Medical, Pharma, Engineering, Science, Technology and Business

^{1}Department of Statistics and Operations Research, College of Science, King Saud University, Saudi Arabia

^{2}Department of Mathematics, College of Science, El-Mansoura University, El-Mansoura, Egypt

- *Corresponding Author:
- Ahmed Saeed Abd-Allah

Prince Sattam Bin Abdul-Aziz University

Hotat Bani Tamim, Kingdom of Saudi Arabia

**Tel:**00966552057393

**E-mail:**[email protected]

**Received date: ** January 03 , 2017; **Accepted date:** April 21, 2017; **Published date:** April 28, 2017

**Citation: **Abd-Allah AS, Al-Khedhairi A (2017) Initial and Final Characterized Fuzzy and Finer Characterized Fuzzy -Spaces. J Appl Computat Math 6: 350. doi: 10.4172/2168-9679.1000350

**Copyright:** © 2017 Abd-Allah AS, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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 ϕ 12 -space and 1 3, 2 T ϕ 12 -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 12 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.

Fuzzy filter; Fuzzy topological space; Operation;
Characterized fuzzy space; Metriz-able characterized fuzzy space;
Induced characterized fuzzy space; *α*-Level characterized space; *φ*_{1,2}*ψ*_{1,2}-fuzzy continuous; Initial and final characterized fuzzy spaces;
Characterized fuzzy -space; Characterized fuzzy -space; AMS
classification; Primary 54E35, 54E52; Secondary 54A4003E72

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 2X 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 -spaces and characterized fuzzy -spaces
or (characterized Tychonoff spaces) [9]. In this paper, Basic notions
related to the characterized fuzzy and the characterized fuzzy -spaces are introduced and studied. Some of this the metrizable
characterized fuzzy spaces, initial and final characterized fuzzy spaces
and three finer characterized fuzzy -spaces are introduced and
classified by the characterized fuzzy and characterized fuzzy -spaces. The metrizable characterized fuzzy space is introduce as a
generalization of the weaker and stronger forms of the fuzzy metric
space introduced by Gahler and Gahler [3]. For every stratified fuzzy
topological space (X, τ_{d}) generated canonically by an fuzzy metric
d on X, the metrizable characterized fuzzy space (X, *φ*_{1,2}.int_{τd}) is
characterized fuzzy T_{4}-space in sense of Abd-Allah [10] and therefore
it is characterized fuzzy and characterized fuzzy L-space.
The induced characterized fuzzy space (X, *φ*_{1,2}.intω) is characterized
fuzzy and characterized fuzzy -space if and only if the related
ordinary topological space (X, T) is *φ*_{1,2} -space and *φ*_{1}, -space,
respectively, that is, the notions of characterized fuzzy -spaces
and characterized fuzzy -spaces are good extension as in sense of
Lowen [11]. Moreover, the *α*-level characterized space (X, *φ*_{1,2}.int*α*) and
the initial characterized space (X, *φ*_{1,2}.int_{i}) are characterized -space
and characterized -space if the related characterized fuzzy space
(X, *φ*_{1,2}.int_{τ}) is characterized fuzzy -space and characterized fuzzy -space, respectively. We show that the finer characterized fuzzy
space of the characterized fuzzy -space and of the characterized
fuzzy -space is also characterized fuzzy and characterized fuzzy -space, respectively. The categories of all characterized fuzzy and of all characterized fuzzy -spaces will be denoted by CFRSpace
and CRF-Tych, respectively. We show that these categories
are concrete categories and they 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, respectively. That is, all
the initial and final characterized fuzzy -spaces and all the initial
and final characterized fuzzy -spaces are exist in the categories
CFR-Space and CRF-Tych. Moreover, we show that the initial and
final characterized fuzzy spaces of the characterized fuzzy -space
and of the characterized fuzzy -space are characterized fuzzy and characterized fuzzy -spaces, respectively. As an special cases,
the characterized fuzzy subspace, characterized fuzzy product space,
characterized fuzzy quotient space and characterized fuzzy sum space
of the characterized fuzzy -space and of the characterized fuzzy -space are also characterized fuzzy and characterized fuzzy -spaces, respectively. Finally, in section 5, we introduce and study
three finer characterized fuzzy and three finer characterized fuzzy -spaces as a generalization of the weaker and stronger forms of
the completely regular and fuzzy -spaces introduced [1,12,13].
The relations between such new characterized fuzzy -spaces and
our characterized fuzzy -spaces are introduced. More general the
relations between such new characterized fuzzy -spaces and our
characterized fuzzy -spaces are also introduced. Meany special
cases from these finer characterized fuzzy -spaces and from finer
characterized fuzzy -spaces are listed in **Table 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}. Sup*μ* 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 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 . The
fuzzy filter on X [14] is the mapping M: L^{X} →L such that the following
conditions are fulfilled:

(F_{1}) Μ () ≤ *α* for all *α* ∈ L and (1)=1.

(F_{2}) (*μ* ∧ *η*)= (*μ*) ∧ (*η*) for all *μ*, *η* ∈ L^{X}.

The fuzzy filter is said to be homogeneous [14] if M () =*α* for all *α* ∈ L. For each x ∈ X, the mapping : L^{X} → L defined
by *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:

(2.1)

for all *η* ∈ L^{X}. As shown in ref. [15], *μ* ≤ *η* if and only if ≤ 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:

M ≠ L ≥ N implies M ≠ N and M ≥ L ≠ N implies M ≠ N .

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 *Α* is a non-empty set of
fuzzy filters on X, then the supremum exists [2] and given
by for all *μ* = L^{X} but the infimum does not exists, in general. As shown in ref. [16], the infimum of *Α* with respect to the finer relation for fuzzy filters exists if and
only if holds for all finite subset . In this case the infimum is given by:

for all *μ* ∈ L^{X}.

Fuzzy filter bases. A family (B* _{α}*)

(V1) *μ* ∈ * _{α}* implies

V2) For all *α*, β ∈ L_{0} with *α*∧β ∈ L_{0} and all *μ* ∈ * _{α}* and

As shown in ref. [2], each valued fuzzy filter base (* ^{α}*)

(S1) ∈B for every *α* ∈ L.

(S2) For all *μ*, *η* ∈ there is a fuzzy set σ ∈ such that σ ≤ *μ*, σ ≤ *η* and sup σ=sup *μ* ∧ sup *η*.

Each superior fuzzy filter base generated a homogeneous fuzzy
filter on X by sup *η* for all *μ*∈L^{X} and each fuzzy
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 otherwise for all *η* ∈ L^{X}.
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 and [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
S (*μ*) { x X | *μ*(x) } and W (*μ*) { x X | *μ*(x) } *α* *α* = ∈ > *α* = ∈ ≥ *α* 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:

(2.2)

where, *x+y* is the fuzzy real number defined by for all ξ ∈ R.

**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 By the
identity operation on we mean the operation such
that for all *μ* ∈ L^{X}. The constant operation on is the
operation defined by for all *μ* ∈ L^{X}. If ≤ is a partially
order relation on defined as follows: for all *μ* ∈ L^{X}, then is a completely distributive lattice. The
operation *φ*: L^{X} → L^{X} is called:

(i) Isotone if *μ* ≤ *η* implies *φ**μ* ≤ *φ**η*, for all *μ*, *η* ∈ L^{X}.

(ii) Weakly finite intersection preerving (wfip, for short) with
respect to *A ⊆ L ^{X} if η ∧ϕ (μ) ≤ ϕ (η ∧ μ)* holds, for all

(iii) Idempotent if ϕ (*μ*) =ϕ (ϕ (*μ*)), for all *μ* ∈ L^{X}.

The operations *φ*, *ψ* ∈ are said to be dual if *ψ*(*μ*)=co(*φ* (co*μ*))
or equivalently *φ*(*μ*)=co(*ψ* (co*μ*)) for all *μ* ∈ L^{X}, where co*μ* 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 co*μ* is *φ*-open. The operations *φ*, 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

*φ*_{1}, *φ*_{2} ∈ Then the *φ*_{1,2}-interior of the fuzzy set *μ*: X → L is a
mapping *φ*_{1}2.int*μ*: X → L defined by:

(2.3)

That is, the *φ*_{1,2}.int*μ* is the greatest *φ*_{1}-open fuzzy set *η* such that *η**φ*_{2} 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*_{1,2}-
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.

**Proposition**

For each two operations *φ*_{1}, *φ*_{2} ∈ O(L^{X},τ) and for each *μ*, *η*∈*φ*_{1}, *φ*_{2} ∈
L^{X}, the mapping *φ*_{1,2}.int: X → L fulfills the following axioms [7]:

(i) If *φ*_{2} ≥ 1_{L}^{X}, then *φ*_{1,2}.int*μ* ≤ *μ*.

(ii) *φ*_{1,2}.int is isotone, i.e if *μ* ≤ *η*, then *φ*_{1,2}.int*μ* ≤ *φ*_{1,2}.int*η*.

If *φ*_{2} ≥ 1_{L}X is isotone and *φ*_{1} is with respect to *φ*_{1}O (X), then

If *φ*_{2} is isotone and idempotent operation, then

**Proposition**

Let (X, τ) be a fuzzy topological space and *φ*_{1}, *φ*_{2} ∈ Then the
following are fulfilled:

(i) If *φ*_{2} ≥ 1_{L}X, then the class *φ*_{1,2}OF (X) of all *φ*_{1,2}-open fuzzy sets
on X forms an extended fuzzy topology on X [7,21].

If , then the class *φ*_{1,2}OF (X) of all *φ*_{1,2}-
open 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].

If *φ*_{2} ≥ 1_{L}X is isotone and idempotent operation and *φ*_{1} is with
respect to *φ*_{1}OF (X), then *φ*_{1,2}OF (X) is fuzzy topology on X [16,18].

Because of Propositions 2.1 and 2.2, if the fuzzy topological space(X, τ) be fixed and

*φ*_{1}, *φ*_{2} O(L^{X},τ). Then the relation between the class *φ*_{1,2}OF (X) of all *φ*_{1,2}-open fuzzy sets on X and the mapping *φ*_{1,2}.int is given by:

(2.4)

and the following axioms are fulfilled:

(I1) If *φ*_{2} ≥ 1_{L}^{X}, then *φ*_{1,2}.int*μ* ≤ *μ* holds, for all *μ* ∈ L^{X}.

(I2) If *μ* ≤ *η*, then *φ*_{1,2}.int*μ* ≤ *φ*_{1,2}.int*η* for all *μ*, *η* ∈ L^{X}.

(I3)

(I4) If is isotone and *φ*_{1} is with respect to *φ*_{1}OF (X), then *φ*_{1,2}.int*μ* ∧ ∧ *φ*_{1,2}.int*η*=*φ*_{1,2}.int (*μ* ∧ *η*) for all *μ*, *η* ∈ L^{X}. s

(I5) If *φ*_{2} is isotone and idempotent, then *φ*_{1,2}.int (*φ*_{1,2}.int*μ*)=*φ*_{1,2}.
int*μ* for all *μ* ∈ L^{X}.

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 [7] of the *φ*_{1,2}-open fuzzy subsets of X. The characterized fuzzy space (X, *φ*_{1,2}.int)
is said to be stratified if and only if for all *α* ∈ L. As
shown in ref. [7], the characterized fuzzy space (X, *φ*_{1,2}.int) is stratified
if the related fuzzy topology is stratified. Moreover, the characterized
fuzzy space (X, *φ*_{1,2}.int) is said to have the weak infimum property [21], provided . The
characterized fuzzy space (X, *φ*_{1,2}.int) is said to be strongly stratified
[21], provided *φ*_{1,2}.int is stratified and have the weak infimum property.
If (X, *φ*_{1,2}.int) and (X, *ψ*_{1,2}.int) are two characterized fuzzy spaces, then
(X, *φ*_{1,2}.int) is said to be finer than (X, *ψ*_{1,2}.int) and denoted by *φ*_{1,2}.
int ≤ *ψ*_{1,2}.int, provided *φ*_{1}, 2.int*μ* ≥ *ψ*_{1,2}.int*μ* holds for all *μ* ∈ L^{X}. If τ is
a fuzzy topology on the set X and *φ*_{1}, *φ*_{2} ∈ then by the initial
characterized space of (X, τ) we mean the characterized spaces (X,
(*φ*_{1,2}O(X))*α*) and (X, i(*φ*_{1,2}O(X))), respectively where (*φ*_{1,2}O(X))*α* and
i(*φ*_{1,2}O(X)) are defined as follows:

Sometimes we denoted to the *α*-level characterized space and the
initial characterized space of (X, τ) by respectively. If T is an ordinary topology on a set X and then by the induced characterized fuzzy space on
X we mean the characterized fuzzy space which is
defined by:

Sometimes we denoted to the induced characterized fuzzy space for the ordinary topological space (X, T) by

If , then the class of all *φ*_{1,2}-
open fuzzy of X coincide with τ which is defined in [22,23] and hence
the characterized fuzzy space with the fuzzy
topological space (X, τ).

*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 which is defined by:

(2.5)

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*_{1,2}-
fuzzy neighborhood stack with the cutting property at x. The *φ*_{1,2}-fuzzy
neighborhood filters fulfill the following conditions:

*holds for all μ,*

Clearly is the fuzzy set *φ*_{1,2}.int. The
characterized fuzzy space is characterized as the fuzzy filter
pre topology [7], that is, as a mapping such that (N1)
to (N3) are fulfilled.

*φ*_{1,2}*ψ*_{1,2}-Fuzzy continuity

Let now the fuzzy topological spaces (X, τ_{1}) and (Y, τ_{2}) are fixed, *φ*_{1}, and . The mapping f: (X, *φ*_{1,2}.int) →(Y, *ψ*_{1,2}.
int) is said to be *φ*_{1,2}*ψ*_{1,2}-fuzzy continuous if

(2.6)

holds for all *η* ∈ LY [7]. If an order reversing involution′ of L is
given, we have that f is a *φ*_{1,2} *ψ*_{1,2}-fuzzy continuous if and only if holds for all *η* ∈ LY. Here *φ*_{1,2}.cl and *ψ*_{1,2}. cl, mean the closure operators related to *φ*_{1,2}.int and *ψ*_{1,2}.int, respectively
which are defined by *φ*_{1,2}.cl *μ*=co (*φ*_{1,2}.int co*μ*) for all *μ* ∈ L^{X}. Obviously if
f is *φ*_{1,2} *ψ*_{1,2}-fuzzy continuous, then the inverse f−1: (Y, *ψ*_{1,2}.int) → (X, *φ*_{1,2}.
int) is *ψ*_{1,2}*φ*_{1,2}-fuzzy continuous, that is holds for all h ∈L^{X}

By means of characterizing *φ*_{1,2}-fuzzy neighborhoods the *φ*_{1,2}*ψ*_{1,2}-fuzzy continuity
of f can also be characterized. The mapping f: (X, *φ*_{1,2}.int) → (Y, *ψ*_{1,2}.int)
is *φ*_{1,2}*ψ*_{1,2}-fuzzy continuous if N*ψ*_{1,2} (f(x)) ≥ FLf(N*φ*_{1,2} (x)) holds for all x
∈ X. Obviously, in case of L={ 0, 1 }, *φ*_{1}=*ψ*_{1}=int, *φ*_{2}=1_{L}X and *ψ*_{2}=1LY
the *φ*_{1,2}*ψ*_{1,2}-fuzzy continuity coincides with the usual fuzzy continuity.

**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 fi: X → X_{i} is the mapping from X into X_{i}. By the initial *φ*_{1,2}-
fuzzy 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 fi: (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:

(2.7)

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*φ*_{1,2}: 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.

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}.intA:
L^{A} → L^{A} defined by:

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}.intA will be called induced *φ*_{1,2}-
interior operator of *φ*_{1,2}.int on the subset A of X. The triple (A, *φ*_{1,2}.intA)
is said to be characterized fuzzy subspace of (X, *φ*_{1,2}.int) [7].

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 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:

for all *μ* ∈ L^{X}, will be called *φ*_{1,2}-fuzzy product of the δ_{1,2}L-interior
operators δ_{1,2}.int_{i}. The triple (X, *φ*_{1,2}.int) is said to be characterized fuzzy product space [7] of the characterized fuzzy spaces (X_{i}, δ_{1,2}.int_{i}). The *φ*_{1,2}.int will be denoted by and it is initial *φ*_{1,2}-fuzzy interior
operator of (δ_{1,2}.int_{i})_{i}∈_{I} with respect to the family (π_{i})_{i}∈_{I} of projections.
The characterized fuzzy product space (X, *φ*_{1,2}.int) also will be denoted
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 fi: X_{i} → X is the mapping of X_{i} into a set X. The final *φ*_{1,2}-
fuzzy 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

for all x ∈ X and *μ* ∈ L^{X}.

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}.intf: L^{A} → L^{A}, defined by:

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}.intf will be
called quotient *φ*_{1,2}-fuzzy interior operator and the triple (A, *φ*_{1,2}.intf) is
said to be characterized fuzzy quotient space [7].

Note that in this case *φ*_{1,2}.int is idempotent, *φ*_{1,2}.intf 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}.intf is a pre topology which need not be idempotent. An example is
given [25] (p. 234).

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:ei: X_{i} → X
be the canonical injection from X_{i} into X given by ei(xi)=(xi, 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 (ei)i∈_{I}.

(δ_{1,2}.int_{i})_{i}∈_{I} *φ*_{1,2}.int will be called sum *φ*_{1,2}-fuzzy interior operator will
be denoted by Σ δ_{1,2}.int_{i}. The pair (X, *φ*_{1,2}.int) is said to be characterized
fuzzy sum space [7] and it will be denoted also by

The notions of characterized fuzzy T_{s} and of characterized
fuzzy Rk-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} ∈ 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} ∈ 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} ∈ 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 T_{1}-space if for all x, y ∈
X such that x ≠ y there exist *μ*, *η* ∈ L^{X} and *α*, β ∈ L_{0} such that *μ*(x) < *α* ≤ (*φ*_{1,2}.int*μ*)(y) and *η*(y) < β ≤ (*φ*_{1,2}.int*η*)(x) are hold. 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˙ ̸≤ N*φ*_{1,2}(y) and y˙ ̸≤ N*φ*_{1,2}(x).

**Proposition**

Let (X, T) be an ordinary topological space and *φ*_{1}, *φ*_{2} ∈ ∈ O(P(X),T)
such that *φ*_{2} ≥ iP(X) is isotone and idempotent. Then (X, T) is *φ*_{1,2}T_{1}-space
if and only if the induced characterized fuzzy space (X, *φ*_{1}, 2.intω) is
characterized fuzzy T_{1} [27].

**Proposition**

Let (X, τ) be an fuzzy *φ*_{1,2}-T_{1} space and *φ*_{1}, *φ*_{2} ∈ O(L^{X},t) such that *φ*_{2} is isotone and idempotent. Then the *α*-level characterized space (X, *φ*_{1,2}.
int*α*) and the initial characterized space (X, *φ*_{1,2}.int_{i}) are T_{1}-spaces [27].

**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 fi: 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 fi: 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].

**Proposition**

Let the characterized fuzzy space (X, *φ*_{1,2}.int) is characterized
fuzzy T_{1} and δ_{1,2}.int is finer than *φ*_{1,2}.int. Then the characterized
fuzzy space (X, δ_{1,2}.int) is also fuzzy T_{1} [27].

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 [9] (resp. fuzzy R_{3}-space [10] if for all x ∈ X, F ∈ *φ*_{1,2}C(X)
such that x ̸ F (resp. F_{1}, F_{2} ∈ *φ*_{1,2}C(X) such that F_{1} ∩ F_{2}=∅), there exists
an *φ*_{1,2}*ψ*_{1,2}-fuzzy continuous mapping such that

for all y ∈ F (resp. the infimum) does not exist).
Proposition 2.8 [9] Let (X, τ) be a fuzzy topological space, *φ*_{1}, *φ*_{2} ∈ O(X,τ) and Ω is a subbase for the characterized fuzzy space (X, *φ*_{1,2}.int_{τ}).
Then, (X, *φ*_{1,2}.int_{τ}) is characterized fuzzy R_{2} 1_{2}-space if and only if for all
F ∈ Ω′ and x ∈ X such that x ∈/F, there exists a *φ*_{1,2}*ψ*_{1,2}-fuzzy continuous
mapping fuzzy characterized
fuzzy T_{4}-spaces such that f(x) = 1and f ( y) = 0 for all y ∈ F.

Let a fuzzy topological space(X, τ) be fixed and *φ*_{1}, *φ*_{2} ∈ . Then
the characterized fuzzy space (X, *φ*_{1,2}.int) is said to be characterized
fuzzy or characterized Tychonoff fuzzy space [9] (resp. fuzzy
T_{4}-space [10] if and only if it is characterized fuzzy (resp.
characterized fuzzy R_{3}) and characterized fuzzy T_{1}-space. The related
fuzzy topological space(X, τ) is said to be fuzzy *φ*_{1,2}- (resp. fuzzy *φ*_{1,2}-T_{4}) if and only if it is fuzzy *φ*_{1,2}- (resp. fuzzy *φ*_{1,2}-R_{3}) and fuzzy *φ*_{1,2}-T_{1} space.

**Proposition**

Every characterized fuzzy T_{4}-space is characterized fuzzy -space [9].

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:

(1) d(x, y)=0∼ if and only if x=y.

(2) d(x, y)=d(y, x) for all x, y ∈ X.

(3) d(x, y) ≤ d(x, z)+ d(z, y) holds for all x, y, z ∈ 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 ◦ dx: ξ ∈ *μ* and x ∈
X} as a base, where dx: X → R*_{L} is the mapping defied by: dx(y)=d(x, y)
and

Where has the domain is and is the restriction
of R_{δ} on L R . Now, consider *φ*_{1}, *φ*_{2} ∈ O(L^{X},τ_{d}), 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].

**Proposition**

Let (X, τ_{d}) be an stratified fuzzy topological space generated
canonically by an fuzzy metric d on X and *φ*_{1}, *φ*_{2} ∈ then the
metrizable characterized fuzzy space (X, *φ*_{1,2}.int_{τd}) is characterized
fuzzy T_{4}-space.

**Proof:** Let such that = ∅. 1 2 Then for all
x ∈ F_{1} and y ∈ F_{2}, we get d (x, y) ≠ 0 ∼, that is, there exists δ>0 such
that d(x, y)(2δ)>0 and therefore

holds. Consider and then

for all

for all 2 y∈F . Hence, *μ* and *η* are *φ*_{1,2}-fuzzy neighborhoods
in (X, *φ*_{1,2}.int_{τd}) at all x ∈ F_{1} and all y ∈ F_{2}, respectively,
this means Because of
the symmetry and triangle inequality of d and (2.2), we get and therefore holds for all z ∈ X,
that is, sup (*μ* ∧ *η*)<1. Hence, the infimum N*φ*_{1,2} (F_{1}) ∧ N*φ*_{1,2} (F_{2}) does
exists and therefore (X, *φ*_{1,2}.int_{τd}) is characterized fuzzy R_{3}-space. Because
of Theorem 3.1 [27], it is clear that (X, *φ*_{1,2}.int_{τd}) is characterized fuzzy T_{1}-
space. Consequently, (X, *φ*_{1,2}.int_{τd}) is characterized fuzzy T_{4}-space.

**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 Tk-space for

In the following we introduce and study the concepts of
characterized -space and of characterized spaces in the
classical case. Let (X, T) be an ordinary topological space and *φ*_{1}, *φ*_{2} . Then the characterized space (X, *φ*_{1,2}.intT) is said to be
characterized -space if for all x ∈ X, F ∈ *φ*_{1,2}C(X) such that x ̸F,
there exists an *φ*_{1,2}*ψ*_{1,2} continuous mapping f: (X, *φ*_{1,2}.intT) → (I, *ψ*_{1,2}.
intTI) such that f(x)=1 and f(y)=0 for all y ∈ F, where *ψ*_{1,2}.intI is the usual *ψ*_{1,2}-interior operator on the closed unit interval I and *ψ*_{1}, *ψ*_{2} ∈O_{(P (I),TI)}. Moreover, the ordinary characterized space (X, *φ*_{1,2}.intT) is said to be
characterized -space or classical characterized-Tychonoff space if
and only if it is characterized T_{1}-space and characterized -space.

**Proposition**

Let (X, T) be an ordinary topological space and *φ*_{1}, *φ*_{2} ∈ O_{(P (X),T)} such that *φ*_{2} ≥ i_{P (X)} is isotone and idempotent. Then, (X, *φ*_{1,2}.intT) is
characterized -space if and only if the induced characterized fuzzy
space (X, *φ*_{1,2}.intω) is characterized fuzzy -space.

**Proof:** Let (X, *φ*_{1,2}.intT) is characterized -space,such that x ̸F. Then, there exists *φ*_{1,2}δ_{1,2}-continuous mapping g: such that
g(x) =1and g( y) = 0 for all and for all L *α* ∈ 1 , where Hence, the mapping g: is *φ*_{1,2}δ_{1,2}-fuzzy continuous. Consider h: is the map-ping defied by h(z) = for all z ∈ I, then h is δ_{1,2}*ψ*_{1,2}-
fuzzy continuous and there-fore there exists an *φ*_{1,2}*ψ*_{1,2}-fuzzy
continuous mapping such
that for all y **∈**F. Consequently, is
characterized fuzzy -space.

Conversely, let is characterized fuzzy -space,
x **∈** X and such that Then, and Therefore, there exists an fuzzy continuous mapping such that and for all Since then there could be
found the mapping which is -continuous with *and* . Hence, is characterized -space.

**Corollary 3.1**

Let (X, T) be an ordinary topological space and such that is isotone and idempotent. Then, is
characterized -space if and only if the induced characterized fuzzy
space (X, φ_{1,2}.int_{ω}) is characterized fuzzy -space.

**Proof:** Immediate from Propositions 2.3 and 3.2.

Proposition 3.2 and Corollary 3.1, show that the notions of characterized fuzzy and characterized fuzzy -spaces are good extension as in sense of Lowen [11].

In the following proposition for each fuzzy topological space (X, τ), we show that the α-level characterized space and the initial characterized space are characterized -spaces if the characterized fuzzy space is characterized fuzzy .

**Proposition 3.3**

Let (X, τ) be a fuzzy topological space and such that is isotone and idempotent. Then the α-level characterized space and the initial characterized space are characterized -spaces if is characterized fuzzy -space, there exists

**Proof: **Consider is characterized fuzzy -space,
x **∈** X and such that . Then . and . Because of is characterized fuzzy Space,-space, there exists an -fuzzy continuous mapping f: (X, φ_{1,2}.int_{τ}) → (I_{L}, ψ_{1,2}.int_{I}) and f(y)=0 such that and for
all . Since φ_{1,2}.in t_{τ}= φ_{1,2}.int_{α} and then there
could be found the mapping which is
φ_{1,2}ψ_{1,2}-continuous with f_{α}(x)=1 and f_{α}(y)=0 for all y **∈** F. Consequently,
(X, φ_{1,2}.int_{α}) is characterized space. The second case is similarly,
that is, if (X, φ_{1,2}.int_{τ}) is characterized fuzzy -space.

**Corollary 3.2**

Let (X, τ) be a fuzzy topological space and φ_{1}, φ_{2} **∈** O_{(L}X,_{τ)} such that is isotone and idempotent. Then the α-level characterized
space (X, φ_{1,2}.int_{α}) and the initial characterized space (X, φ_{1,2}.int_{i}) are
characterized -spaces if the characterized fuzzy space (X, φ_{1,2}.int_{τ})
is characterized fuzzy .

**Proof:** Immediate from Propositions 2.4 and 3.3.

In the following it will be shown that the finer characterized fuzzy space of a characterized fuzzy -space and of a characterized fuzzy -space is also characterized completely fuzzy -space and characterized fuzzy -space, respectively.

**Proposition**

Let (X, τ) is a fuzzy topological space and φ_{1}, φ_{2}**∈** O(L^{X}, τ). If the
characterized fuzzy space (X, φ_{1,2}.int_{τ}) is characterized fuzzy and
δ_{1,2}.int_{τ} is finer than φ_{1,2}.int_{τ}, then (X, δ_{1,2}.int_{τ}) is also characterized
fuzzy and δ_{1,2}.int_{τ} -space.

**Proof:** Let Ω is a sub base for the characterized fuzzy space and such that Such that Then, there is such that and
therefore or all i ∈ {1,. . ., n}. Because of Proposition
2.8, there exists a φ_{1,2}ψ_{1,2}-fuzzy continuous mappings f_{i}: (X, φ_{1,2}.int_{τ}) → (I_{L}, ψ_{1,2}.int_{I}) such that and is also fulfilled
for all In particular this means that and for all y **∈** F and i **∈** {1,. . ., n}. Since δ_{1,2}.int_{τ} is finer than
φ_{1,2}.int_{τ}, then any one of these mappings gives us the required
δ_{1,2}ψ_{1,2}-fuzzy continuous mappings g: (X, δ_{1,2}.int_{τ}) → (I_{L}, ψ_{1,2}.int_{I}) such that and and f_{i}(y)=0 for all y **∈** F and i **∈** {1,. . .,
n}. Since δ_{1,2}.int_{τ} is finer than φ_{1,2}.int_{τ}, then any one of these mappings gives us the required δ_{1,2}ψ_{1,2}-fuzzy for all y **∈** F. Consequently,
(X, δ_{1,2}.int_{τ}) is characterized fuzzy Space.

**Corollary 3.3** Let (X, τ) be a fuzzy topological space and φ_{1}, φ_{2} ∈
O_{(L}X,_{τ)}. If (X, φ_{1,2}.int_{τ}) is characterized fuzzy -space and δ_{1,2}.int_{τ} is
finer than φ_{1,2}.int_{τ}, then (X, δ_{1,2}.int_{τ}) is also characterized fuzzy -space.

**Proof:** Immediate from Propositions 2.7 and 3.4.

In this section we are going to introduce and study the notion of initial and final characterized fuzzy -spaces and the notions of initial and final characterized fuzzy -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 and fuzzy -spaces. New additional properties for the initial and final characterized fuzzy -spaces and for the initial and final characterized fuzzy -spaces are given. The categories of all characterized fuzzy and of all characterized fuzzy -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 -spaces and all the initial and final characterized fuzzy -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 of all

**Proposition**

Let X be a set and I be a class. For each *i* **∈** *I*, let the characterized
fuzzy space of all *δ*_{1,2}-open fuzzy subsets of *X _{i}* is characterized
fuzzy -space. If is an -closed injective mapping from

**Proof:** Let *x* **∈** *X* and such that *x F*. Since is φ_{1,2}δ_{1,2}-closed injective for some *i* **∈** *I*, then and Because of is characterized fuzzy - space for
all *i ***∈** *I*, then there

exists an -fuzzy continuous mapping such that and for all *y* **∈** *F*. Therefor the composition

fuzzy continuous mapping such that and for all *y F*. Consequently, is
characterized fuzzy -space.

**Corollary 4.1** Let *X* be a set and I be a class. For each *i* ∈ *I*, let the
characterized fuzzy space of all *δ*_{1,2}-open fuzzy subsets of *X _{i}* is characterized fuzzy -space. If is an -closed injective
mapping from

**Proof:** Immediate from Propositions 2.5 and 4.1.

**Corollary 4.2**

The characterized fuzzy subspace and the characterized fuzzy product space of a characterized fuzzy -space (resp. characterized fuzzy -space) are also characterized fuzzy -space (resp. characterized -space)

**Proof:** Follows immediately from Proposition 4.1 and Corollary 4.1. 2

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 ). Then, the representation
of the initial φ_{1,2}-interior operator of the initial
characterized fuzzy space of with respect to as a fuzzy filter pre topology which is defined by:

for all *x* **∈** *X* and *μ* **∈** *L ^{X}* is also characterized fuzzy (resp.
characterized fuzzy ).

Now, if we consider the case of I being a singleton, then we have the following results as special cases from Proposition 4.1 and Corollary 4.1.

**Proposition**

Let (*X, τ _{1}*) and (

**Proof:** Straight forward.

**Corollary 4.4**

Let (*Y, τ _{2}*) be an fuzzy topological spaces and is an
φ

**Proof:** Follows immediately from Proposition 4.2. 2

In the following let *X* be a set and *I* be a class. For each *i ∈ I*, let
the characterized fuzzy space of all δ

**Proof:** Let *x* **∈** *X* and such that *x F*. Since is surjective and -closed for some *i* **∈** *I*, then there exists and for which and such that Because of is characterized fuzzy -space for
all* i ***∈** *I*, then there exists an fuzzy continuous mapping *g:* such that and for all *z ∈ K*, that is and for all

**Corollary 4.5**

Let *X* be a set and *I* be a class. For each* i* **∈** *I*, let the characterized
fuzzy space of all δ_{1,2}-open fuzzy subsets of *X _{i}* is characterized
fuzzy -space. If is an surjective -fuzzy open mapping
from Xi into X and -closed for some

**Proof:** Immediate from Propositions 2.6 and 4.3. 2

**Corollary 4.6**

The characterized fuzzy quotient space (*A, φ*_{1,2}.int_{f}) and the char
characterized fuzzy -space) are also characterized fuzzy (resp.
characterized fuzzy ) *L*-spaces.

**Proof:** Follows immediately from Proposition 4.3 and Corollary 4.5. 2

Now, if we consider the case of I being a singleton, then we have the following results as special cases from Proposition 4.3 and Corollary 4.5.

Proposition 4.4 Let (X, τ_{1}) and (Y, τ_{2}) are two fuzzy topological
spaces, and If is an
subjective δ_{1,2}φ_{1,2}-fuzzy open mapping from *X* into *Y* and closed, then the final characterized fuzzy space (*X, φ*_{1,2}.int) of (*Y, δ*_{1,2}.
int) with respect to *f* is characterized fuzzy (resp. characterized
fuzzy )L-space if (*Y, δ*_{1,2}.int) is characterized fuzzy (resp.
characterized fuzzy ) L-spaces.

**Proof:** Straight forward.

**Corollary 4.7**

Let (*Y, τ _{2}*) be an fuzzy topological spaces and → X is an

**Proof:** Follows immediately from Proposition 4.4. 2.

In this section we are going to introduce and study some finer characterized fuzzy and finer characterized fuzzy -paces as a generalization of the weaker and stronger forms of the completely fuzzy regular and fuzzy -spaces introduced [28,12,13]. The relations between such characterized fuzzy -spaces and our characterized fuzzy -spaces which presented [9] are introduced. More generally, the relations between such characterized fuzzy -spaces and our characterized fuzzy -spaces are also introduced.

Characterized fuzzy H and characterized fuzzy H-spaces.
In the following we introduce and study the concept of characterized
completely fuzzy regular Hutton and characterized fuzzy Huttonspaces
as a generalization of the weaker and stronger forms of the
completely fuzzy regular and fuzzy -spaces in sense of Hutton
[28], respectively. The relation between characterized completely
fuzzy regular Hutton-spaces and the characterized fuzzy -spaces
in our sense is introduced. More generally, the relations between
characterized fuzzy Hutton-spaces and the characterized fuzzy -spaces in our sense is also introduced. Let (*X, τ*) be a fuzzy
topological space and . Then the characterized fuzzy
space (*X, φ*_{1,2}.int) is said to be characterized completely fuzzy regular
Hutton-space or (characterized fuzzy H-space, for short) if for an there exists a collection and an φ_{1,2}ψ_{1,2}-fuzzy
continuous mapping such that and holds for all *y* **∈** *X*. Then characterized fuzzy space (X, φ_{1,2}.int) is said to be
characterized fuzzy Hutton-space or (characterized fuzzy H-space, for short) if and only if it is characterized fuzzy H and
characterized fuzzy -spaces.

In the classical case of *L*={0, 1}, and the -fuzzy continuity of f is up to an identification the
usual fuzzy continuity of f. Then in this case the notions of characterized
fuzzy *H*-spaces and of characterized fuzzy H-spaces are
coincide with the notion of fuzzy completely regular spaces and the
notion fuzzy -spaces defined by Hutton [28], respectively. Another
special choices for the operations φ_{1}, φ_{2}, ψ_{1} and ψ_{2} are obtained (**Table 1**).

In the following proposition, we show that the characterized
fuzzy -spaces which are presented [9] are more general than the
characterized fuzzy *H*-spaces.

**Proposition 5.1**

Let (*X, τ*) be an fuzzy topological space and .

Then every characterized fuzzy *H*-space (*X, φ*_{1,2}.int) is
characterized fuzzy -space.

**Proof:** Let (*X, φ*_{1,2}.int) is characterized fuzzy *H*-space, *x* **∈** *X* and such that . Then, and , therefore holds for all *α ∈ L*. Hence, and
therefore for all , there exists a family in

**Corollary 5.1** Let (X, τ) be an fuzzy topological space and Then every characterized fuzzy H-space is
characterized fuzzy -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.

**Example 5.1.**

Let *X={x, y}* with *x ≠ y* and is an
fuzzy topology on X. Choose Hence, and there is the only case of *x
∈ X, F={y} ∈ φ _{1,2}C(X) *such that Since the mapping which is defined by and for all y≠x is
φ

On other hand, let (*X, φ*_{1,2}.intτ) is characterized fuzzy *H*-space,
then(X, φ_{1,2}.int_{τ}) is characterized fuzzy *H* and characterized fuzzy *T _{1}*-space. Since and then there
exists a collection such that Moreover, for an φ

holds only when *z=y*, but it is not holds when *z=x*, because and this is a contradiction. Hence, is not
characterized fuzzy *H*-space and therefore it is not characterized
fuzzy *H*-space.

Characterized fuzzy K and characterized fuzzy K-spaces. In the following we introduce and study the concept of characterized completely fuzzy regular Katasars spaces and characterized fuzzy Katasars spaces as a generalization of the weaker and stronger forms of the completely fuzzy regular and fuzzy -spaces introduced by Katasars [13], respectively. The relation between characterized fuzzy completely regular Katasars spaces and the characterized fuzzy -spaces in sense Abd-Allah and Khedhairi [9] is introduced. More generally, the relations between characterized fuzzy Katasars spaces and the characterized fuzzy -spaces in sense of [9] is also introduced.

Let (*X, τ*) be an fuzzy topological space and . Then
the characterized fuzzy space (*X, φ*_{1,2}.int) is said to be characterized
completely fuzzy regular Katasars-space or (characterized fuzzy K-space, for short) if for every *x ∈ X* and

In the classical case of L={ 0, 1 }, and fuzzy continuity of f is up to an identification
the usual fuzzy continuity of f. Then in this case the notions of
characterized fuzzy K-space and of characterized fuzzy *K*-spaces are coincide with the notion of completely fuzzy regular spaces and the notion of fuzzy -spaces presented by Katasars [13],
respectively. Another special choices for the operations *φ*_{1}, *φ*_{2}, *ψ*_{1} and *ψ*_{2} are obtained in **Table 1**. In the following proposition we show that the
notion of characterized fuzzy -spaces which are presented [9] are
more general than the characterized fuzzy K-spaces.

**Proposition**

Let (*X, τ*) be an fuzzy topological space and . Then
every characterized fuzzy *K*-space (*X, φ*_{1,2}.int) is characterized
fuzzy space.

**Proof:** Let (*X, φ*_{1,2}.int) is a characterized fuzzy *K*-space, *x X* and such that Then, and , therefore holds for all *α* **∈** *L*. Because of is characterized
fuzzy *K*-space, then there exists a *φ*_{1,2}ψ_{1,2}-fuzzy continuous
mapping such that and are hold for all *y ***∈*** X* and *α ***∈*** L*. In case of *y ***∈*** F*, we
have , that is, for all* t>0, y* **∈** *F* and therefore for all *y ***∈*** F*. In case of *y=x*, we have holds
for all *α* **∈** *L*, and therefore . Hence, there exists a φ_{1,2}ψ_{1,2}-fuzzy
continuous mapping such that and for all *y* **∈** *F*. Consequently, is characterized
fuzzy -space in sense [9].

**Corollary 5.2** Let (X, τ) be an fuzzy topological space and Then every characterized fuzzy K-space is
characterized fuzzy -space.

**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.

**Example 5.2.**

Consider the characterized fuzzy space which is
defined in Example 5.1, then as shown in Example 5.1, is
characterized fuzzy -space in sense [9] and characterized fuzzy *T _{1}*-
space, therefore is characterized fuzzy -space in sense [9].

On other hand, for any -fuzzy continuous mapping such that and for all we shall
consider with , that is, there exists some such that . Therefore, holds only when *z=x* and it is not fulfilled when *z=y*. Moreover, holds only when *z=y* and it is not fulfilled
when *z=x*. Hence, is not characterized fuzzy *K*-space
and therefore it is not characterized fuzzy *K*-space.

In the following we introduce and study the concepts of characterized completely fuzzy regular Kandil and Shafee spaces and of characterized fuzzy Kandil and Shafee spaces as a generalization of the weaker and stronger forms of the completely fuzzy regular and fuzzy -spaces presented by Kandil and Shafee [12], respectively. The relation between characterized completely fuzzy regular Kandil and Shafee spaces and the characterized fuzzy -spaces which are presented [6]. More generally, the relations between characterized fuzzy Kandil El-Shafee-spaces and the characterized fuzzy -spaces in sense [9] is also introduced.

Let (*X, τ*) be an fuzzy topological space and .Then
the characterized fuzzy space is said to be characterized
completely fuzzy regular Kandil and Shafee space or (characterized
fuzzy KE-space, for short) if for every and such that , there exists an -fuzzy continuous mapping such that and are
hold for all *y* **∈** *X* and *α* **∈** *L*. The characterized fuzzy space is said to characterized quasi fuzzy *T _{1}*-space or (characterized QF

In the following proposition we show that the characterized fuzzy - spaces which are presented [9] are more general than the characterized fuzzy KE-spaces.

**Proposition 5.3**

Let (*X, τ*) be an fuzzy topological space and Then
every characterized fuzzy KE-space (*X, φ*_{1,2}.int) is characterized
fuzzy -space.

**Proof:** Let (*X, φ*_{1,2}.int) is a characterized fuzzy KE-space, *x* **∈** *X* and such that Then, and therefore . Because of (*X, φ*_{1,2}.int) is characterized fuzzy KE space, then there exists a φ_{1,2}ψ_{1,2}-fuzzy continuous mapping such that and are
hold for all *y* **∈** *X*. In case of *y* **∈** *F*, we have 0, that is, *f(y)(s)*=0 for all *s*>0and therefore for all *y* **∈** *F*. In
case of *y=x*, we have holds and then *f(x)* (*s*)=1 for all *s* < 1, therefore . Hence, there exists a φ_{1,2}ψ_{1,2}-fuzzy
continuous mapping such that and for all *y* **∈** *F*. Consequently, (*X, φ*_{1,2}.int) is characterized fuzzy -space in sense [9].

**Corollary 5.3**

Let (*X, τ*) be an fuzzy topological space and . Then
every characterized fuzzy KE-space is characterized fuzzy -space.

**Proof:** Follows immediately from Proposition 5.3 and the fact that
every characterized QF*T _{1}*-space is characterized fuzzy

The following example shows that the inverse of Proposition 5.3 and Corollary 5.3 are not true in general.

**Example 5.3.**

Consider the characterized fuzzy space which is
defined in Example 5.1, then as shown in Example 5.1, is
characterized fuzzy -space in sense [9] and characterized fuzzy*T _{1}*-
space, therefore is characterized fuzzy -space in sense [9].

Now, choose and then such that . Hence, for any φ_{1,2}ψ_{1,2}-fuzzy
continuous mapping such that and for all we get holds for all *z* **∈** *X*. But holds only for
z=y and it is not fulfilled for z=x. Consequently, is not
characterized fuzzy KE-space and therefore it is not characterized
fuzzy KE-space.

In this paper, basic notions related to the characterized fuzzy and the characterized fuzzy -spaces which are presented [9]
are introduced and studied. These notions are named metrizable
characterized fuzzy spaces, initial and final characterized fuzzy spaces,
some finer characterized fuzzy and characterized fuzzy -spaces. The metrizable characterized fuzzy space is introduced as a
generalization of the weaker and stronger forms of the fuzzy metric
space introduced by Gahler and Gahler [3]. For every stratified fuzzy
topological space generated canonically by an fuzzy metric we proved
that, the metrizable characterized fuzzy space is characterized fuzzy T_{4}-
space in sense of Abd-Allah [10] and therefore, it is characterized fuzzy and characterized fuzzy -space. The induced characterized
fuzzy space is characterized fuzzy and characterized fuzzy -space if and only if the related ordinary topological space is *φ*_{1,2} -space and *φ*_{1,2} -space, respectively. Hence, the notions of
characterized fuzzy and of characterized fuzzy are good
extension in sense of Lowen [11]. Moreover, the α-level characterized
space and the initial characterized space are characterized -space
and characterized -space if the related characterized fuzzy space
is characterized fuzzy -space and characterized fuzzy -space,
respectively. We shown that the finer characterized fuzzy space of
a characterized fuzzy -space and of a characterized fuzzy -space is also characterized fuzzy and characterized fuzzy -space, respectively. The categories of all characterized fuzzy and
of all characterized fuzzy -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, respectively. That is, all the initial and final characterized
fuzzy -spaces exist in CFR-Space and also all the initial and final
characterized fuzzy -spaces exist in CRF-Tych. We shown that the
initial and final characterized fuzzy spaces of a characterized fuzzy -space and of characterized fuzzy -space are characterized fuzzy and characterized fuzzy -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 -space and of a characterized fuzzy -space are also characterized fuzzy and characterized fuzzy spaces, respectively. Finally, we introduced and studied three finer
characterized fuzzy and three finer characterized fuzzy L-spaces as a generalization of the weaker and stronger forms of the
completely regular and the fuzzy -spaces introduced [28,12,13].
These fuzzy spaces are named characterized fuzzy H, characterized
fuzzy K, characterized fuzzy KE, characterized fuzzy H, characterized fuzzy K and characterized fuzzy KE-spaces.
The relations between characterized fuzzy H, characterized fuzzy K, characterized fuzzy KE-spaces and the characterized fuzzy -space which are presented [9] are introduced. More generally, the
relations between characterized fuzzy H, characterized fuzzy K,
characterized fuzzy KE-spaces and the characterized fuzzy -spaces
are also introduced. Meany special cases from these finer characterized
fuzzy and finer characterized fuzzy -spaces are listed in **Table 1**.

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