tvlATEnfrATUtf ECKUn

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tvlATEnfrATUtf ECKUn
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tvlATEnfrATUtf
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npe1etto*t, a BTopofr.- ttutrcnuil, npe}e,non T e}o.na 9.
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B BrrAy tfu;rrrp xBocToBceTrr, rroJrararor
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ran, uro6rr 6rrno A" =p(r(A))
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or
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(A')) (EA" e V)
( * , A" ) = l .
lIcnonray.fitrpIIEqIITIIIAeaJIvBa\vI\BbIBoAIIM:a,='Yg(f).>
2. 7. I ly c r a X, Y -ro n o rl o ? u a e c v ,u e n p ocrp& H ,crl a)IcX X Y .
7neueaJLeItrtdbt,y r eeprc) eHua,:
(1) | - ornpb\roe cooTsercreue (: corpaHf,eT oTnpb|Tbleil,trduffecrea\;
(2) T ornpatro 6 na,uc7ofiroq,ne (n, y) e l;
O r o n o , n o e u v e c n l r r n o r r , f , . T t l f , r6, n u a n u a n H e n p e p b t l H o c r t l
153.
( 3 ) e c , n t lU c* U , r o l - ' @ ) c l i m i n f s = = f- 1 ( y u );
( 4) ec r lu Ue+ A, ro I-' (y ) c l i m s u p g s s l - ' (yu)
(2) -- (3). flpem4e Bcero BaMerrrM, qro ycJroBlre orxpbrrocrr4 B crano 3 u a q a e r c rre A yl oqee: (V A ' x A ) (l r' = * \
4 a p rnof t r or r ne ( * , A)= T
(r', U') e I. Snaqur, rpu r eT-'(A)
lr rrro6ou E - +*
AJrrr Her(oroporo
s ' x z B b r [ o J r r r e H(or ' , A ) e f ,
r. e. fieliminfc="I-t(y,).
( 3) * ( 4) . O ue B rrAH o .
(4)* (1). Ecnra I ue ornpbrro, ro rrMeercfl or,KpbrroeU B X, AJrfl Koroporo f\f (U) He aaMrHyro. I4r.ar, cyilIecrRyer cerb (A)= y\f (U) ra11a.n,
naft4ercn (cp. 23) cerb (".)u=",
ero /s * A a A e|(U).
finnBcflHororeU
cxoA.Errlaflcfl,r( t u ,o6naAaloulaflteu 'cnoficrBoM,rITo #6 e I-r (ye). -fl'cuo, rrro
AJIfl E = *oo 6yget freeU. flonyuaerc.fl rlporunopeune. F
npocrpz,trcrra u T eX)(Y.
?neuea2.8. Ilycro X, Y - ron\rlo?urdecw,ue
y
perc1
e
ldu.r,
tdbl r ee
:
JLerdr
,
( 1) T - aau, n H ,Arorw
e n u re c re o e XX Y;
( 2) ec r lu Ue+ U , ro l i m s u p 6 = " I-t (y .) - I-' (y) ;
(3) ecrltl Ue* U, ro lim infg=eI-t (ye) - I-t (y) .
neJle, to uafi.qyrc.E rrHAeKc E = ** Ir aJIeMeETfr' x r1 Aqrfl-Koropbrx (t', Ae)e
e I. flocr(oilbxy AeF U,,Mbr Br{ArrM,qro (*, U) e |
(n6o n I nxognr 6ecHoEequo flnuaxufr, n aroft Toqne aneueur)
(2) * ( 3) . O ue ru s n o .
(3)*(t). Ecna (rE,U)eI
u r E * f i , A t * A , r o c yqeroM reopeurr 1.9
6y4erreliminfe==I-l(v).Crarro6rrrr,(r,a)=T.>
2.9. IlpeAJrox(errus.2.7 u 2.8 Sanruuecntr BocxoAflr n }typarononouy [8],
cM. ranlne reopeMw 1.2 u 6.2 B [3].
S 3. Konrnanrnocrb u ey6nenpepbrBHoerb
3.0. B arou uaparpa(fe Aarorcfl craH4apruble u EecrauAaprHbre Hprrreputr. r(oMrranTrrocrrr r4 aHaJIorarIHbIx troHsrnfr. 4nn (punbTpoB, gera.uubnpyron{Ee Kracctrqecfirre Sanrrr HecraHAapruoft o6qeft rotrorrornr [5, 6]. flpune4eubr rrpnJroxterrrrfl K Teopru cy,6nenpepbr'BubrxcoorBercrnufi, paaaurofi n [3, 4].
3.1. (Dunrrp g (n rorloJrorllrlec'KoMrrpocrpaucrBe X\ rraabrBaror noJwn&nrtrbtn, (cn. [Z]), ecJra namgrrft {unrrpl
6onee donxuft, .ter F, urueer
Toqr(y rrpunocuoBeutr.ffB X. CoornercrBerrHo cerb HaabrBalor nonnanrruofi,,
ecJrv.fialfiAa.fl ee rroAceTb uMeeT cxoAflrqyrocfl rroAcerb.
3.2. Cran7aprnutti fiuanrp I e X ae.naerca, noil,nanrH,blJtr6 ToM u ro.nbK,o'Tolrr,cJrAvae,ecJru nagc1aa rovna e?o J)toru,a1at,
ono.nocran0apru,a: p(9\=
c nst (X) .
{ -+ flycrr
x e [email protected]) . Paccuorpuu ynrrpaSr,rfibrp (r),: *{(l c X:
teU|
B rrcxoArroMrrpocrpaHcrBeX. flcno, uro (*)=9
a, craJro 6rrrr, rrMeerc.fl cTalrAaprHafl Toqnare' tana.fi, qro fr x fr' . Tlumur cJroBaMr4,
# - omonocTaHAapTHaflTorrfia.
* Ecnr I = 9, ro p\9)c p(g). flycrr r e p,(9). Tor4a r e nst(X),
T. e. gJrff rrenoropoft r' e'X
6y4er r x rl . flocnegnee oaHaqaer, qto a' ?oqna upuxocHol,ewufr.F.
3.3. Qy.norp I e X ae,naerca noJwn&nrvblJtt,
6 ToJtctr rorlbtto row cJLauae,
ecJLu O"na ,nrc6oeo ornpblTo?o nonpbtrufl, Jw,tdoilcecrlaX ttafi,7erca nordeHLlne
no1nonpbtrue' H,enoropoeoeJLeMeH,Ta
ua g.
HoMtranrerr, ro [email protected])cnst(X).
Vvrarrrnan, qro nst(X) Jre?ntrr B MoHatre
nro6oro crarlAaprHoro tronpbrrrr.fiE, nrtnogav: (tr/e 9) (vr= F) (flE e'6'S
fi e E. B KasecrBe l4cnoMoro .F Morr(HoBg.srb nro6oft '6ecnoneqHoMaJrbrftaneMOHT 9. fIpnMeufl.fi rlocJleAoBareJrbrrotrprrHqr{[br rdAeaJrrraaqurrr4 rrepeuooa,
uoilyquM rpe6yeuoe.
r54
C. C. Ityrare.na?se
* Ilycm 6 - ornpbrroe rror(pblrlle X u p(E) - o6re4znerrlre crauAaprHbrx aJreMerrreB6, r. e. MoHaAa6. flo npnnqu,ny [epeuoca rrMerorcfl crau[apruoe F eg,r4 roEeqHoe cranAapruoe troAMuo]necrBoEo B 6 rattuq qro
Suaunr, p(g)-W(6).
b r, o n s t ( X ) UEo=F=p(g).
O c r a e r c nB c r r o M r r r r rq
aro B Toquocrrr [epeceseHrre Morre/{,crauAaprubrx ornpbrrbrx fronpbrtufr X, V
3.4. CSopruyruporannrrft B 3.3 rptrBHaH Aerraer ecrecrtseuHbrMrroucn
anaJrora Kprrrepr4.trXayc4op,Sa 4nn {ra_nrrpos. B aroft cBfl'Bur6y4en paccMarprdBarbpaBnoMepuoetrpocrpancrBo (X, %).
3.5. @ran}rp I
n X uaebrBaror 6noltH,eo?parduueH,Hblwl
ecrr4 AJrfl Kaffigoro onpyrfieHr4fl U e0L[ rrMeercfl Houerrrrafl (J-ceru Henorqporo aJrer{esra F
$wwpa 9.
3.6. @znrrp I
e X naarrsaror norLrdblwlecilu nam4rrft {nnrrp }torur,
6onee ronnraft, uewrF, cxo4rarcx n X.
3.7. Cran1apru,auil, fiu"oorp fl,BJlfl,ercfl,noJLttbLMI Tort,r,u roJLblto rorw
cJLAH&e) ecrlu nattc)aa npe}cran1aprnaa
roqft,a eao ltolta7at, onoJLocran)aprna.
{ -> flycrr g - norunfi Sramrp v. fre pst(X) n W(g\ - upeAcrauAaprHa.fl Torrxa MoHaAbr9. \peacTaHAaprHocTb / oaHarraer,qro / Jrex(rrr B MoHa}torura 9. flpn eroM V(F\n [email protected])+ g. flcno, qro
i1e rrenoroporo $unrrpa
BepxHflfl rpaub I n I - aro $nnrrp ltonrn r4, craJro 6rrrr, rrMeercfl Torrr(a
.r'eoX,
v, r=nst(X).
O r c r o g ar ' 4 r
A J r f ln o r o p o f ir, ' = W ( 9 ) n W ( g ) .
+- flycrr 9=9
vg-{nnrrp
}topr. Ecnn reW(9),ro
reV(F)=
cnst(X).3rraquT'y9ecrsToqKa[puHocHoBeHIIfl.>
3.8. Cran,}aprnutti $tt,nurp nerlnercfl, Bnorure o?paHulteH,rdblw
6 ToJw u
Totblto roilt cJLAHa,e,
ec"'ttLnauc}aa roqna e?o rLoHa}at npe}crau,6apru,a.
{ -> flo nprauq?rrry[epeHoca AJIfl Ha]r(AorocraHAaprHoro onpy]r(euws.U
uJ 0l,l raMerorcflcTaHAaprnrrr,laner{euT F Srrnrrpa g rr KoHetrrHoe
cranAaprEoe MHotnecrBo,E taxue, .lro f/ (E)= F. Crano 6nmr, Wg)c. U (E). Telr caI r r br r rA r r f l.t = W (g )
n n ro 6 o ro [J e " o l tr 6y4er r= [J(r' )
trpr{ rroA xoA fl qeM
crarrAaprHoM r' . flonornnnr I :: x{[/ (r'): (J = 0lI, r = tl (r')\. flcuo, rrro
9 - 6aawc {ra.nrtpa ltoruu lr r = W(9) lro rocrpoeHr4ro. CnegonareJrbrro,
$(g) c pst(X) .
<- ,{onycruru, qro pacoMarpraeaeMnrit
Snntrp g ranoB, rrro W(F)c pst (X), , rerM He Meuee F ue B,rroJrHe
orpaHl{qen. flo il,pr{Hqrrrryirepeuoca
Ilr{eercfl cTaHAaprrroeonpy}fieHrre U v.s "Q.,1tattoe, rrro AJrflBcflKwx F e"9
tt
nro6oro craHAaprnoro KoHeqnoro MHo)r(ecrBaE uaft4ercfl. r e F, rie ilofraAaroulnii B U (E). flo rprruqlrrly lrAeaJrrraaqrrr4r4Meercfl Drr,eMeurr e p(9)'
raHoft, qro r # tl (A) Arrfl HaffiAoro craHAaprHoro A = X. IIo, ycnonuro r e.
e p(9 ), rAe I - $znrrp }tomr,r.Boerlrelr G e'9 rax, uro6rr 6nrno GX G c
c'U. Tor4a [;-rfl BC.HKoro
Borrper(rrr{cxoAa e G nnruorrHeHor'e V(9)-U(v)
Eorry .AorryqeHr4ro.
3.9. Kplrreprafi Xayc4op{a gu Surnrpon.
@utarp fr,rrrfl,ercfl,r+ownar(,rHbllt
8 TOM U TO,nbHO TOM CJLAUA?I eC&U,-Oru'nOrLOt{,U SnOJLIde OAqAHUUeTL.
{ -> flocrarouuo pa6orarb B crarrAaprHoM auTypaffie. Ecnn I xolrnarrerr, ro IL(g)cnst(X)
no 3.2. V'.rurnrsa.fl,qro nst(X)cpst(X),
aanilroqaeM:
I no;l.on rr BrroJrneorpanrrqeu.
.- Ecnz F anonue orpauzqeu, ro uo 3.8 p(g)cpst(X).
Ectn I
nor I oE , r o lr ( g ) | l p s t(X)= n s t(X ). O rc ro ga B brB oA rrM:
p(g):
pst(X
)0
W (g)
c ns!({). Ocraercs cocJrarrc.Ena 3.2.
3.10. Hafi4enubre rlptrBrrafir4 Moryr 6rmr rroJro)fieubrB ocuoBy r4ayqenv.fl.
paailI4qublx TorloJlorrrueorrrx uon.aruft, 6nuaxlrx H HerrpepbrBrrocrrr. OcranoBrrMcfl BAecr,Ha oArroMnB Hr4x (cv'. [2-a]\.
3.11. CoornercrBlre f, geficrnyroqee ue X n Y, uaebrBaror cy6nenpepu,sItbtJtt e roqn,e fr. vB dom l, ecJrrr o6paa Srnrrpa e.Kpecruocreft roqxn r npw I
flBirflerc.fl noMrraHTHbIMB Y. Coo'rnercrBze l, cy6neupepbr,BHoeB nam4oft
Toqne dom l, HaabrBarorcy6nenpepblrtrblJn.
3.12. CranlaprH,oe cooreercrrue T ua X e Y cy6nenpepblsno I row u
ro.nbnororw c.nAq,ael
ec.nu I (nst (X) )c nst (I).
O r o no,noeuve cnu t no rr,fl,Tnn t, 6.0uanux n He np ep bt,e H,oeru
t55
'
{ flonaaareJrbcrBocJleAyer r{B 7.2 n 3.2, v6o nst(X) rpeAcraBJrfler co6ofio6reF[EeEIIeMoIIaAToqgRcTaIIAapTHofofl[pa.X.>
3.13. Cooreercrlue cyfu,enpepbteno 6 ToM u roJlbno roJtt,cJLAHaelecJLu
oH,onepeeoilur rcoil,nanrnw,e fiu.nbrpbl 6 ttownaw,Ttrbte.
{ IIocnoJID,-r(ySrnrrp oxpecruocreft Toqxz BaBeAoMonoMrrar(Terr,AocraToquocTbtrptrBe4eunoro ycJIoBItfl 6eccmopna. flyctr Terrepb Bapanee rraBecrrro,
ttTo coorBercrBrre cy6neupepbrBrro. Eea ywraneur,rsodqnocrrr Molr(Ho pa6orarr
ts crauAaprrroM aurypaffie. flpznnexarr 3.1,2 u. 3.2, nmpvM, rrro n gannoft crryaqtrtr o6paa crarrAaprrroro noMtraxruoro {uumpa
KoMtranTe.rr.Ocraercs
cocJrarbcfl Ha trptrEqra,truepenoca. D
3.14. B cssau c fiplrrepueu 3.13 cy6ue,npepbrBubrecoorBercrtBtrfi.EaBbraaror rruorAa Horv,nanrrrbltw,u
(cp. [2]).
3.15. Cy6nenpepbtsn.oecoor6erci:r6ue,)efr,creyrou4ee6 rayc\op$oso npocTp aH,cT6o, cor p aH,fl,e
T oTH,ocuTeJLbIIarc noMn&nTH,ocTb.
ro U_ c_ns t ( X ) . Cr aro 6 u rr, I(U )c n s t(Y).
Iz l anecrno[5], uro B aroM cl ryvae | ([/) ornocareJrbrro KoMtraxrHo.
3.16. IIycro T - aaunlraroe cy6nenpepblrru,oecoorlercreue.' Toe\a I
nolLyH,enpe
pblsno c6epfra.
{ flo trpruqrrtry uepeuoca Mornno pa6orarb B craEgaprnoM arrrypame. .
. ?Irar, uyorb A - cranAaprHoe saMnrryroe Muolr(ecr,Borr r e cll-t (A). Tlrrtea ' e ' A 6y4er (fr' , a' )ef.
P as.a' e
e rn ci ifr' N f i, A J r f 'i.Horb p o ro trp rH e r(o ro p o M
€ I (nst (X) ), to naftAercfl craHAaprnoe a n o6paae, Atrrfl r(oroporo a x a' .
&eA. B cr,rJryBaMnuyrocrn I BbrrroJrueno
B cury BaMnHyroww A BbrBoArxM:
(r, o)ef. lfuar\ n= f-'(,4).
Z.tl. flpegromeurre 3.16 $aHtmueoHr ylcranoBJrenoB 147 o6o6rqaet
"c noMoqbro
6onee pau.HgeyrBeplltAenrre o SynHqvrflx vB [3]. B Bar$rroqeHfre
TeopeMbr {..9 gaAuM rrpocroe rrecraH,4aprHoe;(onaaareJrbcrBose6olruroft uo4u{ranar1trn Hptrreprrr HerrpepbrBnocrvr5.1. ua [3].
3.18. Ilycra f: X -. Y - $ynrcr4ua, leficrayrcu4an 6 rayc1op$oeo epocrpaH,crso. Toela I u,enpepblrHa B ToM n roJlbno rolw c,nAu&e1ecr,u 1na, naenOofi rovrcu fr na X uneercfl, eJLewerdr
A ua Y ranoti, t4ToAcJlorue frE-+r 6JLever c7urlecrsolarrueno7ceru ("n)n=*, 0ta noropofr,f (rr)* A.
{ Hyat4aercfl B trpoBepr(e Jruurb Aocraroqrrocrb cSopuynrrpoBaEuoro
trpr,ranana. Ey4eu pa6oraru n crarrAaprnoM arrrypame. IIo ycJroBrrrouMeeM
(Yre-*.n) (syn -* A)
(rr, a)=f.
Ha ocgotsauurr reopeMbr I..9 trocJleAHeecoorrrortreHr4e Mo}r(Eo trepetr[carb
B BITAE
:
(Y*' o r) (sA' N A)
B qacrsocrrr, AJI.EEemoroporoU'xA
Y B a x J l r o q a e fqt rr,o A : f ( r ) .
" q o p S o- n o c r w
(pynnqna. tr
IrerrpepbrBna.fl
r. e. I
(i', A')= f .
B c4ry xaycrrrmollHerroy'[email protected]).
ItpoMe roro, fi'xr"*f(fi')*[email protected]),
JINTEPATYPA
:
'7;Dotecki Sz. Tangencyand differentiation: some applicationsof convergencetheory.Alnali tli Matematicapura ed applicata,1982,v. 13Qil n3-255.
2. Penot .7.-P.Compactnets, filters and relations.- J. Math. Anal. AppI., {9E3, v. 93,
N 2, p. 4m_,4fi.
3. Faller R. V; Relations among continuousantl various .non-continuousfunctions.Pacific f. Math., 1968,v. 25, N 3, p. 495-509.
4. Smithson R. E. Subcontinuity for multifunctions.- Pacific I. Math., 1975,v. 6{, N 4
p. 283-288.
t56
5. Lusemburg W. A. J. A general theory of monads.- In: Applications of model theory
to algebra, ana.Iysisand probabilify. New.York: Holt, Rinehart aud, Winston, 1969,
p. 18-86.
6. Stroyan K. D., Laxembur1 W. .A. I.Introtluction to the theory of infinitesimals.New,York: Acad. Press,1976.
7. Aarnes ,. F.t Anilenaes P. n. Oa nets anil filters.- Math. Scan(L, 1972,.v. 3|', N 2"
p. ?85-292
I
9, frgrare&a|ae C. C. llu(prunreBtrMaJlbubreaacareJlbnbrerouycbr.- (ou6, uar.' *rnt.,
1985,r.26, N 6, c. 67-76.
e. Eoeocu6upcn
Craran.nocrVnulLa
24 aneapa 1.985 e.
I
tf
)

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