SOME WEAKER FORMS OF SMOOTH FUZZY CONTINUOUS FUNCTIONS

In this paper we introduce various notions of continuous fuzzy proper functions by using the existing notions of fuzzy closure and fuzzy interior operators like Rτ -closure, Rτ -interior, etc., and present all possible relations among these types of continuities. Next, we introduce the concepts of α-quasi-coincidence, qα r -pre-neighborhood, qα r -pre-closure and qα r pre-continuous function in smooth fuzzy topological spaces and investigate the equivalent conditions of qα r pre-continuity.


INTRODUCTION
Šostak [28] defined I-fuzzy topology as an extension of Chang's fuzzy topology [2].It has been developed in many directions by many authors.For example see [8,16].Ramadan [23] gave a similar definition of fuzzy topology on a fuzzy set in Šostak's sense and called by the name "smooth fuzzy topological space".
On the other hand, studying different forms of continuous functions in topological space is an interesting area of research which attracts many researchers.In the fuzzy context, after the introduction of fuzzy proper function from a fuzzy set in to a fuzzy set [1], several notions of continuous fuzzy proper functions between Chang's fuzzy topological spaces are defined and their properties are discussed in [3].The concepts of smooth fuzzy continuity, weakly smooth fuzzy continuity, qn-weakly smooth fuzzy continuity, (α,β)-weakly smooth fuzzy continuity of a fuzzy proper function on smooth fuzzy topological spaces and their inter-relations are investigated in [5,23,26,27,10].
Lee and Lee [19] introduced the notion of fuzzy r-interior which is an extension of Chang's fuzzy interior.Using fuzzy r-interior, they define fuzzy r-semiopen sets and fuzzy r-semicontinuous maps which generalize fuzzy semiopen sets and fuzzy semicontinuous maps in Chang's fuzzy topology, respectively.Some basic properties of fuzzy rsemiopen sets and fuzzy r-semicontinuous maps are investigated in [19].In [22], the concepts of several types of weak smooth compactness are introduced and investigated some of their properties.
In [7,20], the notions of fuzzy semicontinuity, fuzzy γ-continuity of a fuzzy proper functions, fuzzy separation axioms, fuzzy connectedness and fuzzy compactness are defined.
Ganguly and Saha [6] introduced the notions of δ-cluster points and θ-cluster points in Chang's fuzzy topological spaces.Kim and Park [15] introduced δclosure in Šostak's fuzzy topological spaces.Kim and Ko [13] introduced fuzzy super continuity, fuzzy δcontinuity, fuzzy almost continuity in the context of Šostak's fuzzy topological spaces.They proved that fuzzy super continuity implies both fuzzy δ-continuity and fuzzy almost continuity.Similar works are discussed by various researchers, see [12,14,18,21].
Further, by introducing the notions α-quasicoincidence,    -pre-neighborhood,    -pre-closure and    -pre-continuity, we investigate the relations between    -pre-continuity and the property F(P  (A, r)) ≤ P  (F(A),r), for every A ≤ µ in smooth fuzzy topological spaces.

The pair (µ,τ) is called a smooth fuzzy topological space.
A fuzzy subset U ∈   is said to be fuzzy open if τ(U) > 0 and fuzzy closed if τ(µ -U) > 0.
A fuzzy set U ∈   is called a q-neighborhood of a fuzzy point    in µ if    [µ] and τ(U) > 0.
Definition 4 [1]: Let F be a fuzzy proper function from µ to ν.If U ∈   and V∈   , then F(U):S → I and F -1 (V) : X →  are defined by The inverse image of a fuzzy subset V under a fuzzy proper function F can be easily obtained as (F - 1 (V))(x) = µ(x) Λ V(s), where s ∈ S is the unique element such that F(x,s)=µ(x).
The statement of the above theorem is not true when F is not one-to-one or F(µ)≠ .The following counterexamples justify our statement.
Since F is one-to-one and ν = F() and by Theorem 2, we have F -1 (ν-V) =  − F -1 (V).Therefore,   ( −F -1 (V), r) =  − F -1 (V).□ The statement of the above theorem is not true when F is not one-to-one or F()≠ .The following counterexamples justify our statement.

Theorem 7: Let F : (𝜇, 𝜏) → (𝜈, 𝜎) be a one-to-one fuzzy proper function with ν = F(𝜇). If F is fuzzy almost continuous, then F is fuzzy almost 𝑟 1 -continuous.
Since the proof of this theorem is similar to that of Theorem 4.7 in [11], we prefer to omit the details.The statement of the above theorem is not true when F is not one-to-one F() ≠ .The following counterexamples justify our statement.Let the fuzzy proper function F : (, ) → (, ) be defined by F(x,s) = 0.7, F(x,t) = 0, F(y,s) = 0.5, F(y,t) = 0.Then, F is not one-to-one and F() [0.7,0]  [,] = .We fix r = 0.5.For the    -neighborhood  1 of any then as in the previous counterexample, we can verify that F is one-to-one, F() [0.7,0.6][,] ≠  and F is fuzzy almost continuous.
The statement of the above theorem fails to be true when F is not one-to-one and F(μ) ≠ .The following counterexamples justify our statement.
forms of continuities such as fuzzy weakly δ-continuity, fuzzy weakly δ-r1-continuity, fuzzy weakly δ-r2-continuity, fuzzy weakly δ-r3-continuity, etc., and inter-relations among them are obtained completely.Further, we have introduced new notion of quasi coincidence namely α-quasi coincidence and then a fuzzy closure operator PClα is introduced.Using this fuzzy closure operator,    -pre-continuous fuzzy proper function is introduced and all properties of this function are obtained.

2 .
If q < r and if F : (, ) → (, ) is fuzzy almost [r,q]1-continuous, then F is fuzzy almost r1-continuous and fuzzy almost q1 continuous.3.If r < q and if F : (, ) → (, ) is fuzzy almost r2-continuous, then F is fuzzy almost [r,q]2-continuous.4. If q < r and if F : (, ) → (, ) is fuzzy almost [r,q]2-continuous, then F is fuzzy almost r2-continuous and F is fuzzy almost q2-continuous.The results obtained in this section are summarized in the following implication diagram.FUZZY    -PRE-CLOSURE AND FUZZY    -PRE-CONTINUOUS MAPS Definition 13: We say that U, V ∈   are said to be α-quasi-coincident referred to  [written as UqαV[μ]] if there exists x ∈ X such that U(x) + V(x) > μ (x) + α.If U is not α-quasi coincident with V, then we write U ̅αV[μ].