Initial and Final Characterized Fuzzy 13 2 T and Finer Characterized Fuzzy 12 2 R-Spaces

Basic notions related to the characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces are introduced and studied. The metrizable characterized fuzzy spaces are classified by the characterized fuzzy 1 2 2 R and the characterized fuzzy T4-spaces in our sense. The induced characterized fuzzy space is characterized by the characterized fuzzy 1 3 2 T and characterized fuzzy 1 3 2 T -space if and only if the related ordinary topological space is 1 2 , 2 R φ1 2 -space and 1 3 , 2 T φ1 2 -space, respectively. Moreover, the α-level and the initial characterized spaces are characterized 1 2 2 R and characterized 1 3 2 T -spaces if the related characterized fuzzy space is characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T , respectively. The categories of all characterized fuzzy 1 2 2 R and of all characterized fuzzy 1 3 2 T -spaces will be denoted by CFR-Space and CRF-Tych and they are concrete categories. These categories are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets and hence all the initial and final lifts exist uniquely in CFR-Space and CRF-Tych. That is, all the initial and final characterized fuzzy 1 2 2 R spaces and all the initial and final characterized fuzzy 1 3 2 T -spaces exist in CFR-Space and in CRF-Tych. The initial and final characterized fuzzy spaces of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. As special cases, the characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are also characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. Finally, three finer characterized fuzzy 1 2 2 R -spaces and three finer characterized fuzzy 1 3 2 T -spaces are introduced and studied. Citation: Abd-Allah AS, Al-Khedhairi A (2017) Initial and Final Characterized Fuzzy 1 3 2 T and Finer Characterized Fuzzy 1 2 2 R -Spaces. J Appl Computat Math 6: 350. doi: 10.4172/2168-9679.1000350


Introduction
Eklund and Gahler [1] introduced the notion of fuzzy filter and by means of this notion the point-based approach to the fuzzy topology related to usual points has been developed. The more general concept for the fuzzy filter introduced by Gahler [2] and fuzzy filters are classified by types. Because of the specific type of the L-filter however the approach of Eklund and Gahler [1] is related only to the L-topologies which are stratified, that is, all constant L-sets are open. The more specific fuzzy filters considered in the former papers are now called homogeneous. The notion of fuzzy real numbers is introduced by Gahler and Gahler [3], as a convex, normal, compactly supported and upper semi-continuous fuzzy subsets of the set of all real numbers R. The set of all fuzzy real numbers is called the fuzzy real line and will be denoted by R L , where L is complete chain.
The operation on the ordinary topological space (X,T) has been defined by Kasahara [4] as a mapping φ from T into 2 X such that A ⊆ A φ , for all A ∈ T. Abd El-Monsef et al. [5], extend Kasahara [4] operation to the power set P (X) of the set X Kandil et al. [6] extended Kasahars's and Abd El-Monsef's operations by introducing operation on the class of all fuzzy sets endowed with an fuzzy topology τ as a 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 fuzzy filters and the operations on the class of all fuzzy sets on X endowed with an fuzzy topology τ are applied in ref.
[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 -interior of the fuzzy set, φ 1,2 -fuzzy convergence and φ 1,2 -fuzzy neighborhood filters are defined. The notion of φ 1,2 -interior operator for the fuzzy sets is also defined as a mapping φ 1,2 .int: L X → L X which fulfill (I1) to (I5). Since there is a one-to-one correspondence between the class of all φ 1,2 -open fuzzy subsets of X and these operators, then the class φ 1,2 OF (X) of all φ 1,2 -open fuzzy subsets of X is characterized by these operators. Hence, 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 of φ 1,2 -open fuzzy subsets. For each characterized fuzzy space (X, φ 1,2 .int) the mapping which assigns to each point x of X the φ 1,2 -fuzzy neighborhood filter at x is said to be φ 1,2 -fuzzy filter pre topology [7]. It can be identified itself with the characterized fuzzy space (X, φ 1,2 .int). The characterized fuzzy spaces are characterized by many of characterizing notions, for example by: φ 1,2 -fuzzy neighborhood filters, φ 1,2 -fuzzy interior of the fuzzy filters and by the set of all φ 1,2 -inner points of the fuzzy filters. Moreover, the notions of closeness and compactness in characterized fuzzy spaces are introduced and studied in ref. [8]. For an fuzzy topological space (X, τ), the operations on (X, τ) and on the fuzzy topological space (I L , I), where I=[0, 1] is the closed unit interval and I is the fuzzy topology defined on the left unit interval I L are applied to introduced and studied the notions of characterized fuzzy 1

Preliminaries
We begin by recalling some facts on fuzzy sets and fuzzy filters. Let L be a completely distributive complete lattice with different least and last elements 0 and 1, respectively. Consider L 0 =L\{0} and L 1 =L\{1}. Recall that the complete distributivity of L means that the distributive law ( ) ( ) 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. The standard example of L is the real closed unit interval I=[0, 1]. For a set X, let L X be the set of all fuzzy subsets of X, that is, of all mappings µ: X → L. Assume that an order-reversing involution α 7→α ′ is fixed. For each fuzzy set µ, let co µ denote the complement of µ defined by: (co µ) (x)=co µ(x) for all x ∈ X. For all x ∈ X and α ∈ L 0 . Supµ means the supremum of the set of values of µ. The fuzzy sets on X will be denoted by Greek letters as µ, η, ρ,. . . etc.
Denote by α the constant fuzzy subset of X with value α ∈ L. The fuzzy singleton x α is an fuzzy set in X defined by x α (x)=α and x α (y)=0 for all y x ≠ , α ∈ L 0 . The class of all fuzzy singletons in X will be denoted by S(X). For every x α ∈ S(X) and µ ∈ L X , we write x α ≤ µ if and only if α ≤ µ(x). The fuzzy set µ is said to be quasi-coincident with the fuzzy set ρ and written µ q ρ if and only if there exists x ∈ X such that µ(x)+ ρ(x)>1.
If µ not quasi-coincident with the fuzzy set ρ, then we write q µ ρ . The fuzzy filter on X [14] is the mapping M: L X →L such that the following conditions are fulfilled: ≤ for all α ∈ L and (1)=1.
The fuzzy filter  is said to be homogeneous [14] if x µ µ x = for all µ ∈ L X is a homogeneous fuzzy filter on X. The homogenous fuzzy filter at the fuzzy set is defined by the same way as follows, for each µ ∈ L X , the mapping µ: L X → L defined by for all σ ∈ L X is also homogenous fuzzy filter on X, called homogenous fuzzy filter at µ ∈ L X . Obviously, the relation between homogenous fuzzy filter µ˙ at µ ∈ L X and the homogenous fuzzy filter x˙ at x ∈ X is given by: for all η ∈ L X . As shown in ref. [15], µ ≤ η if and only if . . µ η ≤ holds for all µ, η ∈ L X . Let  L X and  L X denote to the sets of all fuzzy filters and of all homogeneous fuzzy filters on X, respectively. If  and  are fuzzy filters on the set X, then  is said to be finer than , denoted by  ≤ , provided  (µ) ≥  (µ) holds for all µ ∈ L X . Noting that if L is a complete chain then M is not finer than N, denoted by  ̸ ≤ , provided there exists µ ∈ L X such that  (µ) <  (µ) holds. As shown in ref. [4], if ,  and L are three fuzzy filters on a set X, then we have: The coarsest fuzzy filter  on X is the fuzzy filter has the value 1 at 1 and 0 otherwise. Suprema and infimum of sets of fuzzy filters are meant with respect to the finer relation. An fuzzy filter  on X is said to be ultra [2] fuzzy filter if it does not have a properly finer fuzzy filter. For each fuzzy filter  ∈  L X there exists a finer ultra fuzzy filter U ∈  L X such that U ̸ ≤ . Consider  is a non-empty set of fuzzy filters on X, then the supremum M A M Fuzzy filter bases. A family (B α ) α ∈ L0 of non-empty subsets of L X is called a valued fuzzy filter base [2] if the following conditions are fulfilled: (V2) For all α, β ∈ L 0 with α∧β ∈ L 0 and all µ ∈  α and η ∈  β there are γ ≥ α∧β and σ ≤ µ ∧ η such that σ ∈  γ .
As shown in ref. [2], each valued fuzzy filter base ( α ) α ∈ L defines an fuzzy filter  on X by = ∨ for all µ ∈ L X . Conversely, each fuzzy filter  can be generated by a valued fuzzy filter base, e.g., by (α-pr ) α ∈ L0 with α-pr M={µ∈L X ⃒ α≤(µ)}. (α-pr ) α ∈ L0 is a family of pre filters on X and it is called the large valued filter base of . Recall that a pre filter on X [17] is a non-empty proper subset of  of L X such that (1) µ, η ∈  X implies µ ∧ η ∈  and (2) from µ ∈  and µ ≤ η it follows η ∈ .  subset  of L X is said to be superior fuzzy filter base [2] if the following conditions are fulfilled: (S1) B α ∈ for every α ∈ L.
Each superior fuzzy filter base  generated a homogeneous fuzzy filter  on X by filter  can be generated by a superior fuzzy filter base, e.g., by base where base M will be called the large superior fuzzy filter base of . If X is a non-empty set and µ is an fuzzy subset of X, then is a superior fuzzy filter base of a homogeneous fuzzy filter on X, called superior principal fuzzy filter generated by µ and will be denoted by [µ]. In case L is a complete chain and µ is not constant we have [µ] (η)=sup µ, when µ ≤ η and [ ]( ) For each ordinary subset M of X we have that [ ] where χ M is the characteristic function of M.

Fuzzy topology
By the fuzzy topology on a set X, we mean a subset of L X which is closed with respect to all supreme and all finite infimum and contains the constant fuzzy sets 0 and 1 [16,18]. A set X equipped with an fuzzy topology τ on X is called an fuzzy topological space. For each fuzzy topological space (X, τ), the elements of τ are called the open fuzzy subsets of this space. If τ 1 and τ 2 are fuzzy topologies on a set X, then τ 1 is said to be finer than τ 2 and τ 2 is said to be coarser than τ 1 , provided τ 2 ⊆ τ 1 holds. For each fuzzy set µ ∈ L X , the strong α-cut and the weak α-cut of µ are the ordinary subsets of X respectively. For each complete chain L, the α-level topology and the initial topology [19] of an fuzzy topology τ on the set X are defined as follows: respectively, where inf is the infimum with respect to the finer relation for topologies. On other hand if (X, T) is an ordinary topological space, then the induced fuzzy topology on X is given by Lowen [17] as the following: The fuzzy topological space(X, τ) and also τ are said to be stratified provided α ∈ τ holds for all α ∈ L, that is, all constant fuzzy sets are open [19].

The fuzzy unit interval
The fuzzy unit interval will be denoted by I L an it is defined in [3] as the fuzzy subset: where I=[0, 1] is the real unit interval and is the set of all positive fuzzy real numbers. Note that, the binary relation ≤ is defined on R L as follows: for all x, y ∈ R L , where for all α ∈ L 0 . Note that the family Ω which is defined by: is a base for an fuzzy topology I on I L , where R δ and R δ are the fuzzy subsets of R L defined by for all x ∈ R L and δ ∈ R. 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: where, x+y is the fuzzy real number defined by

Operation on fuzzy sets
In the sequel, let a fuzzy topological space (X, τ) be fixed. By the operation [6] on the 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 (L x , τ) . By the identity operation on O (L X , τ) , we mean the operation 1 L X : for all µ ∈ L X . If ≤ is a partially order relation on O (L X , τ) defined as follows: for all µ ∈ L X , then (O (L X , τ) , ≤) is a completely distributive lattice. The operation φ: L X → L X is called: (ii) Weakly finite intersection preerving (wfip, for short) with respect to ) 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 a set X we mean the mapping φ: P (X) → P (X) such that int A ⊆ A φ for all A ∈ 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) and it defined by: i P (X) (A)=A for all A ∈ P (X).

The φ-open fuzzy sets
Let a fuzzy topological space (X, τ) be fixed and φ ∈ O (L X, τ) . The fuzzy set µ: X → L is said to be φ-open fuzzy set if µ ≤ µ φ holds. We will denote the class of all φ-open fuzzy sets on X by φ of (X). The fuzzy set µ is called φ-closed if its complement coµ is φ-open. The operations φ, ψ ∈ O (L x , τ) are equivalent and written φ ∼ ψ if φ of (X)=ψ of (X).

The φ 1,2 -interior fuzzy sets
Let a fuzzy topological space (X, τ) be fixed and . Then the φ 1,2 -interior of the fuzzy set µ: X → L is a mapping φ 12 .intµ: X → L defined by: .
That is, the φ 1,2 .intµ is the greatest φ 1 -open fuzzy set η such that η φ2 less than or equal to µ [19]. The fuzzy set µ is said to be φ 1,2 -open if and only if µ ≤ φ 1,2 .int µ. The class of all φ 1,2 -open fuzzy sets on X will be denoted by φ 1,2 OF (X). The complement co µ of the φ 1,2 -open fuzzy subset µ will be called φ 1,2 -closed, 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}, 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 φ 1,2 -open if A ⊆ φ 1,2 . int A. The complement of a φ 1,2 -open subset A of X will be called φ 1,2closed. 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.

Proposition
Let (X, τ) be a fuzzy topological space and φ 1 , φ 2 ∈ O (L X , τ) . Then the following are fulfilled: , then the class φ 1,2 OF (X) of all φ 1,2open fuzzy sets on X forms a supra fuzzy topology on X [21].
If φ 2 ≥ 1 L X is isotone and φ 1 is with respect to φ 1 OF (X), then φ 1,2 OF (X) is an fuzzy pre topology on X [21].

Characterized Fuzzy Spaces
Independently on the fuzzy topologies, the notion of φ 1,2 -interior operator for the fuzzy sets can be defined as a mapping φ 1,2 .int: L X → L X which fulfill (I1) to (I5). It is well-known that (2.3) and (2.4) give a one-to-one correspondence between the class of all φ 1,2 -open fuzzy sets and these operators, that is, φ 1,2 OF (X) can be characterized by the φ 1,2 -interior operators. In this case the triple (X, φ 1,2 .int) as well as the triple (X, φ 1,2 OF (X)) will be called characterized fuzzy space for all α ∈ L. As shown in ref.

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 points and at the ordinary subsets of this space. Let (X, τ) be a fuzzy topological space and ( , ) , .
As follows by (I1) to (I5) for each x ∈ X, the mapping 1,2 ( ) : for all µ ∈ L X , is a fuzzy filter on X, called φ 1,2 -fuzzy neighborhood filter at x [7]. If the related φ 1,2 -interior operator fulfill the axioms (I1) and (I2) only, then the mapping , defined by (2.5) is fuzzy stack [21], called φ 1,2 -fuzzy neighborhood stack at x. Moreover, if the φ 1,2 -interior operator fulfill the axioms (I1), (I2) and (I4) such that in (I4) instead of η ∈ L X we take α¯, then the mapping , defined by (2.5) is a fuzzy stack with the cutting property, called φ 1,2fuzzy neighborhood stack with the cutting property at x. The φ 1,2 -fuzzy neighborhood filters fulfill the following conditions: is the fuzzy set , .int µ is characterized as the fuzzy filter pre topology [7], that is, as a mapping , : such that (N1) to (N3) are fulfilled.

Initial characterized fuzzy spaces
In the following let X be a set, let I be a class and for each i ∈ I, let (X i , δ 1,2 .int i ) be a characterized fuzzy space of δ 1,2 -open fuzzy subsets of X i and f i : X → X i is the mapping from X into X i . By the initial φ 1,2fuzzy interior operator of (δ 1,2 .int i ) i ∈ I with respect to (f i ) i ∈ I , we mean the coarsest φ 1,2 -fuzzy interior operator φ 1,2 .int on X for which all mappings f i : (X, φ 1,2 .int) → (X i , δ 1,2 .int i ) are φ 1,2 δ 1,2 -fuzzy continuous. The triple (X, φ 1,2 .int) is said to be initial characterized fuzzy space [7] of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I . The initial φ 1,2 -fuzzy interior operator φ 1,2 .int: L X → L X of (δ 1,2 .int i ) i∈I with respect to (f i ) i∈I always exists and is given by: .
for all µ ∈ L X . For each i ∈ I, let , : is the representation of δ 1,2 .int i as an fuzzy filter pre topology. Then because of (2.5) and (2.7), the mapping N φ1,2 : X → F L X which is defined by: for all x ∈ X and µ ∈ L X , is the representation of the initial φ 1,2 -fuzzy interior operator of (ψ 1,2 .int i ) i∈I with respect to (f i ) i∈I as the fuzzy filter pre topology.

Characterized Fuzzy Subspaces
Let A be a subset of a characterized fuzzy space (X, φ 1,2 .int) and i: A,→ X is the inclusion mapping of A into X. Then the mapping φ 1,2 .int A : for all η ∈ L A is initial φ 1,2 -fuzzy interior operator for φ 1,2 .int with respect to the inclusion mapping i: A,→ X. φ 1,2 .int A will be called induced φ 1,2interior operator of φ 1,2 .int on the subset A of X. The triple (A, φ 1,2 .int A ) is said to be characterized fuzzy subspace of (X, φ 1,2 .int) [7].

Characterized Fuzzy Product Spaces
Assume that (X i , δ 1,2 .int i ) is a characterized fuzzy space for each i I, where I is any class. Let X be the cartesian product i X π of the family (X i ) i∈I and π i : X → X i the related projections. The i∈I, mapping φ 1,2 .int: L X → L X , defined by: .

Final characterized fuzzy spaces
It is well-known (cf. e.g., [11,24]) that in the topological category all final lifts uniquely exist and hence also all final structures exist. They are dually defined. In case of the category CF-Space of all characterized fuzzy spaces the final structures can easily be given, as is shown in the following: Let I be a class and for each i ∈ I, let (X i , δ 1,2 .int i ) be an characterized fuzzy space and f i : X i → X is the mapping of X i into a set X. The final φ 1,2fuzzy interior operator of (δ 1,2 .int i ) i∈I with respect to (f i ) i∈I is the finest φ 1,2 .int on X for which all mappings f i : (X i , δ 1,2 .int i ) → (X, φ 1,2 .int) are δ 1,2 φ 1,2 -fuzzy continuous [7]. Hence, the triple (X, φ 1,2 .int) is the final characterized fuzzy space of ((X i , δ 1,2 .int i )) i∈I with respect to (f) i∈I . The final φ 1,2 L-interior operator φ 1,2 .int: L X → L X of (δ 1,2 .int i ) i∈I with respect to (f i ) i∈I exists and is given by .

Characterized Fuzzy Quotient Spaces
Let (X, φ 1,2 .int) be a characterized fuzzy space and f: X→A is an surjective mapping. Then the mapping φ 1,2  for all a ∈ A and µ ∈ L A , is final φ 1,2 -fuzzy interior operator of φ 1,2 .int with respect to f which is not idempotent. Then the φ 1,2 .int f will be called quotient φ 1,2 -fuzzy interior operator and the triple (A, φ 1,2 .int f ) is said to be characterized fuzzy quotient space [7].
Note that in this case φ 1,2 .int is idempotent, φ 1,2 .int f need not be. Even in the classical case of L={0, 1}, φ 1 =int and φ 2 =1 L X we have the following: If φ 1,2 .int is up to an identification the usual topology, then φ 1,2 .int f is a pre topology which need not be idempotent. An example is given [25] (p. 234).

Characterized Fuzzy Sum Spaces
Assume that (X i , δ 1,2 .int i ) is a characterized fuzzy space for each i ∈, where I is any class. Let X be the disjoint union of the family (X i ) i∈I and for each i ∈ I, let φ 1,2 .int: L X → L X , defined by:e i : X i → X be the canonical injection from X i into X given by e i (x i )=(x i , i). Then the mapping φ 1,2 .int: L X → L X , defined by: ,  for all i ∈ I, of a ∈ X i and µ ∈ L X , is said to be final φ 1,2 -fuzzy interior operator with respect to (e i ) i∈I .

Characterized Fuzzy T 1 And Fuzzy Φ 1,2 T 1 -Spaces
The notions of characterized fuzzy T s and of characterized fuzzy R k -spaces are investigated and studied [9,10,26,27] for all . These characterized spaces depend only on the usual points and the operation defined on the class of all fuzzy subsets of X endowed with an fuzzy topology τ. Let the fuzzy topological space(X, τ) be fixed and φ 1 , φ 2 ∈ O (L X, τ) then the characterized fuzzy space all fuzzy subsets of X endowed with an fuzzy topology τ. Let the fuzzy topological space (X, τ) be fixed and φ 1 , φ 2 ∈ O (L X , τ) then the characterized fuzzy space all fuzzy subsets of X endowed with an fuzzy topology τ. Let the fuzzy topological space (X, τ) be fixed and φ 1 , φ 2 ∈ O (L X, τ) then the characterized fuzzy space (X, φ 1,2 .int) is said to be characterized fuzzy T 1 -space if for all x, y ∈ X such that (X, φ 1,2 .int) is said to be characterized fuzzy The related fuzzy topological space(X, τ) is said to be fuzzy φ 1,2 -T 1 if for all x, y ∈ X such that x y ≠ , we have x˙ ̸ ≤ Nφ 1,2 (y) and y˙ ̸ ≤ Nφ 1,2 (x).

Proposition
Let X be a set, let I be a class and for each i ∈ I, let the characterized fuzzy space (X i , δ 1,2 .int i ) is characterized fuzzy T 1 and f i : X → X i be an injective mapping for some i ∈ I. Then the initial characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I is also characterized fuzzy T 1 -space [10].

Proposition
Let X be a set, let I be a class and for each i ∈ I, let the characterized fuzzy space (X i , δ 1,2 .int i ) is characterized fuzzy T 1 and f i : X i → X be an surjective mapping for some i ∈ I. Then the final characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I (X, φ 1,2 .int) is characterized fuzzy T 1 -space [27].

R and Characterized Fuzzy R 3 -Spaces
Let a fuzzy topological space(X, τ) be fixed and φ 1 , φ 2 ∈ O (L X, τ) . Then the characterized fuzzy space (X, φ 1,2 .int) is said to be characterized fuzzy 1 T and characterized fuzzy T 4 -spaces such that f(x) = 1 ( ) 0 and f y = for all y ∈ F.

Proposition
Every characterized fuzzy T 4 -space is characterized fuzzy

T -Spaces
By the fuzzy metric on the set X [6], we mean that the mapping d: X × X:→ R ⋆ L such that the following conditions are fulfilled: (2) d(x, y)=d(y, x) for all x, y ∈ X.
Where 0 ∼ denotes the fuzzy number which has value 1 at 0 and 0 otherwise. The set X equipped with an fuzzy metric on X will be called fuzzy metric space. Each fuzzy metric on a set X generated canonically a stratified fuzzy topology τ d which has the set B={ξ • d x : ξ ∈ µ and x ∈ X} as a base, where d x : X → R ⋆ L is the mapping defied by: d x (y)=d(x, y) and τd) , then as shown in ref. [20], the characterized fuzzy space (X, φ 1,2 .int τd ) is stratified. The stratified characterized fuzzy space (X, φ 1,2 .int τd ) is said to be metrizable characterized fuzzy space.
In the following proposition we shall prove that every metrizable characterized fuzzy space is characterized fuzzy T 4 -space in sense of Abd-Allah [10].

Example 3.1
From Propositions 2.9 and 3.1, we get that the metrizable fuzzy space in sense of Gahler and Gahler [3] is an example of a metrizable characterized fuzzy T 4 -space and that is also example of a metrizable characterized fuzzy T k -space for
In the following it will be shown that the finer characterized fuzzy space of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy T -space is also characterized completely fuzzy T -space, respectively.

Proof:
Let Ω is a sub base for the characterized fuzzy space

-Spaces
In this section we are going to introduce and study the notion of initial and final characterized fuzzy T -spaces. The characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space are studied as special case from the initial and final characterized fuzzy R -spaces will be denoted by CFR-Space and CRF-Tych, respectively. Note that the categories CFR-Space and CRF-Tych are concrete categories. The concrete categories CFR-Space and CRF-Tych are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets. Hence, all the initial and final lifts exist uniquely in the categories CFR-Space and CRF-Tych, respectively.
This means that they also topological over the category SET. That is, all the initial and final characterized fuzzy T -spaces exist in CFR-Space and CRF-Tych, respectively.
In the following let X be a set, let I be a class and for each i ∈ I, let the characterized fuzzy space (X i , δ 1,2 .int i ) of all δ 1,2 -open fuzzy subsets of X i is characterized fuzzy is φ 1,2 δ 1,2 -closed injective mapping from X into X i . Then we show in the following that the initial characterized fuzzy space (X,φ 1,2 .int) of ((X i , δ 1,2 .int i )) i ∈ I with respect to (f i ) i∈I is also characterized fuzzy

R
-space. More general, we show under the same conditions, that the initial characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I is characterized fuzzy T -spaces, respectively.

Proposition
Let X be a set and I be a class. For each i ∈ I, let the characterized fuzzy space (X i , δ 1,2 .int i ) of all δ 1,2 -open fuzzy subsets of X i is characterized fuzzy 1 2 2 R -space. If f i : X → X i is an φ 1,2 δ 1,2 -closed injective mapping from X into X i for some i ∈ I, then the initial characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I is also characterized fuzzy 1 R -space for all i ∈ I, then there exists an δ 1,2 ψ 1,2 -fuzzy continuous mapping g:  T -space. If f i : X → X i is an φ 1,2 δ 1,2 -closed injective mapping from X into X i for some i ∈ I, then the initial characterized fuzzy space (X, φ 1,2 .int) ((X i , δ 1,2 .int i )) i∈I of with respect to (f i ) i∈I is also characterized fuzzy 1 3 2

Corollary 4.2
The characterized fuzzy subspace (A, φ 1,2 .int A ) and the characterized fuzzy product space As shown in ref.
[7], the characterized fuzzy space (X, φ 1,2 .int) is characterized as a fuzzy filter pre topology, then we have the following result:

Corollary 4.3
For each i ∈ I, let is δ 1,2 .int i as the fuzzy filter pre topology is characterized fuzzy R 2 fuzzy 31 2 T ). Then, the representation of the initial φ 1,2 -interior operator of the initial characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I as a fuzzy filter pre topology which is defined by: for all x ∈ X and µ ∈ L X is also characterized fuzzy

Corollary 4.4
Let (Y, τ 2 ) be an fuzzy topological spaces and δ 1 , δ 2 f: X → Y is an φ 1,2 δ 1,2 -closed injective mapping from X into Y fuzzy δ 1,2 1 then the initial fuzzy topological space (X, f −1 (τ 2 )) of (Y, τ 2 ) with respect to f is fuzzy φ 1,2 1 is surjective mapping from X i into X and f i −1 is φ 1,2 δ 1,2 -closed in the classical sense. Then as in case of the initial characterized fuzzy spaces, we show in the following that the final characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to(f i ) i∈I is also characterized fuzzy 1 2 2 R -space. More general, we show under the same conditions that, the final characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I is characterized fuzzy 1 3 2 T space if each of the characterized fuzzy spaces (X i , δ 1,2 .int i ) is characterized fuzzy T -spaces for all i ∈ I. Moreover, as special cases we show that the characterized fuzzy quotient space and the characterized fuzzy sum space of the characterized fuzzy is φ 1,2 δ 1,2 -closed for some i ∈ I, then the final characterized fuzzy space (X, φ 1,2 .int) of ((X i , δ 1,2 .int i )) i∈I with respect to (f i ) i∈I is also characterized is surjective and f i −1 is φ 1,2 δ 1,2 -closed for some i ∈ I, then there exists K ∈ δ 1,2 C(X i ) and Because of (X i , δ 1,2 .int i ) is characterized fuzzy R -space.
In the following proposition, we show that the characterized T -space.
Proof: Follows immediately from Proposition 5.1.
The following example shows that the inverse of Proposition 5.1 and of Corollary 5.1 is not true in general.
Proof: Follows immediately from Proposition 5.2.
The following example shows that the inverse of Proposition 5.2 and of Corollary 5.2 is not true in general.

T KE-Spaces
In the following we introduce and study the concepts of R -spaces presented by Kandil and Shafee [12], respectively.
In the following proposition we show that the characterized fuzzy R KE-spaces.

Proposition 5.3
Let (X, τ) be an fuzzy topological space and R -space in sense [9].

Corollary 5.3
Let (X, τ) be an fuzzy topological space and T KE-space is characterized fuzzy Proof: Follows immediately from Proposition 5.3 and the fact that every characterizedQFT 1 -space is characterized fuzzy T 1 -space.
The following example shows that the inverse of Proposition 5.3 and Corollary 5.3 are not true in general.