Medical, Pharma, Engineering, Science, Technology and Business

**Abd-Allah AS ^{1} and Al-Khedhairi A^{2*}**

^{1}Department of Mathematics, College of Science, El-Mansoura University, El-Mansoura, Egypt

^{2}Department of Statistics and Operations Research, College of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia

- *Corresponding Author:
- Al-Khedhairi A

Mathematics Department

College of Science and Humanity Studies

Prince Sattam Bin Abdul-Aziz University

PO Box 132012, Hotat Bani Tamim 11941, Saudi Arabia

**Tel:**966 800 116 9528

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

**Received Date:** December 08, 2016; **Accepted Date:** December 31, 2016; **Published Date:** January 26, 2017

**Citation: **Abd-Allah AS, Al-Khedhairi A (2017) The Relations between Characterized Fuzzy Proximity, Fuzzy Compact, Fuzzy Uniform Spaces and Characterized Fuzzy *T _{s}*–Spaces and Fuzzy

**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

In this research work, we study the relations between the characterized fuzzy Ts–spaces and characterized fuzzy Rk–spaces presented in old papers, for s ∈ {0,1,2,3,3 1/2 ,4} and k ∈ {1,2,2 1/2 ,3} and the characterized fuzzy proximity spaces presented. We also study the relations between the characterized fuzzy Ts–spaces, the characterized fuzzy Rk–spaces and the characterized fuzzy compact spaces which is presented in old paper, as a generalization of the weaker and stronger forms of the G–compactness defined by Gähler. Moreover, we show here the relations between these characterized fuzzy Ts–spaces, characterized fuzzy Rk–spaces and the characterized fuzzy uniform spaces introduced and studied by Abd-Allah in 2013 as a generalization of the weaker and stronger forms of the fuzzy uniform spaces introduced by Gähler.

Fuzzy filter; Fuzzy topological space; Operationsl; Isotone
and idempotent; Characterized fuzzy space; φ_{1,2}–fuzzy neighborhood
filters; Fuzzy uniform structure; Characterized fuzzy proximity space;
Characterized fuzzy compact space; Characterized fuzzy uniform space;
Characterized *FTs*–space; Fφ_{1,2}–*T _{s}* space; Characterized

The notion of fuzzy filter has been introduced by Eklund et al.
By means of this notion the point-based approach to fuzzy topology
related to usual points has been developed. The more general concept
for fuzzy filter introduced by Gähler [1] and fuzzy filters are classified by
types. Because of the specific type of fuzzy filter however the approach
of Eklund is related only to the fuzzy topologies which are stratified,
that is, all constant fuzzy sets are open. The more specific fuzzy filters
considered in the former papers are now called homogeneous. The
operation on the ordinary topological space (*X,T*) has been defined by
Kasahara [2] as the mapping φ from *T* into 2^{X} such that A ⊆ A^{φ}, for
all *A T*. In 1983, Abd El-Monsef et al. [3] extend Kasahara operation
to the power set *P (X)* of a set *X*. In 1999, Kandil [4] and the author
extended Kasahars’s and Abd El-Monsef’s operations by introducing
an operation on the class of all fuzzy subsets endowed with an fuzzy
topology t as the mapping *φ : L ^{X} → L^{X}* such that

The notions of the fuzzy filters and the operations on the class of
all fuzzy subsets on X endowed with a fuzzy topology t are applied by
Abd-Allah in [5-7] 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 -fuzzy interior of a fuzzy subset, -fuzzy
convergence and -fuzzy neighborhood filters are defined and applied
to introduced many special classes of separation axioms. The notion of -interior operator for a fuzzy subset is defined as a mapping .int:LX
→ LX which fulfill (I1) to (I5) in Abd-Allah [5]. There is a one-to-one
correspondence between the class of all -open fuzzy subsets of X and
these operators, that is, the class OF(*X*) of all -open fuzzy subsets
of X can be characterized by these operators. Then the triple (X, .
int) as will as the triple (X, OF (*X*)) will be called the characterized fuzzy space [5] of -open fuzzy subsets. The characterized fuzzy
spaces are identified by many of characterizing notions in Abd-Allah
[5-7], for example by the -fuzzy neighborhood filters, -fuzzy
interior of the fuzzy filters and by the set of -inner points of the
fuzzy filters. Moreover, the notions of closeness and compactness
in the characterized fuzzy spaces are introduced and studied by
Abd-Allah in [7]. The notions of characterized FTs-spaces, F -TS
spaces, characterized FRk-spaces and F -Rk spaces are introduced
and studied in Abd-Allah [9-11] for all and . The notions of characterized fuzzy compact spaces,
characterized fuzzy proximity spaces and characterized fuzzy uniform
spaces are introduced and studied by the author in 2004 and 2013
in [7,12]. This paper is devoted to introduce and study the relations
between the characterized *FT _{s}* and

**Preliminaries**

We begin by recalling some facts on fuzzy subsets and on fuzzy
filters. Let *L* be a completely distributive complete lattice with different
least and last elements 0 and 1, respectively. Let *L _{0} = L* \ {0}. Sometimes
we will assume more specially that

**Fuzzy filters**

The fuzzy filter on *X* [1] is the mapping *M: L ^{X}* → L such that the
following conditions are fulfilled:

The fuzzy filter is called homogeneous [14] if for all For each *x X*, the mapping defined by for all *µ L ^{X}* is a homogeneous fuzzy filter on

**Lemma 2.1**

If and are fuzzy filters on a set X. Then the following sentences are fulfilled [1].

**Proposition 2.1**

For all we have if and only if [15].

For each non-empty set of the fuzzy filters on X the supremum exists [1] and given by

for all . Whereas the infimum of A does not exists in general as an fuzzy filter. If the infimum exists, then we have

for all , where *n* is an positive integer, *µ _{1},…,µ_{n}* is a collection such
that and are fuzzy filters from Let

(2.1)

**Fuzzy filter bases**

The family of a non-empty subsets of *L ^{X}* is called a valued fuzzy filter base [1] if the following conditions are fulfilled:

(V2) For all a,ß L_{0} with and all there are

As shown in Gähler [1], each valued fuzzy filter base defines a fuzzy filter for all Conversely, each fuzzy filter M can be generated by a valued fuzzy filter
base, e.g. by with α-p*r* . The is the family of pre filters on *X* and it is called the large
valued fuzzy filter base of Recall that the pre filter on *X* [16] is a
non-empty proper subset of L^{X} such that (1) implies and (2) from

**Valued and superior principal fuzzy filters**

Let a non-empty set *X* be fixed, µ ∈ L^{X} and a ∈ L such that *α* ≤ sup *µ*, the valued principal fuzzy filter [20] generated by *µ* and α, will be
denoted by [*µ,α*], is the fuzzy filter on *X* which has with otherwise as a valued fuzzy filter base.
For all , we have and . Moreover, for each we have if ß = a and otherwise. The superior principal
fuzzy filter [1] generated by *µ*, written [*µ*], is the homogeneous fuzzy
filter on *X* which has as a superior fuzzy
filter base. As shown in Katsaras [18], the superior principal fuzzy filter
[*µ*] is representable by a fuzzy pre filter if and only if sup *µ* = 1.

**Fuzzy filter functors and fuzzy filter monads**

The fuzzy filter functor is the covariant functor from
the category SET of all sets to this category which assigns to each set X
the set and to each mapping the mapping . The
homogeneous fuzzy filter functor is the sub fuzzy filter
functor of FL which assigns to each set X the set and to each mapping *f* the domain-range restriction of the mapping For each set X, let be the mapping defined by for all x∈X, and let be the mapping for which = M (μ) for all μ *L * and *M∈F _{L}X*. Moreover, let be the mapping which assigns to each fuzzy filter the fuzzy
filter with id
the identity set functor and F are
natural transformations. is a monad in the categorical sense,
called the fuzzy filter monad [1], that is, and for each set

**Fuzzy topologies**

By a fuzzy topology on a set *X* [20,21], we mean a subset of *L ^{X}* which is closed with respect to all suprema and all finite infima and
contains the constant fuzzy sets . A set

**Fuzzy proximity spaces**

The binary relation δ on *L ^{X}* is called fuzzy proximity on

is the negation of δ.

(P2)

(P3)

(P4)

(P5)

The set *X* equipped with an fuzzy proximity *δ* on *X* is said to be fuzzy
proximity space and will be denoted by (*X,δ*). Every fuzzy proximity *δ* on a set *X* is associated an fuzzy topology on *X* denoted by *τ _{δ}*. The fuzzy
proximity

**Operation on fuzzy sets**

In the sequel, let a fuzzy topological space (*X,τ*) be fixed. By the
operation [4] on a set *X*, we mean the mapping such that

*int * holds, for all where 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 Also by the constant operation on , we mean the operation such that for al If ≤ is a partially ordered relation on defined as
by for all then obviously, is
a completely distributive lattice. As an application on this partially
ordered relation, the operation will be called:

(i) Isotone

(ii) Weakly finite intersection preserving (wfip, for short) with
respect to holds, for all *ρ ∈A* and *μ ∈ L ^{X}*.

(iii) Idempotent if

The operations are said to be dual if *μ = co ( (coμ))* or equivalently for all where *coμ* denotes the
complement of *μ*. The dual operation of *φ* is denoted by . In the
classical case of L = {0,1}, by the operation on the set X [3], we mean the
mapping such that *int* for all *A* in the power set *P (X)* and the identity operation on the class of all ordinary operations on *X* will be denoted by

**φ-open fuzzy subsets**

Let a fuzzy topological space (*X,τ*) be fixed and . The
fuzzy subset *μ : X → L* is said to be *φ*-open fuzzy subset if holds.
We will denote the class of all *φ*-open fuzzy subsets on *X* by OF (*X*).
The fuzzy subset *μ* is called *φ*-closed if its complement *coμ* is *φ*-open.
The two operations are equivalent and written if
and only if

**φ _{1,2}–interior of fuzzy subsets**

Let a fuzzy topological space (X,τ) be fixed and Then the φ_{1,2}-interior of the fuzzy subset *μ: X → L* is the mapping φ_{1,2}.
int μ : X → L defined by:

(2.2)

As easily seen that is the greatest *φ _{1}*-open fuzzy subset

**Proposition 2.2**

If (*X,τ*) is an fuzzy topological space and . Then, the
mapping fulfills the following axioms [5]:

(i) If , then .int *μ ≤ μ* holds.

(ii) .int is isotone operator, that is, if *ρ* holds for all

(iii)

(iv) If is isotone and *φ _{1}* is wfip with respect to

(v) If _{2} is isotone and idempotent operation, then .

(vi)

**Proposition 2.3**

Let (*X,τ*) be an fuzzy topological space and . Then
the following are fulfilled [5]:

(i) If , then the class OF(X) forms extended fuzzy
topology on *X* [19].

(ii) If and , then the class OF(*X*) forms a
supra fuzzy topology on *X* [19].

(iii) If is isotone and *φ _{1}* is wfip with respect to φ

(iv) If is isotone and idempotent operation and φ1 is
wfip with respect to *φ*_{1}OF(*X*), then OF(*X*) is a fuzzy topology on *X* [20,21].

From Propositions 2.2 and 2.3, if the fuzzy topological space (*X,τ*)
be fixed and . Then

(2.3)

and the following conditions are fulfilled:

(I1) If , then holds, for all *μ ∈ L ^{X}*.

(I2) If

(I3)

(I4) If is isotone and is wfip with respect to then

(I5) If_{1} is isotone and idempotent, then

**Characterized fuzzy spaces**

Independently on the fuzzy topologies, the notion of -interior
operator for fuzzy subsets can be defined as a mapping .int : *L ^{X}* →

**Fuzzy unit interval**

The fuzzy unit interval will be denoted by *I _{L}* and it is defined in
Gähler [24] as the fuzzy subset where

**φ _{1,2}–fuzzy neighborhood filters**

An important notion in the characterized fuzzy space (X,φ_{1,2}.int)
is that of the φ_{1,2}-fuzzy neighborhood filter at the point and at the
ordinary subset in this space. Let (*X,τ*) be a fuzzy topological space and . As follows by (I1) to (I5) for each *x∈X*, the mapping

(2.4)

for all is a fuzzy filter, called *fuzzy neighborhood filter* at *x* [5]. If then the *fuzzy neighborhood filter* at F will be
denoted by and it will be defined by:

Since is fuzzy filter for all *x∈X*, then is also fuzzy filter on *X*. Because of , then we have holds. If the related -interior operator fulfill the axioms (I1) and (I2)
only, then the mapping , defined by (2.4) is an fuzzy
stack, called *fuzzy neighborhood stack* at *x*. Moreover, if the -
interior operator fulfill the axioms (I1), (I2) and (I4) such that in (I4)
instead of we take then the mapping , is
an fuzzy stack with the cutting property, called -*fuzzy neighborhood
stack with the cutting property* at *x*. Obviously, the -fuzzy
neighborhood filters fulfill the following conditions:

Clearly, is the fuzzy subset

The characterized fuzzy space is characterized as the *fuzzy filter pre topology* [5], that is, as a mapping such that the conditions (N1) to (N3) are fulfilled.

**φ _{1,2}ψ_{1,2}–fuzzy continuity**

Let now the fuzzy topological spaces are fixed, The mapping is said to be –*fuzzy continuous* [5] if the inequality

(2.5)

holds for all If an order reversing involution ′ of L is
given, then we have that *f* is a fuzzy continuous if and only if holds for all Note that and means that the closure operators related to respectively which are defined by Obviously if *f* is fuzzy continuous and the inverse *f*^{–1}of *f* exists, then -fuzzy continuous,
that is, holds for all By means
of characterizing the -fuzzy neighborhoods and which are defined by (2.4), the fuzzy continuity of *f* can also be characterized as follows:

The mapping fuzzy continuous
if the inequality holds for each *x*∈*X*. Obviously, in case of , the -fuzzy continuity coincides with the usual fuzzy continuity.

**φ _{1,2}–fuzzy convergence**

Let an fuzzy topological space (*X,τ*) be fixed and . If *x* is a point in the characterized fuzzy space is a fuzzy filter on *X*. Then -*fuzzy convergence* [5] to
x and written , provided is finer than the fuzzy
neighborhood filter . Moreover, is said to be -fuzzy
convergence to F and written provided is finer than
the -fuzzy neighborhood filter for all x∈F, that is, is
finer than the -fuzzy neighborhood filter .

**Internal φ _{1,2}-closure of fuzzy sets and φ_{1,2}-closure operators**

Let a fuzzy topological space (*X,τ*) be fixed and . The
internal -closure of the fuzzy set is the mapping defined by:

(2.6)

(2.6)

for all *x*∈*X*. In (2.6), the fuzzy filter *M* my have additional properties,
e.g, we my assume that is homogeneous or even that is ultra. Obviously, holds for all The mapping which
assigns to each fuzzy filter *M* on *X*, that is,

(2.7)

is called -closure operator [7] of the characterized fuzzy space with respect to the related fuzzy topology *τ*. Obviously, the -closure operator is isotone hull operator, that is, for all

**Lemma 2.2**

Let (*X,τ*) be a fuzzy topological space and . Then for
each *x∈X*, we have [10].

**Characterized fuzzy R_{k} and fuzzy φ_{1,2}R_{k}-spaces**

The notions of characterized fuzzy *R _{k}* and fuzzy

Let a fuzzy topological space (*X,τ*) be fixed and Then the characterized fuzzy space is said to be:

(1) Characterized *FR _{2}*-space (resp. Characterized

(2) Characterized - space if for all *x*∈*X*, such that there exists an -fuzzy continuous mapping such that and for all . The related
fuzzy topological space (*X,τ*) is said to be - space if and only
if is characterized - space.

**Characterized fuzzy T_{s} and fuzzy φ_{1,2}-T_{S} spaces**

The notions of characterized fuzzy *T _{s}* and fuzzy φ

Let a fuzzy topological space (X,τ) be fixed and Then the characterized fuzzy space is said to be:

(1) Characterized *FT0*-space (resp. Characterized *FT1*-space) if for
all *x,y∈X* such that *x≠y* there exist and such that μ (x) < holds or (resp. and) there exist such
that holds. The related fuzzy topological space
(*X,τ*) is said to be space such that *x≠y* we have or (resp. and)

(2) Characterized *FT _{2}*-space if for all

(3) Characterized *FT _{s}* space if and only if it is characterized

**Proposition 2.4**

Let (*X,τ*) be an fuzzy topological space and . Then
the characterized fuzzy space is characterized -space if
and only if [8].

**Proposition 2.5**

If is characterized *FT _{2}*-space and φ

**Proposition 2.6**

Let a fuzzy topological space (*X,τ*) be fixed and Then the following are fulfilled [8,22]:

(1) Every characterized *FT _{i}*-space is characterized

(2) The characterized fuzzy subspace and the characterized
fuzzy product space of a family of characterized *FT _{2}*-spaces are also
characterized

**New Relations between Characterized FT_{s}, Characterized FR_{k} and Characterized Fuzzy Proximity Spaces**

In this section we are going to introduce and study the relations
between the characterized *FT _{s}*-spaces, the characterized

**Proposition 3.1**

Let a fuzzy topological space (*X,τ*) be fixed and such
that is isotone and idempotent and φ_{1} is wfip with respect to . Then the supremum of the φ_{1,2}-fuzzy neighborhood filters at *x∈X* which is given by:

(3.1)

for all is a fuzzy filter on X called a φ_{1,2}-fuzzy neighborhood filter
at *μ*.

**Proof:** Fix an hen because of (2.4) and the condition

and

Thus, condition (F1) is fulfilled. To prove condition then because of Proposition 2.4 and (2.4) we have

Hence, holds for all *x∈X* and holds for all fulfills condition
(N_{1}). For condition (N_{2}), let Because of
Proposition 2.4, we have holds and which implies
that holds for all *y∈X*. Hence and therefore condition (N_{2}) is fulfilled.
Since for any *y*∈*X* we have represents the mapping φ_{1,2}int ρ. Then from Proposition 2.4 we have and then for all Thus, condition (N3) is also fulfilled and therefore neighborhood filters.

Not that in Bayoumi et al. [15], the supremum of the empty set of the fuzzy filters is the finest fuzzy filter. This means for all Because of (2.4) the equations (2.1) and (2.2) can be written as in the following:

(3.2)

(3.3)

for all Here a useful remark is given

**Remark 3.1:** The homogeneous fuzzy filter at the ordinary point
x is nothing that a homogeneous fuzzy filter at the fuzzy point x_{α}, that
is, Moreover, the φ_{1,2}–fuzzy neighborhood
filter at x is itself the *φ*_{1,2}–fuzzy neighborhood filter at x_{α}.

The *φ*_{1,2}–fuzzy neighborhood filter at the fuzzy subset and the homogeneous fuzzy filter fulfill the following
properties.

**Lemma 3.1**

Let (*X,τ*) be a fuzzy topological space and Then for
all the following properties are fulfilled:

(1) implies implies

(2) *μ* ≤ implies

(3)

(4)

(5) implies there is an and

**Proof:** Let . From condition (N1) we have ρ holds
and therefore for all we have holds. Hence, holds also. Thus, and are hold. Similarly, if then from (N1) we have which implies Thus, (1) is fulfilled. Since *μ ≤ ρ* implies *μ (x) ≤ ρ (x)* for
all *xX*, then

Hence, for all and therefore holds. Hence, (2) is fulfilled.

Since then from (2) we have Now, let

Hence, holds and therefore (3) is fulfilled. To prove (4), let holds. Because of (2.1), (3.1) and (N1), we have and then holds. Hence, Proposition 2.1 implies Thus, (4) is fulfilled. Finally, let holds for all Hence, there is such that

This means there is such that and are hold for all Thus, and are also hold. Consequently, (5) is fulfilled.

In the characterized fuzzy space (X,φ_{1,2}.int), the fuzzy proximity
will be identified with the finer relation on the fuzzy filters, specially
with the finer relation on the φ_{1,2}-fuzzy neighborhood filters. This
shown in the following proposition.

**Proposition 3.2**

Let (*X,τ*) be a fuzzy topological space and Then the
binary relation *δ* on *L ^{X}* which is defined by:

for all is fuzzy proximity on X.

**Proof:** Let Because of
(1) in Lemma 3.1, we have and therefore . Hence,
condition (P1) is fulfilled.

Since and are hold for all then and implies Conversely, let then are hold. Hence, (3) in Lemma 3.1 implies holds and therefore Consequently, (P2) is fulfilled. To prove (P3), since holds for all Hence, implies Thus, (P3) is fulfilled.

Let Because of
(1) and (4) in Lemma 3.1, we have and therefore that is, (P4) is fulfilled. Finally, let then Because of (5) in Lemma
3.1, there is an are
hold. Hence, are also hold, that is, Thus, (P5) holds and consequently, *δ* is fuzzy
proximity on *X*.

If a fuzzy topological space (*X,τ*) be fixed and Then
each fuzzy proximity *δ* on *X* is associated a set of all φ_{1,2}-open fuzzy
subsets of *X* with respect to *δ* denoted by In this case the
triple as will as is said to be *characterized
fuzzy proximity space*. The related φ_{1,2}-interior and φ_{1,2}-closure
operators are given by:

(3.5)

respectively, for all Consider the characterized fuzzy proximity
space be fixed and then μ is said to be neighborhood for the point *x∈X* if and only if Moreover, the
mapping -fuzzy
continuous, provided implies for all

In the following we will show that the characterized fuzzy proximity
space is characterized *FT _{0}*-space as in sense of [8] if and
only if δ is separated.

**Proposition 3.3**

Let (*X,τ*) be a fuzzy topological space, and is a fuzzy
proximity on *X*. Then the characterized fuzzy proximity space is characterized *FT _{0}*-space if and only if

**Proof:** Let such that *x≠y*. Then and therefore there is such
that Because of (3.4), we have and hence μ (x) > ρ (y) holds for all that is, *μ (x) >ρ* (*y*) holds for all with Choose then because of Remark 3.1, we get Using Proposition
3.2 we get holds for all Thus, *δ* is
separated.

Conversely, let *δ* is separated fuzzy proximity and let *x,y∈X* such
that *x≠y*. Then, and because of Proposition 3.2 and Remark 3.1,
we have Therefore, holds for
all Consider, we get and Hence, there exists such that that
is, and therefore is characterized *FT _{0}*-space.

In the following proposition, the _{1,2}-closure of the fuzzy subsets in
the characterized fuzzy space (*X,φ*_{1,2}.int_{δ}) are equivalent with the fuzzy
subsets by the fuzzy proximity *δ* on *X*.

**Proposition 3.4**

Let (*X,τ*) be a fuzzy topological space, such that and *δ* is a fuzzy proximity on *X*. Then, if and only if

**Proof:** Let such that then Proposition 3.2 implies is isotone operator, then are hold for all and therefore

Conversely, Let such that Because of Proposition 3.2 we have and is isotone operator, then holds for all From Lemma 3.1, we have . Therefore, holds. Using Lemma 3.1 we get and therefore

In the following proposition, we show that the associated
characterized fuzzy proximity space is characterized *FR _{2}*-
space if the related fuzzy topological space (

**Proposition 3.5**

Let (*X,τ*) be a fuzzy topological space, and is an fuzzy
proximity on *X*. Then the associated characterized fuzzy proximity
space is characterized *FR _{2}*-space if (

**Proof:** Let *xX* and with Because of Proposition
3.2, we have and from (P5), there is such that and Therefore Proposition 3.4 implies and hence are hold.
Hence, implies there is such that and are hold. Since is space,
then from Theorem 3.1 in Abd-Allah [12], we have is
characterized *FR _{2}*-space.

The binary relation << on *L ^{X}* is said to be

(1)

(2) If

(3) If

(4) If _{1} are hold for all

The fuzzy topogeneous order ≪ is said to be *fuzzy topogeneous
structure* if it fulfilled the condition:

(5) If are hold for all

The fuzzy topogeneous structure ≪ is said to be *fuzzy topogenous
complementarily* symmetric if it fulfilled the condition:

(6) If

As shown in Katsaras [23], every fuzzy topogeneous structure ≪
is identify with the mapping such that if and
only if μ <<η holds for all The fuzzy topogeneous structures are classified by these mappings. As is easily seen, each fuzzy topogeneous
order N can be associated a fuzzy pre topology int* _{N}* on a set

(3.5)

for all is a sequence of fuzzy topogenous structure
on the set X and is a sequence of fuzzy topogenous structure on *I _{L}*, then the fuzzy real function is said to be associated with the
sequence if and only if implies that holds for all is the set of all positive
integer numbers.

**Remark 3.2**

Given that are two sequence of
complementarily symmetric fuzzy topogenous structures and on *X* and *I _{L}*, respectively. If

**Lemma 3.2**

Consider are complementarily symmetric fuzzy
topogenous structures on a set *X*. Then, for each *F,G∈P (X)* such that associated with the
sequence for which for all *xF* and for all *yG*′ [23].

Because of equation (3.5), Remark 3.2 and Lemma 3.2, we can easily deduce the following proposition.

**Proposition 3.6**

Let is a characterized fuzzy proximity space and *F,G∈P *(*X*) such that -fuzzy continuous
mappings implies then are -separable.

**Proof:** Let ≪ be a complementarily symmetric fuzzy topogenous
structure identified with δ. Because of (3.5), implies that. Since δ-fuzzy continuous, then because of Remark 3.2, we have
that f is associated with ≪. Hence, Lemma 3.2 implies that are
separated by *f* and therefore are Φ-separable.

**Proposition 3.7**

Let are two characterized fuzzy proximity spaces. If the mapping is is -fuzzy continuous.

**Proof:** Similar to the proof of Proposition 11.2 in Gähler [13].

In the following we are going to show an important relation between the associated characterized fuzzy proximity space and the
characterized *FR _{3}*-space.

**Proposition 3.8**

Let (*X,τ*) be a fuzzy topological space and such that is isotone and φ_{1} is wfip with respect to where *L* is
complete chain. If (*X,τ*) is a fuzzy normal topological space, then the
binary relation *δ* on *X* which is defined by:

(2.6)

for all is a fuzzy proximity on X and (*X,δ*) is a fuzzy proximity
space. On other hand if (*X,δ*) is a fuzzy proximity space with *δ* fulfills (3.6), then the associated characterized fuzzy proximity space is characterized *FR _{3}*-space.

**Proof:** Let (*X,τ*) is fuzzy normal topological space and *δ* a binary relation on *X* defined by (3.6). Then, implies and from Lemma 3.1 part (1) we get and then . Hence, condition (P1)
is fulfilled. For showing condition (P2), let for a fixed
fuzzy subsets Then, Since L is complete chain, is isotone and φ_{1} is wfip with
respect to and
therefore Because of Lemma 3.1 part
(3), we have and . Thus, and are hold and therefore implies . On the other hand let and Then from Lemma 3.1 we have tha the inequalities are hold and
therefore that is, imply Hence, (P2) is fulfilled.
Now, let such that the finest fuzzy filter on X and from the fact we get for all that is,
(P3) is also fulfilled. Since implies which means by the inequality that Because of Proposition 2.1 and the fact that φ_{1,2}.
cl is hull operator we get Thus, (P4) is
fulfilled. Let such that Consider, , hence and therefore holds. Since (*X,τ *) is characterized fuzz normal space, then
from Theorem 3.2 in Abd-Allah [12], there exists with arbitrary
choice such that are
hold. Therefore, there exists *∈L ^{X}* such that and Hence, (P5) is also fulfilled. Consequently,

Conversely, let and therefore Hence because of Lemma 3.1 part (1) we have From (P5), there
exists such that are hold. Because of Lemma 3.1 part (1), we have Hence, holds for all. Consider and for all , then
we get = 0 and Hence, there
exist such that , that is, the infimum does not exists. Consequently, is characterized *FR _{}_{3}*-space.

In the following we are going to show an important relation
between the associated characterized fuzzy proximity space int_{δ}) by the fuzzy proximity δ defined by (3.6) and the associated
characterized fuzzy space that introduced form the fuzzy
normal topological space (*X,τ*).

**Proposition 3.9**

Let (*X,τ*) is a fuzzy normal topological space and such that is isotone and φ_{1} is wfip with respect to φ_{1}OF (*X*). If *δ* is the fuzzy proximity on *X* defined by (3.6) and L is a complete
chain, then is finer than Moreover, if and only if is characterized *FR _{4}*-space.

**Proof:** Let (*X,τ*) is fuzzy normal topological space and μ is φ_{1,2}δ-
fuzzy neighborhood for the point *xX*, then and because
of (3.6), we have Therefore, Because of Proposition 2.1, we get and -fuzzy neighborhood of x and
therefore the family that is,

Now, let and denote for the φ_{1,2}-fuzzy neighborhood filters at *x* in
the characterized fuzzy space and in the associated
characterized fuzzy proximity space respectively.
Then, is characterized *FR _{3}* and

Conversely, let - fuzzy
neighborhood of *x* in Then, and this means that Because
of Proposition 2.1, we get holds for all *x∈X*. Thus, Hence, Proposition
2.4 implies that, is characterized *FR _{1}*-space. Because
of Proposition 3.7, is characterized

In the following we are going to introduce some important relations joining our characterized -spaces, characterized -spaces and the associated characterized fuzzy proximity spaces.

**Proposition 3.10**

Let (*X,τ*) be an fuzzy topological space and If δ
is an fuzzy proximity on X, then the associated characterized fuzzy
proximity space is characterized - space.

**Proof:** Let *xX* and neighborhood of *x*, then . Because of Proposition 3.2, we
get that *x _{1}* and

**Corollary 3.1**

Let (*X,τ*) be a fuzzy topological space, and δ is
a fuzzy proximity on X. Then the associated characterized fuzzy
proximity space is characterized - space.

**Proof :** Immediately from Propositions 2.4 and 3.10.

Now, we introduce an example of an fuzzy proximity *δ* on a set *X* and show that it is induces an associated characterized - space
compatible with the related characterized fuzzy space.

**Example 3.1**

Let is a fuzzy topology
on X. Choose and Hence, *x ≠ y *and there is only two cases, the first is and the
second is We shall consider the first case and
the second case is similar. Consider the mapping -fuzzy
continuous and therefore is characterized - space and
obviously is also characterized *FR _{1}*-space, that is, is
characterized - space. Now, consider δ is a binary relation on

- fuzzy continuous mapping

with

for all Hence obviously, δ is a fuzzy proximity on *X* and that is, the associated characterized fuzzy proximity
space with *δ* is characterized - space and compatible
with

**Some Relations between Characterized FT_{s} and
Characterized Fuzzy Compact Spaces**

The notion of *φ*_{1,2}-fuzzy compactness of the fuzzy filters and of the fuzzy topological spaces are introduced by Abd-Allah in [7] by
means of the *φ*_{1,2}-fuzzy convergence in the characterized fuzzy spaces.
Moreover, the fuzzy compactness in the characterized fuzzy spaces is
also introduced by means of the *φ*_{1,2}-fuzzy compactness of the fuzzy
filters and therefore it will be suitable to study here the relations
between the characterized fuzzy compact spaces and some of our
classes of separation axioms in the characterized fuzzy spaces.

Let (*X,τ*) be an fuzzy topological space, *F ⊆ X* and Then *x∈X* is said to be *φ*_{1,2}-adherence point for the fuzzy filter on *X* [7],
if the infimum exists for all *φ*_{1,2}-fuzzy neighborhood
filters As shown in Abd-Allah [7], the point *x∈X* is said to be *φ*_{1,2}-adherence point for the fuzzy filter on *X* if and
only if there exists an fuzzy filter finer than *M* and that is, are hold for some The
subset *F* of *X* is said to be *φ*_{1,2}-fuzzy closed with respect to *φ*_{1,2}.int if implies *x∈F* for some The subset *F* is said to
be *φ*_{1,2}-fuzzy compact [7], if every fuzzy filter on F has a finer *φ*_{1,2}−fuzzy
converging filter, that is, every fuzzy filter on *F* has *φ*_{1,2}-adherence point
in *F*. Moreover, the fuzzy topological space (*X,τ*) is said to be *φ*_{1,2}-fuzzy
compact if *X* is *φ*_{1,2}-fuzzy compact. More generally, the characterized
fuzzy space (*X*,*φ*_{1,2}.int) is said to be fuzzy compact space if the related
fuzzy topological space (*X,τ*) is *φ*_{1,2}-fuzzy compact.

At first, in the following we shall benefit from these facts. Consider the fuzzy unit interval topological space be given and . Then:

(1) The usual topological space (*I,T _{I}*) and the ordinary characterized
usual space on the closed unite interval I = [0,1] are compact T

(2) The closed unite interval I is identified with the fuzzy number
[0,1]^{~} in Gähler [24] defined by [0,1]^{~} (*α*) = 0 for all *α ∈ I* and [0,1]^{~} (*α*)
= 0 for all *α Ï I*.

(3) The characterized fuzzy unite space is up to an identification the characterized usual space in the classical sense.

In the following proposition, we show that every *φ*_{1,2}-fuzzy compact
subset in the characterized *FT _{2}*–space (

**Proposition 4.1**

Let a fuzzy topological space (*X,τ*) be fixed and Then every *φ*_{1,2}-fuzzy compact subset of the characterized *FT _{2}*-space is

**Proof:** Let is characterized *FT _{2}*-space and

and is characterized *FT _{2}*-space, then and imply that

**Proposition 4.2 **

Let be a fuzzy unit interval topological space and Then the characterized fuzzy unit interval space is characterized fuzzy compact *FT _{2}*-space.

**Proof:** Let be an ordinary characterized usual space.
Then, is characterized compact space in the classical
sense, that is, every filter on -adherence point. Consider the
mapping defined by: for
all then it is easily to seen that fuzzy homeomorphism
between is
characterized fuzzy compact space. Since (*I,T _{I}*) is - space, then is characterized

Now, we are going to prove an important relation between the
characterized compact *FT _{2}*-spaces and the characterized

**Proposition 4.3**

Let (*X,τ*) be n fuzzy topological space and Then
every disjoint *φ*_{1,2}-fuzzy compact subsets *F _{1}* and

**Proof:** Let *F _{1}* and

Secondly, the notion of the fuzzy compactness for the characterized fuzzy spaces fulfills the following property which will be also used in the prove of this important result which given in Proposition 4.4.

**Lemma 4.1**

Let (*X,τ*) be a fuzzy topological space and Then
every *φ*_{1,2}-fuzzy closed subset of the characterized fuzzy compact space -fuzzy compact.

**Proof:** Let *F* is *φ*_{1,2}-fuzzy closed subset of the characterized fuzzy
compact space and let Then, implies that *x∈F*. Since and hence there exists and Since and then Thus, for all we get such that is *φ*_{1,2}-adherence point of that is, *F* is *φ*_{1,2}-fuzzy compact.

**Proposition 4.4**

Let (*X,τ*)be an fuzzy topological space and Then every
characterized fuzzy compact *FT _{2}*-space is characterized

**Proof:** Follows directly from Lemma 4.1 and Proposition 4.3.

One of the application of Proposition 4.4, we have more generally the following result to the characterized fuzzy unit interval space.

**Proposition 4.5**

Let be an fuzzy unit interval topological space and Then the characterized fuzzy unit interval space is characterized -space.

**Proof:** Because of Proposition 4.2, the characterized fuzzy unit
interval space is characterized fuzzy compact *FR _{2}*-space.
Therefore from Proposition 4.4, we get is characterized

The *φ*_{1,2}-fuzzy compactness in the characterized fuzzy spaces is
applied to fulfilled the Generalized Tychonoff Theorem [11] and from
(2) in Proposition 2.6, the characterized fuzzy product space of the
characterized *FR _{2}*-spaces is also characterized

**Proposition 4.6**

Let be a fuzzy unit interval topological space and Then the characterized fuzzy cube is characterized *FR _{2}*-space and it is characterized

**Proof:** Since the characterized fuzzy cube is product of copies of
the characterized fuzzy unit interval space and by means of
Proposition 4.2, is characterized fuzzy compact *FR _{2}*-space.
Then because of Proposition 2.6, part (3) and Generalized Tychonoff
Theorem in Abd-Allah [11], it follows that, the characterized fuzzy
cube is characterized

**Lemma 4.2**

Let (*X,τ*) and (*X,σ*) are two fuzzy topological spaces such that τ
is finer than σ If and is
characterized fuzzy compact space, then s also characterized
fuzzy compact space.

**Proof:** Let -fuzzy neighborhood
and *ψ*_{1,2}-fuzzy neighborhood at *x* with respect to *ψ*_{1,2}.intτ and *ψ*_{1,2}.int_{σ},
respectively. Since *τ* is finer than *σ*, then N *x* for all *x∈X*. Because of is characterized fuzzy compact space, then for
all such that for all *x* *X*. Therefore for all *x X*. Consequently, (X,*φ*_{1,2}.
int_{τ}) is characterized fuzzy compact space.

**Proposition 4.7**

Let (*X,τ*) and (*X,σ*) are two fuzzy topological spaces such that is
finer than is
characterized fuzzy compact space and is characterized *FT _{2}*-
space, then -fuzzy isomorphic.

**Proof:** Since *τ* is finer than *σ*, then Hence, because
of Proposition 2.5, is characterized *FT _{2}*-space. From Lemma
4.2, we have is characterized fuzzy compact space. Hence,
we can find the identity mapping which
is bijective -fuzzy continuous and its inverse is -fuzzy
continuous, that is,

**Proposition 4.8**

Let (*X,τ*) be a fuzzy topological spaces and Then
every characterized fuzzy compact space is characterized *FT _{2}*-space if and only if it is characterized - space.

**Proof:** Let is characterized fuzzy compact *FT _{2}*-space.
Because of Proposition 4.4 we have is characterized FT4-
space and therefor Proposition 4.6 in Abd-Allah S [11], implies that is characterized - space. Conversely, let is characterized - space, then because of Proposition 3.2 in Abd-
Allah [11] and part (1) of Proposition 2.6, it follows that is
characterized fuzzy compact

From Lemma 4.2 and Corollary 3.3 in [22], we can prove the following result.

**Proposition 4.9**

Let (*X,τ*) and (*X,σ*) are two fuzzy topological spaces such that *τ* is finer than *σ*, and If is
characterized fuzzy compact space and is characterized - space, then -fuzzy
isomorphic.

**Proof:** Follows directly from Corollary 3.3 in [22] and Lemma 4.2
similar to the proof of Proposition 4.7.

**Some Relations Between Characterized FT_{s}, Characterized FR_{k} and Characterized Fuzzy Uniform Spaces**

In this section, we are going to investigate and study the relations
between the characterized *FT _{s}*-spaces, the characterized

By the fuzzy relation on the set X, we mean the mapping *R : X×X* → L, that is, any fuzzy subset of *X×X*. For each fuzzy relation *R* on *X*,
the inverse *R*^{-1} of *R* is the fuzzy relation on *X* defined by *R*^{-1} (*x,y*) = *R* (*y,x*) for all be a fuzzy filer on *X×X*. The inverse is a fuzzy filter on *X×X* defined by The
composition *R _{1} _{ᴼ} R_{2}* of two fuzzy relations

for all For each pair (*x,y*) of elements *x* and *y* of *X × X*, the
mapping defined by: for all *R ∈ X × X* is a homogeneous fuzzy filter on are fuzzy filers on *X × X* such that and hold for some *x,y,z∈X*. Then the
composition is a fuzzy filter [13] on X × X defined by:

for all

By the *fuzzy uniform structure* on a set X [13], we mean a fuzzy
filter on *X × X* such that the following axioms are fulfilled:

The pair is called *fuzzy uniform space*. The fuzzy uniform
structure [13] on a set X is said to be *separated* if for all with
xÏy there is such that and *R (x,y)* = 0. In this case
the fuzzy uniform space is called *separated fuzzy* uniform space. Let is a fuzzy uniform structure on a set X such that holds
for all *x∈X* and let which is
defined by:

for all is a fuzzy filter on *X*, called the image of with respect
to the fuzzy uniform structure [13], where such that

Each fuzzy uniform structure on the set *X* is associated a
stratified fuzzy topology then the
set of all *φ*_{1,2}-open fuzzy subsets of *X* related to forms a base for an
characterized stratified fuzzy space on *X* generated by the *φ*_{1,2}-interior
operator with respect to denoted by is the
related stratified characterized fuzzy space. In this case, will
be called the associated *characterized fuzzy uniform space* [12] which is
stratified. The related *φ*_{1,2}-interior operator *φ*_{1,2}.int_{U} is given by:

(5.1)

for all The fuzzy set *μ* is said to be -fuzzy
neighborhood of x∈X in the associated characterized fuzzy uniform
space Because of (2.1), (3.1) and
(5.1) we have that

(5.2)

for all In this case -fuzzy
neighborhood filters of the associated characterized fuzzy uniform
space at *x* and *μ*, respectively.

**Proposition 5.1**

Let *X* be a non-empty set, *U* is a fuzzy uniform structure on *X* and Then the fuzzy uniform space (*X,U*) is separated if and
only if the associated characterized fuzzy uniform space is
characterized *FT _{0}*-space.

**Proof:** Let is separated and let Then,
there exists such that Consider *μ* = *R[y _{1}]* for which

for all Hence, there exists such that *μ (x) < α* that is, is characterized *FT _{0}*-space.

Conversely, let is characterized *FT _{0}*-space and let

**Corollary 5.1**

Let *X* be a non-empty set, is a fuzzy uniform structure on *X* and Then the fuzzy uniform space is separated if
and only if the associated stratified fuzzy topological space is space.

**Proof:** Immediate from Proposition 5.1 and Theorem 2.1 in Abd-
Allah [8].

**Proposition 5.2**

Let *X* be a non-empty set, is a fuzzy uniform structure on *X* and Then the fuzzy uniform space is separated if
and only if the associated characterized fuzzy uniform space is characterized *FT _{1}*-space.

**Proof:** Let is separated and let *x,y∈X* such that *x ≠ y*. Then,
there exists such that and for all *i*∈{1.2}. Consider then we have and Moreover, and for all Hence, there exists are hold. Consequently, is
characterized *FT _{1}*-space.

Conversely, let is characterized *FT _{1}*-space and let

**Corollary 5.2**

Let *X* be a non-empty set, is a fuzzy uniform structure on *X* and Then the fuzzy uniform space is separated if
and only if the associated stratified fuzzy topological space is *F* *φ*_{1,2}-*T*_{1} space.

**Proof:** Immediate from Proposition 5.2 and Theorem 2.2 in Abd-
Allah [8].

For each fuzzy uniform structure on the set *X*, the mapping *h :* which is defined by is global
homogeneous fuzzy neighborhood structure on *X* [13]. The mapping *h* will be called global homogeneous fuzzy neighborhood structure *associated to* the fuzzy uniform structure and will be denoted by The global fuzzy neighborhood structure *h* on the set *X* is said to be *symmetric* [13], provided that exists if and only if As shown in Gähler [13], for each fuzzy
uniform structure , the associated homogenous fuzzy neighborhood
structure is symmetric and both the global homogenous fuzzy
neighborhood structures associated to the fuzzy uniform structures and its homogenization are coincide.

**Proposition 5.3**

Let be an fuzzy uniformly continuous mapping between fuzzy uniform spaces. Then the mapping between the associated global homogeneous fuzzy neighborhood spaces is -fuzzy continuous [13].

**Proposition 5.4**

Let be an fuzzy uniformly continuous mapping between fuzzy uniform spaces, Then the mapping between the associated characterized fuzzy uniform spaces is -fuzzy continuous.

**Proof:** Immediate from Proposition 3.3 in Abd-Allah [11] and
Proposition 5.3.

In the following, we prove that for each fuzzy uniform structure on
a set *X*, there is an induced stratified fuzzy proximity on *L ^{X}*. Moreover,
both the fuzzy uniform structure and this induced stratified fuzzy
proximity are associated with the same stratified characterized fuzzy
uniform space.

**Proposition 5.5**

Let *X* be a non-empty set, is a fuzzy uniform structure on *X* and Then the binary relation which is defined by:

(5.3)

for all is a stratified fuzzy proximity on *X*. Moreover, both the
fuzzy uniform structure and the induced stratified fuzzy proximity are associated with the same stratified characterized fuzzy uniform
space, that is,

**Proof:** Immediate from (5.2), (5.3) and Proposition 3.2.

**Corollary 5.3**

Let are two fuzzy uniform spaces, and Then the mapping is fuzzy uniformly continuous between fuzzy uniform spaces if and only if the mapping fuzzy continuous between the associated stratified fuzzy proximity spaces.

**Proof:** Immediate from Propositions 5.4 and 5.5.

Because of Propositions 3.7 and 5.5 and Corollary 5.3, we can deduce the result.

**Proposition 5.6**

Let be an fuzzy uniform space, *F,G∈P (X)* such that and If Φ is the family of
all fuzzy uniformly continuous functions for which are Φ–separable.

**Proof:** Immediate from Propositions 3.7 and 5.5 and Corollary 5.3.

Now, we shall prove that the stratified characterized fuzzy uniform space which associated with an fuzzy uniform structure is characterized - space in sense of Abd-Allah et al. [11].

**Proposition 5.7**

Let *X* be a non-empty set, is a fuzzy uniform structure on *X* and Then the associated stratified characterized fuzzy
uniform space with the fuzzy uniform structure is
characterized - space.

**Proof:** Let *xX*, such that Since –fuzzy
neighborhood of *x*, that is, On
account of Proposition 5.6, we get that *x _{1}* and

**Corollary 5.4**

Let be a separated fuzzy uniform space and Then the associated stratified characterized fuzzy uniform space with the fuzzy uniform structure is characterized - space.

**Proof:** Immediate from Propositions 5.2 and 5.7.

the following we give an example of a homogeneous fuzzy uniform structure and we show that the associated stratified characterized fuzzy uniform space is characterized fuzzy uniform - space.

**Example 5.1**

The fuzzy metric in sense of S. Gähler and W. Gähler [24] is
canonically generate a homogeneous fuzzy structure as follows:
Consider *X* is non-empty set and d is a fuzzy metric on *X*, then the
mapping which is defined by:

for all is a homogeneous fuzzy uniform structures on X.
Moreover, the associated stratified characterized fuzzy uniform
space is identical with the associated characterized fuzzy
metrizable space that is, . Because
of Proposition 3.1 in Abd-Allah et al. [22], we have, is
characterized *FT _{4}*-space and therefore is also characterized

In this paper, we studied the relations between the characterized
fuzzy *T _{s}*–spaces, the characterized fuzzy

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