Upcoming SlideShare
×

# 1 s2.0-s1574035804702117-main

544 views
487 views

Published on

0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
544
On SlideShare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
11
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 1 s2.0-s1574035804702117-main

1. 1. “DEFINITION BANK” IN FUZZY TOPOLOGY GOVINDAPPA NAVALAGI Abstract. DEFINITION BANK in Fuzzy Topology is a nice collection of all existing deﬁnitions (particularly - weaker forms) w.r.t. fuzzy open sets, fuzzy closed sets,fuzzy mappings(=fuzzy functions), fuzzy separation axioms and their generalized weak forms . Also, it includes some of the newest deﬁnitions w.r.t to fuzzy subsets, fuzzy functions ,fuzzy separation axioms and fuzzy covering axioms. Contents Introduction 2 Part 1. Fuzzy Sets and Fuzzy Neighbourhoods 5 1.1. Fuzzy Regular open and fuzzy regular closed sets 5 1.2. Fuzzy semi (pre, α, semipre)-sets and their neighbourhoods 7 1.3. Fuzzy generalized open (semiopen, preopen, α-open etc.) and fuzzy generalized closed (semiclosed, preclosed, α-closed etc.) sets 9 Part 2. Fuzzy Separation Axions 12 2.1. Fuzzy separation Axioms 12 2.2. Weaker forms of Fuzzy separation Axioms 15 2.3. Fuzzy Regularity Axioms 16 2.4. Fuzzy Normality Axioms 18 2.5. Fuzzy Compactness 19 Date: 28-12-2001. 1991 Mathematics Subject Classiﬁcation. 54A40. Key words and phrases. Fuzzy Preopen, fuzzy semiopen, fuzzy α-open, fuzzy β- open(=fuzzy semipreopen),fuzzy θ-open, fuzzy semi-θ-closed sets, etc.... fuzzy Pre- continuous, fuzzy semicontinuous,fuzzy semiprecontinuous,fuzzy α-continuous ,fuzzy Pre- open,fuzzy semiopen ,fuzzy α-open,fuzzy α-closed , fuzzy weakly semiopen,fuzzy weakly preopen,fuzzy weakly α-open functions, etc...,fuzzy pre-To, fuzzy − pre − T1, fuzzysemi − T2, fuzzyalmostregular, fuzzyp − normal, fuzzyalmostp − normal, fuzzys − closed, fuzzyS − closed, fuzzys − compact, fuzzystronglycompactspaces.. 1
2. 2. 2 GOVINDAPPA NAVALAGI 2.6. Fuzzy S-closed spaces 22 2.7. Fuzzy Connected spaces 23 2.8. Fuzzy Extremally disconnected spaces and fuzzy submaximal spaces 24 Part 3. Fuzzy Mappings 25 3.1. Fuzzy Continuity, fuzzy openness and its allied deﬁnitions 25 3.2. Fuzzy weak forms of continuity, openness and allied deﬁnitions 28 3.3. Fuzzy weak forms of generalized continuity and fuzzy generalized openness and allied deﬁnitions 36 Acknowledgement 40 References 40 Introduction The concept of fuzzy sets was introduced by Prof. L.A. Zadeh in his classi- cal paper[161]. After the discovery of the fuzzy subsets, much attention has been paid to generalize the basic concepts of classical topology in fuzzy setting and thus a modern theory of fuzzy topology is developed. The notion of fuzzy sub- sets naturally plays a signiﬁcant role in the study of fuzzy topology which was introduced by C.L. Chang[31] in 1968. In 1980, Ming and Ming[73], introduced the concepts of quasi-coincidence and q-neighbourhoods by which the extensions of functions in fuzzy setting can very interestingly and eﬀectively be carried out. Since then many concepts of General Topology are being extend to Fuzzy Topol- ogy.Fuzzy semiopen sets were ﬁrst introduced and studied by K.K.Azad[10] in 1981.In 1991, Bin Shahna[19] extended the concepts of preopen and α-open sets in Fuzzy Topology.In 1998,S.S.Thakur and S.Singh [152] introduced the concept of Fuzzy semipreopen sets in Fuzzy Topology.The generalization of fuzzy general- ized open(resp.fuzzy generalized closed) sets and fuzzy generalized continuity was extensively studied in recent years by S.S.Thakur, R.Malviya, H.Maki,T.Fukutaka,
3. 3. “DEFINITION BANK” IN FUZZY TOPOLOGY 3 M.Kojima, H.Harade,R.K.Saraf, M.Caldas and M.Khanna.Fuzzy continuity of Chang in 1968, it has been proved to be of fundamental importance in the realm of fuzzy topology.Along with this, many researchers[10],[?],[169],[?],[47], [80]and[160] have studied fuzzy non-continuity. One of them [169] introduced and studied fuzzy pre semiopen sets and fuzzy pre semicontinuus mappings in fuzzy topological spaces. Deﬁnitions of Fuzzy sets and their fuzzy neighbourhoods are available in Section I. In Section II, collection of deﬁnitions concern with the fuzzy separation axioms is made. Deﬁnitions concern with the fuzzy mappings (fuzzy functions) are available in Section III. In this survey article we made an attempt of bringing all available deﬁnitions in ”Fuzzy Topology” under single umbrella, called “Deﬁnition Bank In Fuzzy Topology” . We also suggest some new deﬁnitions w.r.t the above dis- cussions, marked in the list as DeﬁnitionNEW . Hence, one can observe the ”New Deﬁnitions”.Regarding the “Deﬁnition Bank in Fuzzy Topology ”, suggestions, cor- rections or additions to either list would be gratefully received. Throughout this paper by (X, τ)or simply by X we mean a fuzzy topological space (fts, shorty) due to Chang [31] we give the following deﬁnitions : Deﬁnition 0.0.1. (a) Let X be a non-empty set and I the unit interval [0,1]. A fuzzy set in X is an element of the set IX of all functions from X to I. (b) 0X and 1X denote the fuzzy sets given by 0X(x) = 0 , for all x ∈ X and 1X(x) = 1 , for all x ∈ X . (c) Equality of two fuzzy sets λ and µ on X is determined by the usual equality condition for mappings, which is given by λ = µ ⇒ (for all x ∈ X,λ(x) = µ(x). (d) A fuzzy subset λ on X is said to be a subset of a fuzzy β on X written as λ ≤ β, if λ(x) ≤ β(x), for all x ∈ X (e) The complement of a fuzzy set λ on X is given by Co(λ) or simply λ = 1 − λ (f) A fuzzy topology τ on X is collection of subsets of IX , such that (i)0X,1X ∈ τ , (ii)if λ, β ∈ τ, then β ∈ τ, (iii)if λi ∈ τ for each i ∈ Λ , then i∈Λ λi ∈ τ.
4. 4. 4 GOVINDAPPA NAVALAGI The pair (X, τ) is called a fuzzy topological space (in short, fts or fuzzy space ). (g) Closure of a fuzzy set λ is denoted by Cl(λ) or λ bar , and is given by Cl(λ) = {µ : µisafuzzyclosedsetandλ ≤ µ} The interior of λ is denoted by Int(λ) , and is given by Int(λ) = {ν : νisafuzzyopensetandλ ≥ ν}. (h) A fuzzy set λ in a fts (X, τ) is a neighbourhood ,or nbhd for short, of a fuzzy set µ iﬀ there exists an open fuzzy set β such that µ ⊂ β ⊂ λ. (i) Let λ and β be two fuzzy sets in a fts (X, τ), and let β ⊂ λ.Then β is called an interior fuzzy set of λ iﬀ λ is a nbhd of β.The union of all interior fuzzy sets of λ is called the interior of λ and is denoted by λo . R.H.Warren in 1977 and 1978 , deﬁned the following. Deﬁnition 0.0.2. [156] (a) Let λ be a fuzzy set in a fts (X, τ).A point x ∈ X is called a fuzzy limit point of λ iﬀ whenever λ(x) = 1, then for each x∃y ∈ X − {x} such that x(y)Λa(y) = 0 ; or whenever a(x) = 1, then a−1 (x) > 0 and for each open x satisfying 1−x = a(x)∃y ∈ X − x such that x(y)Λa(y) = 0. The fuzzy derived fuzzy set of a (denoted by a’ ) and deﬁned as : a’(x) = a− (x)ifxisafuzzylimitpointofa, = 0otherwise. (b) Let (X, τ) be a fts and let A ⊂ X .Then the family, τA = {g|A : g ∈ τ} is a fuzzy topology on A, where g|A is the restriction of g to A,called the relative fuzzy topology on A or the fuzzy topology on A induced by the fuzzy topology τ on X.Note that (A, τA) is called a subspace of (X, τ). In 1973,J.A.Goguen et al[44] the following is given: Deﬁnition 0.0.3. Let τ be a fuzzy topology on X and let B, S ⊂ τ .Then B is called a basis for τ iﬀ each element of τ is the supremum of members of B.Also, S is called a subbasis for τ iﬀ the family of all ﬁnite inﬁmums of elements of S is a basis for τ. Due to Chang[31] and Goguen et alGSW1 the following is given:
5. 5. “DEFINITION BANK” IN FUZZY TOPOLOGY 5 Deﬁnition 0.0.4. Let f be a function from X to Y .Let b be a fuzzy set in Y and let a be a fuzzy set in X.Then the inverse image of b under f4isthefuzzysetf−1 (b) in X deﬁned by f−1 (b)(x) = b(f(x)) for x ∈ X i.e., f−1 (b) = bof.The image of a under f is the fuzzy set f(a) in Y deﬁned by f(a)(y) = {a(x) : f(x) = y} for y ∈ Y i.e., f(a)(y) = Sup{a(x) : x ∈ f−1 (y)}. In 1973,G.J.Nazaroﬀ[102]have deﬁned the fuzzy closure of a fuzzy set : Deﬁnition 0.0.5. Let a be a fuzzy set in a fts (X, τ).Then {b : bisaclosedfuzzysetinXandb ≥ a} is called the closure of a and is denoted by a− . In 1977,R.H.Warren[157] have deﬁned the fuzzy boundary set and fuzzy bound- ary operators in the following. Deﬁnition 0.0.6. Let a be a fuzzy set in a fts (X, τ).The fuzzy boundary of a , denoted by ab , is deﬁned as the inﬁmum of all closed fuzzy sets d in X with the property : d(x) ≥ a− (x) for all x ∈ X for which (a− (1 − a)− )(x) > 0 . Note: Clearly,(i) ab is a fuzzy closed set and ab ≤ a− ;(ii)(a− (1 − a)− )(x) = 0 , then ab = {allfuzzyclosedsetsinX} = 0. Note:In a fts (X, τ), if α(a) = ab for each fuzzy set a Part 1. Fuzzy Sets and Fuzzy Neighbourhoods 1.1. Fuzzy Regular open and fuzzy regular closed sets Deﬁnition 1.1.1. A fuzzy set λ in a fts X is called, (1)Fuzzy regular open[10] if λ = Int(Cl(λ)). (2) Fuzzy regular closed[10] if λ = Cl(Int(λ)). A fuzzy point in X with support x ∈ X and value p(0 < p ≤ 1) is denoted by xp. Two fuzzy sets λ and β are said to be quasi-coincident (q-coincident, shorty) denoted by λqβ, if there exists x ∈ X such that λ(x) + β(x) > 1[80] and by − q we denote ” is not q-coincident ” . It is known[80] that λ ≤ β if and only if λq(1 − β).
6. 6. 6 GOVINDAPPA NAVALAGI A fuzzy set λ is said to be q-neighbourhood (q-nbd) of xp if there is a fuzzy open set µ such that xpqµ, and µ ≤ λ if µ(x) ≤ λ(x) for all x ∈ X. The interior, closure and the complement of a fuzzy set λin X are denoted by Int(λ), Cl(λ) and 1 − λ = λc respectively. Recall that a fuzzy point xp is said to be a fuzzy θ-cluster point of a fuzzy set λ[80], if and only if for every fuzzy open q-nbd µ of xp , Cl(µ) is q-coincident with λ. The set of all fuzzy θ-cluster points of λ is called the fuzzy θ-closure of λ and will be denoted by Clθ(λ). A fuzzy set λ will be called θ-closed if and only if λ = Clθ(λ). The complement of a fuzzy θ-closed set is called of fuzzy θ-open and the θ-interior of λ denoted by Intθ(λ) is deﬁned as Intθ(λ) = {xp : for some fuzzy open q-nbd, β of xp, Cl(β) ≤ λ}. Deﬁnition 1.1.2. [161] Let X be a non-empty set and I be the unit interval [0,1]. A fuzzy set in X is a mapping from X into I.The null set 0 (zero) is the mapping from X into I assumes only the value 0 and the whole set 1, is the mapping from X into I which takes the value 1 only. Deﬁnition 1.1.3. [161] A fuzzy set A is contained in a fuzzy set B denoted by A ≤ B iﬀ A(x) ≤ B(x) , for each x ∈ X.The complement Ac of a fuzzy set A of X is (1 − A deﬁned by (1 − A)c (x) = 1 − A(X), for each x ∈ X.If A is a fuzzy set of X and B is a fuzzy set of Y then AxB is a fuzzy set of XxY , deﬁned by (AxB)(x, y) = (A(x), B(y)), for each (x, y) ∈ XxY . Deﬁnition 1.1.4. [31] Let f : X → Y be a mapping .If A is a fuzzy set of Y , then f−1 (A) is a fuzzy set of X , deﬁned by f−1 (A)(x) = A(f(x)) for each x ∈ X. Deﬁnition 1.1.5. [93] (i)Symbole Ao will stand for the support of A in X ; (ii) Symbol, Cλ will stand for the fuzzy set of X having the value λ at each point in X, where λ ∈ (0, 1]. Deﬁnition 1.1.6. [58] A fuzzy set on X is called a fuzzy singleton iﬀ it takes the value 0 for all points x ∈ X except 1.
7. 7. “DEFINITION BANK” IN FUZZY TOPOLOGY 7 Note that the point at which a fuzzy singleton takes the non-zero value is called the support of the singleton and the corresponding of (0,1] , its value. A fuzzy singleton with the value 1 is called crisp singleton. 1.2. Fuzzy semi (pre, α, semipre)-sets and their neighbourhoods Deﬁnition 1.2.1. A fuzzy set λ in a fts X is called, (1) Fuzzy semiopen[10] if λ ≤ Cl(Int(λ)) (in short Fs-open set). (2) Fuzzy semiclosed[10] if Int(Cl(λ)) ≤ λ (in short Fs-closed set). (3) Fuzzy preopen[19] if λ ≤ Int(Cl(λ)). (4) Fuzzy preclosed[19] if Cl(Int(λ)) ≤ λ. (5) Fuzzy α-open[19]if λ ≤ Int(Cl(Int(λ))) (in short Fα-open ). (6) Fuzzy semipreopen[145]if λ ≤ Cl(Int(Cl(λ))) (in short Fsp-open) (7) Fuzzy α-closed[19]if Cl(Int(Cl(λ))) ≤ λ (in short Fα-closed). (8) Fuzzy semipreclosed =β-closed [145]if Int(Cl(Int(λ) ≤ λ (in short Fsp- closed) The family of all fuzzy semiopen (fuzzy preopen, fuzzy α-open and fuzzy semipreopen) sets of X is denoted by FSO(X) (resp. FPO(X), Fα(X) and FSPO(X)). Recall that if, λ be a fuzzy set in a fts X then pCl(λ) = {β : β ≥ λ, β is fuzzy preclosed} (resp. pInt(λ) = {β : λ ≥ β, β is fuzzy preopen}) is called a fuzzy preclosure of λ (resp. fuzzy preinterior of λ)[19]and[140]. Deﬁnition 1.2.2. [160] The fuzzy semiclosure of A , denoted by sCl(A) ,is deﬁned by the intersection of all fuzzy semiclosed sets containing A. [160] The fuzzy semiinterior of A , denoted by sInt(A) , is deﬁned by the union of all fuzzy semiopen sets contained in A. Deﬁnition 1.2.3. [80, 92] A fuzzy subset A of an fts X is said to be fuzzy semi- regular ,if it is both fuzzy semioen set and fuzzy semiclosed set. The family of all fuzzy semi-regular sets of an fts X is denoted by FSO(X)
8. 8. 8 GOVINDAPPA NAVALAGI Deﬁnition 1.2.4. [80, 92] A fuzzy point xβ of X is said to be in the fuzzy semi- θ-closure of A , denoted by sClθ(A), if A ∩ sClU = 0 for every U ∈ FSO(X) containing xβ. Note that A is called fuzzy semi-θ-closed[92] if A = sClθ(A).The complement of a fuzzy semi-θ-closed set is called fuzzy semi-θ-open set. S.S.Thakur and S.Singh[145]have deﬁned semi-preclosure and semi-preinterior of a fuzzy subset A of fts X as follows; Deﬁnition 1.2.5. [145] A fuzzy set λ in a fts X is called: (i)fuzzy semipreopen if there exists a fuzzy preopen set such that ≤ λ ≤ Cl. (ii)fuzzy semipreclosed if there exists a fuzzy preclosed set such that Int ≤ λ ≤. Deﬁnition 1.2.6. [145] spCl(A) = Inf{B : B ≥ A; BisFsp − closedsetinX} spInt(A) = Sup{B : B ≤ A; BisFsp − opensetofX}. Deﬁnition 1.2.7. Let A be a fuzzy set in an fts X and xβ be a fuzzy point in X.Then, (i) A is called a fuzzy pre-q-nbd of xβ[106]if there exists a fuzzy preopen subset B in X such that xβqB ≤ A. (ii)a fuzzy pre-θ-nbd[104]ofxβ if there exists fuzzy preopen pre-q-nbd B of xβ such that pCl(B)q− (1 − A. Deﬁnition 1.2.8. [169] A fuzzy set A in X is said to be (i)fuzzy pre-semi-open if A ≤ (A− )o ; (ii) fuzzy pre-semi-closed if A ≥ (Ao )−. Deﬁnition 1.2.9. [167] A fuzzy set A in X is called a fuzzy pre-semi-nbd of a fuzzy point xα in X iﬀ there exists a fuzzy pre-semiopen set B in X such that xα ∈ B ≤ A. Deﬁnition 1.2.10. [167] A fuzzy set A in X is called a fuzzy pre-semi-q-nbd of a fuzzy point xα in X iﬀ there exists a fuzzy pre semiopen set B in X such that xαqB ≤ A.
9. 9. “DEFINITION BANK” IN FUZZY TOPOLOGY 9 [167] Every fuzzy q-nbd of a fuzzy point is always a fuzzy semi-q-nbd of the fuzzy point and every fuzzy semi-q-nbd of a fuzzy point is always a fuzzy pre semi-q-nbd of the fuzzy point. Deﬁnition 1.2.11. [70] A fuzzy set λ in a fts X is called : (i)fuzzy -β-open if λ ≤ ClIntCl(λ) ; (ii) fuzzy-β-closed if λc is a fuzzy -β-open ,or equivalently, if IntClInt(λ) ≤ λ. Deﬁnition 1.2.12. [76] Let A be a fuzzy set of an fts (X, τ).Then A is called a fuzzy less strongly semiopen set iﬀ there exists B ∈ τ such that B ≤ A ≤ sInt(sClB) Deﬁnition 1.2.13. [76] Let A be a fuzzy set of an fts (X, τ).Then A is called a fuzzy less strongly semiclosed set iﬀ there exists fuzzy closed set B such that sCl(sInt(B) ≤ A ≤ B Deﬁnition 1.2.14. [167] A be a fuzzy set of an fts (X, τ).Then A is called a fuzzy strongly semiopen set iﬀ there exists B ∈ τ such that B ≤ A ≤ Int(ClB) Deﬁnition 1.2.15. [167] A be a fuzzy set of an fts (X, τ).Then A is called a fuzzy strongly semiclosed set iﬀ there exists a fuzzy closed set B such that Cl(Int(B ≤ A ≤ B Deﬁnition 1.2.16. [13] (i) The class consisting of all fuzzy α-sets of fts (X, τ) is called a fuzzy α-structure and is denoted by τα ; (ii) The class consisting of all fuzzy β-open sets of fts (X, τ) is called a fuzzy β-structure and is denoted by τβ . 1.3. Fuzzy generalized open (semiopen, preopen, α-open etc.) and fuzzy generalized closed (semiclosed, preclosed, α-closed etc.) sets For the ﬁrst time the concept of fuzzy generalized closed sets was considered by S.S.Thakur and R.Malviya[150]
10. 10. 10 GOVINDAPPA NAVALAGI Deﬁnition 1.3.1. A fuzzy subset A of a fts X is a fuzzy generalized closed (in short Fg-closed) if Cl(A) ≤ B whenever A ≤ B and B is fuzzy open. The complement of a Fg-closed set is called Fg-open set. In 1998 Maki ,Fukutaka,Kojima and Harada[69] introduced the notion of fuzzy semi -generalized closed (in short Fsg-closed) sets by replacing the closure operator by semi-closure operator and by replacing openness of the superset with semiopen- ness, Deﬁnition 1.3.2. A fuzzy subset A of a fts X is called Fsg-closed if sClA ≤ B whenever A ≤ B and B is fuzzy semiopen . NOTE: Fg-closed and Fsg-closed sets are independent notions and none of them implies the other. In 1998, Maki and others in [69] have also introduced the concepts of Fuzzy generalized semi-closed sets (in short Fgs-closed). Deﬁnition 1.3.3. [150] A fuzzy subset A of a fts X is called Fgs-closed if sClA ≤ B whenever A ≤ B and B is fuzzy open . NOTE:Authors have mentioned that the both Fg-closed and Fsg-closed sets imply Fgs-closedness. Deﬁnition 1.3.4. [?] A fuzzy subset A of X is called fuzzy generalized semipreclosed set (in short Fgsp-closed) if spClA ≤ B whenever A ≤ B and B is fuzzy open set in X NOTE: Every Fsp-closed set is Fgsp-closed and every Fgs-closed set is Fgsp- closed set. Deﬁnition 1.3.5. [?] A fuzzy set A of X is called Fgsp-open if Ac is Fgsp-closed in X. Deﬁnition 1.3.6. [116] A fuzzy subset A of X is said to be Q-set ( in fact has the property Q) if IntClA = ClIntA
11. 11. “DEFINITION BANK” IN FUZZY TOPOLOGY 11 Deﬁnition 1.3.7. [?] Let A be fuzzy set in X.The generalized semi-preclosure of A (denoted by gspCl(A)) is deﬁned as follows: gspCl(A) = Inf{B : B ≤ A; BisFgsp − closedsetofX} . Deﬁnition 1.3.8. [?] Let A be fuzzy set of X.The generalized semi-preinterior of A ( denoted by gspInt(A)) is deﬁned as follows : gspInt(A) = Sup{B : B ≤ A; BisFgsp − opensetofX} Deﬁnition 1.3.9. [?] Let A be a fuzzy set in X and xβ is a fuzzy point of X. A is called fuzzy generalized semi-preneighbourhood of xβ if there exists a Fgsp-open set B in X such that xβ ∈ B ≤ A. Deﬁnition 1.3.10. [?] Let A be a fuzzy set in X and xβ be a fuzzy point in X. A is called generalized semipre-quasi-neighbourhood of xβ if there exists a Fgsp-open set B in X such that xβqB ≤ A. Deﬁnition 1.3.11. [139] A fuzzy subset A of X is said to be fuzzy generalized α-closed (in short Fg-closed) set in X if Fτα Cl(A) ≤ B whenever A ≤ B and B is Fα-open in X .We denote the collection of all these sets by FG(X). The complement of a Fgα-closed set is called Fgα-open set. Deﬁnition 1.3.12. NEW A fuzzy subset A of X is called a Fsg*-closed (resp. Fg*s-closed, Fg ∗ α-closed, Fαg∗-closed, Fg*p-closed, Fg*sp-closed, Fθ-g*-closed and Fδ-g*-closed) set if sclA ≤ U (resp. sclA ≤ U, clα ≤ U, clα ≤ U, pclA ≤ U, spclA ≤ U, clθ ≤ U and clδ ≤ U) whenever A ≤ U and U is Fsg-open (resp. Fgs-open, Fgα-open, Fαg-open, Fgp-open, Fgsp-open, Fθ-g-open and Fδ-g-open) set in X. Deﬁnition 1.3.13. NEW A fuzzy subset A of X is called a Fsg*-open (resp. Fg*s-open, Fg*α-open, Fαg*-open, Fg*p-open, Fg*sp-open, Fθ-g*-open and Fδ- g*-open) if its complement is a Fsg*-closed (resp. Fg*s-closed, Fg*α-closed, Fαg*- closed, Fg*p-closed, Fg*sp-closed, Fθ-g*-closed and Fδ-g*-closed).
12. 12. 12 GOVINDAPPA NAVALAGI Deﬁnition 1.3.14. NEW A fuzzy subset A of a space X is said to be fuzzy semi-pre generalized closed set (Fspg-set) if spcl(A) ≤ U whenever A ≤ U and U is fuzzy semiopen. Deﬁnition 1.3.15. [128] A fuzzy setA of X is said to be (i)Fgα∗ -closed if Fτ∗ Cl(A) ≤ Int(B) whenever A ≤ B and B is Fα-open in X ,(ii)Fgα∗∗ -closed if Fτα (A) ≤ IntCl(H) whenever A ≤ H and H is Fα-open set in X. NOTE: We denote FGα∗ (X) and FGα∗∗ (X) for the collection of all Fgα∗ -closed and Fgα∗∗ -closed sets in X respectively. Recently, Saraf,Caldas and Mishra[131] have deﬁned the concepts of fuzzy α- generalized closed sets in fuzzy setting. Deﬁnition 1.3.16. A fuzzy subset A of X is said to be fuzzy α-generalized closed set(in short Fαg-closed)if τα − Cl(A) ≤ B whenever A ≤ B and B is fuzzy open set in X. Deﬁnition 1.3.17. Let A ≤ B ≤ (X, τ).A fuzzy subset A of B is said to be Fαg-closed relative to B if A is Fαg-closed in its subspace (B, τB) . Deﬁnition 1.3.18. [15] Let X be a fts.A fuzzy set λ in X is called : (i) generalized fuzzy closed (in short gfc) iﬀ Cl(λ) ≤ µ whenever λ ≤ µ and µ is fuzzy open. (ii) generalized fuzzy open (in short gfo) iﬀ 1 − λ is gfc. (iii) Int∗ (λ) = {µ : µ ≤ λandµisgfo}. (iv) a gfc is called regular gf-closed if λ = Cl∗ (Int∗ (λ).Thefuzzycomplementofregulargf− closedsetiscalledregulargf − openset. Part 2. Fuzzy Separation Axions 2.1. Fuzzy separation Axioms Deﬁnition 2.1.1. [93] An fts (X, τ) is said to be FT1 -space in the sence of Ganguly and Shah ,if for two distinct fuzzy points xλ ,yµ in X
13. 13. “DEFINITION BANK” IN FUZZY TOPOLOGY 13 (i)x = y implies that there an fuzzy open nbd of xλ which is not q-coincident with yµ and there is fuzzy open nbd of yµ which is not q-coincident with xλ ; (ii) x = y and λ < µ (say) imply that xλ has an fuzzy open nbd and ,there is a q-nbd of yµ which is not q-coincident with xλ . Deﬁnition 2.1.2. [93] An fts (X, τ) is said to be FT2 -space in the sence of Ganguly and Shah ,if for two distinct fuzzy points xλ ,yµ in X (i)x = y implies xλ ,yµ have fuzzy open nbds which not q-coincident ; (ii) x = y and λ < µ (say) imply that xλ has an fuzzy open nbd and ,yµ has an fuzzy open q -nbd which are not q-coincident. Deﬁnition 2.1.3. [112] An fts (X, τ) is said to be FT2 -space in the sence of Z.Petrivic,if for each pair of points, (i)xt ,yr ,x = y implies that there exist two disjoint fuzzy open sets λ and µ such that xt ∈ λ ,yr ∈ µ ; (ii) x=y and t¡r implies that there exists fuzzy open set λ such that xt ∈ λ ,yrqCl(λ) . In 1981,Malghan and Benchalli[67] have deﬁned the following Hausdorﬀ space which is weaker than that of T.E.Gantner et al. Deﬁnition 2.1.4. [67] A fts (X, ) is called Hausdorﬀ (we denote it by Hausdorﬀ (MB))space if x, y ∈ X with x = y imply that there exist fuzzy open sets a and b with a(x) = 1 = b(y) and a ≤ 1 − b. Similar to the above deﬁnition of Hausdorﬀ space, we deﬁne the following. Deﬁnition 2.1.5. NEW A fts (X, ) is called semi-Hausdorﬀ space if x, y ∈ X with x = y imply that there exist fuzzy semiopen sets a and b with a(x) = 1 = b(y) and a ≤ 1 − b. Deﬁnition 2.1.6. NEW A fts (X, ) is called pre-Hausdorﬀ space if x, y ∈ X with x = y imply that there exist fuzzy preopen sets a and b with a(x) = 1 = b(y) and a ≤ 1 − b.
14. 14. 14 GOVINDAPPA NAVALAGI Deﬁnition 2.1.7. NEW A fts (X, ) is called α-Hausdorﬀ space if x, y ∈ X with x = y imply that there exist fuzzy α-open sets a and b with a(x) = 1 = b(y) and a ≤ 1 − b. Deﬁnition 2.1.8. NEW A fts (X, ) is called β-Hausdorﬀ space if x, y ∈ X with x = y imply that there exist fuzzy β-open sets a and b with a(x) = 1 = b(y) and a ≤ 1 − b. Deﬁnition 2.1.9. [95] A fts (X, τ) is said to be : (i) To iﬀ for any pair of distinct points xλ and yµ of X , either xλ has a q-nbd which is not q-conincident with yµ or yµ has a q-nbd which is not q-coincident with xλ; (ii) T1 iﬀ any pair of distinct points in X are weakly separated ; (iii) T2 iﬀ any pair of distinct points in X are strongly separated ; (iv) regular iﬀ any point xλ and any closed set A in X , such that xλ /∈ A, are strongly separated; (v) normal iﬀ any two closed sets of X , which are not q-coincident , are strongly separated; (vi) A T1 -regular space is called a T3 -space; (vii) A T1-normal space is called a T4 -space; (viii) completely normal iﬀ for any two weakly separated sets are strongly sepa- rated. Deﬁnition 2.1.10. [142] A fts (X, τ) is said to be : (i) fuzzy almost To (in short FATo) if for each pair of fuzzy singletons p and q with diﬀerent supports in X , there exists U ∈ RO(τ) such that p ≤ U ≤ Co(q) or q ≤ U ≤ Co(p); (ii) fuzzy almost T1 (in short FAT1) if for each pair of fuzzy singletons p and q with diﬀerent supports in X , there exists U, V ∈ RO(τ) such that p ≤ U ≤ Co(q) and q ≤ V ≤ Co(p);
15. 15. “DEFINITION BANK” IN FUZZY TOPOLOGY 15 (iii) fuzzy almost T2 (in short FAT2) if for each pair of fuzzy singletons p and q with diﬀerent supports in X , there exists U, V ∈ RO(τ) such that p ≤ U ≤ Co(q) and q ≤ V ≤ Co(p) and U ≤ Co(V ); (iv) fuzzy almost T212 (in short FAT212) if for each pair of fuzzy singletons p and q with diﬀerent supports in X , there exists U, V ∈ RO(τ) such that p ≤ U ≤ Co(q) and q ≤ V ≤ Co(p) and δCl(U) ≤ Co(δCl(V )); (v) fuzzy almost regular (in short FAR) if for each pair consisting of a fuzzy singleton p and a fuzzy regular closed set A in X such that p ≤ Co(A), there exist U, V ∈ τ such that p ≤ U , A≤ V and U ≤ Co(V ); (vi) fuzzy almost normal (in short FAN)if for each pair consisting of a fuzzy closed set A and a fuzzy regular closed set B in X such that A ≤ Co(B), there exist U, V ∈ τ such that B ≤ U ,A ≤ V and U ≤ Co(V ); (vii) fuzzy almost mildly normal (in short FMN)if for each pair consisting of a fuzzy regular closed sets A and B in X such that A ≤ Co(B), there exist U, V ∈ τ such that B ≤ U ,A ≤ V and U ≤ Co(V ). 2.2. Weaker forms of Fuzzy separation Axioms Deﬁnition 2.2.1. [140] A fuzzy topological space X is said to be fuzzy semi-T2 iﬀ for every pair of fuzzy singletons p1 and p2 with diﬀerent supports, there exist two fuzzy semiopen sets U and V such that p1 ≤ U ≤ pc 2 , p2 ≤ V pc 1 and U ≤ V c . Deﬁnition 2.2.2. [169] An fts X is called fuzzy pre-semi-To iﬀ for every pair of distinct fuzzy points xα and yβ, the following conditions are satisﬁed: (i)when x = y, either xα has a fuzzy pre-semi-nbd U such that Uq− yβ or yβ has a pre-semi-nbd V such that V q− xα ; (ii)when x = y and α < β (say), yβ has a fuzzy pre-semi-q-nbd V such that V q− xα. Deﬁnition 2.2.3. [169] An fts X is called fuzzy pre-semi-T1 iﬀ for every pair of distinct fuzzy points xα and yβ, the following conditions are satisﬁed:
16. 16. 16 GOVINDAPPA NAVALAGI (i)when x = y, either xα has a fuzzy pre-semi-nbd U and yβ has a pre-semi-nbd V such that Uq− yβ and V q− xα ; (ii)when x = y and α < β (say), yβ has a fuzzy pre-semi-q-nbd V such that V q− xα. Deﬁnition 2.2.4. [169] An fts X is called fuzzy pre-semi-T1 iﬀ for every pair of distinct fuzzy points xα and yβ, the following conditions are satisﬁed: (i)when x = y , and xα and yβ have fuzzy pre semi-nbds which are not q- coincident ; (ii)when x = y and α < β (say),then xα has a fuzzy pre-semi-nbd U and yβ has a fuzzy pre-semi-nbd V such that Uq− V . Deﬁnition 2.2.5. [15] A fts X is called fuzzy T1/2-space if every generalized fuzzy closed set in X is fuzzy closed in X. 2.3. Fuzzy Regularity Axioms Deﬁnition 2.3.1. [93] An fts (X, τ) is said to be fuzzy regular in the sence of Ganguly and Shah ,iﬀ for a fuzzy point xλ and a fuzzy closed set A in X : (i)A(x) = 0 implies there exist fuzzy open sets U and V such that xλ ∈ V and A ≤ U and Uq− V ; (ii) λ > A(x) > 0 µ implies that there exist fuzzy open sets U and V such that A ≤ U , xλqV and Uq− V . Deﬁnition 2.3.2. [55] An fts (X, τ) is said to be fuzzy regular in the sence of Hutton and Reilly iﬀ every fuzzy open set V can be expressed as union of fuzzy open sets Uα’s such that Cl(Uα ≤ V , for all α . Deﬁnition 2.3.3. [36] An fts (X,τ) is said to be quasi fuzzy regular ( in short q- fuzzy regular) iﬀ for each singleton xα and closed fuzzy set F in X with xαq(1−, thereexistU,V∈ τ such that xαqU ,F ⊂ V and UqV . Deﬁnition 2.3.4. [82] A fts space X is called fuzzy almost regular if foreach fuzzy regular open set U in X with xαqU , there exists a fuzzy regular open set V such that xαqV ≤ ClV ≤ V such that Cl(V ) ≤ U.
17. 17. “DEFINITION BANK” IN FUZZY TOPOLOGY 17 Deﬁnition 2.3.5. MG3 A fts space X is called fuzzy almost regular if for each fuzzy point xα in X and each fuzzy regular open q-nbd U of xα , there exists a fuzzy regular open q-nbd V of xα such that Cl(V ) ≤ U. Deﬁnition 2.3.6. [130] A fts X is said to be fuzzy s-regular if for each closed set F of X and each fuzzy pointxβ∈ 1 − F, there exist disjoint U, V ∈ FSO(X) such that xβ ∈ U and F ≤ V . Deﬁnition 2.3.7. [31] A fts X is said to be S-regular iﬀ each fuzzy open set λ of X is a union of fuzzy semiopen sets λj of X such that λ− j ≤ λ , for j. Deﬁnition 2.3.8. NEW A fts X is said to be fuzzy p-regular (resp. fuzzy α-regular and fuzzy sp-regular ) if for each closed set F of X and each fuzzy point xβ ∈ 1−F, there exist disjoint U, V ∈ FPO(X)(resp.U,V∈ F{αO(X) and U, V ∈ FSPO(X)) such that xβ ∈ U and F ≤ V . Deﬁnition 2.3.9. NEW A fts space X is called fuzzy almost p-regular (resp.fuzzy almost α-regular and fuzzy almost sp-regular )if for each regular closed set F of X and each fuzzy pointxβ∈ 1 − F, there exist disjoint U, V ∈ FPO(X) (resp. U, V ∈ FαO(X) and U, V ∈ FSPO(X) ) such that xβ ∈ U and F ≤ V . Deﬁnition 2.3.10. [104] An fts X is said to be fuzzy p-regular if for each fuzzy point xα in X and each fuzzy preopen pre-q-nbd V of xα, there exists a fuzzy preopen pre-q-nbd U of xα such that pCl(U) ≤ V Deﬁnition 2.3.11. NEW A space X is said to be fuzzy strongly preregular(resp. fuzzy strongly α-regular and fuzzy strongly sp-regular ) iﬀ if for each fuzzy preclosed (resp.fuzzy α-close and fuzzy sp-closed )set F of X and each fuzzy pointxβ∈ 1 −F, there exist disjoint U, V ∈ FPO(X)(resp. U, V ∈ FαO(X) and U, V ∈ FSPO(X) )such that xβ ∈ U and F ≤ V . Deﬁnition 2.3.12. NEW An fts X is said to be fuzzy g-regular (resp. fuzzy semi- g-regular ,fuzzy gs-regular )if for each fuzzy closed (resp. fuzzy sg-closed and fuzzy
18. 18. 18 GOVINDAPPA NAVALAGI gs-closed ) set F of X and each fuzzy pointxβ∈ 1 − F, there exist disjoint U, V ∈ FGO(X) (resp. U, V ∈ FSGO(X) and U, V ∈ FGSO(X) )such that xβ ∈ U and F ≤ V . 2.4. Fuzzy Normality Axioms Recall the following that Deﬁnition 2.4.1. [92] An fts (X, τ) is said to be fuzzy normal in the sence of Ganguly and Shah ,iﬀ (i)for two fuzzy closed sets A and B in X such that AB 0 0 = 0, there exist fuzzy open sets U and V such that A ≤ U , B ≤ V and Uq− V ; (ii) if x ∈ A0 and =yandxµ /∈ A then there exist fuzzy open sets U and V such that xµqU , A ≤ V and Uq− V . Deﬁnition 2.4.2. (i) A fts space X is said to be fuzzy almost normal[?] if every pair of disjoint sets µ and ν , one of which is fuzzy closed and the other is fuzzy regularly closed, can be strongly separated, (ii) a fts X is said to be fuzzy mildly normal[?]if for every pair of disjoint fuzzy regularly closed subsets F1 and F2 of X, there exist disjoint fuzzy open sets U and V such that F1 ⊂ U and F2 ⊂ V . Deﬁnition 2.4.3. NEW An fts (X, τ) is said to be fuzzy s-normal iﬀ (i)for two fuzzy closed sets A and B in X such that AB 0 0 = 0, there exist fuzzy semi open sets U and V such that A ≤ U , B ≤ V and Uq− V ; (ii) if x ∈ A0 and =yandxµ /∈ A then there exist fuzzy semi open sets U and V such that xµqU , A ≤ V and Uq− V . Deﬁnition 2.4.4. NEW An fts (X, τ) is said to be fuzzy p-normal iﬀ (i)for two fuzzy closed sets A and B in X such that AB 0 0 = 0, there exist fuzzy pre-open sets U and V such that A ≤ U , B ≤ V and Uq− V ; (ii) if x ∈ A0 and =yandxµ /∈ A then there exist fuzzy pre -open sets U and V such that xµqU , A ≤ V and Uq− V .
19. 19. “DEFINITION BANK” IN FUZZY TOPOLOGY 19 Deﬁnition 2.4.5. NEW A fts X is said to be fuzzy p-normal(resp.fuzzy α-normal and fuzzy sp-normal ) if for every two disjoint fuzzy closed subsets A and B of X, there exist two disjoint fuzzy preopen(resp. fuzzy α-open and fuzzy semipreopen) sets U and V such that A ≤ U and B ≤ V . Deﬁnition 2.4.6. NEW A fts X is said to be fuzzy semi-normal (resp.fuzzy pre- normal) if for every two disjoint fuzzy semiclosed (resp. fuzzy preclosed) A and B of X, there exist two disjoint fuzzy semiopen(resp.fuzzy preopen ) sets U and V such that A ≤ U and B ≤ V . Deﬁnition 2.4.7. NEW A fts X is said to be fuzzy semi-g-normal if for every pair of disjoint fsg-closed sets A and B of X, there exist disjoint fuzzy semiopen sets U and V of X such that A ≤ U and B ≤ V . Deﬁnition 2.4.8. NEW A fts X is said to be fuzzy generalized s-normal (i.e.,fgs- normal) if for every pair of disjoint fgs-closed sets A and B of X, there exist disjoint fuzzy semiopen sets U and V of X such that A ≤ U and B ≤ V . 2.5. Fuzzy Compactness Deﬁnition 2.5.1. [31] (i) A family of fuzzy sets is a cover of a fuzzy set β iﬀ β ⊂ {v : v ∈ }. It is an open cover iﬀ each member of is an open fuzzy set . A subcover of is a subfamily of which is also a cover. (ii) A fts X is compact iﬀ each open cover has a ﬁnite subcover. Deﬁnition 2.5.2. [37] A fts X is almost compact iﬀ every open cover of X has a ﬁnite subcollection whose closures cover X, or equivalently , every open cover has a ﬁnite proximate subcover. Deﬁnition 2.5.3. [52] A fts X is said to be fuzzy nearly compact if every fuzzy regular open cover of X has a ﬁnite subcover ,or equivalently if every open cover of X has a ﬁnite subcollection such that the interiors of closures of fuzzy sets in this subcollection covers X .
20. 20. 20 GOVINDAPPA NAVALAGI Deﬁnition 2.5.4. [52] A fts X is said to be lightly compact iﬀ every countable open cover of X has a ﬁnite subcollection whose closures cover X. Deﬁnition 2.5.5. [146] In a fts X , a family ν of fuzzy subsets of X is called an α(resp .gs-open,sg-open)-covering of X iﬀ ν covers X and ν ≤ Fα(X)(resp.ν ≤ Fgs(X),ν ≤ Fsg(X)) Deﬁnition 2.5.6. [146],[51] A fuzzy topological space (X, τ) is called α-compact if every α-open cover of X has a ﬁnite subcover. Deﬁnition 2.5.7. [51] A fuzzy topological space (X, τ) is called β-(resp.pre-)compact if every β-(resp.pre-)open cover of X has a ﬁnite subcover. Deﬁnition 2.5.8. NEW A fuzzy topological space (X, τ) is called fgs-compact(resp.fsg- compact) if every fgs-open (resp.fsg-open)cover of X has a ﬁnite subcover. Deﬁnition 2.5.9. NEW A fts (X, τ) is called FGPO-compact if every cover of X by fgp-open sets has a ﬁnite subcover. Deﬁnition 2.5.10. NEW A fts X is said to be FSC-compact if for each fuzzy closed subset A of X and τ-fuzzy semiopen cover u of A, there exists a ﬁnite subfamily of elements of u, say Vi, 1 ≤ i ≤ n with A ⊂ i≤n clVi. We propose the following. Deﬁnition 2.5.11. NEW A fts X is called fsgC-compact (resp. fgsC-compact) if for each fuzzy closed set A of X and τ-fsg-open (resp. τ-fgs-open) cover u of A, there exists a ﬁnite subfamily of elements of u, say, Vi, 1 ≤ i ≤ n with A ⊂ i≤n clVi. Deﬁnition 2.5.12. NEW A fts X is said to be fuzzy semi compact (resp. fuzzy semi-countably compact) if every cover (resp. countable cover) of X by fuzzy semiopen sets has a ﬁnite subcover. Deﬁnition 2.5.13. NEW A fts X is said to be fuzzy β-compact if every cover of X by fuzzy β-open sets has a ﬁnite subcover.
21. 21. “DEFINITION BANK” IN FUZZY TOPOLOGY 21 Note β-open sets are known as semipreopen sets. Deﬁnition 2.5.14. [?] A fts X is said to be fuzzy semi-θ-compact iﬀ every fuzzy semi-θ-open cover of X has ﬁnite subcover. Deﬁnition 2.5.15. [15] A family of gf-open sets in X is called gf-open cover of a fuzzy set β iﬀ β ⊂ {v : v ∈ }. A subcover of is a subfamily of which is also a cover. (ii) A fts X is said to be fg-compact iﬀ each gf-open cover of X has a ﬁnite subcover. (iii) A fuzzy set λ is said to be fg-compact if λ is fg-compact relative to X . Deﬁnition 2.5.16. [?] A fts X is said to be fuzzy RS-compact iﬀ for every fuzzy regular semiopen cover {Uα : α ∈ } of X, there exists a ﬁnite subset o of such that {Intα : α ∈ o} = 1X. Deﬁnition 2.5.17. [?] A fuzzy set A is said to be fuzzy RS-compact(in short FRSC-set) iﬀ for every fuzzy regular semiopen cover {Uα : α ∈ } of X, there exists a ﬁnite subset o of such that {Intα : α ∈ o} ≥ A. Deﬁnition 2.5.18. [95] A collection {Bα : α ∈ } of fuzzy sets in X is said to form a fuzzy ﬁlterbase in X for every ﬁnite subset o of , {Bα : α ∈ o} = 0x Deﬁnition 2.5.19. [103] A fuzzy point xα in a fts X is said to be fuzzy rs- accumalation point of a fuzzy ﬁlterbase {Bα : α ∈ } iﬀ for each fuzzy regular semiopen set U with xαqU and for each Bα , BαqIntU. Deﬁnition 2.5.20. [12] A fuzzy subset A of a fts (X, τ) is said to be FN-closed(=fuzzy N-closed) relative to an fts X if for any fuzzy open cover of A there exists a ﬁnite subcollection the interiors of the closures of which cover A. Deﬁnition 2.5.21. [12] A fuzzy subset A will be called FN-closable relative to τ if A is FN-closed relative to τ. Deﬁnition 2.5.22. [12] A fts X is said to be fuzzy locally nearly compact if each fuzzy point has a fuzzy neighbourhood which is FN-closable relative to τ.
22. 22. 22 GOVINDAPPA NAVALAGI 2.6. Fuzzy S-closed spaces Deﬁnition 2.6.1. [131] A subset A of a space X is S-closed [?](s-closed[?]) rela- tive to X if every cover of A by semiopen sets of X has a ﬁnite subfamily whose closures(resp. semiclosures) cover A. Deﬁnition 2.6.2. [131] A fts (X, τ) is called fuzzy s-closed(resp.fuzzy S-closed) if every cover {Vα : α ∈ } of X by fuzzy semiopen sets of X, there exists a ﬁnite subset o of such that X = {sClVα : α ∈ o} (resp.X = {ClVα : α ∈ o}). We deﬁne the following. Deﬁnition 2.6.3. NEW A fts (X, τ) is called fuzzy countably s-closed(resp.fuzzy countably S-closed) if every countable cover {Vα : α ∈ } of X by fuzzy semiopen sets of X, there exists a ﬁnite subset o of such that X = {sClVα : α ∈ o} (resp.X = {ClVα : α ∈ o}). Deﬁnition 2.6.4. NEW A fuzzy subset A of a fts X is fuzzy SG-closed (fuzzy sg- closed) relative to X if every cover of A by fuzzy s.g-open sets of X has a ﬁnite subfamily whose closures (resp. semiclosures) cover A. Deﬁnition 2.6.5. NEW A space X is fuzzy SG-closed (fuzzy sg-closed) if every cover of X by fuzzy s.g-open sets of X has a ﬁnite subfamily whose closures (resp. semiclosures) cover X. Deﬁnition 2.6.6. NEW A fts X is called weakly s-compact if every countable fuzzy open cover of X has a ﬁnite subfamily the semiclosures of whose members cover X. Deﬁnition 2.6.7. [31] A fts X is called S-closed iﬀ every fuzzy semiopen cover of X has a ﬁnite subcollection whose closures cover X. Deﬁnition 2.6.8. [31] A fuzzy set λ of X is called fuzzy semiregular if it is both fuzzy semiopen and fuzzy semiclosed. Deﬁnition 2.6.9. [31] A fts X is called semi S-closed iﬀ every semiopen cover of X has ﬁnite subcollection whose semiclosures cover X.
23. 23. “DEFINITION BANK” IN FUZZY TOPOLOGY 23 2.7. Fuzzy Connected spaces Deﬁnition 2.7.1. [118] A fuzzy topological space (X, τ) is said to be disconnected if X = A ∪ B, where A and B are non-empty fuzzy open sets in X such that A ∩ B = 0. Deﬁnition 2.7.2. [118] A fuzzy topological space X is said to be connected if X cannot be represented as the union of two non-empty,disjoint fuzzy open sets on X. Deﬁnition 2.7.3. [93] Two non-empty fuzzy sets A and B in an fts X are said to be fuzzy separated if Cl(A)q− B and Cl(B)q− A. Deﬁnition 2.7.4. [93] A fuzzy set A in an fts X is said to be fuzzy connected if A cannot be expressed as the union of two fuzzy separated sets . Deﬁnition 2.7.5. [93] A fuzzy set G in an fts X is said to be fuzzy disconnected iﬀ there are two non-empty sets A1 and A2 such that A1 and A2 are weakly separated and G =A12 . Deﬁnition 2.7.6. [93] A set G in a fts X is said to be connected iﬀ G is not disconnected in X. Deﬁnition 2.7.7. [47] Two non-empty fuzzy sets A and B in an fts X are said to be fuzzy semi-separated if sCl(A)q− B and sCl(B)q− A. Deﬁnition 2.7.8. [47] A fuzzy set A in an fts X is said to be fuzzy semi-connected if A cannot be expressed as the union of two fuzzy semi-separated sets. Deﬁnition 2.7.9. [108] Two non-empty fuzzy sets A and B in an fts X are said to be fuzzy pre-separated if pCl(A)q− B and Aq− pCl(B). Deﬁnition 2.7.10. [108] A fuzzy set which cannot be expressed as the union of two fuzzy pre-separated sets is said to be a fuzzy pre-connected set.
24. 24. 24 GOVINDAPPA NAVALAGI Deﬁnition 2.7.11. NEW Two non-empty fuzzy sets A and B in an fts X are said to be fuzzy α-separated if αCl(A)q− B and αCl(B)q− A. Deﬁnition 2.7.12. NEW A fuzzy set A in an fts X is said to be fuzzy α-connected if A cannot be expressed as the union of two fuzzy α-separated sets. Deﬁnition 2.7.13. NEW Two non-empty fuzzy sets A and B in an fts X are said to be fuzzy semipre-separated if spCl(A)q− B and Aq− spCl(B). Deﬁnition 2.7.14. NEW A fuzzy set which cannot be expressed as the union of two fuzzy semipre-separated sets is said to be a fuzzy semi-connected set. Deﬁnition 2.7.15. [21] A space X is said to be fuzzy hypperconnected if every non-empty fuzzy open subset of X is fuzzy dense in X. Deﬁnition 2.7.16. [15] A fts X is said to be (i) fg-connected iﬀ the only fuzzy sets which are both gf-open and gf-closed are 0x and 1x. (ii) generalized fuzzy supper connected if there is no proper regular gf-open set in X. (iii) generalized fuzzy strongly connected if it has non-zero fuzzy closed sets λ1 and λ2 such that λ1 + λ2 ≤ 1. If X is not generalized fuzzy strongly connected then it is said to be generalized fuzzy weakly connected . 2.8. Fuzzy Extremally disconnected spaces and fuzzy submaximal spaces Deﬁnition 2.8.1. [31] A fts X is called an Extremally Disconnected (E.D.) space if and only if f− ∈ τX for every fuzzy open set f ∈ τX . Deﬁnition 2.8.2. NEW A fts X is called almost fuzzy e.d if bdU = clU −U is ﬁnite for every U ∈ FRO(X).
25. 25. “DEFINITION BANK” IN FUZZY TOPOLOGY 25 Deﬁnition 2.8.3. NEW A fts X is called fuzzy submaximal (resp. fuzzy g-submaximal ) if each fuzzy dense subset is fuzzy open (resp. fuzzy g-open) Deﬁnition 2.8.4. NEW A fts X is called fsg-submaximal if every fuzzy dense set of X is fsg-open. Hence , in [?] it is newly deﬁned the following new class of spaces called strongly sg-submaximal : Deﬁnition 2.8.5. NEW A fts X is called fuzzy strongly sg-submaximal if every fgsp-closed subset of X is fgs-closed. Deﬁnition 2.8.6. A fts X is called semiregular if τ = τs [?]. Deﬁnition 2.8.7. [15] A fts X is said to be fuzzy extremally disconnected if Cl∗ (λ) is gf-open , when every λ is gf-open set. Part 3. Fuzzy Mappings 3.1. Fuzzy Continuity, fuzzy openness and its allied definitions Deﬁnition 3.1.1. Let f : (X, τ1) → (Y, τ2) be a function from a fts (X, τ1) into a fts (X, τ2). The function fis called: (i) fuzzy continuous[31]if f−1 (λ) is fuzzy open set in X for each fuzzy open setλ in Y . (ii) fuzzy open (resp. fuzzy closed )[31] if f(λ) is a fuzzy open(resp. fuzzy closed) set in Y for each fuzzy open(resp. fuzzy closed) set λin X. (iii) fuzzy contra-open (resp. fuzzy contra closed) if f(λ) is a fuzzy closed (resp. fuzzy open) set of Y for each fuzzy open (resp. closed) set λ in X. Deﬁnition 3.1.2. [31] A fuzzy homeomorphism is an F-continuous one-to-one map of a fts X onto a fts Y such that the inverse of the map is also F-continuous. Note: If there exists a fuzzy homeomorphism of one fuzzy space onto another, the two fuzzy spaces are said to be F-homeomorphic and each is a fuzzy homeomorph of the other. Two fts’s are topologically F-equivalent iﬀ they are F-homeomorphic.
26. 26. 26 GOVINDAPPA NAVALAGI Deﬁnition 3.1.3. [10] Let f be a function from a fts X into a fts Y . Then f is said to be fuzzy almost continuous iﬀ f−1 (µ)isfuzzyopen(resp.fuzzyclosed)inXforeveryfuzzyregularopen(resp in Y . Deﬁnition 3.1.4. [159] A function f : X → Y is said to be fuzzy W-almost open iﬀ f−1 (µ− ) ≤ f−1 (µ)− for every fuzzy open set µ in Y . Deﬁnition 3.1.5. [9] A function f : X → Y is said to be fuzzy weakly open (resp.FW- almost open) iﬀ f(V ) ≤ Int(f(Cl(V )))(resp.f−1 (Cl(V )) ≤ Cl(f−1 (V ) for every fuzzy open(resp. fuzzy regular open) set of X(resp. of Y ). Deﬁnition 3.1.6. [159] A function f : X → Y is said to be fuzzy almost continuous of Husain iﬀ f−1 (µ− ) ≤ f−1 (µ)−o for every fuzzy open set µ in Y . Deﬁnition 3.1.7. [10] A function f : X → Y is said to be fuzzy weakly continuous for each fuzzy point xβ ∈ X if for each fuzzy open set V of Y containing f(xβ), there exists an fuzzy open set U in X containing xβ such that f(U) ≤ clV . Deﬁnition 3.1.8. [91] A function f : X → Y is called fuzzy almost open (resp. fuzzy almost closed) if the image of each fuzzy regular open set (resp. fuzzy regular closed set) of X, is fuzzy open (resp. fuzzy closed) set in Y . Deﬁnition 3.1.9. [?] A function f : X → Y is called fuzzy almost open in the sense of S.Nanada (written as F.a.o.N) if for each open set U of X, f(U) ≤ intcl(f(U)). Deﬁnition 3.1.10. [91] A function f : X → Y is called fuzzy almost open in the sense of S.Nanada (written as F.a.o.N) if the image of each fuzzy regular open set of X is fuzzy open set in Y . Deﬁnition 3.1.11. [92] A function f : X → Y is called fuzzy almost open in the sense of Ganguly et al (written as F.a.o.G) if for each open set U of Y , f−1 Cl(A) ≤ Cl(f−1 (A)).
27. 27. “DEFINITION BANK” IN FUZZY TOPOLOGY 27 Deﬁnition 3.1.12. Let f : X → Y be a mapping from fts X to fts Y.Thenfiscalled : (i)fuzzy completely continuous[86] if f−1 (V ) is a fuzzy regular open in X for each fuzzy open set V in Y . (ii)fuzzy strongly continuous[86]if f−1 (V ) is a fuzzy clopen in X for each fuzzy subset V in Y . Deﬁnition 3.1.13. Let f : X → Y be a mapping from fts X to fts Y.Thenfiscalled : (i)fuzzy contra open if f(λ) is a fuzzy closed set of Y , for each fuzzy open set λ in X [26] ; (ii)fuzzy contra closed if f(λ) is a fuzzy open set of Y , for each fuzzy closed set λ in X ; (iii)fuzzy contra θ - openNEW if f(λ) is a fuzzy θ -closed set of Y , for each fuzzy θ -open set λ in X. (iv)fuzzy contra θ -closed NEW if f(λ) is a fuzzy θ -open set of Y , for each fuzzy θ -closed set λ in X. (v)fuzzy contra regular open NEW if f(λ) is a fuzzy regular closed set of Y , for each fuzzy regular open set λ in X. (vi)fuzzy contra regular closed NEW if f(λ) is a fuzzy regular open set of Y , for each fuzzy regular closed set λ in X. Deﬁnition 3.1.14. [?] A function f : X → Y is called quasi-θ-continuous if inverse image of each θ-open set of Y is θ-open set in X. Deﬁnition 3.1.15. [114] Let X = (X, τX) and Y = (Y,Y ) be fts. A function f : X → Y is said to be fuzzy -θ-continuous (resp. fuzzy weakly -θ-continuous) if for each fuzzy point xt and each fuzzy open nbd λ of f(xt) there is an fuzzy open nbd µ of xt such that f(Cl(µ)) ≤ Cl(λ) (resp. f(IntClµ) ≤ Cl(λ). Deﬁnition 3.1.16. [?] Let X = (X, τX) and Y = (Y,Y ) be fts. A function f : X → Y is said to be fuzzy R-map iﬀ f−1 (α) is a fuzzy regular open subset of X for each fuzzy regular open subset α of Y .
28. 28. 28 GOVINDAPPA NAVALAGI Deﬁnition 3.1.17. [78] Let X = (X, τX ) and Y = (Y,Y ) be fts. A function f : X → Y is said to be fuzzy totally continuous if the inverse image of every fuzzy subset of Y is a fuzzy clopen subset of X. Deﬁnition 3.1.18. [81] A mapping f : X → Y from a fts X to another fts Y will be called fuzzy almost continuous (in the sense of Mukherjee and Sinha) iﬀ for each fuzzy point xα of X and each fuzzy open nbd A of f(xα), Cl(f−1 (A)) is a fuzzy nbd. of xα. Note : Above deﬁnition of fuzzy almost continuous function f is independent of fuzzy almost continuous and fuzzy weakly continuos functions . 3.2. Fuzzy weak forms of continuity, openness and allied definitions Deﬁnition 3.2.1. Let f : (X, τ1) → (Y, τ2) be a function from a fts (X, τ1) into a fts (X, τ2). The function fis called: (i)fuzzy semi continuou[10]if f−1 (λ) is a fuzzy semiopen set of X for each fuzzy open set λ in Y . (ii)fuzzy pre continuous[19]if f−1 (λ) is a fuzzy preopen set of X for each fuzzy open set λ in Y (iii)fuzzy semipre continuous[145]if f−1 (λ) is a fuzzy semi-preopen set of X for each fuzzy open set λ in Y (iv) fuzzy preclosed[128],[19] if f(λ) is a fuzzy preclosed set of Y for each fuzzy closed set λ in X. (v)fuzzy preopen[20]if f(λ) is a fuzzy preopen set of Y for each fuzzy open set λ in X. (vi)fuzzy strongly semicontinuous[19]if f−1 (λ) is a fuzzy α-open set of X for each fuzzy open set λ in Y (vii)fuzzy semiopen[10]if f(λ) is a fuzzy semiopen set of Y for each fuzzy open set λ in X (viii)fuzzy α-open[20]if f(λ) is a fuzzy α-open set of Y for each fuzzy open set λ in X
29. 29. “DEFINITION BANK” IN FUZZY TOPOLOGY 29 (ix)fuzzy α-closed[20] if f(λ) is a fuzzy α-closd set of Y for each fuzzy closed set λ in X (x)fuzzy semiclosed[47]if f(λ) is a fuzzy semiclosed set of Y for each fuzzy closed set λ in X. (xi) fuzzy irresolute[81]f−1 (λ) is a fuzzy semiopen set of X for each fuzzy semiopen set λ in Y (xii)a fuzzy semi (resp. strongly )-irresolute iﬀ f−1 (λ) is a fuzzy semiopen (resp. fuzzy semi-θ-open) set of X for each fuzzy semiθ-open (resp.fuzzy semiopen)subset λ in Y [68] Deﬁnition 3.2.2. Let f : (X, τ1) → (Y, τ2) be a function from a fts (X, τ1) into a fts (X, τ2). The function fis called: (i)fuzzy M-preclosed[128] if f(λ) is a fuzzy preclosed set of Y for each fuzzy preclosed set λ in X. (ii)fuzzy α-continuous[146]if the inverse image of each fuzzy open set in Y is fuzzy α-open in X. (iii)fuzzy α-irresolute[146] if the inverse image of eachfuzzyα-open set in Y is fuzzy α-open in X Deﬁnition 3.2.3. Let f : X → Y be a mapping from fts X to fts Y.Thenfiscalled : (i)fuzzy preirresolute[106] if f−1 (V ) is a fuzzy pre- open in X for each fuzzy pre-open set V in Y . (ii)fuzzy weakly preirresolute[104] if f−1 (V ) is a fuzzy pre- open in X for each fuzzy pre θ-open subset V in Y . Deﬁnition 3.2.4. [108] A function f : X → Y is said to be fuzzy completely pre-irresolute if f−1 (V ) is fuzzy regular open in X for each fuzzy preopen subset V in Y or equivalently, f−1 (V )fuzzyregularclosedsetinXforeachfuzzypreclosedsubsetVinY.
30. 30. 30 GOVINDAPPA NAVALAGI Deﬁnition 3.2.5. [108] A function f : X → Y is said to be fuzzy weakly completely pre-irresolute if f−1 (V ) is fuzzy regular open in X for each fuzzy pre θ-open sub- set V in Y or equivalently, f−1 (V )fuzzyregularclosedsetinXforeachfuzzypreθ- closed subset V in Y . Deﬁnition 3.2.6. NEW A function f : X → Y is said to be fuzzy strongly α- irresolute (i.e., f.s.α.i), if for each fuzzy point xβ in X and each fuzzy α-open set L of Y containing f(xβ), there exists a fuzzy open set W in X such that xβqW and f(W) ≤ L. Deﬁnition 3.2.7. NEW A function f : X → Y is called fuzzy almost preirresolute if for each xβ in X and for each fuzzy pre-neighbourhood V of f(xβ), (f−1 (V ))∗ is a fuzzy pre-neighbourhood of xβ. Deﬁnition 3.2.8. NEW A function f : X → Y is said to be always fuzzy β-open if f(V ) ∈ FβO(Y ) for each V ∈ FβO(X) where F(X) denotes the family of all fuzzy β -open sets of X. Deﬁnition 3.2.9. NEW A function f : X → Y is said to be fuzzy strongly β-closed if the image of a fuzzy β-closed set in X is fuzzy β-closed in Y . Deﬁnition 3.2.10. NEW A function f : X → Y is called fuzzy strongly M- precontinuous (=FSMPC) if the inverse image of each fuzzy preopen set is fuzzy open. The FSMPC functions are called fuzzy strongly precontinuous functions.. Deﬁnition 3.2.11. NEW A functionf : X → Y is called fuzzy presemiopen (resp. fuzzy presemiclosed [148] ) if f(F) is fuzzy semiopen (resp.fuzzy semiclosed) set in Y for each fuzzy semiopen set F (resp. fuzzy semiclosed set F) in X. Deﬁnition 3.2.12. [21] A function f : X → Y is said to be contra-pre-semiclosed provided that f(λ) is fuzzy semiopen in Y for each fuzzy semiclosed subset λ of X.
31. 31. “DEFINITION BANK” IN FUZZY TOPOLOGY 31 Deﬁnition 3.2.13. [21] A function f : X → Y is said to satisfy the fuzzy weakly interiority condition if sInt(f(λ)) ≤ f(λ) for each fuzzy open subset λ of X. Deﬁnition 3.2.14. NEW A function f : X → Y is said to be fuzzy faintly semicon- tinuous (resp. fuzzy faintly precontinuous, fuzzy faintly β-continuous) if for each fuzzy point xβ in X and each fuzzy θ-open set V of Y containing f(xβ), there exists a fuzzy semiopen set (resp. fuzzy preopen set, fuzzy β-open set) U of X containing xβsuchthatxβqU and f(U) ≤ V . Deﬁnition 3.2.15. [135] A function f : X → Y is said to be fuzzy strongly α- continuous (=Fsα-continuous) if f−1 (A) is Fα-open in X , for every Fs-open set A in Y . Deﬁnition 3.2.16. [78] A function f : X → Y is called fuzzy totally semicontinu- ous if the inverse image of each fuzzy open subset of Y is a fuzzy semiclopen subset of X. Deﬁnition 3.2.17. [78] A function f : X → Y is called fuzzy almost semiopen if the image of each fuzzy semiclopen subset is fuzzy open . Deﬁnition 3.2.18. NEW A function f : X → Y is called fuzzy semi strongly continuous iﬀ the inverse image of every fuzzy subset of Y is a fuzzy semiclopen subset in X. Deﬁnition 3.2.19. NEW A function f : X → Y is called fuzzy slightly semicontin- uous if for each fuzzy point xβ in X and every fuzzy clopen subset V of Y containing f(xβ), there exists a fuzzy semiopen subset U of X such that xβqU and f(U) ≤ V . Deﬁnition 3.2.20. [169] A mapping f : X → Y is said to be fuzzy pre semicon- tinuous if f−1 (B) is fuzzy pre semiopen in X, for each fuzzy open set B in Y or equivalently, f−1 (D) is fuzzy pre semiclosed in X for each fuzzy closed set D of Y . Deﬁnition 3.2.21. [167] A mapping f : X → Y is said to be :
32. 32. 32 GOVINDAPPA NAVALAGI (i) fuzzy pre semiopen if f(A) is fuzzy pre semiopen in Y for each fuzzy open set A in X; (ii) fuzzy pre semiclosed if f(A) is fuzzy pre semiclosed in Y for each fuzzy closed set A in X (iii) fuzzy pre semi irresolute if f−1 (B) is fuzzy pre semiopen in X for each fuzzy pre semiopen set B in Y . Deﬁnition 3.2.22. NEW A function f : X → Y is called fuzzy quasi precontinuous at xβ ∈ X if for each fuzzy open set V containing f(xβ), there exists a fuzzy preopen set U such that xβqU and f(U) ≤ clV . Deﬁnition 3.2.23. [145] A mapping f : X → Y is called Fuzzy semi-precontinuous if f−1 (A) is Fsp-open in X for every fuzzy open set A of Y . Deﬁnition 3.2.24. NEW A function f : X → Y is called (i) fuzzy strongly α-open (resp. fuzzy strongly semiopen, fuzzy strongly preopen) if the image of each fuzzy α-open (resp.fuzzy semiopen, fuzzy preopen) set in X is a fuzzy α-open (resp. fuzzy semiopen, fuzzy preopen) set in Y . Deﬁnition 3.2.25. NEW A function f : X → Y is called (i) fuzzy quasi α-open (resp. fuzzy quasi semiopen, fuzzy quasi preopen) if the image of each fuzzy α-open (resp. fuzzy semiopen, fuzzy preopen) set in X is fuzzy open set in Y . Deﬁnition 3.2.26. NEW A function f : X → Y is called fuzzy quasi-α-closed (resp. fuzzy quasi semiclosed, fuzzy quasi preclosed) if the image of each fuzzy α-closed (resp. fuzzy semiclosed, fuzzy preclosed) set in X is fuzzy closed set in Y ) Deﬁnition 3.2.27. NEW A function f : X → Y is called fuzzy strongly-α-closed if the image of each fuzzy α-closed set in X is fuzzy α-closed set in Y . Deﬁnition 3.2.28. NEW A function f : X → Y is said to be fuzzy weakly α- irresolute if for each xβ ∈ X and each fuzzy α-open set U of Y containing f(xβ), there is a V ∈ FSO(X) such that xβ ∈ V and f(V ) ≤ U.
33. 33. “DEFINITION BANK” IN FUZZY TOPOLOGY 33 Deﬁnition 3.2.29. NEW A function f : X → Y is said to be fuzzy p-irresolute provided that for each fuzzy point xβ in X and each fuzzy preopen set V of Y such that f(xβ)qV and Y − V is fuzzy connected, there is a fuzzy preopen set U of X such that xβ ∈ U and f(U) ≤ V . Deﬁnition 3.2.30. NEW A function f : X → Y is said to be fuzzy ultra p- continuous if f−1 (V ) is fuzzy open for each fuzzy preopen set V in Y with fuzzy connected complement. Deﬁnition 3.2.31. NEW A function f : X → Y is fuzzy p-open (resp. fuzzy p-closed) if f(U) is fuzzy open (resp. fuzzy closed) for each fuzzy preopen (resp. fuzzy preclosed) set U of X. Deﬁnition 3.2.32. NEW A function f : X → Y is said to be fuzzy ultra p-quotient map provided that f−1 (U) ∈ FPO(X) iﬀ U ∈ FPO(Y ). Deﬁnition 3.2.33. NEW A function f : X → Y is called fuzzy weakly-θ-irresolute, if for each xα ∈ X and each V ∈ FSO(Y, f(xα)), there exists U ∈ FSO(X, xα) such that f(U) ≤ clV . Deﬁnition 3.2.34. NEW A function f : X → Y is called fuzzy strongly-θ-irresolute, if for each xα ∈ X and eachV ∈ FSO(Y, f(xα)), there exists U ∈ FSO(X, xα) such that f(clU) ≤ V . Deﬁnition 3.2.35. NEW A function f : X → Y is said to be fuzzy s-continuous if f−1 (V ) is fuzzy open for each V ∈ FSO(Y ). Deﬁnition 3.2.36. NEW A function f : X → Y is said to be fuzzy weakly-quasi- continuous if for each fuzzy point xβ in X, each fuzzy open set U containing xβ and each fuzzy open set V containing f(xβ) there exists fuzzy open set G ∈ X such that Gq− U and f(G) ≤ clV . Deﬁnition 3.2.37. NEW A function f : X → Y is said to be fuzzy almost quasi continuous if inverse image of each fuzzy regular open set of Y is fuzzy semiopen set in X.
34. 34. 34 GOVINDAPPA NAVALAGI Deﬁnition 3.2.38. [147] A mapping f from a fts X to a fts Y is called fuzzy M-semiprecontinuous if f−1 (λ ∈ FSPO(X) for every fuzzy set λ ∈ FSPO(Y ). Deﬁnition 3.2.39. A function f from a fts X to a fts Y is said to be : (i)fuzzy weakly semiopen[21] if f(λ) ≤ sInt(f(Cl(λ))) for each fuzzy open set λ of X. (ii)fuzzy weakly preopen[23] if f(λ) ≤ pInt(f(Cl(λ))) for each fuzzy open set λ of X. (iii)fuzzy weakly α-open NEW if f(λ) ≤ αInt(f(Cl(λ))) for each fuzzy open set λ of X. (iv)fuzzy weakly β-open NEW if f(λ) ≤ βInt(f(Cl(λ))) for each fuzzy open set λ of X. (v)fuzzy weakly θ-open[25] if f(λ) ≤ Intθ(f(Clλ))) for each fuzzy open set λ of X. (vi)fuzzy weakly semiclosed[22] if sCl(f(Int(β))) ≤ f(β) for each fuzzy closed set β in X. (vii)fuzzy weakly preclosed[24] if pCl(f(Int(β))) ≤ f(β) for each fuzzy closed set β in X. (viii)fuzzy weakly α-closed NEW if αCl(f(Int(β))) ≤ f(β) for each fuzzy closed set β in X. (ix)fuzzy weakly β-closed NEW if βCl(f(Int(β))) ≤ f(β) for each fuzzy closed set β in X. (x)fuzzy weakly θ-closed[26] if Clθ(f(Int(β))) ≤ f(β) for each fuzzy closed set β in X. Deﬁnition 3.2.40. [92] A function f : X → Y is called fuzzy semi α-irresolute function if f−1 (λ) is fuzzy semiopen in X for each fuzzy α-open set λ in Y . Deﬁnition 3.2.41. [93] A function f : X → Y is called fuzzy (θ-s)continuous function if for each fuzzy point xα in X and each fuzzy semiopen λ in Y containing f(xα), there exists fuzzy open set µ in X containing xα such that f(µ) ≤ λ.
35. 35. “DEFINITION BANK” IN FUZZY TOPOLOGY 35 Deﬁnition 3.2.42. [94] A function f : X → Y is called fuzzy α-quasi-irresolute function (in short f.α.q.i) if for each fuzzy point xα in X and each fuzzy semiopen λ in Y containing f(xα), there exists an fuzzy α-open set µ in X containing xα such that f(µ) ≤ λ. Deﬁnition 3.2.43. [31] A mapping f : X → Y from fts X to another fts Y is called fuzzy semi-weakly continuous if for each fuzzy semiopen set λ of Y , we have f−1 (λ) ≤ sInt[f−1 (sClλ)]. Deﬁnition 3.2.44. NEW A mapping f : X → Y from fts X to another fts Y is called fuzzy pre-weakly continuous if for each fuzzy preopen set λ of Y , we have f−1 (λ) ≤ pInt[f−1 (pClλ)]. Deﬁnition 3.2.45. NEW A mapping f : X → Y from fts X to another fts Y is called fuzzy α-weakly continuous if for each fuzzy α-open set λ of Y , we have f−1 (λ) ≤ αInt[f−1 αClλ)]. Deﬁnition 3.2.46. NEW A mapping f : X → Y from fts X to another fts Y is called fuzzy β-weakly continuous if for each fuzzy β-open set λ of Y , we have f−1 (λ) ≤ βInt[f−1 (βClλ)]. Deﬁnition 3.2.47. [29] A map f : X → Y from a fts X to another fts Y is said to be pre-fuzzy -β-closed if the image of every fuzzy -β-closed set of X is fuzzy -β-closed in Y . Deﬁnition 3.2.48. [?] A function f : X → Y from a fts X to another fts Y is said to be a fuzzy completely irresolute function iﬀ f−1 (α) is fuzzy regular open subset of X for every fuzzy semiopen subset α in Y . Deﬁnition 3.2.49. [?] A function f : X → Y from a fts X to another fts Y is said to be a fuzzy weakly completely irresolute function iﬀ f−1 (α) is fuzzy regular open subset of X for every fuzzy semi-θ-open subset α in Y or iﬀ f−1 (β) is fuzzy regular closed subset of X for every fuzzy semi-θ-closed subset α in Y .
36. 36. 36 GOVINDAPPA NAVALAGI 3.3. Fuzzy weak forms of generalized continuity and fuzzy generalized openness and allied definitions Deﬁnition 3.3.1. NEW A function f : X → Y is said to be: (i) a fuzzy semigen- eralized continuous i.e.fuzzy sg-continuous [136](resp. fuzzy generalized semicon- tinuous i.e. fuzzy gs-continuous , fuzzy g-continuos , fuzzy gp-continuous , fuzzy αg-continuous and fuzzy gsp-continuous ) if f−1 (λ) is fsg-closed set (resp. fgs-closed set, fg-closed, fgp-closed set, fαg-closed set and fgsp-closed set) in X for each fuzzy closed subset λ of Y , (ii) fuzzy semigeneralized closed i.e. fsg-closed (fg-closed , fgs-closed ) if f(µ) is fsg-closed (resp. fg-closed, fgs-closed) set in Y for each fuzzy closed set µ in X, (iii) fsg-open (resp. fgs-open , fg-open ) if f(ν) is fsg-open set (resp. fgs-open set and fg-open set) in Y for each fuzzy open set ν in X, (iv) fsg- irresolute (fgp-irresolute , fg-irresolute , fgs-irresolute , fgc-irresolute fgsp-irresolute and fαg-irresolute ) if f−1 (λ) is fsg-closed (resp. fgp-closed, fg-closed, fgs-closed, fg-closed, fgsp-closed, fαg-closed) set in X for each fsg-closed (resp. fgp-closed, fg-closed, fgs-closed, fg-closed, fgsp-closed, fαg-closed) set in Y . Deﬁnition 3.3.2. NEW A function f : X → Y is called frg-continuous (resp. fgpr- continuous if f−1 (λ) is a frg-closed (resp. fgpr-closed) set of X for each fuzzy closed set λ of Y . Deﬁnition 3.3.3. NEW A function f : X → Y is called fuzzy strongly gp- continuous if the inverse image of each fgp-closed set of Y is fuzzy open in X. Deﬁnition 3.3.4. NEW A function f : X → Y is called fuzzy perfectly gp- continuous if the inverse image of each fgp-closed set of Y is fuzzy clopen in X. Deﬁnition 3.3.5. NEW A function f : X → Y is called fuzzy pre-sg-continuous if f−1 (λ) is fsg-closed in X for every fuzzy semi-closed subset λ Y . Deﬁnition 3.3.6. [15] A map f : X → Y is called : (i) generalized fuzzy continuous (in short gf-continuous) if the inverse image of every fuzzy closed set in Y is gf-closed in X.
37. 37. “DEFINITION BANK” IN FUZZY TOPOLOGY 37 (ii) strongly fuzzy continuous if the inverse image of each fuzzy set in Y is both fuzzy open and fuzzy closed set in X. (iii) perfectly fuzzy continuous if the inverse image of each fuzzy open set of Y is both fuzzy open and fuzzy closed set in X. (iv) strongly gf-continuous if the inverse image of each gf- open set of Y is fuzzy open in X. (v) perfectly gf-continuous if the inverse image of every gf-open set in Y is both fuzzy open and fuzzy closed set in X. (vi) fuzzy gc-irresolute if the inverse image of every gf-closed set in Y is gf-closed set in X. Deﬁnition 3.3.7. [136] A mapping f : X → Y is called fuzzy semi-generalized continuous (in short Fsg-continuous)if f−1 (A) is Fsg-closed in X for every fuzzy closed set A of Y . Deﬁnition 3.3.8. [10] A mapping f : X → Y is called Fs-continuous if f−1 (A) is Fs-open set in X for every fuzzy open set A of Y . Deﬁnition 3.3.9. [?] A function f : X → Y is called Fgsp-continuous if f−1 (A) is Fgsp-closed in X for every fuzzy closed set A of Y . Deﬁnition 3.3.10. [?] A function f : X → Y is called Fgsp-irresolute if f−1 (A) is Fgsp-closed in X for every Fgsp-closed set A of Y . Deﬁnition 3.3.11. [128] A mapping f : X → Y is said to be Fgα∗ -continuous (resp. Fgα∗∗ -continuous,Fgα-continuous) if for every fuzzy closed set B of Y , f−1 (B) is Fgα∗ -closed (resp.Fgα∗∗ -closed ,Fgα-closed) in X. Deﬁnition 3.3.12. [?] A mapping f : X → Y is termed fuzzy pre-α-open if the image of every fuzzy α-open set of X is fuzzy α-open in Y . Deﬁnition 3.3.13. NEW A function f : X → Y is called: (i) fuzzy strongly sg- continuous if the inverse image of each fsg-open set of Y is fuzzy open in X. (ii)
38. 38. 38 GOVINDAPPA NAVALAGI fuzzy perfectly sg-continuous if the inverse image of every fsg-open set (fsg-closed) set of Y is fuzzy clopen set in X. (iii)fuzzy strongly gs-continuous if the inverse image of each fgs-open set of Y is fuzzy open in X. (iv) fuzzy perfectly gs-continuous if the inverse image of each fgs-open(fgs-closed) set of Y is fuzzy clopen set in X. (v) fuzzy weakly sg-continuous if the inverse image of each fsg-open set of Y is fuzzy semiopen set in X. (vi) fuzzy weakly gs-continuous if the inverse image of each fgs-open set of Y is fuzzy semiopen set in X. (vii) fuzzy sg*-continuous if the inverse image of each fuzzy semiopen set of Y is fsg-open set in X, and (viii) fuzzy gs*-continuous if the inverse image of each fuzzy semiopen set of Y is fgs-open set in X. Deﬁnition 3.3.14. NEW We say that a mapping f : X → Y is said to be fuzzy pre- semi-continuous (resp. fuzzy semiprecontinuous) if for each fuzzy open set V of Y , f−1 (V ) ∈ FPSO(X) (resp.f−1 (V ) ∈ FSPO(X)), where FPSO(X) is nothing but FSPO(X)semipreopen set. The deﬁnitions of FSPO(X) and FPSO(X) are de- ﬁned as below: (i) spintA = A∩(clintA∪intclA) (ii) FSPO(X) = A ⊂ X : A = spintA (iii) psintA = A ∩ clintclA (iv)FPSO(X) = A ⊂ X : A = psintA. Deﬁnition 3.3.15. NEW A function f : X → Y is called fuzzy α ∗ ∗g-continuous if f−1 (V ) is an fuzzy α ∗ ∗g-closed set of X whenever V is a fuzzy closed set of Y . Deﬁnition 3.3.16. NEW A function f : X → Y is called fuzzy quasi-sg-continuous if the preimage of every fuzzy open set of Y is fsg-closed set in X. Deﬁnition 3.3.17. NEW A function f : X → Y is called fuzzy g*-continuous if f−1 (V ) is a fg*-closed set of X for every fuzzy closed V of Y . Deﬁnition 3.3.18. NEW A function f : X → Y is called fsg*-continuous (resp. fg*s-continuous, fg∗α-continuous, fαg∗-continuous, fg*p-continuous,fg*sp-continuous, fθ-g*-continuous and fδ-g*-continuous) if f−1 (F) is a fsg*-closed (resp. fg*s-closed, fg ∗ α-closed, fαg∗-closed, fg*p-closed, fg*sp-closed, fθ-g*-closed and fδ-g*-closed) set of X for every fuzzy closed F of Y .
39. 39. “DEFINITION BANK” IN FUZZY TOPOLOGY 39 Deﬁnition 3.3.19. NEW A function f : X → Y is called fg*-irresolute if f−1 (V ) is a fg*-closed set of X for every fg*-closed set V of Y . Deﬁnition 3.3.20. NEW A function f : X → Y is called fsg*-irresolute (resp. fg*s-irresolute, fg ∗α-irresolute, fαg∗-irresolute, fg*p-irresolute, fg*sp-irresolute, fθ- g*-irresolute and fδ-g*-irresolute) if f−1 (F) is a fsg*-closed (resp. fg*s-closed, fg∗α- closed, fαg∗-closed, fg*p-closed, fg*sp-closed, fθ-g*-closed and fδ-g*-closed ) set of X for every fsg*-closed (resp. fg*s-closed, fg*α-closed, fαg*-closed, fg*p-closed, fg*sp-closed, fθ-g*-closed and fδ-g*-closed) set F of Y . Deﬁnition 3.3.21. NEW A function f : X → Y is called fδ-g-continuous (resp.f δ-g-irresolute) if f−1 (V ) is fδ-g-closed set in X for each fuzzy closed set V (resp. fδ-g-closed) set of Y . Deﬁnition 3.3.22. A function f : X → Y is called (i) fuzzy θ-g-continuous (resp.fuzzy θ-g-irresolute) if f−1 (V ) is fθ-g-closed set in X for every fuzzy closed (resp. fuzzy θ-g-closed) set V of Y . Deﬁnition 3.3.23. NEW A bijection map f : X → Y is called a fuzzy semi- generalized homeomorphism i.e. fsg-homeomorphism (resp.fuzzy generalized semi homeomorphism i.e., fgs-homeomorphism) if f is both fsg-continuous and fsg-open map (resp. iﬀ is both fgs-continuous and fgs-open). Deﬁnition 3.3.24. NEW A bijection map f : X → Y is called a fsgc-homeomorphism (resp. fgsc-homeomorphism) if f is fsg-irresolute and its inverse f−1 is also fsg- irresolute map (resp. iﬀ is fgs-irresolute and its f−1 is also fgs-irresolute). Finally, we give the following. Deﬁnition 3.3.25. NEW Let X and Y be topological spaces, let f : X → Y be a function, and let p ∈ X. Then f is said to be fuzzy semi generalized C-continuous (= fsgC-continuous) at fuzzy point p provided if U is an fuzzy open subset of Y containing f(p) such that Y − U is FSGO-compact, there is an fsg-open subset V of containing p such that f(V ) ≤ U.
40. 40. 40 GOVINDAPPA NAVALAGI Acknowledgement I am thankful to Professors: 1) Dr. Miguel Caldas, Brasil. (2) Dr. Ratnesh K.Saraf , India (3) Dr. E. E. Kerre for sending many of their re / preprints as soon as I requested. References [1] M.E.Abd El-Hakeim,Some strong separation in fuzzy topological spaces,Fuzzy sets and systems,60(1993)233-239. [2] M.E.Abd El-Monsef and M.H.Ghanim, Almost compact fuzzy topological spaces,Delta J.Sci.5(1981),19 [3] M.E.Abd El-Monsef ,I.M.Hanafy and S.N.El-Deeb,Fuzzy weakly α-continuous func- tions,Bul.of Inst.of Math.Aca.Sinica,19(1),75-85 [4] N.Ajmal, Fuzzy continuity and its pointwise characterizations by dual points and fuzzy nets.(communicated) [5] N.Ajmal and S.K.Azad : Fuzzy almost continuity and its pointwise characterization by dual points and fuzzy nets.Fuzzy Sets and Systems (to appear). [6] N.Ajmal and B.K.Tyagi ,Regular fuzzy spaces and fuzzy almost- regular spaces,Mat.Vesnik 40(1988),97-108. [7] F.M.H.Alla, On fuzzy topological spaces,Ph.D.Thesis,Assiut Univ.(1984). [8] F.M.H.Alla, α-continuous mappings,in fuzzy topological spaces,Bull.Cal.Math.Soc.80(1988)323-329. [9] A.A.Allam and A.A.Zahran, On fuzzy δ-continuity and α-near compactness in fuzzy topological spaces,Fuzzy sets and systems,50(1992)103-112. [10] K.K.Azad,On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity, J.Math.Anal.Appl., 82(1981)14-32. [11] K.K.Azad,On fuzzy Hausdorﬀ spaces and fuzzy perfect map- pings,J.Math.Anal.Appl.82(1981) 297-305. [12] M.Y.Bakier,FN-closed sets and fuzzy locally nearly compact spaces,Fuzzy sets and systems,88(1997)255-259 [13] M.Y.Bakier,On fuzzy α-structure and fuzzy β-structure,Fuzzy sets and system,92(1997)383-386 [14] G.Balasubramanian , On extensions of fuzzy topologies, Kybernetika 28(3)(1992) 239- 244. [15] G.Balasubramanian and P.Sundaram, On some generalizations of fuzzy continuous func- tions,Fuzzy sets and systems,86(1997),93-100. [16] R.Bellman and M.Giertz, On the analytic formation of the theory of fuzzy sets,Information Sci., 5(1973) 149-156. [17] R.N. Bhoumik and A. Mukherjee, Fuzzy weakly completely continuous functions, Fuzzy Sets and Systems, 55(1993), 347- 354. [18] R.N. Bhoumik,A. Mukherjee and A.K.Pal, Fuzzy completely irresolute and fuzzy weakly completely irresolute functions, Fuzzy Sets and Systems, 59(1993), 79-85. [19] A.S.Bin Shahna,On Fuzzy strongly semicontinuity and fuzzy precontinuity,Fuzzy sets and Systems,44(1991)303-308. [20] A.S.Bin Shahna, Mappings in fuzzy topological spaces,Fuzzy sets and systems,61(1994),209-213. [21] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy weakly semiopen func- tions (Submitted) [22] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy weakly semiclosed functions (Submitted) [23] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy weakly preopen func- tions (Submitted)
41. 41. “DEFINITION BANK” IN FUZZY TOPOLOGY 41 [24] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy weakly preclosed func- tions (Submitted) [25] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy weakly θ-closed func- tions (Submitted) [26] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy weakly θ-open func- tions (Submitted) [27] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy semi-θ-irresolute func- tions (submitted) [28] Miguel.Caldas,Govindappa Navalagi and Ratnesh Saraf,On Fuzzy almost precontinuous functions (submitted) [29] Miguel Caldas and R.K.Saraf,characterizations of spaces and maps via fuzzy β-open sets,Preprint. [30] Miguel Caldas and R.K.Saraf,Preserving Fuzzy sg-closed sets,Preprint. [31] C.L.Chang,Fuzzy topological spaces,J.Math.Anal.Appl.24(1968)182-190. [32] J.R.Choi, B.Y.Lee and J.H.Park,On the fuzzy θ-continuous mappings,Fuzzy sets and systems,54(19930107-114. [33] D.Coker and A.H.Es, On fuzzy S-closed spaces (preprint) [34] D.Coker and A.H.Es,On fuzzy RS-compact spaces,Doga Mat.13(1989)34-42. [35] F.Conrad, Fuzzy topological concepts,J.Math.Anal.Appl.74(1980)433-440 [36] Dewan Muslim Ali, Quasi fuzzy regular spaces,Proc.Math.Soc.BHU Vol 2.(1986)103-107. [37] A.Di Concilio and G.Gerla, Almost compact in fuzzy topological spaces, Fuzzy Sets and Systems, 13(1984) 187- 192. [38] A.H.Es, On almost compactness and near compactness in fuzzy topological spaces,Fuzzy sets and systems,22 (1987),289-295. [39] S.Ganguly and S. Saha, A note on semi-open sets in fuzzy topological spaces, Fuzzy Sets and Systems, 18(1986) 83- 96. [40] Ganguly S. and Saha S.,A note on δ-continuity and δ-connected sets in fuzzy set the- ory,Simon Stevin. 62(2) (1988)127-41. [41] Ganguly S. and Saha S.,On separation axioms and separation of connected sets in fuzzy topological spaces,Bull.Cal.Math.Soc.79(1987)215-225. [42] Ganguly S. and Saha S.,On separation axioms and Ti-continuity,Fuzzy sets and systems, 1265-1275. [43] Ganguly S. and Saha S.,A note on compactness in a fuzzy setting,Fuzzy sets and systems,34(1990)117-124. [44] T.E.Gantner,R.C.Steinlage and R.H.Warren,Compactness in fuzzy topological spaces,J.Math.Anal.Appl.(to appear) [45] H.M.Ghanim,E.E.Kerre and A.S.Mashhour,J.Math.Anal.Appl.102(1984)189-202. [46] M.H.Ghanim,O.A.Tantawy and Fawzia M.Selim, On lower separation axioms,Fuzzy sets and systems,85(1997)385-389 [47] B.Ghosh,Semicontinuous and semiclosed mappings,and semi-connectedness in fuzzy sett- ting,Fuzzy sets and systems,35(3)(1990) 345-355. [48] B. Ghosh,Fuzzy extremally disconnected spaces, Fuzzy Sets and Systems,46(1992), 245- 250. [49] J.A.Goguen,The fuzzy Tychonoﬀ theorem,J.Math.Anal.Appl.43(1973)734-742 [50] Gulham,Aslim,Eftal Tan and Oya Bedre, On some generalizations of fuzzy almost con- tinuity,Pure and Applied Mathematika Science,Vol.XXXIII,No.1-2,March (191)125-129. [51] I.M.Hanafy,A class of strong forms of fuzzy complete continuity,Fuzzy sets and systems,90(1997)349-353 [52] A. Haydar Es, Almost compactness and near compactness in fuzzy topological spaces, Fuzzy Sets and Systems, 22(1987), 289- 295. [53] A. Haydar Es, and Dogan Coker,On several types of degree of fuzzy compactness,Fuzzy sets and systems,87(1997)349-359. [54] W.C.Hong,RS-compact spaces, J.Korean Math.Soc.17(1980)39-43 [55] B.Hutton and Reilly I.L.:Separation axioms in fuzzytopological spaces.Department of Mathematics,Univ.of Auckland,Report No.55, March 1974. [56] B.Hutton and Reilyy I.L., Fuzzy sets and Systems , 3(1980)93-104. [57] G.Kilibarda,Some separation axioms in fuzzy topological spaces,Mat.Vesnik 36(1984)271-284.
42. 42. 42 GOVINDAPPA NAVALAGI [58] E.E.Kerre, Simon Stevin Quart.J.Pure Appl.Math.,53,(1979),229-249 [59] E.E.Kerre , Govindappa Navalagi and D.Weken,Fuzzy almost normal and Fuzzy mildly normal spaces (preprint) [60] E.E.Kerre , Govindappa Navalagi and D.Weken,Fuzzy preneighbourhoods (preprint) [61] F.H.Khedr,F.M.Zeyada and O.R.Sayed,α-continuity and cα-continuity in fuzzifying topology,Fuzzy sets and systems,116(2000)325-337 [62] W.Kotze,The fuzzy Hausdorﬀ Property,Glas.Mat.20(40)(1985)409-417. [63] R.Lowen,Fuzzy topological spaces and fuzzy compactness,J.Math.Anal.Appl.56(1976) 521. [64] Biljana Krsteska,A note on the article ”Fuzzy alpha sets and alpha-continuous maps,Fuzzy sets and systems,120(2001)549-550. [65] Biljana Krsteska,A note on the article ”Fuzzy less strongly semiopen sets and fuzzy less strong semicontinuity,Fuzzy sets and systems,107(1999)107-108. [66] S.R.Malghan and S.S.Benchalli,Open maps , closed maps and Local compactness in Fuzzy topological spaces , Jour.Math.Anal.Appl. Vol. 99 (2),(1984),338-349. [67] S.R.Malghan and S.S.Benchalli,Fuzzy topological spaces,, Glasnik Mat. 16(1981)313-325 . [68] S.Malakar, On fuzzy semi-irresolute and strongly irresolute functions,Fuzzy sets and systems,45(1992)239-244. [69] H.Maki,T.Fukutaka,M.Kojima and H.Harade,Generalized closed sets in fuzzy topological spaces,Meeting on topo.Space Theory and Appl.(1998),23–26. [70] A.S. Mashhour, M.H. Ghanim and M.A. Fath Alla, On fuzzy non-continuous mappings, Bull. Cal. Math. Soc.,78(1986),57-69. [71] A.S. Mashhour, M.H. Ghanim and M.A. Fath Alla,-separation axioms and α- compactness in fuzzy topological spaces,Rockey Mountain J.Math.16(1986),591. [72] A.S.Mashhour and M.H.Ghanim ,Fuzzy closure spaces,J.Math. Anal.Appl.106(1985),154-170. [73] P.P. Ming and L.Y. Ming, Fuzzy topology I. Neighborhood structure of fuzzy point and Moore-Smith convergence, J.Math. Anal. Appl., 76(1980), 571-594. [74] P.P.Ming and L.Y.Ming,Fuzzy Topology II-Producct and Quotient spaces,J.Math.Anal.Appl. 77(1980)20-37. [75] H.C.Ming,Fuzzy topological spaces,J.Math.Anal.Appl.110(1985)141-178. [76] F.J.Ming,Fuzzy less strongly semiopen sets and fuzzy less strongly semicontinuity,Fuzzy sets and systems,73(1995)279–290. [77] C.D.Mitri and E. Pascali:Characterization of fuzzy topology from neighbourhoods of fuzzy points.J.Math.Anal.Appl.93(1983),1-14. [78] Anjan Mukherjee,Fuzzy totally continuous and totally semicontinuous functions,Fuzzy sets and systems,107(1999)227-230. [79] M.N. Mukherjee and S.P. Sinha, Some stronger forms of weaker forms of fuzzy continuous mappings on fuzzy topological spaces, Fuzzy Sets and Systems, 36(1990)375-387. [80] M.N. Mukherjee and S.P. Sinha, fuzzy θ -closure operator on fuzzy topological spaces, Internat. J. Math. Math. Sci. 14(1991), 309-314. [81] M.N. Mukherjee and S.P. Sinha, On some weaker forms of fuzzy continuous and fuzzy open mappings on fuzzy topological spaces, Fuzzy Sets and Systems, 32(1989), 103- 114. [82] M.N. Mukherjee and S.P. Sinha, On some nearly fuzzy continuous functions between fuzzy topological spaces,Fuzzy sets and systems ,34(1990)245-254. [83] M.N. Mukherjee and S.P. Sinha,Irresolute and almost open functions between fuzzy topological spaces,Fuzzy sets and system,29(1989)381-386. [84] M.N. Mukherjee and S.P. Sinha, Almost compact fuzzy topological spaces,Mat.Vesnik 41(1989)89-97. [85] M.N. Mukherjee and S.P.Sinha,Almost compact sets in fuzzy topological spaces,Fuzzy sets and systems,38(1990)389-396. [86] M.N. Mukherjee and B.Ghosh,On fuzzy S-closed spaces and FSC sets, ,Bull.Malaysian Math.Soc.12(1989)1-14. [87] M.N. Mukherjee and B.Ghosh, On nearly compact and θ-rigid fuzzy sets, in fuzzy topo- logical spaces,Fuzzy sets and systems,43(1991)57-68. [88] M.N. Mukherjee and B.Ghosh, Some strong forms of fuzzy continuous mappings on fuzzy topological spaces,Fuzzy sets and systems,38(1990)375-387.
43. 43. “DEFINITION BANK” IN FUZZY TOPOLOGY 43 [89] M.N. Mukherjee and R.P.Chakraborty, On fuzzy almost compactness,Fuzzy sets and systems,98(1998)207-210. [90] S.Nanda, On Fuzzy topological spaces, Fuzzy sets and system,19(1986), 193-197. [91] S.Nanda,Strongly compact fuzzy topological spaces,Fuzzy sets and systems,42(1991)259- 262. [92] Govindappa Navalagi and R.K.Saraf,On fuzzy contra continuous functions and fuzzy strongly S-closed spaces,(Preprint) [93] Govindappa Navalagi and R.K.Saraf, On fuzzy contra semicontinuous functions and fuzzy strongly semi S-closed spaces,(preprint) [94] Govindappa Navalagi and R.K.Saraf,On fuzzy strongly α-irresolute functions(under preparation) [95] Govindappa Navalagi and R.K.Saraf,On fuzzy almost irresolute and fuzzy separation axioms,(under preparation) [96] Govindappa Navalagi and R.K.Saraf,On fuzzy semi α-irresolute functions,(under prepa- ration) [97] Govindappa Navalagi and R.K.Saraf,On fuzzy (θ − s)continuousfunctions,(underpreparation) [98] Govindappa Navalagi and R.K.Saraf,On fuzzy α-quasi -irresolute functions,(under preparation) [99] Govindappa Navalagi and R.K.Saraf,On fuzzy strongly α-closed functions,under preparation. [100] Govindappa Navalagi and R.K.Saraf,On fuzzy weakly α-irresolute functions,under preparation. [101] Govindappa Navalagi and R.K.Saraf, On fuzzy quasi α-open functions,under preparation. [102] G.J.Nazaroﬀ, Fuzzy topological polysystems,,J.Math.Anal.Appl.41(1973),478-485. [103] Y.B. Park ,S.J.Cho and J.H.Park,RS-compact fuzzy topological spaces,Fuzzy sets and system,77(1996)241-246. [104] Jin Han Park and H.Y.Ha, Fuzzy weakly preirresolute and fuzzy strongly preirresolute map- pings,J.Fuzzy Math.4(1996)131-140 [105] Jin Han Park,B.Y.Lee and J.R.Choi,Fuzzy θ-connectedness ,Fuzzy sets and systems,59(1993)237- 244. [106] Jin Han Park and B.H.Park,Fuzzy preirresolute mappings,,Pusan-Kyongnam Math.J.10(1995)303- 312. [107] Jin Han Park and B.H.Park,Fuzzy weakly open mappings on fuzzy topological spaces ,Pusan National Univ.Tech.Report,34(1992). [108] Jin Han Park,Yong Beom Park and Sung Jin Cho, Fuzzy completely pre-irresolute and weakly completely preirresolute mappings,Fuzzy sets and systems 97((1998)115-121. [109] J.H. Park, Y.B. Park and J.S Park, Fuzzy weakly open mappings and fuzzy RS-compact sets, Far East J. Math. Sci. Special vol., (1997), Part II, 201-212. [110] J.H.Park,B.Y.Lee and j.R.Choi,Fuzzy θ-connectedness,Fuzzy sets and systems,59(1993)237-244. [111] Z.Petricevic, A comparision of diﬀerent forms of continuity in fuzzy topology , to be published. [112] Z.Petricevic,Ro and R1 axioms in fuzzy topology, Mat.Vesnik 41(1989)21-28. [113] Z.Petricevic,A comparision of diﬀerent forms of continuity in fuzzy topology,(Unpublished). [114] Z.Petricevic,Weak forms of continuity in fuzzy topology,Mat.Vesnik ,42(1990),35-44. [115] R.Prasad ,S.S.Thakur and R.K.Saraf,Fuzzy α-irresolute mappaings,Jour. Fuzzy Math.2(2)(1994)335-339. [116] R.Prasad,S.S.Thakur and R.K.Saraf,Fuzzy α∗-continuous mappings, Proc.Nat.Sem.on Fuzzy Math.and Appl.,(1996)134-144. [117] R.Prasad,S.S.Thakur and R.K.Saraf, Fuzzy almost α-continuous mappings, Mathematical Fourmm,Vol.XII,1998,1-10. [118] K.S.Raja Sethupathy and S.Laksmivarahan,Connectedness in fuzzy topology,Kybernetika,Vol.13,No.3(1977)190-193 [119] S.E.Rodbaugh ,The Hausdorﬀ separation axioms for fuzzy topological spaces,Gen.Topology and its applications,11(1980)319. [120] S.E.Rodbaugh ,Connectivity and the L-fuzzy unit interval,Rocky Mountain J.Math.12(1982)113 [121] S.E.Rodbaugh ,A categorical accommodation of various notions of fuzzy topology,Fuzzy sets and systems,9)1983)241. [122] S.Saha,Fuzzy δ-continuous mappings, J.Math.Anal.Appl.126 (1987),130-142. [123] S.Saha,Separation axioms in fuzzy topological spaces,Fuzzy sets and systems,45(1992)261-270. [124] M.Sarkar,J.Math.Anal.Appl. ,79(1981)384-394. [125] R.K.Saraf ,M.Caldas and M.Khanna,Utilization of gs-closed sets in fuzzy topology (submitted)
44. 44. 44 GOVINDAPPA NAVALAGI [126] R.K.Saraf ,M.Caldas and M.Khanna,Fsg-closed maps and Fgs-closed maps,Jour.Tri.Math.Soc.2(2000) 69-76. [127] R.K.Saraf and M.Khanna,Fuzzy Generalized Semipreclosed sets,Jour.Tri.Math.Soc.,3(2001)59-68. [128] R.K.Saraf and S.Mishra,Fgα-closed sets,Jour.Tri.Math.Soc.2(2000) 27-32. [129] R.K.Saraf and M.Caldas,Fuzzy semipreopen sets ,Math.Education(1999),22-26. [130] R.K.Saraf ,M.C.Caldas and S.Mishra,Results via Fgα-closed sets and Fαg-closed sets,preprint. [131] R.K.Saraf,M.C.Caldas and S.Mishra,Fuzzy s-closed subspaces,Ultra Sci.12(2)(2000)229-233. [132] R.K.Saraf and M.Khanna, Fuzzy semi-generalized continuous mappings(Submitted) [133] R.K.Saraf and S.K.Gupta, δ-semipreopen set in fuzzy topology,The Bulletin,GUMA ,Vol. 591998)89-93(Published in Sept.1999). [134] R.K.Saraf,S.K.Gupta and M.Caldas, Fuzzy semipreopen sets, Applied Science Periodical Vol.1 No.1,Feb.(1999)22-25. [135] R.K.Saraf,S.Mishra and G.B.Navalagi,On Fuzzy strongly α-continuous functions (Submitted) [136] R.K.Saraf, S.Mishra and S.K.Gupta, δ-semiopen sets in fuzzy topology (submitted). [137] S.Saha, J.Math.Anal.Appl.126(1987) 130-142. [138] M.K.Singal and N.Rajavanshi,Fuzzy α-sets and α-continuous mappings,Fuzzy sets and Systems 48,383-390(1992). [139] S.P.Sinha and S .Malakar ,On s-closed fuzzy topological spaces,J.Fuzzy Math.2(1)(1994)95-103. [140] M.K.Singal and N.Prakash,Fuzzy semiopen sets,Jour.of Ind.Math.Soc.63(1997)171-182. [141] M.K.Singal and N.Prakash,Fuzzy preopen sets and fuzzy preseparation axioms,Fuzzy sets and systems,44(1991)273-281. [142] M.K.Singal and N.Prakash,Regularly open sets in fuzzy topological spaces ,Fuzzy sets and systems,50(1992)343-353. [143] S.Srivastava, S.N.Lal and A.K.Srivastava,Fuzzy Hausdroﬀ spaces.J.Math.Anal.Appl.81(1981),497– 506. [144] P.Sundaram and N.Nagaveni, On weakly generalized continuous maps,weakly general- ized closed maps and weakly generalized irresolute maps in topological spaces,Far East J.Math.Sci.6(6)(1998),903-912. [145] S.S.Thakur and S.Singh, Fuzzy semipreopen sets and fuzzy semi- precontinuity,Fuzzy sets and Systems,98(1998)383-391. [146] S.S.Thakur and R.K.Saraf, α-compact fuzzy topological space, Mathematica Bohemica,120(3)(1995),299-303. [147] S.S.Thakur and S .Singh , Fuzzy M-precontinuous mappings J.Indian Acad.Math.Vol.21,No.1(1999)39-44 [148] S.S.Thakur and R.K.Saraf,Fuzzy pre α-open mappings,Acta Ciencia Indica, Vol. XXI,No.4 M,507(1995). [149] S.S.Thakur and R.K.Saraf, Fuzzy pre-semiclosed mappings,(submitted) [150] S.S.Thakur and R.Malviya, Fuzzy gc-irresolute mappings,Proc.Math.Soc.,BHU II(1995),184-186. [151] S.S.Thakur and Kiran Koperiha, A note on semi-preopen sets,Mathematica,Tome 35(58)(1993)89- 92. [152] S.S. Thakur and R.K. Saraf, Fuzzy pre-semiclosed mappings,Ind.Sci.Cong.(CHENNAI) 1999. [153] C.K.Wang :Fuzzy points and local properties of fuzzy topology. J.Math.Anal.Appl.46(1974),316- 328. [154] C.K.Wang,Covering properties and fuzzy topological spaces,J.Math.Anal.Appl.43(1973)697. [155] C.K.Wang,Fuzzy topology: Product and Quotient spaces,J.Math.Anal.Appl.45(1973),697-704. [156] R.H.Warren, Neighbourhoods,bases,and continuity in fuzzy topological spaces,Rocky Mountain J.Math.8(1978)459-470. [157] R.H.Warren, Boundary of a fuzzy set ,Indiana Univ.Mathematical Jour.,Vol.26 ,No.2,(1997)191- 197. [158] R.H.Warren, Continuity of mappings of fuzzy topological spaces,Notices Amer.Math.Soc.21(1974),A-451. [159] T.H.Yalvac,Fuzzy sets and functions in fuzzy spaces,J.math. Anal.Appl.,126(1987),409-423. [160] T.H.Yalvac,Fuzzy semiinterior and fuzzy semiclosure in fuzzy sets,J.math. Anal.Appl.,132(1988),356-364.. [161] L.A.Zadeh,Fuzzy sets ,Inform. and Control. 8(1965) 338-353. [162] A.M.Zahran,Regularly open sets and a good extension on fuzzy topological spaces,Fuzzy sets and systems,116(2000)353-359.
45. 45. “DEFINITION BANK” IN FUZZY TOPOLOGY 45 [163] A.M.Zahran,On some near fuzzy continuous functions on fuzzy topological spaces,J.Fuzzy Math.3(1)(1995)209-219. [164] A.M.Zahran,Fuzzy regular semiopen sets and αs-closed spaces,J.Fuzzy Math.2(3)(1994)579-585. [165] A.M.Zahran,A note on the article ’Fuzzy less strongly semiopen sets and fuzzy less strongly semicontinuity,Fuzzy sets and systems,110(2000)143-144. [166] A.M.Zahran,A note on the article” A class of strong forms of fuzzy complete continuity”,Fuzzy sets and system 121(2001)365-366. [167] B.S.Zhong, Fuzzy strongly semiopen sets and fuzzy strongly semicontinuity, Fuzzy sets and systems,52(1992)345-351. [168] B.S.Zhong, Fuzzy pre-semiopen sets and fuzzy pre semicontinuity,Proc.ICIS’92(1992)918-920 [169] B.S.Zhong,Fuzzy weak semicontinuity, Fuzzy sets and systems 47(1992)93-98. [170] B.S.Zhong and Wang Wan Ling, Fuzzy non-continuous mappings and fuzzy pre-semi-separation axioms, Fuzzy sets and systems,94(1998)261-268. Department of Mathematics, G.H. College, Haveri-581110, Karnataka, India E-mail address: gnavalagi@hotmail.com
46. 46. “DEFINITION BANK” IN FUZZY TOPOLOGY GOVINDAPPA NAVALAGI Abstract. DEFINITION BANK in Fuzzy Topology is a nice collection of all existing deﬁnitions (particularly - weaker forms) w.r.t. fuzzy open sets, fuzzy closed sets,fuzzy mappings(=fuzzy functions), fuzzy separation axioms and their generalized weak forms . Also, it includes some of the newest deﬁnitions w.r.t to fuzzy subsets, fuzzy functions ,fuzzy separation axioms and fuzzy covering axioms. Contents Introduction 2 Part 1. Fuzzy Sets and Fuzzy Neighbourhoods 5 1.1. Fuzzy Regular open and fuzzy regular closed sets 5 1.2. Fuzzy semi (pre, α, semipre)-sets and their neighbourhoods 7 1.3. Fuzzy generalized open (semiopen, preopen, α-open etc.) and fuzzy generalized closed (semiclosed, preclosed, α-closed etc.) sets 9 Part 2. Fuzzy Separation Axions 12 2.1. Fuzzy separation Axioms 12 2.2. Weaker forms of Fuzzy separation Axioms 15 2.3. Fuzzy Regularity Axioms 16 2.4. Fuzzy Normality Axioms 18 2.5. Fuzzy Compactness 19 Date: 28-12-2001. 1991 Mathematics Subject Classiﬁcation. 54A40. Key words and phrases. Fuzzy Preopen, fuzzy semiopen, fuzzy α-open, fuzzy β- open(=fuzzy semipreopen),fuzzy θ-open, fuzzy semi-θ-closed sets, etc.... fuzzy Pre- continuous, fuzzy semicontinuous,fuzzy semiprecontinuous,fuzzy α-continuous ,fuzzy Pre- open,fuzzy semiopen ,fuzzy α-open,fuzzy α-closed , fuzzy weakly semiopen,fuzzy weakly preopen,fuzzy weakly α-open functions, etc...,fuzzy pre-To, fuzzy − pre − T1, fuzzysemi − T2, fuzzyalmostregular, fuzzyp − normal, fuzzyalmostp − normal, fuzzys − closed, fuzzyS − closed, fuzzys − compact, fuzzystronglycompactspaces.. 1
47. 47. 2 GOVINDAPPA NAVALAGI 2.6. Fuzzy S-closed spaces 22 2.7. Fuzzy Connected spaces 23 2.8. Fuzzy Extremally disconnected spaces and fuzzy submaximal spaces 24 Part 3. Fuzzy Mappings 25 3.1. Fuzzy Continuity, fuzzy openness and its allied deﬁnitions 25 3.2. Fuzzy weak forms of continuity, openness and allied deﬁnitions 28 3.3. Fuzzy weak forms of generalized continuity and fuzzy generalized openness and allied deﬁnitions 36 Acknowledgement 40 References 40 Introduction The concept of fuzzy sets was introduced by Prof. L.A. Zadeh in his classi- cal paper[161]. After the discovery of the fuzzy subsets, much attention has been paid to generalize the basic concepts of classical topology in fuzzy setting and thus a modern theory of fuzzy topology is developed. The notion of fuzzy sub- sets naturally plays a signiﬁcant role in the study of fuzzy topology which was introduced by C.L. Chang[31] in 1968. In 1980, Ming and Ming[73], introduced the concepts of quasi-coincidence and q-neighbourhoods by which the extensions of functions in fuzzy setting can very interestingly and eﬀectively be carried out. Since then many concepts of General Topology are being extend to Fuzzy Topol- ogy.Fuzzy semiopen sets were ﬁrst introduced and studied by K.K.Azad[10] in 1981.In 1991, Bin Shahna[19] extended the concepts of preopen and α-open sets in Fuzzy Topology.In 1998,S.S.Thakur and S.Singh [152] introduced the concept of Fuzzy semipreopen sets in Fuzzy Topology.The generalization of fuzzy general- ized open(resp.fuzzy generalized closed) sets and fuzzy generalized continuity was extensively studied in recent years by S.S.Thakur, R.Malviya, H.Maki,T.Fukutaka,
48. 48. “DEFINITION BANK” IN FUZZY TOPOLOGY 3 M.Kojima, H.Harade,R.K.Saraf, M.Caldas and M.Khanna.Fuzzy continuity of Chang in 1968, it has been proved to be of fundamental importance in the realm of fuzzy topology.Along with this, many researchers[10],[?],[169],[?],[47], [80]and[160] have studied fuzzy non-continuity. One of them [169] introduced and studied fuzzy pre semiopen sets and fuzzy pre semicontinuus mappings in fuzzy topological spaces. Deﬁnitions of Fuzzy sets and their fuzzy neighbourhoods are available in Section I. In Section II, collection of deﬁnitions concern with the fuzzy separation axioms is made. Deﬁnitions concern with the fuzzy mappings (fuzzy functions) are available in Section III. In this survey article we made an attempt of bringing all available deﬁnitions in ”Fuzzy Topology” under single umbrella, called “Deﬁnition Bank In Fuzzy Topology” . We also suggest some new deﬁnitions w.r.t the above dis- cussions, marked in the list as DeﬁnitionNEW . Hence, one can observe the ”New Deﬁnitions”.Regarding the “Deﬁnition Bank in Fuzzy Topology ”, suggestions, cor- rections or additions to either list would be gratefully received. Throughout this paper by (X, τ)or simply by X we mean a fuzzy topological space (fts, shorty) due to Chang [31] we give the following deﬁnitions : Deﬁnition 0.0.1. (a) Let X be a non-empty set and I the unit interval [0,1]. A fuzzy set in X is an element of the set IX of all functions from X to I. (b) 0X and 1X denote the fuzzy sets given by 0X(x) = 0 , for all x ∈ X and 1X(x) = 1 , for all x ∈ X . (c) Equality of two fuzzy sets λ and µ on X is determined by the usual equality condition for mappings, which is given by λ = µ ⇒ (for all x ∈ X,λ(x) = µ(x). (d) A fuzzy subset λ on X is said to be a subset of a fuzzy β on X written as λ ≤ β, if λ(x) ≤ β(x), for all x ∈ X (e) The complement of a fuzzy set λ on X is given by Co(λ) or simply λ = 1 − λ (f) A fuzzy topology τ on X is collection of subsets of IX , such that (i)0X,1X ∈ τ , (ii)if λ, β ∈ τ, then β ∈ τ, (iii)if λi ∈ τ for each i ∈ Λ , then i∈Λ λi ∈ τ.
49. 49. 4 GOVINDAPPA NAVALAGI The pair (X, τ) is called a fuzzy topological space (in short, fts or fuzzy space ). (g) Closure of a fuzzy set λ is denoted by Cl(λ) or λ bar , and is given by Cl(λ) = {µ : µisafuzzyclosedsetandλ ≤ µ} The interior of λ is denoted by Int(λ) , and is given by Int(λ) = {ν : νisafuzzyopensetandλ ≥ ν}. (h) A fuzzy set λ in a fts (X, τ) is a neighbourhood ,or nbhd for short, of a fuzzy set µ iﬀ there exists an open fuzzy set β such that µ ⊂ β ⊂ λ. (i) Let λ and β be two fuzzy sets in a fts (X, τ), and let β ⊂ λ.Then β is called an interior fuzzy set of λ iﬀ λ is a nbhd of β.The union of all interior fuzzy sets of λ is called the interior of λ and is denoted by λo . R.H.Warren in 1977 and 1978 , deﬁned the following. Deﬁnition 0.0.2. [156] (a) Let λ be a fuzzy set in a fts (X, τ).A point x ∈ X is called a fuzzy limit point of λ iﬀ whenever λ(x) = 1, then for each x∃y ∈ X − {x} such that x(y)Λa(y) = 0 ; or whenever a(x) = 1, then a−1 (x) > 0 and for each open x satisfying 1−x = a(x)∃y ∈ X − x such that x(y)Λa(y) = 0. The fuzzy derived fuzzy set of a (denoted by a’ ) and deﬁned as : a’(x) = a− (x)ifxisafuzzylimitpointofa, = 0otherwise. (b) Let (X, τ) be a fts and let A ⊂ X .Then the family, τA = {g|A : g ∈ τ} is a fuzzy topology on A, where g|A is the restriction of g to A,called the relative fuzzy topology on A or the fuzzy topology on A induced by the fuzzy topology τ on X.Note that (A, τA) is called a subspace of (X, τ). In 1973,J.A.Goguen et al[44] the following is given: Deﬁnition 0.0.3. Let τ be a fuzzy topology on X and let B, S ⊂ τ .Then B is called a basis for τ iﬀ each element of τ is the supremum of members of B.Also, S is called a subbasis for τ iﬀ the family of all ﬁnite inﬁmums of elements of S is a basis for τ. Due to Chang[31] and Goguen et alGSW1 the following is given:
50. 50. “DEFINITION BANK” IN FUZZY TOPOLOGY 5 Deﬁnition 0.0.4. Let f be a function from X to Y .Let b be a fuzzy set in Y and let a be a fuzzy set in X.Then the inverse image of b under f4isthefuzzysetf−1 (b) in X deﬁned by f−1 (b)(x) = b(f(x)) for x ∈ X i.e., f−1 (b) = bof.The image of a under f is the fuzzy set f(a) in Y deﬁned by f(a)(y) = {a(x) : f(x) = y} for y ∈ Y i.e., f(a)(y) = Sup{a(x) : x ∈ f−1 (y)}. In 1973,G.J.Nazaroﬀ[102]have deﬁned the fuzzy closure of a fuzzy set : Deﬁnition 0.0.5. Let a be a fuzzy set in a fts (X, τ).Then {b : bisaclosedfuzzysetinXandb ≥ a} is called the closure of a and is denoted by a− . In 1977,R.H.Warren[157] have deﬁned the fuzzy boundary set and fuzzy bound- ary operators in the following. Deﬁnition 0.0.6. Let a be a fuzzy set in a fts (X, τ).The fuzzy boundary of a , denoted by ab , is deﬁned as the inﬁmum of all closed fuzzy sets d in X with the property : d(x) ≥ a− (x) for all x ∈ X for which (a− (1 − a)− )(x) > 0 . Note: Clearly,(i) ab is a fuzzy closed set and ab ≤ a− ;(ii)(a− (1 − a)− )(x) = 0 , then ab = {allfuzzyclosedsetsinX} = 0. Note:In a fts (X, τ), if α(a) = ab for each fuzzy set a Part 1. Fuzzy Sets and Fuzzy Neighbourhoods 1.1. Fuzzy Regular open and fuzzy regular closed sets Deﬁnition 1.1.1. A fuzzy set λ in a fts X is called, (1)Fuzzy regular open[10] if λ = Int(Cl(λ)). (2) Fuzzy regular closed[10] if λ = Cl(Int(λ)). A fuzzy point in X with support x ∈ X and value p(0 < p ≤ 1) is denoted by xp. Two fuzzy sets λ and β are said to be quasi-coincident (q-coincident, shorty) denoted by λqβ, if there exists x ∈ X such that λ(x) + β(x) > 1[80] and by − q we denote ” is not q-coincident ” . It is known[80] that λ ≤ β if and only if λq(1 − β).
51. 51. 6 GOVINDAPPA NAVALAGI A fuzzy set λ is said to be q-neighbourhood (q-nbd) of xp if there is a fuzzy open set µ such that xpqµ, and µ ≤ λ if µ(x) ≤ λ(x) for all x ∈ X. The interior, closure and the complement of a fuzzy set λin X are denoted by Int(λ), Cl(λ) and 1 − λ = λc respectively. Recall that a fuzzy point xp is said to be a fuzzy θ-cluster point of a fuzzy set λ[80], if and only if for every fuzzy open q-nbd µ of xp , Cl(µ) is q-coincident with λ. The set of all fuzzy θ-cluster points of λ is called the fuzzy θ-closure of λ and will be denoted by Clθ(λ). A fuzzy set λ will be called θ-closed if and only if λ = Clθ(λ). The complement of a fuzzy θ-closed set is called of fuzzy θ-open and the θ-interior of λ denoted by Intθ(λ) is deﬁned as Intθ(λ) = {xp : for some fuzzy open q-nbd, β of xp, Cl(β) ≤ λ}. Deﬁnition 1.1.2. [161] Let X be a non-empty set and I be the unit interval [0,1]. A fuzzy set in X is a mapping from X into I.The null set 0 (zero) is the mapping from X into I assumes only the value 0 and the whole set 1, is the mapping from X into I which takes the value 1 only. Deﬁnition 1.1.3. [161] A fuzzy set A is contained in a fuzzy set B denoted by A ≤ B iﬀ A(x) ≤ B(x) , for each x ∈ X.The complement Ac of a fuzzy set A of X is (1 − A deﬁned by (1 − A)c (x) = 1 − A(X), for each x ∈ X.If A is a fuzzy set of X and B is a fuzzy set of Y then AxB is a fuzzy set of XxY , deﬁned by (AxB)(x, y) = (A(x), B(y)), for each (x, y) ∈ XxY . Deﬁnition 1.1.4. [31] Let f : X → Y be a mapping .If A is a fuzzy set of Y , then f−1 (A) is a fuzzy set of X , deﬁned by f−1 (A)(x) = A(f(x)) for each x ∈ X. Deﬁnition 1.1.5. [93] (i)Symbole Ao will stand for the support of A in X ; (ii) Symbol, Cλ will stand for the fuzzy set of X having the value λ at each point in X, where λ ∈ (0, 1]. Deﬁnition 1.1.6. [58] A fuzzy set on X is called a fuzzy singleton iﬀ it takes the value 0 for all points x ∈ X except 1.