Simple and compact direct topologies

Topology and its Applications 103 (2000) 111–129

Simple and compact direct topologies✩

Niel Shell1

The City College of New York (CUNY), Convent Avenue at 138th Street, New York, NY 10031, USA

Received 4 March 1998; received in revised form 30 September 1998

Abstract

We characterize the compact and locally compact Hausdorff topological groups and rings thatare completions of groups with a topology induced by a direct system, and we characterize thetopological groups and rings whose topologies are induced by simple direct topologies. The fact thatthep-adic topology on the additive group of the rational field is not a direct topology (for which Zobelgave a long intricate proof) is a special case of an immediate corollary of our characterization oflocally compact completions of direct topologies. We show that a direct topology has a neighborhoodbase at zero consisting of open subgroups if and only if it is induced by a direct system consisting ofsubgroups. 2000 Elsevier Science B.V. All rights reserved.

Keywords:Canonical representation; Condensation; Direct system; Direct topology; Simple directsystem; Extended direct system; Absorbing multiplication; Absorbing product; Absorbing sequence

AMS classification: Primary 54H11, Secondary 54H13; 22A05

Direct topologies (see Definition 2 below) on fields have provided important examples(see [7,8,10]). Every topology in the lattice of Hausdorff group topologies on a fixed groupis a supremum of topologies, each of which is the infimum of a family of direct topologies.

Here we characterize the following:(1) the compact and locally compact Hausdorff topological groups and rings whose

topologies are extended direct topologies (Theorem 10);(2) the direct systems for which the completion of the associated topological group or

ring is compact or locally compact (Corollaries 10.6 and 10.4);(3) the abstract groups on which there exists a direct topology with compact or locally

compact completion (Corollaries 10.5 and 10.2);

✩ Research for this work was supported by the Research Foundation of the City University of New York via GrantNumber 668435.1 Email: [email protected].

0166-8641/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0166-8641(98)00164-3

112 N. Shell / Topology and its Applications 103 (2000) 111–129

(4) the topological groups and rings whose topologies are simple direct or extendedsimple direct topologies (Theorems 5 and 9);

(5) the sequences that determine simple direct topological groups and rings (Theorems 4and 6);

(6) the abstract groups on which there exist simple or non-Archimedean direct orextended direct topologies (Corollaries 4.1 and 11.4).

We show that a direct topology on a group has a neighborhood base at zero consistingof open subgroups if and only if it is induced by a non-Archimedean direct system.

1. Introduction

We useZ, Z>0,Z>0, Q, R, andC to denote the sets of integers, positive integers, non-negative integers, rational numbers, real numbers, and complex numbers, respectively. Wewrite #A for the cardinality of a setA, and we letc denote the cardinality ofR.

We use the word “sequence” for a function whose domain isZ>0× Z>0 as well as forfunctions whose domain is a subset ofZ.

For an Abelian groupG (written additively),A∗ will denote the nonzero elements of asubsetA of G, and[A] will denote the subgroup generated byA. For a collection{Ai} ofsubsets ofG,

∑i Ai will denote the set of all finitely nonzero sums

∑i ai , ai ∈Ai for all i.

We define the empty sum to be 0. The order of an elementx in a group is denoted by ordx.As in [2, p. 444], we define therankof a group to be the largest cardinality of a subsetA ofthe group with the properties that each element ofA has order which is infinite or a positivepower of a prime and that a finite sum

∑nixi , with xi ∈ A andni ∈ Z, for all i, is zero

only if nixi = 0 for all i. If A andB are subsets of a ringR, AB= {ab: a ∈ A, b ∈ B},and〈A〉 will denote the subring generated byA.

If a group topology (i.e., a topology with respect to which addition and negation arecontinuous) is being considered on a groupG, the closure in the completion ofG of asubsetA of G will be denoted byA. With this notationG is the completion ofG. We calla topological groupnon-Archimedeanif it has a neighborhood base at zero consisting ofsubgroups.

A ring topology is an additive group topology on a ring with respect to whichmultiplication is continuous. A topological ring is called non-Archimedean if thetopology on its additive group is non-Archimedean andlocally boundedif there existsa neighborhoodW of zero such that for each neighborhoodU of zero there exists aneighborhoodV of zero such thatVW,WV⊂U .

Recall that a topology isultraregular if it has a base of clopen sets. A collection oftopologies on a set will be considered partially ordered by containment.

We usea ∨ b (a ∧ b) to denote the least upper bound (respectively, the greatest lowerbound) ofa andb, wherea andb come from a lattice (such as the reals with the usualorder or a lattice of group topologies);

∨A (∧A) denotes the least upper bound (greatest

lower bound) of an arbitrary subsetA in a lattice.

N. Shell / Topology and its Applications 103 (2000) 111–129 113

If Gi is an (additively written) Abelian group for eachi ∈ Z>0, then the Cartesianproduct

∏∞i=1Gi is considered a group with pointwise addition. If eachGi is a ring,∏∞

i=1Gi is considered a ring with pointwise addition and multiplication (cf. notation foran absorbing product in the paragraph below). The set of all finitely nonzero sequencesin∏Gi is denoted by

∑∞i=1⊕Gi . For an arbitrary indexed set{Gγ }γ∈Γ , the groups∏

γ∈Γ Gγ and∑γ∈Γ ⊕Gγ are defined analogously. A sequence of subgroups{Gi} of

a group is calledindependentif a finite sum∑i gi , with gi ∈Gi , is equal to zero only if

eachgi is zero. A sequence of subrings of a ring is called independent if the sequence ofadditive groups of the subrings is independent. The symbol

∑⊕Gi is also used for theinternal direct sum of an independent sequence of subgroups{Gi} of a groupG.

A sequence{Mi} of subrings of a ringR will be called absorbing(orthogonal) ifMiMj ⊂ Mi∨j (MiMj = {0}) for all i 6= j . An increasing or orthogonal sequence ofsubrings is obviously absorbing. If{Mi} is an absorbing sequence of subrings, then theproduct of two elements inS =∑∞i=1Mi is again inS. Thus,S is the ring generatedby⋃i Mi .

Let {Mi} be a finite or infinite absorbing sequence of subrings of a ringR. The Cartesianproduct of the{Mi} considered with pointwise addition and with multiplication defined by

{xi} · {yi} ={ ∑j∨k=i

xj yk

}will be called theabsorbing direct productand the rule of multiplication will be calledabsorbing multiplication. An absorbing direct product of an infinite sequence will bedenoted by

∏·∞i=1Mi . The subset of all finitely nonzero sequences in

∏·∞i=1Mi will be

denoted by∑∞i=1 ⊕Mi . A finite absorbing direct product will be denoted by

∏·ni=1Mi or∑n

i=1 ⊕Mi .

Lemma 1.1. For each absorbing sequence{Mi} of subrings of a ringR,∑∞i=1 ⊕Mi has

proper divisors of zero if and only if∏·∞

i=1Mi has proper divisors of zero.

Proof. Certainly any proper divisor of zero of∑∞i=1 ⊕Mi is a divisor of zero in

∏·∞i=1Mi .

If {xi}{yi} = 0 for {xi}, {yi} ∈ (∏·∞i=1Mi)∗, with xm,yn 6= 0, then{x ′i}{y ′i} = 0 for x ′i = xi ,

y ′i = yi if i 6m∨ n, andx ′i = y ′i = 0 otherwise. 2Theorem 1. For each absorbing sequence{Mi} of subrings of a ringR,

∑ni=1 ⊕Mi ,∑∞

i=1 ⊕Mi , and∏·∞

i=1Mi are rings with the following properties:(1) {xi 1} · · · {xi k} = {∑i1∨···∨ik=i

∏kj=1xij j }.

(2) Each of these rings is commutative ifR is commutative.(3) The natural injection ofMi into

∑ ⊕Mi is a ring isomorphism. If{Mi} isindependent, then there is a natural isomorphism from the internal direct sum oftheMi onto the absorbing direct sum of theMi .

(4)∑ ⊕Mi (also

∏·Mi ) has no proper divisors of zero if and only if{Mi} is

independent and the internal direct sum of theMi has no proper divisors of zero.Thus, the absorbing sum of theMi has no proper divisors of zero only if it isisomorphic to the internal direct sum of theMi .


(5) If {Mi} is orthogonal, then absorbing multiplication coincides with pointwisemultiplication.

(6) If {Mi}i∈Z>0 (or {Mi}16i6n) is increasing, then the rings∏·Mi and

∏Mi are

isomorphic.

Proof. (4) If {Mi} is not independent and∑ki=1 xi = 0, with xi ∈Mi andxk 6= 0, then

{xi}ek(xk) = 0, wherexi is defined to be zero fori > k andek(x) is the sequence whosekth entry isx and whose other entries are zero.

(6) The functionτ ({xi}) = {yi}, wherey1 = x1 and yi = xi − xi−1 for i > 1 is anisomorphism from

∏∞i=1Mi onto

∏·∞i=1Mi and from

∏ni=1Mi onto

∏·ni=1Mi . 2

Corollary 1.1. Let {Mi} be an absorbing sequence of subrings in the ringR.(1) If R has an identity1 and1∈M1, then the sequence whose first entry is1 and whose

other entries are zero is the identity of∏·Mi and

∑ ⊕Mi .(2) If {Mi} is orthogonal and eachMi has an identityei , then{ei} is the identity of∏·

Mi .(3) If {Mi} is increasing and eachMi has an identityei , then the sequence whose first

entry ise1 and whoseith entry isei − ei−1, for i > 1, is the identity of∏·Mi .

For a subsetA⊂ RZ>0, let A0 be the set of finitely nonzero sequences inA with sumzero; and letAc be the set of all eventually constant sequences inA.

Corollary 1.2. If {Mi} is an increasing sequence of subrings of a ringR, then

∞∑i=1

⊕Mi∼=( ∞∏i=1

Mi

)c

and∞∑i=1

⊕Mi∼=( ∞∑i=1

⊕Mi

)0

.

2. Direct topologies

Some basic facts about direct topologies are summarized below. The reader is referredto [7,8,10] for most proofs and for further information. Global and product direct systemsand a few elementary observations are considered for the first time here.

Definition 1. A sequence of subsets{Mi n}i,n∈Z>0 of an Abelian groupG will be called adirect systemon (or in) the groupG if

(D1) 0∈Mi n for all i andn andMi n = {0} if i < n;(D2) −Mi n =Mi n for all i andn;(D3) Mi n+1+Mi n+1⊂Mi n for all i andn; and(D4) [∑k−1

i=1 (Mi 1+Mi 1+Mi 1)] ∩ (Mk 1+Mk 1+Mk 1)= {0} for all k > 1.If R is a ring and{Min} is a direct system on the additive group ofR, {Mi n} will be

called a direct system on a ring (or on the ringR) if(D5) for all i andn,

∑j∨k=i xj yk ∈Mi n wheneverxj ∈Mj n+1 andyk ∈Mkn+1 for all

j andk; and


(D6) for all x ∈ R andn > 0, there existsk > 0 such that, for alli, xMi n+k ⊂Mi n andMi n+kx ⊂Mi n.

A direct system consisting of additive subgroups will be called anon-Archimedean directsystem.

For each positive integern, Un =∑∞i=1Mi n. The sequence{Un} will be called theneighborhood base associated with the direct system.

In what follows{Mi n} will be a direct system on either a group or a ring.The statement “

∑xi is acanonical representationof x ∈ Un” will mean xi ∈Mi n for

all i, xi = 0 for all but finitely manyi, andx is the sum of the nonzero elementsxi . We usenotation such as

∑xi = ∑

yi or∑xi ∈ Un to indicate that the equality or membership

holds and all series occurring in the expression are canonical.When sets{Mi n} are defined only fori > n, it will be assumed thatMi n = {0} for i < n.For all n, Un ⊂ U1 and a canonical representation of an element inUn is a canonical

representation when viewed as an element inU1. Condition (D4) readily implies thatcanonical representations of elements inU1 are unique. The functionϕi :U1→ Mi 1

which assigns tox the elementxi of its canonical representation is well-defined. Ifx, y, x + y ∈ U1, condition (D4) impliesϕi(x + y)= ϕi(x)+ ϕi(y) for all i. In particular,ϕi |U2 is an additive function. Forx, y ∈ U1 such thatx − y ∈ Un, one easily sees thatϕi(x)= ϕi(y) for i < n.

Theorem 2. If {Mi n} is a direct system on the group(ring) G, then the associatedneighborhood base is a neighborhood base at zero for a Hausdorff group(ring) topologyonG. The setsUn, n> 2, are clopen in this topology.

Definition 2. A topology induced by some direct system will be called adirect topology.A group (ring) topology on a group (ring)H will be called anextended direct topologyif H contains a dense direct subgroup (subring); or, equivalently, if there exists a directtopology on a group (ring)G such thatG⊂H ⊂ G, and the topology ofH is the subspacetopology fromG.

A direct system on a group or ringG is calledglobal if the associated base setU1

equalsG. A topology is called aglobal direct topologyif it is induced by some globaldirect system.

The closure of a non-Archimedean subgroup is non-Archimedean, and a relativelytopologized subgroup of a non-Archimedean group is non-Archimedean. Thus, a group isa non-Archimedean extended direct topological group if and only if it is an extended non-Archimedean direct topological group. (That is, a group is non-Archimedeanly topologizedand has a dense direct subgroup if and only if the group has a dense non-Archimedeanlytopologized direct subgroup.)

Lemma 3.1. Let {Mi n} be a direct system on the groupG. If xi ∈Mi 1 for each positiveintegeri and the series

∑∞i=1 xi (is convergent and) has sum zero in the topology associated

with the direct system, thenxi = 0 for eachi.


Recall that{Un} is a neighborhood base at zero in the completion,G, of the topologicalgroupG. A series

∑∞i=1 xi is called a canonical representation ofx ∈ Un if xi ∈Mi n, for

all i, andx is the sum of the series.

Theorem 3. Let Un be the closure of the standard base setUn in the completionG of thegroupG with respect to the topology associated with the direct system{Mi n}.

(1) For n> 2,

Un ={ ∞∑i=1

xi : xi ∈Mi n ∀ i, and givenm∃I (m)

such thatxi ∈Mim ∀ i > I (m)}.

(2) Canonical representations inU2 are unique. Thus, the definitionϕi(∑∞j=1xj ) = xi

extendsϕi to U1∪ U2; and(i) ϕi(x + y)= ϕi(x)+ ϕi(y), for x, y, x + y ∈ U2; and, ifG is a ring and{Mi n}

is a direct system on a ring, then, also,(ii) ϕi(xy)=∑j∨k=i ϕj (x)ϕk(y) for x, y, xy ∈ U2.In particular,ϕi |U3

is a “homomorphism”.(3) If

∑∞i=1 xi is a canonical representation of an elementx ∈ U2, thenx ∈ G if and

only if {xi} is finitely nonzero, andx ∈ Un if and only ifxi ∈Mi n for all i.Hence, a nondiscrete direct topology is not complete.(4) For n> 2, Un is clopen.Thus each group with an extended direct topology has an ultraregular completion.(5) −Un =Un and−Un = Un for all n.(6) Un+1+Un+1⊂Un andUn+1+ Un+1⊂ Un for all n.(7) For rings,Un+1Un+1⊂Un andUn+1Un+1⊂ Un for all n.

For i, n > 1, the setsMi n are closed discrete subsets ofG (i.e., these sets have noaccumulation points inG), sincex − y /∈ Un+1 for distinctx, y ∈Mi n.

Definition 3. Given an infinite subsetD ⊂ Z>0, we letD(i) denote theith element ofDlisted as an increasing sequence.

Suppose{Mi n} is a direct system on a group or ring associated with the topologyT .The sequenceM ′i n = MD(i) n, for i > n, andM ′i n = {0}, for n > i, will be called thecondensationof {Mi n} toD. (One readily verifies that this is a direct system on the groupor ring.) The topologyT {Min,D} associated with the condensation will be called thecondensation with respect to{Mi n} of T toD. If a fixed direct system{Min} is understood,T {Mi n,D} will be shortened toTD .

The sequenceM ′′i n = Mi n for i ∈ D and {0} otherwise is another direct systeminducingTD . There are cardinalityc distinct direct topologies which are condensationsof any given nondiscrete direct topology.


If {Gγ }γ∈Γ is a family of groups with direct systems{Mγi n}, then

M(π)i n =

{{xγ } ∈

∏γ∈Γ

Gγ : xγ ∈Mγi n ∀γ ∈ Γ

}is a direct system, which we will call theproduct direct system, with associatedneighborhood base at zero given by

U(π)n ={{xγ } ∈

∏γ

Uγn :∨γ

hgtγ (xγ ) <∞},

where{Uγn } is the base at zero associated with{Mγ

i n}, and

xγ =∑i

xγ i , hgtγ (xγ )=∨{i: xγ i 6= 0}.

Also

M(σ)i n =

{{xγ } ∈

∑γ∈Γ⊕Gγ : xγ ∈Mγ

i n ∀γ ∈ Γ}=M(π)

i n ∩∑γ∈Γ⊕Gγ

is a direct system on∑γ ⊕Gγ with U(π)n ∩∑γ ⊕Gγ as associated neighborhood base at

zero. If eachGγ is a ring and{Mγi n} is direct system on the ringGγ , then{M(σ)

i n } is adirect topology on the ring

∑γ∈Γ ⊕Gγ with pointwise multiplication (but (D6) may fail

for {M(π)i n }). If Γ is finite, the topology associated with the product direct system is the

product topology. Therefore, a finite product of direct topological groups (rings) is a directtopological group (ring).

If S andT are group topologies on a groupG, with S 6 T , then the identity map onGhas a unique extension to a continuous homomorphismτ from the completion(G,T )ˆ intothe the completion(G,S) . In general,τ is not injective. However, ifS andT are induced,respectively, by direct systems{Ni n} and{Mi n} such thatMi n ⊂Ni n, for all i andn, thenone readily verifies thatτ is injective.

3. Simple direct topologies

Definition 4. If there exists a sequence{Mi} of sets such thatMi n =Mi for i > n> 1, wesay{Mi n} is asimpledirect system and{Mi n} (and also its associated neighborhood baseat zero, its associated topology and the associated topological group or ring) isdeterminedby {Mi}. A topology is called asimple direct topologyif it is induced by some simple directsystem. We say a topological group (ring) is a simple direct topological group (ring) if itstopology is a simple direct topology. A topological group (ring) containing a dense simpledirect subgroup (subring) is calledextended simple direct(and its topology is called anextended simple direct topology).


Not every direct system inducing a simple direct topology will be simple. A product ofsimple direct systems is simple.

Theorem 4.(1) A sequence{Mi} of subsets of an Abelian groupG determines a simple direct system

onG if and only ifM1 is a symmetric set containing zero,{Mi}i>2 is an independentsequence of subgroups ofG, and(M1+M1+M1)∩ (∑∞i=2Mi)= {0}.

Consequently, a simple direct topology is non-Archimedean. Also, the associated basesetU1 (U1) is clopen inG (G).

(2) A sequence{Mi} of subgroups determines a simple direct topology if and only if itis independent.

(3) In a simple direct topological groupG with associated base at zero{Un} determinedby a sequence{Mi}, the functionψ(

∑∞i=n xi) = {xi}, xi ∈Mi for all i, defines a

topological isomorphism fromUn, n > 2, onto the Cartesian product∏∞i=n Mi of

the groupsMi with the discrete topology;ψ(Un)=∑∞i=n⊕Mi . (The fact that someUn is an open subgroup whose relative topology is the product topology uniquelydetermines the topology onG.)

Proof. Note thatU1 (U1) is clopen inG (G) because both it and its complement are unionsof cosets of the open subgroupU2 (U2). 2Corollary 4.1 (cf. Corollary 11.4).Let H be an Abelian group. The following areequivalent:

(1) There is a nondiscrete extended simple direct topology onH .(2) There are at least cardinalityc simple direct topologies onH .(3) There is an independent sequence of nonzero subgroups inH .Thus, the discrete topology is the only extended simple direct topology on an Abelian

group of finite rank(in particular,Z or Q).

Proof. Each nondiscrete simple direct topology has cardinalityc condensations.2Example 1. Let p be a prime integer, and letZ∞p be the subgroup{k/pn +Z: n> 0, k ∈Z} of Q/Z. SinceZ∞p has rank one, the discrete topology is the only simple (or non-Archimedean—see Corollary 11.4) direct topology onZ∞p .

If we choose a sequence{mi} of positive integers such thatpmi+1 > 2i (pmi ∨2i+3), then

Mi n ={k

pmi+Z: k ∈ Z, |k|6 2i−n

},

for i > n> 1, is easily seen to be a direct system (which is not simple) onZ∞p .

Theorem 5. A Hausdorff Abelian topological groupH is an extended simple directtopological group if and only ifH contains an independent sequence{Mi} of subgroupssuch that

(1) the closureU of∑∞i=1⊕Mi in H is open;


(2) ψ(∑xi) = {xi}, wherexi ∈Mi for all i, defines a topological isomorphism from

U onto a (relatively topologized) subgroup of the Cartesian product∏Mi of the

groupsMi with the discrete topology; and(3) there exists a complete setA of representatives ofH/U such that[A] ∩ U ⊂∑∞

i=1⊕Mi .A Hausdorff Abelian topological groupH is a simple direct topological group if and

only if it contains an independent sequence{Mi} of subgroups such that∑∞i=1⊕Mi is

open and satisfies(2) above.

Proof. SupposeG⊂H ⊂ G, whereG is topologized by the direct system determined by{Mi} (which has the setsUn =∑∞i=n⊕Mi as associated base) and the relative topology ofH in G coincides with the given topology onH . By omittingM1 if necessary, we assumethat all setsMi are subgroups. Then, forψ defined as above,ψ from U1 onto

∏Mi , with

each factor discretely topologized, is a topological isomorphism. Note thatU = U1 ∩H .ThenU is a neighborhood of zero inH and a subgroup ofH . Thus,U is open inH . SinceG is a dense subset ofH , we may select a setA of representatives ofH/U in G. Then

[A] ∩U ⊂G∩ U1=U1=∞∑i=1

⊕Mi.

Conversely, if{Mi}, U , andA satisfy conditions (1)–(3) andU1 =∑∞i=1Mi , we letG = [A] + U1. We show that the given topologyT on H , restricted toG, and thetopologyT {Mi} onG determined by{Mi} coincide. First, we verify thatG ∩ U = U1.CertainlyU1⊂G∩U , and, conversely, ifx ∈G∩U , then there existsa ∈ [A], n ∈ Z, andxi, yi ∈Mi , with xi = 0 for i > n, such that

x = a +∑

xi =∑

yi;a =

∑yi −

∑xi ∈ [A] ∩U ⊂U1.

By condition (2), {yi − xi} (and hence{yi}) is finitely nonzero, sox = ∑yi ∈ U1.SinceU1 ⊂ U , condition (2) impliesT |U1 = T {Mi}|U1. Since (by condition (1))Uis T -open inH , U1 is T |G-open inG. Also, U1 is T {Mi}|G-open inG. Now, twogroup topologies coincide if they have a common open subgroup with a common relativetopology. (A neighborhood base at the identity for the common subgroup is a neighborhoodbase at the identity for both topologies on the whole group.)

If E denotes the closure ofE in H , thenU1=U , and

G= [A] +U1⊃⋃a∈A

a +U1⊃⋃a∈A

a +U1=A+U =H.

It is easy to establish the characterization of simple direct topological groups.2Condition (2) of Theorem 5 implicitly requires that each element inU have a unique

series representation∑∞i=1 xi , with xi ∈Mi . For this requirement to be met, it is sufficient

that


ψ1 :∑

Mi→∏

Mi,

ψ1

(n∑i=1

xi

)= {x1, . . . , xn,0, . . . ,0}

be a topological isomorphism, sinceψ1 can be extended by continuity to a topologicalisomorphism fromU into

∏Mi .

Theorem 6. A sequence{Mi} determines a simple direct topology on a ringR if and onlyif

(1) M1 is a symmetric set containing0;(2) {Mi}i>2 is an independent absorbing sequence of subrings ofR;(3) (M1+M1+M1)∩∑∞i=2⊕Mi = {0}; and(4) for all x ∈R, xMi,Mix ⊂Mi for all but finitely manyi.

Consequently a simple direct topology on a ring is non-Archimedean and locally bounded.If R is the union of an increasing sequence{Ri} of subrings and{Mi} is an independent

sequence(as additive subgroups) such thatMi is a two-sided ideal ofRi for all i, then{Mi} determines a simple direct system onR.

Proof. We verify that a nondiscrete topology with base{Un} determined by{Mi} is locallybounded by observing thatUnU2,U2Un ⊂Un for eachn> 2. If

x =∑i>n

xi ∈Un; y =∑i>2

yi ∈ U2; xy =∑k>n

zk,

zk =∑i<k

xiyk +∑j<k

xkyj + xkyk,

then the containmentsMiMk,MkMj ⊂Mk imply that each term ofzk (and hencezk itself)belongs toMk . Similarly,U2Un ⊂Un. 2

Let Mr (R) be the ring ofr × r matrices over the ringR, and let{Mi} be a sequenceof rings determining a simple direct topology on the ringR. Then, for eachr ∈ Z>0, thesequence{Mr (Mi)}i>0 determines a simple direct topology onMr (R) and provides avariety of examples of direct ring topologies on noncommutative rings.

Since there are no proper locally bounded ring topologies on a field which is algebraicover a finite field ([4]; or see [6, Theorem 15.1.5]), a portion of Theorem 7 below is animmediate consequence of Theorem 6.

Theorem 7. The discrete topology is the only extended simple direct ring topology on asubring of a field with(the field having) finite transcendence degree over its prime subfield.

Proof. If R is a subring of a fieldK and{Mi} determines a nondiscrete simple direct ringtopology onR, then, by Theorem 6,{Mi} also determines a nondiscrete simple direct ringtopology on the ringR+Z1, whereZ1 is the ring of integers inK. Since any subfield of afield satisfying the hypothesis of the theorem also satisfies the hypothesis of the theorem, it


suffices to show that the discrete topology is the only simple direct topology on a subringR

of a fieldK which has finite transcendence degree over its prime subfieldK0, under theassumption that the identity ofK belongs toR andK0(R) = K. For such a ringR, wechoose a transcendence basis{t1, . . . , tn} ({t1, . . . , tn} = ∅ if n = 0) of K/K0 which isin R [10, Section II.12].

Let {Mi} determine a simple direct ring topology onR. Then there exists an integeri0 such thatZ1[t1, . . . , tn]Mi ⊂Mi for all i > i0. For x ∈M∗i (actually for anyx ∈ K∗),x is algebraic overK0(t1, . . . , tn), sox−1 =∑k hkx

k, where, for allk, hk = fk/gk withfk, gk ∈ Z1[t1, . . . , tn]. Let ri ∈K0[t1, . . . , tn] be a common multiple of allgk . Then

ri = (rix−1)x ∈M∗i ; rirj ∈ (Mi ∩Mj)∗.

ThereforeMi can have a nonzero element for only one indexi > i0, and{Mi} determinesthe discrete topology.2Theorem 8. If {Mi} is an absorbing sequence of subrings of a topological ringR, then∏·∞

i=1Mi with the Cartesian product of the topologies induced byR on the subringsMi , isa topological ring. If{Mi} determines a simple direct ring topology onR with associatedbase setU1, then

∏·∞i=1Mi (

∑∞i=1 ⊕Mi) with the product of the relative, i.e., discrete,

topologies on the setsMi is topologically isomorphic toU1 (U1).If {Mi} is an absorbing sequence of subrings of a ringR, and, forx ∈Mi , ei(x) is the

sequence whoseith entry isx and whose other entries are zero, andM ′i = {ei(x): x ∈Mi},then{M ′i} determines a direct topology on

∑∞i=1 ⊕Mi .

Theorem 9. A Hausdorff topological ringS is an extended simple direct topological ringif and only ifS contains an independent absorbing sequence{Mi} of subrings and a subsetA such that

(1) the closureU of∑∞i=1⊕Mi in S is open;

(2) ψ(∑xi)= {xi}, wherexi ∈Mi for all i, defines a topological isomorphism fromU

onto a(relatively topologized) subring of the absorbing Cartesian product∏·∞

i=1Mi

of the ringsMi with the discrete topology;(3) A is a complete set of representatives ofS/U such that

〈A〉 ∩U ⊂∞∑i=1

⊕Mi and[A

∞∑i=1

⊕Mi

]∪[( ∞∑

i=1

⊕Mi

)A

]⊂ 〈A〉 +

∞∑i=1

⊕Mi,

where〈A〉 is the subring generated byA;(4) for eacha ∈A, there exists an integeri0 such thataMi,Mia ⊂Mi for all i > i0.A Hausdorff topological ringS is a simple direct topological ring if and only if there

exists an independent absorbing sequence{Mi} of subrings ofR such that∑∞i=1Mi is

open,ψ as above is a topological isomorphism into the absorbing product, and(4′) for eachx ∈ S, there exists an integeri0 such thatxMi,Mix ⊂Mi for all i > i0.


Proof. As in Theorem 5, ifR ⊂ S ⊂ R, whereR is a simple direct ring determined bya sequence{Mi} of subrings,U is the closure inS of the sum of theMi andA is acomplete set of representatives ofS/U in R, then all four conditions are satisfied. Weverify the second containment in (3): Ifx ∈ R, thenx − a ∈ U for somea ∈A. Therefore,x − a ∈ R ∩U = U1, andx ∈A+∑Mi . Thus,R = A+∑Mi ⊂ 〈A〉 +∑Mi ⊂ R, and〈A〉 +∑Mi is a ring.

If the conditions are satisfied, then we letR = 〈A〉 +∑Mi . ThenR is a subring ofS,and{Mi} determines a direct system on the additive group ofR. Since{Mi} is absorbing,the direct system satisfies (D5). The fact that (D6) is also satisfied follows from (4) and thefact that{Mi} is absorbing. Now the proof thatS is extended direct is as for Theorem 5.2

4. Compact and locally compact direct topologies

Lemma 10.1. If the completionG of a groupG with a topology induced by a direct system{Min} is locally compact, then there existsn0 such thatMi n is a finite set forn> n0. If Gis compact,Mi n is finite for all i andn.

Proof. Since the completionsUn of the standard base sets form a base of clopenneighborhoods at zero inG, there existsn0 such thatUn0 is compact. ThenMi n is finitefor n> n0, since we observed in Section 2 thatMi n has no accumulation points inG. If Gis compact, the last statement applies withn0= 1. 2Lemma 10.2. If the completionG of a groupG with a topology induced by a direct system{Min} is locally compact, then there existsn0 such that{i: Mim \Mi k 6= ∅} is finite fork >m> n0. If G is compact, then we may choosen0= 1.

Proof. Choosen0 such that the completionUn0 of then0th standard base set is compact.Fix k andm such thatk >m> n0. Choosexi ∈Mim \Mi k for i ∈ I = {i: Mim \Mi k 6= ∅},and let xi = 0 for i /∈ I . SupposeI is infinite. Then the sequence{sj }, where sj =∑j

i=1 xi , has a cluster pointx ∈ G. Thus, there existj, l ∈ I , with, say,j < l, such thatsl − x, x − sj ∈ Uk+1. Therefore,xl = ϕl(sl − sj ) ∈ Ml k , which contradicts the choiceof xl . 2Lemma 10.3. If {Un} is the standard neighborhood base at zero for a simple directtopology on a groupG, thenG/Un ∼=G/Un for n> 2.

Proof. The natural homomorphism fromG/Un to G/Un is injective becauseG∩Un =Unand is surjective becauseG+ Un = G. 2Theorem 10. A Hausdorff topological group(ring)H is a locally compact extended directtopological group(ring) if and only ifH contains an independent(independent absorbing)sequence{Mi} of finite subgroups(subrings) such that

(1) the closureU of∑∞i=1⊕Mi is open;


(2′) ψ(∑xi)= {xi}, wherexi ∈Mi for all i, defines a topological isomorphism fromU

onto the Cartesian product∏∞i=1Mi (with absorbing multiplication ifH is a ring)

of theMi with the discrete topology;(3) there exists a setA of representatives ofH/U such thatB∩U ⊂∑∞i=1⊕Mi , where

B = [A] (B = 〈A〉), and[A

∞∑i=1

⊕Mi

]∪[( ∞∑

i=1

⊕Mi

)A

]⊂ 〈A〉 +

∞∑i=1

⊕Mi;

(4) if H is a ring, then for eacha ∈A, there exists an integeri0 such thataMi,Mia ⊂Mi for all i > i0.

A topological group or ring satisfying these conditions is compact if and only ifH/U isfinite.

A Hausdorff topological group(ring) is a direct topological group(ring) with a locallycompact completion if and only if there exists an independent(independent absorbing)sequence{Mi} of finite subgroups(subrings) such that

∑⊕Mi is open andψ , as above,is a topological isomorphism(into the absorbing product ifH is a ring and satisfyingcondition(4′) in Theorem9 if H is a ring).

Proof. If H is a locally compact extended direct group, thenH is complete, soH = Gfor some subgroup (subring)G with a topology induced by a direct system{Mi n}. ByLemmas 10.1 and 10.2, we may replace{Mi n} by a direct system of the formM ′i n =Mi+i0n+n0, for i > n, and assume thatMi n is finite for all i and n andMi 2 = Mi 1

for all i > 2. ThenMi 1 is a subgroup (subring ifH is a ring) for all i > 2, andMi =Mi+1 1 determines a simple direct system onG. If G is a ring, this simple direct systemsatisfies (D5) because of the choice ofMi and satisfies (D6) as a result of Lemma 10.2 andthe choice ofMi .

Let {Un} and{Vn} be the bases at zero associated with{Mi n} and{Mi}, respectively.ThenUn+1⊂ Vn, sinceMi n ⊂Mi 1 for all i > n> 1. Another application of Lemma 10.2shows thatVk ⊂Un for somek.

Thus, a locally compact extended direct group (ring) is the completion of a simple directtopological group (ring) determined by a sequence{Mi} of finite subgroups (subrings).As in Theorems 5 and 9, conditions (1)–(4) (withU = U1) hold. SinceU = U1, U iscomplete and containsU1, andψ is onto the product. Conversely, if allMi are finite andconditions (1), (2′), (3) and (4) hold, then Theorems 5 and 9 and the Tychonoff theoremimply H is an extended direct topology andU is a compact neighborhood of zero, soH islocally compact. 2Corollary 10.1. If H is a locally compact extended direct topological group(ring), thenthere exists a simple direct system consisting of finite subgroups(subrings) on a subgroup(subring)G for whichH is the completion ofG.


Condition (3) in Theorems 5, 9, and 10 implies that the relative topology and the directtopology on[A]+∑∞i=1⊕Mi are equal. This conclusion is not necessarily valid for choicesof A not satisfying condition (3):

Example 2. Let G be the internal direct sum of a sequence{Mi}i>0 of finite nonzeroAbelian groups such that, for some choice of elementsxi ∈Mi , the sequence{ordxi} isunbounded. SupposeM0 is the cyclic group generated byx0. LetH = G, the completionof G with respect to the simple direct topology determined by{Mi}i>1. Let T denotethe (compact) topology onH . Let Un =∑∞i=nMi be the standard neighborhood baseat zero inG, and let Un be the closure ofUn in H . Let A = {ka: 0 6 k < ordx0},wherea =∑∞i=0 xi , and letG′ = [A] +∑∞i=1⊕Mi . Let T ′ be the direct topology onG′ determined by{Mi}. SinceG is a torsion group anda ∈G′ has infinite order,G andG′are not isomorphic groups. SinceUn ⊂G′ ∩ Un, we haveT ′ > T |G′ . If kn =∏n

i=0 ordxi ,thenkna ∈ (G′ ∩ Un) \U1. Therefore,T ′ > T |G′ .

The completion(G′,T ′)ˆ is not compact, since (see Lemma 10.3)

G′/U1∼= [a]/([a] ∩U1

)= [a]/{0}∼= Z.This example may be modified to makeG a direct topological ring. Let{pi} be an

unbounded sequence of prime integers; letMi be the field withpi elements; and viewGas a ring with pointwise multiplication. ThenT above, restricted toG, is a direct topologyon the ringG. Let a andA be as above, and letG′ = 〈A〉 +∑Mi . Again, the topologyT ′determined by{Mi} (which satisfies (D5) and (D6)) is strictly finer thatT |G′ .

Corollary 10.2. LetH be an Abelian group. The following are equivalent:(1) There exists a nondiscrete extended direct topology onH for which the completion

ofH is locally compact.(2) There exist at least cardinalityc simple direct topologies onH for which the

completion ofH is locally compact.(3) H contains an independent sequence of finite nonzero subgroups.Thus, an Abelian group of finite rank(in particular,Z or Q) or a torsion free group(in

particular, an additive subgroup of a vector space over a field of characteristic zero) doesnot have a nondiscrete direct topology with a locally compact completion.

The completion of a direct topology on a group is ultraregular. The completion of anadditive subgroup ofRn, n > 1 (which is its closure inRn) is discrete or has a nonzeroconnected component [1, Chapter 7, Proposition 1.2.3]. Therefore, no additive subgroup ofRn is a nondiscrete extended direct topological group. This conclusion is also an immediateconsequence of Corollary 10.2. In particular (by lettingn= 1,2,4) we have that nonzeroadditive subgroups of the connected locally compact division rings (viz.,R, C and thefield of quaternions) are not extended nondiscrete direct topological groups. The followinganalogous result is another immediate consequence of Corollary 10.2.


Corollary 10.3. LetK be the completion with respect to a valuation on a finite extensionL ofQ with ap-adic valuation(i.e.,K is an arbitrary locally compact field, other thanRor C, of characteristic zero). If H is a nonzero subgroup of the additive group ofK, thenH with the relative topology fromK is not an extended direct topological group.

Zobel [10, p. 38] gave an intricate direct proof of the special case of Corollary 10.3obtained by lettingH = L =Q and weakening the conclusion “is not an extended directtopological group” to “is not a direct topological group”.

If K is the (locally compact) completion with respect to thex-adic topology of a simpletranscendental extensionF(x) of a finite fieldF , then the relative topology on the additivesubgroupF [x] is the simple direct topology associated with the sequence{Fxi}. It isobvious from Theorem 6 that{Fxi} is not a direct system on thering F [x]. From [10,Theorem 3.7] (also appearing as [7, Theorem 5]) we see that no direct system on the ringF [x] induces the relative topology. Thus, we have an example of a ring topology (withcompact completion) which can be induced by a direct system satisfying (D1)–(D4), butnot by any direct system satisfying (D1)–(D6).

Corollary 10.4. A simple direct group(ring) topology on a group(ring) G determined bya sequence{Mi} of subgroups(subrings) has a locally compact completion if and only ifMi is finite for all but finitely manyi.

Proof. If {Mi} determines the topology ofG and G is locally compact, then, byLemma 10.1,Mi is finite for all but finitely manyi. Conversely, ifMi is finite for alli > n, thenUn is a compact neighborhood of zero by Theorem 4.2Corollary 10.5. There exists a nondiscrete direct topology on an Abelian groupH forwhich the completion ofH is compact if and only if there exists an independent sequence{Mi} of finite nonzero subgroups ofH such that

∑∞i=1Mi has finite index inH .

Corollary 10.6. A simple direct group(ring) topology on a group(ring) G determined bya sequence{Mi} of subgroups(subrings) has compact completion if and only if eachMi isfinite andU1 has finite index inG.

From the elementary theory of uniform spaces, we know that, in general, a uniformtopological space does not uniquely determine the uniformity inducing the topology.However, compact Hausdorff topologies are induced by a unique uniform structure.Analogously, it is obvious that nonzero direct topological groups are never induced byonly one direct system. We might ask, though, whether or not the completionG of a directtopological groupG uniquely determinesG in the following sense. Does there exist adense subgroupG′ of G, distinct fromG, and a direct system{M ′i} onG′ such thatG,the completion ofG, is also the completion ofG′ with respect to the topology induced by{M ′i}? We give below an example for which there exists such a groupG′ and leave open thequestion of whether there exists a direct topological groupG which is uniquely determinedby its completion.


The example below also provides information about two other aspects of directtopologies. First, it is natural to ask whether a subgroup (subring)H of a direct topologicalgroup (ring) with topologyT induced by{Mi n} is again a direct topological group (ring).In particular, one might hope that{Mi n ∩ H }, which one easily sees is a direct system(direct system on a ring) with an associated topologyT ′ > T |H , satisfiesT ′ = T |H . Wecall {Mi n∩H } therelative direct system. In Section 2 we observed that the relative productdirect system on

∑γ ⊕Gγ induced the relative topology. The example below shows that

the relative direct system need not induce the relative topology. We leave open the questionof whether or not at least one direct system inducing a given direct topology has a relativedirect system inducing the relative topology on a given subgroup (subring).

Second, suppose{Ni n} and {Mi n} are direct systems on a groupG with associatedtopologiesS andT , respectively. IfMi n ⊂ Ni n for all i andn, thenS 6 T . One mightask, conversely, if, given direct topologiesS andT such thatS 6 T , there exist directsystems{Ni n} and{Mi n} inducingS andT , respectively, such thatMi n ⊂ Ni n for all iandn.

Example 3. Let G =∑∞i=1⊕Mi (the internal direct sum), where eachMi = {0, ei} is acyclic group of order two. LetG have the direct topologyT determined by{Mi}. Letδ be any strictly increasing function fromZ>0 to itself such thatδ(i) > i for all i. Lethi = ei + eδ(i), and letG′ =∑∞i=1[hi]. Then

G′ ={∑i∈I

hi : I ⊂ Z>0, I finite

}.

SupposeI is a finite nonempty subset ofZ>0. Let x =∑i∈I ei . Define dep(x)=∧ I andhgt(x) =∨ I . Then dep(

∑i∈I hi) =

∧I and hgt(

∑i∈I hi) = δ(

∨I). Thusei /∈ G′ for

anyi, and∑[hi] is a direct sum. LetI (m)= δm(I). Then

dep

(x −

∑i∈I

hi

)= δ

(∧I),

and, inductively,

dep

(x −

n∑m=0

∑i∈I (m)

hi

)= δn+1

(∧I)> n.

Therefore,G′ is dense inG. We letM ′i = [hi], and we letU ′n be the base at zero determinedby {M ′i}. We note

U ′n ={∑i∈I

hi : I ⊂ [n,∞), I finite

}=Un ∩G′,

soG is the completion of both the direct groupsG andG′.CertainlyMi n ∩G′ (=Mi ∩G′)= {0}, so that the discrete topology is associated with

the relative direct system onG′. The relative topology and the topology associated withthe relative direct system are distinct.


We introduce a sequence{Ni} determining a direct topologyS strictly coarser thanT .We letNi = [si], wheresi =∑i

j=1 ej . Fork > n,

ek = sk − sk−1 ∈ Vn :=∞∑i=n

Ni.

Therefore,Un+1⊂ Vn, from which we concludeS 6 T . Now {sn} converges to zero withrespect toS and {sn} converges to the nonzero element with canonical representation∑∞i=1 ei with respect toT . Thus, S 6= T . The extensionτ of the identity onG to

a continuous homomorphism from(G,T )ˆ into (G,S)ˆ (see comments at the end ofSection 2) takes

∑∞i=1 ei into zero. Therefore, there do not exist direct systems{M ′i n}

and{N ′i n} with associated topologiesT andS, respectively, such thatM ′i n ⊂N ′i n for all iandn.

If we view each setMi as the field with two elements and viewG as a ring with pointwisemultiplication, then{Mi} determines a direct system on the ringG.

5. Non-Archimedean direct topologies

Although simple direct topologies are non-Archimedean, not all non-Archimedeandirect topologies are simple. The topologies constructed in [5] (see also comments onthese topologies in [8]) are non-Archimedean but not locally bounded (because they arering topologies on an algebraic extension of a finite field) and, thus, not simple. However,a non-Archimedean direct topology can always be induced by a non-Archimedean directsystem:

Lemma 11.1. Suppose{Mi n} is a direct system on the groupG and{Un} is the associatedbase at zero. IfH is an additive subgroup,k > n > 1, andUk ⊂H ⊂ Un, then[Mi k] ⊂Mi n for all i.

Proof. SinceMi k is symmetric,[Mi k] consists all finite sums fromMi k . If x1 ∈Mi k , thenx1 ∈Mi n. Assume all sums ofm elements fromMi k are inMi n andx1, . . . , xm+1 ∈Mi k .Then

s = x1+ · · · + xm+1 ∈H ⊂Un;s = (x1+ · · · + xm)+ xm+1

= ϕi(x1+ · · · + xm)+ ϕi(xm+1)= ϕi(s) ∈Mi n. 2Corollary 11.1. If a standard base setUk , k > 1, associated with a direct system{Mi n} isan additive group, then so isMi k for eachi.

Proof. In Lemma 11.1, letk = n andH =Uk . 2Corollary 11.2. Each global direct topology on a group is finer than or equal to a simpledirect topology on the group.


Thus, the discrete topology is the only extended global direct topology on a group offinite rank.

Proof. If {Mi n} is a global direct system, the setsMi 1 are subgroups, according toCorollary 11.1. Hence, they determine a simple direct topology which is clearly weakerthan or equal to the topology associated with{Min}. 2Lemma 11.2. If {Mi n} is a direct system on a groupG and {[Mi 1]} is an independentsequence of groups, then{[Min]} is a non-Archimedean direct system for a non-Archimedean direct topology coarser than or equal to the topology induced by{Mi n}.

If G is a ring and{Mi n} satisfies(D5) and(D6), then{[Mi n]} satisfies(D5) and(D6).

Theorem 11. Suppose{Mi n} is a direct system on a group(ring) G and {Un} is theassociated neighborhood base at zero. IfU1 contains an open subgroup(open additivesubgroup) H , then, for somek > 1, the setsM ′i n = [Mi+k n+k] form a non-Archimedeandirect system(on a ring) for a non-Archimedean direct topologyT ′ coarser than or equalto the original direct topologyT . In T ′,H is an open group.

If T is non-Archimedean, thenT ′ = T . Thus, a direct topological group(ring) is non-Archimedean if and only if it is induced by a non-Archimedean direct system(a non-Archimedean direct system which satisfies satisfies(D5) and(D6).

Proof. The hypothesis and Lemmas 11.1 and 11.2 imply that{M ′i n} satisfies the conditionsof the first statement of the theorem fork such thatUk ⊂H .

We let {Vn} be the standard neighborhood base at zero for{M ′i n}. If T is non-Archimedean, then, givenUn, there is an open groupH ′ and an integerm such thatUm ⊂ H ′ ⊂ Un. If n > k, thenVm−k ⊂ Un, since all elements inVm−k are finite sumsof elements inUm ⊂H ′. I.e.,T ′ > T . 2Corollary 11.3. Each non-Archimedean direct topology on a group is finer than or equalto a simple direct topology.

Thus, the discrete topology is the only non-Archimedean extended direct topology on anAbelian group of finite rank.

Corollary 11.4. Let H be an Abelian group. The conditions in Corollary4.1 andconditions(4) and(5) below are all equivalent.

(4) There is a nondiscrete extended non-Archimedean direct topology onH .(5) There are at least cardinalityc non-Archimedean direct topologies onH .

Proof. Conditions (3)–(5) are easily seen to be equivalent.2We call an additive groupsemisimpleif each of its subgroups is a direct summand of

the group. (An Abelian group is semisimple if and only if each of its elements has finitesquare-free order; cf. [3, p. 437].)

Corollary 11.5. Each non-Archimedean direct topology on a semisimple group is global.


Proof. Let {Mi n} be a non-Archimedean direct system on a groupG, and letG be thedirect sum ofH and the standard base setU1. To obtain a global direct system for the sametopology, letM ′i n =Mi−1n−1 for i > n> 2, letM1 1=H , and letM ′i 1=Mi−1 1. 2

Acknowledgement

The author is grateful to José E. Marcos for pointing out an error in the original proof ofTheorem 7.

References

[1] N. Bourbaki, General Topology, Part 2, Addison-Wesley, Reading, MA, 1966.[2] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, New York, 1963.[3] T.W. Hungerford, Algebra, Holt, Rinehart and Winston, New York, 1974.[4] J.O. Kiltinen, Inductive ring topologies, Trans. Amer. Math. Soc. 134 (1968) 149–169.[5] J.E. Marcos, An example of a nontrivial ring topology on the algebraic closure of a finite field,

J. Pure Appl. Algebra 108 (1996) 265–278.[6] N. Shell, Topological Fields and Near Valuations, Dekker, New York, 1990.[7] N. Shell, Direct topologies from discrete rings, in: Eleventh Summer Conference on General

Topology and Applications (Portland, Maine, 1995), Ann. New York Acad. Sci. 806 (1996)364–381.

[8] N. Shell, Direct topologies from discrete rings, II, Comm. Algebra 27 (1999) 189–218.[9] O. Zariski, P. Samuel, Commutative Algebra, Vol. 1, Van Nostrand, Princeton, NJ, 1958.

[10] R. Zobel, Direkte Gruppen- und Ringtopologien, Dissertation, Tech. Univ. Braunschweig, 1973.

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Simple and compact direct topologies