Fuzzy Sets, Fuzzy S-Open and S-Closed Mappings

Abstract

Several properties of fuzzy semiclosure and fuzzy semi-interior of fuzzy sets defined by Yalvac (1988), have been established and supported by counterexamples. We also study the characterizations and properties of fuzzy semi-open and fuzzy semi-closed sets. Moreover, we define fuzzy s-open and fuzzy s-closed mappings and give some interesting characterizations.

1. Introduction

The concept of fuzzy set was introduced by Zadeh in his classical paper [1]. This concept provides a natural foundation for treating mathematically the fuzzy phenomena, which exist pervasively in our real world, and for building new branches of fuzzy mathematics. In the area of Fuzzy Topology, introduced by Chang [2], much attention has been paid to generalize the basic concepts of General Topology in fuzzy setting and thus a modern theory of Fuzzy Topology has been developed.

In recent years, Fuzzy Topology has been found to be very useful in solving many practical problems. Du et al. [3] fuzzified the very successful 9-intersection Egenhofer model [4, 5] for depicting topological relations in Geographic Information Systems (GIS) query. In [6, 7], El Naschie showed that the notion of Fuzzy Topology might be relevant to quantum particle physics and quantum gravity in connection with string theory and theory. Tang [8] used a slightly changed version of Chang’s fuzzy topological space to model spatial objects for GIS databases and Structured Query Language (SQL) for GIS.

Levine [9] introduced the concepts of semi-open sets and semicontinuous mappings in topological spaces. Interestingly, his work found applications in the field of Digital Topology [10]. For example, it was found that the digital line is a -space [11], which is a weaker separation axiom based upon semi-open sets. Fuzzy Digital Topology [12] was introduced by A. Rosenfeld, which demonstrated the need for the fuzzification of weaker forms of notions of Classical Topology. Azad [13] carried out this fuzzification in 1981, and presented some general properties of fuzzy spaces. Several properties of fuzzy semi-open (resp., fuzzy semi-closed), fuzzy regular open (resp., closed) sets have been discussed. Moreover he defined fuzzy semicontinuous (resp., semi-open and semi-closed) functions and studied the properties of fuzzy semicontinuous function in product related spaces. Finally, he defined and characterized fuzzy almost continuous mappings. For related subsequent work in this direction, we refer to [14–27].

In this paper, our aim is to further contribute to the study of fuzzy semi-open and fuzzy semi-closed sets defined by Yalvac [26] by establishing several important fundamental identities and inequalities supported by counterexamples. Cameron and Woods [28] introduced the concepts of s-continuous mappings and s-open mappings. They investigated the properties of these mappings and their relationships to properties of semi-open sets. Khan and Ahmad [29] further worked on the characterizations and properties of s-continuous, s-open and s-closed mappings. We fuzzify the findings of [28, 29]. We define fuzzy s-open and fuzzy s-closed mappings and establish some interesting characterizations of these mappings.

2. Preliminaries

In order to make this paper self-contained, we briefly recall certain definitions and results; for those not described, we refer to [1, 2, 13, 26].

Let be a space of points (objects), with a generic element . A fuzzy set in is characterized by membership function from to the unit interval .

The symbol denotes the empty fuzzy set defined as for all For , the membership function is defined as , for all .

Definition 1 (see [2]). Let be a mapping. Let be a fuzzy set in with membership function . Then the inverse of , written as , is a fuzzy set in whose membership function is defined by Conversely, let be a fuzzy set in with membership function . The image of , written as , is a fuzzy set in whose membership function is given by for all .

Definition 2 (see [2]). A fuzzy topology is a family of fuzzy sets in , which satisfies the following conditions:(1)(2)If , then (3)If for each , then .

is called a fuzzy topology for , and the pair is an fts. Every member of is called -open fuzzy set (or simply an open fuzzy set). A fuzzy set is -closed if and only if its complement is -open.

As in general topology, the indiscrete fuzzy topology contains only and , while the discrete fuzzy topology contains all fuzzy sets.

3. Fuzzy Semi-Open and Fuzzy Semi-Closed Sets

Definition 3 (see [13]). Let be a fuzzy set in an fts . Then is called a fuzzy semi-open set of , if there exists a such that . A fuzzy set is fuzzy semi-closed if and only if its complement is fuzzy semi-open. The class of all fuzzy semi-open (resp., fuzzy semi-closed) sets in is denoted by (resp., ).

Definition 4 (see [26]). Let be a fuzzy set in an fts . Then semi-closure (briefly ) and semi-interior (briefly ) of are defined as and are called the fuzzy semi-closure of and fuzzy semi-interior of respectively.

It is immediate that

It is known [13] that a fuzzy set in an fts is

The following are characterizations of fuzzy semi-closed sets, the proof of Theorem 1 is straightforward.

Theorem 1. A fuzzy set is fuzzy semi-closed if and only if there exists a fuzzy closed set such that

Theorem 2. For a fuzzy set in an fts , is fuzzy semi-closed if and only if .

Proof. If , then .
Suppose . Since is fuzzy semi-closed, so there exists a closed set such that . Thus or . Hence is fuzzy semi-closed and .

Remark 1. It is easily seen that(1)if is fuzzy semi-open (resp., fuzzy semi-closed), then , and are fuzzy semi-open (resp., fuzzy semi-closed);(2)a nonvoid nowhere dense fuzzy set is fuzzy semi-closed and not fuzzy semi-open.

The converse of Remark 1(2) is, in general, not true as is shown by following.

Example 1. Let be a set and be the lattice of membership grades for fuzzy sets in . Let , and be fuzzy sets on and the fuzzy topology generated by , and Then , Calculations give that fuzzy set is both fuzzy semi-closed and fuzzy semi-open but

Theorem 3. For fuzzy sets and in an fts , one has (1)(2)(3)(4)

Proof. (1) and are both fuzzy semi-open . imply and . Combining, or
(2) and imply Conversely and imply and is fuzzy semi-open. But is the largest fuzzy semi-open set such that hence This gives the equality.
and follow easily from

The inequalities and of Theorem 3, are in general irreversible, as is shown by following.

Example 2. Let be a set and be the lattice of membership grades for fuzzy sets on . Let , and be fuzzy sets on and the fuzzy topology generated by , , and Then We choose fuzzy sets and Then calculations give that

It is known [30] that for a fuzzy set in an fts we have

We use this and Theorem 1, and prove the following.

Theorem 4. Let be a fuzzy set in an fts . Then one has the following: (1)(2)(3)(4)

Proof. By the fact that is fuzzy semi-closed and that is fuzzy semi-closed if and only if , it follows immediately.
Since , , so that , since all fuzzy open sets are fuzzy semi-open.
By , , so that . Also , so that and Consequently, we get .
Now so by Theorem 1, is fuzzy semi-closed. Also . Since is the smallest fuzzy semi-closed set with , therefore, . Which implies or .

The inequalites and of Theorem 4 may, in general, not be reversible as is shown in the following.

Example 3. Let be a set and be the lattice of membership grades for fuzzy sets on . Let , , and be fuzzy sets on and the fuzzy topology generated by , , and Then , .
We choose fuzzy sets and , then calculations give that

Theorem 5. For any fuzzy set in an fts , we have (1),(2).

Proof. ()
Similar to .

4. Fuzzy S-Open and Fuzzy S-Closed Mappings

First, we define

Definition 5. A function is said to be fuzzy s-open (resp., fuzzy s-closed) if the image of every fuzzy semi-open (resp., fuzzy semi-closed) set is fuzzy open (resp., fuzzy closed).

Obviously a fuzzy s-open function is fuzzy open.

Definition 6 (see [31]). A fuzzy point is called a boundary point of a fuzzy set if and only if . The union of all the boundary points of is called a boundary of , denoted by . It is clear that

Next, we define

Definition 7. Semiboundary (briefly sBd) of a fuzzy set in an fts is defined as

In the following, we characterize fuzzy s-open mappings in terms of sInt, sCl, and sBd.

Theorem 6. For a function , a fuzzy set in an fts and a fuzzy set in an fts then the following are equivalent: (1) is fuzzy s-open;(2)(3)(4)(5)

Proof. obviously . is fuzzy s-open gives that is fuzzy open in . But is the largest fuzzy open set such that . Therefore for any fuzzy set in . This gives .
For any fuzzy set in , is a fuzzy set in . Then by or or or . This gives .
By we have or or or where , a fuzzy set in . This gives .
For a fuzzy set in , is a fuzzy closed set in . Now . Using we have or This gives .

In the following, we give characterizations of fuzzy s-closed mappings as follows.

Theorem 7. A function is fuzzy s-closed if and only if , for each fuzzy set in an fts .

Proof. Obviously . is fuzzy closed, since is fuzzy s-closed. But is the smallest fuzzy closed set with . Therefore .
Let . We show that is fuzzy closed. By hypothesis, or . This proves that is fuzzy closed.

Theorem 8. If a function is fuzzy s-closed then for each fuzzy set in an fts and each fuzzy semi-open set in an fts with , there exists a fuzzy open set in with such that .

Proof. Let be an arbitrary fuzzy semi-open set in with , where is a fuzzy set in . Clearly (say) is fuzzy open in . Since then straightforward calculations give that . Moreover, we have

Theorem 9. Let be a surjective function from an fts to an fts If for each fuzzy set in and each fuzzy semi-open set in with , there exists a fuzzy open set in with such that , then is s-closed.

Proof. Let be an arbitrary fuzzy semi-closed set in and . Then or . Since is fuzzy semi-open, therefore there exists a fuzzy open set with such that . Since is surjective, we have . Thus is fuzzy open in or is fuzzy closed in . This proves that is s-closed.

Combining Theorems 8 and 9, we have the following.

Theorem 10. A surjective function is fuzzy s-closed if and only if for each fuzzy set in and each fuzzy semi-open set in with , there exists a fuzzy open set in with such that

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Copyright

Copyright © 2009 B. Ahmad and Athar Kharal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.