On Some R1-Properties in Fuzzy Topological Spaces

The concept of fuzzy sets was first introduced, in 1965, by L. A. Zadeh in his new classical paper [1] as an attempt to mathematically handle those phenomena which are inherently vague, imprecise or fuzzy in nature. He interpreted a fuzzy set on a set X as a mapping from X into the closed unit interval [0, 1] I = . Various merits and applications as well as some limitations of fuzzy set theory have since been demonstrated by Zadeh and a large number of subsequent workers. The advent of fuzzy set theory has also led to the development of some new areas of study in mathematics. It has become a concern and a new tool for the mathematicians working in many different areas of mathematics. These have been generally accomplished by replacing subsets, in various existing mathematical structures, by fuzzy sets. In 1968, Chang [2] did ‘fuzzification’ of topology by replacing ‘subsets’ in the definition of fuzzy topology by ‘fuzzy sets’. Since then a large body of concepts and results have been growing in this area which has come to be known as “fuzzy topology”. A major deviation in the definition of fuzzy topology was made by Lowen [3, 4]. He gave a modified definition of fuzzy topology by including all constant fuzzy sets in a fuzzy topology.


Introduction
The concept of fuzzy sets was first introduced, in 1965, by L. A. Zadeh in his new classical paper [1] as an attempt to mathematically handle those phenomena which are inherently vague, imprecise or fuzzy in nature.He interpreted a fuzzy set on a set X as a mapping from X into the closed unit interval [0, 1] I = . Various merits and applications as well as some limitations of fuzzy set theory have since been demonstrated by Zadeh and a large number of subsequent workers.The advent of fuzzy set theory has also led to the development of some new areas of study in mathematics.It has become a concern and a new tool for the mathematicians working in many different areas of mathematics.These have been generally accomplished by replacing subsets, in various existing mathematical structures, by fuzzy sets.In 1968, Chang [2] did 'fuzzification' of topology by replacing 'subsets' in the definition of fuzzy topology by 'fuzzy sets'.Since then a large body of concepts and results have been growing in this area which has come to be known as "fuzzy topology".
A major deviation in the definition of fuzzy topology was made by Lowen [3,4].He gave a modified definition of fuzzy topology by including all constant fuzzy sets in a fuzzy topology.
The concepts of 0 R -type and 1 R -type axioms for fts was first introduced by Hutton and Reilly [5] in 1980.In 1990, Ali et al. [6] introduced some other definitions of fuzzy 0 R -axioms.Srivastava [7], Ali [8, 9], and Azam and Ali [10] also gave some new concepts of 1 R -property in fuzzy topology.
In this paper, we introduce six new concepts of 1 R -properties of fts each of which is shown as the good extension of the topological 1 R -property.We study their interrelations and initialities.In addition, we recall nine concepts of 0 R -properties of fts from [6].In analogy with the well known topological property ( ) we study the relations of this type for fts.It is also shown that, the property ( ) , in general, is also true for fts.

Preliminaries
In this section, we recall some definitions and basic results (which we label as facts) on fuzzy sets and fts.This section is considered as the base and background for the study of subsequent sections.
Definition 2.1 [1]: Let X be a non-empty set and I the unit closed interval [0, 1].A fuzzy set on X is a function : u X I → .∀ x ∈ X, u(x) denotes a degree or the grade of membership of x.The set of all fuzzy sets in X is denoted by X I .Ordinary subsets of X (crisp sets) are also considered as the members of X I which take the values 0 and 1 only.A crisp set which always takes the value 0 is denoted by 0, similarly a crisp set which always takes the value 1 is denoted by 1.

Definition 2.2 [9]: Let
is called the support of u and is denoted by o u or supp(u).By c u , we denote the complement of u which is defined as , by 1 A we denote the characteristics function A. The characteristics function of a singleton set {x} is denoted by 1 x .
Definition 2.4 [9,11]: A fuzzy point x α in X is a special type of fuzzy set in X with the membership function ( ) ( ) x x and x y 0 if x y α α = α = ≠ , where 0 < α < 1 and x, y ∈ X.
The fuzzy point x α is said to have support x and value α.We also write this as 1 x α .
Definition 2.5 [2]: Let : f X Y → be a mapping and u a fuzzy set in X.Then the image ( ) f X Y → be a mapping and u be a fuzzy set in Y. Then the inverse image ( ) is the fuzzy set in X which is defined as Definition 2.7 [2]: Chang C. L. defined a fuzzy topological space as follows: Let X be a set.A class t of fuzzy sets in X is called a fuzzy topology on X if t satisfies the following conditions: The pair (X, t) is then called a fuzzy topological space (in short, fts).The members of t are called t-open sets (or open sets) and their complements are called t-closed set (or closed sets).Definition 2.8: Lowen [3] modified the definition of an fts defined by Chang [2] by adding another condition.In the sense of Lowen R. the definition of an fts is as follows: Let X be a set and t is a family of fuzzy sets in X.Then t is called a fuzzy topology on X if the following conditions hold: and (iv) t contains all constant fuzzy sets in X.The pair (X, t) is called an fts.
We shall use the concept of fts due to Lowen R. unless otherwise stated.Definition 2.9 [6,9]: Let u be a fuzzy set in an fts (X, t).Then the fuzzy closure u and the fuzzy interior o u of u are defined as follows: { } inf : and Fact 2.10 [6,9]: For a fuzzy topological space (X, t) and for X u I ∈ , the following hold: Definition 2.11 [9]: Let (X, t) and (Y, s) be two fts.A function ( ) ( ) (ii) open if and only if ( ) Definition 2.12 [9]: Let ( ) { } , : Cartesian product and : be the projection map.Then the fuzzy topology t on X generated by ∈ is called the product fuzzy topology on X and the pair (X, t) is called the product fts.It can be verified that ( ) above, can be expressed as where if and if .
The product fuzzy topology t is also called the coarsest fuzzy topology on X.
Fact 2.13 [9]: For a family ( ) { } , : i i X t i K ∈ of fts and a fuzzy topology t on the following are equivalent: (i) t is the product of the fuzzy topologies i t 's.
(ii) t is the smallest fuzzy topology on X which makes each projection : ∈ be a family of functions from a set X to fts ( ) , , .
For a topological space (X, T), the l.s.c.fuzzy topology on X associated with T is denoted by ω(T) and is defined as ω(T) Fact 2.17 [9]: Let (X, T) be a topological space.Then (i) (vii) { } 0 1 : and Definition 2.18 [2]: Let P be a property of a topological space and FP its fuzzy topological analogue.Then FP is called a good extension of P if and only if the statement "(X, T) has P if and only if (X, ω(T)) has FP" holds good for every topological space (X, T).

R 1 -properties
In this section we introduce six R 1 -axioms for fts.

FR
-property of (X, t), and 0 u v ∧ = Hence, (X, t) has the property ( ) x ≤ − λ But ( ) ( ) All other proofs are similar.Now we give some counter examples to show the non-implications among the fuzzy 1 R properties mentioned above.
Example 3.1 [9]: Let X be an infinite set and for any x, y ∈ X, we define u xy , a fuzzy set in X, as follows: u xy (x) = 1, u xy (y) = 0 and u xy (z) = 0.

Proof:
Let {(X j , t j ): j∈J} be a family of ( )

FR
-fts, {f j : X→(X j , t j ); j∈J} a family of functions and t the initial fuzzy topology on X induced by the family {f j : j∈J}.Let x, y ∈X, x ≠ y, α∈I 0, 1 and w∈t such that w(x) > α and w(y) = 0. Since w∈t, there exist basic t-open sets, w p such that w = sup {w p : p∈P}.Also each w p must be expressible as As w(x) > α and w(y) = 0 , we can find some k such that 1

Example 3 . 2 [ 9 ]B 3 =
: Let X = I and t be the fuzzy topology on X generated by B = B 1 ∪B 2 ∪B 3 ∪B 4 .Where, B 1 = {1 x : x∈I 0,1 }, B 2 = {u m : m∈N}; u m is a fuzzy set in X defined by u {v n, F : n∈N and F is a finite crisp subset of X }, Where v n, F is a fuzzy set in X

R
For every pair x, y∈X, x ≠ y, ( ) For example, the product fuzzy topology is the initial fuzzy topology induced by the family of projections.Similarly, the subspace topology is also the initial fuzzy topology induced by the inclusion map.
− ∈ ∈ 5 ∀ z∈X, z ≠ x, y.Now consider the fuzzy topology t on X generated by {u xy : x, y ∈ X, x ≠ y}∪ {constants}.It is clear that, say k / such that