© Hindawi Publishing Corp.
COVERING PROPERTIES FOR
LQ
ᐁs WITH NESTED BASES
A. ANDRIKOPOULOS and J. N. STABAKIS
(Received 15 August 1997 and in revised form 10 April 2000)
Abstract.This paper deals with problems on LQᐁspaces which have a nested base; among others we give conditions so that a space (X,τ1,τ2) admits anLQᐁ ᐁ which generates τ1 and ᐁ−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 LQᐁspaces 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,τ1,τ2), find conditions such that there is a local quasi-uniformity with a nested base which generates the topologyτ1and its dual generatesτ2.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 andTheorem 3.1gives an an-swer. The assumptions we put there, easily satisfy the Kopperman-Fox’s demands [7, Theorem 1.1], an alternative approach to the subject (seeRemark 3.2).
Stone’s type theorem for the pairwise paracompactness works well for some defini-tions 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 inTheorem 2.5assures the pairwise paracompactness.
2. A generalization of William’s and Lane’s conditions for metrizability and quasi-metrizability. Consider a bitopological space(X,τ1,τ2)and a filter of neighbornets on X; we call the filter generalized quasi-uniformity (GQᐁ in brief). We also write
[3, Theorem 7.3] and H. P. A. Künzi [8, Theorem 5] which is stated as follows: a space admits a quasi-metricif it admits anLQᐁ ᐁwith a countable base such thatᐁ−1is as well anLQᐁ.
The following preliminary results are essential.
Lemma2.1. Let(X,τ1,τ2)be a bitopological space and consider the classᏮ1
(re-spectively,Ꮾ2) ofτ2×τ1-open (respectively,τ1×τ2-open) entourages such that for any
τ1-neighborhood (respectively,τ2-neighborhood)M1ofx(respectively,M2), there are
τ1-neighborhood (respectively,τ2-neighborhood)N1ofx(respectively,N2) andV1∈Ꮾ1
(respectively,V2∈Ꮾ2) such that V1(N1)⊆M1 (respectively,V2(N2)⊆M2). Then Ꮾ1
(respectively,Ꮾ2) is a subbase for anLQᐁwhich generatesτ1(respectively,τ2).
Proof. We prove theτ1-case only. From the assumptions,Ꮾ1is itself a subbase for aGQᐁᐂ1; in fac t it isτ(ᐂ1)=τ1. Moreover, for anyτ1-neighborhoodU[x]ofx, there is anotherτ1-neighborhoodBofx andV,Welements ofᏮ1such thatV [B]⊆U[x] andW [x]⊆B. Thus(W∩V)◦(W∩V)[x]⊆V [W [x]]⊆V [B]⊆U[x]andᏮ1is an
LQᐁ.
Lemma2.2. AGQᐁfiner than anLQᐁand generating the same topology with it, is anLQᐁas well.
Proof. Ifᐁis theLQᐁandᐂtheGQᐁ, then givenxandV∈ᐂthere isU∈ᐁ
such thatU[x]⊆V [x], whilst there is anotherU∗ ∈ᐁ, such that U∗2[x]⊆U[x]. SinceV is finer thanU, there isV∗∈ᐂ such thatV∗⊆U∗, henceV∗2[x]⊆V [x].
After J. Williams [12, Definition 2.3], we give the following definition.
Definition2.3. We callcofinality of aGQᐁthe least cardinalκfor which the given
GQᐁhas a base of cardinalityκ.
Lemma2.4. If two families{Ꮽα|α∈κ}and{Ꮾβ|β∈κ}of subsets have the same
cardinalityκand are nested, then the family{(Aα×Bα)|α∈κ}is nested and cofinal
to the family{Aα×Bβ|α∈κ, β∈κ}. (Evidently, giveni∈κandj∈κ, there is an
α∈κsuch thatAα×Bα⊆Ai×Bj.)
We now come to one of our basic results. The conditions we have put may be con-sidered as generalizations of the J. William’s and E. P. Lane’s respective assumptions in [12, Theorem 2.9] and [9, Theorem 3.1].
Theorem2.5. Let(X,τ1,τ2)be a bitopological pairwise regular space and{Ꮽα|
α∈I},{Ꮾβ|β∈I}be nested classes of families of subsets ofX. For anyαandβinI
and anyᏭ⊆ᏭαandᏮ⊆Ꮾβ, we put
◦
Ꮽ=A◦ =intτ1A|A∈Ꮽ
, Ꮾ◦ =B◦=intτ2B|B∈Ꮾ
. (2.1)
We assume that(1)∪αᏭ◦and∪βᏮ◦ are bases for the topologiesτ1andτ2, respectively. (2)∩αᏭ◦isτ1-open and∩{X\clτ2A|A∈Ꮽ}isτ2-open.(3)∩β
◦
Ꮾisτ2-open and∩{X\ clτ1A|B∈Ꮾ}isτ1-open.
Proof. Suppose that the collections(Ꮽα)αand(Ꮾβ)βcontainXand∅. For any α∈ I, β ∈ I, and x ∈ X, put α
x = ∩{ ◦
A | x ∈ A◦ andA ∈ Ꮽα}, Λα x = ∩{X\clτ2A|x ∈ X\clτ2A, A ∈Ꮽα}. ᏹβx= ∩{
◦
B|x ∈B◦ andB ∈Ꮾβ} and ᏺβx= ∩{X\clτ1B|x∈X\clτ1B, B∈Ꮾβ}.
We form ᐁα = ∪x(Λαx×αx) and ᐂβ= ∪xᏺβx×ᏹxβ and show that each of the families(ᐁα)αand(ᐂβ)βforms a nested base for anLQᐁcompatible withτ1andτ2, respectively. We prove it for the first family.
The family{ᐁα|α∈I}is nested. In fact, ifα≤β, thenΛβx⊆ΛαxandKxβ⊆Kxα, hence
ᐁα⊆ᐁβ. Next, ifx∈Xand A∈ ∪αᏭα, we can chooseβ∈IandB∈Ꮾβ, such that
x∈B⊆clτ2B⊆A. Moreover, ifγ∈I,B∈Ꮽγ, andA∈Ꮽγ, thenᐁγ[B]= ∪y{γy | Λγy∩B= ∅, y∈X}. If nowy∈A, then Λγy⊆A, hence∪y∈AΛγy⊆A. Ify∉A, then
y∉clτ2Bory∈X\clτ2Band sinceΛ γ
y⊆X\clτ2B, it follows thatΛ γ
y∩B= ∅. Hence
ᐁγ[B]⊆Ain any case and fromLemma 2.1the familyΓ = {ᐁα|α∈I}is a base for anLQᐁsuch thatτ(Γ)=τ1.
It also follows thatᐁ−1
α [x]= ∪y{Λαy:x∈kαy}, hence it isτ2-open andτ(Γ−1)⊆τ2. We also haveE= {ᐂβ|β∈Ᏽ}is anLQᐁsuch thatτ(E)=τ2andτ(E−1)⊆τ1.
LetF=Γ∨E−1. Thenτ(F)=τ1, hence byLemma 2.2,F is anLQᐁ. We now pick up (afterLemma 2.4) a nested familyᐁof entourages of the formᐁα∩ᐂ−1α , where
ᐁα and ᐂα belongs to Γ and E, respectively. This family induces the topology τ1 as well, thusτ(ᐁ)=τ1. It also follows thatτ(ᐁ−1)=τ2, and the proof is complete.
As always, if we say that bitopological space(X,τ1,τ2)is quasi-metricwe mean that there is a quasi-metricdsuch that the topology induced bydcoincides withτ1and that induced byd−1coincides withτ2.
The following corollaries are directly concluded fromTheorem 2.5.
Corollary2.6. If a bitopological pairwise regular space satisfies the assumptions ofTheorem 2.5with the only exception that the given families are countable, then the space is quasi-pseudo-metrizable.
Corollary2.7(see [9, Theorem 3.1]). Let(X,τ1,τ2)be a bitopological space and
(preserving the notation ofTheorem 2.5) we suppose that the two classes of the fam-ilies {Ꮽα}α and {Ꮾβ}β are countable, that ∪{Ꮽ◦α |α ∈I} and ∪{
◦
Ꮾβ |β∈ I} are
bases for the topologies τ1 and τ2, respectively, and that { ◦
Ꮽα |α∈ I} and { ◦
Ꮾβ |
β∈I} are, both of them, τ1 and τ2-locally finite. Then the space is
quasi-pseudo-metrizable.
An immediate consequence ofTheorem 2.5is the following theorem of J. G. Kelly [5, Theorem 2.5].
Theorem2.8. A bitopological pairwise regular space which fulfils the second axiom of countability is quasi-pseudo-metrizable.
In fact, ifᏼnandᏽnare the countable bases ofτ1andτ2, then the familiesᏭn= ∪ᏼn
andᏮn= ∪ᏽnsatisfy the assumptions ofTheorem 2.5and the space is
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.
Theorem2.9. Let(X,τ1,τ2)be a pairwise regular bitopological space,{Ꮽα|α∈I}
and{Ꮾβ|β∈I}be nested collections ofτ1andτ2-open, respectively, families of
sub-sets, each family beingτ1and at the same timeτ2 locally finite and{ ◦
Ꮽα|α∈I}
and∪{Ꮾ◦β|β∈I}beτ1,τ2-open bases, respectively. Then the space is
quasi-pseudo-metrizable.
Proof(cf. [12, Proposition 2.10]). FromTheorem 2.5the space admits anLQᐁ ᐁ
with a nested base such thatᐁ−1is also anLQᐁwith a nested base. IfI=ω, (ωis the ordinal of the natural numbers), then the result comes from the mentioned Fletcher-Lindgren’s theorem. If, on the other hand, clτ2{x}and clτ1{x}areτ1 andτ2-open, respectively, the space(X,τ1,τ2)admits quasi-pseudo-metricsd1andd2defined as follows:
d1(x,y)=
01,, otherwiseify∈clτ2,{x}, d2(x,y)=
01,, ifotherwise.y∈clτ1{x}, (2.2)
From [5, page 86], we deduce thatx∈clτ2{y}impliesy∈clτ1{x}, hence
d−1
1 (x,y)=d1(y,x)=
10,, ifotherwisex∈clτ2,{y},=
01,, otherwiseify∈clτ1,{x},=d2(x,y). (2.3)
So there remains the case cardI > ωat least one of clτ2{x}, c lτ1{x}, say the first, is notτ1-open. We shall derive a contradiction: let{Ꮽαn|n∈ω}be a countable subcol-lection of{Ꮽα|α∈I}such that, for eachn∈ω,Ꮽαn contains aτ1-neighborhood of
xandᏭan+1contains aτ1-neighborhood ofxwhich is strictly smaller than any neigh-borhood of x inᏭan. Such annexists, because any familyᏭα (α∈I) isτ1-locally finite. Let
N= A∈Ꮽan|x∈Aandn∈ω, (2.4) whereNis the countable intersection of open sets and since cardI > ω,Nis an open
τ1-neighborhood ofx. Next, we consider a familyᏭβ in the collection{Ꮽα|α∈I} which contains aτ1-subneighborhoodBofN. The setBdoes not belong to anyᏭαn and we may assume thatᏭαn⊂Ꮽβ, for anyn∈ω. Then, there is anA∈Ꮽan+1\Ꮽan, wherex∈AandA∈Ꮽβ so that everyτ1-neighborhood ofxmeets infinitely many elements ofᏭβ, a contradiction.
3. The necessity of Theorem2.5assumptions. 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 ofTheorem 2.5into a more convenient expression.
Let(X,τ1,τ2) be again a bitopological space. We put for anyx∈X and for any
{y∈X|d−1(x,y) < '}. We also note by int1A(respectively, int2A) the interior with respect tod(respectively, tod−1) of anyA⊆X. Finally, we recall that aprecise
refine-ment(cf. [3, Section 5.4]) of a coverᏯ= {Cα|α∈A}ofXis a coverᏽ= {Rα|α∈A} provided that for eachα∈A,Rα⊆Cα.
Before beginning the theorem, some remarks on aT0nonT1totally ordered quasi-pseudo-metrizable space are necessary. Let(X,d)be such a space. Ifd(x,y)=0, then
x∈cl(y)and vice versa. Moreover, ifd(x,y)=0 andx∈B(α,'), theny∈B(α,')
as well, becaused(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], asxruns throughX,
B(α,')is any sphere andxis larger than any point ofB(α,ε), then any open subset of the formB(x,'), '>0, containsB(α,'), since it contains ally∈]←,x]. This means that in this case it is impossible to refine any open covering ofXin an effective way, which is a necessary presupposition for the demonstration of a Nagata-Smirnov’s-type theorem.
Theorem3.1. AT1topological space(X,τ1,τ2)is quasi-pseudo-metrizable if and
only if there are two countable collections(Ꮽn)n∈ω and(Ꮾn)n∈ω of coverings ofX
consisting ofτ1andτ2-open subsets, respectively, whereᏭn= {Ani|i∈I},Ꮾn= {Bni|
i∈I},Ibeing the same for alln, such that
(1)The family∪Ꮽn(respectively,∪Ꮾn) is aτ1-(respectively,τ2-)open base.
(2)For everyn,Ꮽn+1(respectively,Ꮾn+1) is precise refinement ofᏭn(respectively,Ꮾn). (3)For anynand anyJ⊂I, ∩{Anj|j∈J}(respectively,∩{Bnj|j ∈J}) is aτ1
-(respectively,τ2-) neighborhood of∩{A(n+1)j|j∈J}(respectively,∩{B(n+1)j|j∈J}). (4) Similarly, ∩{X\A(n+1)j |j ∈J} (respectively, ∩{X\B(n+1)j |j ∈J}) is a τ2
-(respectively,τ1-) neighborhood of∩{X\Anj|j∈J}(respectively,∩{X\Bnj|j∈J}).
Proof. For the sufficiency of the statement we follow the demonstration of Theorem 2.5: we construct an LQᐁ ᐁ such thatτ(ᐁ)=τ1 (the construction of a
ᐂsuch thatτ(ᐂ)=τ2is similar), and we arrive, just as inTheorem 2.5, at anLQᐁ ᐃ such thatτ(ᐃ)=τ1andτ(ᐃ−1)=τ2, as desired. We only defineᐁ: for anyxand for anyn∈ωputn
x=int1{∩Ani|x∈ ∩A(n+1)i, i∈I},Λnx=int2{∩(X\A(n+2)i)|x∈ ∩(X\A(n+1)i), i∈I}.
Then the familyᐁ= {Un|n∈ω}, whereUn= ∪{Λnx×nx|x∈X}is a base for anLQᐁcompatible withτ1. More precisely, we show that for anyA∈ ∪Ꮽn, there is another memberB of the family and aUm∈ᐁsuch thatUm[B]⊆A. In fact, given
A=Amithere areB∗=A(m+1)iandB=A(m+2)isuch thatB⊆B∗⊆A. ThenUm[B]= ∪{m
x |Λmx ∩B= ∅, x∈X}. So, ifx∈B∗,mx ⊆ ∩{Ami|i∈I} ⊆A.Ifx∉B∗, then
x∈X\B∗, and sinceX\B∈ {(X\A
(m+2)i)|i∈I},we haveΛmx =int2{∩(X\A(m+2)i)|
i∈I} ⊆X\B, henceΛm
x ∩B= ∅. ThusUm[B]⊆A.
We prove the necessity for the family(Ꮽn)n∈ω. We suppose that there is a quasi-pseudo-metricdsuch thatτ(d)=τ1andτ(d−1)=τ2.
Letᏽ(m)= {B(x,1/m)|x∈X, m∈ω}be a covering ofX. PutS
n[B(x,1/m)]= {t∈
X|B(t,1/n)⊆B(x,1/m)}andEn[B(x,1/m)]= ∪{B(x,1/3m)|x∈Sn[B(x,1/m)]}. Remark that ifm > n, thenEn[B(x,1/m)]= ∅. We suppose thatn≥m.
There holds:
B
x,m(m1+1)
⊆B
x,m(m1+1)
⊆Sm+1
B
x,m1
⊆Em+1
B
x,m1
. (3.1)
In fact, if t∈B(x,1/m(m+1))and λ∈B(t,1/(m+1)), then d(x,λ)≤d(x,t)+
d(t,λ) <1/m, henceB(t,1/(m+1))⊆B(x,1/m)andt∈Sm+1[B(x,1/m)]. We also haveEm(m+1)+1[B(x,1/m(m+1))]⊆B(x,1/m(m+1))⊆B(x,1/m(m+1))⊆Em+1
[B(x,1/m)]. The latter means that the covering{Em(m+1)+1[B(x,1/m(m+1))]|x∈
X}is a precise refinement of{Em+1[B(x,1/m)]|x∈X}. On the other hand, ift∈ ∩xEm(m+1)+1[B(x,1/m(m+1))], then t∈Sm+1[B(x,1/m)] and thusB(t,1/3(m+ 1))⊆Em+1[B(x,1/m)], orB(t,1/3(m+1))⊆ ∩xEm+1[B(x,1/m)]and (3) has been proved.
We now considert∈ ∩x{X\Em+1[B(x,1/m)]}. Ifκ∈Sm+1[B(x,1/m)],d(κ,t) > 1/3(m+1)and sinceEm(m+1)+1[B(x,1/m(m+1))]⊆Sm+1[B(x,1/m)], for anyλ∈
Em(m+1)+1[B(x,1/m(m+1))], it follows thatd(λ,t) >1/m(m+1), hence
Em(m+1)+1
Bx, 1
m(m+1) B−1
t, 1
3(m+1)
= ∅. (3.2)
ThusB−1(t,1/3(m+1))⊆ ∩x{X\E
m(m+1)+1[B(x,1/m(m+1))]}and the statement (4) has been proved.
The required family(Ꮽn)n∈ωis defined as follows:
Ᏹn+1=
En+1
B
x,n1x∈X, n∈ω\{0}
. (3.3)
DefineᏭ2=Ᏹ2and ifᏭn=Ᏹn∗,n,n∗inω, thenᏭn+1=Ᏹn∗(n∗+1) and the proof
is complete.
Remark 3.2. It is evident that for the above-mentioned pairs(Ꮽn,Ꮽn+1), Ꮽn+1 areτ1-cocushioned, τ2-cushioned ofᏭn and for the pairs(Ꮾn,Ꮾn+1),Ꮾn+1areτ 2-cocushioned,τ1-cushioned ofᏮnas well.
In fact, it follows that (i)∩(X\Ani)⊆intτ2∩(X\A(n+1)i)orX\intτ2∩(X\A(n+1)i)⊆ X\∩(X\Ani)= ∪Ani. On the other hand, (ii) clτ2∪(A(n+1)i)⊆X\intτ2∩(X\A(n+1)i). In fact, ift∈intτ2∩(X\A(n+1)i), then there is a τ2-neighborhoodVtτ2 of tsuch thatVτ2
t ⊆X\A(n+1)i for anyiorVtτ2∩A(n+1)i= ∅. ThusVtτ2∩(∪A(n+1)i)= ∅or
t∉clτ2(∪A(n+1)i)or t∈X\clτ2(∪A(n+1)i). Thus clτ2(∪A(n+1)i)⊆Ani. Thus the as-sumptions 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].
4. Some consequences of Theorem2.5.
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 spaceXisparacompact
if and only if, given a coverᏲofX, there is for anyxa sequence{Un[x]:n∈ω}of neighborhoods ofx such that (i)y∈Un[x]x∈Un[y], and (ii) ifx∈X, there is
n∈ωandG∈Ᏺsuch thatU2
n[x]⊆G.
By apair open coverof a bitopological space (X,ᏼ,ᏽ)we mean a family of pairs {(Gα,Hα)|α∈I}such thatGα isᏼ-open,Hαᏽ-open, (ii)Ᏻ= {Gα:α∈I}andᏴ= {Hα:α∈I} are covers ofX and (iii) for each x∈X there is anα∈I such that
x∈Gα∩Hα.
The[11]-pairwise paracompactness. A pairwise regular space(X,ᏼ,ᏽ)is pair-wise paracompact if and only if given a pair cover(Ᏻ,Ᏼ), there is for everyx a se-quence{Un[x]:n∈N}ofᏼ-neighborhoods and a sequence{Vn[x]:n∈N}ofᏽ -neighborhoods ofxsuch that (i)y∈Un[x]x∈Vn[y], (ii) for thatx, there is an
n∈ωand a pair(Gα,Hα)in(Ᏻ,Ᏼ)such thatUn2[x]⊆GαandVn2[x]⊆Hα.
The [4]-pairwise paracompactness. A pairwise regular space (X,ᏼ,ᏽ) is δ-pairwise paracompact if everyᏼor ᏽ-open cover of Xhas aᏼ∨ᏽ(the supremum ofᏼandᏽ) locally finite refinement.
We firstly give (Theorems4.4,4.5, and4.6) conditions under which we may construct on a space quasi-uniformities with nested bases.
Definition4.1. A quasi-uniformity(X,ᐁ)enjoys theneighborhood propertyif for anyx∈Xand anyU∈ᐁ, there is aVx∈ᐁsuch thatVx−1[x]×Vx[x]⊆U.
Proposition4.2. If in(X,ᐁ,ᐁ−1),ᐁandᐁ−1areLQᐁs, thenᐁ2fulfils the neigh-borhood property(ᐁ2= {U∈ᐁ|(∃V∈ᐁ)[V2⊂U]}).
Proof. Let x∈X and U ∈ ᐁ2. Then, there are W ∈ᐁ such thatW2⊆U and
V1x,V2x in ᐁ such thatV1x2 [x]⊆W [x]and V2x−2[x]⊆W−1[x]. Put Vx=V1x∩V2x, thenV−2
x [x]×Vx2[x]⊆W−1(x)×W (x)⊆W2⊆UandVx2∈ᐁ2.
Proposition4.3. If ᐁis anLQᐁ, thenᐁ2is also anLQᐁ, which generates the same topology asᐁ.
Proof. GivenV∈ᐁandx∈X, there is aWx∈ᐁsuch thatWx2[x]⊆V[x],Wx2∈
ᐁ2, henceτ(ᐁ)⊆τ(ᐁ2). If, on the other hand,W∈ᐁ2and x∈X, then there is a
Theorem4.4. Ifᐁandᐁ−1areLQᐁs onX with nested bases, then the set of the
diagonal neighborhoods generates a quasi-uniform topology equivalent toτ(ᐁ). Proof. After Propositions4.2 and4.3, we may suppose that ᐁ is anLQᐁwith a nested baseᏭwhich has the neighborhood property. LetU be a neighborhood of the diagonal andᏮ= {U∩V:V ∈Ꮽ}.Ꮾis a nested class of neighborhoods of the diagonal which generates aGQᐁfiner thanᐁ. Hence, byLemma 2.2,Ꮾis a base for a quasi-uniformityᐃ; furthermore,ᐃfulfils the neighborhood property and induces onXa topology equivalent to that induced byᐁ.
SinceU∈ᐃ, it implies that for anyx∈X there is aVx∈ᐃsuch thatVx−1[x]×
Vx[x]⊆U, and a W ∈ᐃ such that W−3[x]⊆Vx−1[x] and W3[x]⊆Vx[x], hence
W−3
x [x]×Wx3[x]⊆Vx−1×Vx[x]⊆U. Put
W=W−1
x [x]×Wx[x]|x∈X
, (4.1)
whereWis a neighborhood of the diagonal. We will show thatW◦W⊆U.
Let(x,y)∈W◦W, there iszsuch that(x,z)∈W,(z,y)∈W, consequently there areα,βinXsuch thatx∈W−1
α [α],z∈Wα[α]andz∈Wβ−1[β],y∈Wβ[β]. IfWα⊆Wβ, thenx∈W−2
α ◦Wβ−1[β]⊆Wβ−3[β]andy∈Wβ[β], hence(x,y)∈Wβ−3[β]×Wβ[β]⊆U. IfWβ⊆Wα, theny∈Wβ2◦Wα[α]⊆Wα3[α], hence(x,y)∈Wα−1[α]×Wα3[α]⊆U, or
W◦W⊆U.
Theorem4.5. If in a bitopological space(X,τ1,τ2), where both topologies induce
quasi-uniformities with nested bases which have the same cofinalityℵ, a familyᏭof τ2×τ1-neighborhoods of the diagonal has cardinality less thanℵ, then∩Ꮽis a
neigh-borhood of the diagonal.
Proof. It is known (cf. [12, Theorem 2.4]) that in a uniform (as well as in a quasi-uniform) space with a nested base of cofinalityℵ, any collection of open sets and of cardinality less thanℵ, has as intersection an open set. IfᏭis the collection andx∈X, then for anyA∈Ꮽ, there areKA[x]andΛA[x],τ1andτ2-neighborhoods ofx, such thatΛA[x]×KA[x]⊆A. IfK= ∩{KA[x]:A∈Ꮽ}andΛ= ∩{ΛA:A∈Ꮽ}, then from the above statement and the fact thatKandΛareτ1andτ2-neighborhoods ofx, it implies thatΛ×K⊆ ∩Ꮽ. Hence∩Ꮽis aτ2×τ1-neighborhood of the diagonal.
Theorem4.6. Ifᐁandᐁ−1areLQᐁswith nested bases and are of the same
cofi-nality, then there is a quasi-uniformity with a nested base which generates the same, as theᐁ, topology and has the same cofinality.
Proof. Ifℵis the common cofinality ofᐁandᐁ−1, andW
λ(λ∈ ℵ) is any neigh-borhood of the diagonal, then byTheorem 4.4there is a neighborhoodUλ+1of the diagonal such thatUλ+1◦Uλ+1⊆Uλ. Ifλis a limit ordinal number less thanℵand each
Uα, forα < λ, has been chosen, then putUλ= ∩{Uα:α < α},Uλis byTheorem 4.5a neighborhood of the diagonal. The rest are trivial.
Theorem4.7. If for a bitopological space(X,τ1,τ2)there areLQᐁs ᐁ andᐁ−1
with a nested base (both of them) and τ(ᐁ)= τ1, τ(ᐁ−1)=τ2, then the space is
δ-pairwise paracompact.
Proof. AfterTheorem 4.6, there is a quasi-uniformityᐃwhose dualᐃ−1is also quasi-uniformity, both of them have nested bases, they may be extended (Theorem 4.6) until they reach the same cofinalityℵand, finally, they generate topologies onX equiv-alent toτ(ᐁ)andτ(ᐁ−1), respectively. We may also assume that the uniformityᐃ∗ has a nested base with cofinalityℵ.
If ℵ = ω, the space (X,ᐁ) is quasi-metrizable, hence the bitopological space
(X,τ1,τ2)isδ-pairwise paracompact.
Letℵ> ω. From a E. Zakon’s result [13, Theorem 2.1] we may consider that the nested base of the uniformityᐃ∗consists of equivalent relations, which leads—after [12, Theorem 2.7]—to the fact that everyτ1∨τ2=τ(ᐃ∗)-open coverᏭ there is a
τ(ᐃ∗)-discrete refinement such that there is a partition, say{S
αx[x]|x∈X}, of the space, where{Sαx[x]|x∈X}refinesᏭ. Thus, theτ1∨τ2-opening coveringᏭhas a locally finite refinement and theδ-pairwise paracompactness has been proved.
Theorem4.8. If for a bitopological space(X,τ1,τ2)there areLQᐁs ᐁ andᐁ−1
with a nested base (both of them) and τ(ᐁ)= τ1, τ(ᐁ−1)=τ2, then the space is [11]-pairwise paracompact.
Proof. Let(X,τ1,τ2), ᐃand ᐃ−1 be as inTheorem 4.7. If the cofinality of the base equalsω, the space is quasi-metrizable, hence [11]-pairwise paracompact.
Let the cofinality ofℵbe larger thanωand(Ᏻ,Ᏼ)be a(τ(ᐃ),τ(ᐃ−1))-open pair cover ofX, henceᏳ∩Ᏼ= {G∩H|G∈Ᏻ, H∈Ᏼ}is aτ(ᐃ∗)-cover ofX. That cover (as we have referred inTheorem 4.7) has a discrete refinement, say{Sαx[x]|x∈X}. Let
Sαx[x]be the onlyτ(ᐃ∗)-neighborhood ofxwith respect to this refinement. If{Wα∗| ℵ ∈I}and{Wℵ| ℵ ∈I}areτ(ᐃ∗)- andτ(ᐃ)-bases, respectively, then for anyx∈X there is anα∈Isuch thatW∗
α =Wα[x]∩Wα−1[x]⊆Sαx[x]; furthermore ifx∈Gα∩Fα, whereGα∈ᏳandHα∈Ᏼand we preserve for our convenience the sameα, then there is anα∗ such thatW2
α∗[x]⊆Wα−1∗[x]⊆GαandWα−2∗[x]⊆Wα−1∗[x]⊆Hαand we may suppose thatWα∗[x]∩Wα−1∗[x]⊆Sαx[x]⊆Gα∩Hα. On the other hand, we may put
Wα∗[x]=Vn[x]andWα−1∗[x]=Un[x]for everyn∈ω. These two sequences fulfill the requirements of the S. Romaguera [11]-definition of pairwise paracompactness and the proof is complete.
Remark4.9. (1) After Theorems4.7and4.8it is evident that every bitopological space which satisfies the assumptions ofTheorem 2.5isδ-pairwise paracompact as well as [11]-pairwise paracompact.
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A. Andrikopoulos: Department of Mathematics, University of Patras,26500Patras, Greece
J. N. Stabakis: Department of Mathematics, University of Patras, 26500 Patras, Greece
