Weakly continuous functions on mixed fuzzy topological spaces

The notions of continuity was generalized in the fuzzy setting by Chang (1968). Later on Azad (1981) introduced some weaker form of fuzzy continuity like fuzzy almost continuity, fuzzy semi-continuity and fuzzy weak continuity. These are natural generalization of the corresponding weaker forms of continuity in topological spaces. Recently Arya and Singal (2001a and b) introduce another weaker form of fuzzy continuity, namely fuzzy subweakly continuity as a natural generalization of subweak continuity introduced by Rose (1984). In this paper we introduce fuzzy weak continuity in mixed fuzzy topological space.


Introduction
The notion of topological space has been generalized in many ways.The notion of bitopological space and mixed topological space has been introduced and investigated in the resent past.Bitopological spaces have recently been studied by Ganguly and Singha (1984), Tripathy andSarma (2011b, 2012) and others.Mixed topology lies in the theory of strict topology of the spaces of continuous functions on locally compact spaces.The concept of mixed topology is very old.Mixed topology is a technique of mixing two topologies on a set to get a third topology on that set.The works on mixed topology is due to Cooper (1971), Buck (1952), Das and Baishya (1995), Tripathy and Ray (2012), Wiweger (1961) and many others.
In 1965 L. A. Zadeh introduced the concept of fuzzy sets.Since then the notion of fuzziness has been applied for the study in all the branches of science and technology.It has been applied for studying different classes of sequences of fuzzy numbers by Tripathy and Baruah (2010), Tripathy andBorgohain (2008, 2011), Tripathy and Dutta (2010), Tripathy and Sarma (2011a), Tripathy et al. (2012), and many workers on sequence spaces in the recent years.The notion of fuzziness has been applied in topology and the notion of fuzzy topological spaces has introduced and investigated by many researches on topological spaces.Different properties of fuzzy topological spaces have been investigated by Arya and Singal (2001a and b), Chang (1968), Das andBaishya (1995), Ganster et al. (2005), Ganguly andSingha (1984), Ghanim et al. (1984), Katsaras and Liu (1977), Petricevic (1998), Srivastava et al. (1981), Warren (1978), Wong (1974a and b) and many others.Recently mixed fuzzy topological spaces have been investigated from different aspect by Das and Baishya (1995) and others.

Preliminaries
Let X be a non-empty set and I, the unit interval of fuzzy sets in X, to be written as  respectively, are defined as follows: Definition 2.1.A fuzzy topology τ on X is a collection of fuzzy sets in X such that , X τ; if A i τ, iJ then Clearly, cl A (respectively int A) is the smallest (respectively largest) closed(respectively open) fuzzy set in X containing (respectively contained in) A. If there are more than one topologies on X then the closure and interior of A with respect to a fuzzy topology τ on X will be denoted by τ-cl A and τ-int A.
Definition 2.3.A collection Ɓ of open fuzzy sets in a fts X is said to be an open base for xX and for all y Y.Let f be a function from X into Y.Then, for each fuzzy set B in Y, the inverse image of B under f, written as f -1 [B], is a fuzzy set in X defined by Definition 2.5.A fuzzy set A in a fuzzy topological space (X, τ) is called a neighborhood of a point xX if and only if there exists B τ such that B  A and A(x) = B(x)>0.Definition 2.6.A fuzzy point x α is said to be quasi-coincident with A, denoted by x α qA, if and only if α + A(x) > 1 or α > (A(x)) c .Definition 2.7.A fuzzy set A is said to be quasicoincident with B and is denoted by AqB, if and only if there exists a xX such that A(x) + B(x) > 1.
Remark 2.1.It is clear that if A and B are quasicoincident at x both A(x) and B(x) are not zero at x and hence A and B intersect at x. Definition 2.8.A fuzzy set A in a fts (X, τ) is called a quasi-neighbourhood of x λ if and only if The family of all Q-neighbourhood of x λ is called the system of Q-neighbourhood of x λ .Intersection of two quasi-neighbourhood of x λ is a quasineighbourhood.Let (X, τ 1 ) and (X, τ 2 ) be two fuzzy topological spaces and let τ 1 (τ 2 ) = {AI X : for every fuzzy set B in X with AqB, there exists a τ 2 -Q-neighbourhood A α , such that A α qB and τ 1 -closure, A B   }, Then τ 1 (τ 2 ) is a fuzzy topology on X and this is called mixed fuzzy topology, and the space (X, τ 1 (τ 2 )) is called mixed fuzzy topological space.

Proof:
We show that Therefore by definition of mixed topology there exists  continuous, so we have therefore for each fuzzy set A in X and  -open set B with BqA and so by definition of mixed topology, we have -open, and so f : X→Y is continuous.
This completes the proof.
 ) be any two fuzzy bi-topological spaces and f: X→ Y be a mapping such that f is 1   weakly continuous.
Proof.Let B be any * * 1 2 ( ) We show that cl (f -1 (B))  , in this case the closure is with respect to weakly continuous.By definition of weakly continuity we have Taking complement of both side of (3), we get Now from (1) we have f -1 (cl(U))  f -1 (B) Therefore using (2) we have (6) Therefore using ( 4) and ( 6) we have Again from (1) we have Taking closure of both side, we get This completes the proof.Theorem 3.3.Let (X, τ 1 , τ 2 ) and (Y,  1 * ,  2 * ) be two fuzzy bi-topological spaces and weakly continuous, we have weakly continuous.This completes the proof.) open fuzzy set in Y.We show that cl(f -1 (B * ))  f -1 (cl(B * )), the closure is being with respect to  2 .
Let f: X→Y be We know that the mixed topology ).
( 2 This completes the proof.Theorem 3.6.Let (X, 1  weakly continuous.Thus we have for any  This completes the proof.

Conclusion
We have introduced fuzzy weak continuity in mixed fuzzy topological space and have investigated its different properties.The results of this article can be applied for futher investigations and applications in studying different properties weak continuity of functions in mixed fuzzy topological spaces.
The pair (X, τ) is called a fuzzy topological space (fts).Members of τ are called open fuzzy set and the complement of an open fuzzy sets is called a closed fuzzy set.Definition 2.2.If (X, τ) is a fuzzy topological space, then the closure and interior of a fuzzy set A in X, denoted by cl A and int A respectively, are defined as cl A=  { B: B is a closed fuzzy set in X and A  B}.The int A=  {V : V is an open fuzzy in X and V  A}.
f -1 (B * ))  f -1 (cl(B * )), the closure of left hand side is being with respect to 2 the closure of left hand side is with respect to the fuzzy topology 2  .