Normality and Its Variants on Fuzzy Isotone Spaces

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.


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     c    and the isotone axiom is the closure operator on a nonempty set .

Fuzzy Sets
In [2], a fuzzy set in is defined as a function X   : 0, X  

Crisp Fuzzy Sets
Any subset of a set can be identified with its characteristic function defined by; Such characteristic functions are fuzzy sets in .Thus fuzzy sets generalize ordinary sets [3].
For two fuzzy sets  and  in represents the degree of membership of x X  in the fuzzy set  .

5)
The complement of  is 1   .
Let   be fuzzy sets.Then the union of   The intersection of   If i s   are crisp, i.e. they are characteristic functions, then these suprema and infima are actually maxima and minima.

Fuzzy Topology
According to [4], a fuzzy topology on a set is a collection X  of fuzzy sets in satisfying X 1) 0,1   , where is equivalent to the empty set.0 2) If  and  belong to  , then      .
3) If   . The members of  are called open fuzzy sets.
The pair   , X  is called a fuzzy topological space.Fuzzy sets of the form 1   , where  is fuzzy are called closed fuzzy sets.

Functions and Fuzzy Continuity
Let and Y be sets and

Fuzzy Continuity
Given fuzzy topological spaces   , The identity mapping fuzzy topological space   , X  is fuzzy continuous.

Closure and Interior Operation on Fuzzy Sets
Let   , X  be a fuzzy topological space.The closure  and interior o  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 o  is the largest open fuzzy set smaller than  .These definitions coincide with their analogous definitions on ordinary sets.

Let
be the collection of all mappings from to the unit interval 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 The closure operator may also be used to characterize closed sets. 3) for every Similarly, the interior operator may also be used to characterize open sets.
A Cech fuzzy closure operator (or CF-closure operator) on a set is a function satisfying the following three axioms; The pair   , X c is called a fuzzy closure space or fcs.Clearly these axioms can easily be seen to be similar to the Kuratowski axioms in [6].

Results
The following are the main results of this work.

Fuzzy Isotone Space
A fuzzy isotone closure operator on a set is a function satisfying the following two axioms; 2) For every


We would like to modify the definitions of semi-sepa-rated 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   , X c , two fuzzy subsets  and are called semi-separated if The Similarly, hence

Conversely, let  
, Y c Y be a subspace of the fuzzy isotone space   , X c and  and are semi-separated in Thus  and Ф are semi-separated in   , X c .

Normality
A fuzzy isotone space   , X c is normal if for every nonempty pair of fuzzy sets and in such that there exists a fuzzyopen set such that Normality may be characterized via the existence of a fuzzy continuous real-valued function just as in topological spaces.
Let   , X c be a normal fuzzy isotone space.Then for each pair of disjoint fuzzy subsets  and Ф, there exists a fuzzy continuous function   : f X 0  ,1 such that 0 f  on  and 1 f  on .Clearly, this characterization is analogous to the definition of normality via the existence of an Urysohn function on a normal topological space.

Complete Normality
A fuzzy isotone space   , X c is said to be completely normal if every fuzzy subspace of is normal.X Theorem A fuzzy isotone space   , X c is completely normal if and only if for every pair  and of fuzzy subsets with then there exists disjoint fuzzy sets Proof Let be completely normal and Similarly, Therefore, since is completely normal, then is normal and hence there exists and in Therefore, by the hypothesis of the theorem, there exists disjoint fuzzy sets The fuzzy sets U Y  and V are disjoint and contained in and .Hence Y is normal and is therefore completely normal.
Ф   V Y X

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 continuous function Let Y X  be a subspace of  

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 .X ,1 . But is also 0  fuzzy continuous and . Therefore is perfectly normal and hence normal.This implies that X is completely normal and hence perfect normality implies complete normality.

Conclusion
where 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.

 
A fuzzy isotone space   , X c is perfectly normal if is normal and for every fuzzy subset

Theorem
The fuzzy isotone space   , X c is perfectly normal if for every such that ,


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   , X c be perfectly normal fuzzy isotone space.Then for every closed fuzzy set W. J. Thron, "What Results Are Valid on Cech-Closure Spaces," Topology Proceedings, Vol. 6, 1981, pp.135-158.