INTUITIONISTIC FUZZY ᴪ -CONTINUOUS MAPPINGS

In this paper we introduce intuitionistic fuzzy  -continuous mappings and intuitionistic fuzzy  irresolute mappings. Some of their properties are studied.


INTRODUCTION
The concept of fuzzy sets was introduced by Zadeh (1965), is a framework to encounter uncertainity, vagueness and parital truth and it represents a degree of membership for each member Obviously, every fuzzy set A on a nonempty set X is an IFS of the following form A   x,  A (x), 1   A (x) : x  X  2.2. Definition (Atanassov, 1986) of the universe of discourse to a subset of it. By adding the degree of non-membership to fuzzy set and later Atanassov (1986) proposed intuitionistic fuzzy set in 1986 which appeals more accurate to uncertainity quantification and provides the opportunity to precisely model the problem, based on the existing knowledge and observations. On the other hand Coker (1997)

introduced intuitionistic Let A and B be IFSs of the form
fuzzy topological spaces using the notation of intuitionistic fuzzy sets. In this paper we introduced intuitionistic fuzzy  -continuous mappings and studied some of their basic properties. We provide Definition (Atanassov, 1986) Let X be a non empty fixed set and I be the closed interval 0,1. In intuitionistic fuzzy set (IFS)

and B  A;
A is an object of the following form 1~   x, 1, 0, x  X  are respectively the The family of all IFOS (respectively IFSOS, IF  OS, empty set and the whole set of X. (Coker, 1997) An intuitionistic fuzzy topology (IFT in short) on X is a family  of IFSs in X satisfying the following IFSPOS, IFPOS, IFROS) of an IFTS ( X , ) is denoted by IFO(X) (respectively IFSO(X), IF  O(X), IFSPO(X), IFPO(X), IFRO(X)).

cl(int(A)))  A,
The complement A C of an IFOS A in IFTS ( X , ) is (iv) Intuitionistic fuzzy pre closed set (Young Bae called an intuitionistic fuzzy closed set (IFCS in short) in X. (Coker, 1997) and Seok-Zun, 2005) (IFPCS in short) if

cl(int( A))  A.
(v) Intuitionistic fuzzy regular closed set (Joung Kon an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by int(A) = {G / G is an IFOS in X and G  A} IFPC(X), IFRC(X)).

Definition (Young Bae and Seok-Zun, 2005)
Note that for any IFS A in ( X , ) , we have Let A be an IFS in an IFTS ( X , ) . Then

Definition
An IFS A in an IFTS ( X , ) is an (i) Intuitionistic fuzzy generalised closed set (Thakur and Rekha, 2006) U whenever A U and U is an IFOS in X.

Definition
Let f be a mapping from an IFTS ( X , ) into an IFTS (Y , ) . Then f is said to be

Theorem (Parimala et al.,)
Let ( X , ) be an intuitionistic fuzzy topological space. Then the following are hold (i) Every IFCS in X is an IF CS in X.
(ii) Every IFRCS in X is an IF CS in X.
(iii) Every IF  CS and hence IFSCS in X is an IF CS in X.
(iv) Every IF CS in X is an IFSPCS in X.
(v) Every IF CS in X is an IFGSPCS in X.
(vi) Every IF CS in X is an IFGSCS and hence IFSGCS in X.

MAPPINGS
In this section we introduce intuitionistic fuzzy continuous mapping and studied some of its properties.

Theorem
Every IF continuous mapping is an IF -continuous mapping but not conversely.

Proof.
Then f 1 ( A) is IF CS in X but not IFSCS in X. Therefore f is an IF -continuous mapping but not IF semi continuous mapping.

Theorem
Every IF  -continuous mapping is an IFcontinuous mapping but not conversely.

Example
Since every IF  CS is an IF CS in X by Theorem 2.11. Therefore f is an IF  -continuous mapping. X and Y respectively.

Example
Define a mapping Then f 1 ( A) is IF CS in X but not IFCS in X. Therefore f is an IF -continuous mapping but not IF continuous mapping.

Theorem
Every IF semi continuous mapping is an IF continuous mapping but not conversely. Then f 1 ( A) is IF CS in X but not IF  CS in X. Then f is an IF -continuous mapping but not IF  -continuous mapping .

Theorem
Every IF -continuous mapping is an IFSG continuous mapping but not conversely.

Proof.
clearly A  G . Therefore A is an IFGSCS in X. Then f 1 ( A) is IFGSCS in X but not IF CS in X. Then f is an IFGS continuous mapping but not IFcontinuous mapping .
Let f : ( X , )  (Y , ) be an IF  -continuous 3.14. Theorem mapping. Let A be an IFCS in Y. Then by hypothesis f 1 ( A) is an IF CS in X. Since every IF CS is an Every IF -continuous mapping is an IFGSP continuous mapping but not conversely.
IFSGCS by Theorem 2.11 , f 1 ( A) is an IFSGCS in X.

Theorem
Every IF -continuous mapping is an IFGS continuous mapping but not conversely.
clearly A  G . Therefore A is an IFGSPCS in X. Then f 1 ( A) is IFGSPCS in X but not IF CS in X. Then f is an IFGSP continuous mapping but not IF  -continuous mapping . Let f : ( X , )  (Y , ) be an IF  -continuous mapping. Let A be an IFCS in Y. Then by hypothesis f 1 ( A) is an IF CS in X. Since every IF CS is an