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5.2  201904xx Nomoto Eriko

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  – B+1)=-;>=5/C A! :C&*7 $. <9? #8 – 0  %2  – 3=4C&:"@/6(',

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

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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,- 6 5 " 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$E P6F Q' = )* ) ∈ !}  "R  Q! ⊂ . !  " MP6F . 7#"" Q ' ⊂ / '  " NP6F / 7#"RQ! -> BOL1P6 !0 9 ! BO*4=$@ ⋂ !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 ' ⊂ /J I! '39EC)F- !0 0 ⋂ !0 ≠ % I' '39EC)F- '0 0 ⋃ '0 ≠ "J B7.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  $% &  E D $ = ( ∩ " ( ∈ $% }, $′ = ( ∩ " ( ∈ $′% } 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 E D" ⊂ ⋃ #  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., !M 7L"(!) %('$&!M < > %('$&=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 # = ! − " DG, !9 H& ! # (3 1 7 H& @* (  ( ⊂ #   ( ∩ " = +  /.  !9 I & ! # -0  1 7 I, ! " (37 H & ≠ , 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 # = ! − " DG, !9 H& ! # (3 1 7 H& @* (  ( ⊂ #   ( ∩ " = +  /.  !9 I & ! # -0  1 7 I, ! " (37 H & ≠ , 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 #  ! 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