Різні типи квазінеперервності та їх застосування: Автореф. дис... канд. фіз.-мат. наук / В.В. Нестеренко, Львів. нац. ун-т ім. І.Франка.

Euea?anueeee iaoe?iiaeueiee oi?aa?neoao

?iai? ?aaia O?aiea

Ia i?aaao ?oeiieno

Ianoa?aiei Aaneeue Aieiaeeie?iae/

OAeE 517.51

??CI? OEIE EAAC?IAIA?A?AIINO?

OA ?O CANOINOAAIIss

01.01.01 – iaoaiaoe/iee aiae?c

A A O I ? A O A ? A O

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Euea?a – 1999

Aeena?oaoe??th ? ?oeiien

?iaioa aeeiiaia ia eaoaae?? iaoaiaoe/iiai aiae?co *a?i?aaoeueeiai
aea?aeaaiiai oi?aa?neoaoo ?i. TH??y Oaaeueeiae/a.

Iaoeiaee ea??aiee eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe,

aeioeaio

Ianeth/aiei Aieiaeeie? Ee?eeiae/,

*a?i?aaoeueeee aea?aeaaiee oi?aa?neoao

?iai? TH??y Oaaeueeiae/a,

aeioeaio eaoaae?e iaoaiaoe/iiai aiae?co

Io?oe?ei? iiiiaioe aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani?

Iaooi?i TH??e ?aaiiae/,
Ee?anueeee iaoe?iiaeueiee oi?aa?neoao

?iai? Oa?ana Oaa/aiea,

i?ioani? eaoaae?e
ia/enethaaeueii?

iaoaiaoeee;

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe,

aeioeaio

Aaiao Oa?an Iioo??eiae/,

Euea?anueeee iaoe?iiaeueiee
oi?aa?neoao

?iai? ?aaia O?aiea,

aeioeaio eaoaae?e aeaaa?e ?
oiiieia??

I?ia?aeia onoaiiaa ?inoeooo iaoaiaoeee IAI Oe?a?ie,

i. Ee?a

Caoeno a?aeaoaeaoueny “24” ethoiai 2000 ?. i 15 aiaeei? ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae 35.05.07 o Euea?anueeiio
iaoe?iiaeueiiio oi?aa?neoao? ?iai? ?aaia O?aiea ca aae?anith:

290602, i. Euea?a, aoe. Oi?aa?neoaonueea 1, aoae. 377

C aeena?oaoe??th iiaeia iciaeiieoeny o a?ae?ioaoe? Euea?anueeiai
iaoe?iiaeueiiai oi?aa?neoaoo ?iai? ?aaia O?aiea (i. Euea?a, aoe.
Ae?aaiiaiiaa, 5).

Aaoi?aoa?ao ?ic?neaii 17 n?/iy 2000 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee ss.A. Ieeeothe

OA?AEOA?ENOEEA ?IAIOE

Aeooaeueiinoue oaie. Iiiyooy eaaciiaia?a?aiinoi, ui aoei aaaaeaia o 1932

i N.Eaiiinoei, i?e?iaeii aeieea? o cayceo c aeineiaeaeaiiyi noeoiii?
iaia?a?aiinoi ia?icii iaia?a?aieo ooieoeie f(x1,…,xn) aiae n ciiiie

, e

iiaia?a?aiinoue yeeo anoaiiaeee A.Aieueoa??a, ?.Aa? (1898) i A.Aai
(1919) caaei

aei N.Eaiiinoiai.

Aeey ia?icii iaia?a?aieo ooieoeie ooieoeiy f(x1,…,xn)
fxn(x1,…,xn)=f(x1,…,xn-1,xn) /ano

eaaciiaia?a?aiith aiaeiinii noeoiiinoi ciiiieo x1,…,xn-1,xn i oiio a

caaea/a: aea/eoe iiiaeeio oi/ie noeoiii? iaia?a?aiinoi ooieoeie f(x,y),
yei eaa

i aiaeiinii x i iaia?a?aii aiaeiinii y (ooieoei? c eeano KC).

iaeia iaeaieiaiy ?ic?iaeyeanue a i?aoeyo Aeae.A?aeeai?iaeae i O.Iioio?e
(1976), A.Iane

(1990), AE.-I.O?oa

iea (1990) i AE.Aain

oa AE.-I.O?oaeeiea(1992). E.Aa

(1926) iiae iaca

oiiaa (A) oaeiae oaia iaaio oiiao oeio

eaaciiaia?a?aiinoi aeey ooieoeie aeaio ciiiieo, caaeaieo ia aeiaooeao
ia?aeaeaiiiaaeia, i canoinoaaa ?? i?e aeineiaeaeaiii noeoiii?
iaia?a?aiinoi ooieoeie o?ueio aeienieo ciiiieo. A. Aai (1932) ia

oiiao (A) Aa?aey ia aeiaaeie ooieoeie, ui caaeaii ia aeiaooeo iao?e/ieo
i?inoi?ia, iacaaaoe ooieoei?, ui caaeiaieueiythoue oiiai (A)

A-ooieoeiyie, i canoinoaaa A-ooieoei? i?e aea/aiii noeoiii? iaia?aaiinoi
ia?icii iaia?a?aieo ooieoeie aiae aaaaoueio ciiiieo. Ai?iaeiaae
o?eaaeiai /ano oei ?acoeueoaoe E. Aa?eey i A. Aaia caeeoaeeny iica
oaaaith /eneaiieo iaoaiaoeeia, ui aea/aee iiiyooy

Eaaciiaia?a?aiinoi oa eiai aiaeiae. Iai?eeeaae, o aiaeiiiio iaeyaei O.
Iiea?oiia (1987), aea i?

iciaaii aaaaoi ?iaio i?i eaaciiaia?a?aii aiaeia?aaeaiiy, aiie ia
caaaeothoueny. Eeoa a 1996

i oiiaa Aa?aey aoea ia?aianaia iaie ia aeiaaeie aiaeia?aaeaiue
oiiieiai/ieo i?inoi?ia i iacaaia ai?eciioaeueiith eaaciiaia?a?aiin-oth.
O ca’yceo c oeei aeieeei i?e?iaeia ieoaiiy i?i aeineiaeaeaiiy ia noeoiio
iaia?a?aiinoue aiaeia?aaeaiue c eeano KhC,

EMBED Equation.3

wC i KhC, yea i?

aei ieoaiiy i?i iayaiinoue oi/ie neiao?e/ii? eaaciiaia?a?aiinoi o
eaaciiaia?a?aieo ooieoeie, ui ?aiioa aea/aeiny

oieueee aeey ooieoeie c eeano KC a ?iaioao N.Eaiiinoiai (1932), I.Ia?

(1961), Aeae.A?a

iaeaea i O.Iioio?

(1976)

C.Iueio

(1978, 1980), Aeae.Ei i C.Iueio

(1985). E?ii oi

iinoaee iaa?iaii caaea/i, a oaeiae ieoaiiy i?i canoinoaaiiy io?eiaieo
?acoeueoaoia aei ooieoeie aaaaoueio ciiiieo oa iiiaicia/ieo
aiaeia?aaeaiue. ?ic?iaoei oeeo aeooaeueieo i?iaeai i i?enay/aia aeaia
aeena?oaoeiy.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie.

?acoeueoaoe aeena?oaoei? io?eiaii o ?aieao iaoeiaeo aeineiaeaeaiue, yei
i?iaiaeeeenue ia eaoaae?i iaoaiaoe/iiai aiaeico *a?iiaaoeueeiai
aea?aeaaiiai

Oiiaa?neoaoo iiaii TH?iy Oaaeueeiae/a. Oaiaoeea aeena?oaoei? iia’ycaia c
iaoeiai-aeineiaeieie ?iaioaie “Aeineiaeaeaiiy aeanoeainoae
aiaeia?aaeaiue a aano?aeoieo i?inoi?ao” (iiia? ?a?no?aoei? – 6683).

Iaoa i caae

EMBED Equation.3

wC i KhC, ?ica’y

?iciiiaiioieo iaa?iaieo caaea/ iia’ycaieo c eaaciiaia?a?aiinoth, a
oaeiae ia?aianaiiy iniiaieo ?acoeueoaoia aeineiaeaeaiue ia aeiaaeie
ooieoeie aaaaoueio ciiiieo oa iiiaicia/ieo aiaeia?aaeaiue.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoaoia. Ani iaea?aeaii ?acoeueoaoe ?
iiaeie. A aeena?oaoei? anoaiiaeaii oei iiiaeeie oi/ie eaaciiaia?a?aiinoi
aeey ooieoeie aecia/aieo ia R. Iieacaii, ui aiaeia?aaeaiiy aeaio ciiiieo
oi/eiai ?ic?eaia, yeui aiii ai?eciioaeueii eaaciiaia?a?aia i oi/eiai
?ic?eaia aiaeiinii ae?oai? ciiiii?, a oaeiae anoaiiaeaii, ui
aiaeia?aaeaiiy c eeano KhC ia? anthaee uieueio iiiaeeio oi/ie
iaia?a?aiinoi ia eiaeiie ai?eciioaei oa eiaeiie iaia?a?aiie e?eaie.
Anoaiiaeaii, ui eaaciiaia?a?aia ca noeoiiinoth ciiiieo aiaeia?aaeaiiy
f:X(Y(Z ia? o

EMBED Equation.3

wC=KhC. E?ii o

, ?ica’ycaiii iniiaii iaa?iaii caaea/i iia’ycaii c eaaciiaia?a?aiinoth
oa ia?aianaii aeayei ?acoeueoaoe aeineiaeaeaiue ia aeiaaeie ooieoeie
aaaaoueio ciiiieo oa iiiaicia/ieo aiaeia?aaeaiue.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoaoia. ?acoeueoaoe aeena?oaoei?
iinyoue oai?aoe/iee oa?aeoa?. Aiie iiaeooue aooe aeei?enoaii a
caaaeueiie oai?i? ooieoeie, oiiieiai? oa ooieoeiiiaeueiiio aiaeici.

Iniaenoee aianie caeiaoaa/a. Ani iaoeiai ?acoeueoaoe, aeeth/aii a
aeena?oaoeith, iaea?aeaii caeiaoaa/ai iniaenoi. A i?aoeyo [1, 2, 4, 5,
7, 8] A.Iane

iaeaaeaoue iinoaiiaee caaea/ oa a ?iaioi [2]

/anoeia noaooi, a a [1, 7] A.Ieoa

– iaeay aeiaaaeaiiy oai?aie 1 aeey aeiaaeeo aeienieo ciiiieo.

Ai?iaaoeiy ?iaioe. ?acoeueoaoe aeena?oaoei? aeiiiaiaeaeenue ia
Anaoe?a?inueeie iaoeiaie eiioa?aioei?, i?enay/aiie iai’yoi I.
Eaciii?nueeiai, o Eueaiai (1995), iaoe

ie eiioa?aioei?, i?enay/aiie iai’yoi A. Eaaeoeueeiai, o Oa?iiiiei
(1997), ia iiaeia

iaoeiaeo eiioa?aioeiyo “No/anii i?iaeaie iaoaiaoeee” (1998) i “100
?ieia aee

i? ?. Aa?a” (1999) o *a?

, ia iaoeiaie nani? IOO a *a?iiaoeyo (1999), ia na

i c ”

oai?i? ooieoeie oa ooieoeiiiaeueiiai aiaeico”, iaoeiaiio naiiia?i
iaoaiaoe/iiai oaeoeueoaoo o *a?iiaaoeueeiio aea?aeaaiiio oiiaa?neoaoi
(1995 – 1999) oa iaoe

naiiia?ao o Eueaianueeiio oiiaa?neoaoi.

Ioaeieaoei?.Iniiaii ?acoeueoaoe aeena?oaoei? iioaeieiaaii o ainueie
?iaioao,nienie yeeo iiaeaii a eiioei ?aoa?aoo. C ieo o?e iaae?oeiaaii o
aeaeaiiyo c ia?aeieia, caoaa?aeaeaieo AAE Oe?a?ie.

No?oeoo?a i ia’?i ?iaioe. Aeena?oaoeiy neeaaea?oueny c anooio,
/ioe?ueio ?icaeieia, ?icaeoeo ia iiae?icaeiee, aeniiaeia i nieneo
aeei?enoaieo aeaea?ae. Oanya aeena?oaoei? – 107 no

iiie. Nienie aeei?enoaieo aeaea?ae aeeth/a? 65 iaeiaioaaiue.

INIIAIEE CIICO ?IAIOE

O anooii iiaeaii caaaeueio oa?aeoa?enoeeo ?iaioe, aenaioeaii noai
iaoeiai? i?iaeaie, iaa?oioiaaii aeooaeueiinoue oaie, noi?ioeueiaaii
iaoo oa caaea/i aeineiaeaeaiiy.

A ia?oiio ?icaeiei c?iaeaii iaeyae eioa?aoo?e ca oaiith aeena?oaoei?,
aeeeaaeaii iniiaii aeiiiiiaeii iiiyooy oa oai?aie, iia’ycaii c iai?yieii
aeineiaeaeaiiy.

A ae?oaiio ?icaeiei ?ica’ycaia iaa?iaia caaea/a aei oai?aie Ia?eona,
anoaiiaeaii oei iiiaeeie oi/ie eaaciiaia?a?aiinoi i canoiniaaia
ai?eciioaeueia eaaciiaia?a?aiinoue aei oi/eiai ?ic?eaieo aiaeia?aaeaiue.
Ia?oi aeaa iiae?icaeiee 2.1 i 2.2 iiny

iiaeaioia/ee oa?aeoa?.

Iaoae P i Z – oii

i/ii i?inoi?e, G –

nenoaia iiaeiiiaeei i?inoi?o P, p0(P i f:P(Z – aiaeia?

. Ie eaaeaii, ui aiaeia?aaeaiiy f ? G-e

iiaia?a?aiei o oi/oei p0,

aeey eiaeiiai ieieo W oi/ee z0=f(p0) a i?in

i Z i aeey e

ieieo O oi/ee p0 a i?inoi?i P inio? oaea iiiaeeia S(G, ui S(O i
f(S) (W. sseui aiae

f ? G-eaaciiaia?a?aiei o eiaeiie oi/oei p0(P, oi e

, ui f ? G-ea

iiaia?a?aiei. Cae/aeia eaaciiaia?a?aiinoue iaea?aeo?oueny, eiee ie ca G
aa?aii nenoaio anio aiaee?eoeo iaii?iaeiio iiiaeei o P.

Iaoae P=X(Y – oii

i/iee aeiaooie i?inoi?ia X i Y, p0=( x0, y0), ?X:P (X i ?Y : P( Y —

i?ia

i? i?inoi?o P aiaeiiaiaeii ia i?inoi?e X i Y. Iic

/a?ac Gx0 /e, aiaeiiaiaeii, Gy0

anio aiaee?eoeo a P i iaii?iaeiio iiiaeei G, aeey yeeo x0( ?X (G) /e,
aiaeiiaiae

, y0( ?Y (G) . Aiaeia?

iaceaa?oueny neiao?e/ii eaaciiaia?a?aiei aiaeiinii x /e,
aiaeiiaiaeii, y a oi/oei p0, y

aiii Gx0 –eaaciiaia?a?aia /e, aiaeiiaiaeii, Gy0 -e

iiaia?a?aia o oeie oi/oei. Iicia/eii /a?ac Hx nenoaio anio iiiaeei aeaeo
U( y , aea U – aiaee?e

iaii?iaeiy iiiaeei a X, y aeiaieue

oi/ea c Y. Aiaeiai/ii /a?ac Hy iicia/eii nenoaio anio iiiaeei aeaeo
x( V, aea V – aiaee?e

iaii?iaeiy iiiaeeia a Y, x aeiaieue

oi/ea c X. Aiaeia?aaeaiiy f: P (Z iace

ai?eciioaeueii /e, aiaeiiaiaeii, aa?oeeaeueii eaaciiaia?a?aiei a
oi/oei p0,

aiii Hx -eaaciiaia?a?aia /e, aiaeiiaiaeii, Hy -e

iiaia?a?aia a oeie oi/oei. sseui aiaeia?aaeaiiy f neiao?e/ii
eaaciiaia?a?aia aiaeiinii x /e y,

/e aa?oeeaeueii eaaciiaia?a?aia a eiaeiie oi/oei p0(P, oi e

ui f ? ne

eaaciiaia?a?aia aiaeiinii x /e y, ai

/e aa?oeeaeueii eaaciiaia?a?aia. Eaaciiaia?a?aii aiaeia?aaeaiiy f: P
( Z iace

ua noeoiii eaaciiaia?a?aieie.

Iicia/eii /a?ac K_hC(X,Y,Z) eean a

f:X(Y(Z, yei ai?e

eaaciiaia?a?aii i iaia?a?aii aiaeiinii ae?oai? ciiiii?, a /a?ac
ovKC(X,Y,Z) – eean a

f:X(Y(Z, yei eaacii

i aiaeiinii ia?oi? ciiiii? i?e anio cia/aiiyo ae?oai? ciiiii?, ui
i?iaiaathoue aeayeo anthaee uieueio a Y iiiaeeio (naith aeey eiaeiiai
aiaeia?aaeaiiy), i iaia?a?aii aiaeiinii ae?oai? ciiiii?. Aeey
aiaeia?aaeaiiy f:X(Y(Z i oi/e

x in X /a?a

K_x(f) iici

iiiaeeio anio oeo oi/ie y in Y, aeey ye

aiaeia?aaeaiiy f_y:X to Z ? eaacii

o oi/oei x.

ua iaeei eean aiaeia?aaeaiue, yeee ie iicia/aoeiaii ovK_wC(X,Y,Z).
Nthaee i

ani aiaeia?aaeaiiy f:X(Y(Z aeey ye

ovK_x(f)=Y aeey ei

x in X i yei iaia

i aiaeiinii y.

Iaaeaei iiiaei aeey ni?iuaiiy caiinoue K_hC(X,Y,Z), ovKC(X,Y,Z) i
ovK_wC(X,Y,Z) aoaea

ienaoe aiaeiiaiaeii K_hC, ovKC i ovK_wC. Aeey aei

oiiieiai/ieo i?inoi?ia eaaei anoaiiaeoe oaei aeeth/aiiy
ovKC(X,Y,Z)sbs ovK_wC(X,Y,Z)sbs K_hC(X,Y,Z).

Ai?e

eaaciiaia?a?aiinoue, yea ? iaeiei ic iniiaieo iino?oiaioia
aeineiaeaeaiiy, a aeayeeo oai?aiao aeei?enoiao?oueny ia a iiaiie ii?i,
a canoiniao?oueny oieueee oaea ?? aeanoeainoue.

bf Eai

2.2.7. it Iaoa

X, Y i Z — oiii

i/ii i?inoi?e, G — aiae

iiiaeeia, iiiaeeia A uieueia a G i f:X(Y(Z — ai?e

eaaciiaia?a?aia aiaeia?aaeaiiy. Oiaei ia? iinoea aeeth/aiiy f(Gtimes
Y)sbs ovf(A times Y).

Aeia?

oai?aia i?i oa, ui iiiaeeia oi/ie iaia?a?aiinoi C(f) e

iiaia?a?aiiai aiaeia?aaeaiiy f : X to Y, aea X~– oiii

i/iee i?inoi?, Y — iao

, aai caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, ? caeeoeiaith
iiiaeeiith. A iiae?icaeiei 2.3

‘ycaia iaa?iaia caaea/a aei oei?? oai?aie a oiio aeiaaeeo, eiee X=Y=bf
R.

bf Oai?

2.3.7. it Iaoa

B — iiaeiii

bf R ia?o

eaoaai?i? i oeio F_sigma. Oiaei inio? e

iiaia?a?aia ooieoeiy f: bf R to bf R, oaea, ui D(f)= B.

Aeineiae

oeio iiiaeeie K(f)

eaaciiaia?a?aiinoi aiaeia?aaeaiiy f : bf R to bf R i?en

iiae?icaeie 2.4.

anoaiiaeaia oaeee ?acoeueoao.

bf Oa

2.4.1. it Iaoa

A i B iiaeiii

bf R i A=bf Rsetminus B. Aeey oi

uia inioaaea ooieoeiy f:bf Rto bf R, oaea, ui K(f)=A, iaia

, uia aeey eiaeii? aiaee?eoi? iaii?iaeiuei? iiiaeeie G aeeiio?oueny
aeueoa?iaoeaa: aai A ia ? uieueiith a G,

Bcap G — iiiae

ia?oi? eaoaai?i?.

Iiaoaeiaith ooieoeie c aeaieie iiiaeeiaie oi/ie iaia?a?aiinoi,
eaaciiaia?a?aiinoi i eeieiainoi caeiaeenue oaeiae ss.Aaa?o i
ss.Eiiiinueeee (1988). Iai

iaeiinoue a oai?aii 2.4.1 ia

i ia? iinoea a caaaeueiioie neooaoei? i aeieeaa? c oaeiai ?acoeueoaoo.

2.4.2. it Iaoa

X — aa?ian

i?inoi?, ui caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, Y — ia

i?inoi?, f:Xto Y — aeaye

, G — aiae

iiiaeeia a X, A=K(f) i B=L(f). Oiaei aai A ia ? uieuei

a G, aai Bcap G — iiiae

ia?oi? eaoaai?i?.

A eeane/iie ?iaioi N. Eaiiinoiai (1932) ae

iiiyooy eaaciiaia?a?aiinoi caa?oo i cieco, iieacaii, ui ooieoeiy, yea
oi/eiai ?ic?eaia aiaeiinii iaeii?? ciiiii? i eaaciiaia?a?aia aiaeiinii
eiaeii? c iioeo ciiiieo, ? oi/eiai ?ic?eaiith. Aei ?acoeueoaoia, yei
iia’ycaii c oi/eiaith ?ic?eaiinoth neiae aiaeianoe oaeiae ae?oao oai?aio
i?i iaia?a?aiinoue E.Aa?aey. A iiae?icaeiei 2.5 ?

N.Eaiiinoiai i E.Aa?aey ocaaaeueiththoueny, i?e/iio canoiniao?oueny
caanii iioi iaoiaee ii?eoaaiue.

2.5.2. it Iaoa

X — oiii

i/iee i?inoi?, Y — a

ianueeee i?inoi?, yeee ia? cei/aiio inaaaeiaaco, Z — i

i?inoi? i f:X(Y(Z – ai?e

eaaciiaia?a?aia aiaeia?aaeaiiy, yea oi/eiai ?ic?eaia aiaeiinii ae?oai?
ciiiii? i C^x(f)=y in Y: (x,y) in C(f). Oiaei iii

A=x in X: ovC^x(f)=Y — caee

a X.

sseui a iiia?aaeiie oai?aii ia i?inoi? X iaeeanoe ua oiiao aa?iainoi,
oi aiaeia?aaeaiiy f aoaea oi/eiai ?ic?eaia. Iiaeiaii ?acoeueoaoe iioei
niiniaii io?eiaee A.Ianeth/aiei i A.Ieoaeethe.

A oeueiio ae iiae?icaeiei anoaiiaeaia ua iaeia aeanoeainoue
ai?eciioaeueii? eaaciiaia?a?aiinoi.

bf O

2.5.1. it Iaoa

X — oiii

i/iee i?inoi?, i?inoi? Y ia? cei/aiio inaaaeiaaco, Z — i

i?inoi?, f:X(Y(Z – ai?e

eaaciiaia?a?aia aiaeia?aaeaiiy i B_x,varepsilon(f)= y in Y :
omega_f_y(x)0 iiiae

A_varepsilon(f)= x in X : ovB_x,varepsilon(f)=Y –

caee

O?aoie ?icaeie i?ena’y/aiee ieoaiith iayaiinoi oi/ie iaia?a?aiinoi
aiaeia?aaeaiue f:X(Y(Z c eea

K_hC, yei ai?e

eaaciiaia?a?aii i iaia?a?aii aiaeiinii ae?oai? ciiiii?. A iiae?icaeiei
3.1

eaea i?i oi/ee iaia?a?aiinoi aiaeia?aaeaiue c eeano K_hC

, a oaeiae i?i eaaciiaia?a?aiinoue aiaeia?aaeaiue c oeueiai eeano.

Iaoae X, Y i Z — oiii

i/ii i?inoi?e i f:X(Y(Z — aiaeia?

. Aeey iiiaeeie E sbs X times Y i oi/e

y in Y iiee

E_y=~x~in~X~:~(x,y)in~E. Ie eaae

, ui aiaeia?aaeaiiy f ia? it

inoue Aanoiia, yeui aeey eiaeiiai y in Y iii

C_y(f)=E_y, aea E=C(f), caee

a X.

ia?ainiee iaoiae Aa?aey i canoiniaoth/e oai?aio Aaiaoa i?i eaoaai?ith,
ie iaea?aeo?ii ianooiio oai?aio, yea ocaaaeueith? ?acoeueoaoe E.Aa?aey i
A.Aaia.

bf O

3.1.4. it Iaoa

X — oiii

i/iee i?inoi?, Y – caaeiaieueiy? ia?oo aeniiio cei/aiiinoi, Z –
iao?eciaiee i?inoi? i f:X(Y(Z – ai?e

eaaciiaia?a?aia

aiaeia?aaeaiiy, yea iaia?a?aia aiaeiinii ae?oai? ciiiii?. Oiaei f ia?
aeanoeainoue Aanoiia.

A aeena?oaoei? iaaiaeeoueny i iioa aeaaaeaiiy oei?? oai?aie, yea
aaaeaoueny aiae noi?ioeaiiai eaoaai?iei iaoiaeii c aeiiiiiaith
iaiaoiaeieo ni?ooaaiue. Aeei?enoiaoth/e oai?aio 3.1.4, a ?i

i io?eiaia ianooiia oaii?aia i?i eaaciiaia?a?aiinoue aiaeia?aaeaiue c
eeano K_hC.

bf Oai?

3.1.6. it Iaoa

X — aa?ian

i?inoi?, Y – caaeiaieueiy? ia?oo aeniiio cei/aiiinoi, Z – iao?eciaiee
i?inoi? i f:X(Y(Z – ai?e

eaaciiaia?a?aia aiaeia?aaeaiiy, yea iaia?a?aia aiaeiinii ae?oai?
ciiiii?. Oiaei f eaaciiaia?a?aia.

Aiaeiai/ii ?acoeueoaoe i?i neiao?e/io eaaciiaia?a?aiinoue aiaeia?aaeaiue
c eeano KC io?eiaee N.Eaiiinoee, I.Ia?oii, Aeae.A?aeeai?iaeae i
T.Iioio?a oa C.Iueio?ianueeee.

A iiae?icaeiei 3.2

iane/aiinoue iiiaeeie oi/ie iaia?a?aiinoi aiaeia?aaeaiiy f:X(Y(Z c eea

K_hC aa?o

. Aeei?enoiaoth/e ai?eciioaeueio eaaciiaia?a?aiinoue, iaea?aeaii
aeinoaoii oiiae iane/aiinoi iiiaeeie C(f)

, yei oi/iioi iiae oiiae Aeae.Eaeuea?i oa Aeae.O?oaeiea aeey
aiaeia?aaeaiue c eeano overlineCC i

A.Iane

aeey overlineKC.

Aaaae

a ?icaeyae iiiaeeio C_Y(f)=x in X:x times Y subseteq C(f). Ie eaae

, ui aiaeia?aaeaiiy f:X(Y(Z ia? it aean

inoue Aaia, yeui iiiaeeia C_Y(f) ca

a X.

bf Oa

3.2.1. it Iaoa

X — oiii

i/iee i?inoi?, Y caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, Z — ia

i?inoi? i f:X(Y(Z – ai?e

eaaciiaia?a?aia i iaia?a?aia aiaeiinii ae?oai? ciiiii? aiaeia?aaeaiiy.
Oiaei iiiaeeia C_Y(f)=x in X : xtimes Y sbs C(f) caee

a X.

A iiae?icaeiei 3.3

oai?aio 3.1.4 i

iaia?a?aieo e?eaeo i oei naiei ocaaaeueiaii eeane/ii ?acoeueoaoe ?.Aa?a
i N.Eaiiinoiai i?i iayaiinoue oi/ie iaia?a?aiinoi ia?icii iaia?a?aieo
ooieoeie ia iaia?a?aieo e?eaeo i iiaa?oiyo. Aeey iiiaeeie E a aeiaooeo
X times Y i oi/e

x in X iiee

E^x=y in Y: (x,y) in E.

Iaoa

g : X to Y — iaia

aiaeia?aaeaiiy i L=(x,g(x)): x in X~– aiaeiiaiae

a aeiaooeo Xtimes Y. Ie aoae

aiai?eoe, ui e?eaa L ? it cei/a

oeio, yeui inio? iineiaeiaiinoue aiaee?eoeo iiiaeei W_n a X times
Y, oaea, ui ae

eiaeiiai x in X neno

W^x_n: n in bf N ooai

aaco ieieia oi/ee g(x)

Y. sseui i

i? Y iao?eciaiee, oi eiaeia iaia?a?aia e?eaa a X times Y, yea ? a?a

iaia?a?aiiai aiaeia?aaeaiiy g:X to Y, ? e?ea

cei/aiiiai oeio.

bf O

3.3.2. it Iaoa

X i Y — oiii

i/ii i?inoi?e, Z — i

i?inoi?, aiaeia?aaeaiiy f:X(Y(Z – ai?

eaaciiaia?a?aia i iaia?a?aia aiaeiinii ae?oai? ciiiii? i L : y=g(x)
iaia

e?eaa cei/aiiiai oeio a X times Y. Oiaei iiiae

C_L(f)= x in X : (x,g(x)) in C(f) ? oeio G_delta i caee

a X.

iei IV aee

i? anoaiiaeaii ua iaeii aeinoaoii oiiae noeoiii? eaaciiaia?a?aiinoi,
oa?aeoa?ecaoei? neiao?e/ii? eaaciiaia?a?aiinoi oa eaaciiaia?a?aiinoi,
?icaeyiooi ieoaiiy i?i ca’ycee iiae ?icieie eeanaie aiaeia?aaeaiue, a
oaeiae ia?aianaii aeayei ?acoeueoaoe ia aeiaaeie aiaeia?aaeaiue aiae
aaaaoueio ciiiieo oa iiiaicia/ieo aiaeia?aaeaiue.

A iiae?icaeiei 4.1

oai?aio i?i noeoiio eaaciiaia?a?aiinoue aiaeia?aaeaiue c eeano K_hK.

bf Oai?

4.1.2. it Iaoa

X — aa?ian

i?inoi?, Y caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, Z — ?a

i?inoi? i aiaeia?aaeaiiy f:X(Y(Z ai?

eaaciiaia?a?aia i eaaciiaia?a?aia aiaeiinii ae?oai? ciiiii?. Oiaei
aiaeia?aaeaiiy f eaaciiaia?a?aia ca noeoiiinoth ciiiieo.

Oey oai?aia ocaaaeueith? ?acoeueoaoe N.Eaiiinoiai (aeey aeienieo
ciiiieo) i I.Ia?oiia (caiiiaii eaaciiaia?a?aiinoue aiaeiinii ia?oi?
ciiiii? ia ai?eciioaeueio eaaciiaia?a?aiinoue i oiiao iao?e/iinoi
i?inoi?o Z ia ?aaoey?iinoue).

E?ii oiai, iieacaii, ui ai?eciioaeueii i aa?oeeaeueii eaaciiaia?a?aii
ooieoei? aaea ia ciaia’ycaii aooe eaaciiaia?a?aieie (i?eeeaae 4.1.3).
Aieueo

, iaaioue aeiaeaoeiaa oiiaa oi/eiai? ?ic?eaiinoi ooieoei? f:X(Y(Z ia aa

eaaciiaia?a?aiinoi ca noeoiiinoth ciiiieo. Aea yeui ia ai?eciioaeueii i
aa?oeeaeueii eaaciiaia?a?aio ooieoeith iaeeanoe o?ioe neeueiioo oiiao
iiae oi/eiaa ?ic?eaiinoue, yea oaeiae iia’ycaia c iayaiinoth oi/ie
iaia?a?aiinoi ooieoei?, oi oea aeanoue iai eaaciiaia?a?aiinoue ooieoei?
ca noeoiiinoth ciiiieo.

Iicia/eii neiaieii D_y(f) ii

oi/ie x in X, oaee

, u

f:X(Y(Z ?ic

a oi/oei (x,y).

bf Oai?

4.1.4. it Iaoa

X, Y i Z oiii

i/ii i?inoi?e, f:X(Y(Z – ai?e

i aa?oeeaeueii eaaciiaia?a?aia aiaeia?aaeaiiy i iiiaeeia M= y in Y :
D_y(f) — aeanue uie

a X – iiaea ia uieue

Y. Oiaei aiaeia

f eaaciiaia?a?aia ca noeoiiinoth ciiiieo.

?aiioa aaea aoee caaaeaii ?acoeueoaoe N.Eaiiinoiai, I.Ia?oiia,
Aeae.A?aeeai?iaeaea i O.Iioio?e oa C.Iueio?ianueeiai i?i neiao?e/io
eaaciiaia?a?aiinoue aiaeia?aaeaiue c eeano KC. Aeyaey?oueny, ui i i?inoi
noeoiii eaaciiaia?a?aii aiaeia?aaeaiiy oaeiae aoaeooue neiao?e/ii
eaaciiaia?a?aieie o aaaaoueio oi/eao. Aiaeiiaiaeiee ?acoeueoaoe
anoaiiaeth?oueny o iiae?icaeiei 4.2.

bf Oai?

4.2.1. it Iaoa

i? X caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, i?inoi? Y —

aai caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, Z — na

iao?eciaiee oiiieiai/iee i?inoi? i aiaeia?aaeaiiy f:X(Y(Z eaa

. Oiaei inio? caeeoeiaa a Y iiiaeeia B, oaea, ui aiaeia?aaeaiiy f
neiao?e/ii eaaciiaia?a?aia aiaeiinii y

a eiaeiie oi/oei aeiaooeo X times B.

Oey oai

aea? iiaeeeainoue anoaiiaeoe iaiaoiaeii i aeinoaoii oiiae
eaaciiaia?a?aiinoi, yei aeiaiaeyoueny a ianooiiiio iiae?icaeiei. Ia?oa
oai?aia aea? oa?aeoa?ecaoeith noeoiii? eaaciiaia?a?aiinoi.

bf Oa

4.3.2. it Iaoa

i? X caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, Y — aa?ia

i?inoi?, yeee iao?eciaiee ai caaeiaieueiy? ae?oao aeniiio
cei/aiiinoi, Z — nai

iao?eciaiee i?inoi?. Aiaeia?aaeaiiy f:X(Y(Z aoae

iiaia?a?aiei oiaei i oieueee oiaei, eiee f ? aa?oeeaeueii
eaaciiaia?a?aiei i inio? anthaee uieueia a Y iiiaeeia ae?oai? eaoaai?i?
B, oaea, ui aiaeia?aaeaiiy f_y — ea

iiaia?a?aia aeey eiaeiiai y in B.

Iano

?acoeueoao aea? oa?aeoa?ecaoeith neiao?e/ii? eaaciiaia?a?aiinoi.

bf Oa

4.3.5. it Iaoa

X — aa?ian

i?inoi?, i?inoi? Y caaeiaieueiy? ia?oo aeniiio cei/aiiinoi, Y — ia

i?inoi?. Aiaeia?aaeaiiy f:X(Y(Z nei

eaaciiaia?a?aia aiaeiinii y oiaei i oieueee oiaei, eiee aeey eiaeiiai y
in Y aiaeia

f_y : X to Z eaacii

i inio? anthaee uieueia iiiaeeia M_y

G_delta, oaea, ui M_y sbs x in X : y in C(f^x).

Aeae. A?ae

iaeae oa O. Iioio?a (1976) ano

, ui aiaeia?aaeaiiy f:X(Y(Z c KC(X,Y,Z) ia? an

uieueio iiiaeeio oi/ie iaia?a?aiinoi ia eiaeiie ai?eciioaei Xtimes y,
yeui X~– aa?ian

i?inoi?, i?inoi? Y caaeiaieueiy? ia?oo aeniiio cei/aiiinoi, Z — ia

i?inoi?. Iani?aaaei, ye iieacaa A.Ianeth/aiei (1999), oeth oa

iiaeia ?iciianthaeeoe ia oe?oee eean overlineKC(X,Y,Z). sse aiaeci

aeua oaeee ae ?acoeueoao iaea?aeaii a oai?aii 3.1.4, oieuee

aiaeia?aaeaiue c eeano K_hC(X,Y,Z). Oiio i

iinoaei ieoaiiy i?i aca?iica’ycie oeeo aeaio eeania. I/aaeaeii ia?
iinoea oaea aeeth/aiiy overlineKC(X,Y,Z)sbs K_hC(X,Y,Z).

Acaa

, aeeth/aiiy a iioee aie aeey aeiaieueieo oiiieiai/ieo i?inoi?ia ia
i?aaeeueia (i?eeeaaee 4.4.1 i 4.4.2) sse oaea c

, aeey aeiaieueieo oiiieiai/ieo i?inoi?ia i/aaeaeieie ? oaei aeeth/aiiy
ovKC(X,Y,Z)sbs ovK_wC(X,Y,Z)sbs K_hC(X,Y,Z).

I?e ia

oiiaao ia i?inoi?e oei o?e eeane caiaathoueny. Ii?iaiyiith oeeo oa
iioeo eeania i?enay/aiee iiae?icaeie 4.4.

aoei anoaiiaeaii i?iiiaeiee ?acoeueoao, yeee aea? ?iaiinoue eeania
K_hC=ovK_wC .

bf Oai?

4.4.3. it Iaoa

X i Y aa?ian

i i

, ui caaeiaieueiythoue ia?oo aeniiio cei/aiiinoi i Z — iao

i?inoi?. Oiaei ovK_wC(X,Y,Z)=K_hC(X,Y,Z).

Iioii ae

oai?aio 3.1.6 i

eaaciiaia?a?aiinoue aiaeia?aaeaiue c eeano K_hC i oa

4.2.1 i?

inoue eaaciiaia?a?aiiai aiaeia?aaeaiiy anoaiiaeaii ?iaiinoue eeania ovKC
i K_hC.

bf Oai?

4.4.5. it Iaoa

X — aa?ian

i?inoi?, ui caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, Y — aa

i?inoi?, yeee caaeiaieueiy? ae?oao aeniiio cei/aiiinoi aai iao?e/iee
i Z — nai

iao?eciaiee oiiieiai/iee i?inoi?. Oiaei K_hC(X,Y,Z)=ovKC(X,Y,Z).

Iiaeiaii ?iai

i/ii anoaiiaeththoueny aeey eeania ovKC, ovK_wC i K_hC.

bf Oai?

4.4.4. it Iaoa

X i Y — aa?ian

, i?inoi? X caaeiaieueiy? ia?oo aeniiio cei/aiiinoi, i?inoi? Y
caaeiaieueiy? ae?oao aeniiio cei/aiiinoi i Z

i?inoi?. Oiaei K_hK(X,Y,Z)=ovK_wK(X,Y,Z).

bf Oai?

4.4.6. it Iaoa

X i Y — aa?ian

, ui caaeiaieueiythoue ae?oao aeniiio cei/aiiinoi i Z — nai

iao?eciaiee oiiieiai/iee i?inoi?. Oiaei K_hK(X,Y,Z)=ovKK(X,Y,Z).

A iiae?ic

i 4.5 ?ica’y

iaa?iaio caaea/o aei oai?aie 4.2.1 i

neiao?e/ii? eaaciiaia?a?aiinoi eaaciiaia?a?aieo aiaeia?aaeaiue.

4.5.4. it Iaoa

A i B — iiiae

ia?oi? eaoaai?i? i oeio F_sigma ia /en

ie i?yiie bf R. Oiaei inio? e

iiaia?a?aia ooieoeiy f : bf R^2 to bf R, oaea, ui bf Rsm A ? iiiae

oi/ie neiao?e/ii? eaaciiaia?a?aiinoi aiaeiinii x i

iiaia?a?aiinoi aiaeiinii y aiaeia?aaeaiiy f i bf Rsm B ? iiiae

oi/ie neiao?e/ii? eaaciiaia?a?aiinoi aiaeiinii y i

iiaia?a?aiinoi aiaeiinii x aiaeia?aaeaiiy f.

Ia aeiaaeie aiaeia?aaeaiue aiae aaaaoueio ciiiieo ia?aianaii oai?aie
3.1.4, 4.1.3 i 2.5.3 a iiae?ic

i 4.6.

bf Oai?

4.6.1. it Iaoa

X_1times X_2times …times X_n-2 – aa?ian

i?inoi?, X_1, X_2, … , X_n caaei

ia?oo aeniiio cei/aiiinoi, Z — ia

i?inoi? i aiaeia?aaeaiiy f : X_1times X_2times …times X_n to Z iaia

aiaeiinii ciiiieo x_2, … , x_n cie?

f(circ, circ, x_3, … , x_n) ai?e

eaaciiaia?a?aia aeey eiaeiiai iaai?o (x_3, … ,x_n) in X_3times …
times X_n. Oiaei iii

C_x_n=(x_1, … ,x_n-1) in X_1times … times X_n-1 : (x_1, … ,
x_n-1, x_n)~in~C(f)

? caee

a i?inoi?i X_1 times X_2 times … times X_n aeey ei

x_n

X_n. ss

ae aei oiai X_1 times X_2 times … times X_n-1 — aa?ian

i?inoi?, oi aiaeia?aaeaiiy f eaaciiaia?a?aia.

bf O

4.6.2. it Iaoa

X_1, X_2, … , X_n-1 — aa?ian

, i?inoi?e X_2, X_3, … , X_n caaei

ae?oao aeniiio cei/aiiinoi, Z — ?a

i?inoi? i aiaeia?aaeaiiy f : X_1times X_2times …times X_n to Z eaacii

aiaeiinii x_n i aee

1leq k leq n-1 i eiaei

iaai?o (x_1, x_2 … , x_k-1) in X_1 times X_2 times … times X_k-1
aiaeia?

g(x,y)~=~f(x_1, … ,x_k-1, x_k, x_k+1, … , x_n), aea x=x_k,
y=(x_k+1, … , x_n), ai?e

eaaciiaia?a?aia. Oiaei aiaeia?aaeaiiy f eaaciiaia?a?aia ca noeoiiinoth
ciiiieo.

bf O

4.6.3. it Iaoa

X_1 — aa?ian

i?inoi?,X_2, X_3, … , X_n – aa?ian

, yei iathoue cei/aiio inaaaeiaaco, Z — i

i?inoi? i aiaeia?aaeaiiy

f : X_1times X_2times …times X_n to Z oi/e

?ic?eaia aiaeiinii x_n i aee

1leq k leq n-1 i eiaei

iaai?o (x_1, x_2 … , x_k-1)~in~X_1~times~X_2~times … times
X_k-1aiaeia?

g(x,y)=f(x_1, … ,x_k-1, x_k, x_k+1, … , x_n), aea x=x_k, y=(x_k+1,
… , x_n), ai?e

eaaciiaia?a?aia. Oiaei aiaeia?aaeaiiy f oi/eiai ?ic?eaia ca noeoiiinoth
ciiiieo.

A iiae?icaeiei 4.7

iiiaicia/ii aiaeia?aaeaiiy i anoaiiaeaii aeey ieo oai?aio iiaeiaio aei
oai?aie 4.2.1. Ii

aiaeia?aaeaiiyi F : P to Z iace

i?aaeei, ca yeei eiaeiiio aeaiaioo p in P noa

o aiaeiiaiaeiinoue iiiaeeia F(p)sbs Z. Aeey ii

aiaeia?aaeaiue inio? aeaa oeie iaia?a?aiinoi i eaaciiaia?a?aiinoi.
Aiaeia?aaeaiiy F : P to Z iace

iaia?a?aiei caa?oo /cieco/ o oi/oei p_0 in P, yeui ae

aeiaieueii? aiaee?eoi? iiiaeeie W a Z, o

, ui F(p_0)sbs W (F(p_0)cap W not = O) inio? i

O oi/

p_0 a X, oaee

, u

F(p)sbs W (F(p)cap W not = O) aeey anio oi

p in O.

Aiaeia?

F : P to Z iaia

caa?oo /cieco/, yeui aiii iaia?a?aia caa?oo /cieco/ a eiaeiie oi/oei.

Iaoae G — ae

nenoaia iiaeiiiaeei i?inoi?o P. Aiaeia?aaeaiiy F~:~P~to~Z iace

G-eaaciiaia?a?aiei caa?oo /cieco/ o oi/oei p_0~in~P, yeui ae

aeiaieueii? aiaee?eoi? iiiaeeie W a i?inoi?i Z,

, ui F(p_0)sbs W (F(p_0)cap W not = O)

i aeiaieue

ieieo O oi/ee p_0 a P

iiiaeeia O_1 in G, oaea, ui O_1 sbs O i

F(p)sbs W (F(p)cap W not = O) aeey anio p in O_1.

Aiaeia?

F : P to Z

iace

G-eaaciiaia?a?aiei caa?oo /cieco/, yeui aiii G-eaaciiaia?a?aia a

eiaeiie oi/oei.

sse i ?aiioa aa?o/e caiinoue G nenoaio anio aiaee?eoeo iaii?iaeiio
iiiaeei i?inoi?o P,

aiaeiiaiaeio G-eaaciiaia?a?aiinoue caa?oo /cieco/ ie iaceaaoeiaii
i?inoi eaaciiaia?a?aiinoth

caa?oo /cieco/. sseui P~=~Xtimes~Y, oi G^x_0-eaacii

inoue

/cieco/,

G_y_0-eaacii

inoue caa?oo /cieco/, cal H^x-eaacii

inoue

/cieco/ /e

cal H_y-eaacii

inoue caa?oo /cieco/ o oi/oei p_0=(x_0,y_0)in Xtimes Y

iace

aiaeiiaiaeii neiao?e/iith eaaciiaia?a?aiinoth aiaeiinii x caa?oo
/cieco/,

neiao?e/iith eaaciiaia?a?aiinoth aiaeiinii y caa?oo /cieco/,

ai?eciioaeueii eaaciiaia?a?aiinoth caa?oo /cieco/

/e aa?oeeaeueiith eaaciiaia?a?aiinoth caa?oo /cieco/ o oi/oei p_0.

i ?a

, aiaeia?aaeaiiy F : X times Y iace

eaaciiaia?a?aiei aiaeiinii x caa?oo, yeui

aiii ? oaeei o eiaeiie oi/oei p in X times Y.

bf Oai?

4.7.1. it Iaoa

i? X caaeiaieueiy? ae?oao aeniiio cei/aiiinoi,

i?inoi? Y —

aai caaeiaieueiy? ae?oao aeniiio cei/aiiinoi, Z —

i? c ae?oaith aeniuiith cei/aiiinoi i aiaeia?aaeaiiy F:X times Y to Z

eaacii

cieco. Oiaei inio? caeeoeiaa a Y iiiaeeia B,

oaea, ui aiaeia?aaeaiiy F neiao?e/ii eaaciiaia?a?aia cieco aiaeiinii y

a eiaeiie oi/oei aeiaooeo X times B.

AENI

aeena?oaoeieiie ?iaioi aeineiaeaeothoueny aeanoeainoi eaaciiaia?a?aieo
aiaeia?aaeaiue aiae iaeii??, aeaio oa aaaaoueio ciiiieo. Aeei?enoiaoth/e
ai?eciioaeueio eaaciiaia?a?aiinoue iaea?aeaii ?yae ?acoeueoaoia i?i
iayaiinoue oi/ie eaaciiaia?a?aiinoi, neiao?e/ii? eaaciiaia?a?aiinoi oa
iaia?a?aiinoi aeey aiaeia?aaeaiue aiae aeaio ciiiieo. E?ii oiai,
iaea?aeaii iien iiiaeeie oi/ie eaaciiaia?a?aiinoi, oa?aeoa?ecaoeith
eaaciiaia?a?aiinoi oa neiao?e/ii? eaaciiaia?a?aiinoi, a oaeiae
?ica’ycaii ?yae iaa?iaieo caaea/ iia’ycaieo c eaaciiaia?a?aiinoth.

Aeena?oaoeiy iinoeoue iiai iaa?oioiaaii oai?aoe/ii ?acoeueoaoe, yei ?
iaaieie aianeii o caaaeueio oai?ith ooieoeie.

Iniiaii ?acoeueoaoe iioaeieiaaii a i?aoeyo:

1. Ianeth/aiei A.E., Ieoaeethe A. A., Ianoa?aiei A.A. Neiao?e/ia
eaaciiaia?a?aiinoue noeoiii eaaciiaia?a?aieo ooieoeie //

Iao. Nooaei?. – 1999. – O. 11, N2. – N. 204-208.

2. Iane

A.E., Ianoa?aiei A.A. I?i iaia?a?aiinoue ia?icii iaia?a?aieo
aiaeia?aaeaiue ia e?eaeo // Iao.Cooaei?. – 1998. – 9, N 2. – N. 205 –
210.

3. Iano

A. A. I?i iiiaeeio oi/ie eaaciiaia?a?aiinoi //

Iaoeiaee ainiee *a?iiaaoeueeiai oiiaa?neoaoo: Ca. iaoe. i?. – Aei.46.

. – *a?iiaoei: *AeO, 1999. – C. 104 – 106.

4. Iane

A. E., Ianoa?aiei A. A. I?i ?icaeoie iaeiiai ?acoeueoaoo Aa?aey //
Anaoe?a?inueea iaoeiaa eiioa?aioeiy “?ic?iaea

oa canoinoaaiiy iaoaiaoe/ieo iaoiaeia a iaoeiai-oaoii/ieo
aeineiaeaeaiiyo” i?enay/aia 70-?i//th aiae aeiy ia?iaeaeaiiy i?io. I. N.
Eaciii?nueeiai. *.1. – E

ia. – 1995. – N. 80.

5. Iane

A.E., Ianoa?aiei A.A. Ai?eciioaeueia eaaciiaia?a?aiinoue oa ??
canoinoaaiiy. – *a?iiaoei, 1996. – 15 n. – Aeai. a Oe?II

I 01.11.96, N 98 – Oe.96.

6. Iano

A. A. I?i neiao?e/io eaaciiaia?a?aiinoue noeoiii eaaciiaia?a?aieo
ooieoeie // Iaoa?iaee iaoeiai? eiioa?aioei?

i?enay/aii? 125-?i//th aiae aeiy ia?iaeaeaiiy A.Eaaeoeeiai, Oa?iiiieue,
– 1997, – N. 52 – 55.

7. Iane

A.E., Ieoaeethe A. A., Ianoa?aiei A.A. Ca’ycee iiae ?icieie oeiaie
eaaciiaia?a?aiinoi // Iaoa?iaee Iiaeia?iaeii? iaoe. eiio. “Co/anii
i?iaeaie iaoaiaoeee”, – *.3. – *a?iia

i, – 1998. – N. 40 – 42.

8. Iane

A.E., Ianoa?aiei A. A. Iaa?iaia caaea/a aeey eaaciiaia?a?aieo ooieoeie
// Cai?iee iaoeiaeo i?aoeue Eai’yiaoeue-Iiaeieuenueeiai aea?aeaaiiai
iaaeaaiai/iiai oiiaa?neoaooth. Eai’yiaoeue-Iiaeieuenueeee aea?ae. iaae.
oiiaa?., Na?iy oiceei-iaoaiaoe/ia (Iaoaiaoeea). Aeione 4. – 1998, – N.
76 – 79.

Iano

A.A. ?icii oeie eaaciiaia?a?aiinoi oa ?o canoinoaaiiy. ?oeiien.
Aeena?oaoeiy ia caeiaoooy a/aiiai nooiaiy eaiaeeaeaoa
oiceei-iaoaiaoe/ieo iaoe ca niaoeeaeueiinoth 01.01.01~– iaoa

aiaeic, Eueaianueeee iaoeiiiaeueiee oiiaa?neoao iiaii Iaaia O?aiea,
Eueaia, 2000.

Aeineiae

aeanoeainoi eaaciiaia?a?aieo aiaeia?aaeaiue caaeaieo ia oiiieiai/ieo
i?inoi?ao. Iaea?aeaii iien iiiaeeie oi/ie eaaciiaia?a?aiinoi i
anoaiiaeaii iayaiinoue oi/ie iaia?a?aiinoi aiaei?a?aaeaiue aeaio
ciiiieo, yei ai?eciioaeueii eaaciiaia?a?aii i iaia?a?aii aiaeiinii
ae?oai? ciiiii?, a oaeiae oi/ie neiao?e/ii? eaaciiaia?a?aiinoi noeoiii
eaaciiaia?a?aieo aiaeia?aaeaiue. E?ii oiai, iaea?aeaii oa?aeoa?ecaoei?
neiao?e/ii eaaciiaia?a?aieo i noeiii eaaciiaia?a?aieo aiaeia?aaeaiue.

Eeth/iai neiaa: eaaciiaia?a?aii aiaeia?aaeaiiy, ai?eciioaeueia
eaaciiaia?a?aiinoue, neiao?e/ia eaaciiaia?a?aiinoue, ia?icia
eaaciiaia?a?aiinoue.

Nesterenko V.V. Various types of quasicontinuity and their

applications. The manuscript. Thesis for a degree of candidate of
Science (Ph. D.) in Physics and Mathematics, speciality 01.01.01 –
Mathematical Analysis. L’viv national university, L’viv, 2000.

Investigate conditions of quasicontinuous mappings on topological
spaces. A description of the set of quasicontinuity points is given and
the existence of continuity points of maps of two variables which are
horisontally quasicontinuous and quasicontinuous in the second variable
are obtained, and points of symmetrical quasicontinuity of
quasicontinuous maps. Besides a description of symmetrical
quasicontinuity and quasicontinuity are obtained.

Key words: quasicontinuous mappings, horisontal quasicontinuity,
symmetrical quasicontinuity, separate quasicontinuity

Iano

A.A. ?acee/iua oeiu eaaceiai?a?uaiinoe e eo i?eiaiaiea. ?oeiienue.
Aeecna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.01 — iaoa

aiaeec, Eueaianeee iaoeeiiaeueiue oieaa?neoao eiaie Eaaia O?aiea,
Eueaia, 2000.

?aaioa iinayuaia enneaaeiaaieth naienoa ?acee/iuo oeiia
eaaceiai?a?uaiuo ioia?aaeaiee. A ?acaeaea I aeai eeoa?aoo?iue iaci?
?acaeoey eneaaeiaaiey yoeo iiiyoee.

A ?acaeaea II ?anniao?eaathony naienoaa eaaceiai?a?uaiuo ioia?aaeaiee
iaeiie ia?aiaiiie. Caeanue iiaeaii iauaa ii?aaeaeaiea
G-eaaceiai?a?uaiinoe, n eioi?iai, eae /anoiua neo/ae, iieo/aaony
neiao?e/aneay, ai?eciioaeueiay e aa?oeeaeueiay eaaceiai?a?uaiinoue.
Ia?aua aeaa iiae?acaeaea 2.1 e 2.2 iiaea

. Iniiaiui ?acoeueoaoii iiae?acaeaea 2.3 anoue oai?aia 2.3.7, ei

?acaycuaaao ia?aoioth caaea/o e oai?aia i iiiaeanoaa oi/ae
iai?a?uaiinoe eaaceiai?a?uaiiai ioia?aaeaiey a oii neo/aa, eiaaea
ooieoeey caaeaia ia R n cia/aieyie a R. A iiae?acaeaea 2.4 iieo/aii
iienaiea iiiaeanoaa oi/ae eaaceiai?a?uaionoe ioia?aaeaiey f:R(R (oa

2.4.1). A neaae

iiae?acaeaea i?eiaiaia ai?eciioaeueiay eaaceiai?a?uaiinoue e oi/a/ii
?ac?uaiui ioia?aaeaieyi e onoaiiaeaii, /oi ai?eciioaeueii
eaaceiai?a?uaiia ioia?aaeaiea, eioi?ia oi/a/ii ?ac?uaii ioiineoaeueii
aoi?ie ia?aiaiiie aoaeao oi/a/ii ?ic?uaiui io niaieoiiinoe ia?aiaiiuo
i?e iaeioi?uo oneiaeyo ia i?ino?ainoaa (oai?aia 2.5.3).

III i

aii?ino iaee/ey oi/ae iai?a?uaiinoe ioia?aaeaiey f:X(Y(Z c eea

3.1 onoaiiaeaii, /oi ioia?aaeaiea f(KhC e

anthaeo ieioiia iiiaeanoai oi/ae iai?a?uaiinoe ia eaaeaeie ai?eciioaee,
i?e iaeioi?uo oneiaeyo ia i?ino?ainoaa X, Y

Z (oai

3.1.3, 3.1.4, 3.1.7). E?ii

, iieacaii, /oi ai?eciioaeueii eaaceiai?a?uaiua e iai?a?uaiua
ioiineoaeueii aoi?iai ia?aiaiiiai ioia?aaeaiey f:X(Y(Z, aaea X- aa?i

i?ino?ainoai, i?ino?ainoai Y oaeiaeaoai?yao ia?aie aeneiia n/aoiinoe e
Z –

i?ino?ainoai, anoue eaaceiai?a?uaiui io niaieoiiinoe ia?aiaiiuo
(oai?aia 3.1.6). Ian

iiiaeanoaa oi/ae iai?a?uaiinoe ioia?aaeaiey n eeanna KhC iinayuai
iiae?acaeae 3.2.

, /oi aeey ioia?aaeaiey f:X(Y(Z n eea

KhC, aaea X – oiii

i?ino?ainoai, i?ino?ainoai Y oaeiaeaoai?yao aoi?ie aeneiia n/aoiinoe e
Z –

i?ino?ainoai, iiiaeanoai CY(f) a

inoaoi/iui a X (oai?aia 3.2.1). A iiae

3.3 ia?aianaii oai?aio 3.1.4 n a

ia iai?a?uaiua e?eaua (oai?aia 3.3.1).

A ?a

IV onoaiiaeaii oa?aeoa?ecaoeee neiao?e/iie eaaceiai?a?uaiinoe e
eaaceiai?a?uaiinoe, ?anniao?ai aii?in i naycyo iaaeaeo ?acee/iuie
eeannaie ioia?aaeaiee, a oaeaea ia?aianaii iaeioi?ua ?acoeueoaou ia
neo/ae ioia?aaeaiee io ianeieueeeo ia?aiaiiuo e iiiaicia/iuo
ioia?aaeaiee. A iiae?acaeaea 4.1 iaiauaii oai?aio i niaieoiiie
eaaceiai?a?uaiinoe eaaceiai?a?uaiuo ioia?aaeaiee. Onoaiiaeaii, /oi
ai?eciioaeueii eaaceiai?a?uaiia e eaaceiai?a?uaiia ioiineoaeueii aoi?iai
ia?aiaiiiai ioia?aaeaiey f:X(Y(Z, aaea X – aa?i

i?ino?ainoai, i?ino?ainoai Y oaeiaeaoai?yao aoi?ie aeneiia n/aoiinoe e
Z – ?aaoey?iia i?ino?ainoai, aoaeao eaaceiai?a?uaiui io niaieoiiinoe
ia?aiaiiuo (oai?aia 4.1.2). Yoi ae

neacaoue i ai?eciioaeueii e aa?oeeaeueii eaaceiai?a?uaiuo
ioia?aaeaieyo. A iiae?acaeaea 4.2 onoaiiaeaii, /oi eaaceiai?a?uaiia
ioia?aaeaiea f:X(Y(Z eia

oi/ee neiao?e/iie eaaceiai?a?uaiinoe ioiineoaeueii y (oai?aia 4.2.1).
Yoa o

aeaao aiciiaeiinoue onoaiiaeoue iaiaoiaeeiua e aeinoaoi/iua oneiaey
eaaceiai?a?uaiinoe (oai?aia 4.3.2). I?e n

EMBED Equation.3

wC i KhC, eioi

?aoaii a iiae?acaeaea 4.4,

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Різні типи квазінеперервності та їх застосування: Автореф. дис… канд. фіз.-мат. наук / В.В. Нестеренко, Львів. нац. ун-т ім. І.Франка. — Л., 1999. —