**Applied Mathematics** Vol.3 No.9(2012), Article ID:22998,14 pages DOI:10.4236/am.2012.39146

A Unified Theory (I) for Neighborhood Systems and Basic Concepts on Fuzzifying Topological Spaces

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt

Email: o_r_sayed@yahoo.com

Received September 10, 2011; revised August 7, 2012; accepted August 14, 2012

**Keywords:** Łukasiewicz Logic; Semantics; Fuzzifying Topology

ABSTRACT

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [25]. It investigates topological notions defined by means of -open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0, 1]). In this paper we introduce some sorts of operations, called general fuzzifying operations from to , where is a fuzzifying topological space. By making use of them we contract neighborhood structures, derived sets, closure operations and interior operations.

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 L^{X} 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 ;

c);

d);

e);

f);

g);

h);

i);

j);

k).

5) for anya);

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.

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