Fuzzy <i>δ</i>*-Continuity and Fuzzy <i>δ</i>**-Continuity on Fuzzy Topology on Fuzzy Sets

The concept of a fuzzy topology on a fuzzy set has been introduced in [1]. The aim of this work is to introduce fuzzy δ^*-continuity and fuzzy δ^**-continuity in this in new situation and to show the relationships between fuzzy continuous functions where we confine our study to some of their types such as, fuzzy δ-continuity, fuzzy continuity, after presenting the definition of a fuzzy topology on a fuzzy set and giving some properties related to it.

• Fuzzy <i>δ</i>*-Continuity; Fuzzy <i>δ</i>**-Continuity; Quasi-Neighbourhood; Fuzzy <i>δ</i>-Open; Quasi-Coincident

1. Introduction

The concept of a fuzzy topology on a fuzzy set has been introduced by Chakrabarty and Ahsanullah [1]. Neighbourhood systems, quasi-neighbourhood system, subspaces of such fuzzy topology space and quasi-coincidence in this new situation have also been discussed by them. Also, the concepts of fuzzy continuity, Hausdorffness, regularity, normality, compactness, and connectedness have been introduced by Chaudhuri and Das [2]. The concepts of fuzzy δ-closed sets, fuzzy δ-open sets fuzzy regular open, fuzzy regular closed, fuzzy δ- continuity and the relation between fuzzy continuity and fuzzy δ-continuity in this new situation was introduced by Zahran [3]. These functions have been characterized and investigated mainly in light of the notions of quasineighborhood, quasi-coincidence. In our rummage we confined ourselves to the study of some kinds of these functions, the fuzzy continuous function, fuzzy δ-continuity and some types of fuzzy regular. In this paper, we introduce the concepts of a fuzzy δ^*-continuity, fuzzy δ^**- continuity and to show the relationships between types of fuzzy continuous functions in this situation and we examine the validity of the standard results.

2. Preliminaries

Let X and Y be sets and and be two subsets of X, Y respectively. Let I denote the closed unit interval. Let and for By we shall mean the fuzzy subset of X and the value of a fuzzy set at some will be denoted by such that for, and the support of a fuzzy set in X will be denoted by such that for all x in X. If and are fuzzy sets and for all x in X, then is said to be a fuzzy subset of and denoted by. The set of all fuzzy subsets of a nonempty set is denoted by.

Definition 2.1. [2] Let,. A fuzzy set of the form

is called a fuzzy point with support x and value r. is often denoted by.

For a fuzzy point

1).

2).

Definition 2.2. [1] If, the complement of referred to, denoted by is defined by , for each.

Definition 2.3. [2], are said to be quasicoincident (q-coincident, for short) referred to written as if there exists such that

. If and is not quasicoincident referred to, we denoted for this by.

3. Basic Definitions and Properties

In [4,5] fuzzy function have been introduced in a different way considering them as fuzzy relations with special properties. A special kind of fuzzy functions had been called fuzzy proper functions or proper functions that would be the morphisms in the proposed category FUZZY TOP.

Definition 3.1. [1] A fuzzy subset of is said to be a proper function from to if 1), for each

2) For each, there exists a unique such that and if.

Let be a proper function from to.

Definition 3.2. [1] If, then is defined by

for each.

Definition 3.3. [2] If, then is defined by

for each

Proposition 3.4. [2] For a proper function

1), for each.

2), for each.

3) and

Definition 3.5. [2] is said to maximal if for each

Proposition 3.6. [2] If is a maximal fuzzy subset of.

Definition 3.7. [2] Let Then defined by

for each, is said to be the restriction of to.

Proposition 3.8. [2] If then for each,.

Definition 3.9. [1] A collection of fuzzy subsets of a fuzzy set is said to be a fuzzy topology on if 1).

2), then.

3) for each, then.

is said to be a fuzzy topological space (fts, for short). The members of are called fuzzy open sets in. The complement of the members of referred to are called the fuzzy closed sets in. The family of all fuzzy closed sets in will be denoted by.

Definition 3.10. [1] If,

is a fuzzy topology on, is called a subspace of.

Definition 3.11. [1] Let be a fts and then the closure of denoted by is defined by. i.e. is the intersection of all closed fuzzy subsets of containing.

Definition 3.12. [3] Let be a fts and then the interior of denoted by  

. i.e. is the union of all open fuzzy subsets of which contained in.

Definition 3.13. [1] Let be a fts, a fuzzy subset of is called 1) Neighbourhood (nbd, for short) of the fuzzy point if there exists such that 2) Quasi-neighbourhood (q-nbd, for short) of the fuzzy point if there exists such that  

,.

The set of all q-neighbourhood of is called the system of q-nbd of.

Proposition 3.14. [2] If is a maximal subspace of, then, where.

Definition 3.15. [3]

1) is said to be a fuzzy regular open set in a fts.

2) is said to be a fuzzy regular closed set in a fts if is fuzzy regular open.

Definition 3.16. [3] A fuzzy point is said to be a fuzzy δ-cluster (resp. θ-cluster) point of a fuzzy subset of if for each fuzzy regularly open (resp. fuzzy open) q-nbd of

. The set of all fuzzy -cluster

(resp. fuzzy θ-cluster) points of is called fuzzy - cluster (resp. fuzzy θ-closure) and is denoted by

. A fuzzy subset is called a fuzzy δ-closed (resp. θ-closed) if (resp.) and the complement of a fuzzy δ-closed (resp. θ-closed) set is called fuzzy δ-open (resp. θ-open).

Remark 3.17. [3] It is clear that fuzzy regular open (fuzzy regular closed) implies fuzzy δ-open (fuzzy δ- closed) implies fuzzy open (fuzzy closed) but the converses are not true in general.

In this paper, the family of all fuzzy regular open (resp. fuzzy regular closed, fuzzy δ-open, fuzzy δ-closed, fuzzy open, fuzzy closed) sets in will be denoted by

.

4. Fuzzy δ * -Continuity

Unless otherwise mentioned are two fuzzy topologies on, respectively, and a proper function from to.

Definition 4.1. A proper function

is called fuzzy -continuous if

for each.

Example 4.2. Let

and

,.

Consider the fuzzy topologies on, resp.

and. Let the proper function defined by,

one may notice that the only fuzzy open sets in are, and but, , and, ,. Hence is fuzzy δ^*- continuous.

Theorem 4.3. If be fuzzy δ^*- continuous and, then

is fuzzy δ^*-continuous .

Proof: Let such that.

Then there exists fuzzy open such that .

Now

but be fuzzy δ^*-continuous such that . Therefore

.

Hence is fuzzy δ^*-continuous.

Definition 4.4. [2] is said to satisfy property if, for each.

Henceforth such functions will be called fuzzy continuous proper function.

Theorem 4.5. If a proper function is fuzzy δ^*-continuous then, it is fuzzy continuous.

Proof: Let, but is fuzzy δ^*-continuous. Hence and by (Remark (3.17)) every fuzzy δ-open implies fuzzy open. (i.e.). Hence is fuzzy continuous.

We can see from Example (4.2) such that

and, ,

but, ,.

Definition 4.6. [3] A proper function is called fuzzy δ-continuous if

for each.

Remark 4.7. [3] The concepts of fuzzy δ-continuous and fuzzy continuous are independent to each other .

Theorem 4.8. If be fuzzy δ- continuous and, then

is fuzzy δ-continuous.

Proof: Let such that. [by Prop. 3.8]. But is fuzzy δ-continuous such that. Therefore. Hence is fuzzy δ-continuous.

Theorem 4.9. If a proper function is fuzzy δ^*-continuous, then it is fuzzy δ-continuous.

Proof: Let. And by Remark 3.17 every fuzzy regular open implies fuzzy δ-open implies fuzzy open. (i.e. and but is fuzzy δ^*-continuous). Hence . Therefore is fuzzy δ-continuous.

5. Fuzzy δ ** -Continuity

Definition 5.1. A proper function is called fuzzy -continuous if for each.

Example 5.2. Let

and

Consider the fuzzy topologies on and resp. and. Let the proper function defined by_. One may notice that the only fuzzy δ-open sets in are, and and

Hence is fuzzy δ^**-continuous.

Theorem 5.3. If be fuzzy δ^**- continuous and, then

is fuzzy δ^**-continuous.

Proof: Let such that.

[by Prop. 3.8]. But is fuzzy δ^**-continuous such that. Therefore Hence is fuzzy δ^**-continuous. 

Theorem 5.4. If a proper function is fuzzy δ-continuous, then it is fuzzy δ^**-continuous.

Proof: Let and (by Remark 3.17 every fuzzy regular open implies fuzzy δ-open), i.e.

But is fuzzy δ-continuous. Hence

and (by Remark 3.17 every fuzzy δ-open implies fuzzy open). Therefore, (i.e. is fuzzy δ^**-continuous).

Theorem 5.5. If a proper function is fuzzy continuous, then it is fuzzy δ^**-continuous.

Proof: Let and (by Remark 3.17 every fuzzy δ-open implies fuzzy open), i.e.

But is fuzzy continuous. Hence Therefore is fuzzy δ^**-continuous.

We can see from Example (5.2.).

Remark 5.6. It is clear that not every fuzzy δ^**-continuous may be fuzzy δ^*-continuous and we can see from example.

Example 5.7. Let

and

Consider the fuzzy topologies on and resp.

and. Let the proper function defined by, is fuzzy δ^**-continuous but not fuzzy δ^*-continuous such that the only fuzzy δ-open sets in are, and

but.

From what we have deduced so far, we now obtain:

Fuzzy continuous ® Fuzzy δ^**-continuous;

Fuzzy δ-continuous ® Fuzzy δ^**-continuous;

Fuzzy δ^*-continuous ® Fuzzy continuous;

Fuzzy δ^*-continuous ® Fuzzy δ-continuous.

6. Conclusion

The main purpose of this paper introduces a new concept in fuzzy set theory, namely that of a fuzzy δ^*-continuity and fuzzy δ^**-continuity. On the other hand, fuzzy topology on a fuzzy set is a kind of abstract theory of mathematics. First, we present and study fuzzy δ^*-continuity and fuzzy δ^**-continuity from a fuzzy topological space on a fuzzy set into another. Then, we present the relationships between types of fuzzy continuous functions.

7. Acknowledgements

The author is thankful to the referee for his valuable suggestions.

References

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3. A. M. Zahran, “Fuzzy δ-Continuous, Fuzzy almost Regularity (Normality) on Fuzzy Topology No Fuzzy Sets,” Fuzzy Mathematics, Vol. 3, No. 1, 1995, pp. 89-96.

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