Pairwise $k$-Semi-Stratifiable Bispaces and Topological Ordered Spaces

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by  Kedian Li

PAIRWISE k-SEMI-STRATIFIABLE BISPACES AND
TOPOLOGICAL ORDERED SPACES
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1
KEDIANLIANDJILINGCAO∗
0
2

n Abstract. In this paper, we continue to study pairwise(k-semi-)stratifiable
a bitopologicalspaces. Somenewcharacterizationsofpairwisek-semi-stratifiable
J bitopological spaces areprovided. Relationshipsbetween pairwisestratifiable
1 and pairwise k-semi-stratifiable bitopological spaces are further investigated,
1 andanopenquestionrecentlyposedbyLiandLinin[18]iscompletelysolved.
We also study the quasi-pseudo-metrizability of a topological ordered space
] (X,τ,4). It is shown that if (X,τ,4) is a ball transitive topological ordered
N C- and I-space such that τ is metrizable, then its associated bitopological
space (X,τ♭,τ♮) is quasi-pseudo-metrizable. This result provides a partial
G
affirmativeanswertoaproblemin[15].
.
h
t
a
m
[ 1. Introduction

1 Undoubtedly, topology and order are not only important topics in mathematics
v
but also applicable in many other disciplines. For example, nonsymmetric notions
5
of distance are needed for mathematical modelling in the natural, physical and
4
8 cybernetic sciences and the corresponding topological notion is that of a quasi-
2 metric or a quasi-pseudo-metric. The study of quasi-metrizable spaces naturally
0 leads to the concepts ofquasi-uniformitiesandbitopologicalspaces. Inthis aspect,
1. Kelly’s seminal paper [14] made pioneer contributions. On the other hand, the
0 notions of a sober space and the Scott topology, in align with the investigation of
7 partialordersandpre-orders,areusefulintheoreticalcomputerscienceinthestudy
1
of algorithms which act on other algorithms. Moreover, finite topological spaces
:
v (i.e., finite pre-orders) can be used to construct a mathematical model of a video
i monitor screen which may be useful in computer graphics.
X
r Since Kelly’s work in [14], bitopological spaces have attracted the attention of
a
many researchers. For example, Reilly [29] explored separation axioms for bitopo-
logical spaces, Cooke and Reilly [4] discussed the relationships between six def-
initions of bitopological compactness appeared in the literature, Raghavan and
Reilly [28] introduced a notion of bitopological paracompactness and established
2000 Mathematics Subject Classification. Primary54E55;Secondary06F99, 54E20,54F05.
Key words and phrases. Bitopological spaces; C-spaces; I-spaces; Pairwisek-semi-stratifiable
spaces;Pairewisestratifiablespaces;Quasi-pseudo-metrizable;Topological orderedspaces.
The first-author is supported by the National Nature Science Fundation of China, grant No.
61379021,11471153) and the Natural Science Foundation of Fujian Province, China, grant No.
2013J01029). ThesecondauthorthanksthesupportoftheNationalNaturalScienceFoundationof
China,grantNo. 11571158,andthepaperwaswrittenwhenhevisitedMinnanNormalUniversity
inApril2016asMinJiangScholarGuestProfessor.
*Correspondingauthor.
1
2 KEDIANLIANDJILINGCAO∗
a bitopological version of Michael’s classical characterization of regular paracom-
pact spaces. In addition to separation and covering properties, generalized metric
properties have also been considered in the setting of bitopological spaces. In this
direction, Fox[7]discussedthe quasi-metrizabilityofbitopologicalspaces,pairwise
stratifiablebitopologicalspacesandtheirgeneralizationshavebeenintroducedand
studied by [12], [21] and [22].
Interplay between topology and order has been a very interesting area. In 1965,
Nachbin’s book [26] was published. This book is one of general references on the
subjectavailabletoday,anditcoversresultsobtainedby the authorinhis research
on spaces with structures of order and topology. Among many topics in this line,
McCartan[23]studiedbicontinuous(hereincalledC-spaceandI-space)pre-ordered
topological spaces and investigated the relationships between the topology of such
a space and two associated convex topologies, and Faber [6] studied metrizability
in generalizedorderedspaces. Recently, there have been some renewedinterests in
the study of generalized metric properties in bitopological spaces and topological
orderedspaces. Ku¨nziandMushaandjainvestigatedthequasi-pseudo-metrizability
of a topological ordered space in [15] and [25], respectively. They obtained some
results related to the upper topology τ♮ and the lower topology τ♭ of a metrizable
ordered space (X,τ,4) which is both a C- and an I-space in the sense of Priestle
[27]. Moreover, Li [16] as well as Li and Lin [17] carried on the study of pairwise
(semi-)stratifiable bispaces and established some new characterizations for these
classes of bitopological spaces. In a very recent paper [18], Li and Lin further
introduced and studied the class of k-semi-stratifiable bitopological spaces.
The main purpose of this paper is to continue the study of pairwise k-semi-
stratifiable bitopological spaces and their relationships with and applications to
the quasi-pseudo-metrizabilityofa topologicalorderedspaces. For the sakeofself-
completeness,inSection2,weintroducethenecessarydefinitionsandterminologies.
In Section 3, we provide some new characterizationsof pairwise k-semi-stratifiable
bitopological spaces in terms of g-functions σ-cushioned pair k-networks and cs-
networks. InSection4,weconsidersomeconditionsunderwhichapairwisek-semi-
stratifiable bitopological space is pariwise stratifiable. An open question posed in
[18] is completely solved and results in [2] and [3] are extended to the setting of
bitopologicalspaces. Inthelastsection,weconsiderthequasi-pseudo-metrizability
of a topologicalorderedspaces and providea partialaffirmativeanswerto anopen
problem of Ku¨nzi and Mushaandja in [15].
Our notations in this paper are standard. For any undefined concepts and ter-
minologies, we refer the reader to [5] or [10].
2. Preliminaries and notations
A quasi-pseudo-metric d on a nonempty set X is a non-negative real-valued
functiond:X×X →R+suchthat(i)d(x,x)=0and(ii)d(x,z)≤d(x,y)+d(y,z),
for all x,y,z ∈X. If d is a quasi-pseudo-metric on X, then the orderedpair (X,d)
is called a quasi-pseudo-metric space. Every quasi-pseudo-metric d on X induces
a topology τ(d) on X which has as a base the family {B (x,ǫ) : x ∈ X,ǫ > 0},
d
where B (x,ǫ)={y ∈X :d(x,y)<ǫ}. Every quasi-pseudo-metric d on X induces
d
a conjugate quasi-pseudo-metric d−1 on X, defined by d−1(x,y) = d(y,x) for all
x,y ∈ X. A bitopological space [14] (for short, bispace [12]) is a triple (X,τ1,τ2),
PAIRWISE k-SEMI-STRATIFIABLE BISPACES AND TOPOLOGICAL ORDERED SPACES 3
whereX isanonemptyset,topologiesτ1 andτ2 aretwotopologiesonX. Abispace
(X,τ1,τ2) is called quasi-pseudo-metrizable, if there is a quasi-pseudo-metric d on
X such that τ(d)=τ1 and τ(d−1)=τ2.
Let (X,τ1,τ2) be a bispace. For i = 1,2, let Fi(X) denote the family of all
τ -closedsubsetsofX. ForasubsetAofX,letAτi andint (A)denotetheclosure
i τi
andinteriorofA withrespectto τ , i=1,2,respectively. Fori,j =1,2with i6=j,
i
(X,τ1,τ2) is called τi-semi-stratifiable with respect to τj if there exists an operator
G : N × F (X) → τ satisfying (i) H = G (n,H) for all H ∈ F (X),
ij i j n∈N ij i
(ii) if H,K ∈ F (X) with H ⊆ K, then G (n,H) ⊆ G (n,K) for all n ∈ N.
i ij ij
Furthermore, if G satisfies (ii) and (i)’ H =T G (n,H)τi for all H ∈F (X),
ij n∈N ij i
then (X,τ1,τ2) is called τi-stratifiable with respect to τj. Moreover, (X,τ1,τ2) is
T
called pairwise (semi-)stratifiable [7], [12] and [22], if it is both τ1-semi-stratifiable
with respect to τ2 and τ2-semi-stratifiable with respect to τ1.
Recently,LiandLin[18]introducedtheconceptofapairwisek-semi-stratifiable
bispace,whichisanaturalextensionofak-semi-straitifiablespaceintroducedin[19]
tothesettingofbispaces. Fori,j =1,2withi6=j,abispace(X,τ1,τ2)iscalledτi-
k-semi-stratifiablewithrespecttoτ ifthereexistsanoperatorG :N×F (X)→τ
j ij i j
satisfying (i) H = G (n,H) for all H ∈ F (X), (ii) if H,K ∈ F (X) with
n∈N ij i i
H ⊆ K, then G (n,H) ⊆ G (n,K) for all n ∈ N, (iii) if K ⊆ X is τ -compact
ij ij i
and H ∈F (X) sucTh that H∩K =∅, then K∩G (n,H)=∅ for some n∈N. In
i ij
addition, (X,τ1,τ2)is calledpairwise k-semi-stratifiable [18]if itis both τ1-k-semi-
stratifiable with respect to τ2 and τ2-k-semi-stratifiable with respect to τ1.
The next lemma, which gives an important dual characterization of pairwise
k-semi-stratifiable bispaces, will be used in the sequel.
Lemma 2.1 ([18]). A bispace (X,τ1,τ2) is pairwise k-semi-stratifiable if, and only
if, for anyi,j =1,2withi6=j, thereisan operator F :N×τ →F (X)satisfying
ij i j
(1) U = F (n,U) for all U ∈τ ;
n∈N ij i
(2) if U,V ∈τ with U ⊆V, then F (n,U)⊆F (n,V) for all n∈N;
i ij ij
S
(3) if K ⊆ X is τ -compact and U ∈ τ with K ⊆ U, then K ⊆ F (n,U) for
i i ij
some n∈N.
In addition, the operator F can be required to be monotone with respect to n, that
ij
is, F (n,U)⊆F (n+1,U) for all n∈N and all U ∈τ .
ij ij i
By definition, every pairwise stratifiable bispace is pairwise k-semi-straitifable,
and every pairwise k-semi-stratifiable bispace is pairwise semi-stratifiable. Recall
that a bispace (X,τ1,τ2) is said to be pairwise monotonically normal [21] if to
each pair (H,K) of disjoint subsets of X such that H ∈ F (X) and K ∈ F (X)
i j
(i,j =1,2 and i6=j), we can assign a set D (H,K)∈τ such that (i)
ij j
H ⊆D (H,K)⊆D (H,K)τi ⊆XrK,
ij ij
(ii)ifthepairs(H,K)and(H′,K′)satisfyH ⊆H′ andK′ ⊆K,thenD (H,K)⊆
ij
D (H′,K′).
ij
The following result, established by Mar´ın and Romaguera in [21], extends the
celebrated result of Heath et al. in [13] on monotonically normal spaces.
Theorem 2.2 ([21]). A bispace (X,τ1,τ2) is pairwise stratifiable if, and only if, it
is a pairwise monotonically normal and pairwise semi-stratifiable bispace.
4 KEDIANLIANDJILINGCAO∗
Corollary 2.3. A pairwise monotonically normal and pairwise k semi-stratifiable
bispace is pairwise stratifiable.
Atopological ordered space (X,τ,4)is anonempty setX endowedwitha topol-
ogy τ and a partial order 4. A subset A of X is said to be an upper set of X if
x 4 y and x ∈ A imply that y ∈ A. Similarly, we say that a subset A of X is a
lower set of X if y 4 x and x ∈ A imply that y ∈ A. Let τ♭ denote the collection
of τ-open lower sets of X and τ♮ denote the collection of τ-open upper sets of X.
Then, τ♭ and τ♮ are two topologies on X and thus (X,τ♭,τ♮) is a bispace. For any
subset A of X, i(A) (resp. d(A)) will denote the intersection of all upper (lower)
sets of X containing A. Note that i(A) (resp. d(A)) is the smallest upper (resp.
lower)setcontainingA. ItiseasytoseethatA=i(A)if,andonlyif,Aisanupper
set. Similarly, A = d(A) if, and only if, A is a lower set. Following Priestley [27],
we recall that a topological ordered space (X,τ,4) is said to be a C-space if d(F)
and i(F) are closed whenever F is a closed subset of X. Similarly, a topological
ordered space (X,τ,4) is called an I-space if d(G) and i(G) are open whenever G
is an open subset of X.
3. Some new characterizations of pairwise
k-semi-stratifiable bispaces
In[18],LiandLincharacterizedpairwisek-semi-stratifiablebispacesintermsof
pairwise g-functions and extensions of semi-continuous functions. In this section,
wecontinuetoinvestigatehowtocharacterizepairwisek-semi-stratifiablebispaces.
Our first two results can be regarded as either improvements or extensions of a
theoremin[18]. Inaddition,wealsousecushionedpairk-networksandcs-networks
to characterize pairwise k-semi-stratifiable bispaces.
Let (X,τ1,τ2) be a bispace. A pairwise g-function on (X,τ1,τ2) is a pair of
functions (g1,g2) such that for i = 1,2, gi : N × X → τi satisfies x ∈ gi(n,x)
and g (n +1,x) ⊆ g (n,x) for all n ∈ N and x ∈ X. A pairwise family B =
i i
B1,B2 :α∈∆ of subsets of X is called τ -cushioned, where j = 1,2, if for
α α j
any ∆′ ⊆∆,
(cid:8)(cid:0) (cid:1) (cid:9) {B1 :α∈∆′}τj ⊆ B2 :α∈∆′ .
α α
Furthermore, if B is[a countable union of[τj-(cid:8)cushioned fa(cid:9)milies, then it is called
σ-τ -cushioned. A pairwise family B = B1,B2 :α∈∆ is called a pair τ -k-
j α α i
network if for any τ -compact set K and any set U ∈ τ with K ⊆ U, there is a
i i
finite subset ∆′ of ∆ such that (cid:8)(cid:0) (cid:1) (cid:9)
K ⊆ {B1 :α∈∆′}⊆ {B2 :α∈∆}⊆U.
α α
[ [
Our first result improves the equivalence of (1) and (2) in [18, Theorem 2.1].
Theorem 3.1. Let (X,τ1,τ2) be a bispace such that (X,τi) is T1-space for i=1,2.
Then (X,τ1,τ2) is pairwise k-semi-straitifiable if, and only if, there is a pairwise
g-function (g1,g2) such that for i,j =1,2 with i6=j, if K is a τi-compact set and
H is a τ -closed set with K∩H =∅, then
i
K∩ {g (m,x):x∈H} =∅
j
for some m∈N. (cid:16)[ (cid:17)
PAIRWISE k-SEMI-STRATIFIABLE BISPACES AND TOPOLOGICAL ORDERED SPACES 5
Proof. Necessity. Suppose that (X,τ1,τ2) is pairwise k-semi-stratifiable. For i,j =
1,2 and i 6= j, let G : N×F (X) → τ be an operator satisfying the definition
ij i j
of a pairwise k-semi-stratifiable bispace. Without loss of generality, G can be
ij
requiredto be monotonewith respectto n. Define a function g :N×X →τ such
j j
that gj(n,x) = Gij(n,{x}) for all n ∈N and x∈ X. Clearly, (g1,g2) is a pairwise
g-function. IfK isaτ -compactsubsetandH isaτ -closedsubsetwithK∩H =∅,
i i
then K∩G (m,H)=∅ for some m∈N. Note that
ij
{g (m,x):x∈H}⊆G (m,H).
j ij
It follows that [
K∩ {g (m,x):x∈H} =∅.
j
Sufficiency. Let(g1,g2)be(cid:16)a[pairwiseg-functions(cid:17)atisfyingtheassumptioninthe
theorem. For each τ -closed subset H and n∈N, define
i
G (n,H)= {g (n,x):x∈H}.
ij j
We shall verify that G is an operato[r satisfying conditions (i) in the definition of
ij
apairwisek-semi-stratifiablebitopologicalspace,as(ii)and(iii)holdtrivially. Itis
clear that H ⊆ G (n,H). If p6∈H, as {p} is compact, then the assumption
n∈N ij
in the theorem implies that there must be some m∈N such that p6∈ {g (m,x):
j
T
x∈H}. It follows that p6∈G (m,H). Thus, H = G (n,H). (cid:3)
ij n∈N ij S
Theorem3.2. Let(X,τ1,τ2)beabispacesuchthat(TX,τi)isHausdorfffori=1,2.
Then the following statements are equivalent.
(1) (X,τ1,τ2) is pairwise k-semi-straitifiable.
(2) There is a pairwise g-function (g1,g2) such that for i,j = 1,2 with i 6= j,
if {x :n∈N} is a sequence τ -convergent to p and H is a τ -closed subset
n i i
with ({p}∪{x :n∈N})∩H =∅, then
n
({p}∪{x :n∈N})∩ {g (m,x):x∈H} =∅
n j
for some m∈N. (cid:16)[ (cid:17)
(3) There is a pairwise g-function (g1,g2) such that for i,j = 1,2 with i 6= j,
if {x :n∈N} and {y :n∈N} are two sequences in X with {x :n∈N}
n n n
τ -convergent to p and x ∈ g (n,y ) for all n ∈ N, then {y : n ∈ N} is
i n j n n
τ -convergent to p.
i
Proof. (1) ⇒ (2) follows directly from Theorem 3.1, as {p} ∪ {x : n ∈ N} is
n
compact.
(2)⇒(3). Let (g1,g2) be a pairwise g-function satisfying (2). Let {xn :n∈N}
and {y : n ∈ N} be two sequences in X such that {x : n ∈ N} is τ -convergent
n n i
to p and x ∈ g(n,y ). Assume that {y : n ∈ N} is not τ -convergent p. Then
n n n i
{y : n ∈ N} has a subsequence {y : k ∈ N} such that p 6∈ {y :k ∈N}τi. Put
n nk nk
H ={y :k ∈N}. Since {x :k ∈N} is τ -convergent to p, then we can assume
that x nk6∈Hτi for all k≥1.nTkhus, by (2), tihere must be an m∈N such that
nk
({p}∪{x :k∈N})∩ {g (m,x):x∈H} =∅.
nk j
On the other hand, (cid:16)[ (cid:17)
x ∈g (n ,y )⊆g (m,y )⊆ {g (m,x):x∈H}.
nm j m nm j nm j
[
6 KEDIANLIANDJILINGCAO∗
A contradiction occurs.
(3)⇒(1). A proof has been given in [18]. (cid:3)
In [8], k-semi-stratifiable spaces are defined in terms of σ-cushioned pair k-
networks, which is different from (but equivalent to) that given in [19]. Our next
result, which just confirms that the same thing holds in the setting of bispaces,
providescharacterizationsofapairwisek-semi-stratifiablebispaceintermsofcush-
ioned pair k-networks.
Theorem 3.3. A bispace (X,τ1,τ2) is τ1-k-semi-stratifiable with respect to τ2 if,
and only if, it has a σ-τ2-cushioned pair τ1-k-network.
Proof. Necessity. Let F12 : N×τ1 → F2(X) be an operator satisfying conditions
(1)-(3) in Lemma 2.1 such that F12 is also monotone with respect to n. For each
n∈N, define ∆n =τ1 and
Bn = BU1,BU2 :BU1 =F12(n,U),BU2 =U and U ∈∆n .
We claim that Bn(cid:8)i(cid:0)s τ2-cush(cid:1)ioned. Indeed, if U ⊆∆n, by condition(cid:9)(2) in Lemma
2.1, F12(n,U)⊆F12(n, U) for any U ∈U. It follows that
τ2 τ2
{BU1 :SU ∈U} = {F12(n,U):U ∈U}
[ ⊆ [F12 n, U
⊆ {B(cid:16)2 [:U ∈(cid:17)U},
U
whichimpliesthateachBnisτ2-cushioned[. Thus,B = n∈NBnisσ-τ2-cushioned.
Let K be a τ1-compact subset of X and U ∈ τ1 with K ⊆ U. By condition (3)
in Lemma 2.1, there must be some m ∈ N such that KS⊆ F12(m,U) ⊆ U. Then
∆′ ={U}⊆∆ and
m
K ⊆ {B1 :U ∈∆′}⊆ {B2 :U ∈∆′}⊆U,
U U
which implies that B i[s also a pair τ1-k-ne[twork.
Sufficiency. Let B = n∈NBn be a σ-τ2-cushioned pair τ1-k-network, that is,
B is a pair τ1-k-networkSand for each n ∈ N, Bn = Bα1,Bα2 :α∈∆n is a
τ2-cushionedfamily. Withoutlossofgenerality,foreachn∈N,wecanassumethat
Bn ⊆ Bn+1 and Bn is closed under finite union. Defin(cid:8)e(cid:0)F12 : N(cid:1)×τ1 → F(cid:9)2(X)
such that for each n∈N and each U ∈τ1,
τ2
F12(n,U):= {Bα1 :Bα2 ⊆U and α∈∆n} .
First of all, as Bn is τ2-cushion[ed, we have
F12(n,U)⊆ Bα2 :Bα2 ⊆U and α∈∆n ⊆U.
For each x ∈ U, as B is a p[air(cid:8)τ1-k-network and {x} is c(cid:9)ompact, there exist an
m∈Nandanα∈∆m suchthatx∈Bα1 ⊆Bα2 ⊆U. Itfollowsthatx∈F12(m,U),
which implies that n∈NF(n,U) = U. It is clear that if U,V ∈ τ1 with U ⊆ V,
then F12(n,U)⊆ F12(n,V) for any n∈N. Finally, if K is τ1-compact and U ∈ τ1
with K ⊆ U, similaSr to what have done previously, there exist an m ∈ N and an
α ∈ ∆m such that K ∈ Bα1 ⊆ Bα2 ⊆ U. This implies that K ⊆ F12(m,U) ⊆ U.
Therefore, we have checked that F12 is an operator satisfying conditions (1)-(3) in
Lemma 2.1. (cid:3)
PAIRWISE k-SEMI-STRATIFIABLE BISPACES AND TOPOLOGICAL ORDERED SPACES 7
Corollary 3.4. A bispace (X,τ1,τ2) is pairwise k-semi-stratifiable if, and only if,
it has a σ-τ -cushioned pair τ -k-network for each pair of i,j =1,2 with i6=j.
j i
Let (X,τ1,τ2) be a bispace and x ∈ X be a point. A family Px of subsets of
X is called a τ -cs-network at x [11] if for every sequence {x : n ∈ N} that is
i n
τ -convergent to x and an arbitrary open neighborhood U of x in (X,τ ), there
i i
exist an m∈N and an element P ∈P such that
x
{x}∪{x :n>m}⊆P ⊆U.
n
IfeachpointxinX has aτ -cs-networkP ,then P iscalledτ -cs-network
i x x∈X x i
for (X,τ ).
i
S
In [8], Gao characterized k-semi-stratifiable spaces in terms of cs-networks. At
the end of this section, we establish a similar result in the setting of bispaces.
Theorem3.5. Let(X,τ1,τ2)beabispacesuchthat(X,τi)isHausdorfffori=1,2.
Then (X,τ1,τ2)is pairwise k-semi-stratifiableif, andonlyif, for anyi,j =1,2with
i6=j, there is an operator F :N×τ →F (X) satisfying
ij i j
(1) U = F (n,U) for all U ∈τ ;
n∈N ij i
(2) if U,V ∈τ with U ⊆V, then F (n,U)⊆F (n,V) for all n∈N;
i ij ij
(3) for eaSch U ∈τ , {F (n,U):n∈N} is a τ -cs-network at every point of U.
i ij i
In addition, the operator F can be required to be monotone with respect to n, that
ij
is, F (n,U)⊆F (n+1,U) for all n∈N and all U ∈τ .
ij ij i
Proof. The necessity is trivial by Lemma 2.1, as {x}∪{x : n ∈ N} is τ -compact
n i
for any sequence {x :n∈N} in X which is τ -convergentto a point x∈X.
n i
Sufficiency. Suppose that for any i,j = 1,2 with i 6= j, there is an operator
F : N×τ → F (X) satisfying conditions (1)–(3) above and monotonicity. We
ij i j
only need to verify condition (3) in Lemma 2.1. First, note that these conditions
imply thateachpointx isa Gδ-setin both(X,τ1) and(X,τ2). Suppose thatthere
are a τ -compact set K and a U ∈ τ with K ⊆ U, but K 6⊆ F (n,U) for any
i i ij
n ∈ N. Then, there is a sequence {x : n ∈ N} ⊆ K such that x 6∈ F (n,U) for
n n ij
any n ∈N. Since K is τ -compact and points are G in (X,τ ), then {x : n ∈N}
i δ i n
must have a subsequence {x : k ∈ N} which is τ -convergent to a point x ∈ K
nk i
and xnk 6∈Fij(nk,U) for all k ∈N. By condition (3) above, there exist an m0 ∈N
and an n0 ∈N such that
{x}∪{xnk :k >m0}⊆Fij(n0,U)⊆U.
It follows that for any k ∈N with k ≥m0 and nk ≥n0, we have
xnk ∈Fij(n0,U)⊆Fij(nk,U).
Apparently,thiscontradictswiththechoiceofx . Wehaveverifiedthatcondition
nk
(3)inLemma2.1issatisfied,andthus(X,τ1,τ2)ispairwisek-semi-stratifiable. (cid:3)
Note that Theorems 3.3 and 3.5 may shed some light on relationships between
pairwise k-semi-stratifiability and the other generalized metric properties of bis-
paces studied in [25] and other places.
8 KEDIANLIANDJILINGCAO∗
4. When is a pairwise k-semi-stratifiable bispaces
pairwise straitifiable?
In this section, we consider the problem when a pairwise k-semi-stratifiable bis-
pace is pairwise stratifiable. An open question posed in [18] is completely solved,
and some results in [2], [3] and [8] are extended to the setting of bispaces.
Recallthatatopologicalspace(X,τ)issaidtobeFr´echet,ifforeverynonempty
subset A⊆ X and every point x∈ Aτ, there is a sequence {x : n ∈ N}⊆A such
n
that {x :n∈N} converges to x.
n
In a recent paper [18], Li and Lin posed the following open question (see [18,
Question 3.4]).
Question 4.1 ([18]). Is a pairwise k-semi-stratifiable bispace (X,τ1,τ2) pairwise
stratifiable if (X,τ ) is a Fr´echet space for each i=1,2?
i
Our next theorem answers Question 4.1 affirmatively. Note that our theorem
also extends a result in [8] to the setting of bispaces.
Theorem 4.2. Let (X,τ1,τ2) be a pairwise k-semi-stratifiable bispace. If both
(X,τ1) and (X,τ2) are Fr´echet spaces, then (X,τ1,τ2) is pairwise stratifiable.
Proof. In the light of Corollary 2.3, we need to show that (X,τ1,τ2) is pairwise
monotonicallynormal. Foranyfixedi,j =1,2withi6=j,letF :N×τ →F (X)
ij i j
be an operator that is monotone with respect to n and satisfies (1)-(3) in Lemma
2.1. For each pair (H,K) of disjoint subsets of X such that H ∈ F (X) and
i
K ∈F (X), define D (H,K) by
j ij
D (H,K):=int (F (n,X rK)rF (n,X rH))
ij τj ji ij
n∈N !
[
Next, we shall verify that D (·,·) satisfies all conditions in the definition of a
ij
pairwise monotonically normal bispace.
Clearly, D (H,K)∈τ and D (·,·) satisfies condition (ii) in the definition of a
ij j ij
pairwise monotonically normal bispace. Also note that D (H,K)⊆X rK holds
ij
trivially.
Claim 1. H ⊆D (H,K).
ij
Proof of Claim 1. Suppose that there is a point x0 ∈H rDij(H,K). Then
x0 ∈ Xrintτj (Fji(n,XrK)\Fij(n,X rH))
n∈N !
[
τj
= Xr (F (n,X rK)rF (n,X rH)) .
ji ij
n∈N
[
Since (X,τ ) is a Fr´echet space, there is a sequence {x :n∈N} such that
j n
{x :n∈N}⊆X r (F (n,X rK)rF (n,X rH))
n ji ij
n∈N
[
and{xn :n∈N}isτj-convergenttox0. Notethatx0 ∈H impliesthatx0 ∈XrK.
Thus, there is an m∈N such that
C ={x0}∪{xn :n≥m}⊆X rK.
PAIRWISE k-SEMI-STRATIFIABLE BISPACES AND TOPOLOGICAL ORDERED SPACES 9
By (3) in Lemma 2.1, there is an m′ ≥ m such that C ⊆ F (m′,X rK). On the
ji
other hand, note that there must be some p ≥m′ such that x 6∈ F (m′,X rH).
p ij
Otherwise, as F (m′,X rH) is τ -closed, we conclude that
ij j
x0 ∈Fij(m′,X rH)⊆XrH,
which contradicts with the fact x0 ∈H. It follows that
x ∈ F (m′,X rK)rF (m′,X rH)
p ji ij
⊆ (F (n,X rK)rF (n,X rH)).
ji ij
n∈N
[
This certainly contradicts with the selection of {x : n ∈ N}. Hence, Claim 1 has
n
been verified. (cid:3)
Claim 2. D (H,K)τi ⊆X rK.
ij
Proof of Claim 2. Suppose that there is a point x0 ∈ Dij(H,K)τi ∩ K. Since
(X,τ ) is a Fr´echet space, there is a sequence {x : n∈N}⊆D (H,K) such that
i n ij
{xn :n∈N} is τi-convergentto x0. Note that x0 ∈K implies x0 ∈X rH. Thus,
there exists an m∈N such that
{x0}∪{xn :n≥m}⊆XrH.
By condition (3) in Lemma 2.1, there exists some p≥m such that
{x0}∪{xn :n≥m}⊆Fij(p,XrH).
By the selection of {x :n∈N}, we know that
n
{x :n≥m}⊆ (F (n,XrK)rF (n,XrH)),
n ji ij
n∈N
[
which implies that
p−1
{x :n≥m}⊆ (F (n,X rK)rF (n,X rH)).
n ji ij
n=1
[
It follows that there are a q ∈N with 1≤q <p and a subsequence {x :k ≥m}
nk
of {x :n∈N} such that
n
{x :k ≥m}⊆F (q,X rK)rF (q,X rH).
nk ji ij
We concludethatx0 ∈Fji(q,XrK)⊆XrK. Thiscontradictswiththe selection
of x0, and thus Claim 2 has been verified. (cid:3)
CombiningClaims 1 and2, wesee thatD (·,·) alsosatisfiescondition(i) inthe
ij
definition of a pairwise monotonically normal bispace. (cid:3)
Let (X,τ) be a topological space, and let x ∈ X be a point. The collection
of neighborhoods of x in (X,τ) is denoted by N (τ,x). We shall consider the
following G(τ,x)-game played in (X,τ) between two players: α and β. Player
α goes first and chooses a point x1 ∈ X. Player β then responds by choosing
U1 ∈ N (τ,x). Following this, α must select another (possibly the same) point
x2 ∈U1 and in turn β must again respond to this by choosing (possibly the same)
U2 ∈N (τ,x). Theplayersrepeatthisprocedureinfinitelymanytimes,andproduce
aplay(x1,U2,x2,U2,··· ,xn,Un,···)intheG(τ,x)-game,satisfyingxn+1 ∈Un for
all n ∈ N. We shall say that β wins a play (x1,U2,x2,U2,··· ,xn,Un,···) if the
10 KEDIANLIANDJILINGCAO∗
sequence {x : n ∈ N} has a cluster point in X. Otherwise, α is said to have won
n
the play. By a strategy s for β, we mean a ‘rule’ that specifies each move of β in
every possible situation. More precisely, a strategy s for β is an N (τ,x)-valued
function. We shall call a finite sequence {x1,x2,...,xn} or an infinite sequence
{x1,x2,...} an s-sequence if xi+1 ∈ s(x1,x2,...,xi) for each i such that 1 ≤ i < n
or xn+1 ∈ s(x1,x2,...,xn) for each n ∈ N. A strategy s for player β is called a
winning strategy if each infinite s-sequence has a cluster point in X. Finally, we
call x a G(τ)-point if player β has a winning strategy for the G(τ,x)-game. In
addition, if every point of X is a G(τ)-point, then (X,τ) is called a G-space [1].
The notion of G-spaces is a common generalization of the concepts of q-spaces in
[24] and W-spaces in [9].
The next result extends [2, Theorem 2.2.8] and [3, Theorem 3.2] to the setting
of bispaces.
Theorem 4.3. Let (X,τ1,τ2) be a pairwise k-semi-stratifiable bispace. If both
(X,τ1)and(X,τ2)areregular andG-spaces, then (X,τ1,τ2)is pairwise stratifiable.
Proof. Let(g1,g2)beapairwiseg-functionasdescribedincondition(3)ofTheorem
3.2. For i,j = 1,2 with i 6= j, we define G : N×F (X) → τ such that for each
ij i j
n∈N and each H ∈F (X),
i
G (n,H)= {g (n,x):x∈H}.
ij j
Clearly, if H,K ∈ F (X) with H ⊆[K, then G (n,H) ⊆ G (n,K) for all n ∈ N.
i ij ij
Furthermore, it is easy to see that H ⊆ G (n,H)τi for all H ∈F (X).
n∈N ij i
Supposethatthereareapointx∈X andaτ -closedsubsetH inX withx6∈H,
T i
butx∈G (n,H)τi foreveryn∈N. First,wechoosesomeτ -openneighborhoodU
ij i
ofxsuchthatUτi∩H =∅. Since(X,τ )isaG-space,β hasawinningstrategysfor
i
the G(τi,x)-game. Let α’s first move be x1. By our assumption and the definition
of Gij(·,·), there must exist some point y1 ∈H suchthat s(x1)∩U ∩gj(1,y1)6=∅.
Inductively, we can obtain two sequences {x :n∈N} and {y :n∈N} in X such
n n
that for each n∈N, y ∈H and
n
xn+1 ∈U ∩gj(n+1,yn+1)∩ s(xi1,...,xij).
1≤j≤n,
1≤i1≤\...≤ij≤n 
It follows that each subsequence of {x :n ∈ N} is an s-sequence in (X,τ ), and
n i
thus has an cluster point in (X,τ ). Since each point of X is a G -point in (X,τ ),
i δ i
then {x : n ∈ N} must have a convergent subsequence, saying {x : k ∈ N},
in (X,τin). Suppose that {xnk : k ∈ N} is τi-convergent to some poinnkt x∗ ∈ Uτi.
Then, by condition (3) in Theorem 3.2, and the construction of {x : n ∈ N} and
n
x{y∗n∈:Hn.∈ItNfo}lloinwsththeaatbxo∗ve∈,U{yτink∩H: k. W∈eωh}aviseadlesroivτeid-caoncvoenrtgreandticttoionx.∗,Tahnedreftohrues,
x 6∈ G (n,H)τi for some n ∈ N. We have verified that H = G (n,H)τi for
ij n∈N ij
all H ∈Fi(X) and thus (X,τ1,τ2) is pairwise stratifiable. (cid:3)
T
5. Quasi-pseudo-metrizability of topological ordered spaces
In [15], Ku¨nzi and Mushaandja posed the following open problem (refer to [15,
Problem 1]).