!"#$ %&' $()*+, -./0"/* !! !"# $%&'%( &)*)()+, !"#" $%&'()*()% " (X, ρ) #$% &'(%$) *+,- ./0*1" 2'% (x, y) ∈ X×X $3$1 δ(x, y) = inf{1, ρ(x, y)} 4$-'/$&" 5* 4*%*&4, ρ $/' δ &'(%$)/'%$1$1 4'1) ./4*6*1*7 -,1$ Tρ = Tδ ./4*6*1* 890('%$1$+" !"#$% :3;) -*<,%/,% &'(%'/ (.=./.>$/'%$1 (,#,1;4;%" ?1'%&' @" "A *-,%;1B, Bρ (x, r) ⊂ Bδ (x, p) ⊂ Bρ (x, q) ./,B,) C')$/4' r, p, q 0,-;/,%;1;1 <,%/;6;1; 890('%&') -'('B')($%" r < 1 $0' ρ(x, y) = r ⇒ δ(x, y) = inf{1, ρ(x, y)} = ρ(x, y) = r 4$%" D E"FG 'C$(/$6$ 4$)),(' ,/;1;%0,H I'% p $3$1 r = p 2 ).1*&*-/, D E"EG Bρ (x, r) ⊂ Bδ (x, p) '/4' '4$/$%" J,%C;( ./,%,)H I'% q < 1 $3$1 p = q 2 D E"FG ).1*&*-/,H D E"FG 4,1 Bδ (x, p) ⊂ Bρ (x, q) D E"!G '/4' '4$/$%" D E"EG <' D E"!G 8'%'6$1B' ρ <' δ &'(%$)/'%$ 4'1)($%/'%7 -,1$ X K+'%$14' ,-1; (.=./.>$-$ #'/$%/'%/'%L Tρ = Tδ " A" ρ $/' δ #$% X )K&'0$ K+'%$14' (,1;&/; $)$ &'(%$) ./0*1" M6'% I'% (x, y) ∈ X × X $3$1 aδ(x, y) ≤ ρ(x, y) ≤ bδ(x, y) D E"NG 'C$(0$+/$)/'%$ 0,6/,1,B,) #$3$&4' =.+$($O a <' b 0,-;/,%; <,%0,H ρ $/' δ 1;1 4'1) ./4*6*1* 890('%$1$+" !"#$% ρ(x, y) = r ∧ δ(x, y) = p ⇒ ap ≤ r ≤ bp ./4*6*1* 4$)),(' ,/;%0,)H p p Bδ (x, ) ⊂ Bρ (x, r) ⊂ Bδ (x, ) a b ./4*6*1* 89%K%K+" P I,/4'H 91B')$ =%.#/'&4' $O,4' '4$/'1 1'4'1/' Tρ = Tδ ./,B,)(;%" Q" !"#$ %&'('% 4' (,1;&/,1,1 δ &'(%$6$1$ X -'%$1' R2 ).-,%,) 4KCK1K1K+" 5,C/,18;3 &'%)'+/$ ,3;) <' ),=,/; #$%$& -*<,%/,%; #*/*1*+" :3;) #$%$& -*R <,%;1 ),=/,&; ),=,/; #$%$& -*<,%, 'C$( &$4$%S 5$%$& -*<,%;1 )'1,% )K&'0$ 1'4$%S !"#$% ?%1') E"A" 4')$ (,1;& *-,%;1B,H 4K+/'&4')$ I'% x, y 1.)(,0; $3$1 δ &'(%$) O.1)0$-.1* ( 0, x = y δ(x, y) = 1, x 6= y !"#" $%&'%( &)*)()+, !" #$%$&$'() *+',&-+',./ 01'+ 23.)4 O = (0, 0) #+5-+'2,% '67*+8,',' r 9+.,%+:-, +%,7 91;+., Bδ (O, r) $-) 238*).$-$.8)4 δ ',' *+',&,'(+' ( {O}, r<1 Bδ (O, r) = R2 − {O}, r ≥ 1 %,7+./ 01.+(+' +%,7%+ 23.<-(<=< 2$#$4 +%,7 #$.$& 91;+. Bδ (O, 1) = {x | δ(O, x) < 1} = {O} (,./ >+:+-, 91;+. $8) Dδ (O, 1) = {x | δ(O, x) ≤ 1} = R2 ($./ Tδ &)*.)- *6:6-6?$8$'$' +%,7 7<&)-).$ 9+-',@A+ (3.* *+')($.B Tδ = {∅, {O}, {R2 − {O}}, R2 } 01'-+. +9', @+&+'(+ &)*.)- *6:6-6?$'$' 7+:+-, 7<&)-).$($./ C9-)98) Bδ (O, 1) = {0} +%,7 #$.$& 91;+.,',' 7+:-+&, 7)'($8$($.D (6-+9,8,9-+ 7+:+-, #$.$& 91E ;+.+ )5$* ()=$-($./ 0)'@). 6-+.+74 Dδ (O, 1) 7+:+-, #$.$& 91;+.,',' 7+:-+&, 2)') 7)'($8$($.D 9+'$ R2 ($./ R2 = {O} ∪ (R2 − {O}) 6-(1=1'(+'4 &)*.)1@+9,' F). '67*+8, #$. $% '67*+(,./ G6-+9,8,9-+ #$.$& 91;+.,' 7)'+. '67*+8, 967*1./ H/ (a, b) +%,7 +.+-,=,',' 7+:-+&,',' I+4#J 7+:+-, +.+-,=, 6-(1=1'1 238*).$'$@/ G+F+ 2)')- 6-+.+74 Rn 9+ (+ Cn 1@+9-+.,'(+4 C7-$( &)*.$=$') 23.) B(a, r) +%,7 91;+.-+.,',' 7+:-+&-+.,',' D(a, r) 7+:+-, 91;+.-+., 6-(1=1'1 238E *).$'$@/ !"#$% (a, b) '$' 7+:-+&,4 (a, b) 9$ 7+:8+9+' #<*<' 7+:+-, 7<&)-).$' +.+7)8$*$($./ I+4#J 7+:+-,(,. ;) (a, b) ⊂ [a, b] ($. C9-)98)4 (a, b) ⊂ [a, b] ($./ C*) 9+'(+' F). x ∈ [a, b] $%$' (x − ǫ, x + ǫ) ∩ [a, b) 6= ∅ ($./ G6-+9,8,9-+4 [a, b] ⊂ (a, b) ($./ 01 $7$ 7+:8+&+ #+=,'*,8, [a, b] = (a, b) 6-&+8,', 2).)7E *$.$./ Rn 1@+9,'(+7$ (1.1& $%$'4 (a, b) 9).$') B(a, r) +%,7 91;+.,4 [a, b] 9).$') D(a, r) 7+:+-, 91;+., 76'1-+.+74 917+.,(+7$ 18+;1.&+ +9')' *)7.+.-+'E +#$-$./ K/ !"#$ %&'('% ()7$ δ &)*.$=$'$' X 7<&)8$ <@).$'() *+',&-+(,=, &)*.)*6:6-6?$'$' +9.,7 *6:6-6?$ 6-(1=1'1 238*).$'$@/ !"#$% L3@ 76'181 *+',& 19+.,'A+4 F). x, y ∈ X $%$' δ &)*.$7 M6'78$96'1 ( 0, x = y δ(x, y) = 1, x 6= y !" !"#$ %&' $()*+, -./0"/* #$%$&$'() *+',&-+',./01 23'+ 450)6 7)0 7+'4$ #$0 x ∈ X '/8*+9,',' r .+0,%+:-, +%,8 .3;+0, Bδ (x, r) $-) 459*)0$-$09)6 δ ',' *+',&,'(+' ( {x}, r<1 Bδ (x, r) = X − {x}, r ≥ 1 %,8+01 230+(+' +%,8%+ 450<-(<=< 4$#$6 0 < r1 $%$' +%,8 #$0$& .3;+0 Bδ (O, r) = {y | δ(x, y) < r} = {x} ($01 23 ()&)8*$0 8$ 7)0 x ∈ X '/8*+9, $%$' *)8 5=)-$ {x} 8<&)9$ +%,8 #$0 8<&)($01 > 7+-() (X, Tδ ) 3?+., +.0,8 #$0 3?+.(,01 @1 A'B)8$ #5-<&() *+',&-+'+' δ, p, m ;) sn &)*0$8-)0$'$ R2 $%$' <?)0$'() (<C<'<'<?1 D)08)?-)0$ (<?-)&() 0 = (0, 0) '/8*+9, ;) .+0,%+:-+0, r = 1 /-+' +%,8 .3;+0-+0,6 7)0#$0$9$ $%$'6 %$?$'$?1 !"#" $%&'%( &)*)()+, !