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5.2 201904xx Nomoto Eriko

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/ -6 – #34< (", $) – 89 & " :(>*5A,>* ∪ & " '$!7 ( ) 9A ) ( 89 – ;89 & " 89 & )& " ;>*?& ⊂ $@ – 2=89 & " 89 2=%>*1?card & < ℵ1 @ – " "0;89. " 2=89:(>*+ #34< S :(>*5 &

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" + " + ()2 ! &". (C) '/6 (C) | " ! #1,-65 " 0! %34 5%34$ 4* "# S ,- 6

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% – & % <> $'&( – Euclid % !" #' #$ , &$ × ⋯×[#" , &" ]

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*7%$0! #34: ! =! 10! (C)’ (C)’’ 2 (6 (C)’ ?" ! 9<'-=/; &+. ",<'5 ⋂ " ≠ % (C)’’?" /; &+. ! ",8$<'5 ⋂&∈" Χ ≠ % /; &+ ! 8$<'5 " > " ",/;8$<' ") ) ⋂ ") ≠ % = " /; &+.

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&)(%' " P6F # ⋂ ! ≠ % [(C) ⇒ (C)’] ! $ (C)’ +8$E P6FQ' = )* ) ∈ !} "R Q! ⊂ . ! " MP6F . 7#"" Q ' ⊂ / ' " NP6F / 7#"RQ! -> BOL1P6 !0 9! BO*4=$@ ⋂ !0 ≠ % " Q' ->BOL1P6 '0 9 de Morgan D2 ⋃ '0 ≠ "R; (C) 9/' K: BO.IG '0$5!0⋃ '0 = " " " Q⋃ ' = " H#!Q⋃ ' ≠ "R de Morgan D2! ⋂ ! ≠ %R

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#"! 1A+):& (C)’ K! " DF-4I9E$,25 '3F-? ⋂ ! ≠ % [(C)’ ⇒ (C)]G6>8@)= JH 0(= J ! (C)’ %/< F-?I ' = )* ) ∈ !} J I! ⊂ . I ' ⊂ /JI! '39EC)F- !0 0 ⋂ !0 ≠ % I' '39EC)F- '0 0 ⋃ '0 ≠ "JB7.JI ⋃ ' = " %/ I' ¬(C) < J ⋃ ' = " de Morgan ;* ⋂ ! = % ¬(C)’ J

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,7&$3! (C)’’?! 2; '-0 " ".9$<(6 ⋂$∈! Χ ≠ ( <81> [(C)’ ⇒ (C)’’] ! 2; '-0 =!* = , - - ∈ !} :%=#4 )54 2; '-0= !* ⊂ 0 = (C)’ ⋂ !* ≠ (>+ ⋂ !* = ⋂1∈! - ≠ (> [(C)’’ ⇒ (C)’] ! " :<(/=2; '-0 ". <(6>:<( - * - = - (C)’’ ⋂1∈! - = ⋂ ! ≠ (>

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', "#) ! ', " -"" * " "#) . <" ! (%&> # ! (, ,$-⋃ # ⊃ " - # " ! (%& . <" ! !+%&> # ! !+', ,$ ⋃ # ⊃ " - # " ! !+%& .

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<%@' ! * [8A] $24> ! <%@' " ! D 1 (CM) ,5 )9&% E (CM) | " ! #+=7: " ! 0?7:(E <;.>B-3/ 6% " C [(CM ) ⇒ (C)] $% " ! #+=7: E(CM ) $% " ! 0?7:(D $% & ED $ = ( ∩ " ( ∈ $% }, $′ = ( ∩ " ( ∈ $′% } E $ " =7: D" 0?7: $& (E [(C) ⇒ (CM ) ] .E

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"! 8 4*. *.10 $/07 ! -9%5&:' "# , "% ⋯ , "' "! ;"# ∪ "% ∪ ⋯ ∪ "' "! < <3,> ") (+ = 1, 2, ⋯ /) ! 612 1) < #* 1) ") -912(; 1) 2 < ⋃) 1) ⊃ ⋃) ") ; 6:'):'6:' ⋃) 1) ⊂ 6<+ ⋃) 1) ⋃) ") 612 < ;⋃) 1) 2 ⊃ ⋃) ") ⋃) 1) 2 -9%:' ; ⋃) 1) 2 ⋃) ") -912 <

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#"! A ;*3 *311 &45@ ! #"!D! %/>C) " #"! <:1> # " ! %/?89 ED" ⊂ ⋃ # D ! = " ∪ ! − " = ) # ∪ ! − " . ! − " ! ?C)D# ! − " $(- C)7 #+ ! ?89E D! #" !.D# <,2B='C) #, + ! = ) #, ∪ ! − " 06E " ⊂ ⋃ #,E " #"!E

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%('$& =G CA, 2;12 ! #%('$&*<=G L" # ! *<=G !# CA.,!M7L"(!) %('$&!M < > %('$&=GCA,%('$& "(!) ⊂ ⋃(∈* +( # J+( # !# FI1K"L ! = - (∈* "./(+( # ) L" CA!L "./(+( # ) ! FI1! p186 2; 23 #09M! %('$&!L "./(+( # ) D38H+ ! = "./(+(0 # ) ∪ ⋯ ∪ "./(+(3 # ) 4 >M "(!) ⊂ +(0 # ∪ ⋯ ∪ +(3 # "(!) %('$&!M p186 2; 23 J5?K I1 ! *<=G (!#, +#)L., ": ! → !#)" !M L 7# - +# " !:, "./(+#) 4! ! E/I1@ ! !*<!M

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%* (' #12"& & !, !# $%* ,$ ! !# (' - !,% ! )+ $(%) !# )+- <> )%*('

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.2 +- $* 13 (Tychonoff) ,.2' ("# )#∈& +-.2 " = ∏ #∈& "# 3" 3 ) ∈ Λ % "# &0 4 <1(> #" 4/! p212~p214 )

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$ Hausdroff #( <&> "#( ! "! 2 #, % & ∩ ( = * & ∈ ,(#)* ( ∈ ,(%) + <> ! "! 2 %' )+

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Hausdroff !" <> #, % !" 2 ' #, % = ) * #; ⁄ ) 2 ∩ * %; ⁄ ) 2 = .

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Hausdroff .3 '0 ( Hausdroff .3 ! 5 1 +) 4 {$} ! 24 6 *> & $ - ! +5$ ! & 1 ' 5& & {$} #+ & ∉ {$} 6 & $ + ",%6 $ (S − $ ) = ( $ ). $ = $ ", + #+

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#&%"$2 Hausdroff :E 08 14 Hausdroff :E ! B+G, " #&%"$H " ! CG,I 5> # = ! − " DG, !9H& ! # (3 1 7H& @* ( ( ⊂ # ( ∩ " = + /. !9I & ! # -0 1 7I, ! " (37H & ≠ , I ! Hausdroff :EH&, , D@* '</.H ! (/ , 0 / I & ∈ (/ , , ∈ 0 / , (/ ∩ 0 / = +I 0 / , ∈ " 5 " ! D=>I" #&%"$H " A16F)7 ,2 , ⋯ ,4 !H M ⊂ 6 782 4 0 /9 !4; I & , # "

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#&%"$2 Hausdroff :E 08 14 Hausdroff :E ! B+G, " #&%"$H " ! CG,I 5> # = ! − " DG, !9H& ! # (3 1 7H& @* ( ( ⊂ # ( ∩ " = + /. !9I & ! # -0 1 7I, ! " (37H & ≠ , I ! Hausdroff :EH&, , D@* '</.H ! (/ , 0 / I & ∈ (/ , , ∈ 0 / , (/ ∩ 0 / = +I 0 / , ∈ " 5 " ! D=>I" #&%"$H " A16F)7 ,2 , ⋯ ,4 !H M ⊂ 6 782 4 0 /9 !4; I & , (/ # " 0 /

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# Hausdroff '- !& 14 Hausdroff '- ! *. " / " ! +. 0 <(%> # = ⋂&'( ) #*+ /# , ,)/$ - = 1, ⋯ , 1 " /# ⊂ #*+ , #*+ ∩ 4 *+ = 5 / # ∩ 4 *+ = 5 # ∩ 6 &'( ) 4 *+ = 6 &'( ) # ∩ 4 *+ = 5 # ∩ " = 5 , 7& # 8 " 4 *+

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* Hausdroff 19 (. 14 Hausdroff 19 ! 7%:' " ; " ! 8:'< 2 19 ! Hausdroff 19 !# 63$" $ 8$"< <)4| 8$"> 8:'0#8:' $" <5,> " ! 8:';(. 11 " ; (. 12 2 $(") !# 7%:'<&(. $(") !# 8:'< (. 11 !/19 ! ; ! +8:' " (. 12 2 !, !# !/19;$ ! !# 63$"< -;" ! 7%:' $(") !# 7%:'<

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"! . Hausdroff 29 +0 15 ! "! 29:!" Hausdroff 29: #: ! → !" 74&(,:# )1'% ; <-5| )1'%> 2 $129 !, !" #: ! → !" &(, # #() 74 '% # <6/> #+:# 74&(, ; +0 14 3 # 8'% ; p186 * 7 # 74&(, 8'% # )1'% :# )1'% ; +0 14 3 "! 29 ! Hausdroff 29 !" 74'% # 8'% ;

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!"% !" $& # !" $(&; () ( # ) '& ! $& #

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019 !" 6:" $. 16 Euclid 19 !" 6:" # ; # !" */7:" '4 < < 5)> # ⇒ # */7:" # !" 6:" < %" Hausdroff 19 ;$. 14 # 7:"< &' !" #$, ; &' & +!% 8-(:"2 ( &' ; * * > 0} 3 # !" &'

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+,7 !" 4!8# %) 16 Euclid ,7 !" 4!8# # 9 # !" (*58# &1"! : < 3'> # ⇒ # (*58# # !" 4!8#: %" Hausdroff ,7 9%) 14 # 58# : ' 8#-$8# !" /9 # !" 602.: # !" &'

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-.4 !" 2#6% &+ 16 Euclid .4 !" 2#6% # 7 # !" ),36%'/$#8 < 1(> # ⇒ # ),36% # !" 2#6% 8 %" Hausdroff .4 7&+ 14 # 36%8 M M *! &(() ; +) )5" 08 # ⊂ ⋃/01 2 &( () ; +/ )8 # !" ()

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(). !" ,/! "& 16 Euclid ). !" ,/! # 0 # !" %'-/! #* 1 < +$> # ⇒ # %'-/! # !" ,/!1 %" Hausdroff ). 0"& 14 # -/! 1 max{*+ , ⋯ *. } = * B(34 ; *6 ) ⊂ 9(34 ; *) # ⊂ 9(34 ; *)1 # %' 1 # !" 34

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!" 16 Euclid !" # # !" ] < [ > # ] ⇒ # # !" ] # !" ] % = '( , *( × ⋯×['" , *" ] % [ 11 M ! ] [', *] [ 13 J [ ! ] / = [', *] [ 11 0 0 ] # 13 (Tychonoff) (02 )2∈5 0 = ∏ 2∈5 02 0 7 ∈ Λ 02

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!" 16 Euclid !" # [ # !" < > # ⇒ # % & = [), +] ! [ - ∈ & [ % [), -] [ % [), -] & - &/ ) + %

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!" 16 Euclid !" # [ # !" < > # ⇒ # % & = [), +] ! [ - ∈ & [ % [), -] [ % [), -] & - &/ ) + % - &/ % %

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[ !" ] 16 Euclid [ !" ] # # !" ] < > # ] ⇒ # % & = [), +] ! - ∈ & % [), -] % [), -] & - ] &/ + ∈ &/ ! [ & = [), +] ) + % - % &/

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!" 16 Euclid !" # # !" [ < > # ⇒ # % ∈ '( '( ≠ *[ + '( sup '( = 0 0 ≦ +[ 0 ∈ ' 0 ∈ 2 3 2 2 4 > 0 [0 − 4, 0 + 4] ⊂ 2 [ 0 − 4 ∈ '( [%, 0 − 4] 3 2= , ⋯ , 2? [ %, 0 − 4 ∪ 0 − 4, 0 + 4 = [%, 0 + 4] 2= , ⋯ , 2? , 2 [ 0 ∈ '([ + 2 '( 0 0 − 4 0 + 4

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!" 16 Euclid !" # # !" [ < > # ⇒ # % − ' ∈ )* [,, % − '] / 01 , ⋯ , 03 [ ,, % − ' ∪ % − ', % + ' = [,, % + '] 01 , ⋯ , 03 , 0 [ % ∈ )*[ % < 8 % + ' ∈ )* ' % )* [ % = 8, 8 ∈ )* 8 0 )* % % − ' % + '

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!" 16 Euclid !" # # !" [ < > # ⇒ # % − ' ∈ )* [,, % − '] / 01 , ⋯ , 03 [ ,, % − ' ∪ % − ', % + ' = [,, % + '] 01 , ⋯ , 03 , 0 [ % ∈ )*[ % < 8 % + ' ∈ )* ' % )* [ % = 8, 8 ∈ )* 8 0 )* % % − ' % + '

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./4 !" 2"5% ', 16 Euclid /4 !" 2"5% # 6 # !" *-35% (1#" 7 +)6 p5 0&$ 7 /4 !8 Euclid /4 !" *-35% $% , '% × ⋯×[$" , '" ]

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! ;$?& *3 16 Euclid 6= !" ;$?& # @ # !" 04& -9%$ A 2 ! (&8@17 1, 2,A 71 # #56= ! ;$?&@ max #, min # )' <:/> # ! 04;$?& @# !>"> )'A .@# ! & @/ !>"> # +A max #, min # )'A

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! !$# " J*N, 2? 16 Euclid BL !" J*N, # !$# "Q # !" <@KN,6G+*R > ! /,FQ=C 1, 25R C2O;0'Q;4'2?P $: & → !!$# "&ABL & %2E 3IDM8 Q '. $(&) ;0';4'1- R 2? 12 $(&) !$# "R $(&) 2? 16 C 1 ;0';4'7 R 2? 12 & !$# "&ABLQ$ & &ABL &* ID)( R :Q$(&) !$# "R 2? 16 C 1 # &ABL ! !$# " J*N, Q max #, min # 1- R

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%) 06 <"2> 3 9/06 ! (- # 5 $! ,9! %) : <> '.741+8 *& <> "

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.GHQ" %('$& HQ 7N :F 17 ! = (S, &) ".GHQ S! S + 1 D )* " -2R5" !∗ = ! ∪ {)* } T S !∗ O>.G &∗ "<0 .GHQ !∗ = (!∗, &∗) "/S C (i), (ii) @I S! Hausdroff HQ ?L31 T (i) | !∗ %('$& Hausdroff HQ T (ii) | *! ! !∗P1HQ T S&∗ ! G;.GS ! *!.G & )J T <> =A%('$& Hausdroff HQ%('$&HQ7N 98 #TK6 , p219~p222 "4E