Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Sign up for free
Menu
Search
Features
All features
Private URLs
Password Protection
Custom URLS
Scheduled publishing
Remove Branding
Restrict embedding
Deck Collections
Notes
Features
All features
Private URLs
Password Protection
Custom URLS
Scheduled publishing
Remove Branding
Restrict embedding
Deck Collections
Notes
Explore
Featured decks
Featured speakers
Programming
Technology
Storyboards
Explore
Featured decks
Featured speakers
Programming
Technology
Storyboards
Pricing
Search
Sign in
Sign up for free
集合・位相入門/松坂和夫 5.2 節:コンパクト性
Search
NomotoEriko
April 25, 2019
Technology
160
1
Share
Embed
Copy iframe code
Copy JS code
Copy link
Start on current slide
集合・位相入門/松坂和夫 5.2 節:コンパクト性
阪大の学生で開催した『位相・集合・多様体勉強会』の発表スライドです.『集合・位相入門/松坂和夫』の 5.2 節:コンパクト性を取り扱っています.
NomotoEriko
April 25, 2019
More Decks by NomotoEriko
See All by NomotoEriko
プレゼンのすゝめ
nomotoeriko
7
4.3k
集合・位相入門/松坂和夫 3.2 節:整列集合とその比較定理
nomotoeriko
0
640
集合・位相入門/松坂和夫 3.1 節:順序集合
nomotoeriko
0
380
レトリバインターン成果報告
nomotoeriko
0
1.9k
Other Decks in Technology
See All in Technology
hackers_champloo_2026_tidb
makocchi
0
170
パッチ更新でterraform initが落ちた日
grimoh
1
140
スキマバイトという雇用形態は存在しないので、雇用期間1日のアルバイト契約を事業者とワーカーが結ぶ手続きを支えるのがタイミーなんです
akitoshiga
0
210
M5Stackで作る BLEマウス GPSログの世界
tomokusaba
PRO
0
440
大型犬好きによるPugの紹介
yamanoku
1
160
USB PDを使ってスマホを労わりたい
raspython3
0
180
二歩目のフロントエンド・ウェブアクセシビリティ
camiha
2
650
S3 Filesで挑む! 八戸せんべい汁の食べ頃問題
usanchuu
0
110
Gemini Intelligence 入門 ― AppFunctions と Gemini Nano
takuroabe
0
290
1つのリクエストを追って理解する - llm-dのP/D分離
akhino
1
160
初海外で注意すべきこと12選!!!安全快適なre:invent生活にするために
yuuuuuuu168
1
180
ブラウザとサーバーから考えるAI時代のフロントエンド監視
shogo4131
3
1k
Featured
See All Featured
Balancing Empowerment & Direction
lara
6
1.3k
SEO Brein meetup: CTRL+C is not how to scale international SEO
lindahogenes
2
2.9k
Mind Mapping
helmedeiros
1
400
ピンチをチャンスに:未来をつくるプロダクトロードマップ #pmconf2020
aki_iinuma
128
56k
The Organizational Zoo: Understanding Human Behavior Agility Through Metaphoric Constructive Conversations (based on the works of Arthur Shelley, Ph.D)
kimpetersen
PRO
0
500
Keith and Marios Guide to Fast Websites
keithpitt
413
23k
How to optimise 3,500 product descriptions for ecommerce in one day using ChatGPT
katarinadahlin
PRO
3
3.9k
The agentic SEO stack - context over prompts
schlessera
0
970
Building a Modern Day E-commerce SEO Strategy
aleyda
45
9.3k
Color Theory Basics | Prateek | Gurzu
gurzu
1
490
Rails Girls Zürich Keynote
gr2m
96
14k
Bash Introduction
62gerente
615
220k
Transcript
5.2 201904xx Nomoto Eriko
B+1)=-;>=5/C A! :C&*7 $. <9? #8
0 %2 3=4C&:"@/6(',
#34< (", $) 89 & " :(>*5A
,>* ∪ & " '$!7 " & 9A & " 89 ;89 & " 89 & )& " ;>*?& ⊂ $@ 2=89 & " 89 2=%>*1?card & < ℵ/ @ " "0;89. " 2=89:(>*+ / -6 z #34< S :(>*5 &
#34< (", $) 89 & " :(>*5A
,>* ∪ & " '$!7 " & 9A & " 89 ;89 & " 89 & )& " ;>*?& ⊂ $@ 2=89 & " 89 2=%>*1?card & < ℵ/ @ " "0;89. " 2=89:(>*+ / -6
/ -6 #34< (", $) 89
& " :(>*5A ,>* ∪ & " '$!7 ( ) 9A ) ( 89 ;89 & " 89 & )& " ;>*?& ⊂ $@ 2=89 & " 89 2=%>*1?card & < ℵ1 @ " "0;89. " 2=89:(>*+ #34< S :(>*5 &
" + " + ()2 ! &". (C) '/
6 (C) | " ! #1,- 6 5 " 0! %34 5%34$ 4* "# S ,- 6
% & % <> $'&(
Euclid % !" #' #$ , &$ × ⋯×[#" , &" ] <!">
*7%$0! #34: ! =! 10! (C)’
(C)’’ 2 (6 (C)’ ?" ! 9<'-=/; &+. ",<'5 ⋂ " ≠ % (C)’’?" /; &+. ! ",8$<'5 ⋂&∈" Χ ≠ % /; &+ ! 8$<'5 " > " ",/;8$<' ") ) ⋂ ") ≠ % = " /; &+.
&)(%' " <I31C, (C)’ S! $ " MP6 ?!Q BO*4=$@
->P6F # ⋂ ! ≠ % <JA> [(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$5!0 ⋃ '0 = " " " Q⋃ ' = " H #!Q⋃ ' ≠ "R de Morgan D2! ⋂ ! ≠ %R
#"! 1A+):& (C)’ K! " DF-4I9E$,25 '3F-?
⋂ ! ≠ % <B7> [(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
,7&$3! (C)’’?! 2; '-0 " ".9$<(6
⋂$∈! Χ ≠ ( <81> [(C)’ ⇒ (C)’’] ! 2; '-0 =!* = , - - ∈ !} :%=#4 )54 2; '-0= !* ⊂ 0 = (C)’ ⋂ !* ≠ (>+ ⋂ !* = ⋂1∈! - ≠ (> [(C)’’ ⇒ (C)’] ! " :<(/=2; '-0 ". <(6 >:<( - * - = - (C)’’ ⋂1∈! - = ⋂ ! ≠ (>
', "#) ! ', " -""
* " "#) . <" ! (%&> # ! (, ,$ -⋃ # ⊃ " - # " ! (%& . <" ! !+%&> # ! !+', ,$ ⋃ # ⊃ " - # " ! !+%& .
<%@' ! * [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
"! 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 <
#"! A ;*3 *311 &45@ ! #"!D! %/>C)
" #"! <:1> # " ! %/?89 E D" ⊂ ⋃ # D ! = " ∪ ! − " = ) # ∪ ! − " . ! − " ! ?C)D# ! − " $(- C)7 #+ ! ?89E D! #" !.D# <,2B='C) #, + ! = ) #, ∪ ! − " 06E " ⊂ ⋃ #,E " #"!E
%('$& =G CA, 2;12 ! #%('$&*<=G L" # !
*<=G !# CA., !M 7L"(!) %('$&!M < > %('$&=GCA,%('$& <B6> "(!) ⊂ ⋃(∈* +( # 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
%* (' #12" & & !, !# $%*
,$ ! !# (' - !,% ! )+ $(%) !# )+- <> )%* ('
.2 +- $* 13 (Tychonoff) ,.2' ("# )#∈&
+-.2 " = ∏ #∈& "# 3" 3 ) ∈ Λ % "# &0 4 <1(> #" 4/! p212~p214 )
$ Hausdroff #( <&> "#( ! "! 2
#, % & ∩ ( = * & ∈ ,(#)* ( ∈ ,(%) + <> ! "! 2 % ' )+
Hausdroff !" <> #, %
!" 2 ' #, % = ) * #; ⁄ ) 2 ∩ * %; ⁄ ) 2 = .
Hausdroff .3 '0 ( Hausdroff .3 ! 5
1 +) 4 {$} ! 24 6 </*> & $ - ! +5$ ! & 1 ' 5& & {$} #+ & ∉ {$} 6 & $ + ",%6 $ (S − $ ) = ( $ ). $ = $ ", + #+
#&%"$2 Hausdroff :E 08 14 Hausdroff :E ! B+G,
" #&%"$H " ! CG,I <?5> # = ! − " 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 & , # "
#&%"$2 Hausdroff :E 08 14 Hausdroff :E ! B+G,
" #&%"$H " ! CG,I <?5> # = ! − " 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" #&%"$H " A16F)7 ,2 , ⋯ ,4 !H M ⊂ 6 782 4 0 /9 !4; I & , (/ # " 0 /
# Hausdroff '- !& 14 Hausdroff '- ! *.
" / " ! +. 0 <(%> # = ⋂&'( ) #*+ /# , ,)/$ - = 1, ⋯ , 1 " /# ⊂ #*+ , #*+ ∩ 4 *+ = 5 / # ∩ 4 *+ = 5 # ∩ 6 &'( ) 4 *+ = 6 &'( ) # ∩ 4 *+ = 5 # ∩ " = 5 , 7& # 8 " 4 *+
* 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%:' <
"! . 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'% ;
!"% !" $& <!" $& # > #
!" $(&; () ( # ) '& ! $& #
019 !" 6:" $. 16 Euclid 19 !"
6:" # ; # !" */7:" '4 < < 5)> # ⇒ # */7:" # !" 6:" < %" Hausdroff 19 ;$. 14 # 7:" < &' !" #$, ; &' & +!% 8-(:"2 ( &' ; * * > 0} 3 # !" &'
+,7 !" 4!8# %) 16 Euclid ,7 !"
4!8# # 9 # !" (*58# &1"! : < 3'> # ⇒ # (*58# # !" 4!8# : %" Hausdroff ,7 9%) 14 # 58# : ' 8#-$8# !" / 9 # !" 602.: # !" &'
-.4 !" 2#6% &+ 16 Euclid .4 !"
2#6% # 7 # !" ),36%'/$#8 < 1(> # ⇒ # ),36% # !" 2#6% 8 %" Hausdroff .4 7&+ 14 # 36%8 M M *! &(() ; +) )5" 08 # ⊂ ⋃/01 2 &( () ; +/ )8 # !" ()
(). !" ,/! "& 16 Euclid ). !"
,/! # 0 # !" %'-/! #* 1 < +$> # ⇒ # %'-/! # !" ,/! 1 %" Hausdroff ). 0"& 14 # -/! 1 max{*+ , ⋯ *. } = * B(34 ; *6 ) ⊂ 9(34 ; *) # ⊂ 9(34 ; *)1 # %' 1 # !" 34
!" 16 Euclid !" # # !" ] < [
> # ] ⇒ # # !" ] # !" ] % = '( , *( × ⋯×['" , *" ] % [ 11 M ! ] [', *] [ 13 J [ ! ] / = [', *] [ 11 0 0 ] # 13 (Tychonoff) (02 )2∈5 0 = ∏ 2∈5 02 0 7 ∈ Λ 02
!" 16 Euclid !" # [ # !" < >
# ⇒ # % & = [), +] ! [ - ∈ & [ % [), -] [ % [), -] & - &/ ) + %
!" 16 Euclid !" # [ # !" < >
# ⇒ # % & = [), +] ! [ - ∈ & [ % [), -] [ % [), -] & - &/ ) + % - &/ % %
[ !" ] 16 Euclid [ !" ] # #
!" ] < > # ] ⇒ # % & = [), +] ! - ∈ & % [), -] % [), -] & - ] &/ + ∈ &/ ! [ & = [), +] ) + % - % &/
!" 16 Euclid !" # # !" [ < >
# ⇒ # % ∈ '( '( ≠ *[ + '( sup '( = 0 0 ≦ +[ 0 ∈ ' 0 ∈ 2 3 2 2 4 > 0 [0 − 4, 0 + 4] ⊂ 2 [ 0 − 4 ∈ '( [%, 0 − 4] 3 2= , ⋯ , 2? [ %, 0 − 4 ∪ 0 − 4, 0 + 4 = [%, 0 + 4] 2= , ⋯ , 2? , 2 [ 0 ∈ '([ + 2 '( 0 0 − 4 0 + 4
!" 16 Euclid !" # # !" [ < >
# ⇒ # % − ' ∈ )* [,, % − '] / 01 , ⋯ , 03 [ ,, % − ' ∪ % − ', % + ' = [,, % + '] 01 , ⋯ , 03 , 0 [ % ∈ )*[ % < 8 % + ' ∈ )* ' % )* [ % = 8, 8 ∈ )* 8 0 )* % % − ' % + '
!" 16 Euclid !" # # !" [ < >
# ⇒ # % − ' ∈ )* [,, % − '] / 01 , ⋯ , 03 [ ,, % − ' ∪ % − ', % + ' = [,, % + '] 01 , ⋯ , 03 , 0 [ % ∈ )*[ % < 8 % + ' ∈ )* ' % )* [ % = 8, 8 ∈ )* 8 0 )* % % − ' % + '
./4 !" 2"5% ', 16 Euclid /4 !" 2"5%
# 6 # !" *- 35% (1#" 7 +) 6 p5 0&$ 7 /4 !8 Euclid /4 !" *- 35% $% , '% × ⋯×[$" , '" ]
! ;$?& *3 16 Euclid 6= !" ;$?&
# @ # !" 04<?& -9%$ A 2 ! (&8@17 1, 2,A 71 # #56= ! ;$?&@ max #, min # )' <:/> # ! 04;$?& @# !>"> )'A .@# ! <?& @/ !>"> # +A max #, min # )'A
! !$# " J*N, 2? 16 Euclid BL !"
J*N, # !$# " Q # !" <@KN,6G+*R > ! /,FQ=C 1, 25R C2O;0'Q;4'2?P $: & → !!$# "&ABL & %2E 3IDM8 Q '. $(&) ;0';4'1- R <H9> 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
%) 06 <"2> 3 9/06 ! (- #
5 $ ! ,9! %) : <> '.741+8 *& <> "
.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 <MB> 98 #TK6 , p219~p222 "4E