The Relations between Characterized Fuzzy Proximity, Fuzzy Compact, Fuzzy Uniform Spaces and Characterized Fuzzy Ts–Spaces and Fuzzy Rk–Spaces

In this research work, we study the relations between the characterized fuzzy Ts–spaces and characterized fuzzy Rk–spaces presented in old papers, for 1 {0,1,2,3,3 ,4} 2 s∈ and 1 {1,2,2 ,3} 2 k ∈ and the characterized fuzzy proximity spaces presented. We also study the relations between the characterized fuzzy Ts–spaces, the characterized fuzzy Rk–spaces and the characterized fuzzy compact spaces which is presented in old paper, as a generalization of the weaker and stronger forms of the G–compactness defined by Gähler. Moreover, we show here the relations between these characterized fuzzy Ts–spaces, characterized fuzzy Rk–spaces and the characterized fuzzy uniform spaces introduced and studied by Abd-Allah in 2013 as a generalization of the weaker and stronger forms of the fuzzy uniform spaces introduced by Gähler. Citation: Abd-Allah AS, Al-Khedhairi A (2017) The Relations between Characterized Fuzzy Proximity, Fuzzy Compact, Fuzzy Uniform Spaces and Characterized Fuzzy Ts–Spaces and Fuzzy Rk–Spaces. J Appl Computat Math 6: 337. doi: 10.4172/2168-9679.1000337


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:L X → 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 1 1 2 2 {0,1, 2, 2 ,3,3 , 4} ∈ s and 1 2 {0,1, 2, 2 ,3} ∈ k . 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 .int δ ) 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 .int δ ). 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 .int τ ) from the fuzzy normal topological space (X,τ) is finer than the associated characterized fuzzy proximity space (X,φ 1,2 .int δ ) by the fuzzy proximity δ defined by (3.6) and they identical if and only if (X,φ 1,2 .int τ ) 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 1 2 2 -FR space and therefore it is characterized 1 2 3 -FT space. There is a good notion of φ 1,2 -fuzzy compactness of the fuzzy filters and of the fuzzy topological spaces introduced and studied by Abd-Allah et al. [7]. This notion fulfills many properties, for example, it fulfills the Tychonoff Theorem. In section 4, we used this notion to study the relations between the characterized fuzzy compact spaces and our classes of the characterized FT s -spaces and of the characterized FR k -spaces. It will be shown that every φ 1,2 -closed subset of a characterized fuzzy compact space is φ 1,2 -fuzzy compact and each φ 1,2 -fuzzy compact subset of the characterized FT 2 -space is φ 1,2 -closed. Also, it will be shown that each characterized fuzzy compact FT 2 -space is characterized FT 4 -space. Specially, we prove that the characterized fuzzy unit interval space (I L ,ψ 1,2 .int I ) is characterized fuzzy compact FT 2 -space and characterized 1 2 3 -FT space. Generally, we show that every characterized fuzzy compact space is characterized FT 2 -space if and only if it is characterized 1 2 3 -FT space. We show that, if (X,ψ 1,2 . int σ ) is characterized fuzzy compact space finer than the characterized FT 2 -space (X,φ 1,2 .int τ ), then (X,φ 1,2 .int τ ) is φ 1,2 ψ 1,2 -fuzzy isomorphic to (X,ψ 1,2 .int σ ). Moreover, if τ is finer than σ, (X,φ 1,2 .int τ ) is characterized fuzzy compact space and (X,ψ 1,2 .int σ ) is characterized 1 2 3 -FT space, then (X,ψ 1,2 .int τ ) and (X,ψ 1,2 .int σ ) are φ 1,2 ψ 1,2 -fuzzy isomorphic. The notion of fuzzy uniform structure had been introduced and studied by Gähler et al. [13]. This notion with the notion of the operations on the class of all fuzzy subsets are applied to introduce and study the notion of characterized fuzzy uniform spaces. In section 5, we introduce and study the relations between the characterized fuzzy uniform spaces and our classes of the characterized FT s -spaces and of the characterized FR kspaces. We show that the fuzzy uniform space (X,U) is separated if and only if the associated characterized fuzzy uniform space (X,φ 1,2 .int U ) is characterized FT i -space but the fuzzy uniform space (X,U) is separated if and only if the associated stratified fuzzy topological space (X,τ U ) is F 1,2 -T i space for all i∈{0,1}. For each fuzzy uniform structure on a set X, we prove that there is an induced stratified fuzzy proximity on L X . Moreover, both the fuzzy uniform structure and this induced stratified fuzzy proximity are associated with the same stratified characterized fuzzy uniform space. Finally, for each fuzzy uniform space (X,U) we prove that the associated stratified characterized fuzzy uniform space (X,φ 1,2 .int U ) with the fuzzy uniform structure U is characterized 1 2 2 -FR space and it 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).

Fuzzy filters
The fuzzy filter on X [1] is the mapping M: L X → L such that the following conditions are fulfilled: 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
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 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 X X X e L 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 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 for all homogeneous fuzzy filters L on F L X, where 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 for all sets X.

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 ( , ) O τ ≤ is 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 A ⊆ L X if ρ ∧ µ φ ≤ (ρ ∧ µ) φ holds, for all ρ ∈A and µ ∈ L X .
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  Then the φ 1,2 -interior of the fuzzy subset µ: X → L is the mapping φ 1,2 .
int µ : X → L defined by: As easily seen that φ 1,2 .int µ 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 .int µ. 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 .int µ is the classical one. Hence, in this case the ordinary subset A of X is The complement of the φ 1,2 -open subset A of X will be called φ 1,2 -closed. 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 .cl T F = F.

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: 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: 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 .int I 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 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 1,2 (F) ϕ N and it will be defined by: 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 :

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 , provided M is finer than the φ 1,2 -fuzzy neighborhood filter

Lemma 2.2
Let (X,τ) be a fuzzy topological space and 1 2

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] for all 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.

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 1 2

Proposition 2.6
Let a fuzzy topological space (X,τ) be fixed and 1 2 ( , ) , Then the following are fulfilled [8,22]: (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 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 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   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) ϕ µ ≤  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 The φ 1,2 -fuzzy neighborhood filter 1,2 ( ) ϕ µ N at the fuzzy subset µ∈L X and the homogeneous fuzzy filter µ  fulfill the following properties.  holds. Hence, x y x y holds also. Thus, x y 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 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.  N are also hold, that is, µ δ η′ and η δ ρ . Thus, (P5) holds and consequently, δ is fuzzy proximity on X.
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 .int δ ) is said to be characterized fuzzy proximity space. The related φ 1,2 -interior and φ 1,2 -closure operators φ 1,2 .int δ and φ 1,2 .cl δ are given by: In the following we will show that the characterized fuzzy proximity space (X,φ 1,2 .int δ ) is characterized FT 0 -space as in sense of [8] if and only if δ is separated.

Proposition 3.3
Let (X,τ) be a fuzzy topological space, 1 2 ( , ) ϕ ϕ ∈ and is a fuzzy proximity on X. Then the characterized fuzzy proximity space (X,φ 1,2 . int δ ) is characterized FT 0 -space if and only if δ is separated.
Proof: Let (X,φ 1,2 .int δ ) is characterized FT 0 -space and let x,y∈X such that x≠y. Then 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 .int δ ) 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  In the following proposition, we show that the associated characterized fuzzy proximity space (X,φ 1,2 .int δ ) 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 1,2  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 .
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  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 with the complementarily symmetric fuzzy topogenous structures ≪ is φ 1,2 ψ 1,2 δfuzzy continuous, because from (3.5) we get that

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

Proposition 3.6
Let (X,φ 1,2 .int δ ) is a characterized fuzzy proximity space and F,G∈P (X) such that F G χ δχ . If Φ is the family of all 1,2 ψ 1,2 δ-fuzzy continuous Proof: Let ≪ be a complementarily symmetric fuzzy topogenous structure identified with δ. Because of (3.5), F G χ δχ implies that. Since f ∈ Φ is φ 1,2 ψ 1,2 δ-fuzzy continuous, then because of Remark 3.2, we have that f is associated with ≪. Hence, Lemma 3.2 implies that χ F and χ G are separated by f and therefore χ F and χ G are Φ-separable.
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.
holds for all. Consider we get sup(µ∧) = 0 and does not exists. Consequently, 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

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 1 2 , 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.
At first, in the following we shall benefit from these facts. Consider the fuzzy unit interval topological space (I L ,ℑ) be given and In the following proposition, we show that every φ 1,2 -fuzzy compact subset in the characterized FT 2 -space (X,φ 1,2 .int τ ) is φ 1,2 -fuzzy closed with respect to the φ 1,2 -interior operator φ 1,2 .int τ . imply that x = y. Therefore, y∈F for some K ∈ F L F.

Proposition 4.2
Let (I L ,ℑ) be a fuzzy unit interval topological space and 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 .int τ ) have two disjoint φ 1,2 -fuzzy neighborhood filters Hence, for every L ∈F L X we get 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 1 2
Proof: Follows directly from Lemma 4.1 and Proposition 4.3.
One of the application of Proposition 4.4, we have more generally the following result to the characterized fuzzy unit interval space.

Proposition 4.5
Let (I L ,ℑ) be an fuzzy unit interval topological space and   Proof: Since the characterized fuzzy cube is product of copies of the characterized fuzzy unit interval space (I L ,ψ 1,2 .int ℑ ) and by means of Proposition 4.2, (I L ,ψ 1,2 .int ℑ ) is characterized fuzzy compact FT 2 -space. Then because of Proposition 2.6, part (3) and Generalized Tychonoff Theorem in Abd-Allah [11], it follows that, the characterized fuzzy cube is characterized FT 2 -space. Moreover, Proposition 5.1, it follows that the characterized fuzzy cube is characterized FT 4 -space.

Proposition 4.8
Let (X,τ) be a fuzzy topological spaces and 1 2

Some Relations Between Characterized FT s , Characterized FR k and Characterized Fuzzy Uniform Spaces
In this section, we are going to investigate and study the relations between the characterized FT s -spaces, the characterized FT k -spaces and the characterized fuzzy uniform spaces presented in Abd-Allah [12]. For this, we applied the notion of homogeneous fuzzy filter at the point and at the fuzzy set which is defined by (2.1), the superior principal fuzzy filter [µ] generated by µ∈L X and the φ 1,2 -fuzzy neighborhoods at the fuzzy set µ which is defined by (3.1) in the characterized fuzzy space (X,φ 1,2 .int τ ). Specially, the relation between the separated fuzzy uniform spaces, the associated characterized fuzzy uniform FT s -spaces, the associated characterized uniform By the fuzzy relation on the set X, we mean the mapping R : X×X → L, that is, any fuzzy subset of X×X. For each fuzzy relation R on X, the inverse R -1 of R is the fuzzy relation on X defined by R -1 (x,y) = R (y,x) for all x,y∈X. Let U be a fuzzy filer on X×X. The inverse U -1 of U is a fuzzy filter on X×X defined by U -1 (R) = U (R -1 ) for all RL X×X . The composition R 1 ᴼ R 2 of two fuzzy relations R 1 and R 2 on the set X is a fuzzy relation on X defined by: for all x,y∈X. For each pair (x,y) of elements x and y of X × X, the mapping (x,y) ⋅ : L X×X → X defined by: (x,y) ⋅ (R) = R (x,y) for all R ∈ X × X is a homogeneous fuzzy filter on X × X. Let U and V are fuzzy filers on X × X such that (x,y) . ≤ U and (y,z) . ≤ V hold for some x,y,z∈X. Then the composition V ᴼ U of V and U is a fuzzy filter [13] on X × X defined by: By the fuzzy uniform structure U on a set X [13], we mean a fuzzy filter on X × X such that the following axioms are fulfilled: (U1) (x,x) . ≤ U for all x ∈ X.
The pair (X,U) is called fuzzy uniform space. The fuzzy uniform structure U [13] on a set X is said to be separated if for all x,y∈X with xÏy there is R ∈ L X×X such that U (R) = 1 and R (x,y) = 0. In this case the fuzzy uniform space (X,U) is called separated fuzzy uniform space. Let U is a fuzzy uniform structure on a set X such that (x,x) . ≤ U holds for all x∈X and let M ∈ F L X then the mapping U[M]: L X → L which is defined by: Each fuzzy uniform structure U on the set X is associated a stratified fuzzy topology τ U on X. Consider 1 2 ( , ) , τ ϕ ϕ ∈ X L O U , then the set of all φ 1,2 -open fuzzy subsets of X related to τ U forms a base for an characterized stratified fuzzy space on X generated by the φ 1,2 -interior operator with respect toτ U denoted by φ 1,2 .int U and (X,φ 1,2 .int U ) is the related stratified characterized fuzzy space. In this case, (X,φ 1,2 .int U ) will be called the associated characterized fuzzy uniform space [12] which is stratified. The related φ 1,2 -interior operator φ 1,2 .int U is given by: (φ 1,2 .int U µ) (x) = U[ẋ](µ) (5.1) for all x∈X and µ∈L X . The fuzzy set µ is said to be φ 1,2 U-fuzzy neighborhood of x∈X in the associated characterized fuzzy uniform space (X,φ 1,2 .int U ), provided [ ] µ ≤   x U . Because of (2.1), (3.1) and (5.1) we have that

Corollary 5.1
Let X be a non-empty set, U is a fuzzy uniform structure on X and

Corollary 5.2
Let X be a non-empty set, U is a fuzzy uniform structure on X and For each fuzzy uniform structure U on the set X, the mapping h : F L X → F L X which is defined by h (M) = [M] U for all M ∈ F L X is global homogeneous fuzzy neighborhood structure on X [13]. The mapping h will be called global homogeneous fuzzy neighborhood structure associated to the fuzzy uniform structure U and will be denoted by h U . Then the mapping f : (X,φ 1,2 .int U ) → (X,φ 1,2 .int U ) between the associated characterized fuzzy uniform spaces is φ 1,2 ψ 1,2 -fuzzy continuous.
In the following, we prove that for each fuzzy uniform structure on a set X, there is an induced stratified fuzzy proximity on L X . Moreover, both the fuzzy uniform structure and this induced stratified fuzzy proximity are associated with the same stratified characterized fuzzy uniform space.

Proposition 5.5
Let X be a non-empty set, U is a fuzzy uniform structure on X and