REGULAR L-FUZZY TOPOLOGICAL SPACES AND THEIR TOPOLOGICAL MODIFICATIONS

For L a continuous lattice with its Scott topology, the functor ιL makes every regular L-topological space into a regular space and so does the functor ωL the other way around. This has previously been known to hold in the restrictive class of the socalled weakly induced spaces. The concepts of H-Lindelöfness (á la Hutton compactness) is introduced and characterized in terms of certain filters. Regular H-Lindelöf spaces are shown to be normal.


Introduction.
The two functors that provide a working link between the category TOP(L) of L-(fuzzy)-topological spaces and TOP are the Lowen functors ι L and ω L .For a wide class of lattices L's, ι L is a right adjoint and left inverse of ω L .Therefore, it is of interest to know how various L-topological invariants behave with respect to these functors.
In this paper, we show that when L is a continuous lattice with its Scott topology then ι L maps the category Reg(L) of L-regular spaces onto the category Reg of regular spaces.This improves upon and extends a result of Liu and Luo [6] which showed (with different but equivalent terminology) that ι L maps weakly induced L-regular spaces to regular spaces (with L a completely distributive lattice with its upper topology).As a consequence, we have that ω L (Reg) consists precisely of L-regular spaces of ω L (TOP).Some generalities about L-regular spaces are included and stated in a slightly more general situation, viz.for L-topologies that admit a certain type of approximating relation.This captures complete L-regularity and zero-dimensionality.
We also introduce the concept of H-Lindelöfness (compatible with compactness in the sense of Hutton [2]) and characterize it in terms of closed filters.Finally, we prove that H-Lindelöf and L-regular spaces are L-normal.
2. Notation and some terminology.All the fuzzy topological concepts that concern us are standard.We nevertheless recall some of them.
Let L = (L, ) be a complete lattice (bottom denoted 0) endowed with an orderreversing involution .Thus L satisfies the de Morgan laws.For X a set, L X is the set of all maps from X to L (called L-sets).Then (L X , ) is a complete lattice under pointwisely defined ordering and the order-reversing involution.The de Morgan laws are also inherited by L X .An L-topology on X is a family of elements of L X (called open L-sets) such that any supremum and any finite infimum of open L-sets are open.The L-topology of an L-topological space (L-ts Ifis a family of L-topologies on X, then the supremum L-topologyis generated by -.In particular, j∈J π ← j (o(X j )) is the product L-topology on j∈J X j (π j being the jth projection).The set of all restrictions Given α, β ∈ L we let α β whenever for any B ⊂ L with β ≤ B there is a finite α} for every α ∈ L. We write α = {β ∈ L : β α} and dually for α.Each continuous L has the interpolation property: α β implies α γ β for some γ ∈ L. The Scott topology σ (L) on a continuous L is one which has { α : α ∈ L} as a base.We write ΣL for (L, σ (L)) (see [1] for details).
We also recall that L is a frame provided α ∧ B = {α ∧ β : β ∈ B} for every α ∈ L and B ⊂ L.
Given a ∈ L X and α ∈ L, we let [a α] = {x ∈ X : a(x) α}, [a α] = {x ∈ X : a(x) α}, etc.The constant member of L X with value α is denoted α as well, and α1 A = α ∧ 1 A , where 1 A is the characteristic function of A ⊂ X.If Ꮽ ⊂ L X , we let Ꮽ = {a : a ∈ Ꮽ}, Ꮽ = {a : a ∈ Ꮽ}, and similarly for Int Ꮽ.We include for record.
3. L-topologies with approximating relation.Let L = (L, ) be a complete lattice.An L-ts X is called L-regular [3] if for every u ∈ o(X) there exists ᐂ ⊂ o(X) such that u = ᐂ and v ≤ u for all v ∈ ᐂ.This is the case if and only if u = ᐂ = ᐂ.
It is clear that X is L-regular if and only if for every basic open u one has To avoid repetitions of some argument used in [5], we introduced an auxiliary relation ≺ on the L-topology o(X) of an L-ts X. Definition 3.1.Let ≺ be a binary relation on o(X) satisfying the following conditions for all u, v, w 1 ,w 2 ∈ o(X): (1) 0 ≺ u; We say X is ≺-regular if for each open u there exists ᐂ ⊂ o(X) such that u = ᐂ and v ≺ u for all v ∈ ᐂ.

Examples.
(1) X is L-regular if and only if it is ≺-regular with v ≺ u defined by v ≤ u.
(2) X is completely L-regular [3] if and only if it is ≺-regular, where v ≺ u if and only [5] for details and notice that ( 4) and ( 5) of Definition 3.1 require L to be meet-continuous (cf.Section 5).
(3) X is zero-dimensional if and only if it is ≺-regular and v ≺ u, whenever v ≤ w ≤ u for some closed and open w (cf.[9]).

Proposition 3.2. Let L be a complete lattice and let X be any of ≺-regular spaces of Example 3. The following hold
(1 (5) ≺-regularity is preserved by arbitrary products.
Proof.The argument given in [5, Remark 2.5 and Lemma 2.3] for the case (2) of Example 3 goes unchanged in the remaining cases.Proposition 3.3.Let L be a continuous lattice.For X an L-topological space, the following are equivalent: (1) X is ≺-regular.
(2) With L a complete chain without elements isolated from below (e.g., with L = [0, 1]), conditions (3) and ( 4) coincide.When expressed in terms of fuzzy points (these are L-sets of the form α1 {x} ) and with v ≺ u if and only if v ≤ u, these conditions become the definitions of fuzzy regularity given by numerous authors, e.g., [10], thereby showing that all those definitions are equivalent to the one of Hutton-Reilly [3].
(3) For L a frame, the open L-set u in conditions ( 3) and ( 4) of Proposition 3.3 can be assumed to be in any family that generates the L-topology (on account of Proposition 3.2(3)); cf.[8,Lemma 3(iii)].Now we show that the regularity axiom of Liu and Luo [6] is equivalent to the Lregularity for any complete L in which primes are order generating.We recall that p ∈ L is called prime whenever α∧β ≤ p implies α ≤ p or β ≤ p.The set of all primes is order generating if α = {p ≥ α : p is prime} for every α ∈ L. The dual concept is that of a coprime element.In our case, i.e., in (L, ), an element q ∈ L is coprime if and only if q is prime.We have the following.Remark 3.5.Let L be a complete lattice in which primes are order generating.For X an L-ts, the following are equivalent: (1) X is L-regular.
(2) (Liu and Luo [6]) for every x ∈ X, coprime q, and k ∈ κ(X), whenever k(x) q, there exists h ∈ κ(X) such that h(x) q and k ≤ Int h.Now, for X an L-topological space, let ι ΣL X be the topological space with X as the underlying set and with the weak topology generated by o(X) and ΣL, i.e., ι ΣL X has {u ← (σ (L)) : u ∈ o(X)} as a topology.It is called the topological modification of X.
We have o(X) ⊂ o(ω ΣL ι ΣL X) and ι ΣL ω ΣL = id TOP .Hence ω ΣL is an injection.We also recall that if Y is a topological space, then χY denotes the set Y endowed with the L-topology Sometimes it may be more convenient to write (X, ω ΣL (T )) for the space topologically generated from (X, T ), and similarly for ι ΣL .Lemma 4.1.Let L be a continuous lattice.For every L-regular space X, ι ΣL X is a regular topological space.
Now it suffices to note that, by Remark 2.1, Now it is more convenient to write (X, T ) for an L-ts X with the L-topology T .In [6], (X, T ) is said to be weakly induced if 1 [u α] ∈ T for every u ∈ T and α ∈ L.
In what follows, we write "L-regular" on account of Remark 3.5.Corollary 4.2 [6].Let L be completely distributive.If (X, T ) is a weakly induced L-regular space, then (X, [T ]) is regular.
Proof.First, recall that a completely distributive L is continuous and the sets {β ∈ L : β α} (α ∈ L) form a subbase for its Scott topology (see [1, e.g Proof. (1) That ι ΣL maps Reg(L) into Reg is stated in Lemma 4.1.The mapping is onto since for any topological regular X, χX is L-regular and ι ΣL χX = X.
(2) If X is a regular topological space and u is open in ω ΣL X, then for every α ∈ L there is a family ᐃ α of open subsets of X such that By Remark 2.1 and the first equality of (4.5), we obtain (4.6) (Note that there is no distributivity used in arriving at the third equality: always α ∧ B = {α ∧ β : β ∈ B} provided B ⊂ {0, 1} as is the case above).Since α1 W ≤ α1 W , the same argument shows, by using the second equality of (4.5), that we actually have Remark 4.4.
(1) Let L be a continuous frame (then it becomes completely distributive on account of the order reversing involution; cf.[1, Chapter I, Theorem 3.15]).Then the inclusion ω ΣL (Reg) ⊂ Reg(L) obviously follows from Proposition 3.2(4).Indeed, for X a regular space, the L-topology of ω ΣL X is the supremum of two L-regular L-topologies: the one of χX and the one consisting of all constant L-sets (cf.[5, Proposition 1.5.1(7)]).
(2) The equality (4.4) of Theorem 4.3 is available in [12] with L = [0, 1] and in [6] with L completely distributive.Theorem 4.3 is also a supplement to the discussion about regularity in fuzzy topology given in [7].
(3) We recall that an L-ts X is an L-T 3 space if and only if it is L-regular and points of X can be separated by open L-sets.By [5,Remark 8.4], we obtain: ι ΣL (L-T 3 ) = T 3 and ω ΣL (T 3 ) = L-T 3 ∩ ω ΣL (TOP).
We close this section with some remarks about maximal L-regular spaces.Following [11], we say that X is maximal L-regular if the only L-regular L-topology on the set X which is stronger than the original one is L X (the discrete L-topology).Proposition 4.5.Let L be a continuous lattice.Every maximal L-regular space with a nondiscrete topological modification is topologically generated (from a maximal regular space).

H-Lindelöfness.
An L-ts X is called H-Lindelöf if for every k ∈ κ(X), whenever k ≤ ᐁ with ᐁ ⊂ o(X), there exists a countable subfamily ᐁ 0 ⊂ ᐁ such that k ≤ ᐁ 0 .If ᐁ 0 is finite, then X is called H-compact [2].It is clear that H-Lindelöfness is pre- served under continuous surjections.Also, the characterizations of H-compactness in terms of certain filters have their counterparts for H-Lindelöf spaces.Definition 5.1 (cf.[2]).Let Ᏺ ⊂ L X be nonempty and let a ∈ L X .We say that: (1) Ᏺ has the countable intersection property relative to a if Ᏺ 0 a for every countable Ᏺ 0 ⊂ Ᏺ, (2) Ᏺ is a filter if it is closed under finite infima and such that if Theorem 5.2.Let L be a complete lattice and let X be an L-ts.The following are equivalent: (1) X is H-Lindelöf.
(2) Every family ⊂ κ(X) with the countable intersection property relative to an open u satisfies u.
(3) Every closed filter with the countable intersection property relative to an open u satisfies u.
Proof.(1) ⇒( 2).Assume ≤ u.Then u ≤ and there is a countable Ꮿ ⊂ such that u ≤ Ꮿ, a contradiction with the countable intersection property of .
(3) ⇒( 1).Let k ≤ ᐁ.Assume that ᐁ does not have a countable subfamily which covers k.Let ᐁ be the closed filter generated by ᐁ , i.e., let We claim that ᐁ has the countable intersection property relative to k .Suppose that this is not the case.Then for some countable Ᏺ ⊂ ᐁ one has Ᏺ ≤ k .Thus and f ∈Ᏺ Ꮿ f is a countable subfamily of ᐁ, a contradiction with our assumption about ᐁ.Therefore ᐁ has the countable intersection property relative to k , i.e., ᐁ k .Hence k ᐁ and since ᐁ ≤ ᐁ , we conclude that k ᐁ.This contradiction completes the proof.

Remark 5.3.
There is no counterpart of Theorem 4.3 for H-Lindelöfness and Lindelöfness: (1) The set as each open cover of a nonzero closed L-set must contain 1 X ), while ι ΣL X is an uncountable discrete space.
(2) An L-ts topologically generated from a Lindelöf space need not be H-Lindelöf.Indeed, let X be an uncountable Lindelöf topological space.Put L = ᏼ(X) with usual complement as its order-reversing involution (note that ᏼ(X) is a continuous lattice).Then the cover of 1 X consisting of all constant L-sets having values {x} with x ∈ X (these are all open in ω ΣL X) does not have a countable subcover.Therefore ω ΣL X fails to be H-Lindelöf.
(3) However, if ω ΣL X is H-Lindelöf, then X is Lindelöf.Indeed, χX carries a weaker L-topology than ω ΣL X, so that χX is H-Lindelöf, and the latter is equivalent to the statement that X is a Lindelöf space.
(4) All the above discussion applies unchanged to the case of H-compactness and compactness.
It is clear that for any complete L, every H-compact and L-regular space X is Lnormal, i.e., whenever k ≤ u (k is closed and u is open), there exists an open v with k ≤ v ≤ v ≤ u [3].In what follows we show that H-compactness can be replaced by H-Lindelöfness provided L is meet-continuous, i.e., for every α ∈ L and every directed subset Ᏸ ⊂ L there holds: α ∧ Ᏸ = {α ∧ δ : δ ∈ Ᏸ}.We recall that every continuous L is meet-continuous [1].Also, on account of the order-reversing involution, the dual law is valid too.Theorem 5.4.Let L be a meet-continuous lattice.Then every L-regular and H-Lindelöf space is L-normal.
Proof.Let k be closed, u be open, and k ≤ u in an L-regular H-Lindelöf space X.By L-regularity there exist ᐁ ⊂ o(X) and ⊂ κ(X) such that u = ᐁ = ᐁ and k = = Int (the latter on account of the de Morgan laws).By H-Lindelöfness, there exist two countable subfamilies ᐁ 0 ⊂ ᐁ and 0 ⊂ such that k ≤ ᐁ 0 and (again by the de Morgan laws) 0 ≤ u.Thus k ≤ ᐁ 0 ≤ ᐁ 0 and k ≤ Int 0 ≤ 0 ≤ u. (5. 3) The rest of the proof is exactly the same as that of [5,Theorem 9.11] which shows that second countability plus L-regularity implies L-normality.Note that the proof in [5] uses a result holding for L a meet-continuous lattice.

Proof of Remark 3 . 5 ( 2 ). 4 .
Observe that condition (4) of Proposition 3.3 (cf.also Remark 3.4(1)) can be written as follows (with v ≺ u if and only if v ≤ u) : [u p] = v≤u [v p] for every open u and each prime p.And this is just the dual form of (2).The topological modifications of L-regular spaces.The main topic of this paper requires the lattice L to carry a topology such that C(Y , L) is an L-topology for every topological space Y .Among examples of such lattices are the continuous lattices with their Scott topologies.If L is a continuous lattice, then ΣL is a topological lattice (see [1, Chapter II, Corollary 4.16, Proposition 4.17]).The family [Y , ΣL] of all continuous functions from a topological space Y to ΣL is, therefore, closed under finite suprema and finite infima (both formed in L Y ).However, by using the interpolation property of the relation , for every α ∈ L and ᐁ ⊂ [Y , ΣL] one has [ ᐁ α] = {[ ᐂ α] : ᐂ ⊂ ᐁ is finite}, an open subset of Y .Thus [Y , ΣL] is an L-topology on the set Y .For every topological space Y , ω ΣL Y denotes the set Y provided with the L-topology [Y, ΣL].One then says that ω ΣL Y is topologically generated from Y .

Proof.
It suffices to show that every point of an arbitrary subbasic open set of ι ΣL Y has an open neighborhood whose closure is in the set (this is Proposition 3.2(3) with L = {0, 1}).So, let u be open in X, α ∈ L, and let x ∈ [u α].By Proposition 3.3(3) there is an open ., Chapter IV, Exercise 2.31 and Chapter III, Exercise 3.23]).Thus (X, T ) is weakly induced if and only if ι ΣL (T ) ⊂ [T ].Finally, notice that [T ] ⊂ ι ΣL (T ) always since [1 U α] ∈ { , U, X} for every α ∈ L. Let L be a continuous lattice.Then the following hold: