A Unified Theory (I) for Neighborhood Systems and Basic Concepts on Fuzzifying Topological Spaces ()
1. Introduction
In the last few years fuzzy topology, as an important research field in fuzzy set theory, has been developed into a quite mature discipline [1-6]. In contrast to classical topology, fuzzy topology is endowed with richer structure, to a certain extent, which is manifested with different ways to generalize certain classical concepts. So far, according to Ref. [2], the kind of topologies defined by Chang [7] and Goguen [8] is called the topologies of fuzzy subsets, and further is naturally called L-topological spaces if a lattice L of membership values has been chosen. Loosely speaking, a topology of fuzzy subsets (resp. an L-topological space) is a family 
of fuzzy subsets (resp. L-fuzzy subsets) of nonempty set X, and 
satisfies the basic conditions of classical topologies [9]. On the other hand, Höhle in [10] proposed the terminology L-fuzzy topology to be an L-valued mapping on the traditional powerset 
of X. The authors in [4,5,11,12] defined an L-fuzzy topology to be an L-valued mapping on the L-powerset LX of X. In 1952, Rosser and Turquette [13] proposed emphatically the following problem: If there are many-valued theories beyond the level of predicates calculus, then what are the detail of such theories? As an attempt to give a partial answer to this problem in the case of point set topology, Ying in 1991 [14,15] used a semantical method of continuousvalued logic to develop systematically fuzzifying topology. Briefly speaking, a fuzzifying topology on a set X assigns each crisp subset of X to a certain degree of being open, other than being definitely open or not. In factfuzzifying topologies are a special case of the L-fuzzy topologies in [11,12] since all the t-norms on I are included as a special class of tensor products in these paper. Ying uses one particular tensor product, namely Łukasiewicz conjunction. Thus his fuzzifying topologies are a special class of all the I-fuzzy topologies considered in the categorical frameworks [11,12]. Roughly speaking, the semantical analysis approach transforms formal statements of interest, which are usually expressed as implication formulas in logical language, into some inequalities in the truth value set by truth valuation rules, and then these inequalities are demonstrated in an algebraic way and the semantic validity of conclusions is thus established. So far, there has been significant research on fuzzifying topologies [16-21]. In 1979, several characterizations of compactness are unified by the operation introduced by Kasahara [22]. Also, he studied the concept of 
-continuity (where 
is an operation) and defined some types of spaces by using this operation. In 1981, the concept of other type of continuity which generalizes the 
-continuity [22] was introduced by Jankoviĉ [23]. In 1983, Abd El-Monsef, et al. [24] introduced an operation 
on the family 
of all closed sets in the topological space 
which is dual to the operation 
. In 1991, Kerre et al. [25] introduced an extension of the concept of an operation on the class of all fuzzy sets on X endowed with Chang fuzzy topology [7]. It was shown that a lot of characterizations and properties of many concepts and stronger forms can be unified by using this notion. In 1991, Kandil et al. [26] applied the concept of the operation defined in [25] to unify and generalize several characterizations and properties of a lot of already existing weaker and stronger forms of fuzzy continuity. A basic structure of this paper is as follows: First, in Section 2 we offer some definition and results which will be needed in this paper. In Section 3 the concepts of fuzzifying ∆-open sets, C∆-open sets, ∆-closed sets and C∆-closed sets are introduced and some of their properties are discussed. In Section 4 the fuzzifying ∆- and C∆-neighborhood systems are presented and a fuzzifying topology induced by C∆- neighborhood system is introduced. In Section 5 the concepts of fuzzifying ∆- and C∆-derived sets, ∆- and C∆-closure operations and ∆- and C∆-interior operations were established and some of their properties are studied. Finally, in Section 6, we summarize the main results obtained and raise some related problems for further study. Thus we fill a gap in the existing literature on fuzzifying topology.
Note: All corollaries in this paper are results in [14- 21].
2. Preliminaries
We present the fuzzy logical and corresponding set theoretical notations [14,15] since we need them in this paper.
For any formula
, the symbol 
means the truth value of
, where the set of truth values is the unit interval [0, 1]. We write 
if 
for any interpretation. Also, 
is the family of all fuzzy sets in X. The truth valuation rules for primary fuzzy logical formulae and corresponding set theoretical notations are:
1) a)
;
b)
;
c)
.
2) If 
3) If X is the universe of discourse, then
.
In addition the truth valuation rules for derived formulae are:
1)
;
2)
;
3) 
4)
;
5)
;
6) If
, then
a) 
b)
;
c)
.
We give now the following definitions and results in fuzzifying topology [14-21] which are used in the sequel.
Definition 2.1 [14]. Let X be a universe of discourse, and 
satisfy the following conditions:
1)
;
2) for any 
3) for any
Then 
is a fuzzifying topology and 
is a fuzzifying topological space.
Note: In the rest of this paper 
(or briefly X) is always fuzzifying topological space.
Definition 2.2 [14] The family of all fuzzifying closed sets, denoted by 
is defined as 
where 
is the complement of A.
Definition 2.3 [14] The neighborhood system 
of 
is defined as
Definition 2.4 [15] The interior 
or 
of 
is defined as 
Definition 2.5 (Lemma 5.2. [14]). The closure 
or 
of A is defined as
. In Theorem 5.3 [14], Ying proved that the closure
sssis a fuzzifying closure operator since its extension 
, 
where 
is the 
-cut of A and 
satisfies the following Kuratowski closure axioms:
1) 
2) for any 
3) for any
4) for any 
Definition 2.6 [18] For any
,
.
Theorem 2.1 [18] For any 
1) 
2) 
3) 
4) 
Theorem 2.2 [18] For any
, if 
then
1) 
2) 
3) 
4) 
5) 
6) 
Theorem 2.3 For any
1) 
[19];
2) 
[18];
3) 
[18];
4) 
[16].
Theorem 2.4
1) 
2) 
3) 
4) 
Theorem 2.5 For any
1) 
[19];
2) 
3) 
4) 
5) 
Theorem 2.6 For any
1) 
2) 
[17];
3) 
[16].
Definition 2.7 Let X be a non-empty set.
1) By the symbol 
we denote the set of all functions from 
into
. Each member of 
will be called a general fuzzifying operation.
2) Let
.
a) We say that
, if 
for each 
b) We say that 
and 
are dual if
equivalently
for each 
3) A general fuzzifying operation 
is said to be monotone if 
4) A general fuzzifying operation 
is said to be of type 
if
; equivalently
, where 
and 
are dual.
5) A general fuzzifying operation 
is said to be of type 
if 
for any 
Example 2.1
1) From Theorem 2.4 we have 
, 
and 
are of type 
and each member of them is monotone from Theorem 2.2.
2) The fuzzifying operations
, 
and 
and the fuzzifying operations
,
and 
are dual respectively (see Theorem 2.3).
3) From Theorem 2.1 (3), 
is of type
.
4) From Theorem 2.5, one can easily deduce that:
a) 
b) 
c) 
d) 
e) 
f) 
g) 
Note: In the rest of this paper always
.
3. Fuzzifying Open Sets
In this section the concepts of fuzzifying ∆-open sets, C∆-open sets, ∆-closed sets and C∆-closed sets are introduced and some of their properties are discussed.
Definition 3.1 1) The family of all fuzzifying ∆-open sets, denoted by 
is defined as follows:
i.e.,
2) The family of all fuzzifying C∆-open sets, denoted by 
is defined as follows:
i.e.,
3) The family of all fuzzifying ∆ (resp. C∆-closed sets, denoted by 
(resp.
) 
is defined as follows:
Definition 3.2 1) If 
(resp.
, 
, the notion of fuzzifying ∆-open sets coincides with the notion of fuzzifying open (resp. 
- open, semi-open, pre-open, 
-open, 
-open) sets and will be denoted by 
(resp.
);
2) If 
(resp.
, the notion of fuzzifying C∆-open sets coincides with the notion of fuzzifying 
(resp. csemi, cpre, 
,
)- open sets and will be denoted by 
(resp.
,
);
3) If 
(resp.
, the notion of fuzzifying 
-closed sets coincides with the notion of fuzzifying closed (resp. 
-closed, semiclosed, pre-closed, 
-closed, 
-closed) sets and will be denoted by 
(resp.
);
4) If 
(resp.
, the notion of fuzzifying C∆-closed sets coincides with the notion of fuzzifying 
(resp. csemi, cpre, 
,
)-closed sets and will be denoted by 
(resp.
);
Theorem 3.1 1) If ∆ is of type O1, then
a) 
b) 
2) If ∆ is monotone, then a) for any 
b) for any 
Proof. 1) a) 
b) 
2) a) Since 
is monotone, then
b) For any 
(see Lemma 1.1 (1) [14]). Since ∆ is monotone, then from Theorem 2.1 (3) we have
Corollary 3.1 1) a) 
b) for any 
2) a) 
b) for any 
Theorem 3.2 1) If ∆ is of type O1, then
a) 
b) 
2) If ∆ is monotone, then a) for any 
b) for any 
Proof. It is immediate from Theorem 3.1.
Corollary 3.2 1) a) 
b) 
2) a) for any 
b) for any 
Theorem 3.3 1) 
2)
, where 
is the dual of 
Proof.
1) 
2) 
Corollary 3.3
1) 
2) 
Theorem 3.4 Let 
1) If
, a) 
b) 
c) 
d) 
2) a) 
b) 
Proof.
1) a) 
b) From a) above, we have
c) 
d) From (c) above, we have 
2) a) 
b) From a) above, we have
Corollary 3.4 1) a) i) 
ii) 
iii) 
iv) 
(v) 
(vi) 
b) i) 
ii) 
iii) 
iv) 
v) 
vi) 
c) i) 
ii) 
iii) 
iv) 
v) 
d) i) 
ii) 
iii) 
iv) 
v) 
2) a) 
b) 
Corollary 3.5
1) a) 
b) 
c) 
d) 
2) a) 
b) 
c) 
d) 
Theorem 3.5 Let 
for each 
1) 
2) 
Proof. 1) Since 
and 
then 
Thus
2) Since for every 
then one can deduce that 
for every 
Also, since 
one can have that 
From Theorem 3.3 (2) we have
Corollary 3.6 For any
1) 
2) 
Remark 3.1 In crisp setting, i.e., if the underlying fuzzifying topology is the ordinary topology, one can have
Of course the implication “®” in (*) is either the Łukaciewicz’s implication or the Boolean’s implication since these implications are identical in crisp setting. But in fuzzifying setting the statement (*) may not be true as illustrated by the following example.
Example 3.1 Let 
and 
be a fuzzifying topology on X defined as follows:'
;
and 
From the definitions of the interior and the closure of a subset of X and the definitions of the interior and the closure of a fuzzy subset of X and some calculations we have:
and
So,
Theorem 3.6 Let 
for each 
Then
1) 
2) 
Proof. 1) It is obtained from Theorem 2.4 (1)(a) and (2)(a).
2) 
Corollary 3.7
1) 
2) 
Theorem 3.7 Let 
for each 
If for every
, 
or 
then
1) 
2) 
Proof. Using Theorem 3.6 it remains to prove the following
1) Suppose that 
Then for each 
we have 
So
Now, suppose that 
Since
, then
For each 
we have 
Thus,
2) The proof is similar to 1).
Corollary 3.8 If for every 
or 
(resp. 
or 
or 
or 
or 
then
1) 
2) 
Theorem 3.8 Let 
be a monotone. Then
Proof.
First, we have 
On the other hand, Suppose that 
Then for any 
we have 
Since 
is monotone, then by Theorem 3.1 (2)(a) we have
By completely distributive law we have
Corollary 3.9
Remark 3.2 The following are valid in crisp setting:
1) 
2) 
3) 
but in fuzzifying setting these statement may not be true by the following example.
Example 3.2 From Example 3.1 we have
1) 
2) 
3) 
4. Fuzzifying Neighborhood Structure of a Point
In this section the concepts of ∆-neighborhood system and C∆-neighborhood system of a point are presented and a fuzzifying topology induced by C∆-neighborhood system is obtained.
Definition 4.1 The fuzzifying ∆-(resp. C∆-) neighborhood system of
, denoted by 
(resp.
)
, is defined as follows:
i.e.,
.
Definition 4.2 Let
.
1) If 
(resp.
, the fuzzifying ∆-neighborhood system of 
coincides with the fuzzifying (resp. fuzzifying 
-, fuzzifying semi-, fuzzifying pre-, fuzzifying 
-, fuzzifying 
-) neighborhood system of 
and will be denoted by 
(resp.
);
2) If 
(resp.
, the fuzzifying C∆-neighborhood system of 
coincides with the fuzzifying 
- (resp. fuzzifying csemi-, fuzzifying cpre-, fuzzifying 
-, fuzzifying 
-) neighborhood system of 
and will be denoted by 
(resp.
).
Theorem 4.1 Let ∆ be a monotone.
1) 
2) 
Proof. Using Theorem 3.8 we have
2) From 1) the proof is immediate.
Corollary 4.1 Let 
be a fuzzifying topological space.
1) 
1) 
Theorem 4.2 The mapping
, 
, has the following properties:
1) If 
is of type
, then 
is normal for any
;
2) For any 
3) For any 
4) If 
is monotone, then for any 
5) If 
is of type
, then
Proof. 1) Since 
is of type
, then
2) If
, then the results holds. Suppose
. Then there exists 
such that
. Now, we have 
Thus 
holds always.
3) If
, then the result holds. Now, suppose that 
Then we have
.
4) Since 
is monotone, then from Theorem 4.1 a) we have
5) 
Corollary 4.2 The mapping N
has the following properties:
1) For any
, 
is normal;
2) For any 
3) For any 
4) For any 
5) For any 
Theorem 4.3 The mapping 
, has the following properties:
1) If 
is of type
, then 
is normal for any
;
2) For any 
3) For any 
4) If 
is monotone, then or any 
Conversely, if a mapping 
satisfies 1), 3) and 4), then it assigns a fuzzifying topology on X, denoted by
, is defined as follows:
Proof. Since 
is normal and satisfies properties (2) and (3) in Theorem 4.2, then 
is a fuzzifying topology on
. The rest of the proof is similar to the proof of Theorem 4.2.
Corollary 4.3 The mapping 
has the following properties:
1) 
is normal for any
;
2) For any 
3) For any 
4) If 
is monotone, then or any 
5) The mapping 
assigns a fuzzifying topology on X, denoted by 
, is defined as follows:
Theorem 4.4 If ∆ is of type 
and monotone, then 
Proof. Let 
Then
Corollary 4.4 1) 
2) 
3) 
4) 
(5) 
5. Closure and Interior Operations in Fuzzifying Topology
The purpose of this section is to establish the concepts of fuzzifying ∆- and C∆-derived sets, fuzzifying ∆- and C∆-closure operation and fuzzifying ∆- and C∆-interior operation and study some of their properties.
Definition 5.1 The fuzzifying ∆- (resp. C∆-) derived set 
(resp.
) 
of 
is defined as follows:
i.e.,
.
Definition 5.2 For
1) If 
(resp.
, the notion of fuzzifying ∆-derived set of 
coincides with the notion of fuzzifying derived (resp. 
- derived, semi-derived, pre-derived, 
-derived, 
-derived) set and will be denoted by
;
2) If 
(resp.
, the notion of fuzzifying c∆-derived set of 
coincides with the notion of fuzzifying derived (resp. 
-derived, csemi-derived, cpre-derived, 
-derived, 
-derived) set and will be denoted by
.
Theorem 5.1 For every 
we have
1) 
2) 
Proof. 1) Using Theorem 4.3 3) we have
2) It is similar to the proof. of 1).
Corollary 5.1
1) a) 
b) 
c) 
d) 
e) 
f) 
2) a) 
b) 
c) 
d) 
e) 
Theorem 5.2 For every 
we have
1) If ∆ is monotone, then 
2) If ∆ is of type 
and monotone, then
.
Proof. Using Theorem 4.1 (2) we have 1) 
2) It is similar to 1).
Corollary 5.2
1) a) 
b) 
c) 
d) 
e) 
f) 
2) a) 
b) 
c) 
d) 
e) 
Definition 5.3 The fuzzifying ∆-(resp. C∆-) closure 
(resp.
) 
of 
is defined as follows:
i.e.,
.
Definition 5.4 1) If 
(resp.
, the notion of fuzzifying ∆-closure of 
coincides with the notion of fuzzifying closure (resp. 
-closure, semi-closure, pre-closure, 
-closure, 
-closure) operation and will be denoted by
;
2) If 
(resp.
, the notion of fuzzifying C∆-closure of 
coincides with the notion of fuzzifying 
(resp. csemi, cpre, 
,
)-closure operation and will be denoted by
.
Theorem 5.3 For every 
we have
1) 
2) 
Proof.
1) 
2) It is similar to the proof of (1).
Corollary 5.3
1) a) 
b) 
c) 
d) 
e) 
f) 
2) a) 
b) 
c) 
d) 
e) 
Theorem 5.4 For every 
and 
we have
1) If ∆ is of type O1, then a) 
b) 
2) a) 
b) 
3) a) 
b) 
4) a) 
b) 
5) a) 
b) If ∆ is of type O1 and monotone, then
Proof. We prove only a) of each statements since b) is similar.
1) a) 
2) a) It is clear that 
for any 
and 
in case of 
Now, suppose that 
Then 
Therefore
3) a) Using Theorem 5.1 (1) and (2) above, we have
4) a) 
5) a) Since 
then from Theorem 5.2 (1) and (3) (a) above we have
because 
for any 
Corollary 5.4
1) a)
; b)
; c)
; d)
; e)
; f)
; g)
; h)
; i)
; j)
; k)
;
2) for any
, a)
; b)
; c)
; d)
; e)
; f)
; g)
; h)
; i)
; j)
; k)
.
3) for any
, a)
; b) 
; c)
; d)
; e)
; f)
; g)
; h)
; i)
; j)
; k)
.
4) for any 
and
a)
;
b)
;
c)
;
d)
;
e)
;
f)
;
g)
;
h)
;
i)
;
j)
;
k)
.
5) for any
a)
;
b)
;
c)
;
d)
;
e)
;
f)
;
g)
;
h)
;
i)
;
j)
;
k)
.
Theorem 5.5 For every 
we have
1) 
2) 
Proof. 1) Suppose 
Then 
and the results holds. Now, suppose 
Then 
and 
So
Let 
Then 
For any 
, i.e., there exists 
such that 
and
. Now we want to prove
. If not, then there exist 
with 
Hence
and this is a contradiction. Therefore
Since t is arbitrary, it holds that 
2) The proof is similar to 1).
Corollary 5.5 For any A and B.
1) a)
;
b)
;
c)
;
d)
;
e)
;
e)
;
2) a)
;
b)
;
c)
;
d)
;
e)
.
Definition 5.5 The fuzzifying ∆-(resp. C∆-) interior 
(resp.
) 
of 
is defined as follows:
.
Definition 5.6 1) If 
(resp.
, the notion of fuzzifying 
-interior of 
coincides with the notion of fuzzifying interior (resp. 
-interior, semi-interior, pre-interior, 
-interior, 
-interior) operation and will be denoted by
;
2) If 
(resp.
, the notion of fuzzifying 
-interior of 
coincides with the notion of fuzzifying 
(resp. csemi, cpre, 
,
)-interior operation and will be denoted by
.
Theorem 5.6 For every 
we have
1) a) 
b) if 
is of type O1 and monotone, then
2) a) 
b) if 
is of type O1 and monotone, then
3) a) 
b) if 
is of type O1 and monotone, then
4) a) 
b) 
5) a) 
b) 
Proof. We prove only a) of each statements since b) is similar.
First, we prove 2) a) If 
then 
Now, suppose that 
Then we have
3) a) 
4) a) If 
If 
Thus 
5) a) Follows from Theorem 5.3 (1).
Finally, we prove 1) a). From 5) a) and Theorem 5.5 1) we have
Corollary 5.6 For any 
1) a)
;
b)
;
c)
;
d)
;
e)
;
f)
;
g)
;
h)
;
i)
;
j)
;
k)
;
2) a)
;
b)
;
c)
;
d)
;
e)
;
f)
;
g)
;
h)
;
i)
;
j)
;
k)
;
3) a)
;
b)
;
c)
;
d)
;
e)
;
f)
;
g)
;
h)
;
i)
;
j)
;
k)
;
4) a)
;
b)
;
c)
;
d)
;
e)
;
f)
;
g)
;
h)
;
i)
;
j)
;
k)
;
5) a)
;
b)
;
c)
;
d)
;
e)
;
f)
;
g)
;
h)
;
i)
;
j)
;
k)
.
6. Conclusions
The present paper investigates topological notions when these are planted into the framework of Ying’s fuzzifying topological spaces (in semantic method of continuous valued-logic). It continue various investigations into fuzzy topology in a legitimate way and extend some fundamental results in general topology to fuzzifying topology. An important virtue of our approach (in which we follow Ying) is that we define topological notions as fuzzy predicates (by formulae of Łukasiewicz fuzzy logic) and prove the validity of fuzzy implications (or equivalences). Unlike the (more wide-spread) style of defining notions in fuzzy mathematics as crisp predicates of fuzzy sets, fuzzy predicates of fuzzy sets provide a more genuine fuzzification; furthermore the theorems in the form of valid fuzzy implications are more general than the corresponding theorems on crisp predicates of fuzzy sets. The main contributions of the paper are to study some sorts of operations, called general fuzzifying operations. There are some problems for further study:
1) Apply the general fuzzifying operation to convergence theory, continuity, separation axioms etc.
2) What is the justification of these concepts in the setting of (2, L) topologies.