L-FUZZIFYING PROXIMITY, L-FUZZIFYING UNIFORM SPACE AND L-FUZZIFYING STRONG UNIFORM SPACE

A BSTRACT . In this paper the concept of proximity in L -fuzzifying topology is established and some of its properties are discussed. Furthermore we introduce and study the concepts of L -fuzzifying uniform space and L -fuzzifying strong uniform space.


PRELIMINARIES
In 1993, M. Ying [11] introduced and studied the uniformity in [0, 1]−fuzzifying topology as a fuzzy concept, i.e., as a fuzzy subset of P (X × X) for an ordinary set X.In 2003, H. F. Kheder [5], introduced and studied concepts of proximity and strong uniformity in fuzzifying topology as fuzzy concepts.In this paper we introduce and study the concept of proximity, uniformity and strong uniformity in L-fuzzifying topology.In section 2, we extend the concept of fuzzifying proximity due to (Kheder, et al (2003) [5]) into L-fuzzifying setting.Some of basic properties of this extenstion are studied.
Section 3, is devoted to extend and study the concept of uniformity in the sense of (Ying (1993) [10])in L-fuzzifying topology.Finally, the notion of fuzzifying strong uniform space (Kheder, et al (2003) [5]) is generalized by introducing the concept of L-fuzzifying strong uniform spaces.Some results concerning this concept are obtained.In the present paper L is assumed to be a completely residuated lattice such that the following conditions are satisfied: (1) L is totally ordered as a poset.(i.e. for each a, b ∈ L, a < b, or b < a. ) (2) L satisfies that , ∧ , is disributive over arbitrary joins.
Definition 1.2.[9].A structure (L,∨, ∧, * , →, ⊥, ⊤) is called a complete residuated lattice iff (1) (L,∨, ∧, ⊥, ⊤) is a complete lattice whose greatest and least element are ⊤, ⊥ respectively, (2) (L, * , ⊤) is a commutative monoid, i.e., (a) * is a commutative and associative binary operation on L, and (a) * is isotone, (b) → is a binary operation on L which is antitone in the first and isotone in the second variable, (c) → is couple with * as: The basic operations on the family L X of all L-sets on a non-empty set X was defined as follows: Definition 1.3.[1].A complete lattice L is called completely distributive if the following law is satisfied: ∀{A j |j ∈ J } ⊆ P (L), where P (L) is the power subset of L we have, Definition 1.4.(Csa′ sza ′ r (1978) [2]).A binary relation δ on P (X) × P (X) is called a proximity on a set X if it satisfies the following conditions: The pair (X, δ) is said to be a proximity space.
Definition 1.7.(Csa ′ sza ′ r (1978) [2]).A uniform structure U on a set X is a family of subsets of X × X, called entourage, which satisfies the following properties: The pair (X, U ) is said to be a uniform space.
The following results are given in [Ying (1992) [11]).Definition 1.8.Let X be a set and Then, U is called Fuzzifying uniformity and (X, U) is called fuzzifying uniform space.
Lemma 1.1.Let (X, U) be a fuzzifying uniform space and ℑ ∈ I P (X) defined by: Then ℑ is a fuzzifying topology on X and called the fuzzifying (uniform) topology of U.
The following concepts are given in (Kheder, et. al. ( 2003) [5]).Definition 1.9.Let X be a set and let U : Then, U is called a strong fuzzifying uniformity and (X, U) is called a strong fuzzifying uniform space.

Conclusions
in this paper, the notion of fuzzifying strong uniform space (Kheder, et al (2003) [5]) is generalized by introducing the concept of L-fuzzifying strong uniform spaces.Some results concerning this concept are obtained.In the present paper L is assumed to be a completely residuated lattice such that the following conditions are satisfied: (1) L is totally ordered as a poset.(i.e. for each a, b ∈ L, a < b, or b < a. ) (2) L satisfies that , ∧ , is disributive over arbitrary joins.
In the future, we will study topological notions defined by means of regular open sets when these are planted into the framework of Ying's fuzzifying topological spaces (in Lukasiewicz fuzzy logic).We used fuzzy logic to introduce almost separation axioms (almost Hausdorff)-, (almost-regular)-and (almost-normal).we gave the relations of these axioms with each other as well as the relations with other fuzzifying separation axioms.