Разложимость топологических групп
Доведено, що кожну зчислеппу абелеву групу з скінченним числом елементів порядку 2 можна разбита на зчисленпе число підміюжип щільних у будь-якій недискретній груповій топології....
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| Дата: | 1999 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Russian |
| Опубліковано: |
Інститут математики НАН України
1999
|
| Назва видання: | Український математичний журнал |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/164281 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Разложимость топологических групп / Е.Г. Зеленюк // Український математичний журнал. — 1999. — Т. 51, № 1. — С. 41–47. — Бібліогр.: 6 назв. — рос. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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dspace |
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irk-123456789-1642812020-02-10T01:28:06Z Разложимость топологических групп Зеленюк, Е.Г. Статті Доведено, що кожну зчислеппу абелеву групу з скінченним числом елементів порядку 2 можна разбита на зчисленпе число підміюжип щільних у будь-якій недискретній груповій топології. We prove that every countable Abelian group with finitely many second-order elements can be decomposed into countably many subsets that are dense in any nondiscrete group topology. 1999 Article Разложимость топологических групп / Е.Г. Зеленюк // Український математичний журнал. — 1999. — Т. 51, № 1. — С. 41–47. — Бібліогр.: 6 назв. — рос. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164281 512.546 ru Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
Russian |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Зеленюк, Е.Г. Разложимость топологических групп Український математичний журнал |
| description |
Доведено, що кожну зчислеппу абелеву групу з скінченним числом елементів порядку 2 можна разбита на зчисленпе число підміюжип щільних у будь-якій недискретній груповій топології. |
| format |
Article |
| author |
Зеленюк, Е.Г. |
| author_facet |
Зеленюк, Е.Г. |
| author_sort |
Зеленюк, Е.Г. |
| title |
Разложимость топологических групп |
| title_short |
Разложимость топологических групп |
| title_full |
Разложимость топологических групп |
| title_fullStr |
Разложимость топологических групп |
| title_full_unstemmed |
Разложимость топологических групп |
| title_sort |
разложимость топологических групп |
| publisher |
Інститут математики НАН України |
| publishDate |
1999 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/164281 |
| citation_txt |
Разложимость топологических групп / Е.Г. Зеленюк // Український математичний журнал. — 1999. — Т. 51, № 1. — С. 41–47. — Бібліогр.: 6 назв. — рос. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT zelenûkeg razložimostʹtopologičeskihgrupp |
| first_indexed |
2025-07-14T16:51:56Z |
| last_indexed |
2025-07-14T16:51:56Z |
| _version_ |
1837641936358342656 |
| fulltext |
Y~K 512.546
E. F. 3edleHIOK (Y[yZ~K. ml~ycap, nil-r)
P A 3 J I O ~ K H M O C T b T O I I O J I O F H q E C K H X I T Y H H
We prove that every countable Abelian group with a finite number of second order elements can be
decomposed into countable number of subsets which are dense in any nondiscrete group topology.
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�9 E. F. 3EJIEHIOK. 1999
I$SN 0041-6053. Y~p. ~tam. ~.'ypn., 1999, m. 51, N ~ 1 41
42 E.F. 3EJIEHIOK
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ISSN 0041-6053. Y~'p. ~lam. a.'vpn., 1999, m. 51, N ~ I
PA3YIO)KHMOCTb TOI'IOJIOFHqECK!dX FPYIIFI 43
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l'lpo~o.rl~a.a TaKHM 06pa3oM, nOCTpOHM B03pacTalOl.~He rlOCJIe]IOBaTeJIbHOCTH
ISSN 0041-6053. Yrp. ~tam. ~.'vpn., 1999, m. 51, N ~ 1
44 E.F. 3EdlEHIOK
MHO:~eCTB { A ~ : rt < t o } . . . . . {Amt: n < c o } , O6"be/~HrleHH~I KOTOpblX A 1 =
= [.} { A~ : n < co } . . . . . A m = [.J { Ar~ : n < co } H 6y/lyT Tpe6yeMraMn MHO~XecT~aZ4ri.
,~IoKaaame.at,cm~o m e o p e ~ t u 1, I I y c T s F ~ n o z y r p y n n a CZOB S a~qbaBrtTe
(m + 1 ) c nycTsr~ CZOSOM e, L n ~ ~ H o ~ e c r s o s c e x c z o s a s F ~ H H S t n, S~
MHO~KeCTBO BCeX C.rlOB H3 L n, B KOTOpblx nepBble j ~yKB HyncBble, a OCTaglbHble
HCHyJIOBI:,IC,
0 _< j < n, S~: = {On},
FIo/xCTaHOBKa g Ha ~ (m + 1)
By/~eM O603HaqaT~, ee raK:~Ke g.
S,,-= U{S'j : j<n}, S = IJ {S,,: n<co } .
eCTeCTBeHHO HHgytlripyeT no / Ic raHo~Ky Ha F .
OHCBH~HO, qTO Bee MHO)KeCTBa S; , S n , L ,,, S
HHBapHaHTHbZ 0THOCHTeJ[bHO g H op6HTbl BCeX HenycTbIX C~OB H 3 F, OTJIHHHhIX OT
O n, m - - ~JIeMeHTbl. l ' lycTb S - - FIpOH3BOglbHOe CJIOBO 1,13 F . Ec~Irl s E S, TO
nO~OZ~HM s ' = s s* = s. E c z a m e ~ H F \ S , TO s O}XHO3HaqHO pacK~a~ ,mae - r cz B
npoH3Be/~eHne s I "'" s k + 1 , r ~ e
,r S':k§ O < j l < n l , 0 < J 2 < n 2 . . . . . s I E ~Jl ' "'" ' Sk+l E Jk+l ~
0 < j k < n ~, 0 < j k § �9
B aTOM c a y q a e rtoao:~KnM S" = S 1 .. . S k, s* = Ont+...+ntSk+l.
FlyCrb X = { e, x t, x2 . . . . } . K a ~ o M y c z o B y s H F conocTamiM HenycToe
OTXphrro-aaMKHyTOe rlO~MI-IO:,,KeCTBO X ( s ) C X H r o q K y X(S) H X ( s ) TaKt.'Ie, HTO
X ( e ) = X, x ( a ) = e H aaJ t KazcdIoro n > 1 BblI-IOYlHRtOTC~I cae~tymtuHe yCJIOBH.q:
1) n a a a Kazxaoro s H Sn+ l MHO*ecTsa X ( s - 1 ) , i < m, o6paayroT paa6rter tne
X(s), x(sO)= x(s),
2),, ~a~a Bcex s e S , ,_ l , y ~ X ( O n ) n p o a a a e a e H H a x ( s ) y ortpei~e~eHbi H
f ( X ( S ) y ) = f ( x ( s ) ) f ( y ) ,
3) n /X~a Kazc.aoro s H S n f ( X ( s ) ) = X ( g ( s ) ) , f ( x ( s ) ) = x ( g ( s ) ) ,
4)~ ~x~a Kaac, a o r o s H L,, X ( s ) = x ( s ' ) X ( s * ) , x ( s ) = x ( s ' ) x ( s * ) ,
5) . x .e {x(s)sH L.}.
B~a6epeM OTXp~rro-aa~KHyTyu~ nuBapnanTnyIo (OTHOCHTe~bHO f ) oKpecTHOCTb
ez~vmmtu U 1 raxyu~, qTO x 1 ~ U l . T o r ~ a Mno~eCTBO X \ U l TaK>Ke OTKpr~TO-
aaMICayTO n aaBapriaaTnO. C o r z a c a o JaeMMe 3 e r o MO~HO paz6nTb Ha OTKpbITO-
3aMKHyThle nOaMHO~IC.eC'I'Ba A 1 . . . . . A m TaKHe, qTO f (A1) -- A 1, f ( A 2) = A 3 . . . . .
f ( A m ) = A 1 . Bn6epeM ~neMeHT a e A l TaXOtL qTO Xt ~ O ( a ) . l'Io.rlOhK1,IM
X ( 0 ) = U t, X ( 1 ) = A 1, X ( 2 ) = A 2 . . . . . X ( m ) = A m ,
x(0) = e, X(1) = a, X(2) = f(a) . . . . . x(m) = f z - t ( a ) .
(I~4KcrrpyeM n > 1 a npe~tno~o~-nM, , f ro y ~ e onpe/xe~eara X ( s ) , x ( s ) Z ~ t Bcex
S r U { L j : j < n }, npa~eM BbmO.rlH~nOTC.q y c ~ o ~ n ~ I j -- 5j / ~ 1 ~ c e x j < n.
0rrpeaearrM X ( s ) , x ( s ) ZIaa s H L , §
I'IoKa~KCM, qTO B y c z o s n a x 1 n - 5 n MO~aqO S 3aMCltl4Tb Ha L:
1) x ( s O ) = x ( ( s O ) ' ) x ( ( s O ) * ) = x(~)x(O n) = x ( s ) e = x ( s ) (~ - - c z o ~ o , n o z y -
qe rmor na s y~;aaermet4 r tyJte~oro x s o e r a ) ,
ISSN 0041-60.$3. Yxp. ~tam. ~.'Vpn., 1999, m. 51, N ~ 1
PA3~IO)KHMOCTb TOFIOJIOFHqECKHX FPYl'IH 45
X(sO) = x ( ( s O ) ' ) X ( ( s O ) * ) = x ( s ) X ( O . ) = x ( s ' ) x ( s * ) X ( O . ) = x ( s ' )X( s*O) ,
X ( s i ) = x ( ( s i ) ' ) X ( ( s i ) * ) = x ( s ' ) X ( s * i ) , I < i < m ;
2) f ( x ( s ) y ) = f ( x ( s ' ) x ( s * ) y ) = f ( x ( s ' ) ) f ( x ( s * ) y ) = f ( x ( s ' ) ) f ( x ( s * ) ) f ( y ) =
= f ( x ( s ' ) x ( s * ) ) f ( y ) = f ( x ( s ) ) f ( y ) ;
3) f ( x ( s ) ) = f ( x ( s ' ) x ( s * )) = f ( x ( s ' ) ) f ( x ( s * )) = x ( g ( s ' ) ) x ( g (s*)) = x ( g (s)) .
PaCCMOTpHM pa36HeHae MH0mecTBa X MHoxKecTBaMH X ( s ) , s E L n . OLIHO rI3
x(s O)X(s 0), cy- HHX, cKameM X(so) , co~epmnT x,,+l. H0CKO~bKy X ( S o ) = " * TO
mecT~yeT Yn+t e X(s~) TaKO~, wro Xn+ l = x(s~)Yn+ I. B~6epeM oaxpraT0-aaMK-
HyTyIO HnBaprtaHTHyIO oKpeCTHOCTr~ e/InnHILbl Un+ 1 TaKyIO, qTO/~Jlat Bcex S E S n,
y e U,, + 1 rlponasezerlrla x ( s ) y orlpe/~eaem,l, f ( x ( s ) y ) = f ( x ( s ) ) f ( y ) , x ( s ) U n + 1 c
c X ( s ) H Yn+l ~ x(s~)Un+l, ec~a Yn+l ~ x(s~).
Paa6epeMca sHa~Ia:Ie c MHOmeCTBOM X (0 n)" Paao6beM MHOXeCTBO X (0 n) \ Un + 1
Ha OTKp~To-aamKHyTtae no~Mnoxecama B 1 . . . . . B m TaKae, ~TO f (B i ) = B2,
f ( B 2 ) = B 3 . . . . . f ( B m ) = B t (~eMMa 3). Bta6epeM ~:mMenT b e B 1 raI<oll, ~TO
Yn+l E O(b) , ec~14 s~ = 0 n. 1-loJaomHM
X(OnO ) = U,,+i, X ( 0 , 1 ) = B 1, X(0n2 ) = B 2 . . . . . X (Onm) = B m,
X(OnO ) = e, x ( 0 n l ) = b, X(0n2) = f ( b ) . . . . . X(Onm) = f m - l ( b ) .
I lyca~ renepb p - - npon3ao~bHoe c~oao na S n , 0T:mqHoe OT O n, O (p ) - - e ro
op6HTa. Paao6beM MHoxecamo X(p ) \ x ( p ) U n + 1 rrporlaBo:mao aa OTKpUTo-aaMKHy-
Tue nO~MHOmeCTBa C l . . . . . C m rl ma6epeM 3~eMena-ta c I e C l . . . . . c m ~ C m TaKl~e,
UT0 Y,*+I ~ O(Cl) [ - J"" [.JO(Cm)' ecart Y,,+I e t . J { O ( x ) : x r
I'IOJIO:~XHM
X(pO) = x (p )V , ,+ l , X ( p l ) = C 1 . . . . . X ( p m ) = C m ,
x(pO) = x (p ) , x ( p l ) = cl . . . . . x ( p m ) = c~.
~ n a s e O ( p ) \ { p } X ( s ) , x ( s ) o n p e a e ~ M y c ~ o s a e M 3 ,+ I.
I l o c z e TOrO, KaK X ( s ) , x ( s ) onpe~e~en~ ~ a ~cex s e Sn+ l , onpe~e~rtM X ( s ) ,
x ( s ) ~ a ~cex s e Ln+I \Sn+ 1 yc~om~eM 4n+ 1. 3a~eTrtM, qTO e c ~ a Xn+ 1
{ x ( s ) : s e L n } , TO X,,+l = X(So)Y,+ l = X(So)X(So0 = X(Soi) ~aa aeKoToporo
i ~ 0 .
Hs ycao~HJt 5 n c~le~yeT, arO nocTpoeHHoe OTo6pameHne F ~s~---> x ( s ) e X
cyp~eKraSHO. Oao HmiytmPyeT 6rmKtmm h : X --~ @ ~ ( m + 1) ( h ( x ( s ) ) no~Iy-
~aeTcz ~ s npHnHc~marmeM Hy~e~oro xsoera) .
IIocKo~m~y vmomecT~a X ( s ) , X ( sO) , X ( s O 0 ) . . . . - - o~pecamocr~ rO,tKH X (S),
TO 6aeKlmJ~ h Herrpepbm~a. YC~O~HJt 1, 2 c~e~ymT Ha 4 n, 3n, S qacamocTa, h
a3oMopqbnsM. Ec~n X c a e r n o r o xapaKTepa, r o noc~ez~osarez~nOCT~ X(0~) ,
X(02) . . . . ~OmHO c~e~aa~ 6aaofl 0KpecTaocre l l e~HHattr~ H r o r ~ a h 6 y ~ e r
FOM~OMOp~H3MOM.
2. OCHOBHIde pe~ya~,TaT~. TonoJIorHqeCKOC np0CTpaHCTBO Ha3MBaC'fC.q pa3-
o z o ~ a ~ ( R 0"pazJm~HMUM)' ccJm c ro MO~KH0 pa~6HTb Ha ~Ba (Ha ~ 0) IIYIOTHHX
n o ~ o x e c a m a . 0TMeTx4M c~c~aymttmc rrpocT~C y'l'BCp:h~eHH~I O pa~o~d4~OCam.
ISSN 0041-6053. Yrp. ~ m . ~.'vpx., 1999, m. 51, N ~ 1
46 E.F. 3EdlEHtOK
I. HenpepuBnUfl 14H~eKTHBHbI~ o6pa3 paano~KnMoro ( l~ o 'pazno~r~Moro) npocT-
pancraa paaJaomnM (ll 0 -pa3JIOmrlM).
2. 3aMhtKaurm pa3nox<t4Moro (~0-PaaJIOXriMoro) IIO/~npocTpaHcTaa paa~o:C~HMO
( g O'Paeua~176
3. Eclat o~rlopo/xHoe npocTpaHcamo coRep)KnT paa~ox<nMoe (R0-Paarloml4Moe)
no/mpoc'rpaHc'mo, To OHO ~ caMo paa~qom~iMo (R 0-paa~IomHM~
l-lepB~ae RBa yTaepxrd~enria OqeBn/~Hbl, TpeTbe JlerKO cJIe/lyeT H3 ~eMMbl KypaToa-
cKoro ~ IIoplta, yraepx/xeHria 2 a O~mOpO~aOCTn.
TeopeMa 2. llycm~, ( X, x) ~ o~emnaa ueOuctcpemnaa pezynapnaa ao~a.~,naa
.aeeomono.~ozut~ecraa zpynna, f - - nempu6ua.abta,tft OOnOpOOnbtft aemomopqbu3~t na
(X, x) Kone~nozo nopaOtca. ToeOa cyu~ecmeyem pa35uenue ~momecmea X na
cuemnoe t~uc.ao noano.v, cecme, nnomwoLr e mo6oft neauc~cpemnoa mono,aozuu "c" Ha X
maKo12, t~mo
1) ( X, x ' ) ~ .aora.a~,naa .aeeomono.aoeuqecKaa zpynna,
2) f - -nenpepb tonoe omo@amenue a monono~uu x',
3) ~r orpecmnocmb eOunut~bt o mono~o~uu "g nempuoua.abno nepecetcaem
r.a,xOyto o~pecmnocmb eOunu~bt a mono.aoeuu x.
,Roraaame,a~mao. 1-IycT~ m ~ nopmxoK a~To~opdprta~a .1:, h: X .-~@ ~ ( m +
0)
+ 1 ) -- Henpep~a~Ht,lfl naoMop~naM, npe~ocTa~aeM~tl~ TeopeMol~t 1. ~l,.na Kam~oro
a~eMerlTa x ~ X qepe3 ~ (x) o6oaHaqrivt KOdlHqeCTB0 Hap coce~HHX 3$1eMeHTOB B
nocne~oBaTeamHOCT~ aeuystesI,tX KOOp/~rlnaT h(x), OVdlnqahlX OT nap (a, a ) , r~e
a r {1 . . . . ,m} .
3aMeTnM aTO zma mo6oro x ~ X 5(x) = ~( f ( x ) ) ri a~a z m 6 u x x, y ~ X TaKHX,
qYO r(h(x)) + 1 < l(h(y)) ,
~(xy) = - ~ i ( x ) + 8 ( y ) , p(x) = ~,(y), ecJIH
~(X) + 8(y) + 1, ecam p(x) ~ ~.(y),
rae ~.(x), p ( x ) - nepBaa H a o c ~ e a a a a HeHy~em~e Koop~riHaTt,t h(x) . C~eaona-
TeamHo, ~ I a ~Ia~%tx X, y ~ X TaKrtX, qTO r(h(x)) + 1 < l (h(y)) , cymeca~yeT i < m
TaK0e, qTO
g(xfi(y)) = ~ (x )+~(y ) , ~(xfi+l(y)) = ~ ( x ) + ~ ( y ) + 1,
Y I o J ~ o ~
X . = { x e X : ~(x)r ~5(x)-2n(mod2n+l)} .
I/IHHMH C~OBaMH, X n - - MHOTKeCTBO BCeX r a d i x x e X, q-to ~ ( x ) * 0 ~
paazo~KeHae qr~c~la 5(x) Ha npoca~le MaOmnxem~ co~epmrrr n 2-eK. Oqemi~HO,
~ITO Mnomecama X , o6paaylOT pa36neHrle ~uomec ' raa X \ { x ~ X, ~(x) = 0} .
FloKameu, ~rro Kam~oe X , nzoTao a (X, x ' ) . PaCCMOTpaM npoaaaoamma~ a~eMerrr
x c X ri oKpeerH0Ca~ e~taHrima U a TOrlo~ornr~ X'. Fl~omr~M k = 2 n+l - 1 H
au6epeM B U a~leMerrna x t, . . . . x k TaKne, qTO
1) r(h(x)) + 1 < l (h(x t ) ) , r(h(xj)) + 1 < l(h(xj+~)),
2) y i . . . . . yk r U ILrI~ ~m6tax y~ ~ O(xj)= {xj, f ( x j ) . . . . . fra-l(xj)}.
T o r a a cpe~m aaxe~aeuToa xy t ... x y t r xU, r~e yj ~ O(xj), O6.q3aTe.qbHO
rlMeeTc~ ~eMerrr na X , .
Ha .rleMMt,I 2 n TeOpeMbl 2 BraTeKaeT cae~tylomee yraepx~aenHe.
[SSN 0041-6053. Yrp. ,uam, ~.'ypn., 1999 . m. 51, N" I
PA3~O)KHMOCTb TOHOYIOFHqECKHX FPYFIH 47
C.aeOcmoue 1. Ec.au na cr neOuclcpemno~ peeyn.~pno~ nora.nbnoti neoo-
mono.aoeu,~ecro~ epynne cyu4ecmoyem nempuoua.abnbt~t aomo;~topqbu3.~t tcone~noeo
nop.~OKa, mo ona l~ o-Pazno.~u~ta.
~acr~epcnoI~mara xapaKTepOM TOqKH B TOrlOJ-IOFHqeCKOM rlpocTpaHCTBe Ha3bI-
BaeTca nam~ermtuaa I~3 MomHoCTefl 6an ee oKpecTriOCTett.
H. B. HpoTacoB ~toKa3a.a, wro KaZcJ~y10 necqeTny lo a6e~eBy r p y n n y MO:,KHO pa3-
6n-rb Ha CHeTHOe qrlC~O nO~MHO~Kec'rB; n~oTmax a :no6o~ He/IHcKpeTHO/.i rpynnoBot~
TOIIOJIOFHH HecqeTHOFO/IHCrlepCHOHHOFO xapaKTepa. C.rle/]oBaTe.rlbHO, Ka3K]~aH R 0"
nepaa~oatrIMaa a6eneBa r p y m m r ~ e e T cqeTmata ~ncrIepcHoHn~tt x a p a x T e p [4]. OT-
c~o~a, I43 .neMMb~ 1 i4 H3 cJTe]~c'I'BHH 1 BblTeKaeT cnpaBe]~.ritinocTb T a K o r o yTBep-
~ e H n H .
C.aeacmoue 2. B K.aacce a6e.aeobtX u o tc.aacce c,~emnbtx nepuoOu~tecKux epynn
Ka.maa~ neauc~pemna.~ R o-nepa3no~u~ta,~ epynna coaep.~um om~pbtmy~o c~temnyto
6y.aeoy noOzpynny.
Bonpoc 1." BepHo An, HTO Kazcztyto HecHeTay~o r p y n n y M o ~ a o paa6r rn , ria CHeT-
Hoe qr~C~tO nO~HOateCTB, n~OTH~X B Zm60fl ae~nc~peT~of l rpynnoBoI~ Tono .aor tm
necne ' r r loro RrlcnepcaoHrxoro xapa~Tepa ?
Bonpoc 2. BepHo ~Ia, nXO ~a~K~aH cneTnaH rie~rIcKpeTHaH R0-Hepaa.nO~H~aH
r p y n n a co~ep:~rIT OTKpt,~Ty~O 6y~eBy n o ~ r p y n n y ?
F p y n n a txaataBaeTca a6CO.nIOTHO paa~IOata~ofl (a6co.moTHo R0-paa.ao:~nMoit) ,
ec.rlH ee ~OaCHO paa6nTr~ Ha ~Ba (Ha R0 ) n o ~ a o a < e c T B a , n.aOTHb~e B .nzo6ott He~HCK-
peTHO~ rpynnoao t t Tono.norr~a.
C.aeOcmoue 3. Ka~Daa cqemnaa a6eneoa epynna c nonet~nbt.~t quc.~o.~t 9.~ezten-
moo nopaDtza 2 a6co.a~omno R o-Pa3.ao~u~ta.
~[oga3ameabcmoo. l ' IycTb G - - npoHaBO.qbHa~ CHeTHaH a6e.neBa r p y n n a c
KOHeHHbtM qHCJIOM D.neMeHWOB nopH~Ka 2. CHa6l~n~ ee KaKOfl-TO rpynnoBot t npe~;-
KoMnaKTHO~t TononorHet~ X. I-lycTb X~nO~MHOmeCTBO Ha (G, X) a c e x aaeMeHTOB
nopH~moa #: 2, f ~ oTo6paaceHHe Ha X, aa.aaHHoe npaBr~oM x ~ - x . YIpHMeHHM K
nomam,HO~ ~emoxono.norr~aecKo~ r p y n n e X x,x o~;Hopo~ao~,iy aBToMopdpria~y f Ha X
2 - r o nopHaxa TeopeMy 2. l'lo.nyqnM paa6HeHHe X Ha cneTrlOe HHCnO nOa~HO:~eCTB.
Ka~/~oe Ha aTrIX nO~MH0:,KeCTB n.aOTaO a (G, x ' ) / Inn :no6ofi neancmpeTnOtt r p y n -
IIOBOIVI TOIIOJIOFHH "U' Ha G. ~.rI~ ~TOFO ,/~OCTaTOHHO ~OKaaaTb, qTO I<a:~.l~aa oK-
pecT~oc'rb HyaZ B TOrI03IOFHH '~' HeTpHBHaJIbHO nepecemaeT maacay~o oKpeCTHOCTb
Hy.rIH B Tono.norHr~ ~. ~onycTrIM npoTaaHoe: ]Xaa HegOTOptax oKpeCTHOCTe~ Hy~Ia
U, U ' a Tono:~orHHx ~:, I:" CoOTBeTCTBeHHO U N U" = {0}. Bta6epeM ompecamocTH
nyaH V, W B T o n o n o r a z x % x ' TaKae, HTO V - V CU, V" - V" C U'. Tor/~a ~:~H
:no6~,ix pa3:iriHmax a, b ~ V" (a + V ) ~ (b + V ) = 0 (eC~H 6 u aTO 6tarO He TaK, TO
rI~e~ta 6 u ( V ' - V ' ) ["1 ( V - V)~: {0} ) . Cae]~oBaTenbnO, X He npe~KoMnaxT~a.
Bonpoc 3. CymecTByeT : m 6eCKOHeqHaH (cneTHaa) r p y n n a c KOHeHHbX~ H n c a 0 ~
~JIeMeHTOB n o p z ~ a 2, He H a : ~ m t u a a c H a6CO~OTHO R 0 - p a 3 ~ o x a ~ o t a ?
1. Comfort W. W., van Mill J. Groups with only resolvable group topologies//Proc. Amer. Math.
Soc . - 1994. - 120, N ~ 3. - P. 687 - 696.
2. ManbtXUn B.H. 3xc'rper4a.abHo i~ecn~am~e n 6.nHaKne K nH~ rpynma //,/2OK~. AH CCCP. - 1975.
- 2 2 0 , N ~ t. - C . 27 - 3 0 .
3. llpomacoo H. B. A6comomo paa~o~HMtae rpynma/ /Yzp. ~4aT. ~ypH. -- 1996. -- 48, N-" 3. -
C. 383 - 392.
4. l'ipomaco~ H. B. Paa6aeaHa npaM~x npoaaar rpynn//TaM ace. - 1997. - 49, N -~ 10. -
C. 1385 - 1395.
5. llpomacoo H. B. A6coJnoTIma pa3210,X<HMOeTb rpynnu paRHOIla~Ibnblx qHeeJl / / TaM ~e. -
1996.- 48. N -~ 12.-C. 1953- 1956.
6. 3enemox E. F. KoneqH~r rpynnu B I~H TpHaHaJlbrll~. - KneB, 1996. - 12 c. -- (IIpenpHnT /
HAH YKpaHHu. Hn-'r MaTeMaTHml; N ~ 96.3).
l-[o~yqcnO 05.11.96
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