Download On sigma-Induced L-Fuzzy Topological Spaces

Int. J. Pure Appl. Sci. Technol., 16(1) (2013), pp. 61-68
International Journal of Pure and Applied Sciences and Technology
ISSN 2229 - 6107
Available online at www.ijopaasat.in
Research Paper
On
-Induced L-Fuzzy Topological Spaces
Baby Bhattacharya1,* and Jayasree Chakraborty1
1
Department of Mathematics, NIT Agartala, India
*Corresponding author, e-mail: ([email protected])
(Received: 5-3-13; Accepted: 8-4-13)
Abstract: The aim of this paper is to study the concept of
-induced L-fuzzy
topological spaces. Some properties of the functions (namely Scott- -continuous
function) are studied. The Scott- -continuous functions turn out to be the natural
tool for studying the -induced L-fuzzy topological space. We also discuss the
connections between some separations, countability and covering properties of an
ordinary topological space and its corresponding -induced L-fuzzy topological
space.
Keywords: Fuzzy lattice, prime element, regular
-subset, Scott- -continuous
function, -induced L-fuzzy topological space.
1. Introduction
The concept of induced fuzzy topological space was first introduced by Weiss in 1975 [16]. In 1976,
Lowen [10] called these spaces as topologically generated spaces. In 1999, Aygun, Warner, Kudri [2]
introduced the concept of completely induced L-fuzzy topological space. The concept of
-subset
was first introduced by E.P. Lane in [9] and the concept of regular -subset was introduced by Mack
in 1970 [11]. The complement of a regular -subset is called a regular -subset. We introduced a
new class of functions from a topological space ( X, T ) to a fuzzy lattice L with its Scott topology
called Scott- -continuous functions [3]. In this paper we study some of their properties and
characterizations. We also prove that the set (T) of Scott- -continuous functions from a topological
space ( X, T ) to L with its Scott topology, is an L-fuzzy topology. Scott- -continuous functions turn
out to be the natural tool for studying -induced L-fuzzy topological space. Then we discuss the
connection between several properties of an ordinary topological space ( X, T ) and its corresponding
-induced L-fuzzy topological space( X, (T )). Throughout this work X and Y will be non empty
ordinary sets and L = L ( ˅, ˄,′) will denote a fuzzy lattice i.e. a complete completely distributive
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68
62
lattice with a smallest element 0 and a largest element 1 (0 ≠ 1) and with an order-reversing involution
a→a′ (a∈L). L is therefore a continuous lattice. Also LX will denote the lattice of L-fuzzy subsets of X.
We will denote by 1A the characteristic function of the ordinary subset A of X.
We State Some Definitions and Results for Ready References:
Definition 1.1.[11]: A subset H of a topological space X is called a regular
-subset if H is an
intersection of a sequence of closed sets whose interiors contain H. Equivalently,
if H = Gi = clXGi for i∈N ,where each Gi is open in X, then H is regular -subset of X . The
complement of a regular -subset is called a regular -subset.
Definition 1.2.[4]: Let ( X,T ) be a topological space and ∈X. A function :( X,T )→I is called a
Scott continuous (or lower semi continuous) at ∈X iff for every α∈[0,1] with α <f ( ) there is a
neighbourhood U of ʽ ʼ such that α <f (x) for every x∈U. Then f is called Scott continuous (or
lower semi continuous) on X iff f is Scott continuous (or lower semi continuous) at every point of
X.
Definition 1.3.[14]: The set (T) of Scott-continuous functions from a topological space ( X,T ) to L
with its Scott-topology is an L- fuzzy topology, called the induced L-fuzzy topology(IL-FT).
Definition 1.4.[5]: An element p of L is called prime iff p ≠ 1 and whenever a,b∈L with a b≤ p then
a≤ p or b≤ p. The set of all prime elements of L will be denoted by pr (L).
In [15] Warner has determined the prime elements of the fuzzy lattice LX. Here
pr(LX) = {xp: x∈X and p∈pr(L)}
Where for each x∈X and each p∈pr(L) , xp: X→L is the fuzzy set defined by
xp(y) =
These xp are called the L-fuzzy points of X and we have xp is a member of an L-fuzzy set g and we
write xp∈g iff g(x) ≰ p.
Definition 1.5[5]: Let L be a complete lattice and x,y∈L. We say that x is way below y, in symbols
x<<y, if and only if for every directed subset D of L with y≤∨D, there exists a d∈D such that x≤d.
Proposition 1.6[15]: The set of the form {l∈L : δ<<l} are Scott-open.
Result 1.7[12]: The set of the form { ∈L :
p} where p∈pr(L) generates the Scott topology of L.
Proposition 1.8[7]: An L-fts (X,τ) is said to be fuzzy regular iff for every p∈pr(L), for each x∈X and
each closed L-fuzzy set f such that there is y∈X with yp∉f′ and f(x)=0,there exist open L-fuzzy sets
g,h such that xp∈g for every yp∉f′, yp∈h and (∀z∈X) g(z)=0 or h(z)=0.
Definition 1.9[1]: A topological space ( X,T ) is said to be almost regular iff for each non empty
regular closed subset F of X and for each point x∈F′ , there exist disjoint open sets U and V such that
x∈U and F⊂V.
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68
63
Definition 1.10[2]: A topological space ( X,T ) is Hausdroff iff for every x, y∈X (x≠y),there exists
regular open sets U and V such that x∈U, y∈V and U∩V=φ.
Definition 1.11[13]: A topological space ( X,T ) is completely Hausdroff iff every distinct points x, y
of X, there are open sets U and V such that x∈U, y∈V and Cl(U)∩Cl(V) = φ.
Definition 1.12[15]: Let (X,τ) be an L-fuzzy topological space. A non empty sub family β of τ is a
basis for τ iff for every xp∈pr(LX) and every f∈τ with xp∈f , there is a g∈β such that xp∈g≤f.
Definition 1.13[8]: An L-fts (X,τ) is first countable C1,iff for every fuzzy point xp∈pr(LX), there exist
a family of open L-fuzzy sets (fi)i∈N with xp∈ fi ,such that for every g∈τ with xp∈g, there is an i∈N
with fi ≤g.
2. Scott- -Continuous Functions & their Properties
Definition 2.1.[3]: Let ( X,T ) be a topological space and
∈X. A function :( X, T )→L where L
has its Scott topology is said to be Scott- -continuous at ∈X iff for every p∈pr(L) with ( ) p ,
there is a regular
-neighbourhood N of ʽ ʼ in ( X,T ) such that (x)
p , x∈N . i.e.
({ ∈L:
p}). Then is called Scott- -continuous on X iff is Scott- -continuous at every
N⊂
point of X.
Proposition 2.2: The characteristic function of every regular
Proof: Let A be a regular
A is defined as
-subset is Scott- -continuous.
-subset in a topological space ( X,T ) and the characteristic function 1A of
1A(x) =
We have to show that 1A is Scott- -continuous i.e. 1A(x) p , for every x∈A and p∈pr(L). Let ∈X,
p∈pr(L) with1A( ) p . Then
A and A is a regular -neighbourhood of . We also have 1A(x)
p, for all x∈A. Hence 1A is Scott- -continuous at ∈X.
Proposition 2.3: If { j: j∈ } is an arbitrary family of Scott- -continuous functions from a
topological space ( X, T ) L , then =˅j∈Λ j is also Scott- -continuous.
Proof: Let p∈pr (L) and ∈X with ( ) =˅j∈Λ j( ) p , then there is a j∈Λ such that j ( ) p ,
since j is Scott- -continuous at ʽ ʼ , there is regular -neighbourhood N of ʽ ʼ such that j(x) p
, x∈N. Hence (x) =˅j∈Λ j(x) p, x∈N. Thus is Scott- -continuous at ∈X.
Proposition 2.4: Let (X,T) be a topological space. If
, : ( X,T )
functions then
: ( X,T ) L is Scott- -continuous as well.
L are Scott- -continuous
Proof: Let ∈X and p∈pr (L) with
)( ) p. Then ( ) p and ( ) p. Since and are
Scott- -continuous at ʽ ʼ, there are regular -neighbourhoods U and V of ʽ ʼ such that (x) p
for all x∈U and (x)
p for all x∈V. Let W = U V. Then W is a regular
-neighbourhood of ʽ ʼ,
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68
since the intersection of two regular
(
(x) p for all x∈W. Hence
64
-subsets is regular
-subset and p is prime. We have
is Scott- -continuous at ∈X .
Proposition 2.5: Let ( X,T ) be a topological space. The function :( X,T ) L is Scott- -continuous
iff for every p∈pr (L), -1({ ∈L:
p}) can be expressed as a union of some regular -subsets in
( X,T ).
-1
Proof: Necessity: Let p∈pr(L) and x
({ ∈L: p}). Then (x) p. Since
is Scott- continuous at x, then there exist a regular
-subset Nx in (X,T) such that x∈Nx and
-1
-1
Nx⊂ { ∈L:
p}.Hence
{ ∈L: p}= Nx where Nx is regular -subset.
Sufficiency: Let ∈X and p∈pr(L) with ( ) p .Then ∈ -1({ ∈L:
is a regular -subset N in ( X,T ) such that ∈N and N⊂ -1({ ∈L:
Scott- -continuous.
Theorem 2.6: For a topological space ( X,T ) the set
continuous} is an L-fuzzy topology on X.
p}). By the hypothesis, there
p}) , which implies that
(T) ={ ∈LX:
:( X,T )
is
L is Scott- -
Proof: From the proposition 2.2, 2.3 and 2.4, the proof is obvious.
Remark: Since every Scott- -continuous function from a topological space ( X,T ) to a fuzzy lattice
L is Scott continuous, we have (T) ⊂ (T), where (T) is the L-fuzzy topology of Scott continuous
functions from (X,T) to L.
Definition 2.7: The L-fuzzy topology defined above is called a -induced L-fuzzy topology ( -ILFT) and the space (X, ( T )) is called -induced L-fuzzy topological space and the members of
(X, ( T )) are called fuzzy open sets in(X, ( T )) .
Proposition 2.8: For a topological space ( X,T ), the following family forms a base for
(β) = {Zβ : Z is regular
where Zβ : X
(T)
-subset in ( X,T ), β ∈L}
L is defined by Zβ(x)=β if x∈Z and Zβ(x)=0 otherwise.
Proof: Let f∈ (T) and xp∈pr (LX) with (x) p.Then is Scott- -continuous and (x) p .By the
continuity of L[5] there exists η∈L and η<<f(x) and η p. We take β∈L with η <<β <<f(x). Hence
(x)∈{ ∈L: β <<l}. Since { ∈L: β <<l }is Scott open [by proposition1.6], there is a q∈pr(L) such
that (x)∈{ ∈L: l q}⊂{ ∈L: β <<l} by proposition 1.7. Then (x)
q. From the Scott- continuity of , there exist a regular -subset Z in (X,T) such that x∈
∈Z and f(z) q,∀z∈Z. Thus
β<<f(z) for every z∈Z and hence β ≤ f(z) for every z∈Z. Moreover, β
p because β>>η p.
β
β
β
Consequently xp∈Z . Z ∈ (β) and Z ≤ f. Thus by proposition1.12
(β) is a base for
(T).
Lemma 2.9: If A is a regular
-subset in a topological space ( X,T ) then 1A
Proof: By proposition 2.2, the lemma follows immediately.
(T)
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68
65
Definition 2.10: A function f :( X,T1 )→( Y,T2 ) is called -irresolute if the inverse image f -1(A) is
regular -subset in (X, T1) for every regular -subset in Y.
Definition 2.11[3]: A function :(X, (T1))
space is said to be fuzzy continuous if -1( )
Theorem 2.12: A function :(X,
-irresolute.
Proof: A function :(X,
Then 1A
regular
(T1))
(T1))
(Y,
(T2). By fuzzy continuity
(Y, (T2)) from a -IL-FT space to another
(T1) for every
(T2).
(Y,
(T2)) is fuzzy continuous iff
(T2)) is fuzzy continuous and A is regular
-1
(1A) =1 f −1 ( A)
-subset in (X,T1). Let p∈pr(L) and x∈
-1
-IL-FT
:(X,T1)
(Y,T2) is
-subset in (Y,T2).
-1
(T1). Now we shall show that
(A). Then 1 f −1 ( A) (x)
p . Since 1 f −1 ( A)
-subset N in (X,T1) such that x∈N and N⊂1 f −1 ( A) ({ ∈L:
(T1),
p}) =
-1
-1
we have for each x
(A) there exist a regular -subset N in (X,T1) such that x∈N⊂
-1
shows that (A) is regular -subset in (X,T1). Hence :(X,T1) (Y,T2) is -irresolute.
-1
there exist a regular
(A) is
(A). Thus
(A). This
Conversely: Suppose :(X,T1) (Y,T2) is -irresolute. Let λ∈ (T2). We shall show that
-1
( )
(T1). i.e. -1( ):( X,T ) L is Scott- -continuous. Let ∈X and p∈pr(L) with
-1
( )( ) p. Then ( (a)) p. Since :(Y, T2) L is Scott- -continuous at ( )∈Y, there exist a
regular
-subset N in (Y,T2) such that ( )∈N and (y) p for all y∈N. Since N is regular
-1
-subset in (Y,T2), -1(N) is regular -subset in (X,T1) by the hypothesis. Now we have
(N),
-1
-1
which implies there is a regular
-subset B in (X,T1) such that
B⊂ (N). Hence
( )(x)
-1
= ( (x)) p for every x∈B. This shows that
( ) is Scott- -continuous. Consequently
:(X, (T1)) (Y, (T2)) is fuzzy continuous.
3. Connection between Properties of a Topological Space and Its
Corresponding - Induced L-Fuzzy Topological Space:
In this section, we study the connections between separation, countability and covering properties of
an ordinary topological space ( X,T ) and its corresponding -induced L-fuzzy topological space
(X, ( T )) .
Definition 3.1: A topological space ( X, T ) is called
points x,y of X there are regular
-completely Hausdroff iff for any distinct
-subsets A and B such that x∈A , y∈B and Cl(A) Cl(B) = φ.
Definition 3.2: A -ILFT space (X,
(T)) is said to be fuzzy completely Hausdroff iff for every
distinct points x,y of X and every p,q∈ pr (L) there exists L-fuzzy sets f and g such that xp∈f ,yq∈g
and ∀z∈X , Cl(f)(z) = 0 or Cl(g)(z) = 0.
Theorem 3.3: The topological space ( X, T ) is
(X, (T )) is fuzzy completely Hausdroff.
-completely Hausdroff iff the
-IL-FT space
Proof: Let x, y∈X(x y) and p, q∈pr(L). By -complete Hausdroffness of ( X,T ), there exists two
regular -subsets U and V in (X,T), such that x∈U and y∈V and Cl(U) Cl(V) = φ.Then 1U,1V
(T)
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68
because U and V are regular -subsets in ( X,T ). We also have 1U(x) p, 1V(y)
Cl(1U)(z) = 1Cl(U)(z) = 0 or Cl(1V)(z) = 1Cl(V)(z) =0 because Cl(U) Cl(V) =φ.
( X, ( T )) is a fuzzy completely Hausdroff.
66
q and z∈X.
Consequently
Conversely, let x,y∈X(x y) and p,q∈pr(L). From the fuzzy complete Hausdrofness of (X, ( T )),
there exists basic open L-fuzzy sets , which are defined by respectively (z) = if z∈U, (z)=0
otherwise and (z)= if z V, (z)=0 otherwise, where U and V are regular -subsets in ( X,T ) and
∈L such that xp∈ , yq
and ( z∈X), Cl( )(z) =0 or Cl( )(z) =0. Hence we have x∈U and
y∈V where U and V are regular -subsets in ( X,T ) and Cl(U) Cl(V) =φ. Hence ( X,T ) is completely Hausdroff.
Since the intersection of two regular -subsets is regular -subset, then the family of all regular
-subsets in ( X,T ) forms a base for a smaller topology Tσ on X, called the -semi-regularization of
T. A topological space( X,T ) is sad to be -semi-regular iff T = Tσ .i.e. (X, Tσ) space is the -semiregularization topological space. Obviously Tσ⊂T.
Analogous to the almost regular [1] space we define the following:
Definition 3.4: A topological space ( X,T ) is said to be -almost regular iff for each non empty
regular -subset F of X and each point x∈F′, there exists disjoint open sets U and V such that x∈U
and F⊂V.
Obviously, every -almost regular topological space is almost regular.
Proposition 3.5: A topological space ( X,T ) is -almost regular if for each non empty regular subset F of X and each point x∈F′, there exists disjoint regular -subset U and V such that x∈U and
F⊂V.
Proof: It is obvious from the definition of -almost regular. Since regular -subset implies open set.
Lemma 3.6: Let ( X,T ) be a topological space and (X,Tσ) be its -semi-regularization topological
space. Then (X,T ) is -almost regular iff (X, Tσ) is regular.
Theorem 3.7: A topological space ( X,T ) is -almost regular iff the -induced L-fts (X,
fuzzy regular.
( T )) is
Proof: Let p∈pr (L), x∈X and let f be a closed L-fuzzy set in (X, ( T )) such that there is a y∈X
with yp∉ f ′ and f(x) = 0. Then xp f ′ and f′∈ (T). So by the proposition2.8 there is a basic open
L-fuzzy set g∈LX defined by g(z) = γ if z∈A and g(z) = 0 otherwise, where A is regular -subset in
(X,T ) and γ∈L such that xp∈g≤ f ′ . Then, we have g(x) p which implies that g(x)= γ p and hence
x∈A. Since A′ is regular -subset in ( X,T ) and ( X,T ) is -almost regular, then there are regular
-subsets U,V in ( X,T ) such that x∈U, A′⊂V and U∩V=φ. Let η=1U and θ=1V . Then η,θ∈ (T),
η(x) = 1 p and for every yp∉ f . yp∈θ because yp∈ f ′ implies that g(y)≤p and hence g(y)=0 because
γ p. This means that y∈A′.Thus y∈V and θ(y)=1 p. In addition, (∀z∈X) η(z)=0 or θ(z)=0.
Consequently (X, ( T )) is fuzzy regular.
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68
67
Conversely, let K be a non-empty regular
-subset in ( X,T ) and let x∈K. Then 1K is closed in
(X, ( T )) . 1K(x) = 0 and there exist a y∈X with yp∉ 1′K because K is non-empty set. From the fuzzy
regularity of (X, ( T )) , there exist basic open L-fuzzy sets g,h∈LX defined by g(z) = γ if z∈A and
-subsets in
g(z) = 0 otherwise and h(z)=β if z∈B and h(z) = 0 otherwise, where A and B are regular
( X,T ) and γ,β∈L such that xp∈g, yp∈h for every yp∉ 1′K and (∀z∈X) g(z) = 0 or h(z) = 0. Hence
x∈A, K⊂ B and A∩B = φ. Thus by the proposition 3.5 ( X,T ) is -almost regular.
Definition 3.8: A topological space ( X,T ) is called -nearly compact iff every regular
has a finite subcover.
-cover of X
Definition 3.9.[12]: An L-fts ( X,T ) is said to be fuzzy compact iff for every prime element p of L
and every collection( fi)i∈J of open L–fuzzy sets with (˅i∈J fi)(x)
p for all x∈X, there is a finite
subset F of J such that (˅i∈F fi)(x) p) for all x∈X.
Lemma 3.10: A topological space ( X,T ) is -nearly compact iff (X,
definition 3.9[12] we prove the following theorem.
Theorem 3.11: A
-nearly compact.
) is compact .Using the
-ILFT space is fuzzy compact iff the corresponding topological space ( X,T ) is
Proof: Let us assume that ( X,T ) is -nearly compact. Let p∈pr (L) and = { fj : j
} be a family
of basic open L- fuzzy sets in (X, ( T )) with ( fj
(x) p for all x∈X; where fj(x) = if x∈Aj
and fj(x)= 0 otherwise, then Aj are regular
each x∈X there is j∈ such that fj(x)
-subset in ( X,T ) and
p i.e.
p and let
L for every j
= { Aj : there is an j
. Then for
such that
p and fj
}. Then is a family of regular -subsets in ( X,T ) covering X. In other words, is a
regular
-cover of X. From the -compactness of ( X,T ), there exists a finite subfamily of say
where = {A1,A2,…,.An} such that X =
fj) (x) p for all x∈X and thus
n. Hence (
(X,
( T )) is fuzzy compact.
Conversely, suppose (X,
1A
(T) for every j
L-fuzzy sets in (X,
(X,
(T )) is fuzzy compact. Let {Aj: j
and Aj is regular
( T )) with
Hence X =
-subset in ( X,T ). Thus {1 A j : j
1 A j )(x)
( T )) there exist a finite subset
} be a regular
of
-cover of X. Then
} is a family of open
p for all x∈X. From the fuzzy compactness of
such that
1 A j )(x)
p
for all x∈X .
Aj. Thus ( X,T ) is -nearly compact.
Theorem 3.12: Let ( X,T ) be a topological space. Then the -semi regularization topological space
(X,Tσ) is first countable (C1) iff the
1.13(8)].
-induced L-fts ( X,
( T )) is first countable.[Definition
Proof: Assume that (X,Tσ) is first countable. Let xp be any fuzzy point and Γ be a countable local base
of x in (X,Tσ ). Let = { Aγ : A∈ Γ ,γ∈L with γ p}, where Aγ (z) = γ if z∈A and Aγ (z) = 0 otherwise.
We shall show that is a countable local base of xp in (X, ( T )) . Let g∈ (T) and xp∈g. Then, by
proposition 2.8, there exists a basic open L-fuzzy set Zβ such that xp∈Zβ ≤ g, where Z is regular
β
subset in ( X,T ). Thus, Z (x) = β p and hence x∈Z.
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 61-68
68
Since Z Tσ , x∈Z and Γ is a local base of x in (X,Tσ), there exists an A∈ Γ such that x∈A⊂Z. Then,
we have Aβ
and xp ≤ Aβ≤ Zβ ≤ g and therefore is a local base of xp. Moreover, is a countable
local base because is countable. Consequently, (X, (T)) is first countable.
be the countable local
Now assume that (X, ( T )) is first countable. Let x∈X and p∈pr(L) and
-1
base of xp in (X, (T)) . Consider Γ = { ({ ∈L:
p}): f ∈ Γ }. We will prove that Γ is countable
local base of x in (X,Tσ). Let O∈ Tσ and x∈O. Then, there is a regular -subset A in (X,T ) such that
x∈A⊂O. Hence, 1A
(T) and 1A (x) = 1 p since
is a local base of xp, there is f∈ such that
-1
xp∈f ≤ 1A and so x
({ ∈L:
p})⊂A⊂O. Thus Γ is a local base of x in (X,Tσ). Since
is a
countable, Γ is countable and therefore (X,Tσ) is first countable.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
M.K. Singhal and S.P. Arya, On almost regular spaces, Glasnik Mat, 24(4) (1969), 89-99.
H. Aygun, M.W. Warmer and S.R.T. Kudri, Completely induced l- fuzzy topological spaces,
Fuzzy Sets and Systems, 103(1999), 513-523.
B. Bhattacharya and J. Chakraborty, L-fuzzy topological spaces induced by Scott- continuous functions, 2nd International Conference on Rough Sets, Fuzzy Sets and Soft
Computing, ICRFSC’12, January 17th-19th, Tripura, India, 2013.
N. Bourbaki, Elements of Mathematics: General Topology: Part-I, Addssion-Wesely,
Reading, M.A., 1996.
G. Gierz et al, A Compendium of Continuous Lattice, Springer, Berlin, 1980.
W. Gujon, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47(1992), 351376.
S.R.T. Kudri, Compactness in l-fuzzy topological spaces, Fuzzy Sets and Systems, 67(1994),
329-336.
S.R.T. Kudri, Countability in l-fuzzy topological spaces, Fuzzy Sets and Systems, 71(1995),
241-249.
E.P. Lane, Weak C insertion of a continuous function, Notices, Amer. Math. Soc., 26(1979),
A-231.
R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal Appl., 56(1976),
621-633.
J.E. Mack, Countably paracompactness, weak normality properties, Trans. Amer. Math. Soc.,
148(1970), 256-272.
M.W. Warner and R.G.M. Lean, On compact hausdroff Ll-fuzzy spaces, Fuzzy Sets and
Systems, 56(1993), 103-110.
L.A. Steen and J.A. Seebach, Counter Example in Topology, New York, 1970.
M.W. Warner, Fuzzy topology with respect to continuous lattices, Fuzzy Sets and Systems,
35(1990), 85-91.
M.W. Warner, Frame-fuzzy points and membership, Fuzzy Sets and Systems, 42(1991), 335344.
M.D. Weiss, Fixed points, separation and induced topologies for fuzzy sets, J. Math. Anal.
Appl., 50(1975), 142-150.