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PURE MATHEMATICS | RESEARCH ARTICLE

Graded fuzzy topological spaces Ismail Ibedou1,2*

Received: 13 August 2015 Accepted: 01 January 2016 First Published: 07 January 2016

Abstract: In this paper, graded fuzzy topological spaces based on the notion of neighbourhood system of graded fuzzy neighbourhoods at ordinary points are introduced and studied. These graded fuzzy neighbourhoods at ordinary points and usual subsets played the main role in this study.

*Corresponding author: Ismail Ibedou, Faculty of Science, Department of Mathematics, Benha University, 13518 Benha, Egypt; Faculty of Science, Department of Mathematics, Jazan University, KSA E-mail: [email protected]

Subjects: Advanced Mathematics; Foundations & Theorems; Mathematics & Statistics; Science

Reviewing editor: Hari M. Srivastava, University of Victoria, Canada

AMS Subject classification: 54A40

Additional information is available at the end of the article

Kubiak (1985) and Sǒstak (1985) introduced the fundamental concept of a fuzzy topological structure as an extension of both crisp topology and fuzzy topology Chang (1968), in the sense that both objects and axioms are fuzzified and we may say they began the graded fuzzy topology. Bayoumi and Ibedou (2001, 2002, 2002b, 2004) introduced and studied the separation axioms in the fuzzy case in Chang’s topology (1968) using the notion of fuzzy filter defined by Gähler (1995a,1995b).

Keywords: neighbourhood systems; fuzzy filters; fuzzy neighbourhood filters; fuzzy topological spaces; separation axioms

1. Introduction

Now, we will try to investigate fuzzy topological spaces in sense of Sǒ stak, not using fuzzy filters but starting from a neighbourhood system of graded fuzzy neighbourhoods at ordinary points and usual sets. From that neighbourhood system, we can build a fuzzy topology in sense of Sǒ stak and moreover, this fuzzy topology is itself the fuzzy topology in sense of Chang associated with the fuzzy neighbourhoterest Satementod filter x (Gähler, 1995b) at ordinary point x ∈ X defined by Gähler. Interior operator and closure operator are defined using these graded fuzzy neighbourhoods; also

Ismail Ibedou

ABOUT THE AUTHOR

PUBLIC INTEREST STATEMENT

My research interests are: Fuzzy Topology, Fuzzy Topological Groups, Fuzzy Sets, Soft Sets, Soft Topological Spaces, and their applications. My research in before was concerning separation axioms in Chang’s Fuzzy Topology, and its relations with Fuzzy Compactness, Fuzzy Proximity, Fuzzy Uniformities and other types of Fuzzy separation axioms. Also, a wide research was done for Fuzzy Topological Groups and studying its uniformizability and metrizability. These separation axioms in Fuzzy Bitopological spaces are introduced. My research was mainly done with Professor Fatma Bayoumi, [email protected] yahoo.com. In this paper a continuation to my research dealing with the graded fuzzy separation axioms. The start step is defining a fuzzy neighbourhood system of fuzzy graded neighbourhoods at ordinary points and usual subsets.

Separation axioms depend on the concept of neighbourhoods and so, for the fuzzy case, fuzzy neighbourhoods or valued fuzzy neighbourhoods means neighbourhoods with some degree in [0, 1] . These grades to be a fuzzy neighbourhood forced the fuzzy separation axioms to be graded. In the fuzzy case, separation axioms are not sharp concepts. For example, there is no T0 topological space, but there are (𝛼, 𝛽) − T0 topological spaces depending on the existence of the fuzzy neighbourhood with grade 𝛼 at a point or the existence of the fuzzy neighbourhood with grade 𝛽 at the other distinct point. In this paper, I introduced these graded fuzzy separation axioms. The main section was for defining a fuzzy neighbourhood system of fuzzy graded neighbourhoods at ordinary points and usual subsets.

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

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their associated fuzzy topologies coincide with this fuzzy topology in sense of Chang associated with the fuzzy neighbourhood filter of Gähler. Fuzzy continuous, fuzzy open and fuzzy closed mappings are defined with grades according to these graded fuzzy neighbourhoods. Separation axioms in the fuzzy case are introduced based on these graded fuzzy neighbourhoods and thus, axioms are graded. These axioms satisfy common results and implications. These graded axioms are a good extension in sense of Lowen (1978). In Fuzzy neuro systems for machine learning for large data sets (2009) and DCPE Co-Training for Classification (2012), there are some applications based on fuzzy sets.

2. Preliminaries Throughout the paper, let I0 = (0, 1] and I1 = [0, 1). X

A fuzzy topology 𝜏:I → I is defined by Kubiak (1985) and Sǒstak (1985):

𝜏 (0) = 𝜏(1) = 1, (1) X

𝜏 (f ∧ g) ≥ 𝜏(f ) ∧ 𝜏(g) for all f , g ∈ I , (2) 𝜏( (3)

⋁ j∈J

𝜇j ) ≥

⋀ j∈J

𝜏(𝜇j ) for any family of (𝜇j )j∈J ∈ IX.Let 𝜏1 and 𝜏2 be fuzzy topologies on X. Then, 𝜏1 X

is finer than 𝜏2 (𝜏2, which is coarser than 𝜏1), denoted by 𝜏1 ≤ 𝜏2, if 𝜏2 (𝜇) ≤ 𝜏1 (𝜇) for all 𝜇 ∈ I . X

For each fuzzy set f ∈ I , the weak 𝛼 cut-off f is given by w𝛼 f = {x ∈ X ∣ f (x) ≥ 𝛼}; the strong 𝛼 cut-off f is the subset of X, s𝛼 f = {x ∈ X ∣ f (x) > 𝛼}. If T is an ordinary topology on X, then the induced fuzzy topology on X is given by

𝜔(T) = {f ∈ IX ∣ s𝛼 f ∈ T for all 𝛼 ∈ I1 }. fuzzy filters. Let X be a non- empty set. A fuzzy filter on X (Eklund, 1992; Gähler, 1995a) is a mapX ping :I → I such that the following conditions are fulfilled:

(𝛼) ≤ 𝛼 holds for all 𝛼 ∈ I and (1) = 1; (F1) X

(f ∧ g) = (f ) ∧ (g) for all f , g ∈ I . (F2) If and are fuzzy filters on X, the.n is said to be finer than , denoted by ≤ , proX vided that (f ) ≥ (f ) for every f ∈ I . By ≰ we mean that is not finer than . X ≰ ⟺ there is f ∈ I such that (f ) < (f ). X

A non-empty subset of I is called a prefilter on X (Lowen, Lowen), provided that the following conditions are fulfilled:

0 ∉ ; (1) f , g ∈ implies f ∧ g ∈ ; (2) f ∈ and f ≤ g imply g ∈ . (3) For each fuzzy filter on X, the subset

𝛼-pr of IX defined by: 𝛼-pr = {f ∈ IX ∣ (f ) ≥ 𝛼} is a prefilter on X. Proposition 2.1 (Gähler, 1995a) There is a one-to-one correspondence between fuzzy filters on X Page 2 of 13

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and the families (𝛼 )𝛼∈I of prefilters on X which fulfill the following conditions: 0

f ∈ 𝛼 implies 𝛼 ≤ supf . (1) 0 < 𝛼 ≤ 𝛽 implies 𝛼 ⊇ 𝛽. (2) ⋂ ⋁ 𝛽 = 𝛼, we have 𝛽 = 𝛼.This correspondence is given by 𝛼 = (3) For each 𝛼 ∈ I0 with 0<𝛽<𝛼

0<𝛽<𝛼

𝛼-pr for all 𝛼 ∈ I0 and (f ) =

⋁

𝛼 for all f ∈ IX.

g∈𝛼 , g≤f

Proposition 2.2 (Eklund, 1992) Let A be a set of fuzzy filters on X. Then, the following are equivalent. (1) The infimum

⋀

of A with respect to the finer relation of fuzzy filters exists,

∈A

{1 , … , n } (2) For each non-empty finite subset 1 (f1 ) ∧ ⋯ ∧ n (fn ) ≤ sup(f1 ∧ ⋯ ∧ fn ) for all f1 , … , fn ∈ IX,

(3) For each 𝛼 ∈ I0 and each non-empty finite subset f1 , … , fn of 𝛼 ≤ sup(f1 ∧ ⋯ ∧ fn ). Recall that (

⋀

)(f ) =

∈A

⋁

(f ) and (

∈A

⋁

)(f ) =

∈A

⋀

A,

of ⋃

we

have

𝛼-pr , we have

∈A

(f ) for all f ∈ IX.

∈A

Fuzzy neighbourhood filters. For each fuzzy topological space (X, 𝜏) and each x ∈ X , the mapping

x : IX → I defined by (Gähler, Gahler): x (𝜆) = int𝜏 𝜆(x) for all 𝜆 ∈ IX is a fuzzy filter on X, called the fuzzy neighbourhood filter of the space (X, 𝜏) at the point x, and for ̇ X → I is defined by x(𝜆) ̇ short is called a fuzzy neighbourhood filter at x. The mapping x:I = 𝜆(x) for X all 𝜆 ∈ I . The fuzzy neighbourhood filters fulfil the following conditions: (1) ẋ ≤ x holds for all x ∈ X ; X

(x )(int𝜏 f ) = (x )(f ) for all x ∈ X and f ∈ I . (2) A fuzzy filter is said to converge to x ∈ X , denoted by → x, if ≤ x (Gähler, 1995b). 𝜏

The fuzzy neighbourhood filter F at an ordinary subset F of X is the fuzzy filter on X defined in Bayoumi and Ibedou (2002b), by means of x, x ∈ F as:

F =

⋁

x .

x∈F

The fuzzy filter Ḟ is defined by

Ḟ =

⋁

̇ x.

x∈F

Ḟ ≤ F holds for all F ⊆ X . Also, recall that the fuzzy filter 𝜆̇ and the fuzzy neighbourhood filter 𝜆 at a fuzzy subset 𝜆 of X are defined by ⋁ ⋁ x , 𝜆̇ = ẋ and 𝜆 = (1) 0<𝜆(x)

0<𝜆(x)

X respectively. 𝜆̇ ≤ 𝜆 holds for all 𝜆 ∈ I (Bayoumi & Ibedou, 2004).

For each fuzzy topological space (X, 𝜏) the closure operator cl which assigns to each fuzzy filter on X, the fuzzy filter cl is defined by

cl (f ) =

⋁ cl𝜏 g≤f

(g).

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cl is called the closure of . cl is isotone, hull and idempotent operator, that is for all fuzzy filters and on X, we have (Gähler, 1995b): ≤ implies cl ≤ cl ,

(3)

≤ cl,

(4)

3. Neighbourhood systems Definition 3.1 A family (x𝛼 )x∈X of fuzzy sets x𝛼 is said to be a neighbourhood system with grade 𝛼 ∈ I0 on X if it satisfies the following conditions: (Nb1) For all f ∈ x𝛼, we have 𝛼 ≤ f (x), (Nb2) 1 ∈ x𝛼, (Nb3) f , g ∈ x𝛼 implies that f ∧ g ∈ x𝛼, (Nb4) f ∈ x𝛼, f ≤ g imply that g ∈ x𝛼, (Nb5) If f ∈ x𝛼, then there is g ∈ x𝛼, such that for all y ∈ X with 0 < g(y), we have f ∈ y𝛼. Lemma 3.1 These families of prefilters (x𝛼 )𝛼∈I at x ∈ X satisfy the following conditions: 0

f ∈ (Pr1)

x𝛼

implies that 𝛼 ≤ supf ,

0 < 𝛽 ≤ 𝛼 implies that x𝛼 ⊆ x𝛽, (Pr2) ⋂ ⋁ 𝛽 = 𝛼, we have x𝛽 = x𝛼. (Pr3) For every 𝛼 ∈ I0 with 0<𝛽<𝛼

0<𝛽<𝛼

Proof Clear. □ Remark 3.1 For any subset A of X, let us define A𝛼 by A𝛼 =

⋂

x∈A

𝛼 , x𝛼 ∩ x𝛽 = x𝛼∨𝛽, x𝛼 ∪ x𝛽 = x𝛼∧𝛽, iff 𝛼 ≤ A (f ). x𝛼 = {x}

x𝛼, that is f ∈ A𝛼 iff 𝛼 ≤ ⋂ j

𝛼 x j

= x

⋁ j

𝛼j

,

⋃ j

𝛼 x j

⋀

x∈A

x (f )

= x

⋀ j

𝛼j

. For all 𝛼 ≠ 𝛽 in I0, we have x𝛼 ≠ x𝛽. For any 𝛼, 𝛽, 𝛾 ∈ I0, we have x𝛼 ⊆ x𝛼, x𝛼 ⊆ x𝛽 and x𝛽 ⊆ x𝛼 implies that 𝛼 = 𝛽, x𝛼 ⊆ x𝛽 and x𝛽 ⊆ x𝛾 implies that x𝛼 ⊆ x𝛾. Also, for all 𝛼 ≠ 𝛽 ∈ I0, we have either x𝛼 ⊆ x𝛽 or x𝛽 ⊆ x𝛼.

(𝛼) fuzzy open sets, fuzzy open sets. Let us define an (𝛼) fuzzy open set as follows:

𝛼 ≤ 𝜏(f ) 𝗂𝖿 𝖿 for all x ∈ X there is 𝛼 ∈ I0 such that f ∈ x𝛼 and f (x) ≤ 𝛼.

(5)

An (𝛼) fuzzy closed set is the complement of an (𝛼) fuzzy open set. X

A set f ∈ I is said to be fuzzy open if it is (𝛼) fuzzy open for all 𝛼 ∈ I0. In other words, if for all x ∈ X and for all 𝛼 ∈ I0, we have f ∈ x𝛼 and f (x) ≤ 𝛼. It is called a fuzzy closed if it is the complement of a fuzzy open set. These notations are restricted to the usual open and closed sets in fuzzy topology and usual topology. 𝛼

Starting from a neighbourhood system (x )x∈X with grade 𝛼 ∈ I0, we can define an interior operator and a closure operator as follows:

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⋁

intf (x) =

⋀

(6)

f (y),

g∈x𝛼 0

clf (x) =

⋀

⋁

f (y).

(7)

g∈x𝛼 0

For every x ∈ X , x satisfying (Nb1) to (Nb4) is exactly a prefilter on X of all neighbourhoods of x ∈ X with grade 𝛼 ∈ I0. That is, (x𝛼 )x∈X is a family of prefilters with grade 𝛼 ∈ I0 at every x ∈ X constructing after adding condition (Nb5) a neighbourhood system on X with grade 𝛼 ∈ I0. The pair (X, (x𝛼 )x∈X ) is called a neighbourhood space with a grade 𝛼 ∈ I0. 𝛼

From Lemma 3.1 and from the correspondence given in Proposition 2.1 between the fuzzy filters 𝛼 and the families satisfying the conditions (Pr1) to (Pr3), we can say this family (x )𝛼∈I is a represen0 tation of the fuzzy neighbourhood filter x as a family of prefilters. This is given by the following two conditions (Nb) and (Pr):

x (f ) = (Pr)

⋁

𝛼 for all f ∈ IX.

g∈x𝛼 , g≤f X

X

x = {f ∈ I ∣ 𝛼 ≤ x (f )}.Denote the subset x ⊆ I as the fuzzy neighbourhoods with (Nb) grade 𝛼 ∈ I0 of x ∈ X . 𝛼

𝛼

Clearly, both the interior operator and closure operator satisfy the common axioms of interior operator and closure operator, respectively. A fuzzy topology on X could be generated by this interior 𝛼 operator given by (2.2) or this closure operator given by (2.3), using the properties of x stated in (Nb1)—(Nb5). That fuzzy topology is exactly the fuzzy topology 𝜏 associated with the fuzzy neighbourhood filters x given by an interior operation as in (2.2) so that

x (f ) = int𝜏 f (x) for all f ∈ IX . Also, we can consider (8)

x𝛼 = {f ∈ IX ∣ 𝛼 ≤ int𝜏 f (x)} and then, (2.1) for an (𝛼) fuzzy open set could be rewritten as

𝛼 ≤ 𝜏(f ) 𝗂𝖿 𝖿 for all x ∈ X, there is 𝛼 ∈ I0 so that 𝛼 ≤ int𝜏 f (x) , f (x) ≤ 𝛼.

(9)

That is, from a neighbourhood system of graded neighbourhoods, we can deduce interior operation by which it is introduced a graded fuzzy topology and the converse is true. From (1.2) and (1.4) for all x ∈ X and all 𝛼 ∈ I0, we can define clx by 𝛼

clx𝛼 = {f ∈ IX ∣ 𝛼 ≤ clx (f )},

(10)

and equivalently,

clx𝛼 = {f ∈ IX ∣ there is h ∈ x𝛼 , clh ≤ f }.

(11)

For all x ∈ X and all 𝛼 ∈ I0, we have clx ⊆ x . 𝛼

𝛼

Definition 3.2 Let (X, 𝜏1 ) and (Y, 𝜏2 ) be fuzzy topological spaces, and f :X → Y a map. Then, for some 𝛼 ∈ I0, f is called (𝛼) fuzzy continuous if for all (𝛼) fuzzy open set 𝜇 with respect to 𝜏2, we have f −1 (𝜇) is an (𝛼) fuzzy open set with respect to 𝜏1 for all 𝜇 ∈ IY .

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f is called fuzzy continuous if for all fuzzy open set 𝜇 with respect to 𝜏2, we have f −1 (𝜇) is a fuzzy open set with respect to 𝜏1 for all 𝜇 ∈ IY . Definition 3.3 Let (X, 𝜏1 ) and (Y, 𝜏2 ) be fuzzy topological spaces. Then, the mapping f :(X, 𝜏1 ) → (Y, 𝜏2 ) is called (𝛼) fuzzy open ((𝛼) fuzzy closed) mapping if the image f(g) of the (𝛼) fuzzy open ((𝛼) fuzzy closed) set g with respect to 𝜏1 is (𝛼) fuzzy open ((𝛼) fuzzy closed) set with respect to 𝜏2. The mapping f :(X, 𝜏1 ) → (Y, 𝜏2 ) is called fuzzy open (fuzzy closed) mapping if the image f(g) of the fuzzy open (fuzzy closed) set g with respect to 𝜏1 is fuzzy open (fuzzy closed) set with respect to 𝜏2. Now, we define the continuity locally at a point x0 ∈ X between two fuzzy topological spaces using these graded neighbourhoods. Definition 3.4 Let (X, 𝜏) and (Y, 𝜎) be two fuzzy topological spaces. Then, the mapping f : (X, 𝜏) → (Y, 𝜎) is called (𝛼) fuzzy continuous at a point x0 provided that for all g ∈ f𝛼(x ), 0

there exists h ∈ x𝛼 such that h ≤ f −1 (g) for some 𝛼 ∈ I0. f is (𝛼) fuzzy continuous if it is (𝛼) fuzzy con0 tinuous at every x ∈ X . f is an fuzzy continuous if it is (𝛼) fuzzy continuous for all 𝛼 ∈ I0. This is an equivalent definition with Definition 3.2 for the (𝛼) fuzzy continuous mapping and fuzzy continuous mapping.

4. (𝜶, 𝜷)T0-spaces and (𝜶, 𝜷)T1-spaces

This section is devoted to introduce the notions of T0-spaces and T1-spaces using the notion of 𝛼-neighbourhoods at ordinary points. We will introduce different equivalent definitions, and we show that these notions are good extensions in sense of Lowen (1978]). Definition 4.1 A fuzzy topological space (X, 𝜏) is called an (𝛼, 𝛽)T0-space if for all x ≠ y in X, there exists f ∈ x𝛼 such that f (y) < 𝛼; 𝛼 ∈ I0 or there exists g ∈ y𝛽 such that g(x) < 𝛽; 𝛽 ∈ I0. Definition 4.2 A fuzzy topological space (X, 𝜏) is called an (𝛼, 𝛽)T1-space if for all x ≠ y in X there exist f ∈ x𝛼 and g ∈ y𝛽 such that f (y) < 𝛼 and g(x) < 𝛽; 𝛼, 𝛽 ∈ I0. Example 4.1 Let X = {x, y}, and ⎧ 1 ⎪ 𝜏(f ) = ⎨ 13 ⎪ ⎩ 0

at 0 or 1 at x 1 2

otherwise.

Taking 𝛼 = 13, we get that there is f = x 1 in x𝛼 such that f (y) < 𝛼. For all 𝛼 ∈ I0, we can not find any f 2 in y𝛼 such that f (x) < 𝛼. That is, (X, 𝜏) is an (𝛼, 𝛽)T0-space. Example 4.2 Let X = {x, y}, and

𝜏(f ) =

{

1

at 0 or 1

0

otherwise.

Only there is f = 1 which is a graded neighbourhood but for both of x, y. Hence, for all 𝛼 ∈ I0, x𝛼 = y𝛼 and therefore, (X, 𝜏) is not (𝛼, 𝛽)T0-space. Proposition 4.1 Every (𝛼, 𝛽)T1-space is an (𝛼, 𝛽)T0-space. Proof Clear. □

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Example 4 is an (𝛼, 𝛽)T0-space but not (𝛼, 𝛽)T1-space. Example 4.3 Let X = {x, y}, and Taking 𝛼 = 13 and 𝛽 = 13, we get that there is f = x 1 in x𝛼 and g = y 1 in y𝛽 such that f (y) < 𝛼 and 2 2 g(x) < 𝛽, for some 𝛼, 𝛽 ∈ I0. Hence, (X, 𝜏) is an (𝛼, 𝛽)T1-space. ⎧ 1 ⎪ 1 ⎪ 𝜏(f ) = ⎨ 31 ⎪ 3 ⎪ ⎩ 0

at 0 or 1 at x 1 2

at y 1 2

otherwise.

In the following theorems, there will be introduced some equivalent definitions for the (𝛼, 𝛽)T0-spaces and (𝛼, 𝛽)T1-spaces.

Theorem 4.1 Let (X, 𝜏) be a fuzzy topological space. Then, the following statements are equivalent. (X, 𝜏) is (𝛼, 𝛽)T0. (1)

(2) For all x ≠ y in X and for all 𝛼 ∈ I0, x𝛼 ≠ y𝛼. (3) For all x ≠ y in X, there exists f ∈ IX such that f (y) < 𝛼 ≤ int𝜏 f (x); 𝛼 ∈ I0 or there exists g ∈ IX such that g(x) < 𝛽 ≤ int𝜏 g(y); 𝛽 ∈ I0. (4) For all x ≠ y in X, there exists f ∈ IX such that f (y) > cl𝜏 f (x) or there exists g ∈ IX such that g(x) > cl𝜏 g(y). Proof (1) ⇒ (2): From (1), there is f ∈ IX such that int𝜏 f (y) ≤ f (y) < 𝛼 ≤ int𝜏 f (x); 𝛼 ∈ I0 and then, f ∈ x𝛼 and f ∉ y𝛼. Hence, x𝛼 ≠ y𝛼; 𝛼 ∈ I0 and thus, (2) holds. (2) ⇒ (3): There exists f ∈ IX such that int𝜏 f (y) < 𝛼 ≤ int𝜏 f (x); 𝛼 ∈ I0 and then, for g = int𝜏 f , we can say g(y) < 𝛼 ≤ int𝜏 g(x); 𝛼 ∈ I0. The other case is similar and thus, (3) is satisfied. (3) ⇒ (4): From Equation 7, we get that cl𝜏 f (x) < f (y) for all int𝜏 f (y) ≥ 𝛼 > f (x), then (4) holds. (4) ⇒ (1): Since f (y) <

⋀

⋁

h∈x𝛼 0

f (z) = cl𝜏 f (x) implies that z could not be y with 0 < h(y) for all h ∈ x𝛼;

𝛼 ∈ I0, which means that there is h ∈ x𝛼 such that h(y) = 0 < 𝛼 ≤ int𝜏 h(x); 𝛼 ∈ I0. The other case is simi-

lar and thus, (1) holds.

□

Theorem 4.2 Let (X, 𝜏) be a fuzzy topological space. Then, the following statements are equivalent. (X, 𝜏) is (𝛼, 𝛽)T1. (1)

(2) For all x ∈ X , we have cl𝜏 x1 = x1. (3) For all x ≠ y in X, there exist f , g ∈ IX such that f (y) < 𝛼 ≤ int𝜏 f (x) and g(x) < 𝛽 ≤ int𝜏 g(y); 𝛼, 𝛽 ∈ I0. (4) For all x ≠ y in X, there exist f , g ∈ IX such that f (y) > cl𝜏 f (x) and g(x) > cl𝜏 g(y). Proof (1) ⇒ (2): Let y ≠ x in X. Then, cl𝜏 x1 (y) =

⋀

⋁

h∈y𝛼 0

x1 (z), which means for all h ∈ y𝛼, if

x1 (z) > 0 whenever h(z) > 0, then cl𝜏 x1 (y) > 0. From (1), we get that z could not be x with 0 < h(x),

that is, cl𝜏 x1 (y) = 0 for all y ≠ x. At x, it is clear that cl𝜏 x1 (x) = 1. Hence, cl𝜏 x1 = x1 for all x ∈ X , and (2) is fulfilled.

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(2) ⇒ (3):

For

all

cl𝜏 x1 (y) = x1 (y) = 0 =

x ≠ y in ⋀ ⋁ h∈y𝛼 0

X,

we

have

cl𝜏 x1 = x1

x1 (z), which means for all h ∈

and y𝛼,

cl𝜏 y1 = y1.

(2)

means

that

z could not be x with 0 < h(x), that

is there is 𝛼 ∈ I0 and there is h ∈ y𝛼 such that h(x) = 0 < 𝛼 and then, h(x) < 𝛼 ≤ int𝜏 h(y). The other case is similar and therefore, (3) is fulfilled. (3) ⇒ (4): As in Theorem 4.1. (4) ⇒ (1): As in Theorem 4.1.

□

The next proposition shows that the separation axioms (𝛼, 𝛽)T0 and (𝛼, 𝛽)T1 are good extensions in sense of Lowen (1978). Proposition 4.2 A topological space (X, T) is a T0-space (T1-space) if and only if the induced fuzzy topological space (X, 𝜔(T)) is an (𝛼, 𝛽)T0-space ((𝛼, 𝛽)T1-space). Proof Let (X, T) be T0 (T1) and let x ≠ y . Then, there is a neighbourhood y ∈ T such that x ∉ y. Taking f ∈ IX such that y = s𝛼 f ∈ T for some 𝛼 ∈ I1, we get f (x) ≤ 𝛼 < int𝜔(T) f (y), That is, f (x) < 𝛼 ≤ int𝜔(T) f (y)

for some 𝛼 ∈ I0. Similarly, if there is a neighbourhood x ∈ T such that y ∉ x, we can find g ∈ IX such that x = s𝛽 g ∈ T and g(y) ≤ 𝛽 < int𝜔(T) g(x) for some 𝛽 ∈ I1, That is, g(y) < 𝛽 ≤ int𝜔(T) g(x) for some 𝛽 ∈ I0. Hence, (X, 𝜔(T)) is an (𝛼, 𝛽)T0-space ((𝛼, 𝛽)T1).

Conversely, let (X, 𝜔(T)) be an (𝛼, 𝛽)T0-space ((𝛼, 𝛽)T1) and x ≠ y . Then, there exists f ∈ IX such that f (y) < 𝛼 ≤ int𝜔(T) f (x) for some 𝛼 ∈ I0, which means f (y) ≤ 𝛼 < int𝜔(T) f (x) for some 𝛼 ∈ I1, that is there is int𝜔(T) f ∈ IX such that s𝛼 int𝜔(T) f = x ∈ T and y ∉ x. Similarly, the other case is proved. Hence, (X, T) □ is a T0-space (T1). Proposition 4.3 Let (X, 𝜏) be an (𝛼, 𝛽)T0-space ((𝛼, 𝛽)T1) and let 𝜎 be a fuzzy topology on X finer than 𝜏 . Then, (X, 𝜎) is also (𝛼, 𝛽)T0-space ((𝛼, 𝛽)T1-space). Proof (X, 𝜏) is an (𝛼, 𝛽)T0-space ((𝛼, 𝛽)T1) implying that there is f ∈ IX such that 𝛼 ≤ int𝜏 f (x) and f (x) < 𝛼 or (and) there is g ∈ IX such that 𝛼 ≤ int𝜏 g(x) and g(x) < 𝛼, which implies that 𝛼 ≤ 𝜏(f ) or

(and)𝛼 ≤ 𝜏(g). Since 𝜎 is finer than 𝜏 , then 𝛼 ≤ 𝜎(f ) or (and)𝛼 ≤ 𝜎(g), and thus, 𝛼 ≤ int𝜎 f (x) and f (x) < 𝛼 or (and) 𝛼 ≤ int𝜎 g(x) and g(x) < 𝛼. Hence (X, 𝜎) is an (𝛼, 𝛽)T0-space ((𝛼, 𝛽)T1).

□

5. (𝜶, 𝜷)T2-spaces

Here, we introduce and study the Hausdorff separation axiom in fuzzy topological spaces. Definition 5.1 An fuzzy topological space (X, 𝜏) is called an (𝛼, 𝛽)T2-space if for all x ≠ y in X there exist f ∈ x𝛼 and g ∈ y𝛽 such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0. Proposition 5.1 Every (𝛼, 𝛽)T2-space is an (𝛼, 𝛽)T1-space. Proof Let (X, 𝜏) be an (𝛼, 𝛽)T2-space but not (𝛼, 𝛽)T1-space. That is, for x ≠ y , we get for all f ∈ x𝛼 that f (y) ≥ 𝛼 for all 𝛼 ∈ I0. Since for any g ∈ y𝛽 we have g(y) ≥ 𝛽, then (f ∧ g)(y) = f (y) ∧ g(y) ≥ (𝛼 ∧ 𝛽) and thus, sup(f ∧ g) ≥ (𝛼 ∧ 𝛽) which contradicts the axiom (𝛼, 𝛽)T2. Hence, (X, 𝜏) is an (𝛼, 𝛽)T1-space. □

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Example 5.1 Let X = {x, y}, and There are f = x1 ∈ IX and g = x 1 ∨ y1 ∈ IX such that, for 𝛼 = 15 and 𝛽 = 45 in I0, we get that f = x1 ∈ x𝛼 3 and g = x 1 ∨ y1 ∈ y𝛽 such that f (y) = x1 (y) = 0 < 15 = 𝛼 and g(x) = (x 1 ∨ y1 )(x) = 31 < 45 = 𝛽. That is, 3 3 (X, 𝜏) is an (𝛼, 𝛽)T1-space. But for all fuzzy sets f ∈ x𝛼 and g ∈ y𝛽, we get that (𝛼 ∧ 𝛽) ≤ sup(f ∧ g) and thus, (X, 𝜏) is not (𝛼, 𝛽)T2-space.

⎧ 1 ⎪ 1 ⎪ 𝜏(f ) = ⎨ 45 ⎪ 5 ⎪ ⎩ 0

at 0 or 1 at x1 at x 1 ∨ y1 3

otherwise.

Theorem 5.1 Let (X, 𝜏) be an fuzzy topological space. Then, the following statements are equivalent. (X, 𝜏) is (𝛼, 𝛽)T2. (1)

(2) For all fuzzy ultrafilter on X and for all x ≠ y , there is f ∈ x𝛼 such that (f ) < 𝛼; 𝛼 ∈ I0 or there is g ∈ y𝛽 such that (g) < 𝛽; 𝛽 ∈ I0. (3) For all fuzzy filter on X and for all x ≠ y , there is f ∈ x𝛼 such that (f ) < 𝛼; 𝛼 ∈ I0 or there is g ∈ y𝛽 such that (g) < 𝛽; 𝛽 ∈ I0. Proof (1) ⇒ (2): Suppose that there is an fuzzy ultrafilter on X such that (f ) ≥ 𝛼 and (g) ≥ 𝛽 for all f ∈ x𝛼 and g ∈ y𝛽. That is, (f ∧ g) = (f ) ∧ (g) ≥ 𝛼 ∧ 𝛽, but in common we know that (h) ≤ suph for all h ∈ IX, which means that for all f ∈ x𝛼 and g ∈ y𝛼, we have sup(f ∧ g) ≥ (𝛼 ∧ 𝛽) and therefore, (1) implies (2) is satisfied. (2) ⇒ (3): Since for any fuzzy filter on X we find a finer fuzzy ultrafilter on X, that is (f ) ≤ (f ) for all f ∈ IX, then (2) implies that there is f ∈ x𝛼 such that (f ) ≤ (f ) < 𝛼; 𝛼 ∈ I0 or there is g ∈ y𝛽 such that (g) ≤ (g) < 𝛽; 𝛽 ∈ I0. Thus, (3) holds. (3) ⇒ (1): Suppose for all f ∈ x𝛼 and g ∈ y𝛽; 𝛼, 𝛽 ∈ I0 that (𝛼 ∧ 𝛽) ≤ sup(f ∧ g) and (3) is fulfilled. Then, for all fuzzy filter on X, we have (f ) < 𝛼 or (g) < 𝛽; 𝛼, 𝛽 ∈ I0. Hence, (f ∧ g) < (𝛼 ∧ 𝛽) ≤ sup(f ∧ g), which means a contradiction to the common result that □ (f ∧ g) ≤ sup(f ∧ g) and therefore, (f ∧ g) ≤ sup(f ∧ g) < (𝛼 ∧ 𝛽). Thus, (1) is satisfied.

Example 5.2 Let X = {x, y}, and ⎧ 1 ⎪ 1 ⎪ 𝜏(f ) = ⎨ 41 ⎪ 4 ⎪ ⎩ 0

at 0 or 1 at x 1 3

at y 1 3

otherwise.

There are f = x 1 ∈ IX and g = y 1 ∈ IX such that for 𝛼 = g = y 1 in 3

3

y𝛽

such that (𝛼 ∧ 𝛽) =

3

1 4

1 4

and 𝛽 =

1 4

in I0, we get that f = x 1 in x𝛼 and

> sup(x 1 ∧ y 1 ) = 0 and thus, (X, 𝜏) is an (𝛼, 𝛽)T2-space. 3

3

3

Proposition 5.1 A topological space (X, T) is a T2-space if and only if the induced fuzzy topological space (X, 𝜔(T)) is an (𝛼, 𝛽)T2-space. Proof Let x ≠ y in X. Then, there are x , y ∈ T such that x ∩ y = �. Taking f , g ∈ IX int𝜔(T) f (x) > 𝛼 and int𝜔(T) g(y) > 𝛽; such that s𝛼 f = x , s𝛽 g = y for some 𝛼, 𝛽 ∈ I1, then 𝛼, 𝛽 ∈ I1, that is int𝜔(T) f (x) ≥ 𝛼 and int𝜔(T) g(y) ≥ 𝛽; 𝛼, 𝛽 ∈ I0 and then, f ∈ x𝛼 and g ∈ y𝛽 such that s𝛼 f ∩ s𝛽 g = x ∩ y = �, which means that there is no element z ∈ X such that Page 9 of 13

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(f ∧ g)(z) = f (z) ∧ g(z) ≥ int𝜔(T) f (z) ∧ int𝜔(T) g (z) > (𝛼 ∧ 𝛽); 𝛼, 𝛽 ∈ I1, which means for all z ∈ X , we have (f ∧ g)(z) ≤ (𝛼 ∧ 𝛽); 𝛼, 𝛽 ∈ I1. Hence, sup(f ∧ g) ≤ (𝛼 ∧ 𝛽); 𝛼, 𝛽 ∈ I1 and then, sup(f ∧ g) < (𝛼 ∧ 𝛽); 𝛼, 𝛽 ∈ I0 and thus, (X, 𝜔(T)) is an (𝛼, 𝛽)T2-space. f ∈ x𝛼 implies that there are and such that x≠y g ∈ y𝛽 Conversely, int𝜔(T) f (x) ∧ int𝜔(T) g (y) ≥ (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0. That is, for 𝛾 = sup(f ∧ g) ∈ I1, we can say int𝜔(T) f ∈ 𝜔(T), x ∈ s𝛾 int𝜔(T) f and int𝜔(T) g ∈ 𝜔(T), y ∈ s𝛾 int𝜔(T) g, which means that s𝛾 int𝜔(T) f = x ∈ T, s𝛾 int𝜔(T) g = y ∈ T and moreover, x ∩ y = � and thus, (X, T) is a T2-space. (because if there is z ∈ (x ∩ y ), □ then (f ∧ g)(z) ≥ int𝜔(T) f (z) ∧ int𝜔(T) g (z) > 𝛾 = sup(f ∧ g) which is a contradiction).

Proposition 5.2 Let (X, 𝜏) be an (𝛼, 𝛽) T2-space, and let 𝜎 be an fuzzy topology on X finer than 𝜏 . Then, (X, 𝜎) is also an (𝛼, 𝛽)T2-space. Proof Let x ≠ y ∈ X . Then, there are f ∈ x𝛼 and g ∈ y𝛽 such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0, that is 𝛼 ≤ int𝜏 f (x), 𝛽 ≤ int𝜏 g (y) and (𝛼 ∧ 𝛽) > sup(f ∧ g), which means that 𝛼 ≤ int𝜎 f (x), 𝛽 ≤ int𝜎 g (y) and (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0 and thus, f ∈ x𝛼 and g ∈ y𝛽 in (X, 𝜎) such that (𝛼 ∧ 𝛽) > sup(f ∧ g); □ 𝛼, 𝛽 ∈ I0. Hence, (X, 𝜎) is an (𝛼, 𝛽)T2-space.

6. (𝜶, 𝜷)T3-spaces and (𝜶, 𝜷)T4-spaces

In this section, we use fuzzy neighbourhood filters at ordinary sets to define the notions of (𝛼, 𝛽)T3 -spaces and (𝛼, 𝛽)T4-spaces. Definition 6.1 A fuzzy topological space (X, 𝜏) is called (𝛼, 𝛽) regular if for all F = cl𝜏 F in P(X) and x ∉ F, there exist f ∈ x𝛼 and g ∈ F𝛽 such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0. Definition 6.2 A fuzzy topological space (X, 𝜏) is called (𝛼, 𝛽)T3-space if it is regular and (𝛼, 𝛽)T1. Definition 6.3 A fuzzy topological space (X, 𝜏) is called normal if for all F1 = cl𝜏 F1 , F2 = cl𝜏 F2 ∈ P(X) with F1 ∩ F2 = �, there exist f ∈ F𝛼 and g ∈ F𝛽 such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼 ∧ 𝛽 ∈ I0. 1

2

Definition 6.4 A fuzzy topological space (X, 𝜏) is called (𝛼, 𝛽)T4 if it is normal and (𝛼, 𝛽)T1. Proposition 6.1 Every (𝛼, 𝛽)T3-space is an (𝛼, 𝛽)T2-space. Proof Let x ≠ y in X. (X, 𝜏) is an (𝛼, 𝛽)T1-space meaning that cl𝜏 {x} = {x} for each x ∈ X . Now, cl𝜏 {y} = {y}, x ∉ {y}, and (X, 𝜏) is regular implying that there are f ∈ x𝛼 , g ∈ y𝛽 such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0. Hence, (X, 𝜏) is an (𝛼, 𝛽)T2-space. □ Theorem 6.1 For each fuzzy topological space (X, 𝜏), the following are equivalent. (X, 𝜏) is regular. (1)

(2) For all y ∈ F = cl𝜏 F and x ∉ F, we have x𝛼 ⊈ cly𝛼 and y𝛽 ⊈ clx𝛽 for all y ∈ F; 𝛼, 𝛽 ∈ I0. (3) For all x ∈ X and all 𝛼 ∈ I0, we have clx𝛼 = x𝛼. (4) For all x ∈ X , for all fuzzy filter on X, for all f ∈ x𝛼, and for all 𝛼 ∈ I0, we have (f ) ≥ 𝛼 implies cl(f ) ≥ 𝛼. Proof (1) ⇒ (2): Let f ∈ x𝛼; 𝛼 ∈ I0. Suppose that f ∈ cly𝛼 for some y ∈ F, that is, there is h ∈ y𝛼 with cl𝜏 h ≤ f , which means that f (y) ≥ 𝛼. Since for all g ∈ y𝛼, we have g(y) ≥ 𝛼 for all y ∈ F; 𝛼 ∈ I0, then sup(f ∧ g) ≥ (f ∧ g)(y) ≥ 𝛼 = (𝛼 ∧ 𝛼) for some f ∈ x𝛼 for all x ∉ F, and for all g ∈ y𝛼 for some y ∈ F; 𝛼 ∈ I0, which contradicts (1) and therefore, f ∉ cly𝛼 for all y ∈ F. Thus, x𝛼 ⊈ cly𝛼 for all y ∈ F. The other case is similar and hence, (2) is satisfied.

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(2) ⇒ (3): From (2) we deduce that for all f ∈ x𝛼 andg ∈ y𝛽, we have f ∈ cly𝛼 or g ∈ clx𝛽 for all 𝛼, 𝛽 ∈ I0 implies x = y . Hence, for all f ∈ x𝛼, x ∈ X , and all 𝛼 ∈ I0, we get that f ∈ clx𝛼, which means that x𝛼 ⊆ clx𝛼, but from that clx𝛼 ⊆ x𝛼 for all 𝛼 ∈ I0 and for all x ∈ X , we get that clx𝛼 = x𝛼 for all 𝛼 ∈ I0 and for all x ∈ X and thus, (3) holds. (3) ⇒ (4): Let be a fuzzy filter on X with (f ) ≥ 𝛼 for all f ∈ x𝛼 and 𝛼 ∈ I0. From (3), (f ) ≥ 𝛼 for all f ∈ clx𝛼 and 𝛼 ∈ I0 and then, cl(f ) ≥ 𝛼 for all f ∈ x𝛼 and 𝛼 ∈ I0 and thus, (4) is fulfilled. (4) ⇒ (1): Consider = x in (4), we get that clx𝛼 = x𝛼 for all x ∈ X and all 𝛼 ∈ I0. Now, for y ∈ F = cl𝜏 F and x ≠ y, we get for all f ∈ x𝛼 and g ∈ y𝛽 that f ∈ clx𝛼 and g ∈ cly𝛽, which means there are h ∈ x𝛼 withcl𝜏 h ≤ f and k ∈ y𝛽 withcl𝜏 k ≤ g. Choose f = cl𝜏 𝜒F c ∈ x1 and g = cl𝜏 (int𝜏 𝜒F ) ∈ F1, then h = 𝜒F c ∈ x1 k = int𝜏 𝜒F ∈ F1 and such that we can find (𝛼 ∧ 𝛽) = 1 > 0 = sup(𝜒F c ∧ int𝜏 𝜒F ) = sup (h ∧ k), and thus, for all F = cl𝜏 F ∈ X a nd x ∉ F, there exist □ h ∈ x𝛼 and k ∈ F𝛽 such that (𝛼 ∧ 𝛽) > sup(h ∧ k); 𝛼, 𝛽 ∈ I0, and therefore, (1) is satisfied.

Theorem 6.2 Let (X, 𝜏) be a fuzzy topological space. Then, the following are equivalent. (X, 𝜏) is normal. (1)

(2) For all F1 = cl𝜏 F1 , F2 = cl𝜏 F2 ∈ P(X) with F1 ∩ F2 = �, we have x𝛼 ⊈ cly𝛼 and y𝛽 ⊈ clx𝛽 for all x ∈ F1 andy ∈ F2; 𝛼, 𝛽 ∈ I0. (3) For all F = cl𝜏 F ∈ P(X), and all 𝛼 ∈ I0, we have clF𝛼 = F𝛼. (4) For all F = cl𝜏 F ∈ P(X), for all fuzzy filters on X, for all f ∈ F𝛼, and for all 𝛼 ∈ I0, we have (f ) ≥ 𝛼 implies cl(f ) ≥ 𝛼. Proof Similar to the Theorem 6.1.

□

Proposition 6.2 Every (𝛼, 𝛽)T4-space is an (𝛼, 𝛽)T3-space. Proof Let x ∉ F = cl𝜏 F in X. Since (X, 𝜏) is (𝛼, 𝛽)T4, then it is (𝛼, 𝛽)T1, which means that cl𝜏 {x} = {x} for all x ∈ X , which implies that we have F1 = {x} = cl𝜏 {x} and F2 = F with F1 ∩ F2 = �. Hence, there are f ∈ x𝛼 and g ∈ F𝛽 such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0 and thus, (X, 𝜏) is regular and it is (𝛼, 𝛽)T1. □ Therefore, (X, 𝜏) is (𝛼, 𝛽)T3. Example 6.1 Let X = {x, y}, and ⎧ 1 ⎪ 1 ⎪ 𝜏(f ) = ⎨ 12 ⎪ 3 ⎪ ⎩ 0

at 0 or 1 at x1 at y 1 2

otherwise.

We notice that {y} is a closed set and x ∉ {y}. Then, there are f = x1 ∈ IX and g = y 1 ∈ IX such 𝛽 2 such that that for 𝛼 = 12 and 𝛽 = 13 in I0, we get that f = x1 in x𝛼 and g = y 1 in {y} 2 (𝛼 ∧ 𝛽) = 31 > sup(x1 ∧ y 1 ) = 0 and thus, (X, 𝜏) is an (𝛼, 𝛽) regular space. Also, it is an (𝛼, 𝛽)T1-space. 2 Hence, (X, 𝜏) is an (𝛼, 𝛽)T3-space Example 6.2 Let X = {x, y}, and ⎧ 1 ⎪ 1 ⎪ 𝜏(f ) = ⎨ 12 ⎪ 2 ⎪ 0 ⎩

at 0 or 1 at x1 at y1 otherwise.

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We see that {x} and {y} are disjoint closed subsets of X. Then, there are f = x1 ∈ IX and g = y1 ∈ IX 𝛽 𝛼 and g = y1 in {y} such that such that for 𝛼 = 12 and 𝛽 = 12 in I0, we get that f = x1 in {x} 1 (𝛼 ∧ 𝛽) = 2 > sup(x1 ∧ y1 ) = 0 and thus, (X, 𝜏) is an (𝛼, 𝛽) normal space. Also, it is an (𝛼, 𝛽)T1-space. Hence, (X, 𝜏) is an (𝛼, 𝛽)T4-space

Proposition 6.3 A topological space (X, T) is T3 if and only if the induced fuzzy topological space (X, 𝜔(T)) is (𝛼, 𝛽)T3. Proof (X, T) is T1 iff (X, 𝜔(T)) is (𝛼, 𝛽)T1 is proved in Proposition 4.2. Let F = cl𝜏 F and x ∉ F in X. Then, there are x , F ∈ T such that x ∩ F = �. Taking f = 𝜒F c and g = 𝜒 F in 𝜔(T), we get that int𝜔(T) f (x) ∧ int𝜔(T) g (F) = 1 > 0 = sup(f ∧ g). Hence, there are x1 and F1 of x and F respectively, such that 1 > sup(f ∧ g) and thus, (X, 𝜔(T)) is an (𝛼, 𝛽)T3-space. Conversely, F = cl𝜏 F and x ∉ F imply there are f ∈ x𝛼 and g ∈ y𝛽 for all y ∈ F such that int𝜔(T) f (x) ∧ int𝜔(T) g (F) ≥ (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼 ∧ 𝛽 ∈ I0. That is, int𝜔(T) f ∈ 𝜔(T), x ∈ s𝛼 int𝜔(T) f and int𝜔(T) g ∈ 𝜔(T), F ∈ s𝛽 int𝜔(T) g, which means that s𝛼 int𝜔(T) f = x ∈ T, s𝛽 int𝜔(T) g = F ∈ T and more□ over, x ∩ F = �, and thus, (X, T) is a T3-space. Proposition 6.4 A topological space (X, T) is T4 iff the induced fuzzy topological space (X, 𝜔(T)) is an (𝛼, 𝛽)T4. Proof Similar to Proposition 6.3.

□

Proposition 6.5 Let (X, 𝜏) be an (𝛼, 𝛽)T3-space, and let 𝜎 be an fuzzy topology on X finer than 𝜏 . Then, (X, 𝜎) is also an (𝛼, 𝛽)T3-space. Proof Let x ∈ X and F be a closed subset of X with x ∉ F. Then, there are f ∈ x𝛼 and g ∈ F𝛽 such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0, that is 𝛼 ≤ int𝜏 f (x), 𝛽 ≤ int𝜏 g (y) for all y ∈ F and (𝛼 ∧ 𝛽) > sup(f ∧ g), which means that 𝛼 ≤ int𝜎 f (x), 𝛽 ≤ int𝜎 g (y) for all y ∈ F and (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0 and thus, f ∈ x𝛼 and g ∈ F𝛽 in (X, 𝜎) such that (𝛼 ∧ 𝛽) > sup(f ∧ g); 𝛼, 𝛽 ∈ I0. Hence, (X, 𝜎) is an (𝛼, 𝛽) regular space. Proposition □ 4.3 states that (X, 𝜎) is an (𝛼, 𝛽)T1-space, and thus, it is an (𝛼, 𝛽)T3-space. Proposition 6.6 Let (X, 𝜏) be an (𝛼, 𝛽)T4-space and let 𝜎 be a fuzzy topology on X finer than 𝜏 . Then, (X, 𝜎) is also an (𝛼, 𝛽)T4-space. Proof Similar to the proof in Proposition 6.5. Funding The author received no direct funding for this research. Author details Ismail Ibedou1,2 E-mail: [email protected] 1 Faculty of Science, Department of Mathematics, Benha University, 13518 Benha, Egypt. 2 Faculty of Science, Department of Mathematics, Jazan University, KSA. Citation information Cite this article as: Graded fuzzy topological spaces, Ismail Ibedou, Cogent Mathematics (2016), 3: 1138574. References Bayoumi, F., & Ibedou, I. (2001). On GTi-spaces. Journal of Institute of Mathematics & Computer Sciences, 14, 187–199.

□ Bayoumi, F., & Ibedou, I. (2002a). Ti-spaces I. The Journal of the Egyptian Mathematical Society, 10, 179–199. Bayoumi, F., & Ibedou, I. (2002b). Ti-spaces II. The Journal of the Egyptian Mathematical Society, 10, 201–215. Bayoumi, F., & Ibedou, I. (2004). The relation between the GTispaces and fuzzy proximity spaces, G- compact spaces, fuzzy uniform spaces. The Journal of Chaos, Solitons and Fractals, 20, 955–966. Chang, C. I. (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications, 24, 182–190. DCPE Co-Training for Classification (2012). Neuro computing, 86, 75–85. Eklund, P., & Gähler, W. (1992). Fuzzy filter functors and convergence. Applications of Category Theory to fuzzy Subsets (pp. 109–136). Dordrecht: Kluwer. Fuzzy neuro systems for machine learning for large data sets. (2009, March, 6–7). Proceedings of the IEEE International Advance Computing Conference, IEEE Explore (pp. 541–545). Patiala.

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