Linearly ordered compact sets and co-Namioka spaces
It is proved that for any Baire space $X$, linearly ordered compact $Y$, and separately continuous mapping $f:\, X \times Y \rightarrow \mathbb{R}$, there exists a $G_{\delta}$-set $A \subseteq X$ dense in $X$ and such that $f$ is jointly continuous at every point of the set $A \times Y$, i.e., any...
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509439671402496 |
|---|---|
| author | Mykhailyuk, V. V. Михайлюк, В. В. |
| author_facet | Mykhailyuk, V. V. Михайлюк, В. В. |
| author_sort | Mykhailyuk, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:14Z |
| description | It is proved that for any Baire space $X$, linearly ordered compact $Y$, and separately continuous mapping $f:\, X \times Y \rightarrow \mathbb{R}$, there exists a $G_{\delta}$-set $A \subseteq X$
dense in $X$ and such that $f$ is jointly continuous at every point of the set $A \times Y$, i.e., any linearly ordered compact is a co-Namioka space. |
| first_indexed | 2026-03-24T02:41:07Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.51
V. V. Myxajlgk (Çerniv. nac. un-t)
LINIJNO VPORQDKOVANI KOMPAKTY
I KONAMIOKOVI PROSTORY
It is proved that for any Baire space X, linearly ordered compact Y, and separately continuous mapping
f : X × Y → R, there exists a Gδ-set A ⊆ X dense in X and such that f is jointly continuous at every
point of the set A × Y, i.e., any linearly ordered compact is a co-Namioka space.
Dokazano, çto dlq proyzvol\n¥x prostranstva Bera X, lynejno uporqdoçennoho kompakta Y y
razdel\no neprer¥vnoho otobraΩenyq f : X × Y → R suwestvuet plotnoe v X Gδ-mnoΩestvo
A ⊆ X takoe, çto funkcyq f neprer¥vna po sovokupnosty peremenn¥x v kaΩdoj toçke mno-
Ωestva A × Y, t.'e. proyzvol\n¥j lynejno uporqdoçenn¥j kompakt qvlqetsq konamyokov¥m
prostranstvom.
1. DoslidΩennq masyvnosti mnoΩyny toçok sukupno] neperervnosti narizno ne-
perervnyx funkcij, vyznaçenyx na dobutku berivs\koho i kompaktnoho prosto-
riv, zajmagt\ osoblyve misce v teori] narizno neperervnyx vidobraΩen\. Kla-
syçnyj rezul\tat Namioky [1] stav poßtovxom do intensyfikaci] danyx doslid-
Ωen\ i pryviv, zokrema, do vynyknennq nastupnyx ponqt\, qki buly vvedeni v [2].
Nexaj X, Y — topolohiçni prostory. KaΩut\, wo narizno neperervna funk-
ciq f : X × Y → R ma[ vlastyvist\ Namioky, qkwo isnu[ wil\na v X Gδ-mno-
Ωyna A ⊆ X taka, wo f neperervna za sukupnistg zminnyx v koΩnij toçci
mnoΩyny A × Y.
Kompaktnyj prostir Y nazyva[t\sq konamiokovym, qkwo dlq dovil\noho be-
rivs\koho prostoru X koΩne narizno neperervne vidobraΩennq f : X × Y → R
ma[ vlastyvist\ Namioky.
Najbil\ß zahal\ni rezul\taty u naprqmku vyvçennq vlastyvostej konamio-
kovyx prostoriv oderΩano v [3, 4], de vstanovleno, wo klas kompaktnyx kona-
miokovyx prostoriv zamknenyj vidnosno dobutku i mistyt\ kompakty Valdivia.
Krim toho, v [3] pokazano, wo linijno vporqdkovanyj kompakt [ 0, 1 ] × { 0, 1 } z
leksykohrafiçnym porqdkom takoΩ [ konamiokovym i peredovedeno rezul\tat z
[5] pro konamiokovist\ dovil\noho cilkom vporqdkovanoho kompaktu. Takym
çynom, pryrodno vynyka[ pytannq: çy obov’qzkovo dovil\nyj linijno vporqdko-
vanyj kompakt [ konamiokovym prostorom?
U danij statti my pokaΩemo, wo vidpovid\ na dane pytannq [ pozytyvnog.
2. Spoçatku nahada[mo deqki oznaçennq i dovedemo dopomiΩni tverdΩennq.
Nexaj X — topolohiçnyj prostir i f : X → R. Dlq dovil\no] neporoΩn\o]
mnoΩyny A ⊆ X çerez ω f A( ) poznaçatymemo kolyvannq sup ( ) ( )f x f x′ − ′′{ :
′ ′′ ∈ }x x A, funkci] f na mnoΩyni A, a dlq dovil\no] toçky x X0 ∈ çerez
ω f x( )0 poznaçatymemo kolyvannq inf ( ):ω f U U ∈{ }U funkci] f v toçci x0 ,
de U — systema vsix okoliv toçky x0 u prostori X.
Nexaj X, Y — topolohiçni prostory, f : X × Y → R , x X0 ∈ i y Y0 ∈ .
VidobraΩennq f x0
i fy0
oznaçymo takym çynom: f yx0 ( ) = f xy0
( ) = f x y( , ) dlq
dovil\nyx x X∈ i y Y∈ .
Dlq linijno vporqdkovano] mnoΩyny ( , )X < i toçok ′x , ′′ ∈x X , ′x < ′′x ,
poklademo [ , ]′ ′′x x = x X∈{ : ′x ≤ x ≤ ′′}x , ′ ′′[ )x x, = x X∈{ : ′x ≤ x ≤ ′′}x ,
© V. V. MYXAJLGK, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7 1001
1002 V. V. MYXAJLGK
′ ′′( ]x x, = x X∈{ : ′x < x ≤ ′′}x i ′ ′′( )x x, = x X∈{ : ′x < x < ′′}x . Toçky ′x ,
′′ ∈x X , ′x < ′′x , nazyvatymemo susidnimy, qkwo ′ ′′( )x x, = ∅. Nahada[mo, wo
bazu topolohi] na X, uzhodΩeno] z linijnym porqdkom, utvorggt\ vsi neporoΩni
intervaly ′ ′′( )x x, i promiΩky a x,[ ) i x b,( ], de a i b — vidpovidno naj-
menßyj i najbil\ßyj elementy v X (qkwo vony isnugt\). Lehko baçyty, wo
linijno vporqdkovanyj prostir X kompaktnyj vidnosno topolohi], porodΩeno]
linijnym porqdkom, todi i til\ky todi, koly koΩna neporoΩnq mnoΩyna A ⊆ X
ma[ v X toçnu verxng i toçnu nyΩng meΩi.
Topolohiçnyj prostir X nazyva[t\sq zv’qznym, qkwo A B∪ ≠ X dlq do-
vil\nyx dyz’gnktnyx neporoΩnix vidkrytyx v X mnoΩyn A i B. ZauvaΩymo,
wo linijno vporqdkovanyj kompakt X zv’qznyj todi i til\ky todi, koly v X
nema[ susidnix toçok.
TverdΩennq 1. Nexaj ( , )X < — linijno vporqdkovanyj kompakt, f : X → R
— neperervna funkciq i ε > 0. Todi isnu[ take n ∈N , wo dlq dovil\nyx ele-
mentiv a1, a2, … , an, b1, b2 , … , b Xn ∈ takyx, wo a1 < b1 ≤ a2 < b2 ≤ …
… ≤ an < bn , isnu[ k ≤ n take, wo f a f bk k( ) ( )− < ε.
Dovedennq. Rozhlqnemo take skinçenne pokryttq U i Ii : ∈( ) kompaktnoho
prostoru X promiΩkamy Ui, wo ω f iU( ) < ε dlq koΩnoho i I∈ . Poklademo
n = I + 1. Todi dlq dovil\nyx 2n toçok a1 < b1 ≤ a2 < b2 ≤ … an < bn z
prostoru X isnu[ i I0 ∈ take, wo promiΩok Ui0
mistyt\ prynajmni try z cyx
toçok. Vraxuvavßy, wo Ui0
— promiΩok, oderΩymo, wo isnu[ k ≤ n take, wo
ak , b Uk i∈
0
. Todi f a f bk k( ) ( )− ≤ ω f iU( )
0
< ε.
TverdΩennq 2. Nexaj ( , )X < — linijno vporqdkovanyj zv’qznyj kompakt,
a = min X, b = max X, f : X → R — neperervne vidobraΩennq i f a( ) ≠ f b( ) . Todi
isnu[ toçka c a b∈( , ) taka, wo f c( ) = 1
2
f a f b( ) ( )+( ).
Dovedennq. Poklademo y0 = 1
2
f a f b( ) ( )+( ), A = f y− −∞1
0(( , )) i B =
= f y− +∞1
0(( , )) . Oskil\ky f neperervne, to mnoΩyny A i B [ vidkrytymy v X.
Zi zv’qznosti prostoru X vyplyva[, wo mnoΩyna C = X A B\ ( )∪ neporoΩnq.
Zalyßylos\ vybraty dovil\nu toçku c C∈ .
3. Perejdemo do vykladu osnovnyx rezul\tativ.
Teorema. Dovil\nyj linijno vporqdkovanyj zv’qznyj kompakt [ konamiokovym
prostorom.
Dovedennq. Nexaj X — berivs\kyj prostir, ( , )Y < — linijno vporqdkova-
nyj kompakt i f : X × Y → R — narizno neperervna funkciq. Dovedemo, wo
vidobraΩennq f ma[ vlastyvist\ Namioky.
Zafiksu[mo ε > 0 i pokaΩemo, wo vidkryta mnoΩyna Gε = x X∈{ : ω f x y( , ) <
< 9ε dlq koΩnoho y Y∈ } [ wil\nog v X.
Nexaj U — dovil\na neporoΩnq vidkryta v X mnoΩyna. Dlq koΩnoho
x U∈ poznaçymo çerez Nx mnoΩynu takyx nomeriv n ∈N , wo isnugt\ a1, …
… , an, b1, … , b Yn ∈ taki, wo a1 < b1 ≤ a2 < b2 ≤ … ≤ an < bn i f x ai( , ) –
– f x bi( , ) > ε dlq koΩnoho i = 1, … , n. Z tverdΩennq 1 vyplyva[, wo vsi mno-
Ωyny Nx obmeΩeni zverxu. Dlq koΩnoho x U∈ poklademo ϕ( )x = max Nx ,
qkwo Nx [ neporoΩn\og, i ϕ( )x = 0, qkwo Nx = ∅. Z neperervnosti f
vidnosno perßo] zminno] vyplyva[, wo dlq koΩnoho ciloho nevid’[mnoho n
mnoΩyna x U∈{ : ϕ( )x n> } [ vidkrytog v U, tobto funkciq ϕ : U → Z [
napivneperervnog znyzu na berivs\komu prostori U. Tomu (dyv. [6]) funkciq ϕ
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
LINIJNO VPORQDKOVANI KOMPAKTY I KONAMIOKOVI PROSTORY 1003
[ toçkovo rozryvnog. Todi isnugt\ vidkryta v U neporoΩnq mnoΩyna U0 i
nevid’[mne cile çyslo n ∈Z taki, wo ϕ( )x = n dlq koΩnoho x U∈ 0 .
Qkwo n = 0, to f x a( , ) – f x b( , ) ≤ ε dlq dovil\nyx x U∈ 0 i a , b Y∈ .
Vzqvßy dovil\nu toçku y Y0 ∈ i vidkrytu neporoΩng mnoΩynu U U1 0⊆ taku,
wo ω fy
U
0
1( ) < ε, oderΩymo, wo ω f x y( , ) < 3ε dlq dovil\nyx x U∈ 1 i y Y∈ .
Zokrema, U G1 ⊆ ε.
Teper rozhlqnemo vypadok, koly n ∈N . Viz\memo dovil\nu toçku x U0 0∈ i
vyberemo toçky a1, … , an, b1, … , b Yn ∈ taki, wo a1 < b1 ≤ a2 < b2 ≤ … ≤ an <
< bn i f x ai( , )0 – f x bi( , )0 > ε pry 1 ≤ i ≤ n.
Vykorystovugçy neperervnist\ funkci] f vidnosno perßo] zminno], vyberemo
vidkrytyj okil U U1 0⊆ toçky x0 v U takyj, wo f x ai( , ) – f x bi( , ) > ε dlq
dovil\nyx x U∈ 1 ta i ∈ {1, … , n}. PokaΩemo, wo dlq dovil\noho y Y0 ∈ isnu[
vidkrytyj okil V toçky y0 v Y takyj, wo ω
f x V( ) ≤ 4ε dlq koΩnoho x U∈ 1.
Nexaj y G0 ∈ =
Y a b
i
n
i i\ [ , ]
=1
∪ . Oskil\ky mnoΩyna G vidkryta v Y, to isnu[
vidkrytyj v Y promiΩok V takyj, wo V ⊆ G . Todi dlq dovil\nyx a, b V∈ z
a < b ma[mo [ , ]a b ∩ [ , ]a bi i = ∅ dlq koΩnoho i = 1, … , n. Vraxuvavßy, wo
ϕ( )x = n i f x ai( , ) – f x bi( , ) > ε dlq dovil\nyx x U∈ 1 ta i ∈ {1, … , n},
oderΩymo f x a( , ) – f x b( , ) ≤ ε, tobto ω
f x V( ) ≤ ε dlq koΩnoho x U∈ 1.
Nexaj ai < y0 < bi dlq deqkoho i ∈ {1, … , n}. Todi poklademo V = ( , )a bi i .
ZauvaΩymo, wo f x a( , ) – f x b( , ) ≤ 2ε dlq dovil\nyx toçok a, b ∈ ( , )a bi i i
x U∈ 1. Spravdi, prypustymo, wo isnugt\ a, b ∈ ( , )a bi i i x U∈ 1 taki, wo
f x a( , ) – f x b( , ) > 2ε. Todi zhidno z tverdΩennqm 2 isnu[ toçka c a b∈( , ) taka,
wo f x a( , ) – f x c( , ) = f x c( , ) – f x b( , ) > ε, a ce supereçyt\ tomu, wo ϕ( )x =
= n. OtΩe, ω
f x V( ) ≤ 2ε.
Zalyßylos\ rozhlqnuty vypadok, koly y0 ∈ ai{ , bi : 1 ≤ i ≤ n} . Nexaj a0 =
= min Y, b0 = max Y i a0 < y0 < b0 . Poklademo y1 = max ai{( , bi : 0 ≤ i ≤ n} ∩
∩ a y0 0,[ )), y2 = min ai{( , bi : 0 ≤ i ≤ n} ∩ y b0 0,( ]) i V = ( , )y y1 2 . Z tverdΩen-
nq'2 vyplyva[, wo dlq dovil\nyx a ∈ y y1 0,( ], b y y∈[ )0 2, i x U∈ 1 vykonugt\sq
nerivnosti f x a( , ) – f x y( , )0 ≤ 2ε i f x y( , )0 – f x b( , ) ≤ 2ε. Tomu ω
f x V( ) ≤ 4ε
dlq koΩnoho x U∈ 1. U vypadku, koly y0 = a0 abo y0 = b0 , dosyt\ poklasty
V = y y0 2,[ ) abo V = y y1 0,( ].
Dovedemo teper, wo U G1 ⊆ ε. Nexaj ( , )x y U0 0 1∈ × Y . Viz\memo vidkrytyj
okil V toçky y0 u prostori Y takyj, wo ω
f x V( ) ≤ 4ε dlq koΩnoho x U∈ 1.
Vykorystavßy neperervnist\ f vidnosno perßo] zminno], vyberemo okil U U2 1⊆
toçky x0 v X takyj, wo ω fy
U
0
2( ) < ε. Todi ω f U V( )2 × < 9ε.
OtΩe, dlq koΩnoho ε mnoΩyna Gε [ wil\nog v berivs\komu prostori X.
Todi Gδ-mnoΩyna A =
n n
G
=
∞
1
1∩ takoΩ [ wil\nog v X, pryçomu funkciq f
neperervna v koΩnij toçci mnoΩyny A × Y, tobto f ma[ vlastyvist\ Namioky.
Naslidok. Dovil\nyj linijno vporqdkovanyj kompakt [ konamiokovym pros-
torom.
Dovedennq. Nexaj ( , )Y < — dovil\nyj linijno vporqdkovanyj kompakt.
Qkwo mnoΩyna D vsix par susidnix v Y toçok poroΩnq, to zhidno z dovedenog
teoremog prostir Y [ konamiokovym.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
1004 V. V. MYXAJLGK
Nexaj D ≠ ∅. Dlq koΩno] pary d D∈ susidnix toçok v Y poznaçymo çerez
ad i bd vidpovidno livu i pravu susidni toçky pary d, tobto d = a bd d,{ } i
ad < bd . Poklademo Z = Y D∪ ×( )( , )0 1 i oznaçymo linijnyj porqdok na Z,
qkyj [ prodovΩennqm linijnoho porqdku na Y. Nexaj ′ ∈z Y , d D∈ , t ∈( , )0 1 i
′′z = ( , )d t . Todi ′z < ′′z , qkwo ′z ≤ ad , i ′′z < ′z , qkwo bd ≤ ′z . Nexaj ′z =
= ( , )′ ′d t , ′′z = ( , )′′ ′′d t ∈ D × (0, 1). Todi ′z < ′′z , qkwo a ad d′ ′′< abo ′d = ′′d
i ′t < ′′t .
ZauvaΩymo, wo ( , )Z < — kompaktnyj prostir, qkyj ne ma[ susidnix toçok,
pryçomu prostir ( , )Y < [ kompaktnym pidprostorom prostoru ( , )Z < .
Nexaj X — dovil\nyj berivs\kyj prostir i f : X × Y → R — narizno nepe-
rervne vidobraΩennq. Pobudu[mo vidobraΩennq g : X × Z → R, qke [ prodov-
Ωennqm vidobraΩennq f. Dlq dovil\nyx x X∈ i z = ( , )d t ∈ D × (0, 1) pokla-
demo g x z( , ) = ( ) ( , )1 − t f x ad + tf x bd( , ). Lehko baçyty, wo funkciq g takoΩ [
narizno neperervnog. Tomu zhidno z dovedenog teoremog isnu[ wil\na v X Gδ-
mnoΩyna A X⊆ taka, wo g neperervna za sukupnistg zminnyx u koΩnij toçci
mnoΩyny A × Z. Tomu funkciq f neperervna za sukupnistg zminnyx u koΩnij
toçci mnoΩyny A × Y. OtΩe, f ma[ vlastyvist\ Namioky i Y [ konamiokovym.
4. U c\omu punkti my pokaΩemo, wo v navedenyx vywe mirkuvannqx umova
neperervnosti funkci] f vidnosno perßo] zminno] [ istotnog i ne moΩe buty
zaminena na slabßu umovu — kvazineperervnist\.
Nexaj X — topolohiçnyj prostir i f : X → Y. Nahada[mo, wo vidobraΩennq f
nazyva[t\sq kvazineperervnym u toçci x X0 ∈ , qkwo dlq dovil\nyx okoliv U
toçky x0 v X i V toçky y0 = f x( )0 v Y isnu[ vidkryta v X neporoΩnq mno-
Ωyna G U⊆ taka, wo f G V( ) ⊆ . VidobraΩennq f nazyva[t\sq kvazinepererv-
nym, qkwo f kvazineperervne v koΩnij toçci x X∈ .
Pryklad. Nexaj X = (0, 1) i Y = [0, 1] × {0, 1} — linijno vporqdkovanyj
kompakt z leksykohrafiçnym porqdkom, tobto ( , )y i < ( , )z j , qkwo y < z abo
y = z i i < j. Dlq koΩnoho t ∈[ ]0 1, poklademo tl = ( , )t 0 i tr = ( , )t 1 . Funkcig
f : X × Y → R oznaçymo takym çynom: f x y( , ) = 0, qkwo x yr ≤ , i f x y( , ) = 1,
qkwo x yl ≥ .
Dlq koΩnoho x ∈( , )0 1 ma[mo ( ) ( )f x −1 0 = xr r, 1[ ] i ( ) ( )f x −1 1 = 0l lx,[ ],
tomu vsi funkci] f x
[ neperervnymy. Qkwo y l r∈{ }0 0, , to f xy( ) = 1 dlq
koΩnoho x X∈ , i qkwo y l r∈{ }1 1, , to f xy( ) = 0 dlq koΩnoho x X∈ . Nexaj
z ∈( , )0 1 . Todi pry y = zl ma[mo f xy( ) = 0, qkwo x z∈( , )0 , i f xy( ) = 1, qkwo
x z∈[ , )1 . A pry y = zr f xy( ) = 0, qkwo x z∈( , ]0 , i f xy( ) = 1, qkwo x z∈( , )1 .
OtΩe, funkciq f [ kvazineperervnog vidnosno druho] zminno]. Ale funkciq f
rozryvna za sukupnistg zminnyx u koΩnij toçci ( , )x xl i ( , )x xr pry x X∈ .
Avtor vyslovlg[ wyru podqku O.'V.'Maslgçenku za korysni porady, qki
istotno pokrawyly vyklad materialu dano] statti.
1. Namioka I. Separate continuity and joint continuity // Pacif. J. Math. – 1974. – 51, # 2. – P. 515 –
531.
2. Debs G. Points de continuite d’une function separement continue // Proc. Amer. Math. Soc. – 1986.
– 97. – P. 167 – 176.
3. Bouziad A. Notes sur la propriete de Namioka // Trans. Amer. Math. Soc. – 1994. – 344, # 2. –
P. 873 – 883.
4. Bouziad A. The class of co-Namioka spaces is stable under product // Proc. Amer. Math. Soc. –
1996. – 124, # 3. – P. 983 – 986.
5. Deville R. Convergence ponctuelle et uniforme sur un espace compact // Bull. Acad. pol. sci.
Sér. math. – 1989. – 37. – P. 507 – 515.
6. Calbrix J., Troallic J. .P. Applications séparément continues // C. r. Acad. sci. A. – 1979. – 288. –
P. 647 – 648.
OderΩano 26.12.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
|
| id | umjimathkievua-article-3362 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:07Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/75/a75c23dd83ad3181a447a47f24214c75.pdf |
| spelling | umjimathkievua-article-33622020-03-18T19:52:14Z Linearly ordered compact sets and co-Namioka spaces Лінійно впорядковані компакти і конаміокові простори Mykhailyuk, V. V. Михайлюк, В. В. It is proved that for any Baire space $X$, linearly ordered compact $Y$, and separately continuous mapping $f:\, X \times Y \rightarrow \mathbb{R}$, there exists a $G_{\delta}$-set $A \subseteq X$ dense in $X$ and such that $f$ is jointly continuous at every point of the set $A \times Y$, i.e., any linearly ordered compact is a co-Namioka space. Доказано, что для произвольных пространства Бера $X$, линейно упорядоченного компакта $Y$ и раздельно непрерывного отображения $f:\, X \times Y \rightarrow \mathbb{R},$ существует плотное в $X$ $G_{\delta}$ -множество $A \subseteq X$ такое, что функция $f$ непрерывна по совокупности переменных в каждой точке множества $A \times Y$, т. е. произвольный линейно упорядоченный компакт является конамиоковым пространством. Institute of Mathematics, NAS of Ukraine 2007-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3362 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 7 (2007); 1001–1004 Український математичний журнал; Том 59 № 7 (2007); 1001–1004 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3362/3467 https://umj.imath.kiev.ua/index.php/umj/article/view/3362/3468 Copyright (c) 2007 Mykhailyuk V. V. |
| spellingShingle | Mykhailyuk, V. V. Михайлюк, В. В. Linearly ordered compact sets and co-Namioka spaces |
| title | Linearly ordered compact sets and co-Namioka spaces |
| title_alt | Лінійно впорядковані компакти і конаміокові простори |
| title_full | Linearly ordered compact sets and co-Namioka spaces |
| title_fullStr | Linearly ordered compact sets and co-Namioka spaces |
| title_full_unstemmed | Linearly ordered compact sets and co-Namioka spaces |
| title_short | Linearly ordered compact sets and co-Namioka spaces |
| title_sort | linearly ordered compact sets and co-namioka spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3362 |
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