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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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The Relations between Characterized Fuzzy Proximity, Fuzzy Compact, Fuzzy Uniform Spaces and Characterized Fuzzy Ts–Spaces and Fuzzy Rk–Spaces

Abd-Allah AS1 and Al-Khedhairi A2*

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

2Department 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 Ts–Spaces and Fuzzy Rk–Spaces. J Appl Computat Math 6: 337. doi: 10.4172/2168-9679.1000337

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.

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Abstract

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.

Keywords

Fuzzy filter; Fuzzy topological space; Operationsl; Isotone and idempotent; Characterized fuzzy space; φ1,2–fuzzy neighborhood filters; Fuzzy uniform structure; Characterized fuzzy proximity space; Characterized fuzzy compact space; Characterized fuzzy uniform space; Characterized FTs–space; Fφ1,2Ts space; Characterized FRk–space and 1,2Rk space for equation

Introduction

The notion of fuzzy filter has been introduced by Eklund et al. By means of this notion the point-based approach to fuzzy topology related to usual points has been developed. The more general concept for fuzzy filter introduced by Gähler [1] and fuzzy filters are classified by types. Because of the specific type of fuzzy filter however the approach of Eklund is related only to the fuzzy topologies which are stratified, that is, all constant fuzzy sets are open. The more specific fuzzy filters considered in the former papers are now called homogeneous. The operation on the ordinary topological space (X,T) has been defined by Kasahara [2] as the mapping φ from T into 2X such that A ⊆ Aφ, for all A equation 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 : LX → LX such that int μ ≤ μφ for all μ equationLX, where μφ denotes the value of φ at μ.

The notions of the fuzzy filters and the operations on the class of all fuzzy subsets on X endowed with a fuzzy topology 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 equation-fuzzy interior of a fuzzy subset, equation-fuzzy convergence and equation-fuzzy neighborhood filters are defined and applied to introduced many special classes of separation axioms. The notion of equation-interior operator for a fuzzy subset is defined as a mapping equation.int:LX → LX which fulfill (I1) to (I5) in Abd-Allah [5]. There is a one-to-one correspondence between the class of all equation-open fuzzy subsets of X and these operators, that is, the class equationOF(X) of all equation-open fuzzy subsets of X can be characterized by these operators. Then the triple (X, equation. int) as will as the triple (X, equationOF (X)) will be called the characterized fuzzy space [5] of equation-open fuzzy subsets. The characterized fuzzy spaces are identified by many of characterizing notions in Abd-Allah [5-7], for example by the equation-fuzzy neighborhood filters, equation-fuzzy interior of the fuzzy filters and by the set of equation-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 equation-TS spaces, characterized FRk-spaces and F equation-Rk spaces are introduced and studied in Abd-Allah [9-11] for all equation and equation . 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 FTs and FRk-spaces, for equation and equation , the characterized fuzzy proximity spaces and the characterized fuzzy compact spaces. Moreover, we show here the relations between these characterized FTsand FRk-spaces and the characterized fuzzy uniform spaces. In section 2, some definitions and notions related to the fuzzy subsets, fuzzy topologies, fuzzy filters, fuzzy proximity, operations on fuzzy subsets, equation-fuzzy neighborhood filters, characterized fuzzy space, characterized FTs-spaces, F equation-Ts spaces, characterized FRk-spaces and F equation-Rk spaces are given for equation and equation . Section 3, is devoted to introduce and study the relation between the characterized fuzzy proximity spaces and our classes of the characterized FTs-spaces and of the characterized FRk-spaces. It will be shown that in the characterized fuzzy space (X, equation.int), the fuzzy proximity d will be identified with the finer relation on the equation-fuzzy neighborhood filters. Also, we will show that any fuzzy proximity is separated if and only if the associated characterized fuzzy proximity space is characterized FT0 and to each fuzzy proximity is associated a characterized FR2-space in our sense. Generally, it will be shown that the associated characterized fuzzy proximity space (X, equation.intδ) is characterized FR2-space if the related fuzzy topological space (X,τ) is F equation-R2 space. Moreover, for each characterized FR3-space the binary relation on LX defined by means the equation-fuzzy closure operator equation.cl of τ in Equation (3.6), is fuzzy proximity on X and conversely to each fuzzy proximity d which has a equation-fuzzy closure operator fulfills the binary relation given in (3.6) is associated a characterized FR3-space (X, equation.intδ). Moreover, when L is a complete chain, equationis isotone and φ1 is wfip with respect to φ1 OF (X), then we show that the associated characterized fuzzy space (X, equation.intτ) from the fuzzy normal topological space (X,τ) is finer than the associated characterized fuzzy proximity space (X, equation.intδ) by the fuzzy proximity d defined by (3.6) and they identical if and only if (X, equation.intτ) is characterized FT4-space. At the end of this section we prove that the associated characterized fuzzy proximity space (X,f1,2.intδ) is characterized equation - space and therefore it is characterized equation - space. There is a good notion of equation-fuzzy compactness of the fuzzy filters and of the fuzzy topological spaces introduced and studied by Abd-Allah et al. [7]. This notion fulfills many properties, for example, it fulfills the Tychonoff Theorem. In section 4, we used this notion to study the relations between the characterized fuzzy compact spaces and our classes of the characterized FTs-spaces and of the characterized FRk-spaces. It will be shown that every equation-closed subset of a characterized fuzzy compact space is equation-fuzzy compact and each equation-fuzzy compact subset of the characterized FT2-space is equation-closed. Also, it will be shown that each characterized fuzzy compact FT2-space is characterized FT4-space. Specially, we prove that the characterized fuzzy unit interval space equation is characterized fuzzy compact FT2-space and characterized equation - space. Generally, we show that every characterized fuzzy compact space is characterized FT2-space if and only if it is characterized equation - space. We show that, if equation is characterized fuzzy compact space finer than the characterized FT2-space equation is equation -fuzzy isomorphic to equation Moreover, if t is finer than s, (X, equation.intτ) is characterized fuzzy compact space and equation is characterized equation - space, then equation and equation are equation-fuzzy isomorphic. The notion of fuzzy uniform structure had been introduced and studied by Gähler et al. [13]. This notion with the notion of the operations on the class of all fuzzy subsets are applied to introduce and study the notion of characterized fuzzy uniform spaces. In section 5, we introduce and study the relations between the characterized fuzzy uniform spaces and our classes of the characterized FTs-spaces and of the characterized FRk- spaces. We show that the fuzzy uniform space equation is separated if and only if the associated characterized fuzzy uniform space equation is characterized FTi-space but the fuzzy uniform space equation is separated if and only if the associated stratified fuzzy topological space equation is F1,2-Ti space for all i equation{0,1}. For each fuzzy uniform structure on a set X, we prove that there is an induced stratified fuzzy proximity on LX. Moreover, both the fuzzy uniform structure and this induced stratified fuzzy proximity are associated with the same stratified characterized fuzzy uniform space. Finally, for each fuzzy uniform space equation we prove that the associated stratified characterized fuzzy uniform space equation with the fuzzy uniform structure equation is characterized equation - space and it is characterized equation - space if equation is separated.

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 L0 = L \ {0}. Sometimes we will assume more specially that L is a complete chain, that is, L is a complete lattice whose partial ordering is a linear one. For a set X, let LX be the set of all fuzzy subsets of X, that is, of all mappings equation. Assume that an order-reversing involution equation of L is fixed. For each fuzzy subset equation denote the complement of µ and it is given by the relation µ' (x) = µ (x)' for all x equation X. Denote by equation, the constant fuzzy subset of X with value is a equation For all x equation X and for all equation, the fuzzy subset xa of X whose value a at x and 0 otherwise is called a fuzzy point in X. The set of all fuzzy points of a set X will be denoted by S (X).

Fuzzy filters

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

equation

The fuzzy filter equation is called homogeneous [14] if equation for all equation For each x equation X, the mapping equation defined by equation for all µ equation LX is a homogeneous fuzzy filter on X. For each µ equation LX, the mapping equation defined by equation for all equation is a homogeneous fuzzy filter on X, called homogenous fuzzy filter at the fuzzy subset µ equation LX. Let equation and equation be the sets of all fuzzy filters and of all homogeneous fuzzy filters on X, respectively. If equation and equation are fuzzy filters on a set equation is said to be finer than equation, denoted by equation= equation provided equation holds for all µ equation LX. Noting that if L is a complete chain then M is not finer than N, denoted by equation provided there exists µ equation LXsuch that equation holds.

Lemma 2.1

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

equation

Proposition 2.1

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

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

equation

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

equation

for all equation, where n is an positive integer, µ1,…,µn is a collection such that equation and equation are fuzzy filters from equation Let X be a set and equation, then the homogeneous fuzzy filter equation at µ is the fuzzy filter on X given by:

equation (2.1)

Fuzzy filter bases

The family equation of a non-empty subsets of LX is called a valued fuzzy filter base [1] if the following conditions are fulfilled:

equation

(V2) For all a,ß L0 with equation and all equation there are equation

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

Valued and superior principal fuzzy filters

Let a non-empty set X be fixed, µ ∈ LX 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 equation with equation otherwise as a valued fuzzy filter base. For all equation , we have equation and equation. Moreover, for each equation we have equation equation if ß = a and equation otherwise. The superior principal fuzzy filter [1] generated by µ, written [µ], is the homogeneous fuzzy filter on X which has equation 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 equation is the covariant functor from the category SET of all sets to this category which assigns to each set X the set equation and to each mapping equation the mapping equation. The homogeneous fuzzy filter functor equation is the sub fuzzy filter functor of FL which assigns to each set X the set equation and to each mapping f equation the domain-range restriction equation of the mapping equation equation For each set X, let equation be the mapping defined by equation for all x∈X, and let equation be the mapping for which equation = M (μ) for all μ L and M∈FLX. Moreover, let equation be the mapping which assigns to each fuzzy filter equation the fuzzy filter equation with id the identity set functor and equation F are natural transformations. equation is a monad in the categorical sense, called the fuzzy filter monad [1], that is, equation and equation for each set X. Related to the sub functor equation there are analogous natural transformations as η and μ, denoted equation and equation, respectively. ηequation consists of the rangerestrictions equation of the mappings equation is the family of all mappings equation for all homogeneous fuzzy filters equation is the mapping given by equation As has been shown in Gähler et al. [13], equation is a sub monad of equation equation, that is, for the inclusion mappings equation we have equation for all sets X.

Fuzzy topologies

By a fuzzy topology on a set X [20,21], we mean a subset of LX which is closed with respect to all suprema and all finite infima and contains the constant fuzzy sets equation . A set X equipped with an fuzzy topology τon X is called fuzzy topological space. For each fuzzy topological space (X,τ), the elements of τare called open fuzzy subsets of this space. If τ1 and τ2 are two fuzzy topologies on a set X, then τ2 is said to be finer thanτ1 andτ1 is said to be coarser than τ2, provided equation holds. The fuzzy topological space (X,τ) and alsoτ are said to be stratified provided equation holds for all equation hat is, all constant fuzzy subsets are open [17].

Fuzzy proximity spaces

The binary relation δ on LX is called fuzzy proximity on X [18], provided it fulfill the following conditions:

equation is the negation of δ.

(P2) equation

(P3) equation

(P4) equation

(P5) equation

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 τδ. The fuzzy proximity δ on a set X is said to be separated if and only if for all equation such that x ≠ y we have equation

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 equation such that

int equation holds, for all equation where equation denotes the value of φ at μ. The class of all operations on X will be denoted by equation . By the identity operation on equation, we mean the operation equation such that equation for all equation Also by the constant operation on equation, we mean the operation equation such that equation for al equation If ≤ is a partially ordered relation on equation defined as by equation for all equation then obviously, equation is a completely distributive lattice. As an application on this partially ordered relation, the operation equation will be called:

(i) Isotone equation

(ii) Weakly finite intersection preserving (wfip, for short) with respect to equation holds, for all ρ ∈A and μ ∈ LX.

(iii) Idempotent if equation

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

φ-open fuzzy subsets

Let a fuzzy topological space (X,τ) be fixed and equation. The fuzzy subset μ : X → L is said to be φ-open fuzzy subset if equation 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 equation are equivalent and written equation if and only if equation

φ1,2–interior of fuzzy subsets

Let a fuzzy topological space (X,τ) be fixed and equation Then the φ1,2-interior of the fuzzy subset μ: X → L is the mapping φ1,2. int μ : X → L defined by:

equation (2.2)

As easily seen that equation is the greatest φ1-open fuzzy subset ρ such that equation less than or equal to μ [5]. The fuzzy subset μ is said to be equation The class of all φ1,2-open fuzzy subsets of X will be denoted by equation. The complement equation fuzzy subset μ will be called equation -closed and the class of all equation-closed fuzzy subsets of X will be denoted by equationCF(X). In the classical case of L = {0,1}, we note that the fuzzy topological space (X,τ) is up to an identification by the ordinary topological space (X,T) and equation.int μ is the classical one. Hence, in this case the ordinary subset A of X is equation- open if equation. int A. The complement of the equation-open subset A of X will be called equation-closed. The class of all equation-open and the class of all equation-closed subsets of X will be denoted by equation respectively. Clearly, F is equation closed if and only if equation

Proposition 2.2

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

(i) If equation , then equation.int μ ≤ μ holds.

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

(iii) equation

(iv) If equation is isotone and φ1 is wfip with respect to φ1 OF (X), then equation

(v) If 2 is isotone and idempotent operation, then equation. equation

(vi) equation

Proposition 2.3

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

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

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

(iii) If equation is isotone and φ1 is wfip with respect to φ1OF(X), then φ1,2 OF(X) is fuzzy pre topology on X [19].

(iv) If equation is isotone and idempotent operation and φ1 is wfip with respect to φ1OF(X), then equation 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 equation . Then

equation (2.3)

and the following conditions are fulfilled:

(I1) If equation, then equation holds, for all μ ∈ LX.

(I2) If equation

(I3) equation

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

(I5) If1 is isotone and idempotent, then equation equation

Characterized fuzzy spaces

Independently on the fuzzy topologies, the notion of equation -interior operator for fuzzy subsets can be defined as a mapping equation .int : LXLX which fulfill (I1) to (I5). It is well-known that (2.2) and (2.3) give a one-to-one correspondence between the class of all equation -open fuzzy subsets and these operators, that is, equation OF(X) can be characterized by the equation -interior operators. In this case the triple equation as will as the triple equation will be called characterized fuzzy space [5] of the equation-open fuzzy subsets of X. For each characterized fuzzy space equation, the elements of equation of (X) are called equation –open fuzzy subsets of this space. If equation and equation are two characterized fuzzy spaces, then equation is said to be finer than equation and denoted by equation. int provided equation int μ holds for all μ ∈LX. The characterized fuzzy space equation is said to be stratified if and only if equation As shown in Abd-Allah [5], the characterized fuzzy space equation is stratified if the related fuzzy topology is stratified. Moreover, the characterized fuzzy space equation is said to have the weak infimum property [19] provided equation equation The characterized fuzzy space equation is said to be strongly stratified provided equation.int is stratified and have the weak infimum property.

Fuzzy unit interval

The fuzzy unit interval will be denoted by IL and it is defined in Gähler [24] as the fuzzy subset equation where I = [0,1] is the real unit interval and equation is the set of all positive fuzzy real numbers. Note that, the binary relation ≤ is defined on equation as follows: equation for all x, y ∈ equation, where equation and equation Note that the family Ω which is defined by: equation is a base for a fuzzy topology I on IL and the order pair (IL,I) is said to be fuzzy unit interval topological space, where equation and equation are the fuzzy subsets of equation defined by equation for all equation \The restrictions of equation and equationon IL are the fuzzy subsets equation , respectively. Recall that the inequality equation equation holds, where x + y is the fuzzy real number defined by: equation for all equation Consider a fuzzy unit interval topological space (IL,I) be given and equation , then in this work the characterized fuzzy space equation characterized fuzzy unit interval space and we define the cartesian product of a number of copies of the fuzzy unit interval IL equipped with the product of the characterized fuzzy unit interval spaces generated by ψ1,2.intI on it as a characterized fuzzy cube.

φ1,2–fuzzy neighborhood filters

An important notion in the characterized fuzzy space (X,φ1,2.int) is that of the φ1,2-fuzzy neighborhood filter at the point and at the ordinary subset in this space. Let (X,τ) be a fuzzy topological space and equation. As follows by (I1) to (I5) for each x∈X, the mapping

equation

equation (2.4)

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

equation

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

equation

equation

equation

Clearly, equation is the fuzzy subset equation

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

φ1,2ψ1,2–fuzzy continuity

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

equation (2.5)

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

The mapping equation fuzzy continuous if the inequality equation holds for each xX. Obviously, in case of equation, the equation -fuzzy continuity coincides with the usual fuzzy continuity.

φ1,2–fuzzy convergence

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

Internal φ1,2-closure of fuzzy sets and φ1,2-closure operators

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

equation (2.6)

equation (2.6)

for all xX. 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, equation holds for all equation The mapping equation which assigns equation to each fuzzy filter M on X, that is,

equation (2.7)

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

Lemma 2.2

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

Characterized fuzzy Rk and fuzzy φ1,2Rk-spaces

The notions of characterized fuzzy Rk and fuzzy φ1,2Rk- spaces are introduced and studied in Abd-Allah [9,11] for all equation. Moreover, the notion of equation-fuzzy neighborhood filter at the point x and at the ordinary subset of the characterized fuzzy space equation is applied by Abd-Allah [10], to introduced and studied the notions of characterized fuzzy Rk-spaces for k∈{2,3}. However, the notions of fuzzy 1,2Rk-spaces are also given by means of the φ1,2-fuzzy convergence at the point x and at the ordinary subset in the space. We will denote by characterized FRk and Fφ1,2Rk-spaces to the characterized fuzzy Rk and fuzzy φ1,2Rk-spaces for shorts, respectively.

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

(1) Characterized FR2-space (resp. Characterized FR3-space), if for all x∈X, equation such that equation does not exists. The related fuzzy topological space (X,τ) is said to be equation and equation such that equationwe have equation

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

Characterized fuzzy Ts and fuzzy φ1,2-TS spaces

The notions of characterized fuzzy Ts and fuzzy φ1,2-TS spaces are investigated and studied by Abd-Allah and by Abd-Allah and Al-Khedhairi in [8,9,11] for all equation These characterized fuzzy spaces depend only on the usual points and the operation defined on the class of all fuzzy subsets of X endowed with a fuzzy topological space (X,τ). We will denote by characterized equation and equation spaces to the characterized fuzzy equation spaces for shorts, respectively.

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

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

(2) Characterized FT2-space if for all x,y∈X such that x≠y, the infimum equation does not exists. The related fuzzy topological space (X,τ) is said to be equation implies equation

(3) Characterized FTs space if and only if it is characterized FRk- space and characterized FT1-space for equation and equation. The related fuzzy topological space (X,τ) is said to be equation if and only if it is equation

Proposition 2.4

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

Proposition 2.5

If equation is characterized FT2-space and φ1,2.int is finer than ψ1,2. int, then equation is also characterized FT2-space [8].

Proposition 2.6

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

(1) Every characterized FTi-space equation is characterized FTi-1- space for each i∈{2,3,4}.

(2) The characterized fuzzy subspace and the characterized fuzzy product space of a family of characterized FT2-spaces are also characterized FT2-spaces .

New Relations between Characterized FTs, Characterized FRk and Characterized Fuzzy Proximity Spaces

In this section we are going to introduce and study the relations between the characterized FTs-spaces, the characterized FRk-spaces and the characterized fuzzy proximity spaces presented by Abd-Allah in [12]. We make at first the relation between the farness on fuzzy sets and the finer relation on fuzzy filters. So, we show some results for the notion of the φ1,2-fuzzy neighborhood filter equation at the fuzzy subset equation The notion of homogeneous fuzzy filter equationwhich is defined in (2.1) and the notion of equation-fuzzy neighborhood filter equation at the fuzzy subset equation are applied at first to study the relation between the fuzzy proximity δ defined by Katsaras in [18] and our fuzzy separation axioms [8-10]. Moreover, the relations between characterized fuzzy proximity spaces and the characterized FTs-spaces and characterized FRk-spaces are introduced for equation and equation

Proposition 3.1

Let a fuzzy topological space (X,τ) be fixed and equation such that equation is isotone and idempotent and φ1 is wfip with respect to equation. Then the supremum of the φ1,2-fuzzy neighborhood filters equation at x∈X which is given by:

equation (3.1)

for all equation is a fuzzy filter on X called a φ1,2-fuzzy neighborhood filter at μ.

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

equation

and

equation

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

equation

Hence, equation holds for all x∈X and equation holds for all equation fulfills condition (N1). For condition (N2), let equation Because of Proposition 2.4, we have equation holds and which implies that equation holds for all y∈X. Hence equation and therefore condition (N2) is fulfilled. Since for any yX we have equation represents the mapping φ1,2int ρ. Then from Proposition 2.4 we have equation and then equation for all equation Thus, condition (N3) is also fulfilled and therefore equation 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 equation for all equation Because of (2.4) the equations (2.1) and (2.2) can be written as in the following:

equation (3.2)

equation (3.3)

for all equation Here a useful remark is given

Remark 3.1: The homogeneous fuzzy filter equation at the ordinary point x is nothing that a homogeneous fuzzy filter equation at the fuzzy point xα, that is, equation Moreover, the φ1,2–fuzzy neighborhood filter equation at x is itself the φ1,2–fuzzy neighborhood filter equation at xα.

The φ1,2–fuzzy neighborhood filter equation at the fuzzy subset equation and the homogeneous fuzzy filter equation fulfill the following properties.

Lemma 3.1

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

(1) equation implies equation implies equation

(2) μ ≤ implies equation

(3) equation

(4) equation

(5) equation implies there is an equation and equation

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

equation

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

Since equation then from (2) we have equation Now, let equation

equation

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

equation

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

In the characterized fuzzy space (X,φ1,2.int), the fuzzy proximity will be identified with the finer relation on the fuzzy filters, specially with the finer relation on the φ1,2-fuzzy neighborhood filters. This shown in the following proposition.

Proposition 3.2

Let (X,τ) be a fuzzy topological space and equation Then the binary relation δ on LX which is defined by:

equation

for all equation is fuzzy proximity on X.

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

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

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

If a fuzzy topological space (X,τ) be fixed and equation Then each fuzzy proximity δ on X is associated a set of all φ1,2-open fuzzy subsets of X with respect to δ denoted by equation In this case the triple equation as will as equation is said to be characterized fuzzy proximity space. The related φ1,2-interior and φ1,2-closure operators equation are given by:

equation (3.5)

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

In the following we will show that the characterized fuzzy proximity space equation is characterized FT0-space as in sense of [8] if and only if δ is separated.

Proposition 3.3

Let (X,τ) be a fuzzy topological space, equation and is a fuzzy proximity on X. Then the characterized fuzzy proximity space equation equation is characterized FT0-space if and only if δ is separated.

Proof: Let equation such that x≠y. Then equation and therefore there is equation such that equation Because of (3.4), we have equation and hence μ (x) > ρ (y) holds for all equation that is, μ (x) >ρ (y) holds for all equation with equation Choose equation then because of Remark 3.1, we get equation Using Proposition 3.2 we get equation holds for all equation 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 equation Therefore, equation holds for all equation Consider, equation we get equation and equation Hence, there exists equation such that equation that is, equation and therefore equation is characterized FT0-space.

In the following proposition, the 1,2-closure of the fuzzy subsets in the characterized fuzzy space (X,φ1,2.intδ) are equivalent with the fuzzy subsets by the fuzzy proximity δ on X.

Proposition 3.4

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

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

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

In the following proposition, we show that the associated characterized fuzzy proximity space equation is characterized FR2- space if the related fuzzy topological space (X,τ) is equation space.

Proposition 3.5

Let (X,τ) be a fuzzy topological space, equation and is an fuzzy proximity on X. Then the associated characterized fuzzy proximity space equation is characterized FR2-space if (X,τ) is 1,2-R2 space.

Proof: Let xX and equation with equation Because of Proposition 3.2, we have equation and from (P5), there is equation such that equation and equation Therefore Proposition 3.4 implies equation and hence equation are hold. Hence, equation implies there is equation such that equation and . are hold. Since equation is equation space, then from Theorem 3.1 in Abd-Allah [12], we have equation is characterized FR2-space.

The binary relation << on LX is said to be fuzzy topogeneous order on X [23], if the following conditions are fulfilled:

(1) equation

(2) If equation

(3) If equation

(4) If 1 equation are hold for all equation

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

(5) If equation are hold for all equation

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

(6) If equation

As shown in Katsaras [23], every fuzzy topogeneous structure ≪ is identify with the mapping equation such that equation if and only if μ <<η holds for all equation 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 intN on a set X by defining equation for all equation In case of N is fuzzy topogeneous structure, intN is interior operator for fuzzy topology τN on X associated to N. Obviously, there is an identification between the fuzzy proximity δ and the complementarily symmetric fuzzy topogenous structure ≪ on the same set X given by:

equation (3.5)

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

Remark 3.2

Given that equation are two sequence of complementarily symmetric fuzzy topogenous structures equation and equation on X and IL, respectively. If δ and δ* are two fuzzy proximities on X and IL identified with δ and δ* by the equation (3.5), then the associated fuzzy real function equation with the complementarily symmetric fuzzy topogenous structures equation fuzzy continuous, because from (3.5) we get that equation implies equation for all equation

Lemma 3.2

Consider equation are complementarily symmetric fuzzy topogenous structures on a set X. Then, for each F,G∈P (X) such that equation associated with the sequence equation for which equation for all xF and equation 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 equation is a characterized fuzzy proximity space and F,G∈P (X) such that equation-fuzzy continuous mappings equation implies equation then equation are equation -separable.

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

Proposition 3.7

Let equation are two characterized fuzzy proximity spaces. If the mapping equationis equation is equation-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 FR3-space.

Proposition 3.8

Let (X,τ) be a fuzzy topological space and equation such that equation is isotone and φ1 is wfip with respect to equation 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:

equation (2.6)

for all equation 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 equation is characterized FR3-space.

Proof: Let (X,τ) is fuzzy normal topological space and δ a binary relation on X defined by (3.6). Then, equation implies equation and from Lemma 3.1 part (1) we get equation and then equation. Hence, condition (P1) is fulfilled. For showing condition (P2), let equation for a fixed fuzzy subsets equation Then, equation Since L is complete chain, equation is isotone and φ1 is wfip with respect to equation and therefore equation Because of Lemma 3.1 part (3), we have equation and equation . Thus, equation and equation are hold and therefore equation implies equation. On the other hand let equation and equation Then from Lemma 3.1 we have tha the inequalities equation are hold and therefore equation that is, equation imply equation Hence, (P2) is fulfilled. Now, let equation such that equation the finest fuzzy filter on X and from the fact equation we get equation for all equation that is, (P3) is also fulfilled. Since equation implies equation which means by the inequality equation that equation Because of Proposition 2.1 and the fact that φ1,2. cl is hull operator we get equation Thus, (P4) is fulfilled. Let equation such that equation equationConsider, equation , hence equation and therefore equationequationholds. Since (X,τ ) is characterized fuzz normal space, then from Theorem 3.2 in Abd-Allah [12], there exists equation with arbitrary choice such that equation are hold. Therefore, there exists ∈LX such that equation and equation Hence, (P5) is also fulfilled. Consequently, δ is a fuzzy proximity on X.

Conversely, let equation and therefore equation Hence because of Lemma 3.1 part (1) we have equation equation From (P5), there exists equation such that equation equation are hold. Because of Lemma 3.1 part (1), we have equation Hence, equation holds for all. Consider equation and equation for all equation, then we get equation = 0 and equation Hence, there exist equation such that equation, that is, the infimum equation does not exists. Consequently, equation is characterized FR3-space.

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

Proposition 3.9

Let (X,τ) is a fuzzy normal topological space and equation such that equation is isotone and φ1 is wfip with respect to φ1OF (X). If δ is the fuzzy proximity on X defined by (3.6) and L is a complete chain, then equation is finer thanequation Moreover, equationequation if and only if equation is characterized FR4-space.

Proof: Let (X,τ) is fuzzy normal topological space and μ is φ1,2δ- fuzzy neighborhood for the point xX, then equation and because of (3.6), we have equation Therefore, equation Because of Proposition 2.1, we get equation and equation -fuzzy neighborhood of x and therefore the family equation that is, equation

Now, let equation and equation denote for the φ1,2-fuzzy neighborhood filters at x in the characterized fuzzy space equation and in the associated characterized fuzzy proximity space equation respectively. Then, equation is characterized FR3 and FR1-space. Therefore, equation holds for all y ≠ x in X. Hence, equation holds for all x∈X and then equation holds for all y ≠ x in X. Because of Lemma 2.1, we have that equation holds for all y ≠ x in X and therefore equation is characterized FT1-space. Because of Proposition 2.4 and Lemma 2.2, we get equation equation for all xX. Consider μ is the φ1,2-fuzzy neighborhood of x in equation and since equation is a 1,2-fuzzy neighborhood for every y∈X such that equation Thus, equation and hence μ is φ1,2δ-fuzzy neighborhood of x in equation equation that is, equation holds for all x∈X and therefore equation Consequently, equation is characterized FR4-space implies that equation

Conversely, let equation- fuzzy neighborhood of x in equation Then, equation and equation this means that equation Because of Proposition 2.1, we get equation holds for all x∈X. Thus, equation Hence, Proposition 2.4 implies that, equation is characterized FR1-space. Because of Proposition 3.7, equation is characterized FR3-space and the hypothesis that equation is characterized FR3-space. Consequently, equation is characterized FR4-space.

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

Proposition 3.10

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

Proof: Let xX and equation neighborhood of x, then equation . Because of Proposition 3.2, we get that x1 and χF are Φ-separated by the equation fuzzy continuous mapping equation that is, equation Consequently, equation is characterized equation - space.

Corollary 3.1

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

Example 3.1

Let equation is a fuzzy topology on X. Choose equation and equation Hence, x ≠ y and there is only two cases, the first is equation and the second is equation We shall consider the first case and the second case is similar. Consider the mapping equation equation-fuzzy continuous and therefore equation is characterized equation - space and obviously equation is also characterized FR1-space, that is, equation is characterized equation - space. Now, consider δ is a binary relation on LX defied as follows:

equation - fuzzy continuous mapping equation equation

with equation

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

Some Relations between Characterized FTs and Characterized Fuzzy Compact Spaces

The notion of φ1,2-fuzzy compactness of the fuzzy filters and of the fuzzy topological spaces are introduced by Abd-Allah in [7] by means of the φ1,2-fuzzy convergence in the characterized fuzzy spaces. Moreover, the fuzzy compactness in the characterized fuzzy spaces is also introduced by means of the φ1,2-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 equation Then x∈X is said to be φ1,2-adherence point for the fuzzy filter equation on X [7], if the infimum equation exists for all φ1,2-fuzzy neighborhood filters equation As shown in Abd-Allah [7], the point x∈X is said to be φ1,2-adherence point for the fuzzy filter equation on X if and only if there exists an fuzzy filter equation finer than M and equation that is, equation are hold for some equation The subset F of X is said to be φ1,2-fuzzy closed with respect to φ1,2.int if equation implies x∈F for some equation The subset F is said to be φ1,2-fuzzy compact [7], if every fuzzy filter on F has a finer φ1,2−fuzzy converging filter, that is, every fuzzy filter on F has φ1,2-adherence point in F. Moreover, the fuzzy topological space (X,τ) is said to be φ1,2-fuzzy compact if X is φ1,2-fuzzy compact. More generally, the characterized fuzzy space (X,φ1,2.int) is said to be fuzzy compact space if the related fuzzy topological space (X,τ) is φ1,2-fuzzy compact.

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

(1) The usual topological space (I,TI) and the ordinary characterized usual space equation on the closed unite interval I = [0,1] are equation compact T2 space and characterized compact T2-space, respectively in the classical sense.

(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 equation is up to an identification the characterized usual space equation in the classical sense.

In the following proposition, we show that every φ1,2-fuzzy compact subset in the characterized FT2–space (X,φ1,2.intτ) is φ1,2-fuzzy closed with respect to the φ1,2-interior operator φ1,2.intτ.

Proposition 4.1

Let a fuzzy topological space (X,τ) be fixed and equation Then every φ1,2-fuzzy compact subset of the characterized FT2-space is 1,2-closed.

Proof: Let equation is characterized FT2-space and F is φ1,2-fuzzy compact subset of X. Then, for all equation there exists equation such that equation and equationfor some xF. Since equation

and equation is characterized FT2-space, then equation and equation imply that x = y. Therefore, y∈F for some equation Hence, F is φ1,2-fuzzy closed with respect to 1,2.intτ.

Proposition 4.2

Let equation be a fuzzy unit interval topological space and equation Then the characterized fuzzy unit interval space equation is characterized fuzzy compact FT2-space.

Proof: Let equation be an ordinary characterized usual space. Then, equation is characterized compact space in the classical sense, that is, every filter on equation-adherence point. Consider the mapping equation defined by: equation for all equation then it is easily to seen that equationfuzzy homeomorphism between equation is characterized fuzzy compact space. Since (I,TI) is equation- space, then equation is characterized FT2-space and therefore by using the same equation fuzzy homeomorphism, we have for all equation such that equation the infimum equation does not exists. Consequently, equation is characterized FT2-space and therefore equation is characterized fuzzy compact FT2-space.

Now, we are going to prove an important relation between the characterized compact FT2-spaces and the characterized FT4-spaces. For this reason at first, we give a new property for the characterized FT2-spaces by using the φ1,2-fuzzy neighborhood filters for the fuzzy subsets.

Proposition 4.3

Let (X,τ) be n fuzzy topological space and equation Then every disjoint φ1,2-fuzzy compact subsets F1 and F2 of in the characterized FT2-space equation have two disjoint φ1,2-fuzzy neighborhood filters equation for which F1 and F2 are separated by them.

Proof: Let F1 and F2 are two φ1,2-fuzzy compact subsets of the characterized FT2-space equation Then, for all equation for some equation where i∈{1,2}. Since equation for all i∈{1,2},then we can say that equation and therefore there is equation Since equation is characterized FT2-space, then x1 = x2 which contradicts equation or equation which means that the infimum equation does not exists and therefore F1 and F2 can be separated by two disjoint φ1,2-fuzzy neighborhood filters.

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

Lemma 4.1

Let (X,τ) be a fuzzy topological space and equation Then every φ1,2-fuzzy closed subset of the characterized fuzzy compact space equation -fuzzy compact.

Proof: Let F is φ1,2-fuzzy closed subset of the characterized fuzzy compact space equation and let equation Then, equation implies that x∈F. Since equation and hence there exists equation and equation Since equation and equation then equation Thus, for all equation we get equation such that equation is φ1,2-adherence point of equation that is, F is φ1,2-fuzzy compact.

Proposition 4.4

Let (X,τ)be an fuzzy topological space and equation Then every characterized fuzzy compact FT2-space equation is characterized FT4-space.

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

Proof: Because of Proposition 4.2, the characterized fuzzy unit interval space equation is characterized fuzzy compact FR2-space. Therefore from Proposition 4.4, we get equation is characterized FR4- space. Hence, Proposition 4.6 in Abd-Allah [11] gives us that, equation is characterized equation- space.

The φ1,2-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 FR2-spaces is also characterized FR2-space. Hence, by means of Propositions 4.2 and 4.4, the following result goes clear.

Proposition 4.6

Let equation be a fuzzy unit interval topological space and equation Then the characterized fuzzy cube is characterized FR2-space and it is characterized FR4-space.

Proof: Since the characterized fuzzy cube is product of copies of the characterized fuzzy unit interval space equation and by means of Proposition 4.2, equation is characterized fuzzy compact FR2-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 FR2-space. Moreover, Proposition 5.1, it follows that the characterized fuzzy cube is characterized FR4-space.

Lemma 4.2

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

Proof: Let equation -fuzzy neighborhood and ψ1,2-fuzzy neighborhood at x with respect to ψ1,2.intτ and ψ1,2.intσ, respectively. Since τ is finer than σ, then equation N x for all x∈X. Because of equation is characterized fuzzy compact space, then for all equation such that equation for all x X. Therefore equation for all x X. Consequently, (X,φ1,2. intτ) is characterized fuzzy compact space.

Proposition 4.7

Let (X,τ) and (X,σ) are two fuzzy topological spaces such that is finer than equation is characterized fuzzy compact space and equation is characterized FT2- space, then equation-fuzzy isomorphic.

Proof: Since τ is finer than σ, then equation Hence, because of Proposition 2.5, equation is characterized FT2-space. From Lemma 4.2, we have equation is characterized fuzzy compact space. Hence, we can find the identity mapping equation which is bijective equation-fuzzy continuous and its inverse is equation -fuzzy continuous, that is, idX is equation -fuzzy isomorphism. Consequently, equation -fuzzy isomorphic.

Proposition 4.8

Let (X,τ) be a fuzzy topological spaces and equation Then every characterized fuzzy compact space equation is characterized FT2-space if and only if it is characterized equation - space.

Proof: Let equation is characterized fuzzy compact FT2-space. Because of Proposition 4.4 we have equation is characterized FT4- space and therefor Proposition 4.6 in Abd-Allah S [11], implies that equation is characterized equation - space. Conversely, let equation is characterized equation - space, then because of Proposition 3.2 in Abd- Allah [11] and part (1) of Proposition 2.6, it follows that equation is characterized fuzzy compact FT2-space.

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, equation and equation If equation is characterized fuzzy compact space and equation is characterized equation- space, then equation -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 FTs, Characterized FRk and Characterized Fuzzy Uniform Spaces

In this section, we are going to investigate and study the relations between the characterized FTs-spaces, the characterized FTk-spaces and the characterized fuzzy uniform spaces presented in Abd-Allah [12]. For this, we applied the notion of homogeneous fuzzy filter at the point and at the fuzzy set which is defined by (2.1), the superior principal fuzzy filter [μ] generated by equation fuzzy neighborhoods at the fuzzy set μ which is defined by (3.1) in the characterized fuzzy space equation Specially, the relation between the separated fuzzy uniform spaces, the associated characterized fuzzy uniform FTs-spaces, the associated characterized uniform equation - space and the equation space which introduced by Abd-Allah and Abd-Allah et al. in [8,11] are investigated for all equation

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 equation be a fuzzy filer on X×X. The inverse equation is a fuzzy filter on X×X defined by equation The composition R1 R2 of two fuzzy relations R1 and R2 on the set X is a fuzzy relation on X defined by:

equation

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

equation

for all equation

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

equation

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

equation

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

equation

Each fuzzy uniform structure equation on the set X is associated a stratified fuzzy topology equation then the set of all φ1,2-open fuzzy subsets of X related to equation forms a base for an characterized stratified fuzzy space on X generated by the φ1,2-interior operator with respect to equation denoted by equation is the related stratified characterized fuzzy space. In this case, equation will be called the associated characterized fuzzy uniform space [12] which is stratified. The related φ1,2-interior operator φ1,2.intU is given by:

equation (5.1)

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

equation (5.2)

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

Proposition 5.1

Let X be a non-empty set, U is a fuzzy uniform structure on X and equation Then the fuzzy uniform space (X,U) is separated if and only if the associated characterized fuzzy uniform space equation is characterized FT0-space.

Proof: Let equation is separated and let equation Then, there exists equation such that equation Consider μ = R[y1] for which

equation

equation

for all equation Hence, there exists equation such that μ (x) < α equation that is, equation is characterized FT0-space.

Conversely, let equation is characterized FT0-space and let x ≠ y in X. Then, there exists equation andequation such that equation(y). This means that equation holds for all equation Hence, there is equation for which equation if x = y and R (x,y) = μ (x), if x ≠ y such that R (x,y) = 0 and equation = 1. Thus, equation is separated.

Corollary 5.1

Let X be a non-empty set, equation is a fuzzy uniform structure on X and equation Then the fuzzy uniform space equation is separated if and only if the associated stratified fuzzy topological space equation is equation 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, equation is a fuzzy uniform structure on X and equation Then the fuzzy uniform space equation is separated if and only if the associated characterized fuzzy uniform space equation equation is characterized FT1-space.

Proof: Let equation is separated and let x,y∈X such that x ≠ y. Then, there exists equation such that equation and equation for all i∈{1.2}. Consider equation then we have equation equation and equation Moreover,equation and equation for all equation equation Hence, there exists equation equation are hold. Consequently, equation is characterized FT1-space.

Conversely, let equation is characterized FT1-space and let x ≠ y in X. Then, there exists equation and equation such that equation equation and equation are hold. This means that equation are also hold for all equation Hence, there is equationsuch that equationequation if x = y and R1 (x,y) = μ (x) if x ≠ y such that equation andequation andequation if x = y and equation if x ≠ y such that equation and equation Thus, in every case equation is separated.

Corollary 5.2

Let X be a non-empty set, equation is a fuzzy uniform structure on X and equation Then the fuzzy uniform space equation is separated if and only if the associated stratified fuzzy topological space equation is F φ1,2-T1 space.

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

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

Proposition 5.3

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

Proposition 5.4

Let equation be an fuzzy uniformly continuous mapping between fuzzy uniform spaces, equation Then the mapping equation between the associated characterized fuzzy uniform spaces is equation -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 LX. 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, equation is a fuzzy uniform structure on X and equation Then the binary relation equation which is defined by:

equation (5.3)

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

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

Corollary 5.3

Let equation are two fuzzy uniform spaces, equation and equation Then the mapping equation is fuzzy uniformly continuous between fuzzy uniform spaces if and only if the mapping equation 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 equation be an fuzzy uniform space, F,G∈P (X) such that equation and equation If Φ is the family of all fuzzy uniformly continuous functions equation for which equation 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 equation - space in sense of Abd-Allah et al. [11].

Proposition 5.7

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

Proof: Let xX, equation such that equation Since equation –fuzzy neighborhood of x, that is, equation On account of Proposition 5.6, we get that x1 and χF are Φ–separated by the fuzzy uniformly continuous function equation Because of Proposition 5.4, the function equationis equation –fuzzy continuous. Consequently, equation is characterized equation - space.

Corollary 5.4

Let equation be a separated fuzzy uniform space and equation Then the associated stratified characterized fuzzy uniform space equation equation with the fuzzy uniform structure equation is characterized equation - 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 equation - 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 equation which is defined by:

equation

for all equation is a homogeneous fuzzy uniform structures on X. Moreover, the associated stratified characterized fuzzy uniform space equation is identical with the associated characterized fuzzy metrizable space equation that is, equation. Because of Proposition 3.1 in Abd-Allah et al. [22], we have, equation is characterized FT4-space and therefore equation is also characterized FT4-space. Hence from Proposition 4.6 in Abd-Allah et al. [11], we get equation is characterized equation - space.

Conclusion

In this paper, we studied the relations between the characterized fuzzy Ts–spaces, the characterized fuzzy Rk–spaces presented in Abd-Allah and Abd-Allah and Al-Khedhairi [8-10,11] and the characterized fuzzy proximity spaces presented by Abd-Allah [12], for equation and equation We also introduced and studied the relations between our characterized fuzzy Ts–spaces, the characterized fuzzy Rk–spaces and the characterized fuzzy compact spaces presented by Abd-Allah [12] as a generalization of the weaker and stronger forms to the G-compactness defined by Gähler in 1995. Moreover, we shows here the relations between these characterized fuzzy Ts–spaces, the 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 et al. in 1998.

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