O R -CONVERGENCE AND WEAK O R -CONVERGENCE OF NETS AND THEIR APPLICATIONS ∗

In this paper, the theory of O R -convergence and weak O R -convergence of nets is introduced in L -topological spaces by means of neighborhoods and strong neighborhoods of fuzzy points based on Shi’s O -convergence. It can be used to characterize preclosed sets, preopen sets, δ -closed sets, δ -open sets, near compactness and near S ∗ -compactness.


Introduction
As is known now, the Moore-smith convergence theory plays an important role in general topology, it not only is an significantly basic theory of fuzzy topology and fuzzy analysis but also has wide applications in fuzzy inference and some other aspects.In [18], Pu and Liu introduced the concept of Q-neighborhoods and established a systematic Moore-Smith convergence theory of fuzzy nets in [0,1]-topology.It paved a new way for the study of the fuzzy topology.Wang extended this theory to L-fuzzy set theory in [22].Later on, all kinds of convergence theory were presented [2,3,4,7,8,9,12,14].In [19], Shi introduced the O-convergence theory of nets in terms of neighborhoods of fuzzy points in L-space.It overcomes the difficulty which the neighborhood method meets.
In this paper, our aim is to introduce the theory of O R -convergence and weak O R -convergence of nets in L-spaces based on Shi's O-convergence.We shall discuss its properties and use them to characterize preclosed sets, preopen sets, δ-closed sets, δ-open sets, near compactness and near S *compactness.

Preliminaries
Throughout this paper (L, W , V , 0 ) is a completely distributive de Morgan algebra.X a nonempty set.L X is the set of all L-fuzzy sets (or L-sets for short) on X.The smallest element and the largest element in L X are denoted by 0 and 1.
An element a in L is called prime if a ≥ b ∧ c implies that a ≥ b or a ≥ c.An element a in L is called co-prime if a 0 is a prime element [13].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).The set of nonzero co-prime elements in L X is denoted by M (L X ).Members in M (L X ) are also called points.
The binary relation ≺ in L is defined as follows: for a, b ∈ L, a ≺ b if and only if for every subset D ≤ L, the relation b sup D always implies that the existence of d ∈ D with ad [10].In a completely distributive de Morgan algebra L, each member b is a sup of {a ∈ L | a ≺ b}.In the sense of [15,23] For an L-set G ∈ L X , β(G) denotes the greatest minimal family of G ).An L-topological space (or L-space for short) is a pair (X, T ), where T is a subfamily of L X which contains 0, 1 and is closed for any suprema and finite infima.T is called an L-topology on X.Each member of T is called an open L-set and its quasi-complement is called a closed L-set.Definition 2.1.Let (X, T ) be an L-space.A ∈ L X is called (1) regularly open [1] if A −• = A, the complement of a regularly open set is called regularly closed; (2) β-open [16] if (3) preopen [16] if AA −• , the complement of a preopen set is called preclosed.If A is not only preopen, but also preclosed, then we call it preclopen.Definition 2.5.Let (X, T 1 ) and (Y, T 2 ) be two L-spaces.A map f : (2) completely continuous [5,17] Definition 2.8 ( [19,22]).Let {S(n) | n ∈ D} be a net in (X, T ), x λ ∈ M (L X ).S eventually possesses the property P, if there exists n 0 ∈ D such that ∀n ≥ n 0 , S(n) always possesses the property P. S frequently possesses the property P, if for every n ∈ D, there always exists n 0 ∈ D such that n 0 ≥ n and S(n 0 ) possesses the property P. Definition 2.9 ( [19]).
For the sake of convenience, we introduced the following concept.
It can be proved that Definition 2.15 is equivalent to the notion of δclosure in [11] when L = [0, 1].
Obviously we have the following theorem.

O R -convergence and weak
in this case we also say that S O R -converges to x λ , denoted by for each strongly open neighborhood U of x λ , S is eventually in U −• , in this case, we also say that S weakly O R -converges to x λ , denoted by Theorem 3.4.Let S be a net in (X, T ) and x λ ∈ M (L X ).Then the following conditions are equivalent.
(1) x λ is an O R -cluster point of S. ( . By the hypothesis of (1) S is frequently in (2) ⇒ (3) is obvious.
(3) ⇒ (1) Suppose that the given condition hold for a net S and let . By the hypothesis of (3) S is frequently in Analogous to the proof of Theorem 3.4 we can easily obtain the following result.
Theorem 3.5.Let S be a net in (X, T ) and x λ ∈ M (L X ).Then the following conditions are equivalent.
(1) x λ is an O R -limit point of S. ( For weak O R -convergence, we have same conclusions as Theorem 3.4 and Theorem 3.5.We omit them.Theorem 3.6.Let S be a net in (X, T ) and x λ ∈ M (L X ).Then (1) x λ is a weak O R -cluster point of S if and only if for each strongly δ-open neighborhood U of x λ , S is frequently in U. ( Necessity.Suppose that x λ is a weak O R -cluster point of S and U is a strongly δ-open neighborhood of x λ .Then there exists a regularly open L-set C such that CU and x λ ∈ β(C) since By the hypothesis, S is frequently in CU.
(2) This is analogous to the proof of (1). 2 It is easy to prove the following theorem.
Theorem 3.7.Let S be a net in (X, T ), T a subnet of S and x λ , x μ ∈ M (L X ).Then (1) Theorem 3.8.Let x λ ∈ M (L X ), B be β-open.Then the following conditions are equivalent.
(2) There exists a net S quasi-coinciding with B such that ) There exists a net S quasi-coinciding with B such that x λ is an O R -cluster point of S.
(1) ⇔ (3) is analogous to the proof of (1) ⇔ (2). 2 Corollary 3.11.Let (X, T ) be an L-space and A ∈ L X .Then A is preclopen if one of the following conditions is true.
(1) For any net S quasi-coinciding with Proof.Suppose that the condition (1) is satisfied.By Corollary 3.9 A is preclosed.Now we prove that A is preopen, i.e., AA −• .∀x λ A −• = A − 0 − 0 , there exists a net S quasi-coinciding with A − 0 such that S O R −→ x λ .By the hypothesis of (1) and A − 0 = A 0 • , it follows that x λ 6 ≤ A. This implies that The other cases can achieved from the similar progress.2 Theorem 3.12.Let x λ ∈ M (L X ), B ∈ L X .Then the following conditions are equivalent.
(2) There exists a net S quasi-coinciding with B such that ) There exists a net S quasi-coinciding with B such that x λ is a weak O R -cluster point of S.
Proof.(1) ⇒ (2) Suppose that x λ weak quasi-coincides with cl δ (B).Then for each strongly open neighborhood By the hypothesis of (2), S is eventually in A, which contradicts that S quasi-coincides with A 0 .Thus x λ ∈ β(int δ (A)).It implies that Aint δ (A).By Lemma 2.18 we know that int δ (A)A.Therefore A is δ-open.
(1) ⇔ (3) This proof is analogous to the proof of (1) ⇔ (2). 2 Theorem 3.15.Let f : (X, T 1 ) → (Y, T 2 ) be a R-irresolute L-value Zadeh's type mapping.Then (1) For any net S in L X , if (2) This is analogous to the proof of (1). 2 Theorem 3.16.Let f : (X, T 1 ) → (Y, T 2 ) be an almost continuous L-value Zadeh's type mapping.Then (1) For any net (2) This is analogous to the proof of (1). 2 Theorem 3.17.Let f : (X, T 1 ) → (Y, T 2 ) be a completely continuous L-value Zadeh's type mapping.Then x a 6 ≤ G 0 , there exist Since D is a directed set, there exists n 0 ∈ D such that n 0 ≥ n x i for each ik.Thus we can obtain that ∀n ≥ n 0 , S(n) Then T is a constant b-net quasi-coinciding with G. Let x a be an O Rcluster point of T .It is easy to see that x a is also an O R -cluster point of S.
2 Analogous to the proof of Theorem 4.1 and Theorem 4.2 we can easily obtain the following two results.

Lemma 2 . 21 .
Each regular open L-set is δ-open and each regular closed L-set is δ-closed.

2 Theorem 4 . 2 .
then for each finite subfamily Ψ of Φ, there exists S(Ψ) ∈ M (L X ) with height b such that S(Ψ) 6 ≤ G 0 and S(Ψ) 6 ≤ WΨ −• .Take S = {S(Ψ) | Ψ is a finite subfamily of Φ}, then S is a constant b-net quasi-coinciding with G.By b ∈ β * (a) we can take s ∈ β * (a) such that b ∈ β * (s).Then S has an O R -cluster point x s 6 ≤ G 0 .Hence for each finite subfamily Ψ of Φ we have that x s 6 ≤ W Ψ(because if x s W Ψ, then there exists an A ∈ Ψ such that x s A, i.e., A is an open neighborhood of x s , hence there exists a finite subfamily Ψ 0 of Φ such that Ψ ≤ Ψ 0 and S(Ψ 0 )A −• W Ψ −• W Ψ −• 0 , this contradicts the definition of S), in particular x s 6 ≤ B for each B ∈ Φ.But since Φ is an open Q a -cover of G, we know that there exists B ∈ Φ such that x s B, this yields a contradiction withx s 6 ≤ B. So G is nearly compact.An L-set G is nearly compact in (X, T ) if and only if ∀a ∈ M (L), ∀b ∈ β * (a), each b − -net quasi-coinciding with G has an O R -cluster point x a quasi-coinciding with G. Proof.The sufficiency is obvious, we need only to prove the necessity.Let G be nearly compact, a ∈ M (L), b ∈ β * (a) and {S(n) | n ∈ D} be an b − -net quasi-coinciding with G. Then there exists n 0 ∈ D such that ∀n ≥ n 0 , S(n)b.Put E = {n ∈ D | n ≥ n 0 } and T = {T (n) | n ∈ E, V (T (n)) = b, the support point of T (n) is same as S(n)}.

Theorem 4 . 3 .Theorem 4 . 4 .
An L-set G is near S * -compact in (X, T ) if and only if ∀a ∈ M (L), each constant a-net quasi-coinciding with G has a weak O Rcluster point x a / ∈ β(G 0).An L-set G is near S * -compact in (X, T ) if and only if ∀a ∈ M (L), each a − -net quasi-coinciding with G has a weak O R -cluster point x a / ∈ β(G 0 ).