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
![](https://www.scirp.org/html/18-5300308\6959ea48-0cf9-4ee7-b9e7-7032855f6c9a.jpg)
is called a fuzzy point with support x and value r.
is often denoted by
.
For a fuzzy point ![](https://www.scirp.org/html/18-5300308\ee15519d-3c13-4c35-988b-0154b47688c0.jpg)
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 ![](https://www.scirp.org/html/18-5300308\54db25d1-fb32-440f-8d11-8ca5a46be419.jpg)
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
![](https://www.scirp.org/html/18-5300308\fb6de6c6-874f-46ec-b219-cd74ac9cc7ad.jpg)
for each
.
Definition 3.3. [2] If
, then
is defined by
![](https://www.scirp.org/html/18-5300308\87f74a2a-dc4f-47e0-b73d-2dee8d7a7327.jpg)
for each![](https://www.scirp.org/html/18-5300308\51e9b058-77ac-42c0-af26-458b62d0123e.jpg)
Proposition 3.4. [2] For a proper function ![](https://www.scirp.org/html/18-5300308\3fbbf97c-46d7-409e-aa97-91581ad42e26.jpg)
1)
, for each
.
2)
, for each
.
3)
and
![](https://www.scirp.org/html/18-5300308\7ca3af6a-050c-4e54-b1db-e86cfa3430a1.jpg)
Definition 3.5. [2]
is said to maximal if for each ![](https://www.scirp.org/html/18-5300308\09f76903-7bbf-4ba3-9e0c-3eda81c1b043.jpg)
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
, ![](https://www.scirp.org/html/18-5300308\abe7a4ff-0f48-40d0-bf76-fa1272826913.jpg)
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 ![](https://www.scirp.org/html/18-5300308\efbfc336-d5d9-4753-b2c0-6765fce64a39.jpg)
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 ![](https://www.scirp.org/html/18-5300308\bbb81f8b-77a9-435e-8409-f9009dbefcfa.jpg)
. The set of all fuzzy
-cluster
(resp. fuzzy θ-cluster) points of
is called fuzzy
- cluster (resp. fuzzy θ-closure) and is denoted by
![](https://www.scirp.org/html/18-5300308\70c94dc1-a86d-45f9-b960-6b840dd42781.jpg)
. 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
![](https://www.scirp.org/html/18-5300308\cc91c361-5867-49dd-ae76-7190681591b2.jpg)
![](https://www.scirp.org/html/18-5300308\96ee642d-c439-4620-afd9-86097f42220f.jpg)
.
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
![](https://www.scirp.org/html/18-5300308\04923113-489c-4693-9070-d4f7cf16b6db.jpg)
![](https://www.scirp.org/html/18-5300308\6cc37939-19a8-4313-9170-32a6acefcbb9.jpg)
![](https://www.scirp.org/html/18-5300308\4dfb4990-273c-4cf6-970f-04a62f4e2f15.jpg)
![](https://www.scirp.org/html/18-5300308\7d1c88fc-14d1-43aa-931e-aaabe1704041.jpg)
and
,
.
Consider the fuzzy topologies on
,
resp.
and
. Let the proper function
defined by
,
![](https://www.scirp.org/html/18-5300308\46aa5d69-0ec1-435c-8e37-7da7c58b4ae4.jpg)
![](https://www.scirp.org/html/18-5300308\489075f7-0445-4ac2-9ee5-ed1357777eac.jpg)
one may notice that the only fuzzy open sets in ![](https://www.scirp.org/html/18-5300308\85414771-7246-4fca-a96c-303c84739da7.jpg)
are
,
and
but
,
,
and
,
,
. Hence
is fuzzy δ*- continuous.
Theorem 4.3. If
be fuzzy δ*- continuous and
, then
![](https://www.scirp.org/html/18-5300308\6b79f621-e17c-44d0-9b02-b14938425cfd.jpg)
is fuzzy δ*-continuous .
Proof: Let
such that![](https://www.scirp.org/html/18-5300308\b622795c-07aa-4146-beae-fada9c9066b5.jpg)
.
Then there exists fuzzy open
such that
.![](https://www.scirp.org/html/18-5300308\816874b5-6dfb-44c4-8a28-69ce642b8c77.jpg)
Now
![](https://www.scirp.org/html/18-5300308\e52af311-1836-469e-a2de-f0d120ba8f88.jpg)
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
![](https://www.scirp.org/html/18-5300308\378abbf7-4aec-4649-a120-e5b45998a91c.jpg)
![](https://www.scirp.org/html/18-5300308\140812b6-1a5a-4297-a33b-57b9738c2836.jpg)
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
![](https://www.scirp.org/html/18-5300308\3a8e7fc2-87aa-4129-be34-2e424883ea07.jpg)
![](https://www.scirp.org/html/18-5300308\54f36d86-5cdb-416d-99f6-8acd61d168a1.jpg)
and
![](https://www.scirp.org/html/18-5300308\5a39a7c5-040b-40fd-99e5-027f4afe5594.jpg)
Consider the fuzzy topologies on
and
resp.
and
. Let the proper function
defined by![](https://www.scirp.org/html/18-5300308\107dfd96-bdd7-4886-ac5a-1183d35d4584.jpg)
![](https://www.scirp.org/html/18-5300308\1c3f1724-d636-46af-bb29-0f5190341f3d.jpg)
![](https://www.scirp.org/html/18-5300308\386c97c3-c25a-4954-9885-c4fe203cd33b.jpg)
. One may notice that the only fuzzy δ-open sets in
are
,
and
and
![](https://www.scirp.org/html/18-5300308\9d202ffb-eb04-437a-9007-203eb3e27dc4.jpg)
![](https://www.scirp.org/html/18-5300308\e00a3fdb-4e62-4410-a53f-bc61d54f01ec.jpg)
![](https://www.scirp.org/html/18-5300308\c8b1906e-fe8b-4475-ba6c-2f2930fe41f7.jpg)
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
![](https://www.scirp.org/html/18-5300308\835a1796-710c-496d-b9f6-99e3ae9f2573.jpg)
![](https://www.scirp.org/html/18-5300308\3f6b9114-3f57-4ad6-9cea-f335b23140d5.jpg)
and
![](https://www.scirp.org/html/18-5300308\e9b4227c-b683-4525-afa1-278c024114bd.jpg)
Consider the fuzzy topologies on
and
resp.
and
. Let the proper function
defined by
, ![](https://www.scirp.org/html/18-5300308\bf71bad1-1562-4886-8ed3-7e73b41030ea.jpg)
![](https://www.scirp.org/html/18-5300308\d486e7c9-fd6d-4a59-bb90-5b7a61245ded.jpg)
![](https://www.scirp.org/html/18-5300308\3a8e7d41-064b-40a6-8435-5461927dadde.jpg)
is fuzzy δ**-continuous but not fuzzy δ*-continuous such that the only fuzzy δ-open sets in
are
,
and ![](https://www.scirp.org/html/18-5300308\5261e2d2-298e-4980-8ef2-803d407bb417.jpg)
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.