# TheL

Authors: Shui-Li Chen; Jin-Lan Huang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The concepts ofα
ω-remote neighborhood family, γ
ω-cover, and Lω-compactness are defined in Lω-spaces. The characterizations of Lω-compactness are systematically discussed. Some important
properties of Lω-compactness such as ω-closed heredity, arbitrarily multiplicative property, and
preserving invariance under ω-continuous mappings are obtained. Finally, the Alexander
ω-subbase lemma and the Tychonoff product theorem with
respect to Lω-compactness are given.


---

## Body

## 1. Introduction

Compactness is one of the most important notions in general topology, fuzzy topology, andL-topology. Many research workers have presented various kinds
of compactness [1–19] by means of introducing various operators, such as closure operator,
θ-closure operator, δ-closure operator, R-closure operator, S-closure operator, S
R-closure operator, and P
S-closure operator; because the above operators are all order
preserving. That is, they satisfy the following conditions: (i) if A
,
B
∈
L
X and A
⩽
B, then ω
(
A
)
⩽
ω
(
B
); (ii) for any A
∈
L
X, A
⩽
ω
(
A
), where ω
:
L
X
→
L
X can take and of the above operators,
L
X is the family of all L-sets defined on X and with value in L, L is a fuzzy lattice, and 1
X is the greatest L-set of L
X. We introduced a kind of generalized fuzzy space called
L
ω-space in [20] in order
to unify various elementary concepts in L-topological spaces. In the present paper, we will propose and
study a generalized compactness which will be called L
ω-compactness in L
ω-spaces. The L
ω-compactness is a unified form of N-compactness [16, 19], near N-compactness [5], almost
N-compactness [6],
S-compactness [13],
S
R-compactness [1],
P
S-compactness [2],
δ-compactness [9],
θ-compactness [18], and
so forth.


## 2. Preliminaries

Throughout this paper,L denotes a fuzzy lattice, that is, a completely distributive
lattice with order-reserving involution ′, 0 and 1 denote the least and greatest elements of
L, respectively, and M denotes the set that consisting of all nonzero
∨-irreducible elements of L. Let X be a nonempty crisp set, L
X the set of all L-fuzzy sets (briefly, L-sets) on X, and M
*
(
L
X
)
=
{
x
α
:
α
∈
M
,
x
∈
X
} the set of all nonzero ∨-irreducible elements (i.e., so-called molecules [17] or points for short) of L
X. The least and the greatest elements of
L
X will be denoted by 0
X and 1
X, respectively. For any α
∈
M, β
(
α
) is called the greatest minimal set of
α [12], and
β
*
(
α
)
=
β
(
α
)
∩
M is said to be the standard minimal set of
α [17].Definition 1 (Chen and Cheng [20]).
LetX be a nonempty crisp set.(i)
An operatorω: L
X→L
X is said to be an ω-operator if (
1
) for all A, B
∈L
X and A
⩽
B, ω
(
A
)
⩽
ω
(
B
); (
2
) for all A
∈L
X, A
⩽
ω
(
A
).
(ii)
AnL-set A
∈L
X is called an ω-set if ω
(
A
)
=
A.
(iii)
PutΩ
=
{
A
∈
L
X
∣
ω
(
A
)
=
A
}, and call the pair (L
X
,
ω) an L
ω-space.Definition 2 (Chen and Cheng [20]).
Let (L
X, Ω) be an L
ω-space, A
∈
L
X, and x
α
∈
M
*
(
L
X
). If there exists a Q
∈
Ω such that x
α
≰
Q and P
⩽
Q, then call P an ω-remote neighborhood (briefly, ω
R-neighborhood) of x
α. The collection of all ω
R-neighborhoods of x
α is denoted by ω
η
(
x
α
). If A
≰
P for each P
∈
ω
η
(
x
α
), then x
α is said to be an ω-adherence point of A and the union of all ω-adherence points of A is called the ω-closure of A and denoted by ω
cl
(
A
). If A
=
ω
cl
(
A
), then call A an ω-closed set and call A
′ an ω-open set. If P is an ω-closed set and x
α
≰
P, then P is said to be an ω-closed remote neighborhood (briefly,
ω
CR-neighborhood) of x
α and the collection of all ω
CR-neighborhoods of x
α is denoted by ω
η
-
(
x
α
). Note that ω
C
(
L
X
) and ω
O
(
L
X
) are the family of all ω-closed sets and all ω-open sets in L
X, respectively.Definition 3 (Chen and Cheng [20]).
Let (L
X
,
Ω) be an L
ω-space, A
∈
L
X, and ω
int
(
A
)
=
∨
{
B
∈
L
X
∣
B
⩽
A and B is an ω-open set in L
X
}. We call ω
int
(
A
) the ω-interior of A. Obviously, A is ω-open if and only if A
=
ω
int
(
A
).Definition 4 (Huang and Chen [11]).
Let (L
X
,
Ω) be an L
ω-space, let N be a molecular net in L
X, and let x
α
∈
M
*
(
L
X
). If N is eventually not in P for each P
∈
ω
η
-
(
x
α
), then x
α is said to be an ω-limit point of N (or N
ω-converges to x
α). If N is frequently not in P for each P
∈
ω
η
-
(
x
α
), then x
α is said to be an ω-cluster point of N (or N
ω-accumulates to x
α). The union of all ω-limit points (ω-cluster points) of N is written by ω-lim N (ω-a
d
N).Definition 5 (Huang and Chen [11]).
Let (L
X
,
Ω) be an L
ω-space, let I be an ideal in L
X, and let x
α
∈
M
*
(
L
X
). If ω
η
-
(
x
α
)
⊆
I, then x
α is called an ω-limit point of I (or I
ω-converges to x
α). If P
∨
B
≠
1
X for each P
∈
ω
η
-
(
x
α
) and each B
∈
I, then x
α is called an ω-cluster point of I (or I
ω-accumulates to x
α). The union of all ω-limit points (ω-cluster points) of I is denoted by ω-lim I (ω-a
d
I).Definition 6 (Chen and Cheng [20]).
Let (L
X
,
Ω) be an L
ω-space, x
α
∈
M
*
(
L
X
), and β, γ
∈
ω
O
(
L
X
). Then,(i)
β is said to be an ω-base in (L
X
,
Ω) if for each G
∈
ω
O
(
L
X
), there exists a subfamily φ of β such that G
=
∨
B
∈
φ
B;
(ii)
γ is said to be an ω-subbase in (L
X
,
Ω) if the collection consisting of all intersections of
any finite elements in γ is an ω-base in (L
X
,
Ω).Definition 7 (Chen and Cheng [20]).
Assume (L
X, Ω
i) to be an L
ω
i-space (
i
=
1,2
) and f
:
(
L
X
,
Ω
1
)
→
(
L
Y
,
Ω
2
) an L-valued Zadeh’s type function [17]. If f
←
(
B
)
∈
ω
1
O
(
L
X
) for each B
∈
ω
2
O
(
L
Y
), then call f(
ω
1
,
ω
2
)-continuous.


## 3.L
ω-Compact Set and Its Characteristics

In this section, we will introduce the concepts ofα
ω-remote neighborhood family and γ
ω-cover in an L
ω-space first, propose the notion of L
ω-compactness by making use of α
ω-remote neighborhood family next, and then discuss the
characteristics of L
ω-compactness.Definition 8.
Suppose (L
X, Ω) be an L
ω-space, A
∈
L
X, α
∈
M, and Φ
⊆
ω
C
(
L
X
). If there exists a P
∈
Φ such that P
∈
ω
η
-
(
x
α
) for each molecule x
α in A, then Φ is called an α
ω-remote neighborhood family (briefly,
α
ω-RF) of A, in symbol ∧
Φ
<
A
(
α
ω
). If there exists a nonzero ∨-irreducible element λ
∈
β
*
(
α
) with ∧
Φ
<
A
(
λ
ω
), then Φ is said to be an (
α
ω
)
--RF, in symbol ∧
Φ
≪
A
(
α
ω
).Definition 9.
Assume (L
X
,
Ω) be an L
ω-space, A
∈
L
X, γ
′
∈
M, and Γ
⊆
ω
O
(
L
X
). If there is a B
∈
Γ such that B
(
x
)
⩽
γ for each x
∈
τ
γ
′
(
A
) = {
x
∈
X
∣
A
(
x
)
⩾
γ
′
}, then Γ is known as a γ
ω-cover. If there exists a prime element
t
∈
α
*
(
γ
) such that Γ is a t
ω-cover of A, then Γ is said to be a (
γ
ω
)
+-cover of A, where α
*
(
γ
) is the standard maximal set of γ [17].Definition 10.
Assume (L
X
,
Ω) be an L
ω-space and A
∈
L
X. If every α
ω-RF Φ of A has a finite subfamily Ψ such that Ψ is an (
α
ω
)
--RF, where α
∈
M, then call A an αL
ω-compact set. If A is an αL
ω-compact set for any α
∈
M, then call A an L
ω-compact set. Specially, when 1
X is αL
ω-compact, we call (L
X
,
Ω) an αL
ω-compact space, and if (L
X
,
Ω) is αL
ω-compact for each α
∈
M, we say that (L
X
,
Ω) is an L
ω-compact space.Obviously, whenω is the L-closure operator on L
X, the L
ω-compactness is just the N-compactness in [19],
and while ω takes the θ-closure operator (resp., δ-closure operator, R-closure operator, S-closure operator, P
S-closure operator, and S
R-closure operator) on L
X, the L
ω-compactness is just the θ-compactness (resp., δ-compactness, near N-compactness, S-compactness, P
S-compactness, and S
R-compactness). Therefore, the L
ω-compactness is of the universal significance.Example 11.
Let (L
X
,
Ω) be an L
ω-space and A
∈
L
X. If the support σ
0
(
A
)
=
{
x
∈
X
∣
A
(
x
)
>
0
} of A is a finite set, then A is an L
ω-compact set.Proof.
Assume thatσ
0
(
A
)
=
{
x
1
,
x
2
,
…
,
x
n
} and Φ is an α
ω-RF of A. For each i
∈
{
1
,
2
,
…
,
n
} we choose an ω-closed set P
i
∈
Φ with α
⩽
P
i
(
x
i
). Being α
=
sup
β
*
(
α
), there is a λ
i
∈
β
*
(
α
) such that λ
⩽
P
i
(
x
i
). Since β
*
(
α
) is an upper directed set, there is a
λ
∈
β
*
(
α
) with λ
⩾
λ
i for each i
∈
{
1
,
2
,
…
,
n
}, and thus λ
i
⩽
P
i
(
x
i
). Therefore Φ has a finite subfamily Ψ
=
{
P
1
,
P
2
,
…
,
P
n
} which is an (
α
ω
)
--RF of A. By Definition 10, A is an L
ω-compact set.Now we give some characteristics ofL
ω-compactness as follows.Theorem 12.
Let(
L
X
,
Ω
) be an L
ω-space and A
∈
L
X. Then A is an L
ω-compact set if and only if the following conditions hold:(1)
for eachα
∈
M, every α
ω-RF Φ of A has a finite subfamily Ψ with ∧
Ψ
<
A
(
α
ω
);
(2)
for eachα
∈
M, if Φ
=
{
P
} is an α
ω-RF of A, then Φ is also an (
α
ω
)
--RF of A.Proof.
Necessity. Assume that A is L
ω-compact and Φ is an α
ω-RF of A(
α
∈
M
). According to Definition 10, Φ has a finite subfamily Ψ with ∧
Ψ
≪
A
(
α
ω
) and so it certainly holds that ∧
Ψ
<
A
(
α
ω
). Thus (
1
) is satisfied. If Φ = {
P
} is an α
ω-RF of A, then Φ has a finite Ψ with ∧
Ψ
≪
A
(
α
ω
) by the L
ω-compactness of A. Obviously, Ψ
=
Φ, and hence Φ is an (α
ω
)
--RF of A. Therefore (
2
) holds.
Sufficiency. Suppose that conditions (
1
) and (
2
) are satisfied, and Φ is an α
ω-RF of A (α
∈
M). By (
1
), there is a finite subfamily Ψ of Φ such that Ψ is an α
ω-RF of A. Let P
=
∧
Ψ. Then {
P
} is an α
ω-RF of A. According to (
2
), {
P
} is also an α
ω-RF of A; that is, there exists a λ
∈
β
*
(
α
) with λ
⩽
P
(
x
)
=
∧
{
Q
(
x
)
∣
Q
∈
Ψ
} for each molecule x
λ
⩽
A. Since Ψ is finite, we can choose an ω-closed set Q
∈
Ψ with λ
⩽
Q
(
x
); that is, Q
∈
ω
η
-
(
x
λ
). This shows that Ψ is an (
α
ω
)
--RF of A. Therefore A is L
ω-compact.Theorem 13.
Let (L
X
,
Ω) be an L
ω-space and A
∈
L
X. Then A is an L
ω-compact set if and only if for each
γ
′
∈
M, every γ
ω-cover Γ of A has a finite subfamily Ξ such that Ξ is a (
γ
ω
)
+-cover of A.Proof.
Necessity. Suppose that A is an L
ω-compact set and Γ is any γ
ω-cover of A (γ
′
∈
M). Put Φ
=
Γ
′. Then Φ
⊆
ω
C
(
L
X
), and there is an ω-closed set B
′
∈
Φ with B
(
x
)
⩽
γ for each x
∈
τ
γ
′
(
A
); that is, γ
′
⩽
B
′
(
x
); equivalently, B
′
∈
ω
η
-
(
x
γ
′
). This implies that Φ is a γ
′
ω-RF of A. Thus Φ has a finite subfamily Ψ which is a (
γ
′
ω
)
--RF of A; that is, there exists t
′
∈
β
*
(
γ
′
) such that for each x
∈
τ
γ
′
(
A
) we can take an ω-open set B
∈
Ψ
′ with t
′
⩽
B
′
(
x
). In other words, there are t
∈
α
*
(
γ
) and B
∈
Ψ
′
=
Ξ with B
(
x
)
⩽
t for each x
∈
τ
γ
′
(
A
). This means that Ξ is a finite subfamily of Γ and a (
γ
ω
)
+-cover of A.
Sufficiency. Assume that every γ
ω-cover of A has a finite subfamily which is a (
γ
ω
)
+-cover of A (γ
′
∈
M). If Φ is an α
ω-RF of A (α
∈
M), then Γ
=
Φ
′ is a γ
ω-cover of A where γ
=
α
′. Hence Γ has a finite subfamily Ξ which is a (
γ
ω
)
+-cover of A by the hypothesis. Write Ψ
=
Ξ
′. One can easily see that Ψ is a finite subfamily of Φ and is an (
α
ω
)
--RF of A. Therefore A is L
ω-compact.Theorem 14.
Let (L
X
,
Ω) be an L
ω-space and A
∈
L
X. Then A is L
ω-compact if and only if for each α
∈
M and each Φ
⊆
ω
C
(
L
X
) having α-finite intersection property for A (i.e., for each finite subfamily Ψ of Φ and each λ
∈
β
*
(
α
) there exists a molecule x
λ
⩽
A with x
λ
⩽
∧
Ψ), there exists a molecule x
α
⩽
A with x
α
⩽
∧
Φ.Proof.
Necessity. Grant that A is an L
ω-compact set, Φ
⊆
ω
C
(
L
X
), and Φ has α-finite intersection property for A (α
∈
M). If x
α
⩽
∧
Φ for each x
α
⩽
A, then Φ is an α
ω-RF of A by the hypothesis of Φ. Hence Φ has a finite subfamily Ψ which is an (
α
ω
)
--RF of A; that is, there is a λ
∈
β
*
(
α
) satisfying x
λ
⩽
∧
Ψ for each x
λ
⩽
A; in other words, ∨
x
∈
X
(
A
∧
(
∧
Ψ
)
)
(
x
)

λ. It contradicts the fact that Φ has α-finite intersection property for A. Hence the necessity is proved.
Sufficiency. Assume that the condition holds and that
Φ is an α
ω-RF of A. If for any finite subfamily Ψ of Φ, Ψ is not an (
α
ω
)
--RF of A, then for each λ
∈
β
*
(
α
) there exists a molecule x
λ
⩽
A with x
λ
⩽
∧
Ψ; that is, ∨
x
∈
X
(
A
∧
(
∧
Ψ
)
)
(
x
)
⩾
λ. This shows that Φ has α-finite intersection property for A. By the assumption we have x
α
⩽
A satisfying x
α
⩽
∧
Ψ. It contradicts that Φ is an α
ω-RF of A. Therefore Φ has a finite subfamily Ψ which is an (
α
ω
)
--RF of A, and hence A is L
ω-compact.Theorem 15.
Let (L
X
,
Ω) be an L
ω-space and A
∈
L
X. Then A is L
ω-compact if and only if for each α
∈
M, every α-net in A has an ω-cluster point in A with height α.Proof.
Necessity. Suppose that A is an L
ω-compact set and that N
=
{
N
(
n
)
∣
n
∈
D
} is an α-net [16] in
A. If N does not have any ω-cluster point in A with height α, then there exists a P
[
x
]
∈
ω
η
-
(
x
α
) such that N is eventually in P
[
X
] for each x
α
⩽
A; that is, there is a n
(
x
)
∈
D with N
(
n
)
⩽
P
[
x
] whenever n
⩾
n
(
x
). Write Φ
=
{
P
[
x
]
∣
x
α
⩽
A
}. Obviously, Φ is α
ω-RF of A. By the L
ω-compactness of A, Φ has a finite subfamily Ψ
=
{
P
[
x
i
]
∣
i
=
1,2
,
…
,
m
} which is an (
α
ω
)
--RF of A; that is, there is an i
∈
{
1,2
,
…
,
m
} with y
r
⩽
P
[
X
i
] for some r
∈
β
*
(
α
) and each y
r
⩽
A. Take P
=
∧
i
=
1
m
P
[
x
i
]. Then y
r
⩽
P for each y
r
⩽
A. Since D is a directed set, there is an n
0
∈
D, such that n
0
⩾
n
(
x
i
) and N
(
n
)
⩽
P
[
x
i
]
(
i
=
1,2
,
…
,
m
) whenever n
⩾
n
o, and so N
(
n
)
⩽
P. This shows that for each y
r
⩽
A, ∨
(
N
(
n
)
)
⩾
r as long as n
⩾
n
o. It contradicts the fact that N is an α-net. Therefore N has at least an ω-cluster point in A with height α.
Sufficiency. Assume that every α-net in A has at least an ω-cluster point with hight α for each α
∈
M, Φ is an α
ω-RF of A, and 2
(
Φ
) is the set of all finite subfamilies of
Φ. If for each r
∈
β
*
(
α
) and each Ψ
∈
2
(
Φ
), Ψ is not an r
ω-RF of A; that is, x
r
⩽
∧
Ψ for each x
r
⩽
A, and hence there exists a molecule
N
(
r
,
Ψ
)
⩽
A satisfying N
(
r
,
Ψ
)
⩽
∧
Ψ. In β
*
(
α
)
×
2
(
Φ
), we define the relation as follows:
(
r
1
,
Ψ
1
)
⩾
(
r
2
,
Ψ
2
) if and only if r
1
⩾
r
2 and Ψ
1
⊇
Ψ
2, then β
*
(
α
)
×
2
(
Φ
) is a directed set with the relation
“⩾”. Let N
=
{
N
(
r
,
Ψ
)
∣
(
r
,
Ψ
)
∈
β
*
(
α
)
×
2
(
Φ
)
}. One can easily see that N is an α-net in A. We assert that N does not have any ω-cluster point in A with hight α. In fact, for each x
α
⩽
A, we can choose an ω-closed set P
∈
Φ with P
∈
ω
η
-
(
x
α
) by the definition of Φ. Taking r
1
∈
β
*
(
α
) and Ψ
∈
2
(
Φ
), we have P
∈
Ψ according to (
r
,
Ψ
)
⩾
(
r
1
,
{
P
}
), and hence N
(
r
,
Ψ
)
⩽
∧
Ψ
⊆
P. This implies that N is eventually in P, and thus x
α is not an ω-cluster point of N. It is in contradiction with the hypothesis of sufficiency.
Consequently, A is L
ω-compact.Definition 16.
Let (L
X, Ω) be an L
ω-space, let ℱ be an α-filter in L
X; that is, ∨
x
∈
X
(
F
∧
A
)
(
x
)
⩾
α for each F
∈
ℱ and x
α∈
M
*
(
L
X
). If F
⩽
P and for each P
∈
ω
η
-
(
x
α
) and each F
∈
ℱ, then x
α is called an ω-cluster point of ℱ.Theorem 17.
Let (L
X
,
Ω) be an L
ω-space and A
∈
L
X. Then A is L
ω-compact if and only if for each α
∈
M, every α-filter containing A as an element has an ω-cluster point in A with hight α.Proof.
Necessity. Grant that A is an L
ω-compact set and that ℱ is an α-filter containing A as an element. Then F
∧
A
∈
ℱ for each F
∈
ℱ and ∨
x
∈
X
(
F
∧
A
)
(
x
)
⩾
α, and thus there exists a molecule N
(
F
,
r
)
⩽
A with hight r for each r
∈
β
*
(
α
). Define N
=
{
N
(
F
,
r
)
⩽
F
∧
A
∣
(
F
,
r
)
∈
ℱ
×
β
*
(
α
)
} and define a relation in ℱ
×
β
*
(
α
) as follows: (1)
(
F
1
,
r
1
)
⩾
(
F
2
,
r
2
)
iff
F
1
⩽
F
2
,
r
1
⩾
r
2
. Evidently, ℱ
×
β
*
(
α
) is a directed set with the relation
“⩾”, and then N is an α-net in A. By the L
ω-compactness of A and Theorem 15, N has an ω-cluster point in A with hight α, say x
α. We assert that x
α is also an ω-cluster point of ℱ. In reality, N is frequently not in P for each P
∈
ω
η
-
(
x
α
); that is, for each F
∈
ℱ there exist F
1
∈
ℱ with F
1
⩽
F and some r
∈
β
*
(
α
) satisfying N
(
F
1
,
r
)
⩽
P. Hence we have F
⩽
P by virtue of the fact that N
(
F
1
,
r
)
⩽
F
1
⩽
F. This means that x
α is an ω-cluster point of ℱ. Therefore the necessity is proved.
Sufficiency. Suppose that every α-filter containing A as an element has an ω-cluster point in A with hight α for each α∈
M and that Φ is an α
ω-RF of A. If for each Ψ
∈
2
(
Φ
), Ψ is not an (
α
ω
)
--RF of A, then there exists a molecule x
r
⩽
A and x
r
⩽
∧
Ψ for each r
∈
β
*
(
α
). Put ℱ
=
{
F
∈
L
X
∣
∃
Ψ
∈
2
(
Φ
) with (
∧
Ψ
)
∧
A
⩽
F
}. One can easily verify that ℱ is an α-filter containing A as an element, and hence ℱ has an ω-cluster point in A with hight α by the supposition, say x
α. In accordance with Definition 16, we have F
⩽
P for each P
∈
ω
η
-
(
x
α
) and each F
∈
ℱ, specially, ∧
Ψ
⩽
P. Since Φ is an α
ω-RF of A, there exists an ω-closed set Q
∈
Φ with Q
∈
ω
η
-
(
x
α
) for each x
α
⩽
A. Obviously, {
Q
}
∈
2
(
Φ
), so Q
⩽
Q, and this is impossible. Hence there must be a
Ψ
∈
2
(
Φ
) which is an (
α
ω
)
--RF of A. This shows that A is L
ω-compact.Definition 18.
LetI be an ideal in L
X. If ∨
x
∈
X
B
′
(
x
)
⩾
α for each B
∈
I, then I is called an α-ideal (α∈
M).Theorem 19.
Let(
L
X
,
Ω
) be an L
ω-space and A
∈
L
X. Then A is L
ω-compact if and only if every α-ideal I whose A is not in I has an ω-cluster point in A with hight α for each α
∈
M.Proof.
Necessity. Assume that A is an L
ω-compact set, I is an α-ideal whose A is not in I, and N
(
I
)
=
{
N
(
I
)
(
(
b
,
B
)
)
=
b
⩽
A
∣
(
b
,
B
)
∈
D
(
I
)
} where D
(
I
)
=
{
(
b
,
B
)
∣
b
∈
M
*
(
L
X
)
,
B
∈
Iand b
⩽
B
}. Then N
(
I
) is an α-net in A. Hence N
(
I
) has an ω-cluster point in A with hight α by Theorem 15, say x
α. Obviously, x
α is also an ω-cluster point of I. Consequently, the necessity is proved.
Sufficiency. Grant that every α-ideal whose A is not in it has an ω-cluster point in A with hight α for each α∈
M and ℱ is an α-filter containing A as an element. Let I
=
{
F
′
∈
L
X
∣
F
∈
ℱ
}. Evidently, I is an α-ideal whose A is not in I. Now we will prove that ℱ has an ω-cluster point in A with hight α. Actually, by the hypothesis we know that
I has an ω-cluster point in A with hight α, say x
α; that is, F
′
∨
P
≠
1
X; equivalently, F
⩽
P, for each F
∈
ℱ and each P
∈
ω
η
-
(
x
α
). Therefore x
α is an ω-cluster point of ℱ in line with Definition 16, and hence A is an L
ω-compact set by Theorem 17. This implies that the sufficiency holds.


## 4. Some Important Properties ofL
ω-Compactness

In this section, we still further deliberate the properties ofL
ω-compactness in an L
ω-space.Theorem 20.
Let(
L
X
,
Ω
) be an L
ω-space and A
,
B
∈
L
X. If A is L
ω-compact and B is ω-closed, then A
∧
B is L
ω-compact.Proof.
Assume thatN is an α-net in A
∧
B
(
α
∈
M
). Then N is also an α-net in A. Since A is ω-compact, N has an ω-cluster point in A with hight α, say x
α. We assert that x
α
⩽
B. Actually, since N is an α-net in B and N
ω-accumulates x
α, N has an α-subnet T which ω-converges to x
α and so x
α
⩽
ω
cl
(
B
)
=
B. Hence x
α
⩽
A
∧
B, and thus A
∧
B is L
ω-compact in accordance with Theorem 15.This theorem shows that theL
ω-compactness is hereditary with respect to
ω-closed sets.Theorem 21.
LetA and B be both L
ω-compact sets in (
L
X
,
Ω
). Then A
∨
B is also an L
ω-compact set in (
L
X
,
Ω
).Proof.
Suppose thatΦ is an α
ω-RF of A
∨
B (α
∈
M). Then Φ is an α
ω-RF of both A and B. Owing to the L
ω-compactness of A, there are λ
1
∈
β
*
(
α
) and Ψ
1
∈
2
(
Φ
) with ∧
Ψ
1
<
A
(
λ
1
ω
). Similarly, there exist λ
2
∈
β
*
(
α
) and Ψ
2
∈
2
(
Φ
) satisfying ∧
Ψ
2
<
A
(
λ
2
ω
). Take λ
=
λ
1
∧
λ
2 and Ψ
=
Ψ
1
∪
Ψ
2; then λ
∈
β
*
(
α
), Ψ
∈
2
(
Φ
), and ∧
Ψ
<
A
(
λ
ω
); that is, Ψ is an (
α
ω
)
--RF of A
∨
B. Consequently, A
∨
B is L
ω-compact.This theorem indicates that theL
ω-compactness is finitely additive.Theorem 22.
LetL
=
[
0,1
], (
L
X
,
Ω
) be an L
ω-space and let A
∈
L
X be an L
ω-compact set. Then there exists a crisp point
x
∈
X such that A
(
x
)
=
sup
{
A
(
t
)
∣
t
∈
X
}.Proof.
Letα
=
sup
{
A
(
t
)
∣
t
∈
X
}; then α
∈
[
0,1
]. If α
=
0, then A
=
0
X and hence A
(
x
)
=
sup
{
A
(
t
)
∣
t
∈
X
} holds for each x
∈
X. If α>
0, and D is the set of all natural numbers, then we choose
x
n
∈
X with A
(
x
n
)
>
α
-
(
1
/
n
) and N
=
{
x
A
(
x
n
)
n
∣
n
∈
D
}. Obviously, N is an α-net in A, and N has an ω-cluster point x
α in A by virtue of the L
ω-compactness of A. Hence A
(
x
)
⩾
α by x
α
⩽
A. On the other hand, A
(
x
)
⩽
α by the definition of α. Therefore A
(
x
)
=
α
=
sup
{
A
(
t
)
∣
t
∈
X
}.This theorem implies that anL
ω-compact set can reach the maximum at some point in
X as a function.Theorem 23.
Let(
L
X
,
Ω
1
) and (
L
Y
,
Ω
2
) be an L
ω
1-space and an L
ω
2-space, respectively, and let f
:
L
X
→
L
Y be an (
ω
1
,
ω
2
)-continuous L-valued Zadeh’s type function. If
A is an L
ω
1-compact set in (L
X
,
Ω
1), then f
→
(
A
) is an L
ω
2-compact set in (L
Y
,
Ω
2).Proof.
Assume thatΦ is an α
ω
2-RF of f
→
(
A
) and y
α
∈
M
*
(
L
Y
) with y
α
⩽
f
→
(
A
)
(
α
∈
M
). According to the definition of f, there is a molecule x
α∈
M
*
(
L
X
) such that x
α
⩽
A and f
→
(
x
α
)
=
y
α. Thus there is an ω-closed set Q
∈
Φ with f
→
(
x
α
)
⩽
Q; that is, x
α
⩽
f
←
(
Q
). Since f is (ω
1
,
ω
2)-continuous, f
←
(
Q
) is ω-closed in (L
X
,
ω
1), and hence f
←
(
Q
)
∈
ω
1
η
-
(
x
α
). This means that f
←
(
Φ
)
=
{
f
←
(
Q
)
∣
Q
∈
Φ
} is an α
ω
1-RF of A. Therefore Φ has a finite subfamily Ψ
=
{
Q
1
,
Q
2
,
…
,
Q
n
} such that f
←
(
Ψ
) is an (
α
ω
1
)
--RF of A. We assert that Ψ is an (
α
ω
2
)
--RF of f
→
(
A
). In reality, there exists a λ
∈
β
*
(
α
) with ∧
f
←
(
Ψ
)
<
A
(
λ
ω
1
) by virtue of the fact that f
←
(
Ψ
) is an (
α
ω
1
)
--RF of A. Since for each y
λ
⩽
f
→
(
A
) there exists a x
λ
⩽
A satisfying f
→
(
x
λ
)
=
y
λ, and there exists a Q
∈
Ψ with f
←
(
Q
)
∈
ω
1
η
-
(
x
λ
), that is, x
λ
⩽
f
←
(
Q
). Hence y
λ
=
f
→
(
x
λ
)
≰
Q by Lemma  3.1 in [19], and so Ψ is an (
α
ω
2
)
--RF of f
→
(
A
). Consequently, f
→
(
A
) is an L
ω
2-compact set in (L
Y, Ω
2).This theorem means that theL
ω-compactness is topological variant under
(ω
1
,
ω
2)-continuous L-valued Zadeh’s type functions.Definition 24.
Let (X, Ω) be a crisp ω-space, and let 𝒫
(
X
) be the set of all subsets of X, that is, all crisp sets on X and A
∈
L
X, where ω
:
𝒫
(
X
)
→
𝒫
(
X
) is a crisp ω-operator which satisfies the following conditions:
(
1
)ω
(
U
)
⊆
ω
(
V
) for each U
,
V
∈
𝒫
(
X
) and U
⊆
V; (
2
)U
⊆
ω
(
U
) for each U
∈
𝒫
(
X
).(i)
Ifξ
α
(
A
)
=
{
x
∈
X
∣
A
(
x
)
⩽
α
}
∈
ω
C
(
X
), where ω
C
(
X
) denotes the set of all crisp
ω-closed sets on X and α
∈
M, then A is said to be an L-valued lower semicontinuous function on
X.
(ii)
LetΔ
L (Ω) be the set of all L-valued lower semicontinuous functions on
X, and call the pair (L
X, Δ
L
(
Ω
)) the L
ω-space topologically generated by
(X
,
Ω).Theorem 25.
Let (X
,
Ω) be a crisp ω-space and let (L
X
,
Δ
L
(
Ω
)) be the L
ω-space topologically generated by (X
,
Ω). Then A
∈
L
X is L
ω-compact if and only if τ
α
(
A
)
=
{
x
∈
X
∣
A
(
x
)
⩾
α
} is ω-compact for each α
∈
M.Proof.
Necessity. Provided that A
∈
L
X is an L
ω-compact set in (L
X
,
Δ
L
(
Ω
)) and Φ is an ω-open cover of τ
α
(
A
)(
α
∈
M
), let Γ
=
{
χ
G
∣
G
∈
Φ
} and γ
=
α
′, where χ
G is the characteristic function of G. We assert that Γ is a γ
ω-cover of A. In fact, for each x
∈
τ
γ
′
(
A
), there is an ω-open set G
∈
Φ with x
∈
G; that is, χ
G
(
x
)
=
1. Hence χ
G
(
x
)
⩽
γ by virtue of the fact that γ is a prime element in L with γ
≠
1. Thus Φ has a finite subfamily {
G
1
,
G
2
,
…
,
G
m
} such that μ
=
{
χ
G
i
∣
i
=
1,2
,
…
,
m
}
∈
2
(
Γ
) which is a (
γ
ω
)
+-cover of A in line with Theorem 13; that is, there is an i
∈
{
1,2
,
…
,
m
} such that χ
G
i
∈
μ with χ
G
i
(
x
)
⩽
λ for some λ
∈
α
*
(
γ
) and each x
∈
τ
α
(
A
), and so x
∈
G
i. This implies that τ
α
(
A
)
⊆
∪
i
=
1
m
G
i. Hence τ
α
(
A
) is an ω-compact set in (X, Ω).
Sufficiency. Grant that τ
α
(
A
) is an ω-compact set in (X, Ω) for each α
∈
M and that Γ is a γ
ω-cover of A where γ
=
α
′. Then there is an ω-open set B
x
∈
Γ with B
x
(
x
)
⩽
γ for each x
∈
τ
α
(
A
), and hence there exists a prime element
t
(
x
)
∈
α
*
(
γ
) satisfying B
x
(
x
)
⩽
t
(
x
). Put l
t
(
x
)
(
B
x
)
=
{
y
∈
X
∣
B
x
(
y
)
⩽
t
(
x
)
} and Φ
=
{
l
t
(
x
)
(
B
x
)
∣
x
∈
τ
α
(
A
)
}; then Φ is an ω-open cover of τ
α
(
A
) according to x
∈
l
t
(
x
)
(
B
x
) and B
x
∈
Δ
L
(
Ω
). Because of the ω-compactness of τ
α
(
A
), Φ has a finite subfamily Ψ
=
{
l
t
(
x
i
)
(
B
x
i
)
∣
i
=
1,2
,
…
,
m
} which is an ω-open cover of τ
α
(
A
); that is, there exists an i
∈
{
1,2
,
…
,
m
} with x
∈
l
t
(
x
i
)
(
B
x
i
); in other words, B
x
i
(
x
)
⩽
t
(
x
i
) for each x
∈
τ
α
(
A
). Take t
=
∧
i
=
1
m
t
(
x
i
); evidently, t
∈
α
*
(
γ
) and B
x
i
(
x
)
⩽
t. Hence μ
=
{
B
x
i
∣
i
=
1,2
,
…
,
m
} is a (
γ
ω
)
+-cover of A, and thus A is an L
ω-compact set in (L
X, Δ
L
(
Ω
)) by Theorem 13.This theorem indicates that theL
ω-compactness is a good extension in the sense of R. Lowen.Theorem 26.
Let (L
X, Ω) be a stratified ω
T
2 and A
∈
L
X. If A is L
ω-compact, then A is ω-closed.Proof.
We only prove thatx
α
⩽
A for each x
α∈
M
*
(
L
X
) with x
α
⩽
ω
cl
(
A
) by the definition of ω-operator. Actually, if x
α
⩽
ω
cl
(
A
), then there exists a molecular net
N
=
{
x
t
(
n
)
(
n
)
∈
M
*
(
L
X
)
∣
n
∈
D
} in A which ω-converges to x
α in accordance with Theorem  2 in [11]. Write λ
=
∧
m
∈
D
∨
n
⩾
m
t
(
n
); we assert that λ
⩾
α. In fact, if λ
⩾
α, then there is a m
∈
D with ∨
n
⩾
m
t
(
n
)
⩾
α, and let d
=
∨
n
⩾
m
t
(
n
). Since (L
X
,
Ω) is stratified, the constant L-set [
d
] on X is ω-closed and x
α
⩽
[
d
], that is, [
d
]
∈
ω
η
-
(
x
α
). Obviously, N is eventually in [
d
], and it contradicts the fact that N
ω-converges to x
α. Hence λ
⩾
α; that is, ∨
n
⩾
m
t
(
n
)
⩾
α for each m
∈
D. For each r
∈
β
*
(
α
) and each m
∈
D we choose n
(
r
,
m
)
∈
D such that n
(
r
,
m
)
⩾
m and t
(
n
(
r
,
m
)
)
⩾
r, and define the relation “⩾” in β
*
(
α
)
×
D as follows: (2)
(
r
1
,
m
1
)
⩾
(
r
2
,
m
2
)
iff
r
1
⩾
r
2
,
m
1
⩾
m
2
. Then β
*
(
α
)
×
D is a directed set with the relation. Write
S
=
{
x
t
(
n
(
r
,
m
)
)
n
(
r
,
m
)
∣
(
r
,
m
)
∈
β
*
(
α
)
×
D
}; then S
=
N
∘
R, where R
:
β
*
(
α
)
×
D
→
D is defined as R
(
n
(
r
,
m
)
)
=
n
(
r
,
m
). Evidently S is a subnet of N and ω-converges to x
α, and S is an α-net in A. Being the L
ω-compactness of A, S has an ω-cluster point in A with hight α, say z
α. Since (L
X, Ω) is an ω
T
2 space, S
ω-converges to x
α and ω-accumulates to z
α, z
=
x by Theorem  2.7 in [11], and hence x
α
=
z
α
⩽
A. This implies that ω
cl
(
A
)
⩽
A; that is, A is an ω-closed set.The following example shows that the stratified condition in Theorem26 can not be omitted.Example 27.
LetX
=
{
x
} be a single set, L
=
[
0,1
], and let ω
:
L
X
→
L
X be the fuzzy closure operator. Define
ω
O
(
L
X
)
=
{
0
X
,
x
1
/
3
,
1
X
}, where A
:
x
→
[
0,1
] is defined as A
(
x
)
=
x
α, α
∈
[
0,1
] for x
∈
X. Obviously, (L
X
,
Ω) is both an L
ω-compact space and an N-compact space. According to Example 11 we know that A
=
x
1
/
3 is an L
ω-compact set in (L
X
,
Ω), but A is not ω-closed.The following theorems imply that theL
ω-compactness can strengthen ω-seperation properties.Theorem 28.
If (L
X, Ω) is both ω
T
2 and L
ω-compact L
ω-space, then (L
X
,
Ω) is an ω-regular space [11].Proof.
LetG
∈
L
X be an ω-closed pseudocrisp set and let x
λ be a molecule which x is not in supp
G. By Definition  7.1 in [19], there is an α
∈
M such that G
(
x
)
>
0 implies G
(
x
)
⩾
α. For each y
α
∈
M
*
(
L
X
), there are P
y
∈
ω
η
-
(
x
λ
) and Q
y
∈
ω
η
-
(
y
α
) satisfying P
y
∨
P
y
=
1
X by virtue of x
≠
y and the ω
T
2 separation of (L
X
,
Ω). Put Φ
=
{
Q
y
∣
y
α
⩽
G
}; then Φ is an α
ω-RF of G. Since (L
X
,
Ω) is an L
ω-compact space, G is an L
ω-compact set in accordance with Theorem 20, and thus Φ has a finite subfamily Ψ
=
{
Q
y
1
,
Q
y
2
,
…
,
Q
y
n
} which is an (
α
ω
)
--RF of G; that is, there is an r
∈
β
*
(
α
) such that for each molecule z
r
⩽
G we have i
⩽
n with z
r
⩽
Q
y
i. Let Q
=
∧
i
=
1
n
Q
y
i; then z
r
⩽
Q; that is, r
⩽
Q
(
z
) for each z
r
⩽
G. Since G
(
z
)
>
0 implies that G
(
z
)
⩾
α
⩾
r, G
(
z
)
⩽
Q
(
z
) for each z
∈
supp
G, and hence Q
∈
ω
η
-
(
G
). Write P
=
∨
i
=
1
n
P
y
i; then P
∈
ω
η
-
(
x
λ
) and (3)
P
∨
Q
=
(
∨
i
=
1
n
P
y
i
)
∨
(
∧
i
=
1
n
Q
y
i
)
⩾
∨
i
=
1
n
(
P
y
i
∨
Q
y
i
)
=
1
. Consequently, (L
X, Ω) is an ω-regular space.Theorem 29.
Let (L
X
,
Ω) be an L
ω-compact ω
T
2 space. Then (L
X
,
Ω) is an ω-normal space [11].Proof.
Let bothG
,
H be ω-closed pseudocrisp sets in (L
X
,
Ω) with (
supp
G
)
∩
(
supp
H
)
=
ϕ. Then there are λ, μ
∈
M such that G
(
x
)
>
0 if and only if G
(
x
)
⩾
λ, and H
(
x
)
>
0 if and only if H
(
x
)
⩾
μ. According to the proof of Theorem 28, for each molecule y
μ
⩽
G, there is an ω-closed set P
y
∈
ω
η
-
(
G
) satisfying λ
⩽
P
y
(
z
) for each z
∈
supp
G, and there is a Q
y
∈
ω
η
-
(
y
μ
) such that P
y
∨
Q
y
=
1. One can easily see that Φ
=
{
Q
y
∣
y
μ
⩽
B
} is a μ
ω-RF of H. In line with Theorem 20 we know that H is an L
ω-compact set, and so Φ has a finite subfamily Ψ
=
{
Q
y
1
,
Q
y
2
,
…
,
Q
y
n
} such that Ψ is a (
μ
ω
)
--RF of H. Put P
=
∨
i
=
1
n
P
y
i; Q
=
∧
i
=
1
n
Q
y
i; then P
∈
ω
η
-
(
G
), Q
∈
ω
η
-
(
H
) and P
∨
Q
=
1. Therefore (L
X, Ω) is an ω-normal space.


## 5. The Tychonoff Product Theorem

In this section, we will first extend Alexandar’s subbase Lemma in general topology
and give the Alexandar’sω-subbase lemma and next prove that the Tychonoff product
theorem holds in L
ω-spaces.Theorem 30 (Alexandarω-subbase lemma).
Let (L
X, Ω) be an L
ω-space, A
∈
L
X, and let γ be an ω-subbase [20] in
L
X. If for each α
ω-RF  Φ of A where Φ
⊆
γ
′
⊆
ω
C
(
L
X
), there is a finite subfamily Ψ of Φ with ∧
Ψ
≪
A
(
α
ω
)
(
α
∈
M
), then A is L
ω-compact.Proof.
Suppose thatΦ is an arbitrary α
ω-RF of A. We will prove that Φ has a finite subfamily Ψ which is an (
α
ω
)
+-RF of A. In fact, if for each Ψ∈
2
(
Φ
), ∧
Ψ
≪
A
(
α
ω
) does not hold, then H
=
{
Δ
∣
Φ
⊆
Δ
⊆
ω
C
(
L
X
), for all Ψ
∈
2
(
Δ
)
,
∧
Ψ
≪
A
(
α
ω
) does not hold}
≠
∅, and H is a partial-ordered set with respect to the upper bound
and hence there exists a maximal element Δ
0 in H by Zorn’s Lemma. We assert that
Δ
0 satisfies the following conditions:(1)
∧
Δ
0
<
A
(
α
⩾
ω
);
(2)
ifP
∈
Δ
0, then Q
∈
Δ
0 for each Q
∈
ω
C
(
L
X
) with Q
⩾
P;
(3)
ifP
,
Q
∈
ω
C
(
L
X
) and P
∨
Q
∈
Δ
0, then P
∈
Δ
0 or Q
∈
Δ
0.
Actually, since∧
Φ
<
A
(
α
ω
) and Φ
⊆
Δ
0, condition (
1
) holds. If P
∈
Δ
0, Q
∈
ω
C
(
L
X
), Q
⩾
P, and Q is not in Δ
0, then Δ
*
=
Δ
0
∪
{
Q
}
∈
H and Δ
0
⊂
Δ
*. It contradicts the fact that Δ
0 is the maximal element in H thus condition (
2
) holds. Let P
,
Q
∈
ω
C
(
L
X
). If P and Q are both not in Δ
0, then Δ
0
∪
{
P
} and Δ
0
∪
{
Q
} are both not in H by the maximality of Δ
0, and thus there are Ψ
1
,
Ψ
2
∈
2
(
Δ
0
) such that ∧
(
Ψ
1
∪
{
P
}
)
≪
A
(
α
ω
) and ∧
(
Ψ
2
∪
{
Q
}
)
≪
A
(
α
ω
) according to the definition of H; that is, there are s
,
t
∈
β
*
(
α
) with ∧
(
Ψ
1
∪
{
P
}
)
<
A
(
s
ω
) and ∧
(
Ψ
2
∪
{
Q
}
)
<
A
(
t
ω
). Since β
*
(
α
) is upper directed, we can choose r
∈
β
*
(
α
) with r
⩾
s
∨
t. Now we prove ∧
{
Ψ
2
∪
Ψ
2
∪
{
P
∨
Q
}
}
<
A
(
r
ω
). In reality, if Ψ
2
∪
Ψ
2 does not have any ω
R-neighborhood of x
r for each x
r
⩽
A, then Ψ
2
∪
Ψ does not have any ω
R-neighborhood of x
s and x
t, respectively, and hence P
∈
ω
η
-
(
x
s
) and Q
∈
ω
η
-
(
x
t
). Particularly, P
,
Q
∈
ω
η
-
(
x
r
) and so P
∨
Q
∈
ω
η
-
(
x
r
). This shows that ∧
(
Ψ
2
∪
Ψ
2
∪
{
P
∨
Q
}
)
<
A
(
r
ω
). Therefore P
∨
Q is not in Δ
0 by virtue of the definition of Δ
0 and Ψ
1
,
Ψ
2
∈
2
(
Δ
0
). So, condition (
3
) holds.
From(
2
) and (
3
) we have the following result:(4)
IfR
∈
Δ
0, P
i
∈
ω
C
(
L
X
)
(
i
=
1,2
,
…
,
n
) and R
⩽
∨
i
=
1
n
P
i, then there is an i∈
{
1,2
,
…
,
n
} satisfying P
i
∈
Δ
0.
Consider nowγ
′
∩
Δ
0. If γ
′
∩
Δ
0 is an α
ω
-RF of A, then there is a finite subfamily δ of γ
′
∩
Δ
0 which is an (
α
ω
)
--RF of A. Evidently, δ
∈
2
(
Δ
0
); it is in contradiction with Δ
0
∈
H. Hence γ
′
∩
Δ
0 is not an α
ω-RF of A; that is, there is a molecule x
α in A meeting x
α
⩽
∧
(
γ
′
∩
Δ
0
). We now verify that x
α
⩽
∧
Δ
0. In fact, if there is Q
∈
Δ
0 with x
α
⩽
Q, then by Definition  5 in [17] we can take a finite subfamily
{
P
i
j
∣
j
∈
J
i
,
i
∈
I
} of γ
′ satisfying Q
=
∧
i
∈
I
∨
j
∈
J
i
P
i
j, where J
i is a finite set for each i
∈
I. Because of x
α
⩽
Q, we can choose i
∈
I with x
α
⩽
∨
j
∈
J
i
P
i
j. Since Q
⩽
∨
j
∈
J
i
P
i
j, there is a j
∈
J
i such that P
i
j
∈
Δ
0 by (
4
). Hence P
i
j
∈
γ
′
∩
Δ
0 and x
α
⩽
P
i
j; it contradicts the fact that x
α
⩽
∧
(
γ
′
∩
Δ
0
)
⩽
P
i
j; thus x
α
⩽
∧
Δ
0. However, this is in contradiction with
(
1
) again. This implies that Φ has a finite subfamily Ψ with ∧
Ψ
≪
A
(
α
ω
). Therefore A is an L
ω-compact set in (L
X
,
Ω).Theorem 31.
Let{
(
L
X
t
,
Ω
t
)
∣
t
∈
T
} be a collection of L
ω-spaces and let (L
X
,
Ω) be the product space of them. If A
t is an L
ω-compact set in (
L
X
t
,
Ω
t
) for each t
∈
T, then the product A
=
Π
t
∈
T
A
t of all L
ω-compact sets A
t
(
t
∈
T
) is an L
ω-compact set in (L
X
,
Ω).Proof.
Assume thatΦ is an α
ω-RF of A
(
α
∈
M
). By Theorem 30 we can grant that every ω-closed set in Φ is of the form ρ
t
←
(
B
t
) where B
t
∈
ω
C
(
L
X
t
) and ρ
t
:
L
X
→
L
X
t is a protection because {
ρ
t
←
(
U
t
)
∣
U
t
∈
ω
O
(
L
X
)
,
t
∈
T
} is an ω-subbase in (L
X
,
Ω) [20]. Now we
consider the following two cases.
(i) If there exists at
0
∈
T such that no molecule with hight α is contained in A
t
0, then by the L
ω-compactness of A
t
0, there is an r
∈
β
*
(
α
) such that no molecule with hight r is contained in A
t
0. In reality, if there exists a molecule with hight
r in A
t
0 for each r
∈
β
*
(
α
), say N
(
r
), then N
=
{
N
(
r
)
∣
r
∈
β
*
(
α
)
} is an α-net in A
t
0 by the directivity of β
*
(
α
). Since A
t
0 is L
ω-compact, N has an ω-cluster point in A
t
0 with hight α according to Theorem 15. It is in contradiction with the hypothesis of
A
t
0. Thus it can be seen that there exists an
r
∈
β
*
(
α
) with A
t
0
(
x
t
0
)
⩾
r for each x
t
0
∈
X
t
0. Hence for each x
∈
X, we have (4)
A
(
x
)
=
(
Π
t
∈
T
A
t
)
(
x
)
=
∧
t
∈
T
A
t
(
ρ
t
(
x
)
)
⩽
A
t
0
(
ρ
t
0
(
x
)
)
=
A
t
0
(
x
t
0
)
, and hence A
(
x
)
⩾
r for each x
∈
X; that is, no molecule with hight r is contained in A. This shows that for each Ψ∈
2
(
Φ
), Ψ is an (
α
ω
)
--RF of A.
(ii) Suppose that for eacht
∈
T, A
t contains a molecule with hight α, say x
α
t. Since Φ⊆{
ρ
t
←
(
B
t
)
∣
B
t
∈
ω
C
(
L
X
)
,
t
∈
T
}, we can take R
⊆
T such that Φ
=
∪
t
∈
R
Φ
t, where Φ
t
=
{
ρ
t
←
(
B
t
)
∣
B
t
∈
ℬ
t
⊆
ω
C
(
L
X
)
}. Now we prove that there must be s
∈
R with ∧
ℬ
s
<
A
(
α
ω
). In fact, if there is a crisp point
y
t
∈
X
t such that y
t
⩽
A
t
∧
(
∧
ℬ
t
) for each t
∈
R, then we choose a crisp point z in X as follows: if t
∈
R, z
t
=
y
t; if t is not in R, z
t
=
x
t. Taking any ω-closed set ρ
t
←
(
B
t
) in Φ, where t
∈
R and B
t
∈
ℬ
t, we have (5)
ρ
t
←
(
B
t
)
(
z
)
=
B
t
(
z
t
)
=
B
t
(
y
t
)
⩾
(
A
t
∧
(
∧
ℬ
t
)
)
(
y
t
)
⩾
α
, that is, z
α
⩽
ρ
t
←
(
B
t
), and hence z
α
⩽
∧
Φ by the arbitrariness of ρ
t
←
(
B
t
)
∈
Φ. On the other hand, (6)
A
(
z
)
=
∧
t
∈
R
A
t
(
z
t
)
=
(
∧
t
∈
R
A
t
(
y
t
)
)
∧
(
∧
t
∈
R
A
t
(
x
t
)
)
⩾
α
. This implies that z
α is a molecule in A; it contradicts the fact that Φ is an α
ω-RF of A. Consequently, there is s
∈
R with ∧
ℬ
s
<
A
s
(
α
); thus there is a finite subfamily Γ
s of ℬ
s with Γ
s
<
A
s
(
r
ω
) for some r
∈
β
*
(
α
). Put Ψ=
{
ρ
s
←
(
B
s
)
∣
B
s
∈
Γ
s
}; then Ψ∈
2
(
Φ
). We assert that ∧
Ψ
<
A
(
r
ω
). Actually, for any molecule e
r in A with hight r we have A
s
(
e
s
)
⩾
A
(
e
)
⩾
r; that is, e
r
s is a molecule in A
s, where e
=
{
e
t
}
t
∈
T
∈
X. Hence there exists an ω-closed set B
s
∈
Γ
s meeting B
s
∈
ω
η
-
(
e
r
s
) by virtue of the fact that Γ
s is an r
ω-RF of A
s; thus ρ
s
←
(
B
s
)
(
e
)
=
B
s
(
e
s
)
⩾
r; that is, ρ
s
←
(
B
s
)
∈
ω
η
-
(
e
r
). This shows that Ψ is an r
ω-RF of A. Therefore A is an L
ω-compact set in (L
X
,
Ω).Theorem 32 (Tychonoff product theorem).
Let (L
X, Ω) be the product space of a collection of
L
ω-spaces {
(
L
X
t
,
Ω
t
)
∣
t
∈
T
}. Then (L
X, Ω) is L
ω-compact if and only if for each t
∈
T, (L
X
t, Ω
t) is L
ω-compact.Proof.
Necessity. Assume that (L
X, Ω) is an L
ω-compact space. Since ρ
t
:
(
L
X
,
Ω
)
→
(
L
X
t
,
Ω
t
) is an ω-continuous L-valued Zadeh’s type function for each
t
∈
T, (
L
X
t
,
Ω
t
) is an L
ω-compact space by Theorem 23. Therefore the necessity holds.
Sufficiency. It follows from Theorem 31.The following example shows that the inverse theorem of Theorem31 does not hold.Example 33.
LetE
=
{
e
1
,
e
2
,
…
} be a countably infinite set, X
t
=
E for each t
∈
T
=
{
1,2
,
…
}, L
=
[
0,1
], Ω
t
=
[
0,1
]
E and let ω be a fuzzy closure operator. Then (
L
X
t
,
Ω
t
) is a discrete L
ω-space for each t
∈
T. Define A
t
∈
L
X
t (t
∈
T) as follows:ifj
=
1, A
t
(
e
j
)
=
1; if j
⩾
2, A
t
(
e
j
)
=
1
/
t.
Suppose that (L
X
,
Ω) is the product space of {
(
L
X
t
,
Ω
t
)
∣
t
∈
T
} and A
=
Π
t
∈
T
A
t. Now we prove that A is an L
ω-compact set in (L
X, Ω), but A
t is not an ω-compact set in (
L
X
t
,
Ω
t
) for each t
∈
T. In reality, for each x
=
(
x
1
,
x
2
,
…
)
∈
X we put x
t
=
e
j
(
t
)
t, where x
t is a crisp point e
j in X
t; then from the definitions of A
t and fuzzy product set A we know (7)
A
(
x
)
=
(
Π
t
∈
T
A
t
)
(
x
)
=
∧
t
∈
T
A
t
(
x
t
)
=
∧
t
∈
T
A
t
(
e
j
(
t
)
t
)
=
{
0
,
if
there
are
infinite
elements
t
such
that
j
(
t
)
⩾
2
.
1
t
R
,
if
there
is
a
t
R
∈
T
such
that
j
(
t
R
)
⩾
2
and
j
(
t
)
=
1
whenever
t
>
t
R
. Thus it can be seen that A
≠
0
X and if A
(
x
)
⩾
1
/
t
R, then the coordinates x
t
=
e
j
(
t
)
t
=
e
1 of x whenever t
>
t
R. Obviously, points in X satisfying the condition are only finite. Let
α
∈
M, that is, α
>
0, and let Φ be an α
ω-RF of A. Choose t
R
∈
T with 1
/
t
R
<
α. Since there are only finite molecule in
A with hight α, denote the finite crisp points as
x
1
,
x
2
,
…
,
x
n. If (
x
i
)
α
⩽
A for each i
∈
{
1,2
,
…
,
n
}, then there is P
i
∈
Φ with P
i
(
x
i
)
<
α. Put s
=
max
{
P
i
(
x
i
)
∣
P
i
(
x
i
)
<
α
,
i
⩽
n
}; then s
<
α. Taking s
1
∈
(
s
,
α
) and r
=
max
(
s
1
,
1
/
t
R
), we know that A has at most n molecules with hight r, say (
x
i
)
r
(
i
⩽
n
). By the definition of Φ, there is a P
i
∈
Φ such that P
i
∈
ω
η
-
(
(
x
i
)
r
) for each (
x
i
)
r in A. Denote Ψ
=
{
P
i
∈
Φ
∣
P
i
∈
ω
η
-
(
(
x
i
)
r
)
,
i
⩽
n
}; then Ψ
∈
2
(
Φ
) and Ψ is an r
ω-RF of A. This implies that Ψ is an (
α
ω
)
--RF of A by r
∈
β
*
(
α
). Hence A is L
ω-compact in (L
X, Ω). On the other hand, take D
=
T and N
=
{
N
(
m
)
∣
m
∈
D
} where N
(
m
)
=
(
e
m
)
1
/
t for each m
∈
D and each t
∈
T; then N is a (
1
/
t
)-net in A
t. Since (
L
X
t
,
Ω
t
) is discrete, N does not have any ω-cluster point in A
t with hight 1
/
t. Therefore A
t is not L
ω-compact in (
L
X
t
,
Ω
t
) for each t
∈
T according to Theorem 15.



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*Source: 929178-2013-06-12.xml*