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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Initial and Final Characterized Fuzzy 1 3 2 T and Finer Characterized Fuzzy 12 2 R -Spaces

Ahmed Saeed Abd-Allah1* and A Al-Khedhairi2

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

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

*Corresponding Author:
Ahmed Saeed Abd-Allah
Prince Sattam Bin Abdul-Aziz University
Hotat Bani Tamim, Kingdom of Saudi Arabia
Tel: 00966552057393
E-mail: [email protected]

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

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

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

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Abstract

 Basic notions related to the characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces are introduced and studied. The metrizable characterized fuzzy spaces are classified by the characterized fuzzy 1 2 2 R and the characterized fuzzy T4-spaces in our sense. The induced characterized fuzzy space is characterized by the characterized fuzzy 1 3 2 T and characterized fuzzy 1 3 2 T -space if and only if the related ordinary topological space is 1 2, 2 R ϕ 12 -space and 1 3, 2 T ϕ 12 -space, respectively. Moreover, the α-level and the initial characterized spaces are characterized 1 2 2 R and characterized 1 3 2 T -spaces if the related characterized fuzzy space is characterized fuzzy 12 2 R and characterized fuzzy 1 3 2 T , respectively. The categories of all characterized fuzzy 1 2 2 R and of all characterized fuzzy 1 3 2 T -spaces will be denoted by CFR-Space and CRF-Tych and they are concrete categories. These categories are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets and hence all the initial and final lifts exist uniquely in CFR-Space and CRF-Tych. That is, all the initial and final characterized fuzzy 1 2 2 R spaces and all the initial and final characterized fuzzy 1 3 2 T -spaces exist in CFR-Space and in CRF-Tych. The initial and final characterized fuzzy spaces of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. As special cases, the characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are also characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. Finally, three finer characterized fuzzy 1 2 2 R -spaces and three finer characterized fuzzy 1 3 2 T -spaces are introduced and studied.

Keywords

Fuzzy filter; Fuzzy topological space; Operation; Characterized fuzzy space; Metriz-able characterized fuzzy space; Induced characterized fuzzy space; α-Level characterized space; φ1,2ψ1,2-fuzzy continuous; Initial and final characterized fuzzy spaces; Characterized fuzzy -space; Characterized fuzzy -space; AMS classification; Primary 54E35, 54E52; Secondary 54A4003E72

Introduction

Eklund and Gahler [1] introduced the notion of fuzzy filter and by means of this notion the point-based approach to the fuzzy topology related to usual points has been developed. The more general concept for the fuzzy filter introduced by Gahler [2] and fuzzy filters are classified by types. Because of the specific type of the L-filter however the approach of Eklund and Gahler [1] is related only to the L-topologies which are stratified, that is, all constant L-sets are open. The more specific fuzzy filters considered in the former papers are now called homogeneous. The notion of fuzzy real numbers is introduced by Gahler and Gahler [3], as a convex, normal, compactly supported and upper semi-continuous fuzzy subsets of the set of all real numbers R. The set of all fuzzy real numbers is called the fuzzy real line and will be denoted by RL, where L is complete chain.

The operation on the ordinary topological space (X,T) has been defined by Kasahara [4] as a mapping φ from T into 2X such that A ⊆ Aφ, for all A ∈ T. Abd El-Monsef et al. [5], extend Kasahara [4] operation to the power set P (X) of the set X Kandil et al. [6] extended Kasahars’s and Abd El-Monsef’s operations by introducing operation on the class of all fuzzy sets endowed with an fuzzy topology τ as a mapping φ: LX → LX such that int μμφ for all μ ∈ LX, where μφ denotes the value of φ at μ. The notions of fuzzy filters and the operations on the class of all fuzzy sets on X endowed with an fuzzy topology τ are applied in ref. [7] to introduce a more general theory including all the weaker and stronger forms of the fuzzy topology. By means of these notions the notion of φ1,2-interior of the fuzzy set, φ1,2-fuzzy convergence and φ1,2-fuzzy neighborhood filters are defined. The notion of φ1,2-interior operator for the fuzzy sets is also defined as a mapping φ1,2.int: LX → LX which fulfill (I1) to (I5). Since there is a one-to-one correspondence between the class of all φ1,2-open fuzzy subsets of X and these operators, then the class φ1,2OF (X) of all φ1,2-open fuzzy subsets of X is characterized by these operators. Hence, the triple (X, φ1,2.int) as will as the triple (X, φ1,2 OF (X)) will be called the characterized fuzzy space of φ1,2-open fuzzy subsets. For each characterized fuzzy space (X, φ1,2.int) the mapping which assigns to each point x of X the φ1,2-fuzzy neighborhood filter at x is said to be φ1,2-fuzzy filter pre topology [7]. It can be identified itself with the characterized fuzzy space (X, φ1,2.int). The characterized fuzzy spaces are characterized by many of characterizing notions, for example by: φ1,2-fuzzy neighborhood filters, φ1,2-fuzzy interior of the fuzzy filters and by the set of all φ1,2-inner points of the fuzzy filters. Moreover, the notions of closeness and compactness in characterized fuzzy spaces are introduced and studied in ref. [8]. For an fuzzy topological space (X, τ), the operations on (X, τ) and on the fuzzy topological space (IL, ℑ), where I=[0, 1] is the closed unit interval and ℑ is the fuzzy topology defined on the left unit interval IL are applied to introduced and studied the notions of characterized fuzzy -spaces and characterized fuzzy -spaces or (characterized Tychonoff spaces) [9]. In this paper, Basic notions related to the characterized fuzzy and the characterized fuzzy -spaces are introduced and studied. Some of this the metrizable characterized fuzzy spaces, initial and final characterized fuzzy spaces and three finer characterized fuzzy -spaces are introduced and classified by the characterized fuzzy and characterized fuzzy -spaces. The metrizable characterized fuzzy space is introduce as a generalization of the weaker and stronger forms of the fuzzy metric space introduced by Gahler and Gahler [3]. For every stratified fuzzy topological space (X, τd) generated canonically by an fuzzy metric d on X, the metrizable characterized fuzzy space (X, φ1,2.intτd) is characterized fuzzy T4-space in sense of Abd-Allah [10] and therefore it is characterized fuzzy and characterized fuzzy L-space. The induced characterized fuzzy space (X, φ1,2.intω) is characterized fuzzy and characterized fuzzy -space if and only if the related ordinary topological space (X, T) is φ1,2 -space and φ1, -space, respectively, that is, the notions of characterized fuzzy -spaces and characterized fuzzy -spaces are good extension as in sense of Lowen [11]. Moreover, the α-level characterized space (X, φ1,2.intα) and the initial characterized space (X, φ1,2.inti) are characterized -space and characterized -space if the related characterized fuzzy space (X, φ1,2.intτ) is characterized fuzzy -space and characterized fuzzy -space, respectively. We show that the finer characterized fuzzy space of the characterized fuzzy -space and of the characterized fuzzy -space is also characterized fuzzy and characterized fuzzy -space, respectively. The categories of all characterized fuzzy and of all characterized fuzzy -spaces will be denoted by CFRSpace and CRF-Tych, respectively. We show that these categories are concrete categories and they are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets and hence all the initial and final lifts exist uniquely in CFR-Space and CRF-Tych, respectively. That is, all the initial and final characterized fuzzy -spaces and all the initial and final characterized fuzzy -spaces are exist in the categories CFR-Space and CRF-Tych. Moreover, we show that the initial and final characterized fuzzy spaces of the characterized fuzzy -space and of the characterized fuzzy -space are characterized fuzzy and characterized fuzzy -spaces, respectively. As an special cases, the characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space of the characterized fuzzy -space and of the characterized fuzzy -space are also characterized fuzzy and characterized fuzzy -spaces, respectively. Finally, in section 5, we introduce and study three finer characterized fuzzy and three finer characterized fuzzy -spaces as a generalization of the weaker and stronger forms of the completely regular and fuzzy -spaces introduced [1,12,13]. The relations between such new characterized fuzzy -spaces and our characterized fuzzy -spaces are introduced. More general the relations between such new characterized fuzzy -spaces and our characterized fuzzy -spaces are also introduced. Meany special cases from these finer characterized fuzzy -spaces and from finer characterized fuzzy -spaces are listed in Table 1.

Preliminaries

We begin by recalling some facts on fuzzy sets and fuzzy filters. Let L be a completely distributive complete lattice with different least and last elements 0 and 1, respectively. Consider L0=L\{0} and L1=L\{1}. Recall that the complete distributivity of L means that the distributive law . Sometimes we will assume more specially that L is a complete chain, that is, L is a complete lattice whose partial ordering is a linear one. The standard example of L is the real closed unit interval I=[0, 1]. For a set X, let LX be the set of all fuzzy subsets of X, that is, of all mappings μ: X → L. Assume that an order-reversing involution α 7→α′ is fixed. For each fuzzy set μ, let co μ denote the complement of μ defined by: (co μ) (x)=co μ(x) for all x ∈ X. For all x ∈ X and α ∈ L0. Supμ means the supremum of the set of values of μ. The fuzzy sets on X will be denoted by Greek letters as μ, η, ρ,. . . etc. Denote by the constant fuzzy subset of X with value α ∈ L. The fuzzy singleton xα is an fuzzy set in X defined by xα(x)=α and xα(y)=0 for all y ≠ x , α ∈ L0. The class of all fuzzy singletons in X will be denoted by S(X). For every xα ∈ S(X) and μ ∈ LX, we write xαμ if and only if αμ(x). The fuzzy set μ is said to be quasi-coincident with the fuzzy set ρ and written μ q ρ if and only if there exists x ∈ X such that μ(x)+ ρ(x)>1. If μ not quasi-coincident with the fuzzy set ρ, then we write . The fuzzy filter on X [14] is the mapping M: LX →L such that the following conditions are fulfilled:

(F1) Μ () ≤ α for all α ∈ L and (1)=1.

(F2) (μη)= (μ) ∧ (η) for all μ, η ∈ LX.

The fuzzy filter is said to be homogeneous [14] if M () =α for all α ∈ L. For each x ∈ X, the mapping : LX → L defined by μ = μ x for all μ ∈ LX is a homogeneous fuzzy filter on X. The homogenous fuzzy filter at the fuzzy set is defined by the same way as follows, for each μ ∈ LX, the mapping μ: LX → L defined by for all σ ∈ LX is also homogenous fuzzy filter on X, called homogenous fuzzy filter at μ ∈ LX. Obviously, the relation between homogenous fuzzy filter μ˙ at μ ∈ LX and the homogenous fuzzy filter x˙ at x ∈ X is given by:

(2.1)

for all η ∈ LX. As shown in ref. [15], μη if and only if holds for all μ, η ∈ LX. Let LX and LX denote to the sets of all fuzzy filters and of all homogeneous fuzzy filters on X, respectively. If and are fuzzy filters on the set X, then is said to be finer than , denoted by , provided (μ) ≥ (μ) holds for all μ ∈ LX. Noting that if L is a complete chain then M is not finer than N, denoted by ̸≤, provided there exists μ ∈ LX such that (μ) < (μ) holds. As shown in ref. [4], if , and L are three fuzzy filters on a set X, then we have:

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

The coarsest fuzzy filter on X is the fuzzy filter has the value 1 at 1 and 0 otherwise. Suprema and infimum of sets of fuzzy filters are meant with respect to the finer relation. An fuzzy filter on X is said to be ultra [2] fuzzy filter if it does not have a properly finer fuzzy filter. For each fuzzy filter LX there exists a finer ultra fuzzy filter U ∈ LX such that U ̸≤ . Consider Α is a non-empty set of fuzzy filters on X, then the supremum exists [2] and given by for all μ = LX but the infimum does not exists, in general. As shown in ref. [16], the infimum of Α with respect to the finer relation for fuzzy filters exists if and only if holds for all finite subset . In this case the infimum is given by:

for all μ ∈ LX.

Fuzzy filter bases. A family (Bα)α∈L0 of non-empty subsets of LX is called a valued fuzzy filter base [2] if the following conditions are fulfilled:

(V1) μα implies α ≤ sup μ.

V2) For all α, β ∈ L0 with α∧β ∈ L0 and all μα and ηβ there are γ ≥ α∧β and σ ≤ μη such that σ ∈ γ.

As shown in ref. [2], each valued fuzzy filter base (α)α∈L defines an fuzzy filter on X by for all μ ∈ LX. Conversely, each fuzzy filter can be generated by a valued fuzzy filter base, e.g., by (α-pr )αL0 with α-pr M={μ∈LXα(μ)}. (α-pr )αL0 is a family of pre filters on X and it is called the large valued filter base of . Recall that a pre filter on X [17] is a non-empty proper subset of of LX such that (1) μ, ηX implies μη and (2) from μ and μη it follows η. Α subset of LX is said to be superior fuzzy filter base [2] if the following conditions are fulfilled:

(S1) ∈B for every α ∈ L.

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

Each superior fuzzy filter base generated a homogeneous fuzzy filter on X by sup η for all μ∈LX and each fuzzy filter can be generated by a superior fuzzy filter base, e.g., by base where base M will be called the large superior fuzzy filter base of . If X is a non-empty set and μ is an fuzzy subset of X, then is a superior fuzzy filter base of a homogeneous fuzzy filter on X, called superior principal fuzzy filter generated by μ and will be denoted by [μ]. In case L is a complete chain and μ is not constant we have [μ] (η)=sup μ, when μη and otherwise for all η ∈ LX. For each ordinary subset M of X we have that where χM is the characteristic function of M.

Fuzzy topology

By the fuzzy topology on a set X, we mean a subset of LX which is closed with respect to all supreme and all finite infimum and contains the constant fuzzy sets and [16,18]. A set X equipped with an fuzzy topology τ on X is called an fuzzy topological space. For each fuzzy topological space (X, τ), the elements of τ are called the open fuzzy subsets of this space. If τ1 and τ2 are fuzzy topologies on a set X, then τ1 is said to be finer than τ2 and τ2 is said to be coarser than τ1, provided τ2 ⊆ τ1 holds. For each fuzzy set μ ∈ LX, the strong α-cut and the weak α-cut of μ are the ordinary subsets S (μ) { x X | μ(x) } and W (μ) { x X | μ(x) } α α = ∈ > α = ∈ ≥ α of X respectively. For each complete chain L, the α-level topology and the initial topology [19] of an fuzzy topology τ on the set X are defined as follows:

respectively, where inf is the infimum with respect to the finer relation for topologies. On other hand if (X, T) is an ordinary topological space, then the induced fuzzy topology on X is given by Lowen [17] as the following:

The fuzzy topological space(X, τ) and also τ are said to be stratified provided α ∈ τ holds for all α ∈ L, that is, all constant fuzzy sets are open [19].

The fuzzy unit interval

The fuzzy unit interval will be denoted by IL an it is defined in [3] as the fuzzy subset:

where I=[0, 1] is the real unit interval and is the set of all positive fuzzy real numbers. Note that, the binary relation ≤ is defined on RL as follows:

for all x, y ∈ RL, where for all α ∈ L0. Note that the family Ω which is defined by:

is a base for an fuzzy topology I on IL, where Rδ and Rδ are the fuzzy subsets of RL defined by for all x ∈ RL and δ ∈ R. The restrictions of Rδ and Rδ on IL are the fuzzy subsets Rδ IL and Rδ IL, respectively. Recall that:

(2.2)

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

Operation on fuzzy sets

In the sequel, let a fuzzy topological space (X, τ) be fixed. By the operation [6] on the set X we mean the mapping φ: LX → LX such that int (μ) ≤ μφ holds for all μ ∈ LX, where, μφ denotes the value of φ at μ. The class of all operations on X will be denoted by By the identity operation on we mean the operation such that for all μ ∈ LX. The constant operation on is the operation defined by for all μ ∈ LX. If ≤ is a partially order relation on defined as follows: for all μ ∈ LX, then is a completely distributive lattice. The operation φ: LX → LX is called:

(i) Isotone if μη implies φμφη, for all μ, η ∈ LX.

(ii) Weakly finite intersection preerving (wfip, for short) with respect to A ⊆ LX if η ∧ϕ (μ) ≤ ϕ (η ∧ μ) holds, for all ηΑ and μ ∈ LX.

(iii) Idempotent if ϕ (μ) =ϕ (ϕ (μ)), for all μ ∈ LX.

The operations φ, ψ are said to be dual if ψ(μ)=co(φ (coμ)) or equivalently φ(μ)=co(ψ (coμ)) for all μ ∈ LX, where coμ denotes the complement of μ. The dual operation of φ is denoted by φ˜. In the classical case of L={0, 1}, by the operation on a set X we mean the mapping φ: P (X) → P (X) such that int A ⊆ Aφ for all A ∈ P (X) and the identity operation on the class of all ordinary operations O(P (X),T) on X will be denoted by iP (X) and it defined by: iP (X)(A)=A for all A ∈ P (X).

The φ-open fuzzy sets

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

The φ1,2-interior fuzzy sets

Let a fuzzy topological space (X, τ) be fixed and

φ1, φ2 Then the φ1,2-interior of the fuzzy set μ: X → L is a mapping φ12.intμ: X → L defined by:

(2.3)

That is, the φ1,2.intμ is the greatest φ1-open fuzzy set η such that ηφ2 less than or equal to μ [19]. The fuzzy set μ is said to be φ1,2-open if and only if μφ1,2.int μ. The class of all φ1,2-open fuzzy sets on X will be denoted by φ1,2OF (X). The complement co μ of the φ1,2-open fuzzy subset μ will be called φ1,2-closed, the class of all φ1,2-closed fuzzy subsets of X will be denoted by φ1,2CF (X). In the classical case of L={0, 1}, the fuzzy topological space (X, τ) is up to an identification by the ordinary topological space (X, T) and φ1,2.int μ is the classical one. Hence in this case the ordinary subset A of X is φ1,2-open if A ⊆ φ1,2. int A. The complement of a φ1,2-open subset A of X will be called φ1,2- closed. The class of all φ1,2-open and the class of all φ1,2-closed subsets of X will be denoted by φ1,2O(X) and φ1,2C(X), respectively. Clearly, F is φ1,2-closed if and only if φ1,2.clT F=F.

Proposition

For each two operations φ1, φ2 ∈ O(LX,τ) and for each μ, ηφ1, φ2 ∈ LX, the mapping φ1,2.int: X → L fulfills the following axioms [7]:

(i) If φ2 ≥ 1LX, then φ1,2.intμμ.

(ii) φ1,2.int is isotone, i.e if μη, then φ1,2.intμφ1,2.intη.

If φ2 ≥ 1LX is isotone and φ1 is with respect to φ1O (X), then

If φ2 is isotone and idempotent operation, then

Proposition

Let (X, τ) be a fuzzy topological space and φ1, φ2 Then the following are fulfilled:

(i) If φ2 ≥ 1LX, then the class φ1,2OF (X) of all φ1,2-open fuzzy sets on X forms an extended fuzzy topology on X [7,21].

If , then the class φ1,2OF (X) of all φ1,2- open fuzzy sets on X forms a supra fuzzy topology on X [21].

If φ2 ≥ 1LX is isotone and φ1 is with respect to φ1OF (X), then φ1,2OF (X) is an fuzzy pre topology on X [21].

If φ2 ≥ 1LX is isotone and idempotent operation and φ1 is with respect to φ1OF (X), then φ1,2OF (X) is fuzzy topology on X [16,18].

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

φ1, φ2 O(LX,τ). Then the relation between the class φ1,2OF (X) of all φ1,2-open fuzzy sets on X and the mapping φ1,2.int is given by:

(2.4)

and the following axioms are fulfilled:

(I1) If φ2 ≥ 1LX, then φ1,2.intμμ holds, for all μ ∈ LX.

(I2) If μη, then φ1,2.intμφ1,2.intη for all μ, η ∈ LX.

(I3)

(I4) If is isotone and φ1 is with respect to φ1OF (X), then φ1,2.intμ ∧ ∧ φ1,2.intη=φ1,2.int (μη) for all μ, η ∈ LX. s

(I5) If φ2 is isotone and idempotent, then φ1,2.int (φ1,2.intμ)=φ1,2. intμ for all μ ∈ LX.

Characterized Fuzzy Spaces

Independently on the fuzzy topologies, the notion of φ1,2-interior operator for the fuzzy sets can be defined as a mapping φ1,2.int: LX → LX which fulfill (I1) to (I5). It is well-known that (2.3) and (2.4) give a one-to-one correspondence between the class of all φ1,2-open fuzzy sets and these operators, that is, φ1,2OF (X) can be characterized by the φ1,2-interior operators. In this case the triple (X, φ1,2.int) as well as the triple (X, φ1,2OF (X)) will be called characterized fuzzy space [7] of the φ1,2-open fuzzy subsets of X. The characterized fuzzy space (X, φ1,2.int) is said to be stratified if and only if for all α ∈ L. As shown in ref. [7], the characterized fuzzy space (X, φ1,2.int) is stratified if the related fuzzy topology is stratified. Moreover, the characterized fuzzy space (X, φ1,2.int) is said to have the weak infimum property [21], provided . The characterized fuzzy space (X, φ1,2.int) is said to be strongly stratified [21], provided φ1,2.int is stratified and have the weak infimum property. If (X, φ1,2.int) and (X, ψ1,2.int) are two characterized fuzzy spaces, then (X, φ1,2.int) is said to be finer than (X, ψ1,2.int) and denoted by φ1,2. int ≤ ψ1,2.int, provided φ1, 2.intμψ1,2.intμ holds for all μ ∈ LX. If τ is a fuzzy topology on the set X and φ1, φ2 then by the initial characterized space of (X, τ) we mean the characterized spaces (X, (φ1,2O(X))α) and (X, i(φ1,2O(X))), respectively where (φ1,2O(X))α and i(φ1,2O(X)) are defined as follows:

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

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

If , then the class of all φ1,2- open fuzzy of X coincide with τ which is defined in [22,23] and hence the characterized fuzzy space with the fuzzy topological space (X, τ).

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 points and at the ordinary subsets of this space. Let (X, τ) be a fuzzy topological space and . As follows by (I1) to (I5) for each x ∈ X, the mapping which is defined by:

(2.5)

for all μ ∈ LX, is a fuzzy filter on X, called φ1,2-fuzzy neighborhood filter at x [7]. If the related φ1,2-interior operator fulfill the axioms (I1) and (I2) only, then the mapping , defined by (2.5) is fuzzy stack [21], called φ1,2-fuzzy neighborhood stack at x. Moreover, if the φ1,2-interior operator fulfill the axioms (I1), (I2) and (I4) such that in (I4) instead of η ∈ LX we take α¯, then the mapping , defined by (2.5) is a fuzzy stack with the cutting property, called φ1,2- fuzzy neighborhood stack with the cutting property at x. The φ1,2-fuzzy neighborhood filters fulfill the following conditions:

holds for all μ, and

Clearly is the fuzzy set φ1,2.int. The characterized fuzzy space is characterized as the fuzzy filter pre topology [7], that is, as a mapping such that (N1) to (N3) are fulfilled.

φ1,2ψ1,2-Fuzzy continuity

Let now the fuzzy topological spaces (X, τ1) and (Y, τ2) are fixed, φ1, and . The mapping f: (X, φ1,2.int) →(Y, ψ1,2. int) is said to be φ1,2ψ1,2-fuzzy continuous if

(2.6)

holds for all η ∈ LY [7]. If an order reversing involution′ of L is given, we have that f is a φ1,2 ψ1,2-fuzzy continuous if and only if holds for all η ∈ LY. Here φ1,2.cl and ψ1,2. cl, mean the closure operators related to φ1,2.int and ψ1,2.int, respectively which are defined by φ1,2.cl μ=co (φ1,2.int coμ) for all μ ∈ LX. Obviously if f is φ1,2 ψ1,2-fuzzy continuous, then the inverse f−1: (Y, ψ1,2.int) → (X, φ1,2. int) is ψ1,2φ1,2-fuzzy continuous, that is holds for all h ∈LX

By means of characterizing φ1,2-fuzzy neighborhoods the φ1,2ψ1,2-fuzzy continuity of f can also be characterized. The mapping f: (X, φ1,2.int) → (Y, ψ1,2.int) is φ1,2ψ1,2-fuzzy continuous if Nψ1,2 (f(x)) ≥ FLf(Nφ1,2 (x)) holds for all x ∈ X. Obviously, in case of L={ 0, 1 }, φ1=ψ1=int, φ2=1LX and ψ2=1LY the φ1,2ψ1,2-fuzzy continuity coincides with the usual fuzzy continuity.

Initial characterized fuzzy spaces

In the following let X be a set, let I be a class and for each i ∈ I, let (Xi, δ1,2.inti) be a characterized fuzzy space of δ1,2-open fuzzy subsets of Xi and fi: X → Xi is the mapping from X into Xi. By the initial φ1,2- fuzzy interior operator of (δ1,2.inti)iI with respect to (fi)iI, we mean the coarsest φ1,2-fuzzy interior operator φ1,2.int on X for which all mappings fi: (X, φ1,2.int) → (Xi, δ1,2.inti) are φ1,2δ1,2-fuzzy continuous. The triple (X, φ1,2.int) is said to be initial characterized fuzzy space [7] of ((Xi, δ1,2.inti))i∈I with respect to (fi)iI. The initial φ1,2-fuzzy interior operator φ1,2.int: LX → LX of (δ1,2.inti)iI with respect to (fi)iI always exists and is given by:

(2.7)

for all μ ∈ LX. For each i ∈ I, let is the representation of δ1,2.inti as an fuzzy filter pre topology. Then because of (2.5) and (2.7), the mapping Nφ1,2: X → FLX which is defined by:

for all x ∈ X and μ ∈ LX, is the representation of the initial φ1,2-fuzzy interior operator of (ψ1,2.inti)iI with respect to (fi)iI as the fuzzy filter pre topology.

Characterized Fuzzy Subspaces

Let A be a subset of a characterized fuzzy space (X, φ1,2.int) and i: A,→ X is the inclusion mapping of A into X. Then the mapping φ1,2.intA: LA → LA defined by:

for all η ∈ LA is initial φ1,2-fuzzy interior operator for φ1,2.int with respect to the inclusion mapping i: A,→ X. φ1,2.intA will be called induced φ1,2- interior operator of φ1,2.int on the subset A of X. The triple (A, φ1,2.intA) is said to be characterized fuzzy subspace of (X, φ1,2.int) [7].

Characterized Fuzzy Product Spaces

Assume that (Xi, δ1,2.inti) is a characterized fuzzy space for each i I, where I is any class. Let X be the cartesian product of the family (Xi)i∈I and πi: X → Xi the related projections. The i∈I, mapping φ1,2.int: LX → LX, defined by:

for all μ ∈ LX, will be called φ1,2-fuzzy product of the δ1,2L-interior operators δ1,2.inti. The triple (X, φ1,2.int) is said to be characterized fuzzy product space [7] of the characterized fuzzy spaces (Xi, δ1,2.inti). The φ1,2.int will be denoted by and it is initial φ1,2-fuzzy interior operator of (δ1,2.inti)iI with respect to the family (πi)iI of projections. The characterized fuzzy product space (X, φ1,2.int) also will be denoted by

Final characterized fuzzy spaces

It is well-known (cf. e.g., [11,24]) that in the topological category all final lifts uniquely exist and hence also all final structures exist. They are dually defined. In case of the category CF-Space of all characterized fuzzy spaces the final structures can easily be given, as is shown in the following:

Let I be a class and for each i ∈ I, let (Xi, δ1,2.inti) be an characterized fuzzy space and fi: Xi → X is the mapping of Xi into a set X. The final φ1,2- fuzzy interior operator of (δ1,2.inti)iI with respect to (fi)iI is the finest φ1,2.int on X for which all mappings fi: (Xi, δ1,2.inti) → (X, φ1,2.int) are δ1,2φ1,2-fuzzy continuous [7]. Hence, the triple (X, φ1,2.int) is the final characterized fuzzy space of ((Xi, δ1,2.inti))i∈I with respect to (f)i∈I. The final φ1,2L-interior operator φ1,2.int: LX→ LX of (δ1,2.inti)iI with respect to (fi)iI exists and is given by

for all x ∈ X and μ ∈ LX.

Characterized Fuzzy Quotient Spaces

Let (X, φ1,2.int) be a characterized fuzzy space and f: X→A is an surjective mapping. Then the mapping φ1,2.intf: LA → LA, defined by:

for all a ∈ A and μ ∈ LA, is final φ1,2-fuzzy interior operator of φ1,2.int with respect to f which is not idempotent. Then the φ1,2.intf will be called quotient φ1,2-fuzzy interior operator and the triple (A, φ1,2.intf) is said to be characterized fuzzy quotient space [7].

Note that in this case φ1,2.int is idempotent, φ1,2.intf need not be. Even in the classical case of L={0, 1}, φ1=int and φ2=1LX we have the following: If φ1,2.int is up to an identification the usual topology, then φ1,2.intf is a pre topology which need not be idempotent. An example is given [25] (p. 234).

Characterized Fuzzy Sum Spaces

Assume that (Xi, δ1,2.inti) is a characterized fuzzy space for each i ∈, where I is any class. Let X be the disjoint union of the family (Xi)i∈I and for each i ∈ I, let φ1,2.int: LX → LX, defined by:ei: Xi → X be the canonical injection from Xi into X given by ei(xi)=(xi, i). Then the mapping φ1,2.int: LX → LX, defined by:

for all i ∈ I, of a ∈ Xi and μ ∈ LX, is said to be final φ1,2-fuzzy interior operator with respect to (ei)i∈I.

1,2.inti)iI φ1,2.int will be called sum φ1,2-fuzzy interior operator will be denoted by Σ δ1,2.inti. The pair (X, φ1,2.int) is said to be characterized fuzzy sum space [7] and it will be denoted also by

Characterized Fuzzy T1 And Fuzzy φ1,2T1-Spaces

The notions of characterized fuzzy Ts and of characterized fuzzy Rk-spaces are investigated and studied [9,10,26,27] for all . These characterized spaces depend only on the usual points and the operation defined on the class of all fuzzy subsets of X endowed with an fuzzy topology τ. Let the fuzzy topological space(X, τ) be fixed and φ1, φ2 then the characterized fuzzy space all fuzzy subsets of X endowed with an fuzzy topology τ. Let the fuzzy topological space (X, τ) be fixed and φ1, φ2 then the characterized fuzzy space all fuzzy subsets of X endowed with an fuzzy topology τ. Let the fuzzy topological space (X, τ) be fixed and φ1, φ2 then the characterized fuzzy space (X, φ1,2.int) is said to be characterized fuzzy T1-space if for all x, y ∈ X such that (X, φ1,2.int) is said to be characterized fuzzy T1-space if for all x, y ∈ X such that x ≠ y there exist μ, η ∈ LX and α, β ∈ L0 such that μ(x) < α ≤ (φ1,2.intμ)(y) and η(y) < β ≤ (φ1,2.intη)(x) are hold. The related fuzzy topological space(X, τ) is said to be fuzzy φ1,2-T1 if for all x, y ∈ X such that x ≠ y , we have x˙ ̸≤ Nφ1,2(y) and y˙ ̸≤ Nφ1,2(x).

Proposition

Let (X, T) be an ordinary topological space and φ1, φ2 ∈ ∈ O(P(X),T) such that φ2 ≥ iP(X) is isotone and idempotent. Then (X, T) is φ1,2T1-space if and only if the induced characterized fuzzy space (X, φ1, 2.intω) is characterized fuzzy T1 [27].

Proposition

Let (X, τ) be an fuzzy φ1,2-T1 space and φ1, φ2 ∈ O(LX,t) such that φ2 is isotone and idempotent. Then the α-level characterized space (X, φ1,2. intα) and the initial characterized space (X, φ1,2.inti) are T1-spaces [27].

Proposition

Let X be a set, let I be a class and for each i ∈ I, let the characterized fuzzy space (Xi, δ1,2.inti) is characterized fuzzy T1 and fi: X → Xi be an injective mapping for some i ∈ I. Then the initial characterized fuzzy space (X, φ1,2.int) of ((Xi, δ1,2.inti))i∈I with respect to (fi)iI is also characterized fuzzy T1-space [10].

Proposition

Let X be a set, let I be a class and for each i ∈ I, let the characterized fuzzy space (Xi, δ1,2.inti) is characterized fuzzy T1 and fi: Xi → X be an surjective mapping for some i ∈ I. Then the final characterized fuzzy space (X, φ1,2.int) of ((Xi, δ1,2.inti))i∈I with respect to (fi)iI (X, φ1,2.int) is characterized fuzzy T1-space [27].

Proposition

Let the characterized fuzzy space (X, φ1,2.int) is characterized fuzzy T1 and δ1,2.int is finer than φ1,2.int. Then the characterized fuzzy space (X, δ1,2.int) is also fuzzy T1 [27].

Characterized Fuzzy and Characterized Fuzzy R3- Spaces

Let a fuzzy topological space(X, τ) be fixed and φ1, φ2 ∈ O(LX,τ). Then the characterized fuzzy space (X, φ1,2.int) is said to be characterized fuzzy [9] (resp. fuzzy R3-space [10] if for all x ∈ X, F ∈ φ1,2C(X) such that x ̸ F (resp. F1, F2φ1,2C(X) such that F1 ∩ F2=∅), there exists an φ1,2ψ1,2-fuzzy continuous mapping such that

for all y ∈ F (resp. the infimum) does not exist). Proposition 2.8 [9] Let (X, τ) be a fuzzy topological space, φ1, φ2 ∈ O(X,τ) and Ω is a subbase for the characterized fuzzy space (X, φ1,2.intτ). Then, (X, φ1,2.intτ) is characterized fuzzy R2 12-space if and only if for all F ∈ Ω′ and x ∈ X such that x ∈/F, there exists a φ1,2ψ1,2-fuzzy continuous mapping fuzzy characterized fuzzy T4-spaces such that f(x) = 1and f ( y) = 0 for all y ∈ F.

Characterized

Let a fuzzy topological space(X, τ) be fixed and φ1, φ2. Then the characterized fuzzy space (X, φ1,2.int) is said to be characterized fuzzy or characterized Tychonoff fuzzy space [9] (resp. fuzzy T4-space [10] if and only if it is characterized fuzzy (resp. characterized fuzzy R3) and characterized fuzzy T1-space. The related fuzzy topological space(X, τ) is said to be fuzzy φ1,2- (resp. fuzzy φ1,2-T4) if and only if it is fuzzy φ1,2- (resp. fuzzy φ1,2-R3) and fuzzy φ1,2-T1 space.

Proposition

Every characterized fuzzy T4-space is characterized fuzzy -space [9].

Metrizable Characterized Fuzzy Spaces and Characterized -Spaces

By the fuzzy metric on the set X [6], we mean that the mapping d: X × X:→ R*L such that the following conditions are fulfilled:

(1) d(x, y)=0∼ if and only if x=y.

(2) d(x, y)=d(y, x) for all x, y ∈ X.

(3) d(x, y) ≤ d(x, z)+ d(z, y) holds for all x, y, z ∈ X.

Where 0∼ denotes the fuzzy number which has value 1 at 0 and 0 otherwise. The set X equipped with an fuzzy metric on X will be called fuzzy metric space. Each fuzzy metric on a set X generated canonically a stratified fuzzy topology τd which has the set B={ξ ◦ dx: ξ ∈ μ and x ∈ X} as a base, where dx: X → R*L is the mapping defied by: dx(y)=d(x, y) and

Where has the domain is and is the restriction of Rδ on L R . Now, consider φ1, φ2 ∈ O(LXd), then as shown in ref. [20], the characterized fuzzy space (X, φ1,2.intτd) is stratified. The stratified characterized fuzzy space (X, φ1,2.intτd) is said to be metrizable characterized fuzzy space.

In the following proposition we shall prove that every metrizable characterized fuzzy space is characterized fuzzy T4-space in sense of Abd-Allah [10].

Proposition

Let (X, τd) be an stratified fuzzy topological space generated canonically by an fuzzy metric d on X and φ1, φ2 then the metrizable characterized fuzzy space (X, φ1,2.intτd) is characterized fuzzy T4-space.

Proof: Let such that = ∅. 1 2 Then for all x ∈ F1 and y ∈ F2, we get d (x, y) ≠ 0 ∼, that is, there exists δ>0 such that d(x, y)(2δ)>0 and therefore

holds. Consider and then

for all

for all 2 y∈F . Hence, μ and η are φ1,2-fuzzy neighborhoods in (X, φ1,2.intτd) at all x ∈ F1 and all y ∈ F2, respectively, this means Because of the symmetry and triangle inequality of d and (2.2), we get and therefore holds for all z ∈ X, that is, sup (μη)<1. Hence, the infimum Nφ1,2 (F1) ∧ Nφ1,2 (F2) does exists and therefore (X, φ1,2.intτd) is characterized fuzzy R3-space. Because of Theorem 3.1 [27], it is clear that (X, φ1,2.intτd) is characterized fuzzy T1- space. Consequently, (X, φ1,2.intτd) is characterized fuzzy T4-space.

Example 3.1

From Propositions 2.9 and 3.1, we get that the metrizable fuzzy space in sense of Gahler and Gahler [3] is an example of a metrizable characterized fuzzy T4-space and that is also example of a metrizable characterized fuzzy Tk-space for

Characterized and characterized -spaces

In the following we introduce and study the concepts of characterized -space and of characterized spaces in the classical case. Let (X, T) be an ordinary topological space and φ1, φ2 . Then the characterized space (X, φ1,2.intT) is said to be characterized -space if for all x ∈ X, F ∈ φ1,2C(X) such that x ̸F, there exists an φ1,2ψ1,2 continuous mapping f: (X, φ1,2.intT) → (I, ψ1,2. intTI) such that f(x)=1 and f(y)=0 for all y ∈ F, where ψ1,2.intI is the usual ψ1,2-interior operator on the closed unit interval I and ψ1, ψ2 ∈O(P (I),TI). Moreover, the ordinary characterized space (X, φ1,2.intT) is said to be characterized -space or classical characterized-Tychonoff space if and only if it is characterized T1-space and characterized -space.

Proposition

Let (X, T) be an ordinary topological space and φ1, φ2 ∈ O(P (X),T) such that φ2 ≥ iP (X) is isotone and idempotent. Then, (X, φ1,2.intT) is characterized -space if and only if the induced characterized fuzzy space (X, φ1,2.intω) is characterized fuzzy -space.

Proof: Let (X, φ1,2.intT) is characterized -space,such that x ̸F. Then, there exists φ1,2δ1,2-continuous mapping g: such that g(x) =1and g( y) = 0 for all and for all L α ∈ 1 , where Hence, the mapping g: is φ1,2δ1,2-fuzzy continuous. Consider h: is the map-ping defied by h(z) = for all z ∈ I, then h is δ1,2ψ1,2- fuzzy continuous and there-fore there exists an φ1,2ψ1,2-fuzzy continuous mapping such that equation for all y F. Consequently, equation is characterized fuzzy equation -space.

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

Corollary 3.1

Let (X, T) be an ordinary topological space and equation such that equation is isotone and idempotent. Then, equation is characterized equation -space if and only if the induced characterized fuzzy space (X, φ1,2.intω) is characterized fuzzy equation -space.

Proof: Immediate from Propositions 2.3 and 3.2.

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

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

Proposition 3.3

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

Proof: Consider equation is characterized fuzzy equation -space, x X and equation such that equation . Then equation . and equation . Because of equation is characterized fuzzy equation Space,-space, there exists an equation -fuzzy continuous mapping f: (X, φ1,2.intτ) → (IL, ψ1,2.intI) and f(y)=0 such that equation and equation for all equation . Since φ1,2.in tτ= φ1,2.intα and equation then there could be found the mapping equation equation which is φ1,2ψ1,2-continuous with fα(x)=1 and fα(y)=0 for all y F. Consequently, (X, φ1,2.intα) is characterized equation space. The second case is similarly, that is, if (X, φ1,2.intτ) is characterized fuzzy equation -space.

Corollary 3.2

Let (X, τ) be a fuzzy topological space and φ1, φ2 O(LX,τ) such that equation is isotone and idempotent. Then the α-level characterized space (X, φ1,2.intα) and the initial characterized space (X, φ1,2.inti) are characterized equation -spaces if the characterized fuzzy space (X, φ1,2.intτ) is characterized fuzzy equation .

Proof: Immediate from Propositions 2.4 and 3.3.

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

Proposition

Let (X, τ) is a fuzzy topological space and φ1, φ2 O(LX, τ). If the characterized fuzzy space (X, φ1,2.intτ) is characterized fuzzy equation and δ1,2.intτ is finer than φ1,2.intτ, then (X, δ1,2.intτ) is also characterized fuzzy and δ1,2.intτ equation -space.

Proof: Let Ω is a sub base for the characterized fuzzy space equation and equation such that equation Such that equation Then, there is equation such that equation and therefore equation or all i ∈ {1,. . ., n}. Because of Proposition 2.8, there exists a φ1,2ψ1,2-fuzzy continuous mappings fi: (X, φ1,2.intτ) → (IL, ψ1,2.intI) such that equation and equation is also fulfilled for all equation In particular this means that equation and equation for all y F and i {1,. . ., n}. Since δ1,2.intτ is finer than φ1,2.intτ, then any one of these mappings equation gives us the required δ1,2ψ1,2-fuzzy continuous mappings g: (X, δ1,2.intτ) → (IL, ψ1,2.intI) such that equation and equation and fi(y)=0 for all y F and i {1,. . ., n}. Since δ1,2.intτ is finer than φ1,2.intτ, then any one of these mappings equation gives us the required δ1,2ψ1,2-fuzzy for all y F. Consequently, (X, δ1,2.intτ) is characterized fuzzy equation Space.

Corollary 3.3 Let (X, τ) be a fuzzy topological space and φ1, φ2 ∈ O(LX,τ). If (X, φ1,2.intτ) is characterized fuzzy equation -space and δ1,2.intτ is finer than φ1,2.intτ, then (X, δ1,2.intτ) is also characterized fuzzy equation -space.

Proof: Immediate from Propositions 2.7 and 3.4.

Initial and Final Characterized Fuzzy equation and Fuzzy equation 2-Spaces

In this section we are going to introduce and study the notion of initial and final characterized fuzzy equation -spaces and the notions of initial and final characterized fuzzy equation -spaces. The characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space are studied as special case from the initial and final characterized fuzzy equation and fuzzy equation -spaces. New additional properties for the initial and final characterized fuzzy equation -spaces and for the initial and final characterized fuzzy equation -spaces are given. The categories of all characterized fuzzy equation and of all characterized fuzzy equation -spaces will be denoted by CFR-Space and CRF-Tych, respectively. Note that the categories CFR-Space and CRF-Tych are concrete categories. The concrete categories CFR-Space and CRF-Tych are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets. Hence, all the initial and final lifts exist uniquely in the categories CFR-Space and CRF-Tych, respectively.

This means that they also topological over the category SET. That is, all the initial and final characterized fuzzy equation -spaces and all the initial and final characterized fuzzy equation -spaces exist in CFR-Space and CRF-Tych, respectively.

In the following let X be a set, let I be a class and for each i I, let the characterized fuzzy space equation of all δ1,2-open fuzzy subsets of Xi is characterized fuzzy equation -space. For some i I, let equation is φ1,2δ1,2-closed injective mapping from X into Xi. Then we show in the following that the initial characterized fuzzy space (X,φ1,2.int) of equation with respect to equation is also characterized fuzzy equation -space. More general, we show under the same conditions, that the initial characterized fuzzy space (X, φ1,2.int) of equation with respect to equation is characterized fuzzy equation -space if all the characterized fuzzy spaces equation are characterized fuzzy equation -spaces for all i I. Moreover, as special cases we show that the characterized fuzzy subspace, characterized fuzzy product space and characterized fuzzy filter pre topology of a characterized fuzzy equation -space and of a characterized fuzzy equation -space are characterized fuzzy equation -spaces and characterized fuzzy equation -spaces, respectively.

Proposition

Let X be a set and I be a class. For each i I, let the characterized fuzzy space equation of all δ1,2-open fuzzy subsets of Xi is characterized fuzzy equation -space. If equation is an equation -closed injective mapping from X into Xi for some i I, then the initial characterized fuzzy space (X, φ1,2.int) of equation with respect to equation is also characterized fuzzy equation -space.

Proof: Let x X and equation such that x F. Since equation is φ1,2δ1,2-closed injective for some i I, then equation and equation Because of equation is characterized fuzzy equation - space for all i I, then there

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

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

Corollary 4.1 Let X be a set and I be a class. For each iI, let the characterized fuzzy space equation of all δ1,2-open fuzzy subsets of Xi is characterized fuzzy equation -space. If equation is an equation -closed injective mapping from X into Xi for some i I, then the initial characterized fuzzy space equation equation of with respect to equation is also characterized fuzzy equation -space.

Proof: Immediate from Propositions 2.5 and 4.1.

Corollary 4.2

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

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

As shown in ref. [7], the characterized fuzzy space (X, φ1,2.int) is characterized as a fuzzy filter pre topology, then we have the following result:

Corollary 4.3

For each i I, let equation is δ1,2.inti as the fuzzy filter pre topology is characterized fuzzy R2 fuzzy equation ). Then, the representation of the initial φ1,2-interior operator equation of the initial characterized fuzzy space equation of equation with respect to equation as a fuzzy filter pre topology which is defined by:

equation

for all x X and μ LX is also characterized fuzzy equation (resp. characterized fuzzy equation ).

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

Proposition

Let (X, τ1) and (Y, τ2) are two fuzzy topological spaces, equation and equation . If the mapping equation is an φ1,2δ1,2-closed injective from X into Y and (Y, δ1,2.int) is characterized fuzzy equation terized fuzzy equation ) L-space, then the initial characterized fuzzy space equation with respect to f is also characterized fuzzy equation (resp. fuzzy equation ) L-space.

Proof: Straight forward.

Corollary 4.4

Let (Y, τ2) be an fuzzy topological spaces and equation is an φ1,2δ1,2-closed injective mapping from X into Y fuzzy equation -space), then the initial fuzzy topological space equation of equation with respect to f is fuzzy equation − space (resp. fuzzy equation -space) for all equation

Proof: Follows immediately from Proposition 4.2. 2

In the following let X be a set and I be a class. For each i I, let the characterized fuzzy space equation of all δ1,2-open fuzzy subsets of Xi is characterized fuzzy equation -space. For some i I, let equation is surjective mapping from Xi into X and equation is equation -closed in the classical sense. Then as in case of the initial characterized fuzzy spaces, we show in the following that the final characterized fuzzy space (X, φ1,2.int) of equation with respect to equation is also characterized fuzzy equation -space. More general, we show under the same conditions that, the final characterized fuzzy space equation of equation with respect to equation is characterized fuzzy equation space if each of the characterized fuzzy spaces equation is characterized fuzzy equation -spaces for all i I. Moreover, as special cases we show that the characterized fuzzy quotient space and the characterized fuzzy sum space of the characterized fuzzy equation -space and of the characterized fuzzy equation -space are characterized fuzzy equation -spaces and characterized fuzzy equation -spaces, respectively. Proposition 4.3 Let X be a set and I be a class. For each i I, let the characterized fuzzy space equation of all equation -open fuzzy subsets of Xi is characterized fuzzy equation -space. If equation equation is an subjective equation -fuzzy open mapping from Xi into X and equation is equation -closed for some i I, then the final characterized fuzzy space equation of equation with respect to equation is also characterized fuzzy equation -space.

Proof: Let x X and equation such that x F. Since equation is surjective and equation -closed for some i I, then there exists equation and equation for which equation and equation such that equation Because of equation is characterized fuzzy equation -space for all i I, then there exists an equation fuzzy continuous mapping g: equation such that equation and equation for all z K, that is equation and equation for all sF. Therefore, there exists a mapping equation equation such that equation and equation for all sF. Since equation fuzzy open, then equation equation holds for all equation which means that equation is equation -fuzzy continuous. Hence, the composition equation equation -fuzzy continuous mapping and therefore the final characterized fuzzy space (X, φ1,2.int) is characterized fuzzy equation -space.

Corollary 4.5

Let X be a set and I be a class. For each i I, let the characterized fuzzy space equation of all δ1,2-open fuzzy subsets of Xi is characterized fuzzy equation -space. If equation is an surjective equation -fuzzy open mapping from Xi into X and equation -closed for some i I, then the final characterized fuzzy space (X, φ1,2.int) of equation with respect to equation is also characterized fuzzy equation -space.

Proof: Immediate from Propositions 2.6 and 4.3. 2

Corollary 4.6

The characterized fuzzy quotient space (A, φ1,2.intf) and the char characterized fuzzy equation -space) are also characterized fuzzy equation (resp. characterized fuzzy equation ) L-spaces.

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

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

Proposition 4.4 Let (X, τ1) and (Y, τ2) are two fuzzy topological spaces, equation and equation If equation is an subjective δ1,2φ1,2-fuzzy open mapping from X into Y and equation closed, then the final characterized fuzzy space (X, φ1,2.int) of (Y, δ1,2. int) with respect to f is characterized fuzzy equation (resp. characterized fuzzy equation )L-space if (Y, δ1,2.int) is characterized fuzzy equation (resp. characterized fuzzy equation ) L-spaces.

Proof: Straight forward.

Corollary 4.7

Let (Y, τ2) be an fuzzy topological spaces and equation → X is an δ1,2φ1,2-fuzzy open surjective mapping from Y into X and f−1 φ1,2δ1,2-closed, then the final fuzzy topological space equation with respect to f is fuzzy equation -space (resp. fuzzy equation )-space if (Y, τ2) is fuzzy equation -space (resp. fuzzy equation )-space for all equation .

Proof: Follows immediately from Proposition 4.4. 2.

Finer Characterized Fuzzy equation and Finer Characterized Fuzzy equation -Spaces

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

Characterized fuzzy equation H and characterized fuzzy equation H-spaces. In the following we introduce and study the concept of characterized completely fuzzy regular Hutton and characterized fuzzy equation Huttonspaces as a generalization of the weaker and stronger forms of the completely fuzzy regular and fuzzy equation -spaces in sense of Hutton [28], respectively. The relation between characterized completely fuzzy regular Hutton-spaces and the characterized fuzzy equation -spaces in our sense is introduced. More generally, the relations between characterized fuzzy equation Hutton-spaces and the characterized fuzzy equation -spaces in our sense is also introduced. Let (X, τ) be a fuzzy topological space and equation . Then the characterized fuzzy space (X, φ1,2.int) is said to be characterized completely fuzzy regular Hutton-space or (characterized fuzzy equation H-space, for short) if for an equation there exists a collection equation and an φ1,2ψ1,2-fuzzy continuous mapping equation such that equation and equation holds for all y X. Then characterized fuzzy space (X, φ1,2.int) is said to be characterized fuzzy equation Hutton-space or (characterized fuzzy equation H-space, for short) if and only if it is characterized fuzzy equation H and characterized fuzzy equation -spaces.

In the classical case of L={0, 1}, equation equation and equation the equation -fuzzy continuity of f is up to an identification the usual fuzzy continuity of f. Then in this case the notions of characterized fuzzy equation H-spaces and of characterized fuzzy equation H-spaces are coincide with the notion of fuzzy completely regular spaces and the notion fuzzy equation -spaces defined by Hutton [28], respectively. Another special choices for the operations φ1, φ2, ψ1 and ψ2 are obtained (Table 1).

Table

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

Proposition 5.1

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

Then every characterized fuzzy equation H-space (X, φ1,2.int) is characterized fuzzy equation -space.

Proof: Let (X, φ1,2.int) is characterized fuzzy equation H-space, x X and equation such that equation . Then, equation and equation , therefore equation holds for all α L. Hence, equation and therefore for all equation , there exists a family equation in LX such that equation and equation holds for all y X. In case of y F, we get equation holds for all yF and therefore equation for all y F. In case of y=x, we get xα(x)=α equation holds for all α ∈ L and this means that g(x) (s)=1 for all s < 1 and therefore equation Consequently, equation is characterized fuzzy equation -space in sense [9].

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

Proof: Follows immediately from Proposition 5.1.

The following example shows that the inverse of Proposition 5.1 and of Corollary 5.1 is not true in general.

Example 5.1.

Let X={x, y} with x ≠ y and equation is an fuzzy topology on X. Choose equation Hence, equation and there is the only case of x ∈ X, F={y} ∈ φ1,2C(X) such that equation Since the mapping equation equation which is defined by equation and equation for all y≠x is φ1,2ψ1,2-fuzzy continuous, then equation is characterized fuzzy equation -space in sense [9]. Obviously, equation is characterized fuzzy T1- space, therefore equation is characterized fuzzy equation -space.

On other hand, let (X, φ1,2.intτ) is characterized fuzzy equation H-space, then(X, φ1,2.intτ) is characterized fuzzy equation H and characterized fuzzy T1-space. Since equation and equation then there exists a collection equation such that equation Moreover, for an φ1,2ψ1,2-fuzzy continuous mapping equation equation such that equation and equation for all y ≠ x, we get the inequality

equation

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

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

Let (X, τ) be an fuzzy topological space and equation . Then the characterized fuzzy space (X, φ1,2.int) is said to be characterized completely fuzzy regular Katasars-space or (characterized fuzzy equation K-space, for short) if for every x X and μ LX such that equation L0, there exists an equation -fuzzy continuous mapping equation equation such that equation and equation are holds for all y X and α L0. The characterized fuzzy space equation is said to be characterized fuzzy equation Katasars-space or (characterized fuzzy equation K-space, for short) if and only if it is characterized fuzzy equation K-space and characterized fuzzy T1-space.

In the classical case of L={ 0, 1 }, equation equation and equation fuzzy continuity of f is up to an identification the usual fuzzy continuity of f. Then in this case the notions of characterized fuzzy equation K-space and of characterized fuzzy equation K-spaces are coincide with the notion of completely fuzzy regular spaces and the notion of fuzzy equation -spaces presented by Katasars [13], respectively. Another special choices for the operations φ1, φ2, ψ1 and ψ2 are obtained in Table 1. In the following proposition we show that the notion of characterized fuzzy equation -spaces which are presented [9] are more general than the characterized fuzzy equation K-spaces.

Proposition

Let (X, τ) be an fuzzy topological space and equation . Then every characterized fuzzy equation K-space (X, φ1,2.int) is characterized fuzzy equation space.

Proof: Let (X, φ1,2.int) is a characterized fuzzy equation K-space, x X and equation such that equation Then, equation and equation , therefore equation holds for all α L. Because of equation is characterized fuzzy equation K-space, then there exists a φ1,2ψ1,2-fuzzy continuous mapping equation such that equation and equation are hold for all y X and α L. In case of y F, we have equation , that is, equation for all t>0, y F and therefore equation for all y F. In case of y=x, we have equation holds for all α L, and therefore equation . Hence, there exists a φ1,2ψ1,2-fuzzy continuous mapping equation such that equation and equation for all y F. Consequently, equation is characterized fuzzy equation -space in sense [9].

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

Proof: Follows immediately from Proposition 5.2.

The following example shows that the inverse of Proposition 5.2 and of Corollary 5.2 is not true in general.

Example 5.2.

Consider the characterized fuzzy space equation which is defined in Example 5.1, then as shown in Example 5.1, equation is characterized fuzzy equation -space in sense [9] and characterized fuzzy T1- space, therefore equation is characterized fuzzy equation -space in sense [9].

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

Characterized Fuzzy equation KE and Characterized Fuzzy equation KE-Space

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

Let (X, τ) be an fuzzy topological space and equation .Then the characterized fuzzy space equation is said to be characterized completely fuzzy regular Kandil and Shafee space or (characterized fuzzy equation KE-space, for short) if for every equation and equation such that equation , there exists an equation -fuzzy continuous mapping equation equation such that equation and equation are hold for all y X and α L. The characterized fuzzy space equation is said to characterized quasi fuzzy T1-space or (characterized QFT1- space, for short) if for all x, y X such that x ≠ y we have equation and equation for all α, β L. As easily seen that every characterized QFT1-space is characterized fuzzy T1-space. The characterized fuzzy space (X, φ1,2.int) is said to be characterized fuzzy equation Kandil El-Shafeespace or (characterized fuzzy equation KE-space, for short) if and only if it is characterized fuzzy equation KE and characterized QFT1-spaces. Obviously, every characterized fuzzy equation KE-space is characterized fuzzy equation K-space. In the classical case of L={0, 1}, equation and equation -fuzzy continuity of f is up to an identification the usual fuzzy continuity of f. Hence, the notions of characterized fuzzy equation KE-spaces and of characterized fuzzy equation KE-spaces are coincide with the notion of completely fuzzy regular spaces and the notion fuzzy fuzzy equation -spaces presented by Kandil and Shafee [12], respectively. Another special choices for the operations φ1, φ2, ψ1 and ψ2 are obtained in Table 1.

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

Proposition 5.3

Let (X, τ) be an fuzzy topological space and equation Then every characterized fuzzy equation KE-space (X, φ1,2.int) is characterized fuzzy equation -space.

Proof: Let (X, φ1,2.int) is a characterized fuzzy equation KE-space, x X and equation such that equation Then, equation and equation therefore equation . Because of (X, φ1,2.int) is characterized fuzzy equation KE space, then there exists a φ1,2ψ1,2-fuzzy continuous mapping equation equation such that equation and equation are hold for all y X. In case of y F, we have equation 0, that is, f(y)(s)=0 for all s>0and therefore equation for all y F. In case of y=x, we have equation holds and then f(x) (s)=1 for all s < 1, therefore equation . Hence, there exists a φ1,2ψ1,2-fuzzy continuous mapping equation such that equation and equation for all y F. Consequently, (X, φ1,2.int) is characterized fuzzy equation -space in sense [9].

Corollary 5.3

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

Proof: Follows immediately from Proposition 5.3 and the fact that every characterized QFT1-space is characterized fuzzy T1-space.

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

Example 5.3.

Consider the characterized fuzzy space equation which is defined in Example 5.1, then as shown in Example 5.1, equation is characterized fuzzy equation -space in sense [9] and characterized fuzzyT1- space, therefore equation is characterized fuzzy equation -space in sense [9].

Now, choose equation and equation then equation such that equation . Hence, for any φ1,2ψ1,2-fuzzy continuous mapping equation such that equation and equation for all equation we get equation holds for all z X. But equation holds only for z=y and it is not fulfilled for z=x. Consequently, equation is not characterized fuzzy equation KE-space and therefore it is not characterized fuzzy equation KE-space.

Conclusion

In this paper, basic notions related to the characterized fuzzy equation and the characterized fuzzy equation -spaces which are presented [9] are introduced and studied. These notions are named metrizable characterized fuzzy spaces, initial and final characterized fuzzy spaces, some finer characterized fuzzy equation and characterized fuzzy equation -spaces. The metrizable characterized fuzzy space is introduced as a generalization of the weaker and stronger forms of the fuzzy metric space introduced by Gahler and Gahler [3]. For every stratified fuzzy topological space generated canonically by an fuzzy metric we proved that, the metrizable characterized fuzzy space is characterized fuzzy T4- space in sense of Abd-Allah [10] and therefore, it is characterized fuzzy equation and characterized fuzzy equation -space. The induced characterized fuzzy space is characterized fuzzy equation and characterized fuzzy equation -space if and only if the related ordinary topological space is φ1,2 equation -space and φ1,2 equation -space, respectively. Hence, the notions of characterized fuzzy equation and of characterized fuzzy equation are good extension in sense of Lowen [11]. Moreover, the α-level characterized space and the initial characterized space are characterized equation -space and characterized equation -space if the related characterized fuzzy space is characterized fuzzy equation -space and characterized fuzzy equation -space, respectively. We shown that the finer characterized fuzzy space of a characterized fuzzy equation -space and of a characterized fuzzy equation -space is also characterized fuzzy equation and characterized fuzzy equation -space, respectively. The categories of all characterized fuzzy equation and of all characterized fuzzy equation -spaces will be denoted by CFR-Space and CRF-Tych and they are concrete categories. These categories are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets and hence all the initial and final lifts exist uniquely in CFR-Space and CRF-Tych, respectively. That is, all the initial and final characterized fuzzy equation -spaces exist in CFR-Space and also all the initial and final characterized fuzzy equation -spaces exist in CRF-Tych. We shown that the initial and final characterized fuzzy spaces of a characterized fuzzy equation -space and of characterized fuzzy equation -space are characterized fuzzy equation and characterized fuzzy equation -spaces, respectively. As special cases, the characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space of a characterized fuzzy equation -space and of a characterized fuzzy equation -space are also characterized fuzzy equation and characterized fuzzy equation spaces, respectively. Finally, we introduced and studied three finer characterized fuzzy equation and three finer characterized fuzzy equation L-spaces as a generalization of the weaker and stronger forms of the completely regular and the fuzzy equation -spaces introduced [28,12,13]. These fuzzy spaces are named characterized fuzzy equation H, characterized fuzzy equation K, characterized fuzzy equation KE, characterized fuzzy equation H, characterized fuzzy equation K and characterized fuzzy equation KE-spaces. The relations between characterized fuzzy equation H, characterized fuzzy equation K, characterized fuzzy equation KE-spaces and the characterized fuzzy equation -space which are presented [9] are introduced. More generally, the relations between characterized fuzzy equation H, characterized fuzzy equation K, characterized fuzzy equation KE-spaces and the characterized fuzzy equation -spaces are also introduced. Meany special cases from these finer characterized fuzzy equation and finer characterized fuzzy equation -spaces are listed in Table 1.

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