tıkızlık

!"#$ %&
!"!#$!"
!"
#$%$& '&()*(+
!" "
#$%&'()'($
! "#$ %&'() *&+&(&,-. )/01 *2.2/32$!
4! 51$2. 6-$ *&+&(&,-. )/012' *2.2/ &(70%2 -8-' 9#$#.(- :# 1#*#$(- .&;)( %&'()
&(70%232$!
<! 51'2 6-$ .=7# =/#$-'3# S ⊂ T .&;)()') %0>(010' -.- *&+&(&,- :#$-(%-'!
?>#$ (X, T ) *2.2/ -%# (X, S ) 3# *2.2/32$!
!"#$% (X, S ) )/012'2' @#$ 082. A$*=%= 01'2 /070'30 (X, T ) )/012'2'
30 6-$ A$*=%=3=$! B2.2/ &(3)>) -8-'C 6) A$*='='C (X, T ) )/012'30 %&'()
6-$ 0(* A$*=%= :0$32$! D) 0(* A$*=C (X, S ) '-' 3# %&'() 6-$ A$*=%=3=$!
D)')' .0$;2*2 9#'#( &(0$0. 9#8#$%-/3-$! E$'#>-'C [0, 1] 0$0(2>2 =/#$-'3#.%0(* *&+&(&,-1# S C 01$2. *&+&(&,-1# -%# A 3-1#(-7! S ⊂ A 32$! ([0, 1], S
'-' *2.2/ &(3)>)') 6-(-1&$)/! 570 ([0, 1], A ) *2.2/ 3#>-(3-$!
F! (X, S ) :# (X, T ) )/01(0$2 *2.2/ -%# 10 S = T 3-$ 10 30 7).01#%# #3-(#G
7#/(#$H 10'- 6-$-%- A*#.-'3#' -'I# 10 30 .060 3#>-(3-$! !"#"$ %$&'"( )&%*+
,-../ -0,%1%(% #&2%0 3,4'-.%$56
!"#$% S ⊂ T &(3)>)') :0$%010(27! I : (X, T ) → (X, S ) A/3#;(-.
3A'=;=7= 66A :# %=$#.(-3-$! J @0(3# 6-$ *&+&(&,-. #;10+2 3A'=;=7=3=$!
K&(012%1(0C T = S &(0I0.*2$!
L! X %012(070/ %&'%)/ 6-$ .=7# &(%)' D) .=7# =/#$-'3#
C = {T : 10 T ′ %&'()3)$ 10 30 T = X}
M L! N
0-(#%- 6-$ *&+&(&,-3-$! D)'0 O012(06-(-$ B=7(#1#'(#$ B&+&(&,-%- 3#'-(-$! D-$
0(*G.=7#'-' 6) *&+&(&,-1# 9A$# *2.2/ &(70%2 -8-' %&'() &(70%2 9#$#.(- :#
1#*#$(-3-$!
FP
!"#$ %&' ()*)+")*
!"
!"#$% #$%&' (&)*+,-.&./0% )1+12 $&(3(41 (51+)1/6 7(/81) $&(/(+9 :$%&'
$&-(;(% <0/ +,-.%0% )1+12 $&(-(;(3(41%1 =>:)./.&0-6 ?./ @(%=0 <0/ A =
{a1 , a2 , . . . , an , . . .} :(;1&(<0&0/ :$%:'2 +,-.:0%0 A,8,%.&0-6 ?./ n = 1, 2, . . . , n, . . .)
050% An = A′ ∪{a1 , a2 , . . . , an } +,-.:0 (51+)1/B 5,%+, A′ ∈ C C. {a1 , a2 , . . . , an } ∈
C A0/6
Sn
7$&(;3( =>/,&.3.40 =0<0 A ⊂ n=1 An A0/B ;(%0 {An : n ∈ N} (0&.:0 A %1%
(51+ <0/ >/),:,A,/6 D' >/),A.% :$%&' <0/ (&)*>/), :.50&.-.26 E @(&A. A
)1+12 A.40&A0/6 F.-.+ +0 :(;1&(<0&/ :$%:'2 +,-.&./9 #(;1&(<0&0/ G,-&.;.%&./
G$H$&$I0:0%. =>/. )1+12 $&(-(2&(/6
?./ :$%:'2 +,-.A.% :(;1&(<0&0/ <0/ (&) +,-. :.50&.<0&.3.40%. =>/.9 :$%:'2
+,-.&./ A. #(;1&(<0&0/ G,-&.;.%&./ G$H$&$I0:0%. =>/. )1+12 $&(-(2&(/6
J6 D0/ X +,-.:0 ,2./0%A. <0/ ?(':A$/K )$H$&$I0:0 )1+12 <0/ )$H$&$I0A.% A(@(
+(<( $&(-(26 LD(8+( <0/ A.;08&. <0/ X +,-.:0 ,2./0%A. )1+12 <0/ )$H$&$I0
<0/ ?(':A$/K )$H$&$I0:0%A.% 0%3. $&(-(26M
!"#$% (X, T ) )1+12 '2(;9 (Y, H ) ?(':A$/N '2(;1 $&-(+ ,2./. H ⊂ T
$&A'4'%' C(/:(;(&1-6 D' C(/:(;1- (&)1%A( H ′ ⊂ T ′ $&(3(+)1/B ;(%0 H
)$H$&$I0:0%A. +(H(&1 $&(% @./ +,-. T )$H$&$I0:0%A. A. +(H(&1 $&-(&1A1/6
7(/81) $&(/(+ T )$H$&$I0:0%A. )1+12 $&(% @./ +,-. H )$H$&$I0:0%A. A.
)1+12 $&(3(+)1/6 O;/13(9 G.$/.- !6 6P ';(/1%3(9 )1+12 <0/ '2(;A( <0/ (&)
+,-.%0% )1+12 $&-(:1 050% +(H(&1 $&-(:1 =./.+&0 C. ;.)./&0A0/6 D,),% <'%&(/1
<0/ (/(;( =.)0/0/:.+ 8>;&. A0;.<0&0/02Q
A ∈ T ′ ⇒ A +,-.:0 T )$H$&$I0:0%. =>/. )1+12A1/
⇒ A +,-.:0 H )$H$&$I0:0%. =>/. )1+12A1/
⇒ A +,-.:0 H )$H$&$I0:0%. =>/. +(H(&1A1/
⇒A∈H′
E @(&A. T ′ ⊂ H ′ A,/B ;(%0 T )$H$&$I0:0%A. +(H(&1 $&(% @./ +,-. H
)$H$&$I0:0%A. A. +(H(&1A1/6 R;&.;:. H = T $&-(&1A1/6
S6 Q ∩ (R − Q) = ∅ A0/9 (-( Q ∩ (R − Q) = R A0/6
T6 Q '2(;1 C. Q = (R − Q) '2(;1 (;/1&(<0&0/ 0+0 '2(;A1/6 F$&(;1:1;&( U+0%30
#(;1&(<0&-. O+:0;$-'%' :(4&(/&(/6
′
V6 [0, 1] ∩ Q )(-(-.% :1%1/&1A1/9 (-( )1+12 A.40&A0/6
!" "#
$%&'()*()%
6 W./5.& .+:.%0% (51+ ;( A( ;(/1*(51+ @05<0/ (/(&141%1% )1+12 $&-(A141%1 =>:*
)./0%026
!"#$%
!" " #$%$& '&()*(+
!
"#$ {(−n, +n) : n ∈ N} #%&'(% R )'*+'& (#,-&#* ./0'(%1%1 #+-. 2%* 3*4/(/5
6/*7 80# 29 3*4/6'1 (:1&9 2%* #&4 3*4/ ('+%&'0';7 <,&',('= R )'*+'&
(#,-&#* ./0'(% 4-.-; 6'>%&6%*7
"2$ {( n1 , 1) : n ∈ N} #%&'(% (0, 1) #*#&->-1-1 #+-. 2%* 3*4/(/6/*7 80# 29
3*4/6'1 (:1&9 2%* #&4 3*4/ ('+%&'0';7 <,&',('= (0, 1) #+-. #*#&->- 4-.-;
6'>%&6%*7
?7 (X, T ) 4-.-; 9;#,-16#= 4-.-; %.% #&45./0'1%1 2%&'@%0%1%1 A' #*#.'(%4%1%1
4-.-; :&69>919 )3(4'*%1%;7
!"#$% (X, T ) 9;#,-1- 4-.-; .#29& '40'.&' :191 B#9(6:*C :&69>919
.#29& '6%,:*9;7 D':*'0 !7 7E 9,#*-1F#= 4-.-; 9;#,-1 4-.-; #&4 ./0'&'*%
.#G#&-6-*7
A, B ⊂ X #&4 ./0'&'*% 4-.-; %(' .#G#&-6-*&#*7 H I#&6' A ∪ B 2%&'@%0% A'
A ∩ B #*#.'(%4% .#G#&- ./0'&'*6%*7
A ∪ B 2%&'@%0%1%1 2%* #+-. 3*4/(/ U :&(917 X − A ∪ B #+-. :&69>916#1
U ∪ (X − A ∪ B) #%&'(% X 9;#,-1-1 #+-. 2%* 3*4/(/6/*7 H I#&6' (:1&9 2%*
#&4 3*4/(/ ('+%&'2%&%*7 J9 (:1&9 3*4/6'1 X − A ∩ B ./0'(%1% #4#*(#.= )'*%
.#&#1 #%&' A ∩ B #*#.'(%4%1%1 (:1&9 2%* 3*4/(/ :&#F#.4-*7
A ∩ B #*#.'(%4%1%1 4-.-;&->-1- )3(4'*0'. %+%1 6' 2'1;'* 6/@/1F',% 9,)95
&#,#2%&%*%;7 A ∩ B #*#.'(%4%1%1 2%* #+-. 3*4/(/ V :&(917 X − A ∩ B #+-.
:&69>916#1 V ∪ (X − A ∩ B) #%&'(% X 9;#,-1-1 #+-. 2%* 3*4/(/6/*7 H I#&6'
(:1&9 2%* #&4 3*4/(/ ('+%&'2%&%*7 J9 (:1&9 3*4/6'1 X − A ∩ B ./0'(%1%
#4#*(#.= )'*% .#&#1 #%&' A ∩ B #*#.'(%4%1%1 (:1&9 2%* 3*4/(/ :&#F#.4-*7
&'()*%
J9 .%4#G4# 4-.-;-0(- "K9#(%F:0G#F4$ B#9(6:*C 9;#,&#*-1# 4-.-;
"F:0G#F4$ 9;#, 6%,:*9;7 J#;- .#,1#.&#*= B#9(6:*C :&0# .:@9&919 .:,0#;7
H 69*906#= ,9.#*-6#.% %(G#4 )'+'*(%; :&9*L +/1./ 4-.-; #&45./0'&'*%1 .#G#&:&0#(- #1F#. B#9(6:*C 9;#,&#*-16# )'+'*&%6%*7
E7 M:1(9; 2%* ./0'= .'16% /;'*%16'.% ,-./0 #12/343./35 #-6-/-7898.3 )3*'
4-.-;6-*7 N'6'1O
!"#$% (X, S ) (:1&9 4/0&','1&'* "F:P1%4'$ 4:G:&:Q%(% :&(917 O = {Uı :
Uı ∈ S , ı ∈ I} 2%* #+-. 3*4/ :&(917 J9 3*4/6'1 *#()'&' 2%* U0 ∈ O #+-.
′
./0'(%1% ('+'&%07 J9191 4/0&','1% (:1&969*L ,#1% U0 = {x1 , x2 , . . . , xn }
′
#%4 I'* xi 1:.4#(-1- %+'*'1 2%* Ui ∈ O ('+%&'2%&%*7
:&9*7 U0 4/0&','1%1'
Sn
J3,&'F'= X = i=0 Ui :&#F#.4-*7
R7 8,*-. 4:G:&:Q%,' )3*' 2%* ./0'1%1 4-.-; :&0#(- %+%1 )'*'.&% A' ,'4'*&% .:@9&
(:1&9 :&0#(-6-*7 N'6'1O
!"#$% (X, A ) #,*-. 9;#,
S :&(917 B'* x ∈ X %+%1 4'. 3>'&% {x} ./0'(%
#+-.4-*7 H I#&6'= X = {{x} : x ∈ X} :&69>916#1 O = {{x} : x ∈
X} #+-. 2%* 3*4/6/*7 J9 3*4/6'1 (:1&9 2%* #&4 3*4/ ('+%&'2%&0'(% %+%1 X
./0'(%1%1 (:1&9 :&0#(- )'*'.&% A' ,'4'*&%6%*7
!7 M#,-&#2%&%* D/0&','1&'* D:G:&:Q%(%1' )3*' 2%* #&45./0'1%1 4-.-; :&0#(- %+%1
)'*'.&% A' ,'4'*&% .:@9& (:1&9 :&0#(-6-*7 N'6'1O
!"#$ %&' ()*)+")*
!"
!"#$% (X, T ) #$%&' ()*+,-+./ T .+. 0%&',%1+,+* 234,)&).,)* 2565,57+-+
5,8#9#.# (%*-%&%,'4/ X +:+.8);+ -5.,# %,<=;34),)*+. <';'$ 5,%>%9' %:';<'*/
?%*@'< 5,%*%;A -5.,# 5,4%&%. 1+* ;34).+. <';'$ 5,%4%&%>%9'.' BC-<)*),+4/
D)* E%.B+ 1+* A = {a1 , a2 , . . . , an , . . .} -%&',%1+,+* -5.-#$ ;34)-+.+ 83@3.)=
,+4/ D)* n = 1, 2, . . . , n, . . .) +:+. An = A′ ∪ {a1 , a2 , . . . , an } ;34)-+ %:';<'*F
:3.;3 A′ ∈ C () {a1 , a2 , . . . , an } ∈ C 8+*/
Sn
?5,%&>% BC*3,)>)9+ B+1+ A ⊂ n=1 An 8+*F &%.+ {An : n ∈ N} %+,)-+ A .'.
%:'; 1+* C*<3-383*/ G# C*<38). -5.,# 1+* %,<=C*<3 -):+,)4)$/ H E%,8) A
<';'$ 8)9+,8+*/ I)4); ;+ -%&',%1+,* -5.-#$ ;34),)*A 0%&',%1+,+* 234,)&).,)*
2565,57+-+.) BC*) <';'$ 5,%4%$,%*/
D)* -5.-#$ ;34)8). -%&',%1+,+* 1+* %,< ;34) -):+,)1+,)>)9+.) BC*)A -5.-#$
;34),)* 8) 0%&',%1+,+* 234,)&).,)* 2565,57+-+.) BC*) <';'$ 5,%4%$,%*/
J/ 2';'$ 1+* #$%&8%. 1+* D%#-85*K #$%&' 3$)*+.) <%.'4,' 1+*)=1+* -3*);,+ 1+*
L5.;-+&5.#. 1+* <565,57+; )@&%6' 8C.3@343 5,%>%9'.' BC-<)*+.+$/
!"#$% (X, T <';'$ #$%&A (Y, H ) D%#-85*L #$%&' () f : X → Y L5.;-+=
&5.# -3*);,+ () 11C 5,-#./
f −1 (H ) = {f −1 (H) : H ∈ H }
%+,)-+ X 3$)*+.8) 1+* <565,57+8+* M1;$/ 2)*- *)-+4 <565,57+-+A +$83@), <5=
65,57+A 6*57)><+() <565,57+N/ O$83@), <565,57+A f &+ -3*);,+ ;',%. ). ;%1%
<565,57+ 5,8#9#.8%.
f −1 (H ) ⊂ T
8+*/ P<) &%.8%.A X ;34)-+ 3$)*+.8);+ 1+* D%#-85*K <565,57+-+ 1+* <';'$
<565,57+8). ;%1% 5,%4%$ M1;$ !/ / M N Q*51,)4N/ H E%,8)
f −1 (H ) = T
8+*/ G# +-) f −1 : Y → X <)*- L5.;-+&5.#.#. 8% -3*);,+ 5,4%-' 8)4);<+*/
P&,)&-)A P.)*4) "/ / #&%*'>% f 1+* )@&%6' 8C.3@34383*/
!"# $%&%' ()*)+ ,+-$'-&
!". *-&/- 0&12'%/'%&
/ (X, T ) <';'$ () (Y, H ) D%#-85*K +-) -3*);,+ f :→ Y L5.;-+&5.# ;%6%,'8'*/
RC-<)*+.+$/
!"#$% A ∈ T ′ +-) A ;34)-+ T ′ &) BC*) <';'$8'*/ H E%,8) -3*);,+ L5.;=
-+&5. %,<'.8%;+ *)-4+ f (A) ⊂ Y <';'$8'*/ (Y, H ) D%#-85*K 5,8#9#.8%.
f (A) ⊂ Y ;%6%,'8'*/
"/ Y <';'$ +-) π1 : X × Y → X +$83@343 ;%6%,' 1+* 8C.3@3483*/ RC-<)*+.+$/
!"#$% (Y, Y ) <';'$ 1+* #$%& 5,-#./ S%*6'4 <565,57+-+.+ (X × Y, P) +,)
BC-<)*),+4/ A ⊂ X × Y ;%6%,' +-) X − π1 (A) .'. X +:+.8) %:'; 5,8#9#.#
!"#" $%&'% (&)*+,'+,&
!"
/A
#$%&'()'*+,+-. /+( x ∈ X − π1 (A) $0'%+1+ 23431'*+). 5'( y +6+1 (x, y) ∈
27(. A 89:9*7 ;*2<0<1291 X × Y − A 9678&7(. =,*',%' >'( y +6+1 $,*' ?+(
Uxy ∈ B(x) @' Vy ∈ B(y) 8;)4<*<8*9(7 @9(27( 8+ (x, y) ∈ Uxy × Vy ⊂
X × Y − A ;*9A98&7(. /< 2<(<)29 {Vy : y ∈ Y } 9+*'%+ Y 1+1 ?+( 9678
$(&3%3 ;*<(. Y &787- ;*2<0<1291 ?<1<1 %;1*< ?+( 9*& $(&3%3 @9(27(. /<
%;1*< $(&3,' {Vyi : i = 1, 2, . . . , n} 2+,'*+). B+)2+
U = Uxy1 ∩ Uxy2 ∩ . . . ∩ Uxyn
83)'%+1+ &917)*9,9*7). U ∈ B(x) @' U × Y ⊂ X × Y − A ;*9A98&7(.
U ∩ π1 (A) = ∅ 2+(. 5'( x ∈ X − π1 (A) +6+1 ?< ,9:7*9?+*'A'0+1' #$('
X − π1 (A) 171 X +6+12' 9678 ;*2<0< ;(&9,9 6789(.
C9(:7) &;:;*;D+%+ +-2343) E;18%+,;1*9(717 %3('8*+ 87*91 '1 89?9 2;8<*<
&;:;*;D+2+( F?8-. =(1'8 G. . H. C9(:7) &;:;*;D+%+1+ X ×Y, P +*' #$%&'('*+).
I<89(7298+ 2343134*'J
pi−1
2 (Y ) ⊂ P
?9071&7%7171 @9(*70717 %$,*','?+*+(+-.
−1
K9?++ pi−1
2 (Y ) &;:;*;D+%+ &787-27(. K ⊂ Y 83)'%+ &787- +%' pi2 (K) 83L
−1
)'%+ pi2 (Y ) &;:;*;D+%+1' #$(' &787- ;*9A98&7(. M >9*2' 9,17 &;:;*;D+,'
#$(' 89:9*727(. =,*',%' P &;:;*;D+%+1' #$( 2' 89:9*7 ;*<(. =&' ,91291
π2 opi−1
2 (K) = K 27(.
". Y &787- @' >9<%2;(N +%' f : X → Y 2$1343)3131 %3('8*+ ;*)9%7 +6+1
#'('8*+ @' ,'&'(*+ 8;4<*
Gf = {(x, f (x)) : x ∈ X}
#(9O0+1+1 X × Y +6+12' 89:9*7 ;*)9%727(.
!"#$%
/ G +%' y 6= f (x)
!"!#$%&%' G′ 131 9678 ;*2<0<1< #$%&'()'*+,+-. (x, y) ∈
2+(. Y 59<%2;(N ;*2<0<1291 y +*' f (x) 1;8&9*9(7171 8'%+4)','1 ?+('( 8;)L
4<*<0< @9(27(. /<1*9(7 y ∈∈ B(y) @' f (x) ∈ W ∈ B(f (x)) +*' #$%&'('*+). f
%3('8*+ ;*2<0<1291 f (U ) ⊂ W ⊂ (Y −V ) ;*9A98 ?+6+)2' ?+( x ∈ U ∈ B(x)
8;)4<*<0< @9(27(. /<(9291 (x, y) ∈ U × V ∩ G = ∅ ;*2<0< #$(3*3(. 5'(
(x, y) ∈
/ G +6+1 ?< $-'*+8 @9( ;*2<0<19 #$(' G′ 9678&7(.
(!#!)$%&%' G ⊂ X × Y #(9O0+ 89:9*7 ;*%<1. V ∈ B(f (x)) +%' G ∩ (X ×
(Y − V )) ∈ X × Y 89:9*727(. =1A'8+ :(;?*') #'('0+1A'J π1 [G ∩ (X × (Y −
V )) ⊂ X 89:9*727( @' x 1;8&9%717 +6'()'-. f (U ) ∩ (Y − V = ∅ ;*91 ?+(
x ∈ U ∈ B(x) 8;)4<*<0< %'6'*+). f (U ) ⊂ V 2+(. M >9*2' f E;18%+,;1< x
1;8&9%7129 %3('8*+2+(. 5'( x +6+1 ?< $-'*+8 @9( ;*2<0<1291J f E;18%+,;1<
&917) ?$*#'%+12' %3('8*+2+(.
P. f : X → Y 2$1343)3 89:9*7J %3('8*+ @' $(&'1 ;*%<1 @' >'( y ∈ Y +6+1
f −1 ({y}) ⊂ X &787- ;*%<1. Q0'( Y &787- +%'J X <-9,7171 29 &787- ;*2<0<1<
#$%&'(+1+-.
!"#$ %&' ()*)+")*
!"
!"#$% U #$%&'$ X ()*&'$+$+ #,-( .$/ 0/1)') 2%'3+4 5$*6&%&/7& .#'$1%$8$
'#8%#*#( $,$+9 :&/ y ∈ Y 08&'$+$+ ; #%1-+7#($ 1&/' 60/)+1)')+) Ay $%&
60'1&/&%$*< =#+$ f −1 (y) = Ay 2%'3+4 U #$%&'$ Ay +$+ 7& #,-( .$/ 0/1)')7)/4
Ay 1-(-> 2%7383+7#+ U 0/1)')+7&+ '2+%3 .$/ #%1 0/1) '&,$%&.$%$/4 ?3 #%1@
0/1) {Uy1 , Uy2 , . . . , Uyn } 2%'3+4 Uy = Uy1 ∪ Uy2 ∪ . . . ∪ Uyn 7$=&%$* A&
Wy = Y − f (X − Uy ) ()*&'$+$ 1#+-*%#=#%-*4 (X − Uy ) (#B#%-7-/4 f (#B#%.$/ ;2+('$=2+ 2%7383+7#+ f (X − Uy ) 60/)+1)') 7& (#B#%-7-/4
C$*7$ D3 0>&%$(%&/$ (2%#=E# 60/&.$%$/$>4
F#G Wy ()*&'$ #,-(1-/4
F.G y ∈ Wy ⇒ f −1 (y) = Ay ⊂ Uy ⇒ y ∈
/ f (X − Uy )
FEG f −1 (Wy ) = f −1 [Y − f (X − Uy )] = X − f −1 (f (X − Uy )) = X − (X −
Uy ) = Uy
?3/#7#+ 60/)%7)8) 6$.$ Y = {Wy : y ∈ Y #$%&'$ Y ()*&'$+$+ .$/ #,-(
0/1)')7)/4 Y 1-(-> 2%7383+7#+ Y 0/1)')+)+ '2+%3 .$/ #%1@0/1)') A#/7-/4
?3 '+%3 0/1)=& {Wy1 , Wy2 , . . . , Wym } 7$=&%$*4 C$*7$ FEG .#8-+1-'-+- (3%%#@
+-/'#(9
!
m
m
m
[
[
[
Wyi =
f −1 (Wyi ) ⊂
Uyi
X = f −1 (Y ) = f −1
i=1
i=1
i=1
H+ '#87#($ Uyi ()*&%&/$+$+ :&/ .$/$'$ '2+%3 '#=-7# U ∈ U +-+ .$%&D$*$ $7$4
I :#%7&9 X ()*&'$ '2+%3 '#=-7# U ∈ U ()*&'$=%& 0/1)%&.$%$/4 J&*&( ($
X 1-(->7-/4
!4 K&/ #,-( #/#%-( '#%1 12B2%2L$=& 60/& R 3>#=-+7# #,-(1-/4 M0'1&/$+$>4
!"#$%
4N2%O P,-( #/#%-(%#/ #$%&'$ '#%1 12B2%2L$+$+ .$/ 1#.#+-7-/< 72%#=-'-=%# #,-(
#/#%-(%#/ '#%1 12B2%2L$7& #,-(1-/4
Q4N2%O (a, b) ⊂ R A&/$%'$+4 ?$/ x0 ∈ (a, b) +2(1#'- #%#%-*4 ǫ < min{|x0 −
a|, |b − x0 |} 2%*#( )>&/&9 (x − ǫ, x + ǫ) ⊂ (a, b) 7$/4 P=/-E#9 (x − ǫ, x + ǫ) ∈
B(x0 ) 7-/< =#+$9 x0 +2(1#'-+-+ .$/ (2*D3%383 (a, b) 1#/#;-+7#+ (#B'#+-@
=2/4 K&/ x0 ∈ (a, b) +2(1#'- $,$+ .3 0>&%$( A#/ 2%7383+7#+9 R+&/*& !4 4
3=#/-+E#9 (a, b) #/#%-8- #,-( .$/ ()*&7$/4
S4 n ∈ Z 2%*#( )>&/& (n, n + 1) #/#%-(%#/-+-+ .$%&D$*$ #,-(1-/4 M0'1&/$+$>4
!"#$% (n, n + 1) #/#%-(%#/S '#%1 12B2%2L$7& #,-(1-/4 P,-( ()*&%&/$+ :&/
.$%&D$*$ #,-( 2%7383+7#+ {(n, n + 1) : n ∈ N} '#%1 12B2%2L$7& #,-( .$/
()*&7$/4
T4
{0} =
∞
\
1 1
(− , )
n
n
n=1
F !4QG
!"#" $%&'% (&)*+,'+,&
!!
"#$%&$'$($ )*&&+(+,+) -.(-*/ -+012+ +31) )45"&",$( +,+)"-$%$($( +31) .&5+6
0+7$&"8"'$($ 9:-%",$($/;
!"#$% <", n
= 1, 2, 3, . . . $3$( (− n1 , n1 ) +,+&1)&+,1 +31)%1,; =5+ 7*(&+6
,1( > !;?@ +,+)"-$%$ 0+&(1/8+ (.)%+-1(2+( $7+,"% {0} )45"-$2$,; A* )45"
-+&% %.B.&.C$0" 9:," )+B+&121,; D"5") )$ +31) )45"&",$( -.(-*/ %+("-$($(
+,+)"-$%$ +31) .&5+0+7$&$0.,;
E; Q ,+-0.("& -+01&+, )45"-$($(F -+&% %.B.&.C$0" 9:," R */+01(2+ (" +31) ("
2" )+B+&1 .&2*'*(* 9:-%",$($/;
!"#$% G",3"& -+01&+,1( H", +,+&1'1 -.(-*/ -+012+ ,+-0.("& I" $,,+-0.("&
-+01&+, $3",$,; A+#)+ 7$, 2"0$#&"F -+&% %.B.&.C$0" 9:," Q ,+-0.("& -+01&+,
)45"-$ R */+01(2+ 0.'*(2*, >7)/ J(",5" K; ;L@; A"(/", .&+,+)F -+&% %.6
B.&.C$0" 9:," Q′ = R − Q $,,+-0.("& -+01&+, )45"-$ R */+01(2+ 0.'*(2*,;
D.&+01-10&+F -+&% %.B.&.C$($( %+7+(1(+ +$% +31) +,+&1)&+, Q ,+-0.("& -+01&+,
)45"-$ $3$(2" )+&5+/; M H+&2" Q (*( H$3 7$, $3 (.)%+-1 0.)%*,; <$3 7$, 21#
(.)%+-1 2+ 0.)%*,; <", ,+-0.("& -+01 Q ($( 7$, )"(+, (.)%+-121,; A"(/",
#"0 $,,+-0.("& -+01&+, $3$( 2" 9"3",&$2$,;
L; N+&% %.B.&.C$0" 9:," R */+01(2+ 7$, (a, b) +31) +,+&1'1(1( H", x ∈ (a, b)
:'"-$ $3$( 7$, ).5#*&*) .&2*'*(* 9:-%",$($/;
!"#$% O,.7&"5 ! 3:/4&4,)"( 9:-%",$&2$;
P;
(0, 1) =
∞
[
1
1
[ ,1 − ]
n
n
n=2
> !;Q@
.&2*'*(* 9:-%",$($/; A*,+2+( #* -.(*8* 31)+,1(1/R N.(-*/ -+012+ )+B+&1
)45"&",$( 7$&"#$5$ )+B+&1 .&5+0+7$&$,;
!"#$% N$59"&",2" 7+-$%&$'$ -+'&+5+) $3$(F > !;Q@ 7+'1(%1-1(1( -+' 0+(1(1
A $&" 9:-%","&$5; (0, 1) = A .&2*'*(* 9:-%",5") $3$( $)$ )45"($( 7$,7$,$($
)+B-+21'1(1 9:-%",5"&$0$/;
(0, 1) ⊃ A .&2*'* +B+31)%1,; S4()4F H", x ∈ A $3$( x ∈ [ n1 , 1 − n1 ] .&+( 7$,
+,+&1) I+,21,; M H+&2" 0 < x < 1 .&+8+)%1,; M H+&2" x ∈ (0, 1) .&*,;
T+,#1% .&+,+)F x ∈ (0, 1) .&-*(; =,#$5"% )*,+&1 *0+,1(8+ n1 < x < 1 − n1
.&+8+) #")$&2" 7$, n 2.'+& -+01-1 I+,21,; M H+&2"F x ∈ [ n1 , 1 − n1 ] .&*,;
D.&+01-10&+ x ∈ A .&*,;
A )45"-$($ .&*#%*,+( 7$&"#$5" +$% [ n1 , 1 − n1 ] +,+&1)&+,1(1( H"B-$ )+B+&121,;
=5+ "#$%&$'$( -.&*(2+)$ (0, 1) +,+&1'1 +31)%1,; D"5") )$ )+B+&1 +,+&1)&+,1(
-.(-*/ -+012+-1(1( 7$&"#$5$ )+B+&1 .&5+0+7$&$0.,;