In this paper, the concept of compactness measures of (L, M)-fuzzy Q-convergence spaces is introduced. It is used to define compactness measures of (L, M)-fuzzy topological spaces and pointwise (L, M)-fuzzy quasi-uniform spaces. It is shown that the concept of compactness measures satisfies the lattice-valued Tychonoff theorem in the framework of (L, M)-fuzzy Q-convergence spaces and (L, M)-fuzzy topological spaces. Also, it is proved that the compactness measures of a pointwise (L, M)-fuzzy quasi-uniform space is equal to the compactness measures of its induced (L, M)-fuzzy topological space.
Compactness is an important topological property in topological spaces. The concept of compactness of a topological space is usually defined in terms of open covers [10]. Actually, compactness can be also characterized by ultrafilter convergence. Based on this fact, compactness is generalized to axiomatic filter convergence spaces [22]. Concretely, an axiomatic filter convergence space is called compact provided that every ultrafilter converges.
With the development of fuzzy topology, compactness is defined in different ways in the framework of fuzzy topological spaces. Gantner et al. [3] introduced α-compactness for an L-topological space. For an L-subset, Liu [13] introduced Q-compactness, Zhao [29] introduced N-compactness, and Shi [24] introduced S∗-compactness. Recently, Shi [25] introduced a new approach to compactness of an L-topological space by an inequality. In 1985, Šostak [26] introduced a definition of compactness degrees for a fuzzy set by means of the level I-topology. Later, Eş and Çoker [1] defined the degrees of almost compactness, near compactness, countable compactness, light compactness and strong compactness for a fuzzy set. Afterwards, Shi [23] introduced new definitions of countable compactness and the Lindelöf property in L-topological spaces, which do not rely on the structure of the basis lattice L. In 2010, Li and Shi [12] introduced the compactness degrees of L-fuzzy topological spaces. Naturally, this is a motivation to consider compactness in fuzzy convergence spaces. For stratified L-generalized convergence spaces [4], Jäger investigated its compactness [5, 7] and compactification [6, 9]. In the situation of (L, M)-fuzzy convergence spaces [16], Pang and Shi [17] introduced the concept of compactness degrees of an (L, M)-fuzzy convergence space and applied this method to define compactness degrees of an (L, M)-fuzzy topological space. Besides (L, M)-fuzzy convergence structures, there are some other kinds of fuzzy convergence structures in the framework of (L, M)-fuzzy topological spaces, such as (L, M)-fuzzy Q-convergence spaces [14, 15, 19], enriched (L, M)-fuzzy convergence spaces [18, 21, 18, 21] and L-fuzzy N-convergence spaces [20]. Inspired by this, we consider compactness theory in (L, M)-fuzzy Q-convergence spaces.
Not only in fuzzy topological spaces, but also in fuzzy convergence spaces, topological properties are endowed with some degrees or some measures. This requirement seems to be necessary in the development of fuzzy topology. In this paper, we will endow (L, M)-fuzzy Q-convergence spaces with some compactness measures and use it to define compactness measures of (L, M)-fuzzy topological spaces and pointwise (L, M)-fuzzy quasi-uniform spaces.
Preliminaries
Throughout this paper, both L and M denote completely distributive lattices and ′ is an order-reversing involution on L. The smallest element and the largest element in L (M) are denoted by ⊥L (⊥M) and ⊤L (⊤M), respectively. For a, b ∈ L, we say that a is wedge below b in L, in symbols a ≺ b, if for every subset D ⊆ L, ⋁D ⩾ b implies d ⩾ a for some d ∈ D. A complete lattice L is completely distributive if and only if b = ⋁ {a ∈ L ∣ a ≺ b} for each b ∈ L. An element a in L is called coprime if a ⩽ b ∨ c implies a ⩽ b or a ⩽ c. The set of nonzero coprime elements in L (M) is denoted by J (L) (J (M)), respectively. We define a residual implication operation → : L × L → L as the right adjoint for the meet operation ∧ by
This operation plays a particular role in the sequel.
For a nonempty set X, LX denotes the set of all L-subsets on X. The smallest element and the largest element in LX are denoted by and , respectively. LX is also a completely distributive De Morgan algebra when it inherits the structure of the lattice L in a natural way, by defining ⋁, ⋀ , ⩽ and ′ pointwisely. For each x ∈ X and a ∈ L, the L-subset xa, defined by xa (y) = a if y = x, and xa (y) = ⊥ L if y ≠ x, is called a fuzzy point. The set of nonzero coprime elements in LX is denoted by J (LX). It is easy to see that J (LX) = {xλ ∣ x ∈ X, λ ∈ J (L)}. We say that a fuzzy point xλ ∈ J (LX) quasi-coincides with A, denoted by , if λnotleqslantA′ (x) or equivalently xλnotleqslantA′ . The relation “does not quasi-coincide with” is denoted by Let φ : X ⟶ Y be a mapping. Define φ→ : LX ⟶ LY and φ← : LY ⟶ LX by φ→ (A) (y) = ⋁ φ(x)=yA (x) for A ∈ LX and y ∈ Y, and φ← (B) = B ∘ φ for B ∈ LY, respectively.
Definition 2.1. (Yao [27]) A mapping is called an (L, M)-fuzzy filter on X if it satisfies:
;
.
The family of all (L, M)-fuzzy filters on X is denoted by On the set of all (L, M)-fuzzy filters on X, we define an order by if for all A ∈ LX. Then is a partially ordered set.
Example 2.2. (Pang [16]) For each xλ ∈ J (LX) ,
we define as follows:
Then is an (L, M)-fuzzy filter.
Definition 2.3. (Yao [27]) Let and φ : X ⟶ Y be a mapping. Then is an (L, M)-fuzzy filter on Y and is called the image of under φ.
Definition 2.4. (Pang [14] for L = M) An (L, M)-fuzzy Q-convergence structure on X is a mapping which satisfies:
;
implies .
For an (L, M)-fuzzy Q-convergence structure qc on X, the pair (X, qc) is called an (L, M)-fuzzy Q-convergence space.
A mapping φ : (X, qcX) ⟶ (Y, qcY) between (L, M)-fuzzy Q-convergence spaces is called continuous provided that for each and xλ ∈ J (LX). That is, for each . The category of (L, M)-fuzzy Q-convergence spaces with continuous mappings as morphisms will be denoted by (L, M)-FQC.
Definition 2.5. (Kubiak [11] and Šostak [26]) An (L, M)-fuzzy topology on X is a mapping τ : LX ⟶ M which satisfies:
;
τ (A ∧ B) ⩾ τ (A) ∧ τ (B);
τ (⋁ j∈JAj) ⩾ ⋀ j∈Jτ (Aj).
For an (L, M)-fuzzy topology τ on X, the pair (X, τ) is called an (L, M)-fuzzy topological space.
A continuous mapping between (L, M)-fuzzy topological spaces (X, τX) and (Y, τY) is a mapping φ : X ⟶ Y such that τX (φ← (A)) ⩾ τY (A) for each A ∈ LY. The category of (L, M)-fuzzy topological spaces with continuous mappings as morphisms will be denoted by (L, M)-FTop.
Definition 2.6. (Fang [2]) An (L, M)-fuzzy quasi-coincident neighborhood system on X is defined to be a set of mappings satisfying the following conditions:
;
;
.
The pair is called an (L, M)-fuzzy quasi-coincident neighborhood space, and it will be called topological if it satisfies moreover,
.
A continuous mapping between two (L, M)-fuzzy quasi-coincident neighborhood spaces and is a mapping φ : X ⟶ Y such that for each xλ ∈ J (LX) and A ∈ LY. The category of topological (L, M)-fuzzy quasi-coincident neighborhood spaces with continuous mappings as morphisms will be denoted by (L, M)-FTQN.
Proposition 2.7.(Fang [2]) Let be an (L, M)-fuzzy quasi-coincident neighborhood system on X and define by
Then is an (L, M)-fuzzy topology on X.
Proposition 2.8.(Fang [2]) Let τ be an (L, M)-fuzzy topology on X and define by
Then is a topological (L, M)-fuzzy quasi-coincident neighborhood system on X. Moreover, .
Theorem 2.9.(Fang [2]) (L, M)-FTop is isomorphic to (L, M)-FTQN.
Measures of compactness in (L, M)-fuzzy Q-convergence spaces
In order to define compactness measures of (L, M)-fuzzy Q-convergence spaces, we first recall some conclusions with respect to (L, M)-fuzzy ultrafilters in [17].
Lemma 3.1.(Pang [17]) The partially ordered set has maximal elements.
Definition 3.2. (Pang [17]) An (L, M)-fuzzy filter is called an (L, M)-fuzzy ultrafilter provided that it is a maximal element of . The set of all (L, M)-fuzzy ultrafilters is denoted by.
Lemma 3.3.(Pang [17]) For every (L, M)-fuzzy filter , there exists an (L, M)-fuzzy ultrafilter with .
Given a mapping φ : X ⟶ Y and an (L, M)-filter on Y, the mapping defined by
is an (L, M)-fuzzy filter on X if and only if whenever for all B ∈ LY. In the case , it is called the inverse image of under φ. In particular, whenever φ is a surjective mapping.
Lemma 3.4.(Pang [17]) Let φ : X ⟶ Y be a mapping, and let and . Then
provided that φ is surjective.
Lemma 3.5.(Pang [17]) If φ : X ⟶ Y is a mapping and , then .
Now we can define the compactness measures of an (L, M)-fuzzy Q-convergence space by means of (L, M)-fuzzy ultrafilters.
Definition 3.6. For an (L, M)-fuzzy Q-convergence space (X, qc), define the measure Comp (X, qc) to which (X, qc) is compact by
Furthermore, (X, qc) is called compact provided that Comp (X, qc) = ⊤ M.
Whenever L = M = {0, 1}, the equality can be understood as “a convergence space is compact iff every ultrafilter converges”. Hence, this definition can be interpreted as to many-valued logical truth-value of the sentence “a convergence space is compact iff every ultrafilter converges”.
The above notion allows us to talk on compactness of an (L, M)-fuzzy Q-convergence space in some measure even if (X, qc) is not compact. The compactness measure Comp(X,qc) of (X, qc) is a natural characterization for which (X, qc) is a compact (L, M)-fuzzy Q-convergence space. In the sequel, we will show that the compactness measures of an (L, M)-fuzzy Q-convergence space naturally suggests many-valued logical extensions of the properties of compactness in generalized convergence spaces to (L, M)-fuzzy Q-convergence spaces.
Proposition 3.7.If φ : (X, qcX) ⟶ (Y, qcY) is continuous and surjective, then
Proof. It suffices to show that
Take any . Since φ : X ⟶ Y is surjective, we know By Lemma 3.3, we choose an (L, M)-fuzzy ultrafilter such that . Then by Lemma 3.4, we have . Since is maximal, it follows that . Hence by the continuity of φ, we have
By the arbitrariness of , we obtain
as desired. □
In [14], Pang and Fang provided the concrete form of initial structures with respect to L-fuzzy Q-convergence spaces. Since (L, M)-fuzzy Q-convergence spaces are generalizations of L-fuzzy Q-convergence spaces, we can easily give the initial structure of (L, M)-fuzzy Q-convergence spaces. As a special case of initial structures, the product of (L, M)-fuzzy Q-convergence structures can be presented as follows.
Definition 3.8. Let {(Xj, qcj) ∣ j ∈ J} be a family of (L, M)-fuzzy Q-convergence spaces, and pk : X ⟶ Xk be projection mapping. Define by
Then is called the product space of {(Xj, qcj) ∣ j ∈ J}.
Based on the definition of product space of (L, M)-fuzzy Q-convergence spaces, we give the lattice-valued Tychonoff theorem in the framework of (L, M)-fuzzy Q-convergence spaces.
Theorem 3.9.Let {(Xj, qcj) ∣ j ∈ J} be a family of (L, M)-fuzzy Q-convergence spaces. Then
Proof. For convenience, put and . By the definition of product of (L, M)-fuzzy Q-convergence spaces, we know that the projection mapping pk : (X, qcX) ⟶ (Xk, qck) is surjective and continuous for all k ∈ J. Then by Proposition 3.6, it follows that for all j ∈ J. This implies that
Next we prove the inverse, i.e.,
Take any α ∈ J (M) such that
Then for each j ∈ J, and λ ∈ J (L), there exists xj ∈ Xj such that
Next we will show that
By Lemma 3.5, for each , we have for all j ∈ J. Further for each λ ∈ J (L), there exists xj ∈ Xj such that Put x = (xj) ∈ X. Then it follows that
By the arbitrariness of λ and , we obtain
This proves that
as desired. □
Measures of compactness in (L, M)-fuzzy topological spaces
In this section, we define the compactness measures of an (L, M)-fuzzy topological space by convergence of (L, M)-fuzzy ultrafilters. Also, we give the lattice-valued representations of the Tychonoff theorem in (L, M)-fuzzy topological spaces.
Firstly, we study the relations between (L, M)-fuzzy Q-convergence spaces and (L, M)-fuzzy topological spaces from a categorical aspect.
Proposition 4.1. (1) Let (X, qc) be an (L, M)-fuzzy Q-convergence space and define τqc : LX ⟶ M bywhere for each A ∈ LX. Then τqc is an (L, M)-fuzzy topology on X.
(2) If φ : (X, qcX) ⟶ (Y, qcY) is continuous, then so is φ : (X, τqcX) ⟶ (Y, τqcY).
Proof. (1) It suffices to verify that τqc satisfies (LFT1)–(LFT3). In fact,
(LFT1) By the definition of τqc, we obatin
and
(LFT2) Take any A, B ∈ LX. Then
(LFT3) For {Aj ∣ j ∈ J} ⊆ LX, it follows that
This shows that τqc is an (L, M)-fuzzy topology on X.
(2) Since φ : (X, qcX) ⟶ (X, qcX) is continuous, it follows that
Then for each B ∈ LY, we have
Thus, φ : (X, τqcX) ⟶ (Y, τqcY) is continuous. □
Hence, we obtain a functor as follows:
Lemma 4.2.Let (X, τ) be an (L, M)-fuzzy topological space and define as follows:
Then for each xλ ∈ J (LX) and ,
Proof. The sufficiency is obvious. It suffices to show the necessity. Suppose that . By Proposition 2.8, take any A ∈ LX and α ∈ J (M) such that
Then there exists Bα ∈ LX such that and for each , it follows that . Then we have . By the definition of qcτ, there exists zν ∈ J (LX) such that and . Thus,
By the arbitrariness of α, we get
as desired. □
Proposition 4.3. (1) Let (X, τ) be an (L, M)-fuzzy topological space. Then qcτ is an (L, M)-fuzzy Q-convergence structure on X.
(2) If φ : (X, τX) ⟶ (Y, τY) is continuous, then so is φ : (X, qcτX) ⟶ (Y, qcτY).
Proof. (1) It is enough to show that qcτ satisfies (LFQC1) and (LFQC2).
(LFQC1) Take any xλ ∈ J (LX). Then and . Further, it follows that
(LFQC2) Obviously.
(2) Since φ : (X, τX) ⟶ (Y, τY) is continuous, by Theorem 2.9, it follows that is continuous, that is,
Then for each , we have
By the arbitrariness of xλ, we obtain . This shows φ : (X, qcτX) ⟶ (Y, qcτY) is continuous. □
Thus, we obtain another functor as follows:
Theorem 4.4.is a pair of adjoint functors between (L, M)-FQC and (L, M)-FTop.
Proof. It suffices to prove that and . That is to say, for an (L, M)-fuzzy Q-convergence structure qc on X and an (L, M)-fuzzy topology τ on X, it follows that qcτqc ⩾ qc and τqcτ = τ.
On one hand, take any xλ ∈ J (LX) and A ∈ LX. It follows that
Then . Further, take any . Then
By the arbitrariness of xλ, we obtain qc ⩽ qcτqc.
On the other hand, in order to prove τqcτ = τ, by Propositions 2.8 and 4.1, it suffices to show that for each xλ ∈ J (LX). By Lemma 4.2, for each A ∈ LX, we have
as desired. □
Next we give the main concept of this section and study its properties.
Definition 4.5. For an (L, M)-fuzzy topological space (X, τ), define the measure Comp(X, τ) to which (X, τ) is compact by
Proposition 4.6.Let (X, qc) be an (L, M)-fuzzy Q-convergence space. Then
Proof. By Theorem 4.4, we know that . Hence for each , it follows that . Then by Definition 4.5, we have
In general topology, for a surjective and continuous mapping φ : (X, TX) ⟶ (Y, TY) w.r.t. two topological spaces (X, TX) and (Y, TY), the compactness of (X, TX) implies the compactness of (Y, TY). Next we give the lattice-valued characterizations of this conclusion.
Proposition 4.7.If φ : (X, τX) ⟶ (Y, τY) is continuous and surjective, then
Proof. By Proposition 4.3, we know the continuity of φ : (X, τX) ⟶ (Y, τY) implies the continuity of φ : (X, qcτX) ⟶ (Y, qcτY). Furthermore, since φ is surjective, it follows that
In order to show the Tychonoff theorem in (L, M)-fuzzy topological spaces, we first give a lemma in category theory.
Lemma 4.8.[22] Let be a pair of adjoint functors between categories A and B. Then the right adjoint preserves products in B.
Next we give the lattice-valued Tychonoff theorem in (L, M)-fuzzy topological spaces.
Theorem 4.9.(Tychonoff theorem) Let {(Xj, τj) ∣ j ∈ J} be a family of (L, M)-fuzzy topological spaces. Then
Proof. By Theorem 4.4, we know is a pair of adjoint functors between (L, M)-FQC and (L, M)-FTop, i.e., is a right adjoint. Then by Lemma 4.8, we know preserves products in (L, M)-FTop. Therefore, it follows that
Measures of compactness in pointwise (L, M)-fuzzy quasi-uniform spaces
Let be the set of all mappings from J (LX) to LX such that xλnotleqslantd (xλ) for all xλ ∈ J (LX). For any , we define:
f ⩽ g if and only if ∀xλ ∈ J (LX) , f (xλ) ⩽ g (xλ) .
(f ∨ g) (xλ) = f (xλ) ∨ g (xλ) .
(f ⋄ g) (xλ) = ⋀ {f (yμ) ∣ yμnotleqslantg (xλ)} .
Then d0 defined by for all xλ ∈ J (LX) is the smallest element of . We can also prove that and the operations “∨” and “⋄” satisfy the associate law.
Definition 5.1. (Yue and Fang [28]) A pointwise (L, M)-fuzzy quasi-uniformity on X is a mapping such that
;
;
.
For a pointwise (L, M)-fuzzy quasi-uniformity on X, the pair is called a pointwise (L, M)-fuzzy quasi-uniform space. A mapping is called uniformly continuous if , where φ← (f) = φ← ∘ f ∘ φ→.
Theorem 5.2.(Pang [17]) Let be a pointwise (L, M)-fuzzy quasi-uniform space and define as follows:
Then is an (L, M)-fuzzy quasi-coincident neighborhood system on X.
Obviously, is an (L, M)-fuzzy filter. Thus we have the following results.
Theorem 5.3.Let be a pointwise (L, M)-fuzzy quasi-uniform space and define by
Then is an (L, M)-fuzzy Q-convergence structure on X.s
Proof. We need only prove (LFQC1) and (LFQC2).
(LFQC1) Since is an (L, M)-fuzzy quasi-coincident neighborhood system on X, we obtain . Then it follows that
(LFQC2) Take such that . Then
Now we give the definition of compactness measures of a pointwise (L, M)-fuzzy quasi-uniform space as follows.
Definition 5.4. For a pointwise (L, M)-fuzzy quasi-uniform space , define the measure Comp(X, 𝒰) to which is compact by
For a quasi-uniform space (X, U) in the classical case, if (X, U) is compact, then its continuous and surjective image is compact. Next we give the lattice-valued characterization of this conclusion. For this, we give the following two lemmas.
Lemma 5.5.(Pang [17]) If is uniformly continuous, then .
Lemma 5.6.If is uniformly continuous, then is continuous.
Proof. Since is uniformly continuous, by Lemma 5.5, we have for each xλ ∈ J (LX). Then for each , it follows that
This shows the continuity of . □
Proposition 5.7.If is uniformly continuous and surjective, then
Proof. Since is uniformly continuous and surjective, by Lemma 5.6, we know is continuous and surjective. Then
Theorem 5.8.(Pang [17]) Let be a pointwise (L, M)-fuzzy quasi-uniform space and define by
Then is an (L, M)-fuzzy topology on X.
In general topology, a quasi-uniform space is compact if and only if its induced topological space is compact. Now Theorem 5.8 shows a pointwise (L, M)-fuzzy quasi-uniform space can induce an (L, M)-fuzzy topological space . Thus, we will show and have the same measure of compactness. For this, the following lemma is necessary.
Lemma 5.9.(Pang [17]) Let be a pointwise (L, M)-fuzzy quasi-uniform space. Then is a topological (L, M)-fuzzy quasi-coincident neighborhood system on X.
Theorem 5.10.Let be a pointwise (L, M)-fuzzy quasi-uniform space. Then
Proof. By Definitions 4.5 and 5.4, we have
and
In order to prove that , by Lemmas 4.2 and 5.3, it suffices to show that for all xλ ∈ J (LX). For each A ∈ LX, we have
as desired. □
Conclusion
In this paper, we introduce the compactness measures of an (L, M)-fuzzy Q-convergence space, which has nice properties, such as the Tychonoff theorem. We also use this definition to study the compactness of (L, M)-fuzzy topological spaces and pointwise (L, M)-fuzzy quasi-uniform spaces. This paper provides a new way to define the compactness in the framework of (L, M)-fuzzy topological spaces. In the future, we will consider other topological properties of (L, M)-fuzzy topological spaces by using (L, M)-fuzzy Q-convergence structures.
Footnotes
Acknowledgments
The authors would like to express their sincere thanks to the anonymous reviewers and the area editor for their careful reading and constructive comments. This work is supported by the National Natural Science Foundation of China (No. 11501435), the Science and Technology Project of Yulin City (No. 2014CXY-08), the Science and Technology Program of Department of Education in Shaanxi Province (No. 12JK0890).
References
1.
EşH. and
ÇokerD.
, On several types of degree of fuzzy compactness, Fuzzy Sets Syst87 (1997), 349–359.
2.
FangJ.M., Categories isomorphic to L-FTOP, Fuzzy Sets Syst157 (2006), 820–831.
3.
GantnerT.E., SteinlageR.C. and WarrenR.H., Compactness in fuzzy topological spaces, J Math Anal Appl62 (1978), 547–562.
4.
JägerG., A category of L-fuzzy convergence spaces, Quaest Math24 (2001), 501–517.
5.
JägerG., Degrees of compactness in fuzzy convergence spaces, Fuzzy Sets Syst125 (2002), 167–175.