ii)

Proposition 2.12:

Let

First we introduce the main definition in this paper.

Definition 3.1:

Let X be a linear space over field

(FL_{1})

(FL_{2})

(FL_{3})

(FL_{4})

(FL_{5})

Then the triple

Definition 3.2:

Suppose that

tinuous fuzzy set if whenever

Proposition 3.3:

Let

Proof:

Let

1)

2)

3)

4)

Hence

Example 3.4:

Suppose that

Then

Proof:

To prove

(FL_{1}) Since

(FL_{2}) It is clear that

(FL_{3}) If

(FL_{4})

(FL_{5})

Hence

Definition 3.5:

Let

fuzzy

a fuzzy open fuzzy ball of center

We omitted the proof of the next result since it is clear.

Proposition 3.6:

In the fuzzy length space

Definition 3.7:

The sequence

Definition 3.8:

The sequence

Theorem 3.9:

The two Definitions 3.8 and 3.7 are equivalent.

Proof:

Let the sequence

To prove the converse, let

Hence when

So

Lemma 3.10:

Suppose that

Proof:

Let

Lemma 3.11:

If

(a) The operator

(b) The operator

Proof:

If

Hence the operator addition is continuous function.

Now if

And this proves (b).

Definition 3.12:

Suppose that

The proof of the following theorem is easy and so is omitted.

Theorem 3.13:

Any

Definition 3.14:

Suppose that

Definition 3.15:

Suppose that

Lemma 3.16:

Suppose that

Proof:

Let

The fuzzy ball

Conversely assume that

Theorem 4.1:

Suppose that

Proof:

Let

To prove the converse, we must prove that

Definition 4.2:

Suppose that

Theorem 4.3:

If

Proof:

If

for any

Hence

Definition 4.4:

Suppose that

Theorem 4.5:

Suppose that

Proof:

Let

Taking the limit to both sides as

Hence,

Proposition 4.6:

Let

tains a

Proof:

Assume that

Letting

Hence, the sequence

Definition 4.7:

Suppose that

Lemma 4.8:

Assume that

Proof:

Let

Put

Using Remark 2.10 we can find

Hence

Suppose that

Definition 4.9:

Suppose that

The proof of the following lemma is clear and hence is omitted.

Lemma 4.10:

Any

Theorem 4.11:

A fuzzy length space

Proof:

Suppose that

(i) Clear that

(ii) let

(iii) Let

Definition 4.12:

A fuzzy length space

Theorem 4.13:

Every fuzzy length space

Proof:

Assume that

Put

is a contradiction, hence

In this section, we will suppose

Definition 5.1:

Let

Theorem 5.2:

Let

Proof:

Suppose that the operator

then we can find K with

Conversely, assume that every sequence

ment that

Definition 5.3:

Let

Proposition 5.4:

Let

Proof: the argument is similar that of Theorem (5.3) and therefore is not included.

Proposition 5.5:

An operator T of the fuzzy length space

Proof: the operator

Theorem 5.6:

An operator

Proof:

Suppose that T is fuzzy continuous operator and let

Let

Thus, every fuzzy point

Conversely, suppose that

since

Theorem 5.7:

An operator

Proof:

Let

Conversely, suppose that

Theorem 5.8:

Let

Proof:

Let

Theorem 5.9:

Let

(i) T is fuzzy continuous on

(ii)

(iii)

Proof: (i) ⇒ (ii)

Let

[Recall that

Proof: (ii) ⇒ (iii)

Let

Proof: (iii) ⇒ (i)

Let

In this research the fuzzy length space of fuzzy points was defined as a generalization of the definition of fuzzy norm on ordinary points. Based on the defined fuzzy length, most of the properties of the ordinary norm were proved.

Kider, J.R. and Mousa, J.I. (2016) Properties of Fuzzy Length on Fuzzy Set. Open Access Library Journal, 3: e3068. http://dx.doi.org/10.4236/oalib.1103068