COVERING PROPERTIES FOR LQ s WITH NESTED BASES

This paper deals with problems on LQ spaces which have a nested base; among others we give conditions so that a space (X,τ1,τ2) admits an LQ which generates τ1 and −1 τ2, and give necessary and sufficient conditions for a space to be quasi-metrizable, related to concrete covering properties. We also give a Stone’s type characterization of pairwise paracompactness for some categories of LQ spaces with nested bases. 2000 Mathematics Subject Classification. Primary 54E15, 54E55, 54E25.

1. Introduction.J. Williams [12] associated a local uniformity with a nested base, to a certain class of regular spaces which fulfil some covering properties.Some years earlier E. Lane [9,Theorem 3.1] gave some similar covering conditions for a pairwise regular bitopological space to define a quasi-metric on the space, but he left in pending a number of relative questions starting with the one referring to the necessity of the conditions.It is our main purpose to reform the Lane's conditions and establish a local quasi-uniformity with a nested base which gives answers to the questions raised by Lane's paper and for which Williams has responded in the uniform case.
The first problem we confront here can be stated as follows: given a bitopological space (X, τ 1 ,τ 2 ), find conditions such that there is a local quasi-uniformity with a nested base which generates the topology τ 1 and its dual generates τ 2 .Theorem 2.5 solves that problem under conditions which may be considered as generalizations of the ones cited in [12,Theorem 2.9] and [9,Theorem 3.1].The suggestion of necessary and sufficient conditions for a space to be quasi-metrizable of such a form as those which Lane asks for in his paper, is our second point and Theorem 3.1 gives an answer.The assumptions we put there, easily satisfy the Kopperman-Fox's demands [7,Theorem 1.1], an alternative approach to the subject (see Remark 3.2).
Stone's type theorem for the pairwise paracompactness works well for some definitions like, for instance, those introduced in [4,11] whilst it does not for some others, as in [1,2,6,10].The local quasi-uniformity with a nested base which is constructed in Theorem 2.5 assures the pairwise paracompactness.

2.
A generalization of William's and Lane's conditions for metrizability and quasimetrizability.Consider a bitopological space (X, τ 1 ,τ 2 ) and a filter of neighbornets on X; we call the filter generalized quasi-uniformity (GQᐁ in brief).We also write LQᐁ for a locally quasi-uniform space and, as always, we symbolize by τ(ᐁ) the topology generated by a quasi-uniformity ᐁ.The basic result in relation with the quasimetrizability of an LQᐁ space remains the theorem of P. Fletcher and W. F. Lindgren [3,Theorem 7.3] and H. P. A. Künzi [8,Theorem 5] which is stated as follows: a space admits a quasi-metric if it admits an LQᐁ ᐁ with a countable base such that ᐁ −1 is as well an LQᐁ.
The following preliminary results are essential.
Lemma 2.1.Let (X, τ 1 ,τ 2 ) be a bitopological space and consider the class Proof.We prove the τ 1 -case only.From the assumptions, Ꮾ 1 is itself a subbase for a GQᐁ ᐂ 1 ; in fact it is τ(ᐂ 1 ) = τ 1 .Moreover, for any Lemma 2.2.A GQᐁ finer than an LQᐁ and generating the same topology with it, is an LQᐁ as well.
Proof.If ᐁ is the LQᐁ and ᐂ the GQᐁ, then given x and We now come to one of our basic results.The conditions we have put may be considered as generalizations of the J. William's and E. P. Lane's respective assumptions in [12, Theorem 2.9] and [9, Theorem 3.1].
For any α ∈ I, β ∈ I, and x ∈ X, put α We form x and show that each of the families (ᐁ α ) α and (ᐂ β ) β forms a nested base for an LQᐁ compatible with τ 1 and τ 2 , respectively.We prove it for the first family.
The family [B] ⊆ A in any case and from Lemma 2.1 the family Γ = {ᐁ α | α ∈ I} is a base for an LQᐁ such that τ(Γ ) = τ 1 .
As always, if we say that bitopological space (X, τ 1 ,τ 2 ) is quasi-metric we mean that there is a quasi-metric d such that the topology induced by d coincides with τ 1 and that induced by d −1 coincides with τ 2 .
The following corollaries are directly concluded from Theorem 2.5.
Corollary 2.6.If a bitopological pairwise regular space satisfies the assumptions of Theorem 2.5 with the only exception that the given families are countable, then the space is quasi-pseudo-metrizable.
Corollary 2.7 (see [9,Theorem 3.1]).Let (X, τ 1 ,τ 2 ) be a bitopological space and (preserving the notation of Theorem 2.5) we suppose that the two classes of the fam- bases for the topologies τ 1 and τ 2 , respectively, and that An immediate consequence of Theorem 2.5 is the following theorem of J. G. Kelly [5,Theorem 2.5].

Theorem 2.8. A bitopological pairwise regular space which fulfils the second axiom of countability is quasi-pseudo-metrizable.
In fact, if ᏼ n and ᏽ n are the countable bases of τ 1 and τ 2 , then the families Ꮽ n = ∪ᏼ n and Ꮾ n = ∪ᏽ n satisfy the assumptions of Theorem 2.5 and the space is quasi-pseudometrizable.
The last theorem of this section may be considered as a generalization of the E. P. Lane's theorem [9,Theorem 3.1] with respect to the number of the elements which every family has, in other words, with respect to the cofinality of these families.Theorem 2.9.Let (X, τ 1 ,τ 2 ) be a pairwise regular bitopological space, {Ꮽ α | α ∈ I} and {Ꮾ β | β ∈ I} be nested collections of τ 1 and τ 2 -open, respectively, families of subsets, each family being τ 1 and at the same time τ 2 locally finite and open bases, respectively.Then the space is quasi-pseudometrizable.
Proof (cf.[12, Proposition 2.10]).From Theorem 2.5 the space admits an LQᐁ ᐁ with a nested base such that ᐁ −1 is also an LQᐁ with a nested base.If I = ω, (ω is the ordinal of the natural numbers), then the result comes from the mentioned Fletcher-Lindgren's theorem.If, on the other hand, cl τ 2 {x} and cl τ 1 {x} are τ 1 and τ 2 -open, respectively, the space (X, τ 1 ,τ 2 ) admits quasi-pseudo-metrics d 1 and d 2 defined as follows: (2.3)So there remains the case card I > ω at least one of cl τ 2 {x}, cl τ 1 {x}, say the first, is not τ 1 -open.We shall derive a contradiction: let {Ꮽ αn | n ∈ ω} be a countable subcollection of {Ꮽ α | α ∈ I} such that, for each n ∈ ω, Ꮽ αn contains a τ 1 -neighborhood of x and Ꮽ a n+1 contains a τ 1 -neighborhood of x which is strictly smaller than any neighborhood of x in Ꮽ an .Such an n exists, because any family where N is the countable intersection of open sets and since card I > ω, N is an open τ 1 -neighborhood of x.Next, we consider a family Ꮽ β in the collection {Ꮽ α | α ∈ I} which contains a τ 1 -subneighborhood B of N. The set B does not belong to any Ꮽ αn and we may assume that Ꮽ αn ⊂ Ꮽ β , for any n ∈ ω.Then, there is an A ∈ Ꮽ a n+1 \ Ꮽ an , where x ∈ A and A ∈ Ꮽ β so that every τ 1 -neighborhood of x meets infinitely many elements of Ꮽ β , a contradiction.

The necessity of Theorem 2.5 assumptions.
The theorem which is featured in this section answers the question raised by E. P. Lane [9, page 248], whether there are for a bitopological space sufficient and necessary conditions referring to special coverings, at the end the space to be quasi-metric.We give a solution that slightly changes the conditions of Theorem 2.5 into a more convenient expression.
Let (X, τ 1 ,τ 2 ) be again a bitopological space.We put for any x ∈ X and for any We also note by int 1 A (respectively, int 2 A) the interior with respect to d (respectively, to d −1 ) of any A ⊆ X.Finally, we recall that a precise refinement (cf.[3, Section 5.4]) of a cover Before beginning the theorem, some remarks on a T 0 non T 1 totally ordered quasipseudo-metrizable space are necessary.Let (X, d) be such a space.If d (x, y) = 0, then x ∈ cl(y) and vice versa.Moreover, if d(x, y) = 0 and x ∈ B(α, ), then y ∈ B(α, ) as well, because d(α, y) ≤ d(α, x) + d(x, y) < .The space is always ordered.If we suppose the total ordering and consider the subbase of ] ←,x], as x runs through X, B(α, ) is any sphere and x is larger than any point of B(α, ε), then any open subset of the form B(x, ), > 0, contains B(α, ), since it contains all y ∈] ←,x].This means that in this case it is impossible to refine any open covering of X in an effective way, which is a necessary presupposition for the demonstration of a Nagata-Smirnov's-type theorem.
Theorem 3.1.A T 1 topological space (X, τ 1 ,τ 2 ) is quasi-pseudo-metrizable if and only if there are two countable collections (Ꮽ n ) n∈ω and (Ꮾ n ) n∈ω of coverings of X consisting of τ 1 and τ 2 -open subsets, respectively, where (3) For any n and any Proof.For the sufficiency of the statement we follow the demonstration of Theorem 2.5: we construct an LQᐁ ᐁ such that τ(ᐁ) = τ 1 (the construction of a ᐂ such that τ(ᐂ) = τ 2 is similar), and we arrive, just as in Theorem 2.5, at an LQᐁ ᐃ such that τ(ᐃ) = τ 1 and τ(ᐃ −1 ) = τ 2 , as desired.We only define ᐁ: for any x and for any n ∈ ω put n Then the family ᐁ = {U n | n ∈ ω}, where U n = ∪{Λ n x × n x | x ∈ X} is a base for an LQᐁ compatible with τ 1 .More precisely, we show that for any A ∈ ∪Ꮽ n , there is another member B of the family and a We prove the necessity for the family (Ꮽ n ) n∈ω .We suppose that there is a quasipseudo-metric d such that τ(d) = τ 1 and τ(d −1 ) = τ 2 . Let We prove that a subfamily of the family

fulfils the statements (2) and (3). (We have put B instead of cl B.)
There holds: ]} and the statement (4) has been proved.
In fact, it follows that (i)

Some consequences of Theorem 2.5.
The pairwise paracompactness.Since a metrizable space is paracompact, it is a reasonable requirement for a quasi-metrizable space to be pairwise paracompact with respect to any definition of the pairwise paracompactness.Nevertheless, among the relative definitions in M. C. Datta [1], P. Fletcher [2], C. Konstadilaki-Savopoulou and I. L. Reilly [6], T. G. Raghavan [10], S. Romaguera and J. Marín [11] and M. Ganster and I. L. Reilly [4], only the last two satisfy this demand, although all of them coincide with the "paracompactness" in the case where the bitopological spaces are reduced to simple ones.Furthermore, J. Williams [12,Theorem 2.8] demonstrated that locally uniform spaces with nested bases are paracompact.We show that, according to the definitions introduced in [4,11], the pairwise paracompactness is directly derived from quasi-uniformities with a nested base.We will symbolize in the text: [11]-or [4]-pairwise paracompactness, respectively.
For our convenience, we shortly refer to some definitions (cf.mainly in S. Romaguera [11, page 236]).

Junnila's definition of paracompactness.
A regular space X is paracompact if and only if, given a cover Ᏺ of X, there is for any By a pair open cover of a bitopological space (X, ᏼ, ᏽ) we mean a family of pairs The [11]-pairwise paracompactness.A pairwise regular space (X, ᏼ, ᏽ) is pairwise paracompact if and only if given a pair cover (Ᏻ, Ᏼ), there is for every x a sequence x ∈ V n [y], (ii) for that x, there is an n ∈ ω and a pair The [4]-pairwise paracompactness.A pairwise regular space (X, ᏼ, ᏽ) is δpairwise paracompact if every ᏼ or ᏽ-open cover of X has a ᏼ ∨ ᏽ (the supremum of ᏼ and ᏽ) locally finite refinement.
We firstly give (Theorems 4.4, 4.5, and 4.6) conditions under which we may construct on a space quasi-uniformities with nested bases.Definition 4.1.A quasi-uniformity (X, ᐁ) enjoys the neighborhood property if for any x ∈ X and any U ∈ ᐁ, there is a Proposition 4.3.If ᐁ is an LQᐁ, then ᐁ 2 is also an LQᐁ, which generates the same topology as ᐁ.
). If, on the other hand, W ∈ ᐁ 2 and x ∈ X, then there is a and τ(ᐁ 2 ) ⊆ τ(ᐁ).Moreover, evidently, ᐁ 2 is LQᐁ.Proof.After Propositions 4.2 and 4.3, we may suppose that ᐁ is an LQᐁ with a nested base Ꮽ which has the neighborhood property.Let U be a neighborhood of the diagonal and Ꮾ = {U ∩ V : V ∈ Ꮽ}.Ꮾ is a nested class of neighborhoods of the diagonal which generates a GQᐁ finer than ᐁ.Hence, by Lemma 2.2, Ꮾ is a base for a quasi-uniformity ᐃ; furthermore, ᐃ fulfils the neighborhood property and induces on X a topology equivalent to that induced by ᐁ.
Since U ∈ ᐃ, it implies that for any x ∈ X there is a where W is a neighborhood of the diagonal.We will show that Theorem 4.5.If in a bitopological space (X, τ 1 ,τ 2 ), where both topologies induce quasi-uniformities with nested bases which have the same cofinality ℵ, a family Ꮽ of τ 2 × τ 1 -neighborhoods of the diagonal has cardinality less than ℵ, then ∩Ꮽ is a neighborhood of the diagonal.
Proof.It is known (cf.[12,Theorem 2.4]) that in a uniform (as well as in a quasiuniform) space with a nested base of cofinality ℵ, any collection of open sets and of cardinality less than ℵ, has as intersection an open set.If Ꮽ is the collection and x ∈ X, then for any A ∈ Ꮽ, there are : A ∈ Ꮽ} and Λ = ∩{Λ A : A ∈ Ꮽ}, then from the above statement and the fact that K and Λ are τ 1 and τ 2 -neighborhoods of x, it implies that Λ × K ⊆ ∩Ꮽ.Hence ∩Ꮽ is a τ 2 × τ 1 -neighborhood of the diagonal.Theorem 4.6.If ᐁ and ᐁ −1 are LQᐁs with nested bases and are of the same cofinality, then there is a quasi-uniformity with a nested base which generates the same, as the ᐁ, topology and has the same cofinality.
Proof.If ℵ is the common cofinality of ᐁ and ᐁ −1 , and W λ (λ ∈ ℵ) is any neighborhood of the diagonal, then by Theorem 4.4 there is a neighborhood U λ+1 of the diagonal such that U λ+1 •U λ+1 ⊆ U λ .If λ is a limit ordinal number less than ℵ and each U α , for α < λ, has been chosen, then put U λ = ∩{U α : α < α}, U λ is by Theorem 4.5 a neighborhood of the diagonal.The rest are trivial.
Proof.After Theorem 4.6, there is a quasi-uniformity ᐃ whose dual ᐃ −1 is also quasi-uniformity, both of them have nested bases, they may be extended (Theorem 4.6) until they reach the same cofinality ℵ and, finally, they generate topologies on X equivalent to τ(ᐁ) and τ(ᐁ −1 ), respectively.We may also assume that the uniformity ᐃ * has a nested base with cofinality ℵ.
Let the cofinality of ℵ be larger than ω and (Ᏻ, Ᏼ) be a (τ(ᐃ), τ(ᐃ for every n ∈ ω.These two sequences fulfill the requirements of the S. Romaguera [11]-definition of pairwise paracompactness and the proof is complete.Remark 4.9.(1) After Theorems 4.7 and 4.8 it is evident that every bitopological space which satisfies the assumptions of Theorem 2.5 is δ-pairwise paracompact as well as [11]-pairwise paracompact.
(2) The quasi-metrizability is equivalent (according to S. Romaguera and J. Marín [11,Theorem 1]) to the facts of being the space [11]-pairwise paracompact plus of being pairwise developable.The latter property is evident under the assumptions of Theorem 2.5.On the other hand, it is worth seeing that in [11] the authors are not concerned with the case of the cofinality being larger than ω, that is, with the case of the space not being quasi-metrizable.In fact, the pairwise development demands the existence of a sequence of pair open covers the space.

Definition 2 . 3 .Lemma 2 . 4 .
After J. Williams[12, Definition 2.3], we give the following definition.We call cofinality of a GQᐁ the least cardinal κ for which the given GQᐁ has a base of cardinality κ.If two families {Ꮽ α | α ∈ κ} and {Ꮾ β | β ∈ κ} of subsets have the same cardinality κ and are nested, then the family {
Thus the assumptions of Kopperman-Fox's theorem [7, Theorem 1.1] are fulfilled and at the same time give an answer to R. D. Kopperman's question [7, page 106, Question c].

Theorem 4 . 4 .
If ᐁ and ᐁ −1 are LQᐁs on X with nested bases, then the set of the diagonal neighborhoods generates a quasi-uniform topology equivalent to τ(ᐁ).