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.