Graded fuzzy topological spaces

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1 Ibedou, Cogent Mathematics (06), : PURE MATHEMATICS RESEARCH ARTICLE Graded fuzzy topological spaces Ismail Ibedou, * Received: August 05 Accepted: 0 January 06 First Published: 07 January 06 *Corresponding author: Ismail Ibedou, Faculty of Science, Department of Mathematics, Benha University, 58 Benha, Egypt; Faculty of Science, Department of Mathematics, Jazan University, KSA ismailibedou@gmailcom Reviewing editor: Hari M Srivastava, University of Victoria, Canada Additional information is available at the end of the article 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 Subects: Advanced Mathematics; Foundations & Theorems; Mathematics & Statistics; Science Keywords: neighbourhood systems; fuzzy filters; fuzzy neighbourhood filters; fuzzy topological spaces; separation axioms AMS Subect classification: 54A40 Introduction Kubiak (985) and Sǒstak (985) introduced the fundamental concept of a fuzzy topological structure as an extension of both crisp topology and fuzzy topology Chang (968), in the sense that both obects and axioms are fuzzified and we may say they began the graded fuzzy topology Bayoumi and Ibedou (00, 00, 00b, 004) introduced and studied the separation axioms in the fuzzy case in Chang s topology (968) using the notion of fuzzy filter defined by Gähler (995a,995b) Now, we will try to investigate fuzzy topological spaces in sense of Sǒstak, 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 Sǒstak and moreover, this fuzzy topology is itself the fuzzy topology in sense of Chang associated with the fuzzy neighbourhoterest Satementod filter (Gähler, 995b) 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 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, fatma_bayoumi@ yahoocom 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 PUBLIC INTEREST STATEMENT 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, ] 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 T 0 topological space, but there are (α, β) T 0 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 06 The Author(s) This open access article is distributed under a Creative Commons Attribution (CC-BY) 40 license Page of

2 Ibedou, Cogent Mathematics (06), : 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 (978) In Fuzzy neuro systems for machine learning for large data sets (009) and DCPE Co-Training for Classification (0), there are some applications based on fuzzy sets Preliminaries Throughout the paper, let I 0 =(0, ] and I =[0, ) A fuzzy topology τ:i X I is defined by Kubiak (985) and Sǒstak (985): () τ(0) =τ() =, () τ(f g) τ(f ) τ(g) for all f, g I X, () τ( μ ) τ(μ ) for any family of (μ ) J I X Let τ and τ be fuzzy topologies on X Then, τ J J is finer than τ (τ, which is coarser than τ ), denoted by τ τ, if τ (μ) τ (μ) for all μ I X For each fuzzy set f I X, the weak α cut-off f is given by w α f ={x X f (x) α}; the strong α cut-off f is the subset of X, s α 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 I X s α f T for all α I } fuzzy filters Let X be a non- empty set A fuzzy filter on X (Eklund, 99; Gähler, 995a) is a mapping :I X I such that the following conditions are fulfilled: (F) (α) α holds for all α I and () =; (F) (f g) = (f ) (g) for all f, g I X If and are fuzzy filters on X, then is said to be finer than, denoted by, provided that (f ) (f ) for every f I X By we mean that is not finer than there is f I X such that (f ) < (f ) A non-empty subset of I X is called a prefilter on X (Lowen, Lowen), provided that the following conditions are fulfilled: () 0 ; () f, g implies f g ; () f and f g imply g For each fuzzy filter on X, the subset α-pr of I X defined by: α-pr = {f I X (f ) α} is a prefilter on X Proposition (Gähler, 995a) There is a one-to-one correspondence between fuzzy filters on X Page of

3 Ibedou, Cogent Mathematics (06), : and the families ( α ) α I0 of prefilters on X which fulfill the following conditions: () f α implies α supf () 0 <α β implies α β () For each α I 0 with β = α, we have β = α This correspondence is given by α = 0<β<α α-pr for all α I 0 and (f )= g α, g f 0<β<α α for all f I X Proposition (Eklund, 99) Let A be a set of fuzzy filters on X Then, the following are equivalent () The infimum of A with respect to the finer relation of fuzzy filters exists, A () For each non-empty finite subset {,, n } of A, we have (f ) n (f n ) sup(f f n ) for all f,, f n I X, () For each α I 0 and each non-empty finite subset f,, f n of A α-pr, we have α sup(f f n ) Recall that ( )(f )= (f ) and ( )(f )= (f ) for all f I X A A A A Fuzzy neighbourhood filters For each fuzzy topological space (X, τ) and each x X, the mapping : I X I defined by (Gähler, Gahler): (λ) = int τ λ(x) for all λ I X is a fuzzy filter on X, called the fuzzy neighbourhood filter of the space (X, τ) at the point x, and for short is called a fuzzy neighbourhood filter at x The mapping ẋ:i X I is defined by ẋ(λ) =λ(x) for all λ I X The fuzzy neighbourhood filters fulfil the following conditions: () ẋ holds for all x X; () ( )(int τ f )=( )(f ) for all x X and f I X A fuzzy filter is said to converge to x X, denoted by τ x, if (Gähler, 995b) The fuzzy neighbourhood filter F at an ordinary subset F of X is the fuzzy filter on X defined in Bayoumi and Ibedou (00b), by means of, x F as: F = x F The fuzzy filter Ḟ is defined by Ḟ = x F ẋ Ḟ F holds for all F X Also, recall that the fuzzy filter λ and the fuzzy neighbourhood filter λ at a fuzzy subset λ of X are defined by λ = 0<λ(x) ẋ and λ = 0<λ(x), respectively λ λ holds for all λ I X (Bayoumi & Ibedou, 004) 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 )= (g) () cl τ g f () Page of

4 Ibedou, Cogent Mathematics (06), : 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, 995b): implies cl cl, cl, () (4) Neighbourhood systems Definition A family ( α ) of fuzzy sets α x x X x α I 0 on X if it satisfies the following conditions: is said to be a neighbourhood system with grade (Nb) For all f α x, we have α f (x), (Nb) α x, (Nb) f, g α x implies that f g α x, (Nb4) f α x, f g imply that g α x, (Nb5) If f α x, then there is g α x, such that for all y X with 0 < g(y), we have f α y Lemma These families of prefilters ( α ) x α I at x X satisfy the following conditions: 0 (Pr) f α x implies that α supf, (Pr) 0 <β α implies that α β, x x (Pr) For every α I 0 with β = α, we have β = α x x 0<β<α 0<β<α Proof Clear Remark For any subset A of X, let us define α by α = A A α, that is f α iff α x A (f ) x A x A iff α A (f ) α x = α, α {x} x β x = α β x, α x β x = α β x, α α x =, α α x = For all α β in I 0, we have α x β x For any α, β, γ I 0, we have α x α x, α x β x and β x α x implies that α = β, α x β x and β x γ x implies that α x γ x Also, for all α β I 0, we have either α x β x or β x α x (α) fuzzy open sets, fuzzy open sets Let us define an (α) fuzzy open set as follows: α τ(f ) iff for all x X there is α I 0 such that f α x and f (x) α (5) An (α) fuzzy closed set is the complement of an (α) fuzzy open set A set f I X is said to be fuzzy open if it is (α) fuzzy open for all α I 0 In other words, if for all x X and for all α I 0, we have f α x 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 Starting from a neighbourhood system ( α ) x x X with grade α I 0, we can define an interior operator and a closure operator as follows: Page 4 of

5 Ibedou, Cogent Mathematics (06), : intf (x) = g α x 0<g(y) f (y), (6) clf (x) = g α x 0<g(y) f (y) (7) For every x X, α x satisfying (Nb) to (Nb4) is exactly a prefilter on X of all neighbourhoods of x X with grade α I 0 That is, ( α ) x x X is a family of prefilters with grade α I 0 at every x X constructing after adding condition (Nb5) a neighbourhood system on X with grade α I 0 The pair (X, ( α ) ) x x X is called a neighbourhood space with a grade α I 0 From Lemma and from the correspondence given in Proposition between the fuzzy filters and the families satisfying the conditions (Pr) to (Pr), we can say this family ( α x ) α I 0 is a representation of the fuzzy neighbourhood filter as a family of prefilters This is given by the following two conditions (Nb) and (Pr): (Pr) (f ) = g α x, g f α for all f I X (Nb) α x = {f I X α (f )}Denote the subset α x IX as the fuzzy neighbourhoods with grade α I 0 of x X 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 operator given by () or this closure operator given by (), using the properties of α x stated in (Nb) (Nb5) That fuzzy topology is exactly the fuzzy topology τ associated with the fuzzy neighbourhood filters given by an interior operation as in () so that (f )=int τ f (x) for all f I X Also, we can consider α x = {f I X α int τ f (x)} (8) and then, () for an (α) fuzzy open set could be rewritten as α τ(f ) iff for all x X, there is α I 0 so that α int τ 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 () and (4) for all x X and all α I 0, we can define cl α x by cl α x = {f I X α cl (f )}, (0) and equivalently, cl α x = {f I X there is h α x, clh f } () For all x X and all α I 0, we have cl α x α x Definition Let (X, τ ) and (Y, τ ) be fuzzy topological spaces, and f :X Y a map Then, for some α I 0, f is called (α) fuzzy continuous if for all (α) fuzzy open set μ with respect to τ, we have f (μ) is an (α) fuzzy open set with respect to τ for all μ I Y Page 5 of

6 Ibedou, Cogent Mathematics (06), : f is called fuzzy continuous if for all fuzzy open set μ with respect to τ, we have f (μ) is a fuzzy open set with respect to τ for all μ I Y Definition Let (X, τ ) and (Y, τ ) be fuzzy topological spaces Then, the mapping f :(X, τ ) (Y, τ ) 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 τ is (α) fuzzy open ((α) fuzzy closed) set with respect to τ The mapping f :(X, τ ) (Y, τ ) is called fuzzy open (fuzzy closed) mapping if the image f(g) of the fuzzy open (fuzzy closed) set g with respect to τ is fuzzy open (fuzzy closed) set with respect to τ Now, we define the continuity locally at a point x 0 X between two fuzzy topological spaces using these graded neighbourhoods Definition 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 x 0 provided that for all g α, f (x 0 ) there exists h α x 0 such that h f (g) for some α I 0 f is (α) fuzzy continuous if it is (α) fuzzy continuous at every x X f is an fuzzy continuous if it is (α) fuzzy continuous for all α I 0 This is an equivalent definition with Definition for the (α) fuzzy continuous mapping and fuzzy continuous mapping 4 (α, β)t 0 -spaces and (α, β)t -spaces This section is devoted to introduce the notions of T 0 -spaces and T -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 (978]) Definition 4 A fuzzy topological space (X, τ) is called an (α, β)t 0 -space if for all x y in X, there exists f α such that f (y) <α; α I or there exists g β such that g(x) <β; β I x 0 y 0 Definition 4 A fuzzy topological space (X, τ) is called an (α, β)t -space if for all x y in X there exist f α and g β such that f (y) <α and g(x) <β; α, β I x y 0 Example 4 at 0 or at x 0 otherwise Taking α =, we get that there is f = x in α such that f (y) <α For all α I x 0, we can not find any f in α such that f (x) <α That is, (X, τ) is an (α, β)t y 0 Example 4 { at 0 or 0 otherwise Only there is f = which is a graded neighbourhood but for both of x, y Hence, for all α I 0, = α and therefore, (X, τ) is not (α, β)t y 0 α x Proposition 4 Every (α, β)t -space is an (α, β)t 0 Proof Clear Page 6 of

7 Ibedou, Cogent Mathematics (06), : Example 4 is an (α, β)t 0 -space but not (α, β)t Example 4 Taking α = and β =, we get that there is f = x in α and g = y x in β y g(x) <β, for some α, β I 0 Hence, (X, τ) is an (α, β)t such that f (y) <α and at 0 or at x at y 0 otherwise In the following theorems, there will be introduced some equivalent definitions for the (α, β)t 0 -spaces and (α, β)t -spaces Theorem 4 Let (X, τ) be a fuzzy topological space Then, the following statements are equivalent () (X, τ) is (α, β)t 0 () For all x y in X and for all α I 0, α x α y () For all x y in X, there exists f I X such that f (y) <α int τ f (x); α I 0 or there exists g I X such that g(x) <β int τ g(y); β I 0 (4) For all x y in X, there exists f I X such that f (y) > cl τ f (x) or there exists g I X such that g(x) > cl τ g(y) Proof () (): From (), there is f I X such that int τ f (y) f (y) <α int τ f (x); α I 0 and then, f α and f α Hence, α x y x α; α I y 0 and thus, () holds () (): There exists f I X such that int τ f (y) <α int τ f (x); α I 0 and then, for g = int τ f, we can say g(y) <α int τ g(x); α I 0 The other case is similar and thus, () is satisfied () (4): From Equation 7, we get that cl τ f (x) < f (y) for all int τ f (y) α>f (x), then (4) holds (4) (): Since f (y) < h α x 0<h(z) f (z) =cl τ f (x) implies that z could not be y with 0 < h(y) for all h α x ; α I 0, which means that there is h α such that h(y) =0 <α int h(x); α I x τ 0 The other case is similar and thus, () holds Theorem 4 Let (X, τ) be a fuzzy topological space Then, the following statements are equivalent () (X, τ) is (α, β)t () For all x X, we have cl τ x = x () For all x y in X, there exist f, g I X such that f (y) <α int τ f (x) and g(x) <β int τ g(y); α, β I 0 (4) For all x y in X, there exist f, g I X such that f (y) > cl τ f (x) and g(x) > cl τ g(y) Proof () (): Let y x in X Then, cl τ x (y) = h α 0<h(z) y x (z), which means for all h α y, if x (z) > 0 whenever h(z) > 0, then cl τ x (y) > 0 From (), we get that z could not be x with 0 < h(x), that is, cl τ x (y) =0 for all y x At x, it is clear that cl τ x (x) = Hence, cl τ x () is fulfilled = x for all x X, and Page 7 of

8 Ibedou, Cogent Mathematics (06), : () (): For all x y in X, we have cl τ x = x and cl τ y = y () means that cl τ x (y) =x (y) =0 = x (z), which means for all h α y, z could not be x with 0 < h(x), that h α 0<h(z) y is there is α I 0 and there is h α such that h(x) =0 <α and then, h(x) <α int y τh(y) The other case is similar and therefore, () is fulfilled () (4): As in Theorem 4 (4) (): As in Theorem 4 The next proposition shows that the separation axioms (α, β)t 0 and (α, β)t are good extensions in sense of Lowen (978) Proposition 4 A topological space (X, T) is a T 0 -space (T -space) if and only if the induced fuzzy topological space (X, ω(t)) is an (α, β)t 0 -space ((α, β)t -space) Proof Let (X, T) be T 0 (T ) and let x y Then, there is a neighbourhood y T such that x y Taking f I X such that y = s α f T for some α I, we get f (x) α< f (y), That is, f (x) <α f (y) for some α I 0 Similarly, if there is a neighbourhood T such that y, we can find g I X such that = s β g T and g(y) β< g(x) for some β I, That is, g(y) <β g(x) for some β I 0 Hence, (X, ω(t)) is an (α, β)t 0 -space ((α, β)t ) Conversely, let (X, ω(t)) be an (α, β)t 0 -space ((α, β)t ) and x y Then, there exists f I X such that f (y) <α f (x) for some α I 0, which means f (y) α< f (x) for some α I, that is there is f I X such that s α f = T and y Similarly, the other case is proved Hence, (X, T) is a T 0 -space (T ) Proposition 4 Let (X, τ) be an (α, β)t 0 -space ((α, β)t ) and let σ be a fuzzy topology on X finer than τ Then, (X, σ) is also (α, β)t 0 -space ((α, β)t -space) Proof (X, τ) is an (α, β)t 0 -space ((α, β)t ) implying that there is f I X such that α int τ f (x) and f (x) <α or (and) there is g I X such that α int τ 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 int σ f (x) and f (x) <α or (and) α int σ g(x) and g(x) <α Hence (X, σ) is an (α, β)t 0 -space ((α, β)t ) 5 (α, β)t -spaces Here, we introduce and study the Hausdorff separation axiom in fuzzy topological spaces Definition 5 An fuzzy topological space (X, τ) is called an (α, β)t -space if for all x y in X there exist f α and g β such that (α β) > sup(f g); α, β I x y 0 Proposition 5 Every (α, β)t -space is an (α, β)t Proof Let (X, τ) be an (α, β)t -space but not (α, β)t That is, for x y, we get for all f α x that f (y) α for all α I 0 Since for any g β y 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)t Hence, (X, τ) is an (α, β)t Page 8 of

9 Ibedou, Cogent Mathematics (06), : Example 5 There are f = x I X and g = x y I X such that, for α = and β = 4 in I, we get that f = x α x and g = x y β such that f (y) =x (y) =0 < = α and g(x) =(x y y 5 )(x) = < 4 = β That is, 5 (X, τ) is an (α, β)t But for all fuzzy sets f α and g β x y, we get that (α β) sup(f g) and thus, (X, τ) is not (α, β)t at 0 or at x at x y 0 otherwise Theorem 5 Let (X, τ) be an fuzzy topological space Then, the following statements are equivalent () (X, τ) is (α, β)t () For all fuzzy ultrafilter on X and for all x y, there is f α x such that (f ) <α; α I 0 or there is g β y such that (g) <β; β I 0 () For all fuzzy filter on X and for all x y, there is f α such that (f ) <α; α I x 0 or there is g β such that (g) <β; β I y 0 Proof () (): Suppose that there is an fuzzy ultrafilter on X such that (f ) α and (g) β for all f α and g β x y That is, (f g) = (f ) (g) α β, but in common we know that (h) suph for all h I X, which means that for all f α and g α x y, we have sup(f g) (α β) and therefore, () implies () is satisfied () (): Since for any fuzzy filter on X we find a finer fuzzy ultrafilter on X, that is (f ) (f ) for all f I X, then () implies that there is f α x such that (f ) (f ) <α; α I 0 or there is g β y such that (g) (g) <β; β I 0 Thus, () holds () (): Suppose for all f α and g β ; α, β I x y 0 that (α β) sup(f g) and () is fulfilled Then, for all fuzzy filter on X, we have (f ) <α or (g) <β; α, β I 0 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, () is satisfied Example 5 at 0 or 4 4 at x There are f = x I X and g = y at y 0 otherwise g = y in β such that (α β) = > sup(x y 4 I X such that for α = and β = in I, we get that f = x in α and x y )=0 and thus, (X, τ) is an (α, β)t Proposition 5 A topological space (X, T) is a T -space if and only if the induced fuzzy topological space (X, ω(t)) is an (α, β)t Proof Let x y in X Then, there are, y T such that y = Taking f, g I X such that s α f =, s β g = y for some α, β I, then f (x) >α and g(y) >β; α, β I, that is f (x) α and g(y) β; α, β I 0 and then, f α and g β x y such that s α f s β g = y =, which means that there is no element z X such that Page 9 of

10 Ibedou, Cogent Mathematics (06), : (f g)(z) =f (z) g(z) f (z) g (z) > (α β); α, β I, which means for all z X, we have (f g)(z) (α β); α, β I Hence, sup(f g) (α β); α, β I and then, sup(f g) < (α β); α, β I 0 and thus, (X, ω(t)) is an (α, β)t Conversely, x y implies that there are f α and g β x y such that f (x) g (y) (α β) > sup(f g); α, β I 0 That is, for γ = sup(f g) I, we can say f ω(t), x s γ f and g ω(t), y s γ g, which means that s γ f = T, s γ g = y T and moreover, y = and thus, (X, T) is a T (because if there is z ( y ), then (f g)(z) f (z) g (z) >γ= sup(f g) which is a contradiction) Proposition 5 Let (X, τ) be an (α, β) T -space, and let σ be an fuzzy topology on X finer than τ Then, (X, σ) is also an (α, β)t Proof Let x y X Then, there are f α and g β such that (α β) > sup(f g); α, β I, x y 0 that is α int τ f (x), β int τ g (y) and (α β) > sup(f g), which means that α int σ f (x), β int σ g (y) and (α β) > sup(f g); α, β I 0 and thus, f α and g β x y in (X, σ) such that (α β) > sup(f g); α, β I 0 Hence, (X, σ) is an (α, β)t 6 (α, β)t -spaces and (α, β)t 4 -spaces In this section, we use fuzzy neighbourhood filters at ordinary sets to define the notions of (α, β)t -spaces and (α, β)t 4 -spaces Definition 6 A fuzzy topological space (X, τ) is called (α, β) regular if for all F = cl τ F in P(X) and x F, there exist f α x and g β F such that (α β) > sup(f g); α, β I 0 Definition 6 A fuzzy topological space (X, τ) is called (α, β)t -space if it is regular and (α, β)t Definition 6 A fuzzy topological space (X, τ) is called normal if for all F = cl τ F, F = cl τ F P(X) with F F =, there exist f α F and g β F such that (α β) > sup(f g); α β I 0 Definition 64 A fuzzy topological space (X, τ) is called (α, β)t 4 if it is normal and (α, β)t Proposition 6 Every (α, β)t -space is an (α, β)t Proof Let x y in X (X, τ) is an (α, β)t -space meaning that cl τ {x} ={x} for each x X Now, cl τ {y} ={y}, x {y}, and (X, τ) is regular implying that there are f α, g β x y such that (α β) > sup(f g); α, β I 0 Hence, (X, τ) is an (α, β)t Theorem 6 For each fuzzy topological space (X, τ), the following are equivalent () (X, τ) is regular () For all y F = cl τ F and x F, we have α x cl α y and β y cl β x for all y F; α, β I 0 () For all x X and all α I 0, we have cl α x = α x (4) For all x X, for all fuzzy filter on X, for all f α, and for all α I x 0, we have (f ) α implies cl (f ) α Proof () (): Let f α ; α I Suppose that f cl α for some y F, that is, there is h α with x 0 y y cl τ h f, which means that f (y) α Since for all g α, we have g(y) α for all y F; α I, then y 0 sup(f g) (f g)(y) α =(α α) for some f α for all x F, and for all g α for some y F; α I, x y 0 which contradicts () and therefore, f cl α for all y F Thus, α cl α y x y for all y F The other case is similar and hence, () is satisfied Page 0 of

11 Ibedou, Cogent Mathematics (06), : () (): From () we deduce that for all f α andg β, we have f cl α x y y or g cl β x for all α, β I 0 implies x = y Hence, for all f α, x X, and all α I, we get that f cl α x 0 x, which means that α x cl α, but from that cl α α for all α I and for all x X, we get that cl α x x x 0 x = α for x all α I 0 and for all x X and thus, () holds () (4): Let be a fuzzy filter on X with (f ) α for all f α and α I x 0 From (), (f ) α for all f cl α and α I and then, cl (f ) α for all f α and α I x 0 x 0 and thus, (4) is fulfilled (4) (): Consider = in (4), we get that cl α = α for all x X and all α I Now, for y F = cl F x x 0 τ and x y, we get for all f α and g β that f cl α and g cl β x y x y, which means there are h α withcl h f and k β withcl k g Choose f = cl χ and g = cl (int χ ), then x τ y τ τ F c x τ τ F F we can find h = χ F c and k = int χ x τ F F such that (α β) = > 0 = sup(χ F c int τ χ F ) = sup (h k), and thus, for all F = cl τ F X a nd x F, there exist h α and k β such that (α β) > sup(h k); α, β I x F 0, and therefore, () is satisfied Theorem 6 Let (X, τ) be a fuzzy topological space Then, the following are equivalent () (X, τ) is normal () For all F = cl τ F, F = cl τ F P(X) with F F =, we have α cl α and β cl β x y y x for all x F andy F ; α, β I 0 () For all F = cl τ F P(X), and all α I 0, we have cl α F = α F (4) For all F = cl τ F P(X), for all fuzzy filters on X, for all f α, and for all α I F 0, we have (f ) α implies cl (f ) α Proof Similar to the Theorem 6 Proposition 6 Every (α, β)t 4 -space is an (α, β)t Proof Let x F = cl τ F in X Since (X, τ) is (α, β)t 4, then it is (α, β)t, which means that cl τ {x} ={x} for all x X, which implies that we have F ={x} =cl τ {x} and F = F with F F = Hence, there are f α and g β such that (α β) > sup(f g); α, β I and thus, (X, τ) is regular and it is (α, β)t x F 0 Therefore, (X, τ) is (α, β)t Example 6 at 0 or at x at y 0 otherwise We notice that {y} is a closed set and x {y} Then, there are f = x I X and g = y I X such that for α = and β = in I, we get that f = x in α and g = y 0 x in β such that {y} (α β) = > sup(x y )=0 and thus, (X, τ) is an (α, β) regular space Also, it is an (α, β)t Hence, (X, τ) is an (α, β)t -space Example 6 at 0 or at x at y 0 otherwise Page of

12 Ibedou, Cogent Mathematics (06), : We see that {x} and {y} are disoint closed subsets of X Then, there are f = x I X and g = y I X such that for α = and β = in I, we get that f = x in α and g = y in β 0 {x} such that {y} (α β) = > sup(x y )=0 and thus, (X, τ) is an (α, β) normal space Also, it is an (α, β)t Hence, (X, τ) is an (α, β)t 4 -space Proposition 6 A topological space (X, T) is T if and only if the induced fuzzy topological space (X, ω(t)) is (α, β)t Proof (X, T) is T iff (X, ω(t)) is (α, β)t is proved in Proposition 4 Let F = cl τ F and x F in X Then, there are, F T such that F = Taking f = χ F c and g = χ F in ω(t), we get that f (x) g (F) = > 0 = sup(f g) Hence, there are and x F of x and F respectively, such that > sup(f g) and thus, (X, ω(t)) is an (α, β)t Conversely, F = cl τ F and x F imply there are f α and g β x y for all y F such that f (x) g (F) (α β) > sup(f g); α β I 0 That is, f ω(t), x s α f and g ω(t), F s β g, which means that s α f = T, s β g = F T and moreover, F =, and thus, (X, T) is a T Proposition 64 A topological space (X, T) is T 4 iff the induced fuzzy topological space (X, ω(t)) is an (α, β)t 4 Proof Similar to Proposition 6 Proposition 65 Let (X, τ) be an (α, β)t -space, and let σ be an fuzzy topology on X finer than τ Then, (X, σ) is also an (α, β)t Proof Let x X and F be a closed subset of X with x F Then, there are f α and g β x such that F (α β) > sup(f g); α, β I 0, that is α int τ f (x), β int τ g (y) for all y F and (α β) > sup(f g), which means that α int σ f (x), β int σ g (y) for all y F and (α β) > sup(f g); α, β I 0 and thus, f α and x g β in (X, σ) such that (α β) > sup(f g); α, β I F 0 Hence, (X, σ) is an (α, β) regular space Proposition 4 states that (X, σ) is an (α, β)t -space, and thus, it is an (α, β)t Proposition 66 Let (X, τ) be an (α, β)t 4 -space and let σ be a fuzzy topology on X finer than τ Then, (X, σ) is also an (α, β)t 4 Proof Similar to the proof in Proposition 65 Funding The author received no direct funding for this research Author details Ismail Ibedou, ismailibedou@gmailcom Faculty of Science, Department of Mathematics, Benha University, 58 Benha, Egypt Faculty of Science, Department of Mathematics, Jazan University, KSA Citation information Cite this article as: Graded fuzzy topological spaces, Ismail Ibedou, Cogent Mathematics (06), : 8574 References Bayoumi, F, & Ibedou, I (00) On GT i -spaces Journal of Institute of Mathematics & Computer Sciences, 4, Bayoumi, F, & Ibedou, I (00a) T i -spaces I The Journal of the Egyptian Mathematical Society, 0, Bayoumi, F, & Ibedou, I (00b) T i -spaces II The Journal of the Egyptian Mathematical Society, 0, 0 5 Bayoumi, F, & Ibedou, I (004) The relation between the GT i - spaces and fuzzy proximity spaces, G- compact spaces, fuzzy uniform spaces The Journal of Chaos, Solitons and Fractals, 0, Chang, C I (968) Fuzzy topological spaces Journal of Mathematical Analysis and Applications, 4, 8 90 DCPE Co-Training for Classification (0) Neuro computing, 86, Eklund, P, & Gähler, W (99) Fuzzy filter functors and convergence Applications of Category Theory to fuzzy Subsets (pp 09 6) Dordrecht: Kluwer Fuzzy neuro systems for machine learning for large data sets (009, March, 6 7) Proceedings of the IEEE International Advance Computing Conference, IEEE Explore (pp ) Patiala Page of

13 Ibedou, Cogent Mathematics (06), : Gähler, W (995) The general fuzzy filter approach to fuzzy topology I Fuzzy Sets and Systems, 76, 05 4 Gähler, W (995) The general fuzzy filter approach to fuzzy topology II Fuzzy Sets and Systems, 76, 5 46 Kubiak, T (985) On fuzzy topologies (PhD Thesis) A Mickiewicz, Poznan Lowen, R (978) A comparison of different compactness notions in fuzzy topological spaces Journal of Mathematical Analysis and Applications, 64, Lowen, R (977) Initial and final fuzzy topologies and fuzzy Tychonoff theorem Journal of Mathematical Analysis and Applications, 58, Lowen, R (979) Convergence in fuzzy topological spaces General Topology and Applications, 0, Sŏstak, A P (985) On a fuzzy topological structure Rendiconti del Circolo Matematico di Palermo - Springer II, I, The Author(s) This open access article is distributed under a Creative Commons Attribution (CC-BY) 40 license You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially The licensor cannot revoke these freedoms as long as you follow the license terms Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits Cogent Mathematics (ISSN: -85) is published by Cogent OA, part of Taylor & Francis Group Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog with, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA ournal at wwwcogentoacom Page of