HAUSDORFF ABELIAN GROUPS

VOL. 21 NO. (1998) ^9-q6

ALGEBRAIC

OBSTRUCTIONS

TO^SEQUENTIAL

CONVERGENCE

IN

HAUSDORFF ABELIAN GROUPS

BRADD CLARKandSHARON CATES Universityof SouthwesternLouisiana

(Received June 5, 1996 and in revised form October I0, 1996)

ABSTRACT:Given an abeliangroupGand a non-trivial sequence inG,when willitbe possible to construct aHausdroff topology^onGthatallowsthesequenceto

converge?

Asonemight expect ofsuch a naive question, the answer isfar too complicated forasimple response. The purpose of this paper is toprovide someinsights tothis question,especially for the integers, the rationals, and any abelian groups containing thesegroupsassubgroups. Weshowthatthesequenceofsquares inthe integers cannot converge to 0inany Hausdroffgroup topology. Wedemonstratethat any sequenceintherationalsthat satisfiesa "sparseness"condition willconvergeto 0 inuncountably manydifferentHausdorff group topologies.

KEY

WORDS: Abeliantopologicalgroup,Hausdorff,sequentialconvergence AMS CODE:22A05

1. INTRODUCTION

Givenanabeliangroupand a non-trivialsequencein

G,

whenwill itbepossibleto constructa Hausdorff group topologyonGthat allows the sequenceto

converge? As

onemight expect of such anaive question,theanswerisfartoo complicatedforasimpleresponse. Itshould be noted that

"convergence

versus algebra" questionshave beenstudiedelsewhere byFri

[2]

and others. The purposeofthispaperistoprovidesomeinsightsto this question,especiallyin certainwell-known groups and groups thatcontainthese groupsassubgroups.

We

shallassume as additionalhypothesis throughoutthepaperthat Gis anabelian group, and thateach sequence underconsideration is aone-to-onefunctionfromthe natural numbers intoG. SinceGisatopological groupandhence homogeneous,wemay,without lossof generality, assumethat we wishthesequences under consideration toconvergeto0,theidentity element of G. Also weshallusethenotations

7z,

^Q,

R,

and S todenote theintegers, rationals, reals, and circlegroup, respectively.

In

thecourseofthispaper,we willalsoneedto discuss_{1 At2,}the

"meet"

of group topologies and_r2onG.

For

anygroup

G,

thecollectionof all topologies that transformGinto atopological groupformsa latticewhen set inclusion isusedas apartial ordering. Aswas pointed out in

[3],

the intersectionoftwogroup topologiesneed not be agroup topology. Howeveratractablemeet operator was providedforabeliangroupsin

[1].

2.

SPARSE SEQUENCES

Itis a simple observation that ifg 0, then the sequence

(ngln . ^l)

^{cannotconvergeto0 inany}

Hausdorff group topology. The following propositionwillallowusto generalizethisobservation.

Wesaythat the set

{gl,

g2,

g,}

^isann-factorizationof gifand onlyifg

q4 B. CLARK AND S. CATES

PROPOSITION 2.1. If

{xn},eN

^{is theimage set ofa sequencethat} converges to 0 inG and

{xn}nN

^{containsinfinitelymanypairwisedisjoint}

n0-factorizations

^ofxforsomefixed natural numberno,thenx E

{0}.

PROOF. Let 0 E U an opensubset ofG. Wecan find aneighborhood V of0 such that

noV

^C_

^U.

^{Since thesequenceconvergesto0,we can}find a tail of the sequencein V and hence findan

n0-factorization

^ofxcontained inV. Thereforex

. ^noV

^C_

^U,

^{and sinceUwasarbitrary,}

= ^{e (0}.}

Therefereehaspointedoutthatin aHausdorit topological groupproposition 2.1 can be derived by consideringsequentialconvergenceas aspecial caseof

FLUSH-convergence.

This proposition demonstrates an interestingrelationship between topologicalgroupsandnum- ber theory. If the sequence

(1,

4,9, ...,n

2, ...)

were to converge to 0 in

Z,

then the sequence

(1,

-1, -1, 4, -4, -4, 9, -9, -9,

n2, -n2, -n, ...)

must also converge to 0. However in

[4],

^it

wasdemonstrated that thereareinfinitely manypairsof natural numbersn and msatisfying the relationshipn 2m² 1. Henceby thepropositiononlythe indiscretetopologyon^7/.allowsthe sequence ofsquares toconvergeto0.

Nowconsider the sequence

(2,

3, 5,...,p,,

...)

wherep, denotes then

*

primenumber. Ifthis sequence convergesto0, thenthesequence

(2,-2,

3,-3,5,-5, ...,p,,-p,,.. must also converge to0. Thusbythe proposition, if aHausdorff group topologyon

Z

allowsthe sequence toconverge to0, then the twin prime conjectureisfalse.

Let

{g-},e

^bethe imageset ofasequenceinG. Thenthesubgroup

H

generated by must be acountable subgroup. Sinceit is trivial inabeliangroupstoextend topological group structures from asubgroup to allof

G,

it makessensetostudy thequestionof convergenceⁱⁿ specificcountablesubgroups

suc.h

^{as Q. Asaspecialcase,}suppose that

(xk)ke

^isasequence of positive elementsofQ. Withoutlossof generality,we may assumethatthesequenceis astrictly

in(ireasing

^{sequence. Wecall suchasequence a}sparse sequenceifand onlyif:

k

+

^3k

(1.) xk+---! _>

_{foreveryk lI and}

x 2(k + 1)

k

+

^2k

(2.) x+----A

_> for infinitely many k 1.

x

^k+l

PROPOSITION 2.2. If

(x)N

^{is a}sparse sequence in

Q,

then there existuncountably many Hausdorff group topologiesonQthat have

(xk)eN

^convergeto0.

PROOF.

For any real number r, we can definea function

f, R

S by

fr(x)

^rx.

Ifwe place theEuclideantopologyonS

1,

we can use

fr

^{to placea} group topologyon IR. Let

U -,

^{in Thensince}

^{U}eN

^{is a} flmdamental system for the topology ^on S

1,

{f-(Uk)}

Nis afundamental systemforagroup topologyonIR. Whilenoneofthese topologies areHausdorff topologies, we note that whenever r is an irrational number,

f,[Q

is one-to-one, hence, in this situation the relative topologyon Qgenerated by f, will be a Hausdorff group topology.

Let

U f-(U).

^If

x

^U

,

^{it is} necessary for

rx ^e U.

^{Butthis means that either}

1 k-1

m_<rx<m+or(rn-1)+

k <rx_<mforsomemZ. Therefore,wemust have that

rnk

+

^{mk- m}

either--m <r<

or

<r<

x kx kx x

ink-1

mk+l

^{Vk C R. Wewish}

Hence

forxk

^{6 U}

k,

^{it is}necessary thatr 6 U,,ez

kxk kxk

/

tshw that givenanintervalfrm

V’

^say

(

^mk

ink+l)

^{thereexists aninteger}

^m (mo(k ^{+ l) mo(k + l) +} ) (ink-1

^mk

⁺ ).

^{Forthis tohappen, it isnecessary}

that

(k+ 1)x+, (k+ 1)z+,

" ^kxk

andsufficientfor rnk- 1

mo(k + 1) mo(k + 1) +

^rnk

+

kx

^<

(k+ 1)x+,

^and

(k+ 1)x+l

^<

kx----’

^orequivalently that

(ink- 1)z+l

(ink

+ 1)Zk+l

kx +

<

mo<

kz

k

+

Since

<x>eN

^{issparse,wehave that:}

(ink + 1)x+, (ink- 1)xk+,

² (ink

+ 1)(k + 3k) (ink- 1)(k + 3k)

2

kx kx

k

+

>

2k(k + 1) 2k(k + 1)

^k

+

Thuseach intervalusedinformingV contains atleast onefullintervalfrom theconstructionof V

k+l"

Nowinfinitely oftenwehave that

x----!

_>

^k2

^/2k

x

^k

+

^{Usingasimilarargumenttothe above,we}

canshowthatfor infinitely many k, everyinterval inV containstwo intervalsfrom V

+1. But

thismeans that

C=,

V contains aCantorset andhenceis uncountable. Therefore, thereexist

uncountablymanyirrationalnumbers rthat generate Hausdorff group topologiesonQthat have the sequence

(xk>keN

^{convergingto 0.}

If

{x,},eN

^isthe image setofasequence inQ, we candecompose

{x,},eN

^{intotwosubsets,}

{y,},eN

^and

{zj}3

where each y,

>

0and each_zj < 0. Following arearrangement of terms, we can assumethaty,+, >y,andz3+1<_{z3 for all}i, j⁶N.

COROLLARY2.3. If

<x,,>,s

^{is asequenceinQwith}correspondingsequences

(Y,>,N

^and

<lz31)3N

^and

{Y,},es

^t3

{Iz31}3s

^{is theimage ofa}sparse sequence, then there areuncountably

manyHausdorff grouptopologies onQwhichhave

<x,>,eN

^convergeto0.

The methodthat we have used to constructgroup topologiesonQhasitslimitations. Ifwe use ittocreate group topologiesr, and_r2in which thesparse sequences

(x,),es

^and

(y,),

^converge

to0,then we can be sure that the combinedsequence

(xl,

yl,x2,y2,x3,y,

...)

willconvergeto0 in Ar2. Howeverby

[1],

weknowthat

n ⁺

^m

^{+ (-e, e)ln,}

^m

^Z

^ande

^>

^{0 is a}fundamental

7"

rl r2

system forr, A_r2 where r, and _r2 arethe irrationals associated with the topologies r, and

r

respectively. Now

r

^Ar2 will be aHausdorff topology ifandonly if ^rl 6 Q, and will be the r2

indiscretetopologyotherwise.

Supposethat

H

< G^{and that} H Qisanisomorphism intoQ. Wesaythat a sequence

(x,),s

^{inHis asparse in}

^H

^{if andonlyif}

((x))e

^{issparse inQ.}

PROPOSITION

2.4. Let

H <

Gbeadirect sum of cyclicgroups and groups isomorphic toQ. If

(x,),s

^is asequencein H whose associated coordinatesequences areeithersparse or eventually 0, thenthere isaHausdorff group topologyonGthathas

<x,>,es

^{convergingto0.}

ACKNOWLEDGEMENT.

The authors aregratefulto thereferee forhis suggestions in the preparationofthispaper.

96 B. CLARKAND S. CATES

IFEKENCES

[1]

^{B. Clark and}

.

Schneider, "Themeet operator in the latticeof group topologies," Canad.

Math.Bull. 29

(1986),

^478-481.

[2]

^R.Fri.,

"Convergence

andnumbers," TopologyAppl. 70

(1996),

139-146.

[3]

^{P. Samuel,} "Ultrafiltersandcompactifications ofuniform

spaces,"

Trans. Amer.Math.Soc.64

(1948),

100-1.

[4]

^D. Shanks, "Solved and Unsolved Problems inNumber Theory," Second Edition, Chelsea Publishing

Company,

New York, 1978.

Mathematical Problems in Engineering

Special Issue on

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This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

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