FUZZY PARA - LINDELOF SPACES

In this paper we introduce the concept of Para-Lindelof spaces in L-topological spaces by means of locally countable families of L-fuzzy sets. Further some characterizations of fuzzy para-Lindelofness and flintily para-Lindelofness in the weakly induced L-topological spaces are also obtained. More over the behavior of fuzzy para-Lindelof spaces under various types of maps such as fuzzy closed maps, fuzzy perfect maps are also investigated.


Introduction
As a generalization of a set, the concept of fuzzy set was introduced by Zadeh [18].Fuzzy topology comes as the generalization of general topology using the concept of a fuzzy set.In 1968 Chang [6] introduced the concept of fuzzy topology and Lowen [12] introduced a more natural definition of fuzzy topology.
Compactness and metrizability are the heart and soul of general topology.In 1944 J. Dieudonne [7] defined paracompactness as a natural generalization of compactness.Later several other covering properties such as meta-compactness, sub para-compactness, sub meta-compactness, para-Lindelofness etc. have naturally evolved from para compactness.The concept of para-Lindelof spaces was introduced by J. Greever [9] in 1968 and further studies were conducted by Burke ([4,5]), Fleissner-Reed [8].
The concept of paracompactness in fuzzy topology was introduced by Luo [13].Authors have introduced the concept and studied some properties regarding metacompactness, subparacompactness, and submetacompactness in L-topological spaces in [14], [3], [2] respectively.In this paper we define locally countable families and introduce the concept of para-Lindelof spaces in L-topological spaces.Besides getting some characterization for para-Lindelof and flintily para-Lindelof in the weakly induced L-topological spaces, it is also seen that these properties are closed hereditary.Further the invariance of these properties under perfect maps is also proved.
A molecule or co-prime element in a lattice L is a join irreducible element in L and the set of all non zero co-prime elements of L is denoted by M (L) and prime elements by pr(L).A complete lattice L is completely distributive if it satisfies either of the logically equivalent CD1 or CD2 below: CD1: The constant member of L X with value α is denoted by α itself.Usually we will not distinguish between a crisp set and its characteristic function.Wang [15] proved that a complete lattice is completely distributive if and only if for each α ∈ L, there exists B ⊆ L such that (i) a = ∨A and (ii) if A ⊆ L and a ≤ ∨B, then for each b ∈ B, there exists c ∈ A such that b ≤ c.B is called the minimal set of a and β(a) denote the union of all minimal sets of a. Again β * (a) = β(a) ∩ M (L).Clearly β(a) and β * (a) are minimal sets of a.
For α ∈ L and A ∈ L X , we use the following notations.
Clearly L X has a quasi complementation 0 defined point-wisely α 0 (x) = α(x) 0 for all α ∈ L and x ∈ X.Thus the DeMorgan laws are inherited by (L X , 0 ).
Let (L, 0 ) be a complete lattice equipped with an order reversing involution and X be any non empty set.A subfamily τ ⊂ L X which is closed under the formation of sups and finite infs (both formed in L X ) is called an L-topology on X and its members are called open L-sets.The pair (X, τ ) is called an L-topological space (L-ts).The category of all L-topological spaces, together with L-continuous mappings and the composition and identities of set is denoted by L-Top.Quasi complements of open L-sets are called closed L-sets.
We know that the set of all non zero co-prime elements in a completely distributive lattice is ∨-generating.Moreover for a continuous lattice L and a topological space (X, T ), T = i L ω L (T ) is not true in general.By proposition 3.5 in Kubiak [11] we know that one sufficient condition for In [16] Wang extended the Lowen functor ω for completely distributive lattices as follows: For a topological space (X, T ), (X, ω(T )) is called the induced space of (X, T ) where In 1992 Kubiak also extended the Lowen functor ω L for a complete lattice L. In fact when L is completely distributive, ω L = ω.
An L-topological space (X, τ ) is called weakly induced space if ∀α ∈ M (L), ∀A ∈ τ it is true that A (α 0 ) ∈ [τ ] where [τ ] is the set of all crisp open sets in τ .
Based on these facts, in this paper we use a complete, completely distributive lattice L in L X .For a standardized basic fixed-basis terminology, we follow Hohle and Rodabaugh [10].

Definition
[17] Let (X, τ ) be an L-ts.A fuzzy point x α is quasi coincident with D ∈ L X (and write x α ≺ D) if x α 6 ≤ D 0 .Also D quasi coincides with E at x (D q E at x) if D(x) 6 ≤ E 0 (x).We say D quasi coincident with E and write D q E if D q E at x for some x ∈ X.Further D¬ q E means D not quasi coincides with E. We say

Definition
A is called locally finite (*-locally finite) for short, if A is locally finite(*-locally finite) at every molecule x λ ∈ M (L X ).
at most finitely many t ∈ T .And A is *-point finite at x λ if there exists at most finitely many t ∈ T such that The previous notions "locally countable family" is defined for L-ts.They can be also defined for L-subsets: A is called σ-locally countable in B if A is the countable union of sub families which are locally countable in B. A is called σ-locally countable for short, if A is σ-locally countable in >.

Definition
[17] Let (X, τ ) be an L-ts.Then by [τ ] we denote the family of support sets of all crisp subsets in τ .(X, [τ ]) is a topology and it is the background space.(X, τ ) is weakly induced if U ∈ τ is a lower semi continuous function from the background space (X, [τ ]) to L.

Definition
[17] For a property P of ordinary topological space, a property P * of L-ts is called a good L-extension of P , if for every ordinary topological space (X, T ), (X, T ) has the property P if and only if (X, ω L (T )) has property P * .In particular when L = [0, 1] we say P * is a good extension of P .Where ω L (T ) is the family of all lower semi continuous function from (X, T ) to L.

Definition
[17] A collection A refines a collection B(A < B) if for every A ∈ A, there exists B ∈ B such that A ≤ B.

Proposition
[17] Let (X, τ ) be an L-ts.A ⊂ L X is closure preserving.Then for every sub family

Theorem
Every locally countable family of subsets is closure preserving.
for every t ∈ T \ T 0 where T 0 is a countable subset of T .This implies that A t ≤ U 0 for every t ∈ T \ T 0 .If x α 6 ≤ ∨(clA 0 ), then x α 6 ≤ cl A t for every t ∈ T 0 and hence there exist 2.17.Definition [14] A collection U of fuzzy subsets of an L-topological space (X, τ ) is said to be well monotone if the subset relation '<' is a well order on U.

Definition
[14] A collection U of fuzzy subsets of an L-topological space (X, τ ) is said to be directed if U, V ∈ U implies there exists W ∈ U such that U ∨V < W.

Definition
Let (X, τ ) be an L-ts, A ∈ L X , B ⊂ L X .Then st(A, B) = ∨{B ∈ B : B q A} is defined as the star of B about A. If x λ ∈ M (L X ), then st({x λ }, B) is denoted by st(x λ , B).

Definition
Let (X, τ ) be an L-ts.A = {A t : t ∈ T } ⊆ L X is a interior preserving collection if for every subfamily

Para-Lindelof Spaces
there exist an open refinement Ψ of Φ which is σ-locally countable in X and also an α-Q-cover of X.
Para-Lindelof and * -Para-Lindelof are hereditary with respect to closed subsets.

Theorem
Let (X, τ ) be an

Proof.
We need to prove only (i).Suppose that A similar theorem holds for α * -para-Lindelof and * -para-Lindelof spaces also.

Theorem
Let (X, τ ) be a weakly induced L-ts.Then the following conditions are equivalent By the weakly induced property of (X, τ ), O ∈ [τ ].For every V ∈ V, if O ∩ V (α 0 ) 6 = φ, then there exist an ordinary point y ∈ O ∩ V (α 0 ) , and hence B(y) 6 ≤ ⊥, V (y) 6 ≤ α 0 .Therefore V (y) 0 < α and it follows that B(y) 6 ≤ V (y) 0 and thus ) is para-Lindelof, there exist a refinement V of U* which is also a locally countable cover of X.For every is both a refinement of U and an α-Q-cover of >.Now we will prove that W is locally countable.Let x α ∈ M (L X ).Then since V is locally countable, there exist a neighbourhood B of x such that B intersects with V i for countably many V i ∈ V. Now we have χ B ∈ Q(x α ).We will show that χ B q χ V i ∧ U V i for at most countably many i.For if possible χ B q χ V ∧ U V for uncountably many V ∈ V. Then χ B q χ V or χ B q U V for uncountably many V ∈ V.In both cases B intersects with V for uncountably many V ∈ V, which is a contradiction and hence W is locally countable.Therefore (X, τ ) is α-para-Lindelof.This completes the proof. 2

Theorem
Let (X, τ ) be a weakly induced L-ts.Then the following conditions are equivalent . By the weakly induced property of (X, τ ), ) is para-Lindelof, there exist a locally countable and open refinement V of U* which is also a cover of X.For every uncountably many V ∈ V.And hence χ V q B or χ UV (⊥) q B for uncountably many V ∈ V.In both cases V intersects with the neighbourhood of x for uncountably many V ∈ V which is a contradiction that V is locally countable.Hence W is * -locally countable and this completes the proof. 2 3.9.Theorem Let (X, τ ) be an L-ts.Then the following are equivalent (i) (X, τ ) is para-Lindelof; (ii) For every open α-Q-cover A of (X, τ ), there is a locally countable refinement B such that if be a locally countable refinement as given in (ii).Let C be an open α-Q-cover of > such that every element of C intersects at most countably many elements of B. Then for every x α ∈ M (L X ), there is a locally countable refinement D of C such that x α ∈ int(st(x α , D)).
For each B ∈ B, take ) such that W intersects only countably many elements of D. Now since each D ∈ D intersects only countably many elements of B, it follows that W intersects only countably many elements of {st(B, D) : B ∈ B}.Hence G is locally countable and the theorem is proved.2 Similar to Theorem 3.9 we can prove the following result:

Theorem
Let (X, τ ) be an L-ts.Then the following are equivalent (i) (X, τ ) is σ-para-Lindelof; (ii) For any open α-Q-cover A of (X, τ ), there is a σ-locally countable refinement B = ∪B i such that if x α ∈ M (L X ) then x α ∈ int(st(x α , B k )) for some k ∈ N.
4. Flintily Para-Lindelof Spaces and a countable subset T 0 of T such that t ∈ T \ T 0 ⇒ A t ¬ q U.And A is called flintily locally countable in A, if A is flintily locally countable at every molecule x λ ∈ M (↓ A).A is called flintily locally countable for short, if A is flintily locally countable in >.

Theorem
In L-ts the following implications hold Flintily local countable ⇒ * -local countable ⇒ local countable

Proposition
Let (X, τ ) be an L-ts, {A t :

Theorem
Let (X, τ ) be an L-ts, A ∈ L X , A = {A t : t ∈ T } ⊆ L X .If A is flintily locally countable in A, then clA is flintily locally countable in A.

Remark
Clearly flintily local countability is strictly stronger than * -local countability.But in weakly ⊥-induced L-ts they are coincident with each other.

Theorem
Let (X, τ ) be a weakly ⊥-induced L-ts, A ∈ L X , A = {A t : t ∈ T } ⊆ L X .Then A is flintily locally countable in A, if and only if A is * -locally countable in A.

Proof.
By Theorem 4.2, it is enough to prove that * -local countability implies flinty local countability.Suppose A is * -local countable in A. Let x λ ∈ M (↓ A).Then there exist U ∈ Q(x λ ) and a countable subset That is to say χ there exist an open refinement Ψ of Φ which is flintily locally countable in A and Ψ is also an α-Q-cover of A. A is called flintily para-Lindelof if A is flintily α-para-Lindelof for every α ∈ M (L).And (X, τ ) is flintily para-Lindelof if > is flintily para-Lindelof.By Theorem 4.2, the following implications hold: Similar to Theorem 3.5 we can prove that flintily para-Lindelofness is hereditary with respect to closed subsets.

Theorem
In a weakly induced L-ts (X, τ ), the following are equivalent (i) (X, τ ) is flintily para-Lindelof.(ii) There exist α ∈ M (L) such that (X, τ ) is flintily α-para-Lindelof; (ii) it has an open and flintily locally countable refinement Ψ = {A t : t ∈ T } such that Ψ is an α-Q-cover of >.For every t ∈ T , take V t = A t(α 0 ) and V = {V t : t ∈ T }.Then by the weakly induced property of (X, τ ), V is an open cover of (X, [τ ]).Now we will prove V is a locally countable refinement of U.
Let x ∈ X.Since Ψ is flintily locally countable, there exist B ∈ Q(x α )∩ crs(τ ) such that A t q B for only a countable number of members A t s in ψ.Since B ∈ Q(x α ) is crisp, B (⊥) is the neighbourhood of x.For every t ∈ T if A t ¬ q B, then V t ∩ B (⊥) = Φ.So B (⊥) intersects with only a countable members of V, thus V is locally countable.Hence (X, [τ ]) is paraLindelof.(iii) ⇒ (i) suppose α ∈ M (L), A = {A t : t ∈ T } is an open α-Q-cover of >.For every t ∈ T take U t = A t(α 0 ) and U = {U t : t ∈ T }.Since A is an open α-Q-cover of > and (X, τ ) is weakly induced, U is an open cover of (X, [τ ]).Therefore by (iii), there exists an open and locally countable refinement V = {V s : s ∈ S} of U which is also a cover of (X, [τ ]).For every s ∈ S take t(s) ∈ T such that V s ⊂ U t(s) , let W s = A t(s) ∧ χγ s then W s is an open L-set and W s ≤ A t(s) for every s ∈ S. Therefore W = {W s : s ∈ S} is an open refinement of A. Now we will show that W is an open α-Q-cover of >.Let x α ∈ M (L X ) take s ∈ S such that x ∈ V s and hence x ∈ U t(s) .So A t(s) (x) 6 ≤ α 0 , α 6 ≤ A t(s) (x) 0 .Since x ∈ V s , χ V s ∈ Q(x α ), we have W s = A t(s) ∧ χ V s ∈ Q(x α ).Hence W is an open α-Q-cover of >.
Suppose x α ∈ M (L X ), then since V being locally countable in (X, [τ ]), there exist a neighbourhood B of x in (X, [τ ]) such that B intersects with only countably many members of V say V s 0 , V s 1 , V s 2 , • • •.Then for every s ∈ S \ {s 0 , s 1 , s 2 . ..},V s ∩ B = Φ, B ⊂ V s 0 and thus χ B ≤ χ V 0 s ≤ A 0 t(s) ∨ χ V 0 s = W s 0 .That is χ B ¬ q Ws.Hence W is flintily locally countable.This completes the proof. 2

Invariant Theorems
In this section we study the behaviour of para-Lindelof spaces under various types of fuzzy mappings.

Definition
[17] Let (X, τ ), (Y, µ) be L-topological spaces, f : X → Y be an ordinary mapping.Based on this we define the L-fuzzy mapping f → : L X → L Y and its L-fuzzy reverse mapping