Coproducts of abelian topological groups

24 Pages·2002·0.2 MB·English

View metadata, citation and similar papers at core.ac.uk brought to you by CORE
provided by Elsevier - Publisher Connector
TopologyanditsApplications120(2002)403–426
Coproducts of abelian topological groups
PeterNickolas
SchoolofMathematicsandAppliedStatistics,UniversityofWollongong,Wollongong,NSW2522,Australia
Received12September2000;receivedinrevisedform7December2000
Abstract
Thispaperisastudyofcertaintopologicalgrouptopologiesontheweakorrestricteddirectsum
ofafamilyof abeliantopological groups. Amongst thetopologies studiedarethetopology of the
coproduct inthecategoryof abeliantopological groups andKaplan’sasterisktopology. Asimple,
explicitdescriptionofthecoproduct topology isgiven, andadetailedinvestigationismadeofthe
relations between the coproduct topology, the asterisk topology and several other topologies of
interest. The reflexivity properties of the topological coproduct are also analysed. In particular, it
isshownthatthecoproductofafamilyoflocallycompactHausdorffabeliantopologicalgroupsis
reflexiveifandonlyifallbutcountablymanyofthegroupsarediscrete.2002ElsevierScience
B.V.Allrightsreserved.
AMSclassification:Primary22A05;22A99,Secondary22B05;22B99;20K25;54G10;54H11
Keywords:Coproductsofabeliantopologicalgroups;Reflexivityofabeliantopologicalgroups;
Pontrjaginduality;Theboxtopology;Theasterisktopology;Locallycompactabeliantopological
groups;Coproductsofabeliangroups;Weakdirectsumsofabeliangroups;Restricteddirectsums
ofabeliangroups;P-spaces
1. Introduction
The existence and certain fundamental properties of the coproduct in the category
of abelian topological groups can be established by essentially standard arguments of
a general, often categorical, nature. More detailed information, however, requires closer
analysis. In this paper,we begin such an analysis. We have three main aims. The first is
to give a simple, explicit description of the coproducttopology. Recent years have seen
such descriptions given of the topologies of free topological groups and free uniform
groups[9,12],andoffreeabelianuniformgroups[13],andourdescriptionofthecoproduct
topology is a logical addition to this set of results. Our second aim is to investigate a
E-mailaddress:[email protected](P.Nickolas).
0166-8641/02/$–seefrontmatter 2002ElsevierScienceB.V.Allrightsreserved.
PII:S0166-8641(01)00085-2
404 P.Nickolas/TopologyanditsApplications120(2002)403–426
rangeoftopologiesontheabstractcoproductandtheirrelationstooneanotherandtothe
coproduct topology. Apart from the coproduct topology itself, some of these topologies
were introduced specifically in order to study the coproduct [6], others were introduced
in a studyofthe reflexivityof generalabeliantopologicalgroups[7],andstill othersare
introducedhere,partlytoprovideauniformframeworkforthewholediscussion.Ourthird
aimis to examinethereflexivitypropertiesofthe coproduct.Onceagain,workof recent
yearshasbroughtourunderstandingofthereflexivitypropertiesoffreeabeliantopological
groupstoalargelysatisfactorylevel[2,10],anditisnaturaltoseekasimilarunderstanding
in the case of coproducts.From our analysis it follows that the coproductof a family of
locally compact Hausdorff abelian topological groups is reflexive if and only if all but
countablymanyofthegroupsarediscrete.
Let {Ai} = {Ai}i∈I be a family of abelian topological groups. As noted above, the
existence and basic properties of the coproduct of the family {A } in the category of
i
abelian topologicalgroupscan be provedby straightforwardarguments.(A few of these
areoutlinedintheopeningparagraphsof[6].)Someofthepropertiesthatwewilltakefor
grantedandusefreelyhere,alongwiththeexistencestatement,areasfollows.
First, the underlying abstract group of the topological coproduct is the coproduct in
the category of abelian groups of the family of abstract groups underlying the {A }:
(cid:1) i
namely,theweakorrestricteddirectsum A ofthe{A }.(Wedenotetheunderlying
i∈I i i
abstractgroupofA byA again.)Thus,atypicalelementofthe(algebraicortopological)
i i
coproduct has the form (xi)=(xi)i∈I, where xi ∈Ai for all i and xi =0 except for a
finite number of indices i, and we may write the element (x ) if need be in the form
(cid:1) i
x .Second,thecanonicalmappingofeachgroupA intothetopologicalcoproduct
i∈I i i
isatopologicalisomorphismontoitsimage.ThisfactallowsustoidentifyeachA withits
i
imageinthecoproduct,notonlyalgebraicallybutalsotopologically.Third,thetopology
(cid:1)
on the topological coproduct is the strongest topological group topology on A
i∈I i
that induces the original topologies on the {A }. In particular, the coproductof a family
i
of discrete abelian topological groups is discrete. Fourth, there is a natural associativity
in the formation of the coproduct, which we can characterise informally by saying that
coproductsofcoproductsarecoproducts.
Thecoproducttopologyisjustoneofanumberoftopologiesontheabstractcoproduct
(cid:1) (cid:1)
A thatwe willdiscussbelow,andforthisreasonwe reservethenotation A
i∈I i i∈I i
for the abstract coproduct, and specify any associated topology T explicitly, as in
(cid:1)
( A ,T).
i∈I i
2. Topologiesonthecoproduct
2.1. Ageneralconstruction
LetGbeafixedabeliantopologicalgroup.Wewillcallasequence{Un}≡{Un}n∈N of
openneighbourhoodsof 0 in G a descendingsequenceif U ⊇U ⊇U ⊇···. We will
1 2 3
further call a collection U of descending sequences in G a compatible collection if the
followingconditionsaresatisfied.
P.Nickolas/TopologyanditsApplications120(2002)403–426 405
(U-l) ForeachopenneighbourhoodU of0,thereexists{U }∈UsuchthatU ⊆U.
n 1
(U-2) Foreach{Un}∈U,thereexists{Vn}∈UsuchthatVn⊆Un+1 foralln∈N.
(U-3) Foreach{U }∈U,thereexists{V }∈Usuchthat2V ⊆U foralln∈N.
n n n n
(U-4) Foreach{U }∈U,thereexists{V }∈Usuchthat−V ⊆U foralln∈N.
n n n n
(U-5) Forall{U }and{V }∈U,thereexists{W }∈UsuchthatW ⊆U ∩V forall
n n n n n n
n∈N.
ForanyfixeddescendingsequenceU ={U }∈U,andforx∈G,wedefine
n

1, if x∈/U ,
1
(x/U)=1/2n, if x∈Un and x∈/Un+1 forsome n∈N,
0, if x∈U forall n∈N.
n
(cid:1)
Nowlet{Ai}i∈I beafamilyofabeliantopologicalgroups,andwriteA= i∈IAi.Sup-
posethatU isa compatiblecollectionofdescendingsequencesofopenneighbourhoods
i
of0inAi foreachi∈I.Fo(cid:5)reachfamily(Ui)≡(Ui)i∈I ofdescendingsequenceschosen
from the cartesian product U , we define (omitting the inner pair of parentheseson
i∈I i
theleft)
(cid:6) (cid:8)
(cid:7)
N(U )= (x )∈A: (x /U )<1 .
i i i i
i∈I
(This definition is inspired by one of Kaplan [7], which we will reproduceand study in
Section2.2andlaterpartsofthispaper.)
(cid:5)
Theorem 2.1. The collection of all sets N(Ui), as (Ui) runs through i∈IUi, forms an
openneighbourhoodbaseat0inAforatopologywhichmakesAintoatopologicalgroup
andinducestheoriginaltopologyoneachfactorA .
i
Proof. Weusethecriteriaof[5,Theorem4.5]toshowthatthesetsN(U )defineagroup
i
topology.
(i) The first criterionrequiresthatfor each family (U ) thereexists a family (V ) such
i i
that2N(V )⊆N(U ).Consideragiven(U ),withU ={U(i)}foreachi.Usingconditions
i i i i n
(U-2)and(U-3),wecanfinda (V ),withV ={V(i)} foreachi,suchthat2V(i)⊆U(i)
i i n n n+1
for each n. For any fixed i, and for x,y∈A , suppose that max((x/V ),(y/V ))(cid:1)1/2n
i i i
for some n (cid:2) 1; then we have x,y ∈ V(i), and hence x + y ∈ U(i) , and therefore
n n+1
((x+y)/U )(cid:1)1/2n+1.Hencewehave,forsuchanx andy,
i
(cid:9) (cid:10) (cid:9) (cid:10)
(x+y)/U (cid:1) 1max (x/V ),(y/V ) . (1)
i 2 i i
Supposenowthat(x ),(y )∈N(V ),sothatwehave
i i i
(cid:7) (cid:7)
(x /V )<1 and (y /V )<1.
i i i i
i∈I i∈I
Clearly,foreachi thereisann(cid:2)1suchthatmax((x/V ),(y/V ))(cid:1)1/2n.Ifweset
i i
(cid:11) (cid:12) (cid:11) (cid:12)
I = i: (x /V )(cid:2)(y /V ) and I = t: (x /V )<(y <V ) ,
1 i i i i 2 i i i i
then,using(1),wehave
406 P.Nickolas/TopologyanditsApplications120(2002)403–426
(cid:7)(cid:9) (cid:10) (cid:7)(cid:9) (cid:10) (cid:7)(cid:9) (cid:10)
(x +y )/U = (x +y )/U + (x +y )/U
i i i i i i i i i
i∈I i∈I(cid:7)1 (cid:7)i∈I2
(cid:1) 1 (x /V )+ 1 (y /V )
2 i i 2 i i
i∈I1 i∈I2
< 1 + 1 =1,
2 2
andwehave(x )+(y )=(x +y )∈N(U ),asrequired.
i i i i i
(ii)Thesecondcriterionrequiresthatforeach(U )thereexists(V )suchthat−N(V )⊆
i i i
N(U ),andthisisprovedsimilarly,usingcondition(U-4).
i
(iii)Thethirdcriterionrequiresthatforeach(U )andforeach(x )∈N(U ),thereexists
i i i
(V )suchthat(x )+N(V )⊆N(U ).Let
i i i i
(cid:11) (cid:12)
J = i∈I: (x /U )>0 ,
(cid:11) i i (cid:12)
K= i∈I: (x /U )=0andx (cid:12)=0
i i i
and
L={i∈I: x =0}.
i
Since(x )∈N(U ),wehave
(cid:7)i i (cid:7)
(x /U )= (x /U )=s<1,
i i i i
i∈I i∈J
say.Choosep,q∈Nsuchthat
1 1−s 1 1−s
< and < .
2p 2|K| 2q 2
For i ∈J, we have x ∈U(i), where l = −log (x /U ), and we pick an open neigh-
i li i 2 i i
bourhoodV of0inA suchthatx +V ⊆U(i).Fori∈K,wehavex ∈U(i)foralll∈N,
i i i i li i l
andwepickanopenneighbourhoodV of0inA suchthatx +V ⊆U(i).Fori∈L,we
i i i i p
havex =0,andwepickanopenneighbourhoodV of0inA suchthatV ⊆U(i) .We
i i i i q+1
nowconstructV ={V(i)}foralli∈I,asfollows:foralli∈J∪K,weusecondition(U-1)
i n
to choose {V(i)} so that V(i) ⊆V ; and for all i ∈L, we use condition (U-2) to choose
n 1 i
{Vn(i)}sothatVn(i)⊆Uq(i+)n foralln.Let(yi)∈N(Vi).Thenfori∈J,wehave
x +y ∈x +V(i)⊆x +V ⊆U(i),
i i i 1 i i li
sothat
(cid:9) (cid:10)
1
(x +y )/U (cid:1) =(x /U ),
i i i 2li i i
andtherefore
(cid:7)(cid:9) (cid:10) (cid:7) (cid:7)
(x +y )/U (cid:1) (x /U )= (x /U )=s. (2)
i i i i i i i
i∈I i∈J i∈I
Fori∈K,wesimilarlyhavex +y ∈U(i),sothat((x +y )/U )(cid:1)1/2p,andtherefore
i i p i i i
(cid:7)(cid:9) (cid:10) |K| 1−s
(x +y )/U (cid:1) < . (3)
i i i 2p 2
i∈K
P.Nickolas/TopologyanditsApplications120(2002)403–426 407
Fori∈L,wehavex +y =y ,andsince
i i i
(cid:7) (cid:7)
(y /V )(cid:1) (y /V )<1
i i i i
i∈L i∈I
andVn(i)⊆Uq(i+)n foralln,wehave
(cid:7)(cid:9) (cid:10) (cid:7) 1 1−s
(x +y )/U = (y /U )< < . (4)
i i i i i 2q 2
i∈L i∈L
From(2),(3),and(4),wethereforehave
(cid:7)(cid:9) (cid:10)
(x +y )/U <1,
i i i
i∈I
andweconcludethat(x )+N(V )⊆N(U ),asrequired.
i i i
(iv)Thefourthcriterionistrivialforabeliangroups.
(v)The fifth criterionrequiresthatfor each (U ) and (V ), there exists (W ) such that
i i i
N(W )⊆N(U )∩N(V ),andthisisprovedeasilyusingcondition(U-5).
i i i
Thus, the sets N(U ) induce a group topology on A. Finally, we confirm that the
i
factorsA inherittheiroriginaltopologies.Fixi ∈I.IfN =N(U ),thenwehave
i 0 i
(cid:11) (cid:12)
N∩A = (x ): x =0fori(cid:12)=i and(x /U )<1
i0 (cid:11) i i 0 i0 i0 (cid:12)
= (x ): x =0fori(cid:12)=i andx ∈U(i0)
i i 0 i0 1
= U(i0).
1
Therefore,theinducedtopologyonA iscontainedintheoriginaltopology.Conversely,
i0
if U is an arbitraryopenneighbourhoodof 0 in A , then by condition(U-l) there exists
i0
(U ) such that U(i0)⊆U, andso bythe aboveargument,we have N ∩A =U(i0)⊆U,
i 1 i0 1
andtheoriginaltopologyonA iscontainedintheinducedtopology. ✷
i0
(cid:1)
WedenotebyT({U })thetopologydefinedonA= A bythecollections{U }.
i i∈I i i
2.2. Thebox,asteriskandcoproducttopologies
Therearethreecasesoftheaboveconstructionthatwillbeofspecialinterest.
First,takeeachU tobethecollectionofconstantsequencesinA ;thatis,U consistsof
i i i
allsequencesU ={U(i)}inwhichU(i)=U(i)=···=U forsomeopenneighbourhoodU
n 1 2
of0inA .ItiseasytoseethattheU satisfyconditions(U-l)to(U-5).Thenthetopology
i (cid:1) i
T({U })onA= A istheboxorrectangulartopology.Thistopologyismoredirectly
i i∈I i
describedasthesubspacetopologyinducedonAbytheboxtopologyonthefullcartesian
(cid:5) (cid:5)
product A , which has as an open base the sets of the form U , where U is
i∈I i i∈I i (cid:5) i
openin A foreach i.We willdenotethe correspondingrelativelyopenset A∩ U
(cid:1)i i∈I i
inAby U .(Weemphasisethatwhiletheterm“boxtopology”isnormallyusedfor
i∈I i
the topology just mentioned on the full cartesian product, we shall use it from now on
(cid:1)
exclusivelyfortheinducedtopologyonthesubgroupA= A .)Atypicalopenneigh-
i∈I i
bourhoodof 0 (denotedby N(U ) in the generalcase) will be denotedin the case of the
i
408 P.Nickolas/TopologyanditsApplications120(2002)403–426
(cid:1)
boxtopologyby N✷(Ui) (sothat N✷(Ui)= i∈IUi),andthe boxtopology(denotedby
T({Ui})inthegeneralcase)willbedenotedbyT✷.
Second,foranyabeliantopologicalgroupG,andanyopenneighbourhoodU of0inG,
wedefine,followingKaplan[7],
(1/2)U ={x: x∈U and2x∈U}.
(Indeed it is convenientfor later if we permit (1/2)X to denote {x: x ∈Xand2x ∈X}
for an arbitrary subset X of G.) It is easy to see that (1/2)U is again an open neigh-
bourhoodof0,andthat(1/2)U istheunionofallopenneighbourhoodsV of0suchthat
2V ⊆U. (Simple examples show that 2((1/2)U) is not in general contained in U; but
seeLemma4.7,below.)Inductively,wenowdefineadescendingsequenceofopenneigh-
bourhoods(1/2n)U of 0, by setting (1/2n)U =(1/2)(1/2n−1)U, for n=2,3,.... (It is
alsoconvenientattimestopermitn=0intheconstruction,andtointerpret(1/20)U asU.)
Itiseasytocheck(using,forexample,observationsmadebyKaplan)thatthecollectionof
alldescendingsequencesU ={U },whereU =(1/2n)U forsomeopenneighbourhood
n n
U of0inG,formsacompatiblecollectioninG.
(cid:1)
If we formthe topologyon A= A inducedby the collectionsU constructedin
i∈I i i
thiswayoneachgroupA ,thenwehavetheasterisktopology,introducedbyKaplan[7]
i
inhisproofthata(fullcartesian)productofreflexiveHausdorffabeliantopologicalgroups
with the Tychonoff topology is again reflexive. We will denote a typical open neigh-
bourhoodN(Ui) of0intheasterisktopologybyN∗(Ui),andwewilldenotetheasterisk
topologyT({Ui})itselfbyT∗.
Ourthirdcase iswherewe take U tobe thecollectionofalldescendingsequencesof
i
openneighbourhoodsof 0 in A for each i. Again,clearly, U is a compatiblecollection
i i
inA foreachi.Here,wedenotethecorrespondingopenneighbourhoodsN(U )inAby
i i
N (U ),andthetopologyT({U })byT .
(cid:29) i i (cid:29)
(cid:1)
Theorem2.2. Thetopologicalgroup( A ,T )isthecoproductofthefamily{A }in
i∈I i (cid:29) i
thecategoryofabeliantopologicalgroups.
Proof. First,byTheorem2.1,T inducestheoriginaltopologyoneachfactorgroupA .
(cid:29) (cid:1) i
Second, T(cid:29) induces a topology on A = i∈IAi which is at least as strong as the
coproducttopology. To see this, let U be an open neighbourhoodof 0 in the coproduct
topology, choose another open neighbourhood V of 0 such that 2V ⊆U, and, setting
V =V, chooseasequenceV ,V ,V ,...ofopenneighbourhoodsof0inthecoproduct
0 1 2 3
topologysatisfying
2V1⊆V0, 4V2⊆V1, 8V3⊆V2, ..., 2nVn⊆Vn−1, .... (5)
For each i ∈ I and each n ∈ N, set U(i) = V ∩ A , and for each i, write U =
n n i i
{U(i),U(i),...}. Clearly, the U(i) are open in A , and U(i) ⊇ U(i) ⊇ ···, and the set
1 2 n i 1 2
N =N (U )isthereforeintheopenneighbourhoodbaseat0whichdefinesT .Weclaim
(cid:29) i (cid:29)
thatN ⊆U.
P.Nickolas/TopologyanditsApplications120(2002)403–426 409
Fix(x )∈N.Thenx =0exceptwheni liesinsomefinitesetofindices,andthisfinite
i i
setmaybepartitionedintodisjointsetsofindicesJ andKsuchthat(x /U )>0forj ∈J
j j
and(x /U )=0fork∈K.Inparticular,
k k
(cid:7) (cid:7) (cid:7) (cid:7) (cid:7)
(x )= x = x + x and (x /U )= (x /U )<1.
i i j k i i j j
i∈I j∈J k∈K i∈I j∈J
For each j ∈ J, the number (x /U ) has the form 1/2p for some positive integer p
j j
which depends upon j. Let us represent the distinct values amongst these integers p as
n <n <···<n ,andthesevalues’respectivemultiplicitiesas c ,c ,...,c .Thus,we
1 2 r 1 2 r
have
(cid:7)
x ∈c V +c V +···+c V ,
j 1 n1 2 n2 r nr
j∈J
(cid:1)
where,since j∈J(xj/Uj)<1, we have 1(cid:1)cm(cid:1)2nm −1 for m=1,2,...,r. Indeed,
byallowingzerovaluesforthemultiplicities(andinterpreting0V foranyneighbourhood
V tomean{0}),wecanwrite,moresimply,
(cid:7)
xj ∈c1V1+c2V2+···+cN−1VN−1+cNVN, (6)
j∈J
forsomeN ∈N,where0(cid:1)c (cid:1)2m−1form=1,2,...,N.
m
Foreachk∈K,wehavex ∈U(k)=V ∩A ⊆V foralln∈N,soforanynforwhich
k n n k n
2n(cid:2)|K|,wehave
(cid:7)
xk∈(cid:13)Vn+Vn(cid:14)+(cid:15)···+Vn(cid:16)=|K|Vn⊆2nVn⊆Vn−1,
k∈K |K|times
(cid:1)
by(5).Thus,wehave x ∈V ,forallsufficientlylargen.
k∈K k n
Therefore,reusingthenotationof(6)above,wemayextend(6)toread
(cid:7) (cid:7)
xj + xk∈c1V1+c2V2+···+cN−1VN−1+cNVN,
j∈J k∈K
forsomeN ∈N,where0(cid:1)c (cid:1)2m−1form=1,2,...,N.Therefore,
m
(cid:7) (cid:7)
(x ) = x + x
i j k
j∈J k∈K
∈ c(cid:9)1V1+(cid:10)c2V2+(cid:9)···+c(cid:10)N−1VN−1+(cid:9)cNVN (cid:10) (cid:9) (cid:10)
⊆ 21−1 V1+ 22−1 V2+···+ 2N−1−1 VN−1+ 2N −1 VN
⊆ 21V1+22V2+···+2N−1VN−1+2NVN.
Now,byapplying(5)toeachterm,wefind
21V1+22V2+···+2N−1VN−1+2NVN
⊆V0+V1+V2+V3+···+VN−3+VN−2+VN−1
⊆V0+V1+V2+V3+···+VN−3+VN−2+VN−2
⊆V0+V1+V2+V3+···+VN−3+VN−3
410 P.Nickolas/TopologyanditsApplications120(2002)403–426
⊆V0+V1+V2+V3+···+VN−4
.
.
.
⊆V +V +V +V
0 1 2 2
⊆V +V +V
0 1 1
⊆V +V
0 0
=2V
⊆U.
Hence,(x )∈U, whichprovesthe secondassertion.Since the coproducttopologyis the
i
strongestgrouptopologyonAinducingtheoriginaltopologiesonthefactorsA ,wehave
i
theresult. ✷
2.3. Othertopologiesonthecoproduct
(cid:1)
Higgins[6] discusses two familiesof grouptopologieson A= A in additionto
i∈I i
theboxtopologyandtheasterisktopology,whichareingeneraldistinctfromeachother
andfromthecoproducttopology.Thesetopologiesareofinteresttoushere,andweoutline
theconstructions.
First,Higginsintroducesthep-topologies,forp>0,afamilyofvariantsoftheasterisk
topology.Usingthesamedescendingsequences(U )thatwereusedtodefinetheasterisk
i
topology,write
(cid:6) (cid:8)
(cid:7)
N (U )= (x )∈A: (x /U )p<1 ,
p i i i i
i∈I
foreach realnumberp>0. These sets define,for eachfixed p, a grouptopologyon A,
whichHigginscallsthep-topology,andwedenotebyTp.Clearly,T1=T∗.
Second,Higginsintroducesa moregeneralfamilyofvariantsoftheasterisk topology.
Givenafunctionf whichsatisfiestheconditions
(i) f(0)=0,
(ii) f iscontinuousat0,and (7)
(iii) f isstrictlyincreasingon[0,1],
andwiththe(U )asfortheasterisktopology,write
i
(cid:6) (cid:8)
(cid:7)
N{f}(Ui)= (xi)∈A: f(xi/Ui)<1 .
i∈I
Again, these sets define, for each fixed f, a group topology on A, which we denote by
T{f}.Clearly,T{f}=Tp iff isthefunctionf(t)=tp.
Wenowintroduceastillmoregeneralfamilyofvariantsoftheasterisktopology.Given
afamilyoffunctions{fi}≡{fi}i∈I satisfyingtheconditions(7),andwiththe(Ui)again
thesame,write
(cid:6) (cid:8)
(cid:7)
N{fi}(Ui)= (xi)∈A: fi(xi/Ui)<1 .
i∈I
P.Nickolas/TopologyanditsApplications120(2002)403–426 411
Onceagain,thesesetsdefine,foreachfamily{f },agrouptopologyonA,whichwedenote
i
byT{fi}.Clearly,T{fi}=T{f} iffi =f foreachi.
Finally, for completeness, we mention the subspace topology induced on A by the
(cid:5)
Tychonoffproducttopologyonthefullcartesianproduct A .Wedenotethistopology
i∈I i
byT .
$
3. Relationsbetweenthetopologies
Inthissection,weexaminetherelationsamongstthevarioustopologies,andespecially
thequestionofequalitybetweenthecoproducttopologyandtheothertopologies.
3.1. ExtensionstoHiggins’swork
We notethatthereisonlyonegrouptopologyona finitecoproductwhichinducesthe
originaltopologyonallthefactorgroups.Thus,inthecasewheretheindexsetI isfinite,
allofthetopologiesintroducedaboveareequaltothecoproducttopology.
In the case where I is countable, Roelcke and Dierolf [11, p. 61] (see Theorem 3.3
below)proveastatementwhichimpliesthattheboxtopologyandthecoproducttopology
are equal (and are therefore, by Theorem 3.1 below, also equal to the asterisk topology
andthep-topologies).(Theresultof[11]isstatedinaformwhichappliestogroupsthat
are not necessarily abelian, but reduces to our statement in the abelian case.) The case
of a countable coproduct of locally compact Hausdorff abelian topological groups had
been settled earlier, in [1]. The result of [11] also subsumes Theorem 1 of Kaplan [7],
which states that the box topology and the asterisk topology are equal on all countable
coproducts.InTheorem3.10,below,weresolvecompletelythequestionofwhenthebox
andcoproducttopologiescoincide.
Wehavethefollowingstraightforwardrelationsbetweentheabovetopologies(cf.[7,6]
andourearlierremarks).
Theorem3.1. Forallcoproducts:
• T$⊆T✷⊆Tq ⊆Tp⊆T(cid:29),for0<p(cid:1)q;
• T1=T∗;
• T{f}=Tp iff isthefunctionf(t)=tp;and
• T{fi}=T{f} iffi =f foreachi.
Higgins[6]considersthefollowingconditiononanabeliantopologicalgroup:
ThereisanopenneighbourhoodU of0suchthateveryneighbourhoodof0
containsaset(1/2n)U, forsomen∈N. (8)
HeshowsthatiftheindexsetI isuncountable,andifeachA isnon-discreteandHausdorff
i
andhasproperty(8),thenT isstrictlyfinerthanT whenever0<p<q,anditfollows
p q
that noneof the topologiesT$,T✷,T∗ and Tp (forany p>0) is equalto T(cid:29). (It is also
shown in [11]that T✷ is strictly weaker than T(cid:29) on an uncountablecoproductof copies
412 P.Nickolas/TopologyanditsApplications120(2002)403–426
ofR;clearly,Rhasproperty(8).)Moreover,therearefunctionsf forwhichthetopology
T{f} isstrictlyfinerthanthesupremumofallthetopologiesTp,sothissupremumisalso
strictlyweakerthanT .
(cid:29)
Incontrasttothelaststatement,wehavethefollowingresult.
Theorem 3.2. Suppose that each abelian topological group in the family {A } has
(cid:1) i
property (8). Then the coproduct topology T on A= A is the supremum of the
(cid:29) i∈I i
topologiesT{fi},takenoverallfamilies{fi}offunctionssatisfyingtheconditions(7).
Proof. FixadefiningopenneighbourhoodN (U )of0inA,whereU ={U(i),U(i),...}
(cid:29) i i 1 2
for each i. By hypothesis,there is for each i an openneighbourhoodV of 0 in A such
i i
that every neighbourhoodof 0 in A contains a set (1/2n)V for some n. Therefore, for
i i
each fixed i, there are integers n , without loss of generality increasing in j, such that
ij
(1/2nij)Vi ⊆Uj(i+)1 for all j. Define the function fi by linear interpolation between the
values
(cid:9) (cid:10)
f 1/2nij = 1/2j, forj ∈N,
i
f (0) = 0,
i
f (1) = 1.
i
Clearly,f iscontinuous,andbecausethen werechosentobeincreasinginj,itfollows
i ij
that f is strictly increasing,and is thereforea functionsatisfyingthe conditions(7).We
i
claimthat,withVi ={(1/2n)Vi}foralli,wehaveN{fi}(Vi)⊆N(cid:29)(Ui).
Considerafixedelement(xi)∈N{fi}(Vi),sothatwehave
(cid:7)
f (x /V )<1,
i i i
i∈I
andconsiderafixedindexi.Iff (x /V )(cid:12)=0,thenthereisanintegerj suchthat
i i i
(cid:9) (cid:10) (cid:9) (cid:10)
xi∈ 1/2nij Vi and xi ∈/ 1/2ni(j+1) Vi,
sothat
1 1
<(x /V )(cid:1) ,
2ni(j+1) i i 2nij
andtherefore,sincef isstrictlyincreasing,
i
1 1
<f (x /V )(cid:1) .
2j+1 i i i 2j
But since xi ∈ (1/2nij)Vi, we also have xi ∈ Uj(i+)1, and so (xi/Ui) (cid:1) 1/2j+1, giving
(x /U )<f (x /V ). If f (x /V )=0, then clearly (x /V )=0, which implies that x ∈
i i i i i i i i i i i
(1/2n)V for all n, and hence that x ∈U(i) for all j, so that (x /U )=f (x /V )=0.
i i j i i i i i
Therefore,inbothcases,wehave(x /U )(cid:1)f (x /V ).Sincethisholdsforalli,wehave
(cid:1) i i i i i
i∈I(xi/Ui)<1, giving (xi)∈N(cid:29)(Ui), and we conclude that N{fi}(Vi)⊆N(cid:29)(Ui), as
required. ✷