ABSTRACT

The study of fuzzy sets is specifically designed to mathematically represent uncertainty and vagueness by assigning values of membership to objects that belong to a particular set. This notion has been broadly extended to other areas of topology where various topological concepts have been shown to hold on fuzzy topology. Some notions naturally extend to closure spaces without requiring a lot of modification of the underlying topological ideas. This work investigates the variants of normality on fuzzy isotone spaces.

1. Introduction

The idea of a class of sets with a continuum of grade of membership, ranging between zero and one, was first introduced by Zadeh in 1965. A larger degree of membership of an object reflects a stronger sense of belonging to a set. If A is a set in the ordinary sense of the term, then its membership takes only two values, 0 and 1. The notions of inclusion, union, intersection, complement, relation and convexity can be extended to such sets [1].

Fuzzy closure spaces were introduced by [5] in an attempt to show that fuzzy topological spaces do not constitute a natural boundary for the validity of theorems and results. The axioms used to define fuzzy closure spaces are the modified Kuratowski closure axioms that have previously been used to extend the study of the concepts of topological spaces. The class of isotonic spaces is defined using only two Kuratowski closure axioms, namely the grounded axiom and the isotone axiom

where is the closure operator on a nonempty set.

2. Literature Review

2.1. Fuzzy Sets

In [2], a fuzzy set in is defined as a function. Here represents the degree of membership of in the fuzzy set.

2.2. Crisp Fuzzy Sets

Any subset of a set can be identified with its characteristic function defined by;

(1)

Such characteristic functions are fuzzy sets in. Thus fuzzy sets generalize ordinary sets [3].

2.3. Definitions on Fuzzy Sets

Let and be -valued functions defined on a fixed set, i.e fuzzy sets on. Then according to [3];

1) implies for every.

2) implies for every.

3) Maximum function:

.

4) Minimum function:

.

5) Complement function:. These are [0,1]-valued functions.

For two fuzzy sets and in;

1) and are equal if and only if.

2) is contained in if and only if.

3) The union of and is.

4) The intersection of and is.

5) The complement of is.

Let be fuzzy sets. Then the union of is defined by;

(2)

The intersection of is defined by;

(3)

If are crisp, i.e. they are characteristic functions, then these suprema and infima are actually maxima and minima.

2.4. Fuzzy Topology

According to [4], a fuzzy topology on a set is a collection of fuzzy sets in satisfying

1), where is equivalent to the empty set.

2) If and belong to, then.

3) If are fuzzy, then. The members of are called open fuzzy sets.

The pair is called a fuzzy topological space. Fuzzy sets of the form, where is fuzzy are called closed fuzzy sets.

2.5. Functions and Fuzzy Continuity

Let and be sets and be a function. For a fuzzy set in, the inverse image of under is the fuzzy set in defined by;

for. That is.

For a fuzzy set in, the image of under is the fuzzy set in defined for by;

(4)

2.6. Fuzzy Continuity

Given fuzzy topological spaces and, then according to [2], is fuzzy continuous if the inverse image under of any open fuzzy set in is an open fuzzy set in, i.e whenever.

The identity mapping on a fuzzy topological space is fuzzy continuous.

2.7. Closure and Interior Operation on Fuzzy Sets

Let be a fuzzy topological space. The closure and interior of a fuzzy set in are defined respectively by [3] as follows;

It is easily seen that is the smallest closed fuzzy set larger than and that is the largest open fuzzy set smaller than. These definitions coincide with their analogous definitions on ordinary sets.

2.8. Fuzzy Closure Spaces

Let be the collection of all mappings from to the unit interval, i.e is the collection of all fuzzy sets on the non-empty set. Then from [5], an operator is a fuzzy closure operator if and only if

1) constant.

2) for every.

3) for every.

4) for every.

The closure operator may also be used to characterize closed sets. A set is closed if. A fuzzy interior operator is the dual of a closure operator. It is defined by;

1) constant.

2) for every.

3) for every.

4) for every.

Similarly, the interior operator may also be used to characterize open sets. A set is open if.

A Cech fuzzy closure operator (or CF-closure operator) on a set is a function satisfying the following three axioms;

1) constant.

2) for every.

3) for every.

The pair is called a fuzzy closure space or fcs. Clearly these axioms can easily be seen to be similar to the Kuratowski axioms in [6].

3. Results

The following are the main results of this work.

3.1. Fuzzy Isotone Space

A fuzzy isotone closure operator on a set is a function satisfying the following two axioms;

1) constant.

2) For every,.

The pair is called a fuzzy isotone space.

3.2. Semi-Separated and Separated Fuzzy Sets

We would like to modify the definitions of semi-separated and separated sets in order to have their equivalent characterization on fuzzy isotone spaces. This will facilitate the definition of complete normality on fuzzy isotone spaces.

In a fuzzy isotonespace, two fuzzy subsets and are called semi-separated if

.

The fuzzy subsets and are separated if there exists open fuzzy sets, V with and such that. The openness of fuzzy sets on is defined using the dual of the closure operator, i.e the interior operator.

Lemma

Let. Then and are semi-separated in if and only if and are semi-separated in, where is the relativization of into.

Proof

Let be a fuzzy isotone space and be semi-separated fuzzy sets in. Then

.

But hence

.

Similarly, hence

Therefore, and are semi-separated in.

Conversely, let be a subspace of the fuzzy isotone space and and are semi-separated in. Of course and

.

.

Similarly,

.

Thus and are semi-separated in.

3.3. Normality

A fuzzy isotone space is normal if for every nonempty pair of fuzzy sets and in such that there exists a fuzzyopen set such that and.

Normality may be characterized via the existence of a fuzzy continuous real-valued function just as in topological spaces.

Let be a normal fuzzy isotone space. Then for each pair of disjoint fuzzy subsets and Ф, there exists a fuzzy continuous function such that on and on. Clearly, this characterization is analogous to the definition of normality via the existence of an Urysohn function on a normal topological space.

3.4. Complete Normality

A fuzzy isotone space is said to be completely normal if every fuzzy subspace of is normal.

Theorem

A fuzzy isotone space is completely normal if and only if for every pair and of fuzzy subsets with then there exists disjoint fuzzy sets and.

Proof

Let be completely normal and be fuzzy sets with. Denote, a subspace of by. Then since

.

Similarly,

.

Clearly and are closed sets in

such that. Notice

.

Therefore, since is completely normal, then is normal and hence there exists and in such that and.

Conversely, let be a subspace of and such that and with. Then. Similarly,. Therefore, by the hypothesis of the theorem, there exists disjoint fuzzy sets

and.

The fuzzy sets and are disjoint and contained in and. Hence is normal and is therefore completely normal.

3.5. Perfect Normality

Perfect normality has not been defined in fuzzy closure spaces. Therefore, different characterizations are given under this section as modifications from topological spaces. A few basic concepts have to be carried over from general topological spaces before any meaningful definition of perfectly normal isotonic spaces can be given.

3.5.1. Preliminary Definitions

It is known form point-set topology and from fuzzy topology that though the countable union of closed sets need not be closed, and the countable intersection of open sets need not be open, such sets occur frequently in analysis. The occurrence of such sets guarantees perfect normality on a space.

A fuzzy set is called a -set if and only if

where. A set is called an -set if

where.

A fuzzy isotone space is perfectly normal if is normal and for every fuzzy subset of, is a -set. That is for every closed fuzzy set in,. Equivalently, a normal isotonic space is perfectly normal if every open fuzysubset of is an -set. That is, for every fuzzy set, then

.

3.5.2. Theorem

The fuzzy isotone space is perfectly normal if for every such that, and, then a fuzzy continuous function that precisely separates and. That is and

3.5.3. Theorem

Every perfectly normal fuzzy isotone space is completely normal.

Proof

Since a fuzzy isotone space is completely normal if and only if every subspace is normal, then in order to show that perfect normality implies complete normality, it suffices to show the heredity of perfect normality. Let be perfectly normal fuzzy isotone space. Then for every closed fuzzy set there exists a fuzzy continuous function such that.

Let be a subspace of and be closed. Then there exists a fuzzy closed set such that. Since is closed in, then there exists a fuzzy continuous function

such that. But is also fuzzy continuous and. Therefore is perfectly normal and hence normal. This implies that X is completely normal and hence perfect normality implies complete normality.

4. Conclusion

The variants of normality naturally extend to the class of fuzzy isotone spaces and to the fuzzy closure spaces generally. Therefore, on fuzzy isotone spaces, perfect normality implies complete normality which implies normality.

REFERENCES