FU-GUI SHI
Received 8 October 2004 and in revised form 8 June 2005
The concepts of semicompactness, countable semicompactness, and the semi-Lindel¨of property are introduced inL-topological spaces, whereLis a complete de Morgan alge-bra. They are defined by means of semiopenL-sets and their inequalities. They do not rely on the structure of basis latticeLand no distributivity in Lis required. They can also be characterized by semiclosedL-sets and their inequalities. WhenLis a completely distributive de Morgan algebra, their many characterizations are presented.
1. Introduction
The notion of semicompactness [3] was introduced inL-topological spaces by Kudri. In Kudri’s work [6], he followed the lines of his definition of compactness which is equiva-lent to the notion of strong fuzzy compactness in [7,8,13]. However, Kudri’s semicom-pactness relies on the structure ofLandLis required to be completely distributive.
In [10,12], a new definition of fuzzy compactness is presented inL-topological spaces by means of an inequality, which does not depend on the structure of Land no dis-tributivity is required inL. WhenLis a completely distributive de Morgan algebra, it is equivalent to the notion of fuzzy compactness in [7,8,13].
Following the lines of [10,12], we will introduce a new definition of semicompactness inL-topological spaces by means of semiopen L-sets and their inequality, whereLis a complete de Morgan algebra. This definition does not rely on the structure of basis lattice L and no distributivity inL is required. It can also be characterized by semiclosedL -sets and their inequality. WhenLis a completely distributive de Morgan algebra, its many characterizations are presented. Moreover, we also will introduce the notions of countable semicompactness and the semi-Lindel¨of property and research their properties.
2. Preliminaries
Throughout this paper, (L,,,) is a complete de Morgan algebra,X a nonempty set. LX is the set of allL-fuzzy sets (orL-sets for short) onX. The smallest element and the
largest element inLXare denoted by 0 and 1.
Copyright©2005 Hindawi Publishing Corporation
L
An elementainLis called prime element ifa≥b∧cimplies thata≥b ora≥c.a inLis called a coprime element ifais a prime element [5]. The set of nonunit prime elements inLis denoted byP(L). The set of nonzero coprime elements inLis denoted by M(L).
The binary relation≺inLis defined as follows: fora,b∈L,a≺bif and only if for every subsetD⊆L, the relation b≤supD always implies the existence ofd∈Dwith a≤d[4]. In a completely distributive de Morgan algebraL, each elementbis a sup of
{a∈L|a≺b}.{a∈L|a≺b}is called the greatest minimal family ofbin the sense of [7,13], in symbolβ(b). Moreover forb∈L, defineα(b)= {a∈L|a≺b}andα∗(b)=
α(b)∩P(L).
Fora∈LandA∈LX, we use the following notations in [9]:
A(a)=x∈X|A(x)a, A (a)=
x∈X|a∈βA(x). (2.1)
AnL-topological space (orL-space for short) is a pair (X,᐀), where᐀is a subfamily ofLXwhich contains 0, 1 and is closed for any suprema and finite infima.᐀is called anL
-topology onX. Each member of᐀is called an openL-set and its quasicomplementation is called a closedL-set.
Definition 2.1(see [7,13]). For a topological space (X,τ), letωL(τ) denote the family
of all the lower semicontinuous maps from (X,τ) toL, that is,ωL(τ)= {A∈LX|A(a)∈
τ,a∈L}. ThenωL(τ) is anL-topology onX, in this case, (X,ωL(τ)) is topologically
gen-erated by (X,τ).
Definition 2.2(see [7,13]). AnL-space (X,᐀) is weak induced if for all a∈L, for all A∈᐀, it follows thatA(a)∈[᐀], where [᐀] denotes the topology formed by all crisp sets
in᐀.
It is obvious that (X,ωL(τ)) is weak induced.
Lemma2.3 (see [11]). Let(X,᐀)be a weakly inducedL-space,a∈L,A∈᐀. ThenA(a)is
an open set in[᐀].
For a subfamilyΦ⊆LX, 2(Φ)denotes the set of all finite subfamilies ofΦ. 2[Φ]denotes
the set of all countable subfamilies ofΦ.
Definition 2.4(see [10,12]). Let (X,᐀) be anL-space,G∈LXis called (countably)
com-pact if for every (countably) familyᐁ⊆᐀, it follows that
x∈X
G(x)∨
A∈ᐁ A(x)
≤
ᐂ∈2(ᐁ)
x∈X
G(x)∨
A∈ᐂ A(x)
. (2.2)
Definition 2.5(see [10]). Let (X,᐀) be anL-space,G∈LX is said to have the Lindel¨of
property if for every familyᐁ⊆᐀, it follows that
x∈X
G(x)∨
A∈ᐁ A(x)
≤
ᐂ∈2[ᐁ]
x∈X
G(x)∨
A∈ᐂ A(x)
Lemma2.6 (see [10]). LetLbe a complete Heyting algebra, let f :X→Y be a map, fL→: LX→LYis the extension of f, then for any familyᏼ⊆LY,
y∈Y
fL→(G)(y)∧ B∈ᏼ
B(y)
= x∈X
G(x)∧ B∈ᏼ
fL←(B)(x)
. (2.4)
Definition 2.7(see [1]). AnL-setGin anL-space (X,᐀) is called semiopen if there exists A∈᐀such thatA≤G≤cl(A).Gis called semiclosed ifGis semiopen.
Definition 2.8. Let (X,᐀1) and (Y,᐀2) be twoL-spaces. A map f : (X,᐀1)→(Y,᐀2) is
called
(1) semicontinuous [1] if fL←(G) is semiopen in (X,᐀1) for every open L-set G in
(Y,᐀2);
(2) irresolute [2] if fL←(G) is semiopen in (X, !.᐀1) for every semiopen L-set G in
(Y,᐀2).
3. Definition and characterizations of semicompactness
Definition 3.1. Let (X,᐀) be anL-space.G∈LXis called (countably) semicompact if for
every (countable) familyᐁof semiopenL-sets, it follows that
x∈X
G(x)∨
A∈ᐁ A(x)
≤
ᐂ∈2(ᐁ)
x∈X
G(x)∨
A∈ᐂ A(x)
. (3.1)
Definition 3.2. Let (X,᐀) be anL-space.G∈LXis said to have the semi-Lindel¨of property
(or be a semi-Lindel¨ofL-set) if for every familyᐁof semiopenL-sets, it follows that
x∈X
G(x)∨
A∈ᐁ A(x)
≤
ᐂ∈2[ᐁ]
x∈X
G(x)∨
A∈ᐂ A(x)
. (3.2)
Example 3.3. LetXbe any nonempty set and letAbe a [0, 1]-set onXdefined asA(x)=
0.5, for allx∈X. Let᐀= {∅,X,A}. Then the set of all semiopen [0, 1]-sets in (X,᐀) is
᐀. In this case, any [0, 1]-set in (X,᐀) is semicompact, hence it is countably semicompact and has the semi-Lindel¨of property.
Obviously, we have the following theorem.
Theorem3.4. Semicompactness implies countably semicompactness and the semi-Lindel¨of property. Moreover, anL-set having the semi-Lindel¨of property is semicompact if and only if it is countably semicompact.
Since an openL-set must be semiopen, we have the following theorem.
L
From Definitions 3.1 and 3.2, we can obtain the following two theorems by using quasicomplementation.
Theorem3.6. Let(X,᐀)be anL-space.G∈LXis (countably) semicompact if and only if
for every (countable) familyᏮof semiclosedL-sets, it follows that
x∈X
G(x)∧ B∈Ꮾ
B(x)
≥
Ᏺ∈2(Ꮾ)x∈X
G(x)∧ B∈Ᏺ
B(x)
. (3.3)
Theorem3.7. Let(X,᐀)be anL-space.G∈LXhas the semi-Lindel¨of property if and only
if for every familyᏮof semiclosedL-sets, it follows that
x∈X
G(x)∧ B∈Ꮾ
B(x)
≥
Ᏺ∈2[Ꮾ]x∈X
G(x)∧ B∈Ᏺ
B(x)
. (3.4)
In order to present characterizations of semicompactness, countable semicompact-ness and the semi-Lindel¨of property, we generalize the notions ofa–shading anda –R-neighborhood family in [10,12] as follows.
Definition 3.8. Let (X,᐀) be anL-space,a∈L\{1}, andG∈LX. A familyᏭ⊆LXis said
to be
(1) ana–shading ofGif for anyx∈X, (G(x)∨A∈ᐁA(x))a; (2) a stronga–shading ofGifx∈X(G(x)∨A∈ᐁA(x))a;
(3) ana–R-neighborhood family ofGif for anyx∈X, (G(x)∧B∈ᏼB(x))a; (4) a stronga–R-neighborhood family ofGifx∈X(G(x)∧
B∈ᏼB(x))a.
It is obvious that a strong a–shading of G is an a–shading of G, a strong a –R-neighborhood family of Gis ana–R-neighborhood family ofG, andᏼis a stronga– R-neighborhood family ofGif and only ifᏼis a stronga–shading ofG.
Definition 3.9. Leta∈L\{0}andG∈LX. A subfamilyᏭofLXis said to have weaka–
nonempty intersection inGifx∈X(G(x)∧A∈ᏭA(x))≥a.Ꮽis said to have the finite (countable) weaka–intersection property inGif every finite (countable) subfamilyᏲof
Ꮽhas weaka–nonempty intersection inG.
From Definitions3.1,3.2, Theorems3.5and3.6, we immediately obtain the following two results.
Theorem3.10. Let(X,᐀)be anL-space andG∈LX. Then the following conditions are
equivalent.
(1)Gis (countably) semicompact.
(2)For anya∈L\{1}, each (countable) semiopen stronga–shadingᐁofGhas a finite subfamily which is a stronga–shading ofG.
(3)For anya∈L\{0}, each (countable) semiclosed stronga–R-neighborhood familyᏼ ofGhas a finite subfamily which is a stronga–R-neighborhood family ofG.
Theorem3.11. Let(X,᐀)be anL-space andG∈LX. Then the following conditions are
equivalent.
(1)Ghas the semi-Lindel¨of property.
(2)For anya∈L\{1}, each semiopen stronga–shadingᐁofGhas a countable subfamily which is a stronga–shading ofG.
(3)For anya∈L\{0}, each semiclosed stronga–R-neighborhood familyᏼofGhas a countable subfamily which is a stronga–R-neighborhood family ofG.
(4)For anya∈L\{0}, each family of semiclosedL-sets which has the countable weak a–intersection property inGhas weaka–nonempty intersection inG.
4. Properties of (countable) semicompactness
Theorem4.1. LetLbe a complete Heyting algebra. If bothGandHare (countably) semi-compact, thenG∨His (countably) semicompact.
Proof. For any (countable) familyᏼof semiclosedL-sets, byTheorem 3.5we have that
x∈X
(G∨H)(x)∧ B∈ᏼ
B(x)
=
x∈X
G(x)∧ B∈ᏼ
B(x)
∨
x∈X
H(x)∧ B∈ᏼ
B(x)
≥
Ᏺ∈2(ᏼ)x∈X
G(x)∧ B∈Ᏺ
B(x)
∨
Ᏺ∈2(ᏼ)x∈X
H(x)∧ B∈Ᏺ
B(x)
=
Ᏺ∈2(ᏼ)x∈X
(G∨H)(x)∧ B∈Ᏺ
B(x)
.
(4.1)
This shows thatG∨His (countably) semicompact.
Analogously, we have the following result.
Theorem4.2. LetLbe a complete Heyting algebra. If bothGandHhave the semi-Lindel¨of property, thenG∨Hhas the semi-Lindel¨of property.
Theorem4.3. IfGis (countably) semicompact andHis semiclosed, thenG∧His (count-ably) semicompact.
Proof. For any (countable) familyᏼof semiclosedL-sets, byTheorem 3.5we have that
x∈X
(G∧H)(x)∧ B∈ᏼ
B(x)
= x∈X
G(x)∧
B∈ᏼ{H} B(x)
≥
Ᏺ∈2(ᏼ∪{H})x∈X
G(x)∧ B∈Ᏺ
L
=
Ᏺ∈2(ᏼ)x∈X
G(x)∧ B∈Ᏺ
B(x)
∧
Ᏺ∈2(ᏼ)x∈X
G(x)∧H(x)∧ B∈Ᏺ
B(x)
=
Ᏺ∈2(ᏼ)x∈X
G(x)∧H(x)∧ B∈Ᏺ
B(x)
=
Ᏺ∈2(ᏼ)x∈X
(G∧H)(x)∧ B∈Ᏺ
B(x)
.
(4.2)
This shows thatG∧His (countably) semicompact.
Analogously, we have the following result.
Theorem4.4. IfGhas the semi-Lindel¨of property andHis semiclosed, thenG∧Hhas the semi-Lindel¨of property.
Theorem 4.5. Let Lbe a complete Heyting algebra and let f : (X,᐀1)→(Y,᐀2)be an
irresolute map. IfGis a semicompact (resp., countably semicompact, semi-Lindel¨of)L-set in (X,᐀1), then so is fL→(G)in(Y,᐀2).
Proof. We only prove that the theorem is true for semicompactness. Suppose thatᏼis a family of semiclosedL-sets in (Y,᐀2), byLemma 2.6and semicompactness ofG, we have
that
y∈Y
fL→(G)(y)∧ B∈ᏼ
B(y)
= x∈X
G(x)∧
B∈ᏼ
fL←(B)(x)
≥
Ᏺ∈2(ᏼ)x∈X
G(x)∧ B∈Ᏺ
fL←(B)(x)
=
Ᏺ∈2(ᏼ)y∈Y
fL→(G)(y)∧ B∈Ᏺ
B(y)
.
(4.3)
Therefore fL→(G) is semicompact.
Analogously, we have the following result.
Theorem4.6. LetLbe a complete Heyting algebra and let f : (X,᐀1)→(Y,᐀2)be a
semi-continuous map. IfGis a semicompact (resp., countably semicompact, semi-Lindel¨of)L-set in(X,᐀1), then fL→(G)is a compact (countably compact, Lindel¨of)L-set in(Y,᐀2).
Definition 4.7. Let (X,᐀1) and (Y,᐀2) be twoL-spaces. A map f : (X,᐀1)→(Y,᐀2)
is called strongly irresolute if fL←(G) is open in (X,᐀1) for every semiopenL-setG in
(Y,᐀2).
Theorem 4.8. Let L be a complete Heyting algebra and let f : (X,᐀1)→(Y,᐀2) be a
strongly irresolute map. If G is a compact (resp., countably compact, Lindel¨of) L-set in (X,᐀1), then fL→(G)is a semicompact (countably semicompact, semi-Lindel¨of) L-set in
(Y,᐀2).
5. Further characterizations of semicompactness and goodness
In this section, we assume thatLis a completely distributive de Morgan algebra. Now we generalize the notions ofβa–open cover andQa–open cover [10] as follows.
Definition 5.1. Let (X,᐀) be anL-space,a∈L\{0}, andG∈LX. A familyᐁ⊆LXis called
aβa–cover ofGif for anyx∈X, it follows thata∈β(G(x)∨
A∈ᐁA(x)).ᐁis called a strongβa–cover ofGifa∈β(x∈X(G(x)∨
A∈ᐁA(x))).
It is obvious that a strongβa–cover ofGmust be aβa–cover ofG.
Definition 5.2. Let (X,᐀) be anL-space,a∈L\{0}, andG∈LX. A familyᐁ⊆LXis called
aQa–cover ofGif for anyx∈X, it follows thatG(x)∨A∈ᐁA(x)≥a.
It is obvious that aβa–cover ofGmust be aQa–cover ofG.
Analogous to the proof of [10, Theorem 2.9], we can obtain the following theorem.
Theorem 5.3. Let (X,᐀)be anL-space andG∈LX. Then the following conditions are
equivalent.
(1)Gis (countably) semicompact.
(2)For anya∈L\{0}, each (countable) semiclosed stronga–R-neighborhood familyᏼ ofGhas a finite subfamily which is a stronga–R-neighborhood family ofG.
(3)For anya∈L\{0}, each (countable) semiclosed stronga–R-neighborhood familyᏼ ofGhas a finite subfamily which is ana–R-neighborhood family ofG.
(4)For anya∈L\{0}and any (countable) semiclosed stronga–R-neighborhood family
ᏼofG, there exist a finite subfamilyᏲofᏼandb∈β(a)such that Ᏺis a strong b–R-neighborhood family ofG.
(5)For anya∈L\{0}and any (countable) semiclosed stronga–R-neighborhood family
ᏼofG, there exist a finite subfamilyᏲofᏼandb∈β(a)such thatᏲis ab–R-neighborhood family ofG.
(6)For anya∈M(L), each (countable) semiclosed stronga–R-neighborhood familyᏼof Ghas a finite subfamily which is a stronga–R-neighborhood family ofG.
(7)For anya∈M(L), each (countable) semiclosed stronga–R-neighborhood familyᏼof Ghas a finite subfamily which is ana–R-neighborhood family ofG.
(8)For anya∈M(L)and any (countable) semiclosed stronga–R-neighborhood family
ᏼofG, there exist a finite subfamilyᏲofᏼandb∈β∗(a)such thatᏲis a strong b–R-neighborhood family ofG.
(9)For anya∈M(L)and any (countable) semiclosed stronga–R-neighborhood familyᏼ ofG, there exist a finite subfamilyᏲofᏼandb∈β∗(a)such thatᏲis ab–R-neighborhood family ofG.
L
(11)For anya∈L\{1}, each (countable) semiopen stronga–shadingᐁofGhas a finite subfamily which is ana–shading ofG.
(12)For anya∈L\{1}and any (countable) semiopen stronga–shadingᐁofG, there exist a finite subfamilyᐂofᐁandb∈α(a)such thatᐂis a strongb–shading ofG.
(13)For anya∈L\{1}and any (countable) semiopen stronga–shadingᐁofG, there exist a finite subfamilyᐂofᐁandb∈α(a)such thatᐂis ab–shading ofG.
(14)For anya∈P(L), each (countable) semiopen stronga–shadingᐁofGhas a finite subfamily which is a stronga–shading ofG.
(15)For anya∈P(L), each (countable) semiopen stronga–shadingᐁofGhas a finite subfamily which is ana–shading ofG.
(16)For anya∈P(L)and any (countable) semiopen stronga–shadingᐁofG, there exist a finite subfamilyᐂofᐁandb∈α∗(a)such thatᐂis a strongb–shading ofG.
(17)For anya∈P(L)and any (countable) semiopen stronga–shadingᐁofG, there exist a finite subfamilyᐂofᐁandb∈α∗(a)such thatᐂis ab–shading ofG.
(18)For anya∈L\{0}, each (countable) semiopen strongβa–coverᐁofGhas a finite
subfamily which is a strongβa–cover ofG.
(19)For anya∈L\{0}, each (countable) semiopen strongβa–coverᐁofGhas a finite
subfamily which is aβa–cover ofG.
(20)For anya∈L\{0}and any (countable) semiopen strongβa–coverᐁofG, there exist
a finite subfamilyᐂofᐁandb∈Lwitha∈β(b)such thatᐂis a strongβb–cover ofG.
(21)For anya∈L\{0}and any (countable) semiopen strongβa–coverᐁofG, there exist
a finite subfamilyᐂofᐁandb∈Lwitha∈β(b)such thatᐂis aβb–cover ofG.
(22)For anya∈M(L), each (countable) semiopen strongβa–coverᐁofGhas a finite
subfamily which is a strongβa–cover ofG.
(23)For anya∈M(L), each (countable) semiopen strongβa–coverᐁofGhas a finite
subfamily which is aβa–cover ofG.
(24)For anya∈M(L)and any (countable) semiopen strongβa–coverᐁofG, there exist
a finite subfamilyᐂofᐁandb∈M(L)witha∈β∗(b)such thatᐂis a strongβb–cover of
G.
(25)For anya∈M(L)and any (countable) semiopen strongβa–coverᐁofG, there exist
a finite subfamilyᐂofᐁandb∈M(L)witha∈β∗(b)such thatᐂis aβb–cover ofG.
(26)For anya∈L\{0}and anyb∈β(a)\{0}, each (countable) semiopenQa–cover of
Ghas a finite subfamily which is aQb–cover ofG.
(27)For anya∈L\{0}and anyb∈β(a)\{0}, each (countable) semiopenQa–cover of
Ghas a finite subfamily which is aβb–cover ofG.
(28)For anya∈L\{0}and anyb∈β(a)\{0}, each (countable) semiopenQa–cover of
Ghas a finite subfamily which is a strongβb–cover ofG.
(29)For anya∈M(L)and anyb∈β∗(a), each (countable) semiopenQa–cover ofGhas
a finite subfamily which is aQb–cover ofG.
(30)For any a∈M(L)and anyb∈β∗(a), each semiopenQa–cover ofGhas a finite
subfamily which is aβb–cover ofG.
(31)For anya∈M(L)and anyb∈β∗(a), each (countable) semiopenQa–cover ofGhas
a finite subfamily which is a strongβb–cover ofG.
Lemma5.4. Let(X,ω(τ))be generated topologically by(X,τ). IfAis a semiopenL-set in (X,τ), thenχAis a semiopen set in(X,ω(τ)). IfBis a semiopenL-set in(X,ω(τ)), thenB(a)
is a semiopen set in(X,τ)for everya∈L.
Proof. IfAis a semiopen set in (X,τ), then there existsD∈τsuch thatD⊆A⊆cl(D). Thus we have that
χD≤χA≤χcl(D)=cl
χD
. (5.1)
This shows thatχAis semiopen.
IfBis a semiopenL-set in (X,ω(τ)), then there existsE∈ω(τ) such thatE≤B≤cl(E). Thus we have thatE(a)⊆B(a)⊆cl(E)(a).From [9], we can obtain that cl(E)(a)⊆cl(E(a)).
Hence byLemma 2.3, we know thatB(a)is a semiopen set in (X,τ).
The following two theorems show that semicompactness, countable semicompactness and the semi-Lindel¨of property are good extensions.
Theorem5.5. Let(X,τ)be a topological space and let(X,ω(τ))be generated topologically by(X,τ). Then(X,ω(τ)) is (countably) semicompact if and only if(X,τ)is (countably) semicompact.
Proof. Necessity. LetᏭbe a (countable) semiopen cover of (X,τ). Then{χA|A∈Ꮽ}is
a family of semiopenL-sets in (X,ω(τ)) withx∈X(A∈ᐁχA(x))=1. From (countable)
semicompactness of (X,ω(τ)), we know that
ᐂ∈2(ᐁ)
x∈X
A∈ᐂ χA(x)
=
ᐂ∈2(ᐁ)
x∈X
A∈ᐂ χA(x)
=1. (5.2)
This implies that there existsᐂ∈2(ᐁ)such that x∈X(
A∈ᐂχA(x))=1. Hence,ᐂis a
cover of (X,τ). Therefore (X,τ) is (countably) semicompact.
Sufficiency. Let ᐁ be a (countable) family of semiopenL-sets in (X,ω(τ)) and let
x∈X(
B∈ᐁB(x))=a. Ifa=0, then obviously we have that
x∈X
B∈ᐁ B(x)
≤
ᐂ∈2(ᐁ)
x∈X
A∈ᐂ B(x)
. (5.3)
Now we suppose thata=0. In this case, for anyb∈β(a)\{0}, we have that
b∈β
x∈X
B∈ᐁ B(x)
⊆
x∈X
β
B∈ᐁ B(x)
=
x∈X
B∈ᐁ
βB(x). (5.4)
FromLemma 5.4, this implies that{B(b)|B∈ᐁ}is a semiopen cover of (X,τ). From
(countable) semicompactness of (X,τ), we know that there exists ᐂ∈2(ᐁ) such that {B(b)|B∈ᐂ}is a cover of (X,τ). Henceb≤x∈X(B∈ᐂB(x)). Further, we have that
b≤
x∈X
B∈ᐂ B(x)
≤
ᐂ∈2(ᐁ)
x∈X
B∈ᐂ B(x)
L
This implies that
x∈X
B∈ᐁ B(x)
=a= b|b∈β(a)≤
ᐂ∈2(ᐁ)
x∈X
B∈ᐂ B(x)
. (5.6)
Therefore, (X,ω(τ)) is (countably) semicompact.
Analogously, we have the following result.
Theorem5.6. Let(X,τ)be a topological space and let(X,ω(τ))be generated topologically by(X,τ). Then(X,ω(τ))has the semi-Lindel¨of property if and only if(X,τ)has the semi-Lindel¨of property.
Acknowledgment
This work is supported by the National Natural Science Foundation of China (10371079) and the Base Research Foundation of Beijing Institute of Technology.
References
[1] K. K. Azad,On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl.82(1981), no. 1, 14–32.
[2] D. C¸oker and A. H. Es¸,On fuzzyS-closed spaces, Do˘ga Mat.11(1987), no. 3, 145–152. [3] Ch. Dorsett,SemicompactR1and product spaces, Bull. Malaysian Math. Soc. (2)3(1980), no. 1,
15–19.
[4] Ph. Dwinger,Characterization of the complete homomorphic images of a completely distributive complete lattice. I, Nederl. Akad. Wetensch. Indag. Math.44(1982), no. 4, 403–414. [5] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott,A
Com-pendium of Continuous Lattices, Springer, Berlin, 1980.
[6] S. R. T. Kudri,Semicompactness andS∗-closedness inL-fuzzy topological spaces, Fuzzy Sets and Systems109(2000), no. 2, 223–231.
[7] Y.-M. Liu and M.-K. Luo,Fuzzy Topology, Advances in Fuzzy Systems—Applications and The-ory, vol. 9, World Scientific, New Jersey, 1997.
[8] R. Lowen,A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl.64(1978), no. 2, 446–454.
[9] F.-G. Shi,The theory and applications ofLβ-nested sets andLα-nested sets, Mohu Xitong yu Shuxue9(1995), no. 4, 65–72 (Chinese).
[10] , Countable compactness and the Lindel¨of property ofL-fuzzy sets, Iranian Journal of Fuzzy Systems1(2004), no. 1, 79–88.
[11] ,A new notion of fuzzy compactness inL-topological spaces, Inform. Sci.173(2005), 35–48.
[12] ,Fuzzy compactness inL-topological spaces, Acta Math. Sinica, submitted.
[13] G. J. Wang,Theory ofL-Fuzzy Topological Space, Shanxi Normal University Press, Xian, 1988 (Chinese).
Fu-Gui Shi: Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, China