Zhen-Guo Xu and Zi-Qiu Yun
School of Mathematical Science, Suzhou University, Suzhou 215006, P.R. China E-mail: [email protected] and [email protected]
Abstract: In this paper, we shall introduce generalized fuzzy compactness inL-spaces whereLis a complete de Morgan algebra. This definition does not rely on the structure of basis latticeLand no distributivity is required. The intersection of a generalized fuzzy compactL-set and a generalized closed L-set is a generalized fuzzy compactL-set. The generalized irresolute image of a generalized fuzzy compactL-set is a generalized fuzzy compactL-set.
Keywords: L-space; generalized openL-set; generalized closedL-set; generalized fuzzy
compactness
2000 Mathematics Subject Classification: 54A40,54D35.
1. Introduction and preliminaries
In 1976, Lowen first introduced the concepts of fuzzy compactness in [0,1]-spaces in [6]. Subsequently its characterization was given by Wang in terms of α-net in [11]. In 1988, it is again extended toL-spaces [12], whereLis a completely distributive de Morgan algebra (i.e., aF lattice). However the above mentioned definitions of fuzzy compactness seriously depend on the structure of the basis lattice L and complete distributivity was required.
Kubi´ak also extended fuzzy compactness to L-spaces by means of closed L-sets and the way below relation in [4], where complete distributivity was not required. But his definition still depend on the structure of the basis lattice L and can’t be restated in terms of open L-sets by simply using quasi-complementation.
1
In [9, 10], a new definition of fuzzy compactness in presented inL-topological space by means of an inequality, which doesn’t depend on the structure ofL and no distributivity is require in L. When L is a completely distributive de Morgan algebra, it is equivalent to the notion of fuzzy compactness in [5, 7, 12].
The notions of generalized open sets, generalized closed sets and generalized-irresolute mapping were introduced by Balasubramanian and Sundaram in [1].
In this paper, following the lines of [9, 10], we shall introduce a concept of generalized compactness inL-topological spaces in terms of generalized openL-sets and their inequal- ity, whereLis a complete de Morgan algebra. This definition doesn’t rely on the structure of basis lattice L and no distributivity in L is required. It can also be characterized by generalized closed L-sets and their inequality. When L is a completely distributive de Morgan algebra, its many characterizations are presented.
Throughout this paper, (L,W ,V
,0) is a complete de Morgan algebra. 0 and 1 denote the smallest element and the largest element inL, respectively.
A complete lattice L is a complete Heyting algebra if it satisfies the following infinite distributive law: For all a∈L and allB ⊂L,a∧W
B =W{a∧b|b∈B}.
For a nonempty set X, LX denotes the set of all L-topological fuzzy sets (or L-sets for short) on X. 0 and 1 denote the smallest element and the largest element in LX, respectively. An L-space (L-space for short) is a pair (X,T), where T is a subfamily LX which contains 0,1 and is closed for any suprema and finite infima. T is called an L- topology onX. Each member of T is called an openL-set and its quasi-complementation is called a closed L-set. An element ain L is called a prime element if b∧c≤a implies b ≤aor c≤a. ain L is called co-prime element if a0 is a prime element. The set of all nonzero co-prime elements inLis denoted byM(L). It is easy to see thatM(LX) ={xα | x∈X, α∈M(L)} is exactly the set of all nonzero∨-irreducible elements in LX.
According to [12], we know that L is completely distributive if and only if each el- ement a in L has the greatest minimal family(the greatest maximal family), denoted by β(a)(α(a)). Obviously β∗(a) = β(a)T
M(L) is a minimal family of a and α∗(a) = β(a)T
P(L) is a maximal family ofa.
Fora subfamily Φ⊂LX, 2(Φ) denotes the set of all finite subfamily of Φ.
In [1], The notions of generalized open sets, generalized closed sets and generalized- irresolute mapping were introduced in [0,1]-fuzzy set theory by Balasubramanian and Sundaram. They can easily be extended to L-sets as follows:
Definition 1.1. Let (X,T) be anL-space andA∈LX. ThenAis called generalized closed L-set(orgl-closed for short) if cl(A)≤U whenever A≤U and U is openL-set. A is called generalized open (gl-open for short) if A0 is gl-closed.
GLO(X) and GLC(X) will always denote the family of all generalized open L-sets and family of all generalized closedL-sets in X, respectively.
Definition 1.2. Let (X,T1) and (Y,T2) be two L-spaces, f :X → Y be a mapping and fL→:LX →LY be the extension off. Thenf called a generalized irresolute mapping iffL←(B) is generalized open in (X,T1) for each generalized open L-setB in (Y,T2).
Definition 1.3 ([9, 10]). Let (X,T) be anL-space, G∈LX. ThenGis called fuzzy compact if for every family U ⊂ T, it follows that
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
≤ _
V∈2(U)
^
x∈X
G0(x)∨ _
A∈V
A(x)
! .
Lemma 1.4 ([10]). Let (X,T1) and (Y,T2) be two L-spaces, where L is a complete Heyting algebra, f :X → Y be a mapping, fL→ :LX → LY is the extension of f. Then for any P ⊂LY, we have that
_
y∈Y
fL→(G)(y)∧ ^
B∈P
B(y)
!
= _
x∈X
G(x)∧ ^
B∈P
fL←(B)(x)
! .
2. Generalized fuzzy compactness of L-Subsets
Definition 2.1. Let (X,T) be an L-space, G ∈ LX. Then G is called generalized fuzzy compact if for every family U ⊂GLO(X), it follows that
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
≤ _
V∈2(U)
^
x∈X
G0(x)∨ _
A∈V
A(x)
! .
Now we consider characterizations of generalized fuzzy compactness. First we intro- duce the following concept.
Definition 2.2. Let (X,T) be an L-space, a ∈ L \ {1} and G ∈ LX. A family U ⊂GLO(X) is said to be a generalized open a-shading of G if for any x ∈ X with
G(x)≥a0, there exists an A∈ U such that A(x)6≤a. U is said to be a generalized open stronga-shading ofG if V
x∈X
G0(x)∨ W
A∈U
A(x)
6≤afor any x∈X.
Obviously, a generalized open strong a-shading of G is a generalized open a-shading of Gand U is a generalized open a-shading ofG if and only ifG0(x)∨ W
A∈U
A(x)6≤a.
By Definition 2.1 and Definition 2.2 we obtain the following result.
Theorem 2.3. Let (X,T) be an L-space and G ∈LX. Then G is generalized fuzzy compact if and only if for any a∈L\ {1}, each generalized open stronga-shading U of G has a finite subfamily V which is still a generalized open strong a-shading of G.
Proof. Suppose that Gis generalized fuzzy compact and for any a∈ L\ {1},U is any generalized open stronga-shading ofG. Then
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
≤ _
V∈2(U)
^
x∈X
G0(x)∨ _
A∈V
A(x)
! .
and V
x∈X
G0(x)∨ W
A∈U
A(x)
6≤a. So that W
V∈2(U)
V
x∈X
G0(x)∨ W
A∈V
A(x)
6≤a, hence there exists V ∈ 2(U) such that V
x∈X
G0(x)∨ W
A∈V
A(x)
6≤a. Thus V is finite subfamily of U and V is a generalized open stronga-shading ofG.
Conversely, suppose that for any a∈L\ {1}, each generalized open strong a-shading U of G has a finite subfamily V which is still a generalized open strong a-shading of G.
Hence we have that
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
6≤aimplies that ^
x∈X
G0(x)∨ _
A∈V
A(x)
! 6≤a,
therefore
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
≤ ^
x∈X
G0(x)∨ _
A∈V
A(x)
! .
Thus we obtain that
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
≤ _
V∈2(U)
^
x∈X
G0(x)∨ _
A∈V
A(x)
! .
Hence Gis generalized fuzzy compact from Definition 2.1.
Moreover from Definition 2.1 we easily obtain the following theorem by simply using quasi-complementation.
Theorem 2.4. Let (X,T) be an L-space and G ∈LX. Then G is generalized fuzzy compact if and only if for every subfamily P ⊂GLC(X), it follows that
_
x∈X
G(x)∧ ^
B∈P
B(x)
!
≥ ^
F ∈2(P)
_
x∈X
G(x)∧ ^
B∈F
B(x)
! .
Definition 2.5. Let (X,T) be an L-space, a ∈ L \ {1} and G ∈ LX. A family P ⊂GLC(X) is said to be a generalized closed a-remote family of G if for any x ∈ X with G(x) ≥a, there exists a B ∈ P such that B(x) 6≥ a. P is said to be a generalized closed strong a-remote family of Gif W
x∈X
G(x)∧ V
B∈P
B(x)
6≥a.
It is obvious that a generalized closed strong a-remote family of G is a generalized closed a-remote family of G, P is a generalized closed a-remote family of G if and only if G(x)∧ V
B∈P
B(x) 6≥aand P is a generalized closed strong a-remote family of Gif and only if P0 is a generalized open stronga-shading ofG.
From Theorem 2.4 we obtain the following result.
Theorem 2.6. Let (X,T) be an L-space and G ∈LX. Then G is generalized fuzzy compact if and only if for any a∈L\ {0}, each generalized closed strong a-remote family P of G has a finite subfamily F which is still a generalized closed strong a-remote family of G.
Proof. Analogous to the proof of Theorem 2.3.
Theorem 2.7. Let L be a complete Heyting algebra. If both G and H are generalized fuzzy compact, then G∨H is generalized fuzzy compact.
Proof. For any family P ⊂GLC(X), by Theorem 2.4 we have that W
x∈X
(G∨H)(x)∧ V
B∈P
B(x)
=
W
x∈X
G(x)∧ V
B∈P
B(x)
∨
W
x∈X
H(x)∧ V
B∈P
B(x)
≥ (
V
F ∈2(P)
W
x∈X
G(x)∧ V
B∈F
B(x) )
∨ (
V
F ∈2(P)
W
x∈X
H(x)∧ V
B∈F
B(x) )
= V
F ∈2(P)
W
x∈X
(G∨H)(x)∧ V
B∈P
B(x)
. This shows that G∨H is generalized fuzzy compact.
Theorem 2.8. IfGis a generalized fuzzy compactL-set and H is a generalized closed L-set, then G∧H is a generalized fuzzy compact L-set.
Proof. Since G is a generalized fuzzy compact L-set, for any familyP ⊂GLC(X), by Theorem 2.4 we have that
W
x∈X
(G∧H)(x)∧ V
B∈P
B(x)
= W
x∈X
G(x)∧ V
B∈P∪{H}
B(x)
!
≥ V
F ∈2(P∪{H})
W
x∈X
G(x)∧ V
B∈P
B(x)
= (
V
F ∈2(P)
W
x∈X
G(x)∧ V
B∈F
B(x) )
∧ (
V
F ∈2(P)
W
x∈X
G(x)∧
H(x)∧ V
B∈F
B(x) )
= V
F ∈2(P)
W
x∈X
(G∧H)(x)∧ V
B∈P
B(x)
.
This shows that G∧H is a generalized fuzzy compactL-set.
Theorem 2.9. Let(X,T1)and(Y,T2)be twoL-spaces, whereLis a complete Heyting algebra, f :X→Y be a generalized irresolute mapping. IfG is generalized fuzzy compact in (X,T1), then so is fL→(G) is in(Y,T2).
Proof. For any P ⊂GLC(X), by Lemma 1.4 and generalized fuzzy compactness of G, we have that
W
y∈Y
fL→(G)(y)∧ V
B∈P
B(y)
= W
x∈X
G(x)∧ V
B∈P
fL←(B)(x)
≥ V
F ∈2(P)
W
x∈X
G(x)∧ V
B∈F
fL←(B)(x)
= V
F ∈2(P)
W
y∈Y
fL→(G)(y)∧ V
B∈F
B(y)
. ThereforefL→(G) is generalized fuzzy compact.
3. Some characterizations of generalized fuzzy compact
In this section, we assume thatLis a completely distributive de Morgan algebra. We give many characterizations of generalized fuzzy compact.
Theorem 3.1. Let (X,T) be an L-space and G∈LX. Then the following conditions are equivalent:
(1) Gis generalized fuzzy compact;
(2) For any a∈L\ {0}, each generalized closed strong a-remote family P of Ghas a finite subfamily F which is a generalized closed strong a-remote family of G;
(3) For any a∈L\ {0}, each generalized closed strong a-remote family P of Ghas a finite subfamily F which is a generalized closeda-remote family of G;
(4) For any a∈L\ {0}, each generalized closed strong a-remote family P of Ghas a finite subfamily F and b∈β(a) such that F is a generalized closed strongb-remote family of G;
(5) For any a∈L\ {0}, each generalized closed strong a-remote family P of Ghas a finite subfamily F of P and b∈β(a) such that F is a generalized closed b-remote family of G;
(6) For any a∈M(L), each generalized closed strong a-remote family P of G has a finite subfamily F which is a generalized closed strong a-remote family of G;
(7) For any a∈M(L), each generalized closed strong a-remote family P of G has a finite subfamily F which is a generalized closed a-remote family of G;
(8) For any a∈M(L), each generalized closed strong a-remote family P of G has a finite subfamily F of P and b∈β∗(a) such that F is a generalized closed strong b-remote family of G;
(9) For any a∈M(L), each generalized closed strong a-remote family P of G has a finite subfamily F of P and b∈β∗(a) such that F is a generalized closed b-remote family of G.
Proof. By Theorem 2.6 we can obtain (1)⇔(2). (2)⇒(3) is obvious. Now to prove (3)⇒(4), suppose that a ∈ L\ {0} and P is a generalized closed strong a-remote fam- ily of G, then we obtain that W
x∈X
G(x)∧ V
B∈P
B(x)
6≥ a, take c ∈ β(a) such that
W
x∈X
G(x)∧ V
B∈P
B(x)
6≥c, obviously P is a strong generalized closed c-remote family of G, by (3) we know that P has a finite subfamily F which is a generalized closed c- remote family of G. Take b ∈ β(a) such that c ∈ β(b), then F is a generalized closed strongb-remote family ofG. (4) is shown. (4)⇒(5) is obvious, we prove (5)⇒(2). For any a∈L\ {0}, suppose thatP is any generalized closed stronga-remote family ofG, by (5), P has a finite subfamilyF andb∈β(a) such thatF is a generalized closedb-remote family ofG. So that for anyx∈X, G(x)∧ V
B∈F
B(x)6≥b, we obtain W
x∈X
G(x)∧ V
B∈F
B(x)
6≥a, in fact, if W
x∈X
G(x)∧ V
B∈F
B(x)
≥a, then by b∈ β(a), there exists x0 ∈ X such that
G(x0)∧ V
B∈F
B(x0) ≥b, a contradiction. So that W
x∈X
G(x)∧ V
B∈F
B(x)
6≥ a. This im- plies that F is a generalized closed stronga-remote family ofG.
Similarly we can prove that (2)⇒(6)⇒(7)⇒(8)⇒(9)⇒(1).
Now we present some characterizations of generalized fuzzy compactness by means of generalized openL-sets.
Theorem 3.2. Let (X,T) be an L-space and G∈LX. Then the following conditions are equivalent:
(1) Gis generalized fuzzy compact;
(2) For any a∈ L\ {1}, each generalized open strong a-shading U of G has a finite subfamily V which is a generalized open strong a-shading of G;
(3) For any a∈ L\ {1}, each generalized open strong a-shading U of G has a finite subfamily V which is a generalized open a-shading of G;
(4) For any a∈L\ {1}, each generalized open strong a-shading U of G, there exists a finite subfamily V of U and b∈α(a) such thatV is a strong generalized open b-shading of G;
(5) For any a∈L\ {1}, each generalized open strong a-shading U of G, there exists a finite subfamily V of U and b∈α(a) such that V is a generalized open b-shading of G;
(6) For any a ∈ P(L), each generalized open strong a-shading U of G has a finite subfamily V which is a generalized open strong a-shading of G;
(7) For any a ∈ P(L), each generalized open strong a-shading U of G has a finite subfamily V which is a generalized open a-shading of G;
(8) For any a ∈ P(L), each generalized open strong a-shading U of G has a finite subfamily V of U and b∈α∗(a) such that V is a strong generalized openb-shading of G;
(9) For any a ∈ P(L), each generalized open strong a-shading U of G has a finite subfamily V of U and b∈α∗(a) such that V is a generalized open b-shading of G.
Proof. By Theorem 2.3 we can obtain (1)⇔(2).
(2)⇒(3) is obvious.
(3)⇒(4). Suppose that a∈L\ {1} andU is a generalized open stronga-shading of G, then V
x∈X
G0(x)∨ W
B∈U
B(x)
6≤a. Takec∈α(a) such that V
x∈X
G0(x)∨ W
B∈U
B(x)
6≤c, obviously U is a generalized open strong c-shading of G and by (3) we know thatU has
a finite subfamily V which is a generalized open c-shading ofG. Take b∈α(a) such that c∈α(b), then V is a generalized open strongb-shading ofG, (4) is shown.
(4)⇒(5) is obvious.
(5)⇒(2). For any a ∈ L\ {1}, suppose that U is any generalized open strong a- shading of G, by (5), U has a finite subfamily V and b ∈ α(a) such that V is a gen- eralized open b-shading of G. So that for any x ∈ X, G0(x) ∨ W
B∈V
B(x) 6≤ b, we ob- tain V
x∈X
G0(x)∨ W
B∈V
B(x)
6≤ a, in fact, if V
x∈X
G0(x)∨ W
B∈V
B(x)
≤ a, then by b ∈ α(a), there exists x0 ∈ X such that G(x0)∨ W
B∈V
B(x0) ≤ b, a contradiction. So that V
x∈X
G0(x)∨ W
B∈V
B(x)
6≤ a. This implies that V is a generalized open strong a- shading of G.
Similarly we can prove that (2)⇒(6)⇒(7)⇒(9) ⇒(9)⇒(1).
Definition 3.3. Let (X,T) be an L-space, a ∈ L \ {0} and G ∈ LX. A family U ⊂GLO(X) is said to be a generalized open βa-cover of G if for any x ∈ X with a 6∈ β(G0(x)), there exists A ∈ U such that a ∈ β(A(x)). U is said to be a generalized open strong βa-cover ofGifa∈β
V
x∈X
G0(x)∨ W
A∈U
A(x)
.
It is obvious that a generalized open strong βa-cover of G is generalized open βa- cover G and U is a generalized open βa-cover of G if and only if for any x ∈ X, a ∈ β
G0(x)∨ W
A∈U
A(x)
.
Theorem 3.4. Let (X,T) be an L-space and G∈LX. Then the following conditions are equivalent:
(1) Gis generalized fuzzy compact;
(2) For any a ∈ L\ {0}, each generalized open strong βa-cover U of G has a finite subfamily V which is a generalized open strong βa-cover of G;
(3) For any a ∈ L\ {0}, each generalized open strong βa-cover U of G has a finite subfamily V which is a generalized open βa-cover of G;
(4) For any a∈L\ {0}, any generalized open strong βa-cover U of G, there exists a finite subfamily V of U and b ∈L with a∈β(b) such that V is a generalized open strong βa-cover of G;
(5) For any a∈L\ {0}, any generalized open strong βa-cover U of G, there exists a finite subfamilyV of U andb∈L witha∈β(b) such thatV is a generalized openβa-cover of G;
(6) For any a ∈ M(L), each generalized open strong βa-cover U of G has a finite subfamily V which is a generalized open strong βa-cover of G;
(7) For any a ∈ M(L), each generalized open strong βa-cover U of G has a finite subfamily V which is a generalized open βa-cover of G;
(8) For any a∈M(L) and any generalized open strong βa-cover U of G, there exists a finite subfamily V of U and b∈M(L) with a∈β∗(b) such that V is a generalized open strong βa-cover of G;
(9) For any a∈M(L) and any generalized open strong βa-cover U of G, there exists a finite subfamily V of U and b∈M(L) with a∈β∗(b) such that V is a generalized open βa-cover of G.
Proof. We only prove (1)⇔(2).
(1)⇒(2). Suppose that Gis generalized fuzzy compact and for any a∈L\ {0},U is any generalized open strongβa-cover of G. Then
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
≤ _
V∈2(U)
^
x∈X
G0(x)∨ _
A∈V
A(x)
! .
So
β ^
x∈X
G0(x)∨ _
A∈U
A(x)
!!
≤β
_
V∈2(U)
^
x∈X
G0(x)∨ _
A∈V
A(x)
!
.
Bya∈β
V
x∈X
G0(x)∨ W
A∈U
A(x)
, we obtaina∈β W
V∈2(U)
V
x∈X
G0(x)∨ W
A∈V
A(x) !
, therefore
a∈ [
V∈2(U)
β ^
x∈X
G0(x)∨ _
A∈V
A(x)
!!
,
hence there exists aV ∈2(U) such that a∈β ^
x∈X
G0(x)∨ _
A∈V
A(x)
!!
.
Thus V is a generalized open strongβa-cover ofG.
(2)⇒(1). Suppose that for any a∈L\ {0}, each generalized open strongβa-cover U of G has a finite subfamily V which is a generalized open strong βa-cover of G, then we
know that a∈β ^
x∈X
G0(x)∨ _
A∈U
A(x)
!!
implies that a∈β ^
x∈X
G0(x)∨ _
A∈V
A(x)
!!
where V ∈2(U). Hence β ^
x∈X
G0(x)∨ _
A∈U
A(x)
!!
≤β ^
x∈X
G0(x)∨ _
A∈V
A(x)
!!
.
Thus
^
x∈X
G0(x)∨ _
A∈U
A(x)
!
≤ _
V∈2(U)
^
x∈X
G0(x)∨ _
A∈V
A(x)
! .
This prove thatG is generalized fuzzy compact.
Definition 3.5. Let (X,T) be an L-space, a ∈ L \ {0} and G ∈ LX. A family U ⊂GLO(X) is said to be a generalized open Qa-cover of G if for any x∈ X it follows that G0∨ W
A∈U
A(x)≥a.
It is obvious that a generalized open βa-cover of G is a generalized open Qa-cover of G.
Moreover form Definition 2.1 we also can obtain the following result.
Theorem 3.6. Let (X,T) be an L-space and G∈LX. Then the following conditions are equivalent:
(1) Gis generalized fuzzy compact;
(2) For anya∈L\ {0} and any b∈β(a)\ {0}, each generalized open Qa-cover of G, has a finite subfamily which is a generalized open Qb-cover of G;
(3) For anya∈L\ {0} and any b∈β(a)\ {0}, each generalized open Qa-cover of G, has a finite subfamily which is a generalized open βa-cover of G;
(4) For anya∈L\ {0} and any b∈β(a)\ {0}, each generalized open Qa-cover of G, has a finite subfamily which is a generalized open strong βa-cover of G;
(5) For any a∈M(L) and any b∈β∗(a), each generalized open Qa-cover of G, has a finite subfamily which is a generalized open Qb-cover of G;
(6) For any a∈M(L) and any b∈β∗(a), each generalized open Qa-cover of G, has a finite subfamily which is a generalized open βb-cover of G;
(7) For any a∈M(L) and any b∈β∗(a), each generalized open Qa-cover of G, has a finite subfamily which is a generalized open strong βb-cover of G.
Acknowledgments
The authors would like to thank anonymous referees for their valuable comments and suggestions and we also would like to thank professor F.-G. Shi for his profound guide.
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