Journal of Applied & Computational Mathematics

In this research work, we study the relations between the characterized fuzzy T s –spaces and characterized fuzzy R k –spaces presented in old papers, for


Introduction
The notion of fuzzy filter has been introduced by Eklund et al.By means of this notion the point-based approach to fuzzy topology related to usual points has been developed.The more general concept for fuzzy filter introduced by Gähler [1] and fuzzy filters are classified by types.Because of the specific type of fuzzy filter however the approach of Eklund is related only to the fuzzy topologies which are stratified, that is, all constant fuzzy sets are open.The more specific fuzzy filters considered in the former papers are now called homogeneous.The operation on the ordinary topological space (X,T) has been defined by Kasahara [2] as the mapping φ from T into 2 X such that A ⊆ A φ , for all A∈T.In 1983, Abd El-Monsef et al. [3] extend Kasahara operation to the power set P (X) of a set X.In 1999, Kandil [4] and the author extended Kasahars's and Abd El-Monsef's operations by introducing an operation on the class of all fuzzy subsets endowed with an fuzzy topology τ as the mapping φ : L X → L X such that int µ ≤ µ φ for all µ∈L X , where µ φ denotes the value of φ at µ.
The notions of the fuzzy filters and the operations on the class of all fuzzy subsets on X endowed with a fuzzy topology τ are applied by Abd-Allah in [5][6][7] to introduce a more general theory including all the weaker and stronger forms of the fuzzy topology.By means of these notions the notion of φ 1,2 -fuzzy interior of a fuzzy subset, φ 1,2 -fuzzy convergence and φ 1,2 -fuzzy neighborhood filters are defined and applied to introduced many special classes of separation axioms.The notion of φ 1,2 -interior operator for a fuzzy subset is defined as a mapping φ 1,2 .int:LX → L X which fulfill (I1) to (I5) in Abd-Allah [5].There is a one-to-one correspondence between the class of all φ 1,2 -open fuzzy subsets of X and these operators, that is, the class φ 1,2 OF(X) of all φ 1,2 -open fuzzy subsets of X can be characterized by these operators.Then the triple (X,φ 1,2 .int) as will as the triple (X,φ 1,2 OF (X)) will be called the characterized fuzzy space [5] of φ 1,2 -open fuzzy subsets.The characterized fuzzy spaces are identified by many of characterizing notions in Abd-Allah [5][6][7], for example by the φ 1,2 -fuzzy neighborhood filters, φ 1,2 -fuzzy interior of the fuzzy filters and by the set of φ 1,2 -inner points of the fuzzy filters.Moreover, the notions of closeness and compactness in the characterized fuzzy spaces are introduced and studied by Abd-Allah in [7].The notions of characterized FT s -spaces, Fφ 1,2 -T S spaces, characterized FR k -spaces and Fφ 1,2 -R k spaces are introduced and studied in Abd-Allah [9-11] for all {0,1, 2, 2 ,3,3 , 4} ∈ s and . The notions of characterized fuzzy compact spaces, characterized fuzzy proximity spaces and characterized fuzzy uniform spaces are introduced and studied by the author in 2004 and 2013 in [7,12].This paper is devoted to introduce and study the relations between the characterized FT s and FR k -spaces, for introduce and study the relation between the characterized fuzzy proximity spaces and our classes of the characterized FT s -spaces and of the characterized FR k -spaces.It will be shown that in the characterized fuzzy space (X,φ 1,2 .int), the fuzzy proximity δ will be identified with the finer relation on the φ 1,2 -fuzzy neighborhood filters.Also, we will show that any fuzzy proximity is separated if and only if the associated characterized fuzzy proximity space is characterized FT 0 and to each fuzzy proximity is associated a characterized FR 2 -space in our sense.
Generally, it will be shown that the associated characterized fuzzy proximity space (X,φ 1,2 .intδ ) is characterized FR 2 -space if the related fuzzy topological space (X,τ) is Fφ 1,2 -R 2 space.Moreover, for each characterized FR 3 -space the binary relation on L X defined by means the φ 1,2 -fuzzy closure operator φ 1,2 .cl of τ in Equation (3.6), is fuzzy proximity on X and conversely to each fuzzy proximity δ which has a φ 1,2 -fuzzy closure operator fulfills the binary relation given in (3.6) is associated a characterized FR 3 -space (X,φ 1,2 .intδ ).Moreover, when L is a complete chain, 2 1 X L ϕ ≥ is isotone and φ 1 is wfip with respect to φ 1 OF (X), then we show that the associated characterized fuzzy space (X,φ 1,2 .intτ ) from the fuzzy normal topological space (X,τ) is finer than the associated characterized fuzzy proximity space (X,φ 1,2 .intδ ) by the fuzzy proximity δ defined by (3.6) and they identical if and only if (X,φ 1,2 .intτ ) is characterized FT 4 -space.At the end of this section we prove that the associated characterized fuzzy proximity space (X,φ 1,2 .int δ ) is characterized

Preliminaries
We begin by recalling some facts on fuzzy subsets and on fuzzy filters.Let L be a completely distributive complete lattice with different least and last elements 0 and 1, respectively.Let L 0 = L \ {0}.Sometimes we will assume more specially that L is a complete chain, that is, L is a complete lattice whose partial ordering is a linear one.For a set X, let L X be the set of all fuzzy subsets of X, that is, of all mappings f : X → L. Assume that an order-reversing involution α  α' of L is fixed.For each fuzzy subset µ∈L X , let µ' denote the complement of µ and it is given by the relation µ' (x) = µ (x)' for all x∈X.Denote by ᾱ, the constant fuzzy subset of X with value is α ∈ L. For all x ∈ X and for all α ∈ L 0 , the fuzzy subset x α of X whose value α at x and 0 otherwise is called a fuzzy point in X.The set of all fuzzy points of a set X will be denoted by S (X).
The fuzzy filter M is called homogeneous [14] if M(ᾱ) = α for all α∈L.For each x∈X, the mapping ( ) = ( ) for all µ∈L X is a homogeneous fuzzy filter on X.For each µ∈L X , the mapping for all η∈L X is a homogeneous fuzzy filter on X, called homogenous fuzzy filter at the fuzzy subset µ∈L X .Let F L X and F L X be the sets of all fuzzy filters and of all homogeneous fuzzy filters on X, respectively.If M and N are fuzzy filters on a set X, M is said to be finer than N, denoted by M ≤N, provided M (µ) ≥ N (µ) holds for all µ∈L X .Noting that if L is a complete chain then M is not finer than N, denoted by M N, provided there exists µ∈L X such that M (µ) < N (µ) holds.

Lemma 2.1
If M, N and L are fuzzy filters on a set X. Then the following sentences are fulfilled [1].
For each non-empty set A of the fuzzy filters on X the supremum [1] and given by for all µ∈L X , where n is an positive integer, µ 1 ,…,µ n is a collection such that µ 1 ∧…∧µ n ≤ µ and M 1 ,…, M n are fuzzy filters from A. Let X be a set and µ ∈ L X , then the homogeneous fuzzy filter µ  at µ is the fuzzy filter on X given by:
As shown in Gähler [1], each valued fuzzy filter base Conversely, each fuzzy filter M can be generated by a valued fuzzy filter base, e.g. by is the family of pre filters on X and it is called the large valued fuzzy filter base of M. Recall that the pre filter on X [16] is a non-empty proper subset F of L X such that (1) µ, ρ ∈ F implies µ ∧ρ ∈ F and (2) from µ ∈ F and µ ≤ it follows ρ ∈ F.

Valued and superior principal fuzzy filters
Let a non-empty set X be fixed, µ ∈ L X and α ∈ L such that α ≤ sup µ, the valued principal fuzzy filter [20] generated by µ and α, will be denoted by [µ,α], is the fuzzy filter on X which has 0 ( ) otherwise as a valued fuzzy filter base.For all η∈L X , we have The superior principal fuzzy filter [1] generated by µ, written [µ], is the homogeneous fuzzy filter on X which has ß = {µ ∧ ᾱ | α∈L} ∪ {ᾱ | α∈L} as a superior fuzzy filter base.As shown in Katsaras [18], the superior principal fuzzy filter [µ] is representable by a fuzzy pre filter if and only if sup µ = 1.

Fuzzy filter functors and fuzzy filter monads
The fuzzy filter functor F L : SET→SET is the covariant functor from the category SET of all sets to this category which assigns to each set X the set F L X and to each mapping f : X®Y the mapping F L f :F L X →F L Y. The homogeneous fuzzy filter functor F L : SET→SET is the sub fuzzy filter functor of F L which assigns to each set X the set F L X and to each mapping f : X ® Y the domain-range restriction F L f :F L X →F L Y of the mapping F L f :F L X →F L Y.For each set X, let η X :X→F L X be the mapping defined by η X (x) = ẋ for all x∈X, and let L F be the mapping for which e X (µ) (M) = M (µ) for all µ L X and M∈F L X.Moreover, let be the mapping which assigns to each fuzzy filter L on F L X the fuzzy filter the identity set functor and Ob( )

F F F
SET are natural transformations.(F L ,η,µ) is a monad in the categorical sense, called the fuzzy filter monad [1], that is, for each set X. Related to the sub functor F L of F L , there are analogous natural transformations as η and µ, denoted η′ and µ′, respectively.η′ consists of the rangerestrictions of the mappings η X .µ′ is the family of all mappings : ( ) for all homogeneous fuzzy filters L on F L X, where L F is the mapping given by ( )( ) ( ) for all µ∈ L X and M∈F L X.As has been shown in Gähler et al. [13], (F L ,η′,µ′)is a sub monad of (F L ,η,µ) , that is, for the inclusion mappings

Fuzzy topologies
By a fuzzy topology on a set X [20,21], we mean a subset of L X which is closed with respect to all suprema and all finite infima and contains the constant fuzzy sets 0 and 1 .A set X equipped with an fuzzy topology τon X is called fuzzy topological space.For each fuzzy topological space (X,τ), the elements of τare called open fuzzy subsets of this space.If τ 1 andτ 2 are two fuzzy topologies on a set X, thenτ 2 is said to be finer thanτ 1 andτ 1 is said to be coarser than τ 2 , provided τ 1 ⊆ τ 2 holds.The fuzzy topological space (X,τ) and alsoτ are said to be stratified provided ᾱ∈ holds for all α∈L, that is, all constant fuzzy subsets are open [17].

Fuzzy proximity spaces
The binary relation δ on L X is called fuzzy proximity on X [18], provided it fulfill the following conditions: (P1) µ δ ρ implies ρ δ µ for all µ,ρ∈L X , where δ is the negation of δ.
The set X equipped with an fuzzy proximity δ on X is said to be fuzzy proximity space and will be denoted by (X,δ).Every fuzzy proximity δ on a set X is associated an fuzzy topology on X denoted by τ δ .The fuzzy proximity δ on a set X is said to be separated if and only if for all x,y∈X such that x ≠ y we have α β δ x y for all α,ß∈L 0 .

Operation on fuzzy sets
In the sequel, let a fuzzy topological space (X,τ) be fixed.By the operation [4] on a set X, we mean the mapping φ : L X → L X such that int µ ≤ µ φ holds, for all µ ∈ L X , where µ φ denotes the value of φ at µ.The class of all operations on X will be denoted by ( , ) µ µ for all µ∈L X .Also by the constant operation on ( , ) X L O τ , we mean the operation for all µ∈L X , then obviously, a completely distributive lattice.As an application on this partially ordered relation, the operation φ:L X →L X will be called: (ii) Weakly finite intersection preserving (wfip, for short) with respect to , are said to be dual if µ = co ( (coµ)) or equivalently φµ = co (ψ (coµ)) for all µ ∈ L X , where coµ denotes the complement of µ.The dual operation of φ is denoted by ϕ  .In the classical case of L = {0,1}, by the operation on the set X [3], we mean the mapping φ : P (X) → P (X) such that int A ≤ A φ for all A in the power set P (X) and the identity operation on the class of all ordinary operations O (P (X),T) on X will be denoted by i P (X) , where i P (X) (A) = A for all A∈P (X).

φ-open fuzzy subsets
Let a fuzzy topological space (X,τ) be fixed and  int µ : X → L defined by: As easily seen that φ 1,2 .intµ is the greatest φ 1 -open fuzzy subset ρ such that 2 ϕ ρ less than or equal to µ [5].The fuzzy subset µ is said to be φ 1,2 -open if µ ≤ φ 1,2 .intµ.The class of all φ 1,2 -open fuzzy subsets of X will be denoted by φ 1,2 OF(X).The complement coµ of a φ 1,2 -open fuzzy subset µ will be called φ 1,2 -closed and the class of all φ 1,2 -closed fuzzy subsets of X will be denoted by φ 1,2 CF(X).In the classical case of L = {0,1}, we note that the fuzzy topological space (X,τ) is up to an identification by the ordinary topological space (X,T) and φ 1,2 .intµ is the classical one.Hence, in this case the ordinary subset The class of all φ 1,2 -open and the class of all φ 1,2 -closed subsets of X will be denoted by φ 1,2 O (X) and φ 1,2 C (X), respectively.Clearly, F is φ 1,2 -closed if and only if φ 1,2 .clT F = F.
From Propositions 2.2 and 2.3, if the fuzzy topological space (X,τ) be fixed and 1 2 ( , ) , and the following conditions are fulfilled:

Fuzzy unit interval
The fuzzy unit interval will be denoted by I L and it is defined in Gähler [24] as the fuzzy subset is the set of all positive fuzzy real numbers.Note that, the binary relation ≤ is defined on  L as follows: for all x, y ∈  L , where x z for all α ∈ L 0 .Note that the family Ω which is defined by  is a base for a fuzzy topology I on I L and the order pair (I L ,I) is said to be fuzzy unit interval topological space, where R δ and R δ are the fuzzy subsets of  L defined by for all x∈ L and δ ∈.\The restrictions of R δ and R δ on I L are the fuzzy subsets R δ | I L and R δ |I L , respectively.Recall that the inequality R δ (x) ∧ R γ (y) ≤ R δ+γ (x+y) holds, where x + y is the fuzzy real number defined by: x y x y for all ξ ∈.Consider a fuzzy unit interval topological space (I L ,I) be given and 1 2 , then in this work the characterized fuzzy space (I L ,ψ 1,2 .intI ) will be called characterized fuzzy unit interval space and we define the cartesian product of a number of copies of the fuzzy unit interval I L equipped with the product of the characterized fuzzy unit interval spaces generated by ψ 1,2 .intI on it as a characterized fuzzy cube.φ 1,2 -fuzzy neighborhood filters An important notion in the characterized fuzzy space (X,φ 1,2 .int) is that of the φ 1,2 -fuzzy neighborhood filter at the point and at the ordinary subset in this space.Let (X,τ) be a fuzzy topological space and , . As follows by (I1) to (I5) for each x∈X, the mapping which is defined by: for all µ ∈ L X is a fuzzy filter, called φ 1,2 -fuzzy neighborhood filter at x [5].If ∅≠F∈P (X), then the φ 1,2 -fuzzy neighborhood filter at F will be denoted by and it will be defined by: holds.If the related φ 1,2 -interior operator fulfill the axioms (I1) and (I2) only, then the mapping x N , defined by (2.4) is an fuzzy stack, called φ 1,2 -fuzzy neighborhood stack at x.Moreover, if the φ 1,2interior operator fulfill the axioms (I1), (I2) and (I4) such that in (I4) instead of ρ ∈L X we take ᾱ, then the mapping , is an fuzzy stack with the cutting property, called φ 1,2 -fuzzy neighborhood stack with the cutting property at x. Obviously, the φ 1,2 -fuzzy neighborhood filters fulfill the following conditions: holds for all µ, ρ ∈L X and µ ≤ ρ.
, for all x∈X and µ ∈ L X . Clearly, The characterized fuzzy space (X,φ 1,2 .int) is characterized as the fuzzy filter pre topology [5], that is, as a mapping 1,2 :

N F
such that the conditions (N1) to (N3) are fulfilled.

fuzzy continuity
Let now the fuzzy topological spaces (X,τ 1 ) and (Y,τ 2 ) are fixed, holds for all η∈L Y .If an order reversing involution ′ of L is given, then we have that f is a fuzzy continuous if and only if holds for all µ∈L X .By means of characterizing the φ 1,2 -fuzzy neighborhoods int which are defined by (2.4), the fuzzy continuity of f can also be characterized as follows: The mapping f: Obviously, in case of L = {0,1}, , the φ 1,2 ψ 1,2 -fuzzy continuity coincides with the usual fuzzy continuity.

φ 1,2 -fuzzy convergence
Let an fuzzy topological space (X,τ) be fixed and 1 2   ( , )   , x is a point in the characterized fuzzy space (X,φ 1,2 .int),F ⊆ X and M is a fuzzy filter on X.Then M is said to be φ 1,2 -fuzzy convergence [5] to x and written . Moreover, M is said to be φ 1,2 -fuzzy convergence to F and written

Characterized fuzzy R k and fuzzy φ 1,2 R k -spaces
The notions of characterized fuzzy R k and fuzzy φ 1,2 R k -spaces are introduced and studied in Abd-Allah [9,11] Moreover, the notion of φ 1,2 -fuzzy neighborhood filter at the point x and at the ordinary subset of the characterized fuzzy space (X,φ 1,2 .int) is applied by Abd-Allah [10], to introduced and studied the notions of characterized fuzzy R k -spaces for k∈{2,3}.However, the notions of fuzzy 1,2 R k -spaces are also given by means of the φ 1,2 -fuzzy convergence at the point x and at the ordinary subset in the space.We will denote by characterized FR k and Fφ 1,2 R k -spaces to the characterized fuzzy R k and fuzzy φ 1,2 R k -spaces for shorts, respectively.
Let a fuzzy topological space (X,τ) be fixed and ) does not exists.The related fuzzy topological space (X,τ) is said to be Fφ 1,2 R 2 -space (resp.Fφ 1,2 R 3 -space) if for all x∈X (resp.F ∈ φ 1,2 .C (X)) and (2) Characterized f x and ( ) = 0 f y for all y∈F.The related fuzzy topological space (X,τ) is said to be

Characterized fuzzy T s and fuzzy φ 1,2 -T S spaces
The notions of characterized fuzzy T s and fuzzy φ 1,2 -T S spaces are investigated and studied by Abd-Allah and by Abd-Allah and Al-Khedhairi in [8,9,11] for all . These characterized fuzzy spaces depend only on the usual points and the operation defined on the class of all fuzzy subsets of X endowed with a fuzzy topological space (X,τ).We will denote by characterized FT s and Fφ 1,2 -T s spaces to the characterized fuzzy T s and fuzzy φ 1,2 -T S spaces for shorts, respectively.

Proposition 2.4
Let (X,τ) be an fuzzy topological space and
(2) The characterized fuzzy subspace and the characterized fuzzy product space of a family of characterized FT 2 -spaces are also characterized FT 2 -spaces .

New Relations between Characterized FT s , Characterized FR k and Characterized Fuzzy Proximity Spaces
In this section we are going to introduce and study the relations between the characterized FT s -spaces, the characterized FR k -spaces and the characterized fuzzy proximity spaces presented by Abd-Allah in [12].We make at first the relation between the farness on fuzzy sets and the finer relation on fuzzy filters.So, we show some results for the notion of the φ 1,2 -fuzzy neighborhood filter

Proposition 3.1
Let a fuzzy topological space (X,τ) be fixed and 1 2 ( , ) , is isotone and idempotent and φ 1 is wfip with respect to φ 1 OF (X).Then the supremum of the φ 1,2 -fuzzy neighborhood filters 1,2 ( ) at x∈X which is given by: for all µ∈L X is a fuzzy filter on X called a φ 1,2 -fuzzy neighborhood filter at µ.
Proof: Fix an α∈L 0 , then because of (2.4) and the condition

N
Thus, condition (F 1 ) is fulfilled.To prove condition (F 2 ), let ρ,η∈L X , then because of Proposition 2.4 and (2.4) we have the mapping φ 1,2 int ρ.Then from Proposition 2.4 we have
Not that in Bayoumi et al. [15], the supremum of the empty set of the fuzzy filters is the finest fuzzy filter.This means 1,2 (0) N for all µ∈L X .Because of (2.4) the equations (2.1) and (2.2) can be written as in the following: for all ρ∈L X .Here a useful remark is given Remark 3.1: The homogeneous fuzzy filter ẋ at the ordinary point x is nothing that a homogeneous fuzzy filter ẋ at the fuzzy point x α , that is, α  x = ẋ for all x∈X and α∈L 0 .Moreover, the φ 1,2 -fuzzy neighborhood filter 1,2 ( ) The φ 1,2 -fuzzy neighborhood filter 1,

N
at the fuzzy subset µ∈L X and the homogeneous fuzzy filter µ  fulfill the following properties.

Lemma 3.1
Let (X,τ) be a fuzzy topological space and 1 2 ( , ) , . Then for all µ,ρ∈L X the following properties are fulfilled: holds and therefore for all η∈L X we have x y holds.Hence, x y holds also.Thus, Hence, holds for all η∈L X and therefore Since µ, ρ ≤ µ Ú ρ, then from (2) we have

N
. Then x y holds for all λ∈L X .Hence, there is η∈L X such that This means there is η∈L X such that 1,2 ( )( ) ( )

N
are hold for all λ∈L X .Thus, ≤  N are also hold.Consequently, ( 5) is fulfilled.
In the characterized fuzzy space (X,φ 1,2 .int), the fuzzy proximity will be identified with the finer relation on the fuzzy filters, specially with the finer relation on the φ 1,2 -fuzzy neighborhood filters.This shown in the following proposition.
If a fuzzy topological space (X,τ) be fixed and 1 2 ( , ) , . Then each fuzzy proximity δ on X is associated a set of all φ 1,2 -open fuzzy subsets of X with respect to δ denoted by φ 1,2 OF (X) δ .In this case the triple (X,φ 1,2 OF (X) δ ) as will as (X,φ 1,2 .intδ ) is said to be characterized fuzzy proximity space.The related φ 1,2 -interior and φ 1,2 -closure operators φ 1,2 .intδ and φ 1,2 .clδ are given by: .int = and .cl= respectively, for all µ∈L X .Consider the characterized fuzzy proximity space (X,φ 1,2 .intδ ) be fixed and µ∈L X , then µ is said to be φ In the following we will show that the characterized fuzzy proximity space (X,φ 1,2 .intδ ) is characterized FT 0 -space as in sense of [8] if and only if δ is separated.
Conversely, let δ is separated fuzzy proximity and let x,y∈X such that x≠y.Then, 1 1 x y δ and because of Proposition 3.2 and Remark 3.1, we have Hence, there exists
In the following proposition, the 1,2 -closure of the fuzzy subsets in the characterized fuzzy space (X,φ 1,2 .intδ ) are equivalent with the fuzzy subsets by the fuzzy proximity δ on X.

Proposition 3.4
Let (X,τ) be a fuzzy topological space, 1 2 ( , ) , and δ is a fuzzy proximity on X.Then, µδρ if and only if . Thus, In the following proposition, we show that the associated characterized fuzzy proximity space (X,φ 1,2 .intδ ) is characterized FR 2space if the related fuzzy topological space (X,τ) is Fφ 1,2 -R 2 space.

Proposition 3.5
Let (X,τ) be a fuzzy topological space, 1 2 ( , ) , and is an fuzzy proximity on X.Then the associated characterized fuzzy proximity space (X,φ The binary relation << on L X is said to be fuzzy topogeneous order on X [23], if the following conditions are fulfilled: (1) ᾱ  ᾱ for all α∈{0,1}.
The fuzzy topogeneous order ≪ is said to be fuzzy topogeneous structure if it fulfilled the condition: (5) If ≪η, then there is σ∈L X such that <<σ and σ <<η are hold for all µ,η∈L X .The fuzzy topogeneous structure ≪ is said to be fuzzy topogenous complementarily symmetric if it fulfilled the condition: (6) If <<η, then η′ <<µ holds for all µ,η∈L X .
As shown in Katsaras [23], every fuzzy topogeneous structure ≪ is identify with the mapping N: L X → P (L X ) such that η∈N (µ) if and only if µ <<η holds for all µ,η∈L X .The fuzzy topogeneous structures are classified by these mappings.As is easily seen, each fuzzy topogeneous order N can be associated a fuzzy pre topology int N on a set X by defining for all µ ∈L X .In case of N is fuzzy topogeneous structure, int N is interior operator for fuzzy topology τ N on X associated toN.Obviously, there is an identification between the fuzzy proximity δ and the complementarily symmetric fuzzy topogenous structure ≪ on the same set X given by: for all µ,η∈∈L X .If is a sequence of fuzzy topogenous structure on the set X and is a sequence of fuzzy topogenous structure on I L , then the fuzzy real function f : X → I L is said to be associated with the sequence and n∈ + , where  + is the set of all positive integer numbers.

Remark 3.2
Given that are two sequence of complementarily symmetric fuzzy topogenous structures ≪ and  on X and I L , respectively.If δ and δ * are two fuzzy proximities on X and I L identified with δ and δ * by the equation (3.5), then the associated fuzzy real function

Lemma 3.2
Consider ≪ n for n {0,1,…,} are complementarily symmetric fuzzy topogenous structures on a set X.Then, for each F,G∈P (X) such that χ F ≪ 0 χ G there exists a fuzzy real function f : X → I L associated with the sequence
In the following we are going to show an important relation between the associated characterized fuzzy proximity space and the characterized FR 3 -space.

Proposition 3.8
Let (X,τ) be a fuzzy topological space and 1 2 ( , ) , is isotone and φ 1 is wfip with respect to φ 1 OF (X), where L is complete chain.If (X,τ) is a fuzzy normal topological space, then the binary relation δ on X which is defined by: for all µ,ρ∈L X is a fuzzy proximity on X and (X,δ) is a fuzzy proximity space.On other hand if (X,δ) is a fuzzy proximity space with δ fulfills (3.6), then the associated characterized fuzzy proximity space (X,ψ 1,2 ,int δ ) is characterized FR 3 -space.
Hence, (P5) is also fulfilled.Consequently, δ is a fuzzy proximity on X.
In the following we are going to show an important relation between the associated characterized fuzzy proximity space (X,ψ 1,2 .int δ ) by the fuzzy proximity δ defined by (3.6) and the associated characterized fuzzy space (X,ψ 1,2 .int)that introduced form the fuzzy normal topological space (X,τ).
In the following we are going to introduce some important relations joining our characterized

-FT
spaces and the associated characterized fuzzy proximity spaces.

Corollary 3.1
Let (X,τ) be a fuzzy topological space, 1 , and δ is a fuzzy proximity on X.Then the associated characterized fuzzy proximity space (X,φ 1,2 .intδ ) is characterized Proof : Immediately from Propositions 2.4 and 3.10.Now, we introduce an example of an fuzzy proximity δ on a set X and show that it is induces an associated characterized = {1, 0, , } x y is a fuzzy topology on X. Choose φ 1 = int τ ,, φ 2 = cl τ , ψ 1 = int I , and ψ 2 = cl.Hence, x ≠ y and there is only two cases, the first is xÏ F = {y}∈φ 1,2 C(X) and the second is y Ï F = {x} ∈φ 1,2 C (X).We shall consider the first case and the second case is similar.Consider the mapping f : 3 -FT space.Now, consider δ is a binary relation on L X defied as follows: with and ( ) = 0 for all , µ η ≤ ≤ x f y y for all µ,η∈L X .Hence obviously, δ is a fuzzy proximity on X and (X,φ 1,2 .int τ ) = (X,φ 1,2.int δ ), that is, the associated characterized fuzzy proximity space (X,φ 1,2 .intδ ) with δ is characterized 1 2

Some Relations between Characterized FT s and Characterized Fuzzy Compact Spaces
The notion of φ 1,2 -fuzzy compactness of the fuzzy filters and of the fuzzy topological spaces are introduced by Abd-Allah in [7] by means of the φ 1,2 -fuzzy convergence in the characterized fuzzy spaces.Moreover, the fuzzy compactness in the characterized fuzzy spaces is also introduced by means of the φ 1,2 -fuzzy compactness of the fuzzy filters and therefore it will be suitable to study here the relations between the characterized fuzzy compact spaces and some of our classes of separation axioms in the characterized fuzzy spaces.
Let (X,τ) be an fuzzy topological space, F ⊆ X and

M N
implies x∈F for some M∈F L F. The subset F is said to be φ 1,2 -fuzzy compact [7], if every fuzzy filter on F has a finer φ 1,2 −fuzzy converging filter, that is, every fuzzy filter on F has φ 1,2 -adherence point in F.Moreover, the fuzzy topological space (X,τ) is said to be φ 1,2 -fuzzy compact if X is φ 1,2 -fuzzy compact.More generally, the characterized fuzzy space (X,φ 1,2 .int) is said to be fuzzy compact space if the related fuzzy topological space (X,τ) is φ 1,2 -fuzzy compact.

Proposition 4.2
Let (I L ,ℑ) be a fuzzy unit interval topological space and Consequently, (I L ,ψ 1,2 .intI ) is characterized FT 2 -space and therefore (I L ,ψ 1,2 .intI )is characterized fuzzy compact FT 2 -space.Now, we are going to prove an important relation between the characterized compact FT 2 -spaces and the characterized FT 4 -spaces.For this reason at first, we give a new property for the characterized FT 2 -spaces by using the φ 1,2 -fuzzy neighborhood filters for the fuzzy subsets.

Proposition 4.3
Let (X,τ) be n fuzzy topological space and 1 2 ( , ) , . Then every disjoint φ 1,2 -fuzzy compact subsets F 1 and F 2 of in the characterized FT 2 -space (X,φ 1,2 .intτ ) have two disjoint φ 1,2 -fuzzy neighborhood filters Proof: Let F 1 and F 2 are two φ 1,2 -fuzzy compact subsets of the characterized FT 2 -space (X,φ 1,2 .intτ ) such that F 1 ∩ F 2 = ∅.Then, for all M i ∈ F L F i there exists K i ∈F L F i such that K i ≤ M i and 1,2 ( ) x N K for some x i F i , where i∈{1,2}.Since F L F i ≤ F L X for all i∈{1,2}, then we can say that ≤ ≤ i i i x F

N N
K and therefore there is for some x i F i .Since (X,φ 1,2 .intτ ) is characterized FT 2 -space, then x 1 = x 2 which contradicts F 1 ∩ F 2 = ∅.Hence, for every L ∈F L X we get which means that the infimum ∧ F F

N N
does not exists and therefore F 1 and F 2 can be separated by two disjoint φ 1,2 -fuzzy neighborhood filters.
Secondly, the notion of the fuzzy compactness for the characterized fuzzy spaces fulfills the following property which will be also used in the prove of this important result which given in Proposition 4.4.

Lemma 4.1
Let (X,τ) be a fuzzy topological space and

Volume 6 •
Issue 1 • 1000337 J Appl Computat Math, an open access journal ISSN: 2168-9679 subset µ : X → L is said to be φ-open fuzzy subset if µ ≤ µ φ holds.We will denote the class of all φ-open fuzzy subsets on X by OF (X).The fuzzy subset µ is called φ-closed if its complement coµ is φ-open.written φ ~ ψ if and only if φ OF (X) = ψ OF (X).

F
. Hence because of Lemma 3.1 part (1) we Volume 6 • Issue 1 • 1000337 J Appl Computat Math, an open access journal ISSN:

( 1 )
The usual topological space (I,T I ) and the ordinary characterized usual space unite interval I = [0,1] are ψ 1,2compact T 2 space and characterized compact T 2 -space, respectively in the classical sense.

.KK
Then every φ 1,2 -fuzzy compact subset of the characterized FT 2 -space is1,2 -closed.Proof: Let (X,φ 1,2 .intτ ) is characterized FT 2 -space and F is φ 1,2 -fuzzy compact subset of X.Then, for all M ∈ F L F, there exists K ∈ F L F such that K ≤ M and 1for some x∈F.Since K ∈ F L F ≤ F L X and (X,φ 1,2 .intτ ) is characterized FT 2 -space, then1imply that x = y.Therefore, y∈F for some K ∈ F L F.

N
for which F 1 and F 2 are separated by them.
φ 1,2 -fuzzy closed subset of the characterized fuzzy compact space (X,φ 1,2 .intτ ) is φ 1,2 -fuzzy compact.Proof: Let F is φ 1,2 -fuzzy closed subset of the characterized fuzzy compact space (X,φ 1,2 .intτ ) and let M ∈F L F. Then, that x∈F.Since F L F ≤ F L X, then M∈F L X and hence there exists K∈F L X such that K ≤ M and 1,2 ( )ϕ ≤ x N K .Since M ∈ F L F and K ≤ M, then K ∈F L F. Thus, for all M ∈F L F we get K ≤ M such that 1,2 ( ) ϕ ≤ x N K .Therefore, x∈F is φ 1,2 -adherence point of M, that is, F is φ 1,2 -fuzzy compact.
1,2 δ-fuzzy neighborhood for the point x∈X if and only if 1 δ µ′ [7]shown in Abd-Allah[7], the point x∈X is said to be φ 1,2 -adherence point for the fuzzy filter M on X if and only if there exists an fuzzy filter K∈F L X finer than M and The subset F of X is said to be φ 1,2 -fuzzy closed with respect to φ 1,2.int if ϕ x N at x∈X. ϕ ≤ x