Fuzzy C e-I(ec, eo) and Fuzzy Completely C e-I(rc, eo) Functions via Fuzzy e-Open Sets

We introduced the notions of fuzzy C e-I(ec, eo) functions and fuzzy completely C e-I(rc, eo) functions via fuzzy e-open sets. Some properties and several characterization of these types of functions are investigated.


Introduction
With the introduction of fuzzy sets by Zadeh [1] and fuzzy topology by Chang [2], the theory of fuzzy topological spaces was subsequently developed by several fuzzy topologist based on the concepts of general topology. In 2014, the concept of fuzzy e-open sets and fuzzy e-continuity and separations axioms and their properties were defined by Seenivasan and Kamala [3]. In this paper, we introduce the notion of fuzzy -(ec, eo) functions, fuzzy -continuous, fuzzy completely -(rc, eo) functions, and fuzzy e-kernel via fuzzy e-open sets and studied their properties and several characterizations of these types of functions are investigated. In this paper, we denote fuzzy e-open, fuzzy e-closed, and fuzzy regular closed as, eo, ec, and rc, respectively.

Preliminaries
Throughout this paper, ( , ) and ( , ) (or simply and ) represent nonempty fuzzy topological spaces on which no separation axioms are assumed, unless otherwise mentioned.
The intersection of all fuzzy e-closed sets containing is called fuzzy e-closure of and is denoted by fe-cl( ) and the union of all fuzzy e-open sets contained in is called fuzzy e-interior of and is denoted by fe-int( ).

Definition 2. A mapping
: → is said to be fuzzy e * -open [8] if the image of every fuzzy e-open set in is fuzzy e-open set in .

Definition 4.
A fuzzy set is quasicoincident [9] with a fuzzy set denoted by iff there exist ∈ such that ( ) + ( ) > 1. If and are not quasicoincident, then we write and ≤ ⇔ 1 − .

Definition 5.
A fuzzy point is quasicoincident [9] with a fuzzy set denoted by iff there exist ∈ such that + ( ) > 1.

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The Scientific World Journal Definition 6. A fuzzy topological space ( , ) is said to be fuzzy -1 [3] if for each pair of distinct points and of there exist fuzzy e-open sets 1 and 2 such that ∈ 1 and ∈ 2 and ∉ 2 and ∉ 1 .

Definition 7.
A fuzzy topological space ( , ) is said to be fuzzy -2 [3] if for each pair of distinct points and of there exists disjoint fuzzy e-open sets and such that ∈ and ∈ .
Definition 8. A fuzzy topological space is said to be fuzzy weakly Hausdorff [10] if each element of is an intersection of fuzzy regular closed sets.

Definition 9.
A fuzzy topological space is said to be fuzzy e-normal [3] if for every two disjoint fuzzy closed sets and of there exist two disjoint fuzzy e-open sets and such that ≤ and ≤ and ∧ = 0.

Definition 10.
A fuzzy topological space is said to be fuzzy strongly normal [10] if for every two disjoint fuzzy closed sets and of there exist two disjoint fuzzy open sets and such that ≤ and ≤ .
Definition 11. A fuzzy topological space is said to be fuzzy Urysohn [11] if for every distinct points and in there exist fuzzy open sets and in such that ∈ and ∈ and cl( ) ∧ cl( ) = 0.

Definition 13. A function
: → is called fuzzy completely continuous [12] if −1 ( ) is fuzzy regular open in for every fuzzy open set of .

Definition 14.
A fuzzy filter base is said to be fuzzy rcconvergent [10] to a fuzzy point in if for any fuzzy regular closed set in containing there exists a fuzzy set ∈ such that ≤ .
Theorem 24. For a fuzzy function : → , if ( ) , for any fuzzy e-closed set ≤ and for any ∈ , feint( −1 ( )) iff there exists a fuzzy e-open set such that and ( ) ≤ .
Proof. Let ≤ be any fuzzy e-closed set and let ( ) .

and is fuzzy e-open in and
. Conversely, let ≤ be any fuzzy e-closed set and let ( ) . By hypothesis, there exists fuzzy e-open set such that and ( ) ≤ . This implies, ≤ −1 ( ) and then fe-int( −1 ( )).

Fuzzy Completely -(rc, eo) Functions
In this section, the notion of fuzzy completely -(rc, eo) functions is introduced and the relation between other functions is studied and further some structure preservation properties are investigated.  Remark 39. Every fuzzy completely -(rc, eo) function is fuzzy -(ec, eo) and fuzzy -continuous, but the converse is not true, which can be seen in the following example.  Proof. Let ≤ be any fuzzy e-closed set and let ( ) . Then, int ( −1 ( )). Take = int ( −1 ( )); then, ( ) = (int ( −1 ( ))) ≤ ( −1 ( )) ≤ ; is fuzzy -open in and .
(2) ⇔ (4): let be a fuzzy closed set. Since fe-cl( ) is fuzzy e-closed set, then by (2) it follows that −1 (fe-cl( )) is fuzzy regular open. The converse is easy to prove.
Definition 46. A fuzzy filter base is said to be fuzzy econvergent to a fuzzy point in if for any fuzzy e-open set in containing there exists a fuzzy set ∈ such that ≤ .

Theorem 47. If a fuzzy function
: → is fuzzy completely -( , ) for each fuzzy point ∈ and each fuzzy filter base in is fuzzy rc-convergent to , then the fuzzy filter base ( ) is fuzzy e-convergent to ( ).

Proof. Let
∈ and let be any fuzzy filter base in which is fuzzy rc-converging to . Since is fuzzy completely -(rc, eo), then for any fuzzy e-open set in containing ( ), there exists a fuzzy regular closed set in containing such that ( ) ≤ . Since is fuzzy rc-converging to , there exists a ∈ such that ≤ . This means that ( ) ≤ and therefore the fuzzy filter base ( ) is fuzzy e-convergent to ( ).
Theorem 48. If : → is a fuzzy completely -( , ) surjection and is fuzzy S-closed, then is fuzzy e-compact.
Theorem 50. If : → is a fuzzy completely -( , ) injection and is fuzzy e-normal, then is fuzzy strongly normal.