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ℵ0 #$,- :A3'(#$@B? 25654$595 .%-$,(3- .,&$3K,$,(3)3(* C% 1,7(,:$,(3 0,(,',-$,(3- 78()21$8(2 ,)$,( ), .% >/- 1%$$,-)3B3:3F '8(2:A $8()8- @,(1$3 2)2* !" #$%$& '&()*(+ H&,B3),12 ',-3:$,(3 ,-3:5,0,$3:? !"#$ %&'%'%' C2( (X, T ) %F,03 78(2$52-* L 1/:852-2- ,$'1/:8$8(2-)8- #$%A &,- .2( U ,2$852-2- .2$8&2:2 X 1/:852-8 8&2' #$%0#(5,6 U ,2$852-8 X 1/:852-2.2( @)2C-C(C) 6 )8-2$2(* MB8( U ,2$852-8 ,2' 1/:8$8(2- G8J52 ,431 D0, ), 1,J,$3E 2586 U ,2$852-8 X 1/:852-2- !D56 :,! (! 6!>!$5 ? 73) @)2C-C(C) 6 )8-2$2(* MB8( U -%- .2( ,$',2$852 X 1/:852-2- .2( 9('/5/ 2586 .% ,$',2$808 U -%.2( !$2@)2C-C6 )8-2$2(* !"#$ %&'%'(' U 2$8 V6 X 1/:852-2- 212 ,431 9('/5/ #$5%-* MB8( G8( U ∈ U 242- V ⊂ U #$,K,1 &812$)8 .2( V ∈ V 1/:852 7,(5, V 9('/5/-86 U 9('/5/-/.2( 7C4C$8CEC(C)6 )8-2$2(* MB8( ,431 U 9('/5/-/- .2( ./F/$:/&/ 7,(5,6 U 0, 7C4C$#73$3) 73) @)2C(C)6 )208K8B2F* !"#$ %&'%')' C2( (X, T ) %F,03 78(2$52-* MB8( X 1/:852-2- G8( ,431 9('/A 5/-/- 5#-$% .2( ,$'9('/5/ 7,(5,6 (X, T ) %F,03-, 25654 D1*8>!12 E .2( %F,0)3(6 )8-2$2(* !"#$ %&' ()*)+")* !" !"#$ %&'%'%' #"$%&'()* +,-.+,-/0/1. 2&(&3 3,-3)4) 5*,1 67,14)' !"#$%&' 839*9 ()&)* +,!-.%,(/ "*%0 (.7/1. ()&)*)-1) +2"%1',!-.%,(/ 9',( -.7/+/1/ 39*: *,1)7 $ ;,1)+ <$ $= /*. !"#$%&' -,1)+) ,7,4)1>, 2/7 &4-&1*&3 ?,73) (53-97$ @/: -,A 25(91B, C,1D/ -,1)+)1 .4,4 ,*)1>)0) /(/ DE'.-/*>/0/ 4&7.B.F 2/7/4/ E-.3/1. DE7. 1. 35*,(*)3 1. >. '57*93 (,7,-)7$ (")$* %&'%'+' (X, T ) 2/7 9',( 5*491 G. 2/7 H ⊂ X ,*-3&+.4/ G.7/*4/1$ I0.7 (A, TA ) ,*- 9',() -)3)' /4. A (, X /1 ()&)* 2/7 ,*-3&+.4/>/7F >.1/*/7$ ,#)-. %&'%'%' J.7K.* .34.1F &'.7/1>.3/ 4,*- -5A5*5L/(. DE7. -)3)' >.0/*>/7$ J.7K.3-.1F N = {(−n, n)|n ∈ N} ,K)3 ,7,*)3*,7 ,/*.4/ D.7K.* .34.1/1 2/7 ,K)3 E7-&4&>&7F ,1B,3 29 ,/*.>.1 451*9 2/7 ,*-E7-& 4.K/*.+.'$ ,#)-. %&'%'/' M5149' 2/7 3&+.F &'.7/1>.3/ 451*9 -&+*.(.1*.7 -5A5*5L/4/1. DE7. -)3)'>)7$ (")$* %&'%'&' N/7 3&+.*.7 ,/*.4/1/1 C.7 451*9 ,*-,/*.4/1/1 ,7,3.4/-/ 25O >.: 0/*4.F 29 ,/*.(. ,-./0 12134,56 784//595.4 ,1:5;652 (-0#-* %&'%'%' >.1/*/7$ '# (X, T ) "*%0) '3'4 %5%6)7%&' 8*9::'&:9# 957969#7'#; P/Q R',( -)3)'>)7$ P//Q X &'.7/1>.3/ C.7 4&'D.B/1 .1,' 2/7 3,A*,+, 153-,4) G,7>)7$ P///Q @,A,*) 3&+.*.7>.1 5*9O,1 G. 451*9 ,7,3.4/- E'.**/0/1. 4,C/A 5*,1 2/7 ,/*.1/1 ,7,3.4/-/ 25O >.0/*>/7$ P/GQ @,A,*) 3&+.*.7>.1 5*9O,1 G. ,7,3.4/-/ 25O 5*,1 2/7 ,/*.1/1F ,7,3.4/-/ 25O 5*,1 451*9 2/7 ,*-,/*.4/ G,7>)7$ !"#$% (i) ⇒ (ii) S S ,/*.4/ X &'.7/1>. 2/7 4&'D.K 5*491$ S 1/1 C/K2/7 3,A*,+, 153-,4) 5*+,4,(>)F !"#$ =$T$= D.7.0/1B. \ S̄ = ∅ P <$ Q S∈S !"#$%&'( !" (X, T ) #$%&' ()"!*+!,- (i) ./)" X 012)+!,!, 3)" %4'0 5"61+1,1, +7,*# 8!" %*65"61+1 (%"+% (X, T ) #$%&' !"!#!$%! 9&'(%)*+$,(* : 8!" #$%&;'"- (ii) <'0'$'2+' 8!" =%#+;7"> #$%&' !"!# 9*+$,(* : 8!" #$%&;'"- !" " #$%$& '&()*(+ ! "#$%$&'() &* +,-,- ./#.& +$0(-'(1( [ (S̄)′ = X 2 3456 S∈S "#,78,4 9, *1. {(S̄) | S ∈ S } $*#.1*-*- X &:;.1*-*- +*7 $<(& =7':1: "#;$1( 8.;.&'*74 > ?$#8.) ,-. @.7.0*-%.) +, $*#.-*- 1"-#, +*7 {(S̄i )′ | i = 1, 2, . . . , n} $#'=7':1: A$78(7B C$-* ′ n [ (S̄i )′ = X i=1 8*74 9,-,- ./#.0* n \ S̄i = ∅ i=1 "#$%$&'(7) &* +, S -*- 1:D@.< "#;$1(C#$ <.#*/*74 (ii) ⇒ (iii) E X &:;.1*-*- &$F$#( $#'&:;.#.78.- "#,/$- A. 1"-#, $7$&.1*' =D.##*0*-. 1$?*F "#$- +*7 K $*#.1* A.7*#1*-4 K -(- :7.''*0* +*7 S 1:D@.%* A$78(74 2+&D4 !"#$" G454564 > ?$#8. ,--. @.7.0*-%. +, 1:D@.%*- .- $D +*7 &$F#$;$ -"&'$1( A$7"#$%$&'(7 A. K -(- =0.#.7* &$F$#( "#8,0,-8$-) +, &$F#$;$ -"&'$1( K -(- ?.7 =0.1*-. $*' "#$%$&'(7B C$-* K -(- $7$&.1*'* +"/ 8.0*#8*74 (iii) ⇒ (iv) E H$F$#( &:;.#.78.- "#,/$- +*7 K $*#.1*-*- $7$&.1*'* +"/ "#1,-4 I0.7 K $*#.1*-*- ?*< +*7 1"-#, $7$&.1*'* +"/ "#;,C"71$) ,---. @.7.0*-%.) K -(8$ $7$&.1*'* +"/ "#;$C$%$&'(7) &* +,) A.7*#.- A$71$C(;#$ <.#*/*74 (iv) ⇒ (i) E U $*#.1* X &:;.1*-*- +*7 $<(& =7':1: "#1,[ U =X U ∈U +$0(-'(1(-(- ./#.0* "#$7$& \ U′ = ∅ U ∈U +$0(-'(1( .#8. .8*#*74 U #.7 &$F$#( A. $7$&.1*'#.7* +"/ "#8,0,-8$-) ,-/. @.7.J 0*-%. n \ Ui′ = ∅ ′ i=1 "#$%$& /.&*#8. 1"-#, +*7 | i = 1, 2, . . . , n} $#'J$*#.1* A$78(74 9, +$0(-'(-(./#.0*-* C$D$71$& U -,- 1"-#,) {Ui | i = 1, 2, . . . , n} $#'J$*#.1*-*- X &:;.1*-*+*7 $<(& =7':1: "#8,0, @=7:#:74 {Ui′ !"#$ %&' ()*)+")* !" !"#!$ %&'%'(' (X, T ) !"#$ %&%' "(")$*"+% ,!-..%+.-/ -(*-)-/*%/.-/0 #$% X &'()*$+$+ *,-./,0$/$1 2)1 ,3.& 415'*'+'+ *6+/7 0$1 ,/5415'*' 8,19.1: #$$% X &'()*$+$+ 2)1 *6+*7; ,/5&'()*$+$+ )+,; 0$1 -.<./(, +6&5,*. 8,19.1: #$$$% =6> 6/(,-,+ &,?,/. &'()/)19)+ 6/7>,+ A1 ⊃ A2 ⊃ A3 ⊃ . . . ⊃ An ⊃ An+1 ⊃ . . . # @:A% B$0$ $3C$3) ,;,/,+ 2)1 &'()/)1 9$;$*$+$+ ,1,&)*$5$ 06> 9)<$/9$1: !"#$% (i) ⇒ (ii) D E6+*7; 0$1 A &'()*$+$+ 2$30$1 -.<./(, +6&5,*. 6/(,*.+: A +.+F G,1&/. +6&5,/,19,+ 6/7>,+ *,-./,0$/$1 0$1 B = {xn | n ∈ N} ,/5&'()*$+$ ,/,/.(: H,0$$F B +$+ 9) 2$30$1 -.<./(, +6&5,*. 6/(,-,I,&5.1: J-1.I, B +$+ 2)1 4<)*$ 0$1 &)+,1 +6&5,*.9.1K -,+$ ∂B = B 9$1: L)+,1 &'() &,?,/. 6/97<7+9,+ #0&;: ":A: M160/)( N% B &,?,/.9.1: O 2,/9) B ′ ,3.&5.1: L,07/'('; B)1)<$+I)F 2)1 xn ∈ B $3$+ 4-/) 0$1 Vn &6(>7/7<7 07/,0$/$1$; &$ Vn +$+ B $/) xn 9)+ 0,>&, 615,& +6&5,*. 6/(,;: =7 9717(9, {B ′ , Vn′ | n ∈ N} ,$/)*$ X &'()*$+$+ *,-./,0$/$1 0$1 ,3.& 415'*' 6/,I,&5.1 8) 07 ,$/)+$+ *6+/7 0$1 ,/5415'*'+'+ 6/(,-,I,<. ,?,3.&5.1: =7 $*) (i) 8,1*,-.(.+, ,-&.1.9.1: P-/)-*) &,07/'('; -,+/.>5.1K -,+$ A +.+ )+,; 0$1 -.<./(, +6&5,*. 8,19.1: (ii) ⇒ (iii) D Q<)1 0)//$ 0$1 9,(B,9,+ *6+1, An &'()/)1$ )>$5 $*)/)1F 8)1$/)+ &'()/)1 9$;$*$+$+ ,1,&)*$5$ 07 615,& &'()9$1K 96/,-.*.-/, 06> 9)<$/9$1: Q<)1 04-/) 9)<$/*)F B)1)&$-61*, 8)1$/)+ &'()/)1 9$;$*$+$+ 0$10$1/)1$+9)+ G,1&/. 0$1 ,/5 9$;$*$+$ *)3)1)&F 2)1 i ∈ N $3$+ Ai 6= Ai+1 &,07/ )9)0$/$1$;: R)1 n ∈ N $3$+ 0$1 an ∈ An − An+1 4<)*$ *)3)1)& *6+*7; 0$1 S an T 9$;$*$ 6/7>571,/.(: #$$% B)1)<$+I) {an } 9$;$*$+$+ 0$1 -.<./(, +6&5,*. 8,19.1K 07+, a 9$-)/$(: R)1 i ∈ N $3$+ n ≥ i 6/97<7+9, an ∈ Ai 6/,I,<.+9,+ a ∈ Ai 9$1: O 2,/9) a∈ ∞ \ Ai i=1 9$1K -,+$ 8)1$/)+ &'()/)1 9$;$*$+$+ ,1,&)*$5$ 06> 9)<$/9$1: (iii) ⇒ (ii) D V = {Vn | n ∈ N} ,$/)*$ X &'()*$+$+ *,-./,0$/$1 ,3.& 0$1 415'*' 6/*7+: An = {(V1 ∪ V2 ∪ . . . ∪ Vn )′ } n = 1, 2, 3, . . . !" " #$%$& '&()*(+ !" #$%&'()$ %*+*&, %-.$&$/'( '01'0$ *2*&*( 3'/ {An } )'2'4'(' 5&6#76/*&,.8 9$.$( :;/-&$<$=' :'3' ∞ \ Ai = ∅ n=1 5&6/8 > ?*&)$ @'''A :$/$='(<$ 3'/ Ai = ∅ 5&.*&,),/8 B6/*)*( A′i = i \ Vj = X j=1 0,%*/ C$ 36 V ;/7-4-(-( 45(&6 3'/ *&7;/7-4-(-( C*/&,=, )$.$%7'/8 !"#$ %&'%'(' D,%,2 3'/ 62*E 9*64)5/F 5&.*E*3'&'/8 G/($='(H X = {a, b, c} %-.$4' T = {∅, {a}, {a, b}, {a, c}, X} 75+5&5I'4' '&$ )5(*7,&,/4* (X, T ) 62*E, 7,%,2 5&6/8 J(<*%H B = {b} '&$ C = {c} *E/,% C$ %*+*&, %-.$&$/'('( 3'/3'/&$/'()$( *E/,% *0,% %5.#6&6%&*/, E5%76/8 "#!)# %&'%'%' !"#!$ !"#$" %&'()&*")+, -!."'/+0#0 .&1)&'&# *"2 3&0.4/25 06&'( 789797 #"# ,:4,1,2 ;6,))"!),2"#4,# *"2"#" <4/)&'(.(')& =,>."#"? .&1)(@ '/2.& A(!(6 *"2 06&'4(29 B'/ (X, T ) 62*E, '&$ T (6( 3'/ B = {Bn | n ∈ N} 4*E,&*3'&'/ 7*3*(, C$/'&4'(8 U *'&$4' X %-.$4'('( 3'/ *0,% ;/7-4- 5&46(8 9$/ V ∈ U %-.$4' B E$ *'7 %-.$&$/'( 3'/ 3'&$#'.' 5&*/*% E*2,&*3'&'/8 > ?*&)$ X %-.$4' B 7*3*(,(* *'7 %-.$&$/'( 3'/ 3'&$#'.' 5&*/*% E*2,&*3'&'/8 K*/4*E,.,.,2 :$/$='(<$H B ('( 45(&6 3'/ {Bj | j = 1, 2, . . . , n} *'&$4' X %-.$4'(' ;/7$<$%7'/ @3%28 L8 8"A8 G7$ E*()*( ?$/3'/ Bj '0'( Bj ⊂ Vj 5&*<*% #$%'&)$ 3'/ Vj ∈ U C*/ 5&)6=6()*( !"#$% n [ Vj = X j=1 5&*<*%7,/8 "#!)# %&'%'*' B(!(6 *"2 06&'(# !&>&)( =,2 &)A!C+,." A(!(64(29 !"#$% J %-.$4' (X, T ) 62*E,(,( %*+*&, 3'/ *&7%-.$4' 5&46(8 (A, TA ) *&71 62*E,()* %*+*&, 5&*( ?$/ B ⊂ J *&7%-.$4' (X, T ) 62*E, '0'()$ )$ %*+*&, 5&*<*=,()*(H !"#$" !8"8M '&$ %"&#"$ N8 8 )$( '47$($( #$E ?$.$( :;/-&-/8 !"#$ %&' ()*)+")* !" !"#$" %&'%'(' (X, T ) !"#$%&'( A ⊂ X )*+*, -.& !/)+012 %/#"3 42 -.& p∈ /A 42&./#.35 6" $"&"1$! A ⊂ V, p ∈ W, V ∩ W = ∅ %/!7!+ 82+./$2 V 42 W !9*+ +012/2&. 4!&$*&5 #$%&'($) *&+ x ),(-./0 A 10 -.+./0)2 34.1 5.6/7,+8 ,976%6)7.) :$+ x ∈ A &;&) <19$ *&+ Vx ∈ B(x) &9$ <19$ *&+ Wx ∈ B(p) .;0( (,='696(9.+0 >.+70+ (& Vx ∩ Wx = ∅ 70+2 ?.*&& {Vx | x ∈ A} .&9$/& A )0) .;0( *&+ <+-@/@ ,9.A.(-0+2 B :.97$ *6 .&9$)&) !"#$% {Vx1 , Vx2 , . . . , Vxn } C&*& /,)96 *&+ .9-<+-@/@ >.+,9.A.(-0+2 D6 76+6=7. V = n [ V xi i=1 W = n \ Wxi i=1 (@=$9$+& .;0(-0+ >$ ($/&'=$49$+2 E1+0A. W ∈ B(p) A⊂V V ∩W =∅ ,9.A.%0 .;0(-0+2 )"*#"$ %&'%'(' 6.& !"#$%&' ",!:*3*3 ;2& )*+*, !/)+012#. +!<!/*$*&5 F6(.+07. :$+ p ∈ A′ ),(-./0)0) A′ )@) *&+ &;G),(-./0 ,976%6)6 C</-$+7&(H 1.)& A′ .;0(-0+2 B :.97$ A (.I.9070+2 !"#$% )"*#"$ %&'%'+' !"#$%&' 42 )*+*, -.& ",!:*3 -.& !/)+012#.3.3 )*+*, %/1!#* .9.3 =2&2+/. 42 :2)2&/. +%8"/ +!<!/* %/1!#*$*&5 !"#$% J$+$(9&%& <)A$(& K,)6;-.L 1$-$+9&%& &/$ <)A$(& M)$+=$7$ >$+&97&2 )"*#"$ %&'%'&' >*+*, 42 !"#$%&' %/!3 ;2& ",!: 3%&1!/$.&5 !" " #$%$& '&()*(+ !" !"#$% (X, T ) #$%$& '( )*+,-./0 12/ +&*3 .4,+56 X 272-( A 24( B *3/$% '( %*8*4$ 9-.4*3$,$34* #$%$&: ;(/;*5<2 2%2 *4#%=>( .4,+56 ?(@2A%(5 12/ y 5.%#*,$ B %=>(,252 #*/*,$56 )(/ y ∈ B 2725B !"#$" "6 6C +3*/$5D*B A ⊂ Vy , y ∈ Wy , V y ∩ Wy = ∅ .4*D*% A(%24-( *7$% Vy '( Wy %=>(4(/2 '*/-$/6 B 525 {Wy | y ∈ B} *7$% E/#=F ,=5-(5 ,.54+ 12/ {W1 , W2 , . . . , Wm } *4#E/#=,= ,(724(1242/6 G ;*4-( V = m \ Vi , W = i=1 *7$% %=>(4(/2 A ⊂ V, B ⊂ W, m [ 9 "6H: Wi i=1 9 "6": V ∩W =∅ 1*@$5#$4*/$5$ ,*@4*3*D*%4*/-$/B %2 1+B 5./>*4 +&*3 .4>* *%,23.>+5+5 91%&6 %&!'$I JKL *%,23.>+ H6M6 : ,*@4*5>*,$ -(>(%#2/6 N 525 12/ *4#%=>(,2525 %*84*>$ #$%$& 2,( 1+5* <E/(D(42 #$%$& 12/ *4#%=>(-2/ -(5242/6 O4P,2% ,-./01-234546786077 #(./(>2 -(/ %2 5F1.3+#4+ Rn Q%42- +&*3$5$5 #$%$& *4#%=>(4(/2 ;(> ,$5$/4$ ;(> %*8*4$ .4*5 *4#%=>(4(/2-2/R 1+5+5 <E/(D(42 #$%$& *4#%=>(4(/2 2,( 3*45$& ,$5$/4$ .4*5 *4#%=>(4(/2-2/6 S+ %.5+ 94865: '/0;.06<0 -*;* *3/$5#$4$ .4*/*% 25D(4(5(D(%#2/6 S2/ )*+,-./0 +&*3$5-* ,.54+ ;(/ *4#%=>(525 #$%$& .4-+@+ *8*7$%#$/6 T25( ,.54+ #*5( #$%$& *4#%=>(525 124(A2>2525 #$%$& .4-+@+5+ '( 1+ E&(42@25 <E/(D(42 #$%$& *4#%=>(4(/ 2725 -( '*/4$@$5$ <E/>(% %.4*3-$/6 !"#!$ %&'%'(' f : (X, T ) → (Y, S ) '&" 0/1+ & # !"#$% @+5-*5 f (X) ⊂ Y !"#$%& '&" ()*$ &+)* ,# (X, T ) -.$./ "# 2& 3# -.$./3."4 V *24(,2 f (X) *4#%=>(,2525 12/ *7$% E/#=,= .4,+56 f ,=/(%42 .4-+F f −1 (V ) = {f −1 (V )|V ∈ V } *24(,2 X 25 12/ *7$% E/#=,= .4*D*%#$/6 U*/,*3$>$>$& <(/(@25D(B 1+5+5 {f −1 (V1 ), f −1 (V2 ), . . . , f −1 (Vn )} <212 ,.54+ 12/ *4#E/#=,= '*/-$/6S+/*-*5B V1 , V2 , . . . , Vn } ,.54+ *4#F*24(,2525 f (X) %=>(,252 E/#(D(@2 ;(>(5 <E/=4=/6 !"#$ %&' ()*)+")* !" !"#$ %&'%'%' !"#!$ %!&!"$' (! )*"!+$' ,'" f -./+)'0./1 23+34 ,'" +*5! *4!6 "'/%! )3/3"$3%3" (! ,1 +*5! *4!"'/%! 57+)'515 (! 5'/'515 %!&!"$!"'/' 7$3"8 (X, T ) #$%$& '() *&+, -./*0 12 /3)2%.( '() f : (X, T ) → R 4-0%5 /(,-0* 12)(./(06 7*%+)$8+%( 902):2 ;2)2<(0=2> f (X) ;2)?2. 2%/20(0(0 #$%$& '() +.#%3:2/( -.+=+%#$)6 X '-@ -.:+8$<$08+0 f (X) ;9)30#3 %3:2/( '-@ 82<(.8()6 A2)?2. 2%/20(0 /$0$)/$& '() +.#%3:2/( #$%$& -.:+,+=+<$0+ ;9)2> f (X) /$0$).$8$)6 B,)$=+ f (X) %+C+.$8$)6 D'%&6 !"#!$ E6 6FG6 H I+.82 f (X) %3:2/(0(0 ;2)?2. 2%/20 3&2)(082%( ≤ /$)+.+:+/$0+ ;9)2 !/ ,*0*+ 7$2 )3/3"3 ,-./01 2345 12 !/ +*#*+ *)2 )3/3"3 ,-6701 0895 1+)8$)6 !"#$% a = inf{f (X)} , b = sup{f (X)} 8(,2.(:6 J* 8*)*:8+ a = f (x)> b = f (y) -.+=+% @2%(.82 '() x ∈ X (.2 '() y ∈ X 1+) 82:2%#()6 a, b ∈ f (X) -.8*<*0* ;9):2% (?(0 Fn = {z ∈ X : a ≤ f (z) ≤ a + 1 }, n = 1, 2, 3, · · · n (?5(?2 %3:2.2) 8(&(/(0( #+0$:.+,+.$:6 f /3)2%.( -.8*<*08+0 I2) '() Fn %3:2/( %+C+.$8$)6 K,.2,/2> I2) n (?(0 a ∈ Fn -.8*<* 83@303.3)/2> E6 6 D(((G *,+)$0=+> a ∈ ∩Fn ⊂ X -.*)6 L(:8( 82> Kn = {z ∈ X : b − 1 ≤ f (z) ≤ b}, n = 1, 2, 3, · · · n (?5(?2 %+C+.$ %3:2.2) 8(&(/(0( #+0$:.+,+.$:6 7*%+)$8+%(02 '20&2) 83@303@.2 b ∈ ∩Kn ⊂ X -.8*<* /9,.202'(.()6 ()!*)+ %&'%', DB.2M+082)G' (X, T ) ,'" 971)%."- 2.:.$.;'+ 14703 (! S ,16 /1/ <!"<7/=' ,'" 7$2627,7/3 .$)1/8 >1 %1"15%7 7?7&3%7+' '+' '-7%! !?%!&!"%'"@ D(G X *&+,$ #$%$&8$)6 D((G X *&+,$0$0 S +.#5#+'+0$0$0 %(:( 9<2.2)(0820 -.*@+0 I2) 9)#3/30820 /-0.* '() 9)#3 /2?(.2'(.()6 (i) ⇒ (ii) : N$%$& -.:+ #+0$:$08+0 +C+?$% ;9)3.3)6 (ii) ⇒ (i)O H.:+,+0+ 2);( ,90#2:(0( %*..+0+=+<$&6 (ii) 0(0 /+<.+08$<$0$ +:+ (i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∪ {U } -#+')# M -#+')#%' -#. &'8#+&#0D &*+-34)43+- )*%+, 7#0 -+./0.1)1 @-0&406 A/3+' *+2-)-3&4 7, -#+' X 3' -#. *+-?-9.4( 9# 7, &,0,2&- M -#+')#%' -#. *+2-)4 5'0'9#0&#6 U ∈ / M *+&,8,%&-%( 7, *+-%-9)4>&406 E#2&# W = M ∩ S -0-9')#.#%# &1=1%'+#26 A#0 3-%&-%( W ⊂ M *+&,F 8,%&-% W -#+')#%#% X ,>-34%4 /0.'% )*%+, 7#0 -+.-#+')# 3*9.,06 B.' 3-%&-%( W ⊂ S *+&,8,%&-%( W -#+')# X ,>-34%4 /0.'2'>D /0.)'3&# (ii) 9*=,+, ,3-F 04%?- W -#+')#%#% X ,>-34%4 /0.'% )*%+, 7#0 -+./0.1)1 *+-?-9.46 A, #)'( W ⊂ M 9-G)-2-)4 %'&'%#3+'( M -#+')#%#% &' X ,>-34%4 /0.'% )*%+, 7#0 -+./0.13' )-C#G *+&,8, -%+-24%- 5'+#0&#6 H3)- M -#+')# 7, %#.'+#8' )-C#G *+2-3-?-9 ='9#+&' )'$#+2#=.#6 W -#+')# X ,>-34%4 /0.213*0)-( [ x∈X∩ W W ∈W !′ : I6J< *+-?-9 ='9#+&' 7#0 x %*9.-)4 )'$'7#+#0#>6 M 7#0 /0.1 *+&,8,%&-% x ∈ M ∈ M *+-% 7#0 M -$49 912')# @-0&406 S -+.F.-7-% *+&,8,%&-%( M -$49 912')# S -#+')#%' -#. 7->4 912'+'0#% 7#0 7#+'=#2# *+-0-9 3->4+-7#+#06 A,%+-0 -0-)4%&-% x /8')#%# #$'0'% )*%+, )-34&- S1 , S 2 , . . . , S n : I6"< 912'+'0#%# )'$'+#26 x∈ n \ Sj ⊂ M : I6!< [ : I6K< j=1 *+-?-84 -$49.406 x∈ /( W ∈W W) !"#$ %&' ()*)+")* !! "#$%&%'$(') * +,-. /012#234'4' 546 743484 W 464'$2 $2&4#$43, M 1(/841(# 743 93:0 "#$%&%'$(') 523 i = 1, 2, 3, . . . , n 464' X = Si ∪ (Mi1 ∪ Mi2 ∪ Mi3 ∪ . . . ∪ Mir ) * +, ;. "#(<(/ =2/4#$2 8"'#% 8(>?$( Mi1 , Mi2 , Mi3 , . . . , Mir ∈ M (6?/ /012#234 @(3$?3, A23 i = 1, 2, 3, . . . , n 464' Ai = (Mi1 ∪ Mi2 ∪ Mi3 ∪ . . . ∪ Mir ) * +, . $2382/) M∪ n [ j=1 Aj ! ⊃ n \ Sj i=1 ! ∪ n [ Aj j=1 ! =X * +, B. 7(&?':?8? 6?/(3, C9>#2<2 X %D(>? M (4#284'2 (4: 8"'#% 8(>?$( /012#234' 743 74#2=414 "#%3, C% 8"'%6 M (4#284'4' 8264#4=4'2 (>/?3?$?3, C% 743 62#4=/4$43, E 5(#$2) /(7%#010D >('#?=:?3, !"#!$ %&'%'( *F><5"'"G.' !" !#$%&%' (!)!#!*+, -.%&#%/0' 1!" !#$%&%' 1+/ %+#23+'+' 4%/)0$ -.%&0'0' (0,0. !#$%30 +4+' 52/2,#+ 62 &2(2/#+ ,!"-#7 4%/)%' -.%&#%/0' 82/ 1+/+3+'+' (0,0. !#$%3090/: !"#$% {(Xı , Tı ) : ı ∈ I} * +, H. @234#2' :"I"#"J4/ %D(>#(3 (4#284 "#8%', ,-.-/01213 C% %D(>#(3?' (K, P) 6(3I?1 %D(>? :?/?D 482) 4D$0=01 L"'/84>"'M #(3? 8032/#4 "#$%&%'$(') πı (X) = Xı (ı ∈ I) * +, N. 32841#234 $2 :?/?D "#(<(/:?3 *7/D, !"#!$ +, ,O., 4-5-.01213 P#2Q('$23 F2"3214'2 R932) P 6(3I?1 :"I"#"J484'4' S (#: :(7(M '?>#( >(I?#(' (6?/ 93:0#234'4 $0=0'12/ >2:2<2/:43 S7/D, *T,O.U E#1(>('( 23R4 >9':214'4 /%##('(<(&?D, C43 U ⊂ S (#:93:080'0' X 6(3I?1 %D(>?'? 93:2' 546743 8"'#% (#:93:080 "#1($?&?'? @(38(>(#?1, A23 ı ∈ I 464' Uı = {T ∈ Tı : πı−1 (T ) ∈ S} !"#" $%&%' ()*)+ ,+-$'-& !" #$%&'$() U *+&,$-& ./%#010(0+ 23,43%5( 3'65-#37 Uı 385. .*(&'&, 3$'&4$ Xı 65.5+ 0+3%5-5- 9$, :,6*4* /'3(3+) ;'43%#57 Xı -$- 4/-'0 9$, T1 , T2 , . . . , Tn ∈ Tı 3'6:,6*4* 4&8$'&9$'$,#$) <0 #0,0(#3 πı−1 (Tr ), r = 1, 2, . . . , n 3$'&4$ X 83,=5( 0+3%5-5- U #3- 4&8$'&- 4/-'0 9$, 3'6:,6*4* /'0,#0) <0 $4& U %3 %*.'&-&- 23,43%5(3 3%.5,5#5,) >%'&%4& Xı 83,=3- 0+3%5-#3- :%'& 9$, xı :1&4$ 4&8&9$'$,$+ .$7 90 :1& Uı %& 3$6 ?$8 9$, .*(& 63,3@5-#3- $8&,$'&(&+) A&, ı ∈ I $8$- 90 9$8$(#& 4&8$'&B&. :1&'&,$ 9$'&C&-'&,$ 23,'3%3- x = (xı )ı∈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⊂ A ⊂ X /'40-) B -$(A, TA ) -5- 65.5+ 9$, 3'6.*(&4$ /'(345 $8$- G&,&.'$ 2& %&6&,'$ ./C0' 4/-'0 /'(345#5,) L&#&-M Q) E5.5+ 9$, 0+3%#3- 9$, A304#/,R 0+3%5 *+&,$-& 63-5('5 9$,&D9$, 4*,&.'$ 9$, @/-.4$%/-0- 9$, 6/=/'/K$. &C%3=5 #:-*C*(* /'3B315-5 G:46&,$-$+) !"# $%&%' ()*)+ ,+-$'-& !"#$ %&')'%' <$, A304#/,R 0+3%5-5- ?&, -/.6345-5- 65.5+ 9$, ./(C0'010 23,437 90 0+3%3 !"!# $%&%' 9$, 0+3%#5,7 #&-$'$,) !"#$ %&' ()*)+")* !" #$%$& '() *&+, +,-$ &+.+-/+ ,0)01 2$%$&/$)3 45-%56 %0-/(7( 80) -9%2+7$-$'() %9.:*1*;*/*)3 <-=+% '*-*- 20)7( /9;)* /0;(1/()3 >)-0;(-6 79-7*& '() %5.06 5&0)(-/0%( +,)$% 29?919@(,0 AB)0 ,0)01 2$%$&/$)6 +.+ 2$%$& /0;(1/()3 C0-06 A0)D01 0%70-6 5&0)(-/0%( 7+12 29?919@(,0 AB)0 ,0)01 2$%$&/$)6 +.+ 2$%$& /0;(1/()3 C0)D0%20-6 80) x ∈ R -9%2+7$-$- [x − h, x + h], h > 0 :0%1(-/0 2$%$& '() %9.:*1*;* E+)/$)3 !"#$" %&'('%' !"!# $%&%' (')*+) $%&%' ,-" )#$&./!0-0 $%&%' &1/2(#(&#)"%3 ,( &./!0-0 ,-" *!"!# $),)0%0% 1#(2$("("4 E ,0)01 2%$& '() *&+, E0 X 9-*- 2$%$& '() +12%5.07( 917*-3 >-=0 E -(- 2$%$& 91/*;*-* E+)7+,+1$.3 X F(- '() U %9.:*1*;*-* /5:5-01(.3 C0G )0%(,9)7+ (D -9%2+1+)$ %5.07(-( 70D0)0%6 U -*- +D$% '() %9.:*1*% 91/*;*-* %+'*1 0/0)70% A0-011(%20- '():0, ,(2().(: 91.+,$&3 HB,10=0 Y = U ′ %5.07(6 E -(- %+?+1$ '() +12%5.07( 91/*;*-/+-6 2$%$& 91+=+%2$) I E ,( 2$%$& E+)7+,/$;$.$& (D(- '%&3 !"#!$ J3 3KL3 M 8+1/06 X ⊂ V, Y ⊂ W E0 V ∩ W = ∅ 91+- V 6 W %5.010)(-(- E+)1$;$-$ I'%&3 !"#!$ J3 3JL /0- 7B,10,0'(1()(&3 V ⊂ W ′ E0 W +D$% 91/*;*-/+- V̄ ⊂ W ′ 91+=+%2$)3 >20 ,+-/+- W ′ ⊂ U 91/*;*-* /5:5-5)G 70% V̄ ⊂ U ,+&+'(1()(&3 M,7+ V̄ %5.07( X (- %+?+1$ '() %9.:*1*;*/*) E0 U 2+)+N$-/+- %+?7+-$)3 H* B&01 8+1/0 (7?+2$ '(2().0% (D(- +,)$=+ V̄ %5.07(-(2$%$& 91/*;*-+ (:+)02 02.0% ,020=0%2()3 O(./( E -(- ,0)01 2$%$& 91.+7$ 8+1(-/0 (7?+2$ ,+?+1$.3 H*-*- (D(-6 B-=06 X (- 0-+& '() 2+-0 2$%$& U %9.:*1*;*-+ 7+8(? 91/*;*-* AB720)01(.3 x ∈ X (D(- Ux %5.07( x B;07(-(- 2$%$& '() %9.:*1*;* 917*-3 x -9%2+1+)$ X %5.07(-( 2+)+/$;$ &+.+- Ux %5.010)(-(- (D10)( X (- +D$% '() B)2575 91+=+%2$)P B,10,706 B,10 x1 , x2 , . . . , xn ∈ X -9%2+1+)$ E+)/$) %(6 X %5.07( !"#$% U = Ux1 ∪ Ux2 ∪ Ux3 ∪ . . . ∪ Uxn (D(-/0 %+1$) E0 U 2$%$&/$)3 O(./( V %5.07( X (- %0,Q '() %9.:*1*;* 917*-3 U ∩ V %5.07( X (- 2$%$& U *&+,$ (D(-/0%( '() %9.:*1*;* 91/*;*-/+-6 '+:2+%( B&01 8+1 A0)0;(-=06 U *&+,$-/+ X (- B,10 '() W %9.:*1*;* E+)/$) %( W ⊂ U ∩ V 91*)3 >&01 91+)+% W ⊂ V /() E0 W 6E *&+,$-/+ X (- 2$%$& '() %9.:*1*;*/*)3 !"# $%&'% (&)*+,'+,& 3 X 2$%$& E0 Y R+*7/9)S (70 75)0%1( f :→ Y N9-%7(,9-* %+?+1$/$)3 CB720G )(-(&3 !"#" $%&'% (&)*+,'+,& ! "# Y $%&%' ()* π1 : X × Y → X ('+,-,., &/0/1% 2(3 +45,-,.+,3# 64)$*7 3(5('# 8# Y $%&%' 9* :/;)+<3= ()* f : X → Y +45,-,.,5,5 ),3*&1( <1./)% (>(5 ?*3*&1( 9* @*$*31( &<-;1 Gf = {(x, f (x)) : x ∈ X} ?3/AB(5(5 X × Y (>(5+* &/0/1% <1./)%+%3# C# f : X → Y +45,-,., &/0/1%D ),3*&1( 9* 43$*5 <1);5# Y $%&%' <1+;B;5+/D :*3 y ∈ Y (>(5 f −1 ({y}) ⊂ X $%&%' ()*D X ;'/@%5%5 +/ $%&%' <1+;B;5; ?4)$*3(5('# E# F*3 />%& /3/1%& )/1$ $<0<1<G(@* ?43* R ;'/@%5+/ />%&$%3# 64)$*3(5('# H# n ∈ Z <1./& ,'*3* (n, n + 1) /3/1%&1/3%5%5 2(1*-(.( />%&$%3# 64)$*3(5('# I# {0} = ∞ \ 1 1 (− , ) n n n=1 J E# EK *-($1(B(5( &;11/5/3/& )<5);' )/@%+/ />%& &,.*1*3(5 /3/&*)($(5(5 />%& <17 ./@/2(1*L*B(5( ?4)$*3(5('# M# Q 3/)@<5*1 )/@%1/3 &,.*)(5(5D )/1$ $<0<1<G(@* ?43* R ;'/@%5+/ 5* />%& 5* +* &/0/1% <1+;B;5; ?4)$*3(5('# !# N/1$ $<0<1<G(@* ?43* R ;'/@%5+/ 2(3 (a, b) />%& /3/1%B%5%5 :*3 x ∈ (a, b) 4B*)( (>(5 2(3 &<.-;1;& <1+;B;5; ?4)$*3(5('# O# ∞ [ 1 1 (0, 1) = [ ,1 − ] n n n=2 <1+;B;5; ?4)$*3(5('# P;3/+/5 -; )<5;L; >%&/3%5%'Q N<5);' )/@%+/ &/0/1% &,.*1*3(5 2(1*-(.( &/0/1% <1./@/2(1(3#