Download A new definition of fuzzy compactness

Fuzzy Sets and Systems 158 (2007) 1486 – 1495
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A new definition of fuzzy compactness夡
Fu-Gui Shi∗
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China
Received 22 August 2005; received in revised form 12 November 2006; accepted 10 February 2007
Available online 28 February 2007
Abstract
In this paper, a new definition of fuzzy compactness is presented in L-topological spaces when L is a complete DeMorgan algebra.
This definition does not rely on the structure of the basis lattice L and no distributivity in L is required. The intersection of a fuzzy
compact L-set and a closed L-set is fuzzy compact. The continuous image of a fuzzy compact L-set is fuzzy compact. If the set of all
prime elements of L is order generating, then the Alexander Subbase Theorem is true. When L is a completely distributive DeMorgan
algebra, it is equivalent to Lowen’s fuzzy compactness, and in this case, its many characterizations are presented by means of open
L-sets and closed L-sets.
© 2007 Elsevier B.V. All rights reserved.
Keywords: L-topology; Fuzzy compactness; Strong a-shading; Strong a-remote family; a -cover; Strong a -cover
1. Introduction
The concept of compactness is one of the most important concepts in general topology. The concept of compactness
of a [0, 1]-topological space was first introduced by Chang in terms of open cover [2]. Chang’s compactness has been
greatly extended to the variable-basis case by Rodabaugh (see [13]), and it may be regarded as a successful definition
of compactness in poslat topology from the categorical point of view (cf. [12,13]).
Moreover, Gantner et al. [3] introduced -compactness for an L-topological space, Lowen [10,11] introduced fuzzy
compactness (we shall call it Lowen’s fuzzy compactness), strong fuzzy compactness and ultra-fuzzy compactness for
a [0, 1]-topological space, Liu [8] introduced Q-compactness for an L-subset, and Wang and Zhao [17,21] introduced
N-compactness for an L-subset. Recently Shi [15] introduced S ∗ -compactness for an L-subset.
In 1983, Wang [17] presented a characterization of Lowen’s fuzzy compactness in terms of -net and extended it to
L-topological spaces [18], where L is a completely distributive DeMorgan algebra (i.e., a completely distributive lattice
with an order-reversing involution). J.J. Chadwick also gave some characterizations of Lowen’s fuzzy compactness in
[0, 1]-topological spaces [1]. However, the above mentioned notions of fuzzy compactness depend on the structure of
the basis lattice L and L was required to have complete distributivity.
夡 The project is supported by the National Natural Science Foundation of China (10371079) and the Base Research Foundation of Beijing Institute
of Technology.
∗ Tel.: +86 01088583137.
E-mail addresses: [email protected], [email protected] (F.-G. Shi).
0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2007.02.006
F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
1487
Kubiák [6] also extended Lowen’s fuzzy compactness to L-topological spaces by means of closed L-sets and the way
below relation, where complete distributivity was not required. But his definition still depends on the structure of the
basis lattice L and cannot be restated in terms of open L-sets by simply using quasi-complement.
In this paper, our aim is to present a new definition of fuzzy compactness of an L-subset by means of a family of open
L-sets and their inequality, which does not depend on the structure of L and no distributivity in L is required. When L is
a completely distributive DeMorgan algebra, it is equivalent to the notion of Lowen’s fuzzy compactness in [9,18,20].
2. Preliminaries
Throughout this paper, (L, , , ) is a complete DeMorgan algebra, and X is a nonempty set. LX is the set of all
L-fuzzy sets on X. The smallest element and the largest element in LX are denoted by 0 and 1, respectively.
An element a in L is called a prime element if a b ∧ c implies a b or a c. a in L is called a co-prime element
if a is a prime element [4]. The set of nonunit prime elements in L is denoted by P (L). The set of nonzero co-prime
elements in L is denoted by M(L).
According to [19], we know that if L is completely distributive, then each element a in L has the greatest minimal
family (the greatest maximal family), denoted by (a) ((a)). In this case, ∗ (a) = (a) ∩ M(L) is a minimal family
of a and ∗ (a) = (a) ∩ P (L) is a maximal family of a.
An L-topological space (or L-space for short) is a pair (X, T ), where T is a subfamily of LX which contains 0, 1
and is closed for any suprema and finite infima. Each member of T is called an open L-set and its quasi-complement
is called a closed L-set.
For a subfamily ⊆ LX , 2() denotes the set of all finite subfamilies of .
The L-(fuzzy) closed interval [a, b](L) is defined as the set of all equivalence classes of antitone maps : R → L
satisfying
(a−) =
(t) = 1 and (b+) =
(t) = 0,
t<a
t>b
where the equivalence identifies two such maps , if and only if ∀t ∈ I , (t+) = (t+). The canonical L-topology
on R(L) is generated from the subbase {Lt , Rt | t ∈ [a, b]}, where
Lt : I (L) → L
by Lt () = (t−)
Rt : I (L) → L
by Rt () = (t+).
3. A new definition of fuzzy compactness of L-sets
In order to generalize the notion of compactness to L-topological spaces, first let us research compactness in general
topology.
Let (X, ) be a topological space and G ⊆ X. G is said to be compact if each open cover U of G has a finite subfamily
V which is an open cover of G. By the following fact:
G (x) ∨
A(x) = 1,
x∈X
A∈U
⇔ ∀x ∈ X, G (x) ∨
A(x) = 1,
A∈U
⇔ ∀x ∈ X, G (x) = 1
implies
A(x) = 1,
A∈U
⇔ ∀x ∈ X, G(x) = 1
⇔ ∀x ∈ X, G(x) A∈U
implies
A∈U
A(x)
A(x) = 1,
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F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
we know that G is compact if and only if
G (x) ∨
A(x) = 1 implies
x∈X
A∈U
V ∈2(U )
x∈X
G (x) ∨
A(x) = 1.
A∈V
Thus we can naturally generalize the notion of compactness to L-spaces as follows:
Definition 3.1. Let (X, T ) be an L-space. G ∈ LX is called fuzzy compact if for every family U ⊆ T , it follows that
G (x) ∨
G (x) ∨
A(x) A(x) .
x∈X
A∈U
A∈V
V ∈2(U ) x∈X
In fact, “ ” in Definition 3.1 can be replaced by “ = ”.
It is easy to see that if an L-topology T on a set X is finite, then each G ∈ LX is fuzzy compact in (X, T ). Moreover
if X is a singleton set, then for any L-topology on X, G ∈ LX is fuzzy compact. But if X is not a singleton set and L
is not completely distributive, then in general (X, T ) need not be fuzzy compact. This can be seen from the following
example.
Example 3.2. Let L = {(0, 0), (1, 1)} ∪ (0, 1)2 be a subset of the square [0, 1]2 . “ ” is defined as follows:
(m, b)(n, d)
if and only if m n, b d.
It is easy to verify that (L, ) is a complete lattice. ∀x, y ∈ (0, 1), define again (x, y) = (1−x, 1−y), (1, 1) = (0, 0).
Then (L, ) is a complete DeMorgan algebra.
Take X = {x, y} and T = {Aa | a ∈ (0, 1)} ∪ {0, 1}, where Aa (x) = (a, 0) and Aa (y) = (0, a). Then (X, T ) is an
L-space. Obviously for U = T − {0, 1} we have that
A(y) = 10 =
A(x) ∧
A(y)
.
A(x) ∧
A∈U
A∈U
V ∈2(U )
A∈V
A∈V
Therefore (X, T ) is not fuzzy compact.
Remark 3.3. In [7], strong F1 compactness was introduced in L-spaces, which is an extension of Lowen’s strong fuzzy
compactness. It is easy to see that the L-space in Example 3.2 is strong F1 compact. This and [9, Example 10.4.35]
show that there is no implication between strong F1 compactness and fuzzy compactness.
From Definition 3.1 we easily obtain the following theorem by simply using quasi-complement .
Theorem 3.4. Let (X, T ) be an L-space. Then G ∈ LX is fuzzy compact if and only if for every subfamily P ⊆ T , it
follows that
G(x) ∧
G(x) ∧
B(x) B(x) .
x∈X
B∈P
F ∈2(P) x∈X
B∈F
Now we consider other characterizations of fuzzy compactness. First we introduce the following notions.
Definition 3.5. Let (X, T ) be an L-space, a ∈ L\{1} and G ∈ LX . A subfamily U in LX is said to be
(1) an a-shading of G if for any
x ∈ X, it follows
that G (x) ∨ A∈U A(x)a.
(2) a strong a-shading of G if x∈X (G (x) ∨ A∈U A(x))a.
It is obvious that a strong a-shading of G is an a-shading of G. When G = X, the notion of a-shading in Definition
3.5 is the corresponding notion in [6].
F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
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Definition 3.6. Let (X, T ) be an L-space, a ∈ L\{0} and G ∈ LX . A subfamily P in LX is called
(1) an a-remote family of G if for any
x ∈ X, it follows
that G(x) ∧ B∈P B(x)a.
(2) a strong a-remote family of G if x∈X (G(x) ∧ B∈P B(x))a.
It is obvious that a strong a-remote family of G is an a-remote family of G and P is a strong a-remote family of G
if and only if P is a strong a-shading of G.
By Definition 3.1 and Theorem 3.4 we immediately obtain the following result.
Theorem 3.7. Let (X, T ) be an L-space and G ∈ LX . Then the following conditions are equivalent to each other.
(1) G is fuzzy compact.
(2) For any a ∈ L\{1}, each open strong a-shading U of G has a finite subfamily V which is a strong a-shading of G.
(3) For any a ∈ L\{0}, each closed strong a-remote family of G has a finite subfamily F which is a strong a-remote
family of G.
From Theorem 3.7 we can easily obtain the following corollary.
Corollary 3.8. Let (X, T ) be an L-space, P (L) be order generating [4], and G ∈ LX . Then the following conditions
are equivalent to each other.
(1) G is fuzzy compact.
(2) For any a ∈ P (L), each open strong a-shading U of G has a finite subfamily V which is a strong a-shading of G.
(3) For any a ∈ M(L), each closed strong a-remote family of G has a finite subfamily F which is a strong a-remote
family of G.
Theorem 3.9. If G is fuzzy compact and H is closed, then G ∧ H is fuzzy compact.
Proof. Since G is fuzzy compact, for any family P of closed L-sets, by Theorem 3.4 it follows that
(G ∧ H )(x) ∧
B(x)
x∈X
=
B∈P
⎛
⎝G(x) ∧
B(x)⎠
G(x) ∧
F ∈2(P∪{H }) x∈X
⎧
⎨ ⎞
B∈P ∪{H }
x∈X
B(x)
B∈F
⎫ ⎧
⎫
⎬ ⎨ ⎬
=
G(x) ∧
G(x) ∧ H (x) ∧
B(x)
∧
B(x)
⎩
⎭ ⎩
⎭
B∈F
B∈F
F ∈2(P) x∈X
F ∈2(P) x∈X
=
(G ∧ H )(x) ∧
B(x) .
B∈F
F ∈2(P) x∈X
This shows that G ∧ H is fuzzy compact.
Lemma 3.10. Let L be a complete DeMorgan frame, and f : X → Y be a set map. Then for any P ⊆ LX , it follows
that
→
←
fL (G)(y) ∧
G(x) ∧
B(y) =
fL (B)(x) ,
y∈Y
B∈P
x∈X
B∈P
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F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
where fL→ : LX → LY and fL← : LY → LX are defined as follows:
fL→ (A)(y) =
A(x),
fL← (B)(x) = B(f (x)).
f (x)=y
Proof. This can be proved from the following equation.
⎛⎛
⎞
⎞
⎝⎝
B(y) =
G(x)⎠ ∧
B(y)⎠
fL→ (G)(y) ∧
y∈Y
B∈P
y∈Y
=
=
=
⎝
G(x) ∧
y∈Y
x∈f −1 (y)
y∈Y
x∈f −1 (y)
B∈P
x∈f −1 (y)
⎛
B∈P
G(x) ∧
B∈P
G(x) ∧
B∈P
x∈X
⎞
B(f (x)) ⎠
fL← (B)(x)
fL← (B)(x) .
Theorem 3.11. Let L be a complete DeMorgan frame, f : X → Y be a set map, T1 be an L-topology on X, T2 be an
L-topology on Y, and f : X → Y be continuous. If G is fuzzy compact in (X, T1 ), then so is fL→ (G) in (Y, T2 ).
Proof. For any P ⊆ T2 , by Lemma 3.10 and fuzzy compactness of G we have that
G(x) ∧
B(y) =
fL← (B)(x)
fL→ (G)(y) ∧
y∈Y
B∈P
x∈X
F ∈2(P)
x∈X
=
F ∈2(P)
Therefore fL→ (G) is fuzzy compact.
y∈Y
B∈P
G(x) ∧
B∈F
fL← (B)(x)
fL→ (G)(y) ∧
B(y) .
B∈F
The following theorem is a generalization of the Alexander Subbase Theorem.
Theorem 3.12. Let P (L) be order generating [4], R be a subbase for the L-topology T on a set X, and G ∈ LX . If
for any a ∈ P (L), every strong a-shading (contained in R) of G has a finite subfamily V which is a strong a-shading
of G, then G is fuzzy compact.
Proof. Suppose that every strong a-shading (contained in R) of G has a finite subfamily V which is a strong a-shading
of G. Now we prove that every strong a-shading (contained in T ) of G has a finite subfamily which is a strong a-shading
of G. Suppose that U is a strong a-shading of G and it has no finite subfamily which is a strong a-shading of G. Let
= {P | U ⊆ P ⊆ T , P has no finite subfamily which is a strong a-shading of G}.
Then (, ⊆) is a nonempty partially ordered set and each chain has an upper bound, hence by Zorn’s Lemma, has a
maximal element . Now we prove that satisfies the following conditions:
(i) is a strong a-shading of G;
(ii) for every B ∈ T , if C ∈ and C B; then B ∈ ;
(iii) if for any B, C ∈ T , B ∧ C ∈ , then B ∈ or C ∈ .
F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
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We only verify (iii). If B ∈
/ and C ∈
/ , then both {B} ∪ and {C} ∪ are not in . So there exist finite subfamily
1 , 2 ⊆ such that
⎛
⎞
⎝G (x) ∨ B(x) ∨
A(x)⎠ a
A∈
1
x∈X
and
⎛
⎝G (x) ∨ C(x) ∨
⎞
A(x)⎠ a.
A∈
2
x∈X
Let = 1 ∪ 2 . Then
G (x) ∨ B(x) ∨
A(x) a
A∈
x∈X
and
G (x) ∨ C(x) ∨
A(x) a.
A∈
x∈X
When P (L) is order generating, L is a complete DeMorgan frame. Hence we can obtain that
G (x) ∨ (B ∧ C)(x) ∨
A(x) a.
A∈
x∈X
This contradicts B ∧ C ∈ . (iii) is proved.
(iv) From (ii) and (iii) it is immediate that if D ∈ , P1 , P2 , . . . , Pn ∈ T and D P1 ∧ P2 ∧ · · · ∧ Pn , then there
exists i(1 i n) such that Pi ∈ .
Now we consider R ∩ . If R ∩ is a strong a-shading of G, then it has a finite subfamily W which is a strong
a-shading G. Obviously W is also a finite subfamily of , this contradicts the property of . Therefore R ∩ is not a
strong a-shading of G, i.e.,
G (x) ∨
A(x) a.
A∈R∩
x∈X
By (i) we know that is a strong a-shading of G, i.e.,
x∈X
A=
Aij
G (x) ∨
A∈
A(x) a. For each A ∈ , let
where for each i ∈ I , Ji is a finite set and Aij ∈ R,
i∈I j ∈Ji
then A j ∈J
Aij for each i ∈ I , by (iv) we know that there is a j ∈ Ji such that Aij ∈ , thus we obtain that
i
A(x)
A∈
A∈R∩ A(x). Therefore it follows that
G (x) ∨
A(x) a,
x∈X
this contradicts
A∈R∩
x∈X (G
(x) ∨
A∈R∩ A(x))a.
The proof is completed.
Remark 3.13. If P (L) is not order generating, then Theorem 3.12 need not be true. For example, in Example 3.2,
L = {(0, 0), (1, 1)} ∪ (0, 1)2 has only a prime element (1, 1). This shows P (L) = ∅. But (X, T ) is not fuzzy compact.
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F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
Theorem 3.14. Let L be completely distributive, and (X,
T ) be the product of a family {(Xi , Ti )}i∈I of L-spaces. If
for each i ∈ I , Gi is fuzzy compact in (Xi , Ti ), then G = i∈I Gi is fuzzy compact in (X, T ).
The proof can be found in [9,20].
In general, if L is not completely distributive, then the Tychonoff Theorem need not be true. This can be seen from
the following example.
Example 3.15. Let L = {(0, 0), (1, 1)} ∪ {(a, 0), (0, b) | a, b ∈ (0, 1)}. “ ” is defined as follows:
(m, b)(n, d) if and only if m n, b d.
∀x, y ∈ (0, 1), define again (x, 0) = (1 − x, 0), (0, y) = (0, 1 − y), (1, 1) = (0, 0). Then (L, ) is a complete
DeMorgan algebra. Let X = {x, y} and
T1 = {Aa | a ∈ (0, 0.5]} ∪ {0, 1},
T2 = {Ba | a ∈ (0, 0.5]} ∪ {0, 1},
where Aa (x) = Aa (y) = (a, 0), Ba (x) = Ba (y) = (0, a). Then both (X, T1 ) and (X, T2 ) are L-spaces. We can easily
verify that both (X, T1 ) and (X, T2 ) are fuzzy compact.
Now we verify that the product L-space (X × X, T1 × T2 ) is not fuzzy compact. Obviously {P1← (Aa ), P2← (Ba ) |
a ∈ (0, 0.5]} is a subbase of T1 × T2 in the product L-space (X × X, T1 × T2 ). It is easy to see that
P1← (Aa )(x, x) = P1← (Aa )(x, y) = (a, 0),
P1← (Aa )(y, x) = P1← (Aa )(y, y) = (a, 0),
P2← (Ba )(x, x) = P1← (Ba )(y, x) = (0, a),
P2← (Ba )(x, y) = P1← (Ba )(y, y) = (0, a).
Let U = {P1← (Aa ), P2← (Ba ) | a ∈ (0, 0.5]}. We have that
⎛
⎞ ⎛
⎞
D(x, x) = ⎝
P1← (Aa )(x, x)⎠ ∨ ⎝
P2← (Ba )(x, x)⎠
D∈U
a∈(0,0.5]
a∈(0,0.5]
= (0.5, 0) ∨ (0, 0.5) = (1, 1).
Analogously we have that
D(x, y) =
D(y, y) =
D(y, x) = (1, 1).
D∈U
D∈U
D∈U
So we obtain that
D(x, x) ∧
D(x, y) ∧
D(y, x) ∧
D(y, y) = (1, 1).
D∈U
D∈U
D∈U
D∈U
But for each finite subfamily V of U, we easily verify that
D(x, x) ∧
D(x, y) ∧
D(y, x) ∧
D(y, y)
= (1, 1).
V ∈2(U )
D∈V
D∈V
D∈V
D∈V
This shows that (X, T ) is not fuzzy compact.
4. Other characterizations of fuzzy compactness
In this section, we assume that L is a completely distributive DeMorgan algebra.
Now we present some characterizations of fuzzy compactness by means of open L-sets.
F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
1493
Theorem 4.1. Let (X, T ) be an L-space and G ∈ LX . Then the following conditions are equivalent.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
G is fuzzy compact.
For any a ∈ L\{1}, each open strong a-shading of G has a finite subfamily which is a strong a-shading of G.
For any a ∈ L\{1}, each open strong a-shading of G has a finite subfamily which is an a-shading of G.
For any a ∈ L\{1} and any open strong a-shading U of G, there exist a finite subfamily V of U and b ∈ (a) such
that V is a strong b-shading of G.
For any a ∈ L\{1} and any open strong a-shading U of G, there exist a finite subfamily V of U and b ∈ (a) such
that V is a b-shading of G.
For any a ∈ L\{0}, each closed strong a-remote family of G has a finite subfamily which is a strong a-remote
family of G.
For any a ∈ L\{0}, each closed strong a-remote family of G has a finite subfamily which is an a-remote
family of G.
For any a ∈ L\{0} and any closed strong a-remote family P of G, there exist a finite subfamily F of P and b ∈ (a)
such that F is a strong b-remote family of G.
For any a ∈ L\{0} and any closed strong a-remote family P of G, there exist a finite subfamily F of P and b ∈ (a)
such that F is a b-remote family of G.
Proof. By Theorem 3.7 we can obtain (1)⇔ (2). (2) ⇒ (3)
that a ∈ L\{1}
is obvious. To prove (3) ⇒ (4), suppose
(x) ∨
B(x))a,
take
c
∈
(a)
such
that
(G
and
U
is
a
strong
a-shading
of
G,
then
x∈X (G (x) ∨
x∈X
B∈U
B∈U B(x))c, obviously U is a strong c-shading of G, by (3) we know that U has a finite subfamily V which is a
c-shading of G, take b ∈ (a) such that c ∈ (b), then V is a strong b-shading of G, (4) is shown. (4) ⇒ (5) ⇒ (2) is
obvious. Dually we can prove that (1) ⇒ (6) ⇒ (7) ⇒ (8) ⇒ (9) ⇒ (1). Remark 4.2. In Theorem 4.1, a ∈ L\{1} and b ∈ (a) can be replaced by a ∈ P (L) and b ∈ ∗ (a), respectively.
a ∈ L\{0} and b ∈ (a) can be replaced by a ∈ M(L) and b ∈ ∗ (a), respectively.
Remark 4.3. For a completely distributive DeMorgan algebra and a ∈ M(L), it is easy to prove that a closed strong aremote family is exactly equivalent to an a − -R-neighborhood family in [18,20,21]. Thus the notion of fuzzy compactness
in Definition 3.1 is equivalent to Lowen’s fuzzy compactness notion in [14,18,20].
Definition 4.4. Let (X, T ) be an L-space, a ∈ L\{0} and G ∈ LX . A subfamily U in LX is called
(1) a a -cover of G if for any x ∈ X,it follows that a ∈ (G (x) ∨ A∈U A(x)).
(2) a strong a -cover of G if a ∈ ( x∈X (G (x) ∨ A∈U A(x))).
It is obvious that a strong a -cover of G is a a -cover of G.
The following lemma is obvious.
Lemma 4.5. Let a ∈ L\{0} and U ⊆ LX . Then the following conditions are equivalent.
(1) U is a strong a -cover of G.
(2) There exists b ∈ L with a ∈ (b) such that U is a b -cover of G.
(3) There exists b ∈ L with a ∈ (b) such that U is a strong b -cover of G.
By Definition 3.1 and Lemma 4.5 we can obtain the following theorem.
Theorem 4.6. Let (X, T ) be an L-space and G ∈ LX . Then the following conditions are equivalent.
(1) G is fuzzy compact.
(2) For any a ∈ L\{0}, each open strong a -cover of G has a finite subfamily which is a strong a -cover of G.
(3) For any a ∈ L\{0}, each open strong a -cover of G has a finite subfamily which is a a -cover of G.
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F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
(4) For any a ∈ L\{0} and any open strong a -cover U of G, there exist a finite subfamily V of U and b ∈ L with
a ∈ (b) such that V is a strong b -cover of G.
(5) For any a ∈ L\{0} and any open strong a -cover U of G, there exist a finite subfamily V of U and b ∈ L with
a ∈ (b) such that V is a b -cover of G.
Remark 4.7. In Theorem 4.6, a ∈ L\{0}, b ∈ L and b ∈ (a) can be replaced by a ∈ M(L), b ∈ M(L) and a ∈ ∗ (b),
respectively.
Definition 4.8. Let (X, T ) be an L-space, a ∈ L\{0} and G ∈ LX . A subfamily U in LX is called a Qa -cover of G if
for any x ∈ X with G(x)a , it follows that A∈U A(x) a.
It is obvious that a a -cover of G is a Qa -cover of G. If a ∈ M(L), then the notion of open Qa -cover in Definition
4.8 is the notion of Qa -open cover in [16].
Moreover from Definition 3.1 we also can obtain the following result.
Theorem 4.9. Let (X, T ) be an L-space and G ∈ LX . Then the following conditions are equivalent.
(1) G is fuzzy compact.
(2) For any a ∈ L\{0} and any b ∈ (a)\{0}, each open Qa -cover of G has a finite subfamily which is an open
Qb -cover of G.
(3) For any a ∈ L\{0} and any b ∈ (a)\{0}, each open Qa -cover of G has a finite subfamily which is an open
b -cover of G.
(4) For any a ∈ L\{0} and any b ∈ (a)\{0}, each open Qa -cover of G has a finite subfamily which is an open strong
b -cover of G.
Remark 4.10. In Theorem 4.9, a ∈ L\{0} and b ∈ (a)\{0} can be replaced by a ∈ M(L) and b ∈ ∗ (a), respectively.
In this case, condition (2) in Theorem 4.9 is exactly an equivalent condition of fuzzy compactness in [16].
The following two theorems give examples of fuzzy compact L-sets.
Theorem 4.11. The L-closed interval [a, b](L) is fuzzy compact.
Proof. By Theorem 4.1(7) and Remark 4.2 we only need to prove that every closed strong -remote family P of
[a, b](L) consisting of members of subbase {Lt , Rt | t ∈ [a, b]} has a finite subfamily which is -remote family of
[a, b](L). Since P is a strong -remote family of [a, b](L), there exists a ∈ ∗ () such that P is a -remote family
of [a, b](L).
(1) For satisfying , take x = [] ∈ [a, b](L) such that
⎧
⎨ 1, t < a,
(t) = , a t b,
⎩
0, t > b.
This implies that ∀x ∈ [a, b](L), there exists P ∈ P such that x P . If P = Lt , then by P (x) = Lt (x) = (t−)
we know that (t−) = 0, hence t > b, this implies P = 0. If P = Rs , then by P (x) = Rs (x)) = (s+) we know
that (s+) = 1, hence s < a, this implies P = 0. However, we always have P = 0. Therefore {0} is a subfamily of P
and it is a -remote family of [a, b](L).
(2) For satisfying and ∀r ∈ [a, b], take x r = [r ] ∈ [a, b](L) such that
1, t r,
r
(t) =
0, t > r.
Since P is a -remote family of [a, b](L), there exists a P r ∈ P such that P r (x r ). If P r = Lt , then by P r (x r ) =
Lt (x r ) = r (t−) we know that (t−) = 0, hence t > r, this implies r ∈ (−∞, t). If P r = Rs , then analogously we
F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1486 – 1495
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can prove r ∈ (s, +∞). This shows that
= {(−∞, t), (s, +∞) | Lt , Rs ∈ P}
is an open cover of [a, b] in the sense of usual topology. By compactness of [a, b] we know that has a subcover
{(−∞, t), (s, +∞)}. Obviously s < t. Now we prove that F = {Lt , Rs } is a -remote family of [a, b](L).
If F is not a -remote family of [a, b](L), then there exists [] ∈ [a, b](L) such that
Lt ([]) ∧ Rs ([]) = (t−) ∧ (s+) .
This implies that (t−) ∨ (s+) . Further we have that
(t−) ∧ (s+) (s+) ∧ (s+) (s+) ∨ (t−) ,
this yields a contradiction. Therefore [a, b](L) is fuzzy compact.
Theorem 4.12. In L-closed interval [a, b](L), Lr ∧ Rs is fuzzy compact for all r, s ∈ [a, b].
Proof can be obtained from Theorems 3.9 and 4.11.
In [7], strong F2 compactness was introduced in L-spaces, which is an extension of Lowen’s strong fuzzy compactness.
By Theorem 4.2(5) in [7] and Remark 4.2 we can obtain the following result.
Theorem 4.13. Strong F2 compactness implies fuzzy compactness.
In general, fuzzy compactness does not implies strong F2 compactness. This can be seen in [9,18].
Acknowledgments
The author would like to thank S.E. Rodabaugh and referees for their valuable comments and suggestions.
References
[1]
[2]
[3]
[4]
[6]
J.J. Chadwick, A generalized form of compactness in fuzzy topological spaces, J. Math. Anal. Appl. 162 (1991) 92–110.
C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182–190.
T.E. Gantner, R.C. Steinlage, R.H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978) 547–562.
G. Gierz, et al., A Compendium of Continuous Lattices, Springer, Berlin, 1980.
T. Kubiák, The topological modification of the L-fuzzy unit interval, in: S.E. Rodabaugh, E.P. Klement, U. Höhle (Eds.), Applications of
Category Theory to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1992, pp. 275–305, (Chapter 11).
[7] S.-G. Li, L.-L. Zhang, Z.-B. Xu, Strong fuzzy compact sets and ultra-fuzzy compact sets in L-topological spaces, Fuzzy Sets and Systems 147
(2004) 293–306.
[8] Y.M. Liu, Compactness and Tychnoff theorem in fuzzy topological spaces, Acta Math. Sin. 24 (1981) 260–268.
[9] Y.M. Liu, M.K. Luo, Fuzzy Topology, World Scientific, Singapore, 1997.
[10] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621–633.
[11] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446–454.
[12] S.E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40 (1991) 297–345.
[13] S.E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics
of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Dordrecht,
1999, pp. 91–116.
[14] F.-G. Shi, A note on fuzzy compactness in L-topological spaces, Fuzzy Sets and Systems 119 (2001) 547–548.
[15] F.-G. Shi, A new notion of fuzzy compactness in L-topological spaces, Inform. Sci. 173 (2005) 35–48.
[16] F.-G. Shi, C.-Y. Zheng, O-convergence of fuzzy nets and its applications, Fuzzy Sets and Systems 140 (2003) 499–507.
[17] G.J. Wang, A new fuzzy compactness defined by fuzzy nets, J. Math. Anal. Appl. 94 (1983) 1–23.
[18] G.J. Wang, Theory of L-fuzzy Topological Space, Shanxi Normal University Publisher, Xi’an, 1988 (in Chinese).
[19] G.J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems 47 (1992) 351–376.
[20] J.J. Xu, On fuzzy compactness in L-fuzzy topological spaces, Chinese Quart. J. Math. 2 (1990) 104–105.
[21] D.S. Zhao, The N-compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128 (1987) 64–70.