Some Features of Intuitionistic L-R 1 Spaces

In this paper, we have introduced four notions of R1 space in intuitionistic L-topological spaces and established some implications among them. We have also proved that all of these definitions satisfy “good extension” and “hereditary” property. Finally, it has been shown that all concepts are preserved under one-one, onto and continuous mapping.


Introduction
The idea of fuzzy sets and L-fuzzy sets were initially introduced by Zadeh [1] in 1965 and Goguen [2] in 1967 respectively.In 1984, Atanassove [3] defined the concept of intuitionistic fuzzy sets (which take into account both the degree of membership and non membership subject to the condition that their sum does not exceed 1) and many works by the same author and his colleagues appeared in the literature [4][5][6].Later, this concept was generalized to "intuitionistic L-fuzzy sets" by Atanassov and Stoeva [7].Coker [8][9][10] first defines intuitionistic fuzzy topological spaces and some of its properties which is in the sense of C. L. Chang [11].After then, many fuzzy topologists [12][13][14][15][16][17][18] work in separation axioms of fuzzy topological spaces and intuitionistic fuzzy topological spaces, especially Ahmed et al. [19] defines some types of R 1 spaces in intuitionistic fuzzy topological spaces and Islam et al. [20] defines some types of T 2 spaces in intuitionistic L-topological spaces.In this paper, we define some new notions of L-R 1 spaces using intuitionistic Lfuzzy sets and investigate the property of L-R 1 spaces.

Notation and Preliminaries
Through this paper, X will be a nonempty set, ø be the empty set, L be a complete distributive lattice with 0 and 1. A, B, … be intuitionistic L-fuzzy sets, t be the intuitionistic topology, τ be the intuitionistic L-topology, I = [0, 1], and the functions and denote the degree of membership (namely ( )) and the degree of none membership (namely ( )).
Now we recall some basic definitions and known results in intuitionistic L-fuzzy sets and intuitionistic L-topological spaces.Definition 2.1.[1] Let X be a non-empty set and I = [0, 1].A fuzzy set in X is a function which assigns to each element x ε X, a degree of membership u(x) ε I.
Definition 2.2.[21] Let be a function and be fuzzy set in .Then the image ( ) is a fuzzy set in which membership function is defined by Definition 2.3.[12] 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 "( ) has P if and only if (( ( )) has FP" holds good for every topological space( ).
Definition 2.4.[2] Let be a non-empty set and be a complete distributivelattice with 0 and 1.An L-fuzzy set in is a function which assigns to each element , a degree of membership, ( ) .
Remark 2.5.[20] Throughout this paper we consider the complete distributive lattice * + and from above definitions we show that every L-fuzzy set is also a fuzzy set but converse is not true in general.
Example 2.5.1.[20] ( ) then ( ) is the corresponding intuitionistic fuzzy topological space of ( ) Definition 2.11.[20] Let * + and An intuitionistic Lfuzzy point (ILFP for short) ( ) of is an ILFS of defined by In this case, is called the support of ( ) and and are called the value and none value of ( ) respectively.The set of all ILFP of we denoted it by ( )

Definition and Properties of Intuitionistic Lattice Fuzzy R 1 Spaces
In this section, we give four definitions and establish two theorems of R 1 spaces in intuitionistic L-topological spaces.
ILFP ( ) is said to belong to an ILFS Let ( ) be an ILTS.Then we have the following implications: Fig.1.Implications among the properties.