1 s2.0-s1574035804702117-main

“DEFINITION BANK” IN FUZZY TOPOLOGY

GOVINDAPPA NAVALAGI

Abstract. DEFINITION BANK in Fuzzy Topology is a nice collection of allexisting definitions (particularly - weaker forms) w.r.t. fuzzy open sets, fuzzyclosed sets,fuzzy mappings(=fuzzy functions), fuzzy separation axioms andtheir generalized weak forms . Also, it includes some of the newest definitionsw.r.t to fuzzy subsets, fuzzy functions ,fuzzy separation axioms and fuzzycovering 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 weaklypreopen,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..

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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,

“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 .


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 λbar , 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) 6= 0 ; or whenever a(x) 6= 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) 6= 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:


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).


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.


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


(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)


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 6= 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.


[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)


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]


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


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 FταCl(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).


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


(i)x 6= 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 .



(i)x 6= 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 6= 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 6= 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 6= 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 6= y imply that there exist fuzzy preopen sets a and b with a(x) = 1 = b(y) and

a ≤ 1 − b.


Definition 2.1.7. NEW A fts (X, ) is called α-Hausdorff space if x, y ∈ X with

x 6= 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 6= 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.


(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);


(iii) fuzzy almost T2 (in short FAT2) if for each pair of fuzzy singletons p and q


and q ≤ V ≤ Co(p) and U ≤ Co(V );

(iv) fuzzy almost T21\2 (in short FAT21\2) 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 ≤ pc2 , 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 6= 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



(i)when x 6= 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 ;


V q−xα.



(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 .


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 .


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


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 AB0 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


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


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



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 .


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.


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∨


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∨


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} 6= 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 .


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 .


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 .


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.


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.


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).


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 f is 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.


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 .


sense of Ganguly et al (written as F.a.o.G) if for each open set U of Y , f−1Cl(A) ≤Cl(f−1(A)).


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 .


(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 .



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



(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


(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


(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]



(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


(i)fuzzy preirresolute[106] if f−1(V ) is a fuzzy preopen in X for each fuzzy

pre-open set V in Y .

(ii)fuzzy weakly preirresolute[104] if f−1(V ) is a fuzzy preopen 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.


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 .


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 :


(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 .


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 .


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.


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)].


called fuzzy α-weakly continuous if for each fuzzy α-open set λ of Y , we have

f−1(λ) ≤ αInt[f−1αClλ)].

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 .


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 .


(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


(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)


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 .


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 .


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.

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Department of Mathematics, G.H. College, Haveri-581110, Karnataka, IndiaE-mail address: [email protected]

“DEFINITION BANK” IN FUZZY TOPOLOGY

GOVINDAPPA NAVALAGI

Abstract. DEFINITION BANK in Fuzzy Topology is a nice collection of allexisting definitions (particularly - weaker forms) w.r.t. fuzzy open sets, fuzzyclosed sets,fuzzy mappings(=fuzzy functions), fuzzy separation axioms andtheir generalized weak forms . Also, it includes some of the newest definitionsw.r.t to fuzzy subsets, fuzzy functions ,fuzzy separation axioms and fuzzycovering 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 weaklypreopen,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


2.6. 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,


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 .


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 λbar , 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) 6= 0 ; or whenever a(x) 6= 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) 6= 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:


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


(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).


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.


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


(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)


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 6= 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.


[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


strongly semiopen set iff there exists B ∈ τ such that B ≤ A ≤ Int(ClB)


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]


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


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 FταCl(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).


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




(i)x 6= 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 .



(i)x 6= 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 6= 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 6= 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 6= 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 6= y imply that there exist fuzzy preopen sets a and b with a(x) = 1 = b(y) and

a ≤ 1 − b.


Definition 2.1.7. NEW A fts (X, ) is called α-Hausdorff space if x, y ∈ X with

x 6= 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 6= y imply that there exist fuzzy β-open sets a and b with a(x) = 1 = b(y) and

a ≤ 1 − b.


(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.


(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


and q ≤ V ≤ Co(p);


(iii) fuzzy almost T2 (in short FAT2) if for each pair of fuzzy singletons p and q


and q ≤ V ≤ Co(p) and U ≤ Co(V );

(iv) fuzzy almost T21\2 (in short FAT21\2) 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 ≤ pc2 , 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


(i)when x 6= 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 ;


V q−xα.




(i)when x 6= 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 ;


V q−xα.



(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


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 .


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 .


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


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


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


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


Definition 2.4.4. NEW An fts (X, τ) is said to be fuzzy p-normal iff


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



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∨


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 .


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.


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∨


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∨


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∨


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} 6= 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 .


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 .


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 .


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 .


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.



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.


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).


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



(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.


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)).


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 .


sense of Ganguly et al (written as F.a.o.G) if for each open set U of Y , f−1Cl(A) ≤Cl(f−1(A)).



(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 .


(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 .


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 .



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



(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


(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


(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


(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]



(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


(i)fuzzy preirresolute[106] if f−1(V ) is a fuzzy preopen in X for each fuzzy

pre-open set V in Y .

(ii)fuzzy weakly preirresolute[104] if f−1(V ) is a fuzzy preopen 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.


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 .


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 :


(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 .


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 .


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.


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)].


called fuzzy pre-weakly continuous if for each fuzzy preopen set λ of Y , we have

f−1(λ) ≤ pInt[f−1(pClλ)].


called fuzzy α-weakly continuous if for each fuzzy α-open set λ of Y , we have

f−1(λ) ≤ αInt[f−1αClλ)].

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 .


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 .


(ii) strongly fuzzy continuous if the inverse image of each fuzzy set in Y is both


(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


(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)