1 s2.0-s1574035804702117-main

“DEFINITION BANK” IN FUZZY TOPOLOGY
GOVINDAPPA NAVALAGI
Abstract. DEFINITION BANK in Fuzzy Topology is a nice collection of all
existing definitions (particularly - weaker forms) w.r.t. fuzzy open sets, fuzzy
closed sets,fuzzy mappings(=fuzzy functions), fuzzy separation axioms and
their generalized weak forms . Also, it includes some of the newest definitions
w.r.t to fuzzy subsets, fuzzy functions ,fuzzy separation axioms and fuzzy
covering axioms.
Contents
Introduction 2
Part 1. Fuzzy Sets and Fuzzy Neighbourhoods 5
1.1. Fuzzy Regular open and fuzzy regular closed sets 5
1.2. Fuzzy semi (pre, α, semipre)-sets and their neighbourhoods 7
1.3. Fuzzy generalized open (semiopen, preopen, α-open etc.) and fuzzy
generalized closed (semiclosed, preclosed, α-closed etc.) sets 9
Part 2. Fuzzy Separation Axions 12
2.1. Fuzzy separation Axioms 12
2.2. Weaker forms of Fuzzy separation Axioms 15
2.3. Fuzzy Regularity Axioms 16
2.4. Fuzzy Normality Axioms 18
2.5. Fuzzy Compactness 19
Date: 28-12-2001.
1991 Mathematics Subject Classification. 54A40.
Key words and phrases. Fuzzy Preopen, fuzzy semiopen, fuzzy α-open, fuzzy β-
open(=fuzzy semipreopen),fuzzy θ-open, fuzzy semi-θ-closed sets, etc.... fuzzy Pre-
continuous, fuzzy semicontinuous,fuzzy semiprecontinuous,fuzzy α-continuous ,fuzzy Pre-
open,fuzzy semiopen ,fuzzy α-open,fuzzy α-closed , fuzzy weakly semiopen,fuzzy weakly
preopen,fuzzy weakly α-open functions, etc...,fuzzy pre-To, fuzzy − pre − T1, fuzzysemi −
T2, fuzzyalmostregular, fuzzyp − normal, fuzzyalmostp − normal, fuzzys − closed, fuzzyS −
closed, fuzzys − compact, fuzzystronglycompactspaces..
1

2 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 λ
bar , and is given by
Cl(λ) = {µ : µisafuzzyclosedsetandλ ≤ µ}
The interior of λ is denoted by Int(λ) , and is given by
Int(λ) = {ν : νisafuzzyopensetandλ ≥ ν}.
(h) A fuzzy set λ in a fts (X, τ) is a neighbourhood ,or nbhd for short, of a fuzzy
set µ iff there exists an open fuzzy set β such that µ ⊂ β ⊂ λ.
(i) Let λ and β be two fuzzy sets in a fts (X, τ), and let β ⊂ λ.Then β is called
an interior fuzzy set of λ iff λ is a nbhd of β.The union of all interior fuzzy sets of
λ is called the interior of λ and is denoted by λo
.
R.H.Warren in 1977 and 1978 , defined the following.
Definition 0.0.2. [156] (a) Let λ be a fuzzy set in a fts (X, τ).A point x ∈ X is
called a fuzzy limit point of λ iff whenever λ(x) = 1, then for each x∃y ∈ X − {x}
such that x(y)Λa(y) = 0 ; or whenever a(x) = 1, then a−1
(x) > 0 and for each
open x satisfying 1−x = a(x)∃y ∈ X − x such that x(y)Λa(y) = 0.
The fuzzy derived fuzzy set of a (denoted by a’ ) and defined as : a’(x) =
a−
(x)ifxisafuzzylimitpointofa, = 0otherwise.
(b) Let (X, τ) be a fts and let A ⊂ X .Then the family, τA = {g|A : g ∈ τ}
is a fuzzy topology on A, where g|A is the restriction of g to A,called the relative
fuzzy topology on A or the fuzzy topology on A induced by the fuzzy topology τ
on X.Note that (A, τA) is called a subspace of (X, τ).
In 1973,J.A.Goguen et al[44] the following is given:
Definition 0.0.3. Let τ be a fuzzy topology on X and let B, S ⊂ τ .Then B is
called a basis for τ iff each element of τ is the supremum of members of B.Also, S
is called a subbasis for τ iff the family of all finite infimums of elements of S is a
basis for τ.
Due to Chang[31] and Goguen et alGSW1 the following is given:

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
Definition 1.2.1. A fuzzy set λ in a fts X is called,
(1) Fuzzy semiopen[10] if λ ≤ Cl(Int(λ)) (in short Fs-open set).
(2) Fuzzy semiclosed[10] if Int(Cl(λ)) ≤ λ (in short Fs-closed set).
(3) Fuzzy preopen[19] if λ ≤ Int(Cl(λ)).
(4) Fuzzy preclosed[19] if Cl(Int(λ)) ≤ λ.
(5) Fuzzy α-open[19]if λ ≤ Int(Cl(Int(λ))) (in short Fα-open ).
(6) Fuzzy semipreopen[145]if λ ≤ Cl(Int(Cl(λ))) (in short Fsp-open)
(7) Fuzzy α-closed[19]if Cl(Int(Cl(λ))) ≤ λ (in short Fα-closed).
(8) Fuzzy semipreclosed =β-closed [145]if Int(Cl(Int(λ) ≤ λ (in short Fsp-
closed)
The family of all fuzzy semiopen (fuzzy preopen, fuzzy α-open and fuzzy semipreopen)
sets of X is denoted by FSO(X) (resp. FPO(X), Fα(X) and FSPO(X)).
Recall that if, λ be a fuzzy set in a fts X then pCl(λ) = {β : β ≥ λ, β is
fuzzy preclosed} (resp. pInt(λ) = {β : λ ≥ β, β is fuzzy preopen}) is called a
fuzzy preclosure of λ (resp. fuzzy preinterior of λ)[19]and[140].
Definition 1.2.2. [160] The fuzzy semiclosure of A , denoted by sCl(A) ,is defined
by the intersection of all fuzzy semiclosed sets containing A.
[160] The fuzzy semiinterior of A , denoted by sInt(A) , is defined by the union
of all fuzzy semiopen sets contained in A.
Definition 1.2.3. [80, 92] A fuzzy subset A of an fts X is said to be fuzzy semi-
regular ,if it is both fuzzy semioen set and fuzzy semiclosed set.
The family of all fuzzy semi-regular sets of an fts X is denoted by FSO(X)

Definition 1.2.4. [80, 92] A fuzzy point xβ of X is said to be in the fuzzy semi-
θ-closure of A , denoted by sClθ(A), if A ∩ sClU = 0 for every U ∈ FSO(X)
containing xβ.
Note that A is called fuzzy semi-θ-closed[92] if A = sClθ(A).The complement of
a fuzzy semi-θ-closed set is called fuzzy semi-θ-open set.
S.S.Thakur and S.Singh[145]have defined semi-preclosure and semi-preinterior of
a fuzzy subset A of fts X as follows;
Definition 1.2.5. [145] A fuzzy set λ in a fts X is called:
(i)fuzzy semipreopen if there exists a fuzzy preopen set such that ≤ λ ≤ Cl.
(ii)fuzzy semipreclosed if there exists a fuzzy preclosed set such that Int ≤ λ ≤.
Definition 1.2.6. [145] spCl(A) = Inf{B : B ≥ A; BisFsp − closedsetinX}
spInt(A) = Sup{B : B ≤ A; BisFsp − opensetofX}.
Definition 1.2.7. Let A be a fuzzy set in an fts X and xβ be a fuzzy point in
X.Then,
(i) A is called a fuzzy pre-q-nbd of xβ[106]if there exists a fuzzy preopen subset
B in X such that xβqB ≤ A.
(ii)a fuzzy pre-θ-nbd[104]ofxβ if there exists fuzzy preopen pre-q-nbd B of xβ
such that pCl(B)q−
(1 − A.
Definition 1.2.8. [169] A fuzzy set A in X is said to be
(i)fuzzy pre-semi-open if A ≤ (A−
)o ;
(ii) fuzzy pre-semi-closed if A ≥ (Ao
)−.
Definition 1.2.9. [167] A fuzzy set A in X is called a fuzzy pre-semi-nbd of a
fuzzy point xα in X iff there exists a fuzzy pre-semiopen set B in X such that
xα ∈ B ≤ A.
Definition 1.2.10. [167] A fuzzy set A in X is called a fuzzy pre-semi-q-nbd of
a fuzzy point xα in X iff there exists a fuzzy pre semiopen set B in X such that
xαqB ≤ A.

[167] Every fuzzy q-nbd of a fuzzy point is always a fuzzy semi-q-nbd of the fuzzy
point and every fuzzy semi-q-nbd of a fuzzy point is always a fuzzy pre semi-q-nbd
of the fuzzy point.
Definition 1.2.11. [70] A fuzzy set λ in a fts X is called :
(i)fuzzy -β-open if λ ≤ ClIntCl(λ) ;
(ii) fuzzy-β-closed if λc
is a fuzzy -β-open ,or equivalently, if IntClInt(λ) ≤ λ.
Definition 1.2.12. [76] Let A be a fuzzy set of an fts (X, τ).Then A is called a fuzzy
less strongly semiopen set iff there exists B ∈ τ such that B ≤ A ≤ sInt(sClB)
Definition 1.2.13. [76] Let A be a fuzzy set of an fts (X, τ).Then A is called
a fuzzy less strongly semiclosed set iff there exists fuzzy closed set B such that
sCl(sInt(B) ≤ A ≤ B
Definition 1.2.14. [167] A be a fuzzy set of an fts (X, τ).Then A is called a fuzzy
strongly semiopen set iff there exists B ∈ τ such that B ≤ A ≤ Int(ClB)
Definition 1.2.15. [167] A be a fuzzy set of an fts (X, τ).Then A is called a fuzzy
strongly semiclosed set iff there exists a fuzzy closed set B such that Cl(Int(B ≤
A ≤ B
Definition 1.2.16. [13] (i) The class consisting of all fuzzy α-sets of fts (X, τ) is
called a fuzzy α-structure and is denoted by τα
;
(ii) The class consisting of all fuzzy β-open sets of fts (X, τ) is called a fuzzy
β-structure and is denoted by τβ
.
1.3. Fuzzy generalized open (semiopen, preopen, α-open etc.) and
fuzzy generalized closed (semiclosed, preclosed, α-closed etc.)
sets
For the first time the concept of fuzzy generalized closed sets was considered by
S.S.Thakur and R.Malviya[150]

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 Fτα
Cl(A) ≤ B whenever A ≤ B and B is
Fα-open in X .We denote the collection of all these sets by FG(X).
The complement of a Fgα-closed set is called Fgα-open set.
Definition 1.3.12. NEW
A fuzzy subset A of X is called a Fsg*-closed (resp.
Fg*s-closed, Fg ∗ α-closed, Fαg∗-closed, Fg*p-closed, Fg*sp-closed, Fθ-g*-closed
and Fδ-g*-closed) set if sclA ≤ U (resp. sclA ≤ U, clα ≤ U, clα ≤ U, pclA ≤ U,
spclA ≤ U, clθ ≤ U and clδ ≤ U) whenever A ≤ U and U is Fsg-open (resp.
Fgs-open, Fgα-open, Fαg-open, Fgp-open, Fgsp-open, Fθ-g-open and Fδ-g-open)
set in X.
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).

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 = y implies that there an fuzzy open nbd of xλ which is not q-coincident
with yµ and there is fuzzy open nbd of yµ which is not q-coincident with xλ ;
(ii) x = y and λ < µ (say) imply that xλ has an fuzzy open nbd and ,there is a
q-nbd of yµ which is not q-coincident with xλ .
Definition 2.1.2. [93] An fts (X, τ) is said to be FT2 -space in the sence of Ganguly
and Shah ,if for two distinct fuzzy points xλ ,yµ in X
(i)x = y implies xλ ,yµ have fuzzy open nbds which not q-coincident ;
(ii) x = y and λ < µ (say) imply that xλ has an fuzzy open nbd and ,yµ has an
fuzzy open q -nbd which are not q-coincident.
Definition 2.1.3. [112] An fts (X, τ) is said to be FT2 -space in the sence of
Z.Petrivic,if for each pair of points,
(i)xt ,yr ,x = y implies that there exist two disjoint fuzzy open sets λ and µ such
that xt ∈ λ ,yr ∈ µ ;
(ii) x=y and t¡r implies that there exists fuzzy open set λ such that xt ∈ λ
,yrqCl(λ) .
In 1981,Malghan and Benchalli[67] have defined the following Hausdorff space
which is weaker than that of T.E.Gantner et al.
Definition 2.1.4. [67] A fts (X, ) is called Hausdorff (we denote it by Hausdorff
(MB))space if x, y ∈ X with x = y imply that there exist fuzzy open sets a and b
with a(x) = 1 = b(y) and a ≤ 1 − b.
Similar to the above definition of Hausdorff space, we define the following.
A fts (X, ) is called semi-Hausdorff space if x, y ∈ X with
x = y imply that there exist fuzzy semiopen sets a and b with a(x) = 1 = b(y) and
a ≤ 1 − b.
A fts (X, ) is called pre-Hausdorff space if x, y ∈ X with
x = y imply that there exist fuzzy preopen sets a and b with a(x) = 1 = b(y) and
a ≤ 1 − b.

A fts (X, ) is called α-Hausdorff space if x, y ∈ X with
x = y imply that there exist fuzzy α-open sets a and b with a(x) = 1 = b(y) and
a ≤ 1 − b.
A fts (X, ) is called β-Hausdorff space if x, y ∈ X with
x = y imply that there exist fuzzy β-open sets a and b with a(x) = 1 = b(y) and
a ≤ 1 − b.
Definition 2.1.9. [95] A fts (X, τ) is said to be :
(i) To iff for any pair of distinct points xλ and yµ of X , either xλ has a q-nbd
which is not q-conincident with yµ or yµ has a q-nbd which is not q-coincident with
xλ;
(ii) T1 iff any pair of distinct points in X are weakly separated ;
(iii) T2 iff any pair of distinct points in X are strongly separated ;
(iv) regular iff any point xλ and any closed set A in X , such that xλ /∈ A, are
strongly separated;
(v) normal iff any two closed sets of X , which are not q-coincident , are strongly
separated;
(vi) A T1 -regular space is called a T3 -space;
(vii) A T1-normal space is called a T4 -space;
(viii) completely normal iff for any two weakly separated sets are strongly sepa-
rated.
Definition 2.1.10. [142] A fts (X, τ) is said to be :
(i) fuzzy almost To (in short FATo) if for each pair of fuzzy singletons p and q
with different supports in X , there exists U ∈ RO(τ) such that p ≤ U ≤ Co(q) or
q ≤ U ≤ Co(p);
(ii) fuzzy almost T1 (in short FAT1) if for each pair of fuzzy singletons p and q
with different supports in X , there exists U, V ∈ RO(τ) such that p ≤ U ≤ Co(q)
and q ≤ V ≤ Co(p);

(iii) fuzzy almost T2 (in short FAT2) if for each pair of fuzzy singletons p and q
with different supports in X , there exists U, V ∈ RO(τ) such that p ≤ U ≤ Co(q)
and q ≤ V ≤ Co(p) and U ≤ Co(V );
(iv) fuzzy almost T212 (in short FAT212) if for each pair of fuzzy singletons p and
q with different supports in X , there exists U, V ∈ RO(τ) such that p ≤ U ≤ Co(q)
and q ≤ V ≤ Co(p) and δCl(U) ≤ Co(δCl(V ));
(v) fuzzy almost regular (in short FAR) if for each pair consisting of a fuzzy
singleton p and a fuzzy regular closed set A in X such that p ≤ Co(A), there exist
U, V ∈ τ such that p ≤ U , A≤ V and U ≤ Co(V );
(vi) fuzzy almost normal (in short FAN)if for each pair consisting of a fuzzy
closed set A and a fuzzy regular closed set B in X such that A ≤ Co(B), there
exist U, V ∈ τ such that B ≤ U ,A ≤ V and U ≤ Co(V );
(vii) fuzzy almost mildly normal (in short FMN)if for each pair consisting of a
fuzzy regular closed sets A and B in X such that A ≤ Co(B), there exist U, V ∈ τ
such that B ≤ U ,A ≤ V and U ≤ Co(V ).
2.2. Weaker forms of Fuzzy separation Axioms
Definition 2.2.1. [140] A fuzzy topological space X is said to be fuzzy semi-T2 iff
for every pair of fuzzy singletons p1 and p2 with different supports, there exist two
fuzzy semiopen sets U and V such that p1 ≤ U ≤ pc
2 , p2 ≤ V pc
1 and U ≤ V c
.
Definition 2.2.2. [169] An fts X is called fuzzy pre-semi-To iff for every pair of
distinct fuzzy points xα and yβ, the following conditions are satisfied:
(i)when x = y, either xα has a fuzzy pre-semi-nbd U such that Uq−
yβ or yβ has
a pre-semi-nbd V such that V q−
xα ;
(ii)when x = y and α < β (say), yβ has a fuzzy pre-semi-q-nbd V such that
V q−
xα.
Definition 2.2.3. [169] An fts X is called fuzzy pre-semi-T1 iff for every pair of

(i)when x = y, either xα has a fuzzy pre-semi-nbd U and yβ has a pre-semi-nbd
V such that Uq−
yβ and V q−
xα ;
(ii)when x = y and α < β (say), yβ has a fuzzy pre-semi-q-nbd V such that
V q−
xα.
Definition 2.2.4. [169] An fts X is called fuzzy pre-semi-T1 iff for every pair of
(i)when x = y , and xα and yβ have fuzzy pre semi-nbds which are not q-
coincident ;
(ii)when x = y and α < β (say),then xα has a fuzzy pre-semi-nbd U and yβ has
a fuzzy pre-semi-nbd V such that Uq−
V .
Definition 2.2.5. [15] A fts X is called fuzzy T1/2-space if every generalized fuzzy
closed set in X is fuzzy closed in X.
2.3. Fuzzy Regularity Axioms
Definition 2.3.1. [93] An fts (X, τ) is said to be fuzzy regular in the sence of
Ganguly and Shah ,iff for a fuzzy point xλ and a fuzzy closed set A in X :
(i)A(x) = 0 implies there exist fuzzy open sets U and V such that xλ ∈ V and
A ≤ U and Uq−
V ;
(ii) λ > A(x) > 0 µ implies that there exist fuzzy open sets U and V such that
A ≤ U , xλqV and Uq−
V .
Definition 2.3.2. [55] An fts (X, τ) is said to be fuzzy regular in the sence of
Hutton and Reilly iff every fuzzy open set V can be expressed as union of fuzzy
open sets Uα’s such that Cl(Uα ≤ V , for all α .
Definition 2.3.3. [36] An fts (X,τ) is said to be quasi fuzzy regular ( in short q-
fuzzy regular) iff for each singleton xα and closed fuzzy set F in X with xαq(1−, thereexistU,V∈
τ such that xαqU ,F ⊂ V and UqV .
Definition 2.3.4. [82] A fts space X is called fuzzy almost regular if foreach fuzzy
regular open set U in X with xαqU , there exists a fuzzy regular open set V such
that xαqV ≤ ClV ≤ V such that Cl(V ) ≤ U.

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.
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 .
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
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 .
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 AB
0 0 = 0, there exist fuzzy
open sets U and V such that A ≤ U , B ≤ V and Uq−
V ;
(ii) if x ∈ A0 and =yandxµ /∈ A then there exist fuzzy open sets U and V such
that xµqU , A ≤ V and Uq−
V .
Definition 2.4.2. (i) A fts space X is said to be fuzzy almost normal[?] if every
pair of disjoint sets µ and ν , one of which is fuzzy closed and the other is fuzzy
regularly closed, can be strongly separated, (ii) a fts X is said to be fuzzy mildly
normal[?]if for every pair of disjoint fuzzy regularly closed subsets F1 and F2 of X,
there exist disjoint fuzzy open sets U and V such that F1 ⊂ U and F2 ⊂ V .
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 .
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
such that xµqU , A ≤ V and Uq−
V .

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 .
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 .
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 .
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.
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.
A fts (X, τ) is called FGPO-compact if every cover of X
by fgp-open sets has a finite subcover.
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.
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.
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.
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 is a subfamily of which is
also a cover.
(ii) A fts X is said to be fg-compact iff each gf-open cover of X has a finite
subcover.
(iii) A fuzzy set λ is said to be fg-compact if λ is fg-compact relative to X .
Definition 2.5.16. [?] A fts X is said to be fuzzy RS-compact iff for every fuzzy
regular semiopen cover {Uα : α ∈ } of X, there exists a finite subset o of
such that {Intα : α ∈ o} = 1X.
Definition 2.5.17. [?] A fuzzy set A is said to be fuzzy RS-compact(in short
FRSC-set) iff for every fuzzy regular semiopen cover {Uα : α ∈ } of X, there
exists a finite subset o of such that {Intα : α ∈ o} ≥ A.
Definition 2.5.18. [95] A collection {Bα : α ∈ } of fuzzy sets in X is said to
form a fuzzy filterbase in X for every finite subset o of , {Bα : α ∈ o} = 0x
Definition 2.5.19. [103] A fuzzy point xα in a fts X is said to be fuzzy rs-
accumalation point of a fuzzy filterbase {Bα : α ∈ } iff for each fuzzy regular
semiopen set U with xαqU and for each Bα , BαqIntU.
Definition 2.5.20. [12] A fuzzy subset A of a fts (X, τ) is said to be FN-closed(=fuzzy
N-closed) relative to an fts X if for any fuzzy open cover of A there exists a finite
subcollection the interiors of the closures of which cover A.
Definition 2.5.21. [12] A fuzzy subset A will be called FN-closable relative to τ
if A is FN-closed relative to τ.
Definition 2.5.22. [12] A fts X is said to be fuzzy locally nearly compact if each
fuzzy point has a fuzzy neighbourhood which is FN-closable relative to τ.

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.
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}).
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.
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.
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.
Definition 2.7.7. [47] Two non-empty fuzzy sets A and B in an fts X are said to
be fuzzy semi-separated if sCl(A)q−
B and sCl(B)q−
A.
Definition 2.7.8. [47] A fuzzy set A in an fts X is said to be fuzzy semi-connected
if A cannot be expressed as the union of two fuzzy semi-separated sets.
Definition 2.7.9. [108] Two non-empty fuzzy sets A and B in an fts X are said
to be fuzzy pre-separated if pCl(A)q−
B and Aq−
pCl(B).
Definition 2.7.10. [108] A fuzzy set which cannot be expressed as the union of
two fuzzy pre-separated sets is said to be a fuzzy pre-connected set.

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.
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.
Two non-empty fuzzy sets A and B in an fts X are said
to be fuzzy semipre-separated if spCl(A)q−
B and Aq−
spCl(B).
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 .
A fts X is called almost fuzzy e.d if bdU = clU −U is finite
for every U ∈ FRO(X).

A fts X is called fuzzy submaximal (resp. fuzzy g-submaximal
) if each fuzzy dense subset is fuzzy open (resp. fuzzy g-open)
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 :
A fts X is called fuzzy strongly sg-submaximal if every
fgsp-closed subset of X is fgs-closed.
Definition 2.8.6. A fts X is called semiregular if τ = τs [?].
Definition 2.8.7. [15] A fts X is said to be fuzzy extremally disconnected if Cl∗
(λ)
is gf-open , when every λ is gf-open set.
Part 3. Fuzzy Mappings
3.1. Fuzzy Continuity, fuzzy openness and its allied definitions
Definition 3.1.1. Let f : (X, τ1) → (Y, τ2) be a function from a fts (X, τ1) into a
fts (X, τ2). The function fis called:
(i) fuzzy continuous[31]if f−1
(λ) is fuzzy open set in X for each fuzzy open setλ
in Y .
(ii) fuzzy open (resp. fuzzy closed )[31] if f(λ) is a fuzzy open(resp. fuzzy closed)
set in Y for each fuzzy open(resp. fuzzy closed) set λin X.
(iii) fuzzy contra-open (resp. fuzzy contra closed) if f(λ) is a fuzzy closed (resp.
fuzzy open) set of Y for each fuzzy open (resp. closed) set λ in X.
Definition 3.1.2. [31] A fuzzy homeomorphism is an F-continuous one-to-one map
of a fts X onto a fts Y such that the inverse of the map is also F-continuous.
Note: If there exists a fuzzy homeomorphism of one fuzzy space onto another, the
two fuzzy spaces are said to be F-homeomorphic and each is a fuzzy homeomorph
of the other. Two fts’s are topologically F-equivalent iff they are F-homeomorphic.

Definition 3.1.3. [10] Let f be a function from a fts X into a fts Y . Then f is said
to be fuzzy almost continuous iff f−1
(µ)isfuzzyopen(resp.fuzzyclosed)inXforeveryfuzzyregularopen(resp
in Y .
Definition 3.1.4. [159] A function f : X → Y is said to be fuzzy W-almost open
iff f−1
(µ−
) ≤ f−1
(µ)−
for every fuzzy open set µ in Y .
Definition 3.1.5. [9] A function f : X → Y is said to be fuzzy weakly open
(resp.FW- almost open) iff f(V ) ≤ Int(f(Cl(V )))(resp.f−1
(Cl(V )) ≤ Cl(f−1
(V )
for every fuzzy open(resp. fuzzy regular open) set of X(resp. of Y ).
Definition 3.1.6. [159] A function f : X → Y is said to be fuzzy almost continuous
of Husain iff f−1
(µ−
) ≤ f−1
(µ)−o
for every fuzzy open set µ in Y .
Definition 3.1.7. [10] A function f : X → Y is said to be fuzzy weakly continuous
for each fuzzy point xβ ∈ X if for each fuzzy open set V of Y containing f(xβ),
there exists an fuzzy open set U in X containing xβ such that f(U) ≤ clV .
Definition 3.1.8. [91] A function f : X → Y is called fuzzy almost open (resp.
fuzzy almost closed) if the image of each fuzzy regular open set (resp. fuzzy regular
closed set) of X, is fuzzy open (resp. fuzzy closed) set in Y .
Definition 3.1.9. [?] A function f : X → Y is called fuzzy almost open in the sense
of S.Nanada (written as F.a.o.N) if for each open set U of X, f(U) ≤ intcl(f(U)).
Definition 3.1.10. [91] A function f : X → Y is called fuzzy almost open in the
sense of S.Nanada (written as F.a.o.N) if the image of each fuzzy regular open set
of X is fuzzy open set in Y .
Definition 3.1.11. [92] A function f : X → Y is called fuzzy almost open in the
sense of Ganguly et al (written as F.a.o.G) if for each open set U of Y , f−1
Cl(A) ≤
Cl(f−1
(A)).

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 .

Definition 3.1.17. [78] Let X = (X, τX ) and Y = (Y,Y ) be fts. A function
f : X → Y is said to be fuzzy totally continuous if the inverse image of every fuzzy
subset of Y is a fuzzy clopen subset of X.
Definition 3.1.18. [81] A mapping f : X → Y from a fts X to another fts Y will
be called fuzzy almost continuous (in the sense of Mukherjee and Sinha) iff for each
fuzzy point xα of X and each fuzzy open nbd A of f(xα), Cl(f−1
(A)) is a fuzzy
nbd. of xα.
Note : Above definition of fuzzy almost continuous function f is independent of
fuzzy almost continuous and fuzzy weakly continuos functions .
3.2. Fuzzy weak forms of continuity, openness and allied definitions
(i)fuzzy semi continuou[10]if f−1
(λ) is a fuzzy semiopen set of X for each fuzzy
open set λ in Y .
(ii)fuzzy pre continuous[19]if f−1
(λ) is a fuzzy preopen set of X for each fuzzy
open set λ in Y
(iii)fuzzy semipre continuous[145]if f−1
(λ) is a fuzzy semi-preopen set of X for
each fuzzy open set λ in Y
(iv) fuzzy preclosed[128],[19] if f(λ) is a fuzzy preclosed set of Y for each fuzzy
closed set λ in X.
(v)fuzzy preopen[20]if f(λ) is a fuzzy preopen set of Y for each fuzzy open set λ
in X.
(vi)fuzzy strongly semicontinuous[19]if f−1
(λ) is a fuzzy α-open set of X for
each fuzzy open set λ in Y
(vii)fuzzy semiopen[10]if f(λ) is a fuzzy semiopen set of Y for each fuzzy open
set λ in X
(viii)fuzzy α-open[20]if f(λ) is a fuzzy α-open set of Y for each fuzzy open set λ
in X

(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 .
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.
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.
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.
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 .
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..
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.
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 .
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.
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 .
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 .
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 .
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 .
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 )
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 .
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.

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 .
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.
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.
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 ).
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 .
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 .
A function f : X → Y is said to be fuzzy s-continuous if
f−1
(V ) is fuzzy open for each V ∈ FSO(Y ).
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 .
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)].
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)].
A mapping f : X → Y from fts X to another fts Y is
called fuzzy α-weakly continuous if for each fuzzy α-open set λ of Y , we have
f−1
(λ) ≤ αInt[f−1
αClλ)].
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
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 .
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 .
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.
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.
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
fuzzy open and fuzzy closed set in X.
(vi) fuzzy gc-irresolute if the inverse image of every gf-closed set in Y is gf-closed
set in X.
Definition 3.3.7. [136] A mapping f : X → Y is called fuzzy semi-generalized
continuous (in short Fsg-continuous)if f−1
(A) is Fsg-closed in X for every fuzzy
closed set A of Y .
Definition 3.3.8. [10] A mapping f : X → Y is called Fs-continuous if f−1
(A) is
Fs-open set in X for every fuzzy open set A of Y .
Definition 3.3.9. [?] A function f : X → Y is called Fgsp-continuous if f−1
(A)
is Fgsp-closed in X for every fuzzy closed set A of Y .
Definition 3.3.10. [?] A function f : X → Y is called Fgsp-irresolute if f−1
(A)
is Fgsp-closed in X for every Fgsp-closed set A of Y .
Definition 3.3.11. [128] A mapping f : X → Y is said to be Fgα∗
-continuous
(resp. Fgα∗∗
-continuous,Fgα-continuous) if for every fuzzy closed set B of Y ,
f−1
(B) is Fgα∗
-closed (resp.Fgα∗∗
-closed ,Fgα-closed) in X.
Definition 3.3.12. [?] A mapping f : X → Y is termed fuzzy pre-α-open if the
image of every fuzzy α-open set of X is fuzzy α-open in Y .
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.
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.
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 .
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.
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 .
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 .

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 .
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 .
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 .
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).
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.
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, India
E-mail address: gnavalagi@hotmail.com

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