{"id":3760,"date":"2006-07-28T22:21:00","date_gmt":"2006-07-28T13:21:00","guid":{"rendered":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/?p=3760"},"modified":"2021-08-12T12:00:17","modified_gmt":"2021-08-12T03:00:17","slug":"studygroup2006summerchapter2","status":"publish","type":"post","link":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/2006\/07\/28\/studygroup2006summerchapter2\/","title":{"rendered":"StudyGroup2006SummerChapter2"},"content":{"rendered":"

= Basic Topology =<\/code> \ub0b4\uc6a9 \uc694\uc57d: http:\/\/math.korea.ac.kr\/~ywkim\/courses\/2k6rudin\/rudin_ch2.pdf<\/a> <\/p>\n

http:\/\/math.korea.ac.kr\/~ywkim\/courses\/2k6rudin\/rudin_ch2_compactness.pdf<\/a> <\/p>\n

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06\/07\/20\uc5d0 \uacf5\ubd80\ud55c \ub0b4\uc6a9<\/h3>\n
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”’2\uc7a5 26\ubc88”’: \uc774 \ubb38\uc81c\ub294 \uc55e\uc758 23\ubc88, 24\ubc88\uc744 \uc54c\uc544\uc57c \ud55c\ub2e4\uace0 \ub418\uc5b4 \uc788\uc5b4\uc694. \uc774\ub7f4 \ub54c \uba87 \uc2a4\ud15d\uc73c\ub85c \ub098\ub204\uc5b4\uc11c \uc0dd\uac01\ud574 \ubd10\uc57c \ud574\uc694. \uc6b0\uc120 23\ubc88, 24\ubc88\uc758 \ub0b4\uc6a9\uc744 \ubcf4\uba74 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc774\ub7ec\uc774\ub7ec \ud558\uba74, \uc774\ub7ec\uc774\ub7ec\ud55c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4\uace0 \ud588\uc9c0\uc694. \uc774\ub7f0 \uacb0\uacfc\uac00 \uc131\ub9bd\ub418\ub294 \uac70\ub9ac\uacf5\uac04\uc758 \uc608\ub97c \ucc3e\uc544\ubcfc \ud544\uc694\uac00 \uc788\uc5b4\uc694. \uc801\uc5b4\ub3c4 22~24\ubc88\uc740 $ \\mathbb{R}$ \uc774\ub098 \\(\\mathbb{R}^n\\) \uc5d0\uc11c\ub294 \ud56d\uc0c1 \uc131\ub9bd\ud558\uc9c0\uc694.(\ud655\uc778\ud560 \uac83.) \ub530\ub77c\uc11c \uc6b0\uc120 26\ubc88\uc744 $ \\mathbb{R}$ \uc774\ub098 \\(\\mathbb{R}^n\\) \uc5d0\uc11c\ub294 \uc5b4\ub5bb\uac8c \uc99d\uba85\ud560 \uac83\uc778\uac00\ub97c \uc0dd\uac01\ud574 \ubcf4\ub294 \uac83\uc774\uc9c0\uc694. \uc774\uac83\uc774 \uc27d\uace0 \uc774\uac83\ub9cc \uc774\ud574\ud558\uba74 \uc774 \ubb38\uc81c\uac00 \ud558\uace0\uc790 \ud558\ub294 \uc774\uc57c\uae30\uac00 \ubb34\uc5c7\uc778\uc9c0\ub97c \uc54c \uc218 \uc788\uc5b4\uc694. \uadf8\ub7ec\ub2c8\uae4c \uac04\ub2e8\ud558\uac8c $ \\mathbb{R}$ \uc5d0\uc11c \uc5b4\ub5a4 \uc77c\uc744 \ud574\uc57c \ud558\ub294\uc9c0 \uc54c\uc544\ubcf4\uc9c0\uc694. <\/p>\n

\uba3c\uc800 $ X$ \uac00 $ \\mathbb{R}$ \uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc9c0\uc694. \uadf8\ub9ac\uace0 $ X $ \uc758 \uc784\uc758\uc758 \ubb34\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 $ X $ \uc548\uc5d0\uc11c limit point\ub97c \uac16\ub294\ub2e4\uace0 \ud558\uc9c0\uc694. \uadf8\ub9ac\uace0\ub294 \uc774\ub7ec\ud55c \uc870\uac74\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 $ X $ \uac00 compact\uac00 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud569\ub2c8\ub2e4. \uc6b0\ub9ac\ub294 \ubaa8\uc21c\uc744 \ucc3e\uc544\uc57c \ud558\uc9c0\uc694. \uadf8\ub798\uc11c compact\uac00 \uc544\ub2c8\ub77c\ub294 \uc870\uac74\uc744 \uad6c\uccb4\uc801\uc73c\ub85c \uc368 \ubcf4\uc9c0\uc694. \uc774\uac83\uc774 Hint\uc758 $ G_n $ \uc785\ub2c8\ub2e4. \uc989 $ X $ \uc758 open cover\uc778\ub370 finite subcover\ub97c \uac16\uc9c0 \uc54a\ub294 \uadf8\ub7f0 open cover\uc785\ub2c8\ub2e4. \uc774\ub7f0 \uac83\uc744 \uad6c\uccb4\uc801\uc73c\ub85c \uc0c1\uc0c1\ud574 \ubcfc \ud544\uc694\uac00 \uc788\uc5b4\uc694. \uadf8\ub798\uc11c \uc804\uc5d0 \ud574 \ubd24\ub358 \uc608\ub97c \uc0dd\uac01\ud574 \ubd05\ub2c8\ub2e4. \uc989 $ (0,1] $ \uc744 cover\ud558\ub294\ub370 $ G_n=(1\/n,1] $ \ub85c \ud558\uae30\ub85c \ud558\uc9c0\uc694. \uadf8\ub7ec\uba74 $ G_n$ \uc740 finite subcover\uac00 \uc5c6\uc9c0\uc694. \uc774\uc81c hint\uc5d0\uc11c \uc774\uc57c\uae30\ud558\ub294 $ F_n $ \uc740 $ (0,1\/n] $ \uc774 \ub418\uc9c0\uc694. \uc989 $ F_n⊃ F_{n+1} $ \uc774 \ub418\uc9c0\uc694. \uc989 $ F_n$ \uc740 closed \uc774\uace0, \uc810\uc810 \uc791\uc544\uc9c0\ub294(nested)\uc778 sequence of sets \uc774\uc9c0\uc694. \uc774\uc81c \uc774 \uacbd\uc6b0\ub97c \uadf8\ub9bc\uc744 \uadf8\ub824 \ub193\uace0 \ubcf4\uba74 \ubb34\uc5c7\uc774 \ubb38\uc81c\uc778\uc9c0 \uc54c \uc218 \uc788\uc5b4\uc694. \uc218\uc5f4 $ 1\/n $ \uac19\uc740 \uac83\uc744 \uc0dd\uac01\ud574 \ubcf4\uba74 \uc774 \uc810\ub4e4\uc740 $ X $ \uc758 \uc218\uc5f4\uc774\uace0 limit point $ 0 $ \ub97c \uac00\uc9c0\ub294\ub370 \uc774 limit point\uac00 $ X $ \uc5d0 \ud3ec\ud568\ub418\uc5b4 \uc788\uc9c0 \uc54a\uc740 \uac70\uc608\uc694. \uc989 $ F_n $ \ubc16\uc5d0 \ub193\uc774\ub294 $ X $ \uc758 \uc810\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uc7a1\uac8c \ub418\uba74 \uc774 \uc810\ub4e4\uc758 limit point\uac00 $ X $ \ubc16\uc5d0 \ub193\uc774\uac8c \ud560 \uc218 \uc788\uace0 \uc774\ub294 \ubaa8\uc21c\uc744 \uc720\ub3c4\ud55c \uac83\uc774 \ub418\ub294\uac70\uc9c0\uc694. <\/p>\n

\uc5ec\uae30\uc11c \uc870\uc2ec\ud560 \uac83\uc740 \uc6b0\uc120 $ F_n $ \uc758 \ubc16\uc5d0\uc11c \uc810\uc744 \uc7a1\uc73c\uba74\uc11c \ub9e4\ubc88 \uc7a1\ub294 \uc810\uc774 \uc544\uae4c \uc7a1\uc740 \uc810\uacfc \ub2e4\ub974\uac8c \uc7a1\uc544\uc11c \uc11c\ub85c \ub2e4\ub978 \ubb34\ud55c\ud55c \uc810\uc744 \uc7a1\ub294 \uac83\uc744 \ubcf4\uc5ec\uc8fc\ub294 \uac83\uacfc \uc774\uac83\uc758 limit point\uac00 \uc65c $ X $ \uc5d0 \uc5c6\ub294\uc9c0\ub97c \ubcf4\uc774\ub294 \uac83\uc774\uc9c0\uc694. \uc55e\uc758 \uac83\uc740 \uc798 \ud574\ubcf4\uace0\uc694… \ub4b7 \ubd80\ubd84\uc740 \uc774 sequence\uac00 $ F_n $ \uc548\uc5d0 \ub193\uc774\ubbc0\ub85c \uadf8 limit point\ub3c4 $ F_n $ \uc548\uc5d0 \ub193\uc5ec\uc57c \ud558\uc9c0\uc694.( $ F_n $ \uc774 closed \uc774\ub2c8\uae4c) \ub530\ub77c\uc11c limit point\ub294 $ \\bigcap F_n $ \uc5d0 \ub193\uc774\uace0 \uc774 \uc9d1\ud569\uc740 $ X $ \uc758 complement\uac00 \ub418\ub2c8\uae4c(\uc65c?) \uc2e4\uc81c\ub85c \uacf5\uc9d1\ud569\uc774 \ub418\uaca0\uc9c0\uc694. \uadf8\ub9ac\uace0 \uadf8 limit point\ub294 \uacf5\uc9d1\ud569\uc5d0 \ud3ec\ud568\ub418\uac8c \ub418\uc5b4\uc11c \ubaa8\uc21c\uc774 \ub418\uc9c0\uc694. <\/p>\n<\/div>\n<\/div>\n

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\uc9c8\ubb38<\/h2>\n
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Q: \uad50\uc218\ub2d8, \uadfc\ub370 \uc608\ub85c \ub4dc\uc2e0 (0,1] \uc740 compact\uac00 \uc544\ub2c8\uc9c0 \uc54a\ub098\uc694? \uadf8\ub9ac\uace0, Gn\uc740 open, Fn\uc740 closed \uc5ec\uc57c \ud558\ub294\ub370, \uc704\uc758 \uc608\uc5d0\uc11c\ub294 \uadf8\ub807\uc9c0 \uc54a\uc544\uc694. Fn\uc774 closed\uac00 \uc544\ub2cc \uacbd\uc6b0\uc5d0\ub294 limit point\uac00 \uaf2d Fn\uc704\uc5d0 \uc788\ub2e4\ub294 \ubcf4\uc7a5\uc774 \uc5c6\uae30 \ub54c\ubb38\uc5d0, \ubaa8\uc21c\uc744 \uc774\ub04c\uc5b4\ub0bc \uc218 \uc5c6\uc744 \uac83 \uac19\uc544\uc694. (Delete me) <\/p>\n

A: \ub9de\uc544\uc694. \uadf8\ub7ec\ub2c8\uae4c \uba85\uc81c\uc758 \ub300\uc6b0\ub97c \ubcf4\uc778 \uac83\uc774 \ub418\ub098\uc694? compact\uac00 \uc544\ub2c8\uba74 limit point\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0\ub3c4 \uc788\ub2e4\ub294 \uac83\uc744 \ubcf4\uc774\ub294 \uac83\uc774\ub2c8\uae4c… compact\uac00 \uc544\ub2cc \uc608\ub85c (0,1]\uc744 \uc7a1\ub294 \uac83\uc774\uc9c0\uc694. \uc774 \uc608\uc5d0\uc11c \ud2b9\uc218\ud55c $ G_n $ \uc5d0\uc11c \uc5b4\ub5bb\uac8c (0,1] \uc548\uc5d0 limit point\uac00 \uc5c6\uac8c \ub418\ub294\uac00\ub97c \uc0dd\uac01\ud574\uc11c \uadf8\ub300\ub85c \uc77c\ubc18\uc801\uc778 $ G_n $ \uc758 \uacbd\uc6b0\uc5d0 \uc801\uc6a9\ud558\uba74 \uc99d\uba85\uc778 \ub418\ub294 \uac83\uc774\uc9c0\uc694. – \uae40\uc601\uc6b1 <\/p>\n

Q: \uad50\uc218\ub2d8 \uc5ec\uae30 \uc9e4\ub824\uc11c \ubc11\ubd80\ubd84\uc774 \uc548\ubcf4\uc5ec\uc694..\u3161\u315c(2006.7.17.\uae40\uc775\ud658) <\/p>\n

A: \ud760, \uc774\uac74 Internet Explorer\uc758 bug \uc774\uad70\uc694. Mozilla Firefox\uc5d0\uc11c\ub294 \uc798 \ubcf4\uc774\ub294\ub370… (\uae40\uc601\uc6b1) Q: ”’2\uc7a5 21\ubc88”’\uc774 \ucc38 \uac04\ub2e8\ud55c\uac70\uac19\uc740\ub370 \uba38\ub9ac\uac00 \ub098\ube60\uc11c\uadf8\ub7f0\uc9c0 \uc548\ub418\ub124\uc694. 21\ubc88\uc5d0 (a)\ubc88 \ud480\ub54c $ \\overline{A}_0∩ B_0=\\varnothing $ \ub77c\uace0 \uac00\uc815\ud55c\ub4a4\uc5d0 \uc774 \uc9d1\ud569\uc5d0 \uc18d\ud558\ub294 \uc6d0\uc18c $ x $ \uc5d0 \ub300\ud574 $ p(x) $ \uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc74c\uc744 separated \uc131\uc9c8\uc744 \uc774\uc6a9\ud574\uc11c \uc99d\uba85\ud574\uc57c\ub420\uac70\uac19\uc740\ub370 $ p(t) $ \uac00 \uc5f0\uc18d\uc784\uc744 \uc4f0\ub294\uac83 \uc678\uc5d0\ub294 \ubc29\ubc95\uc774 \uc0dd\uac01 \uc548\ub098\ub124\uc694. \ub2e4\ub978 \ud480\uc774 \uac00\uc9c0\uace0\uacc4\uc2e0\ubd84 \uacc4\uc138\uc694? (2006. 7. 7. \uae40\uc775\ud658) <\/p>\n

A: \ubb3c\ub860 $ p(t) $ \uac00 \uc5f0\uc18d\uc784\uc744 \uc4f0\ub294 \uac83\uc774\uc9c0\ub9cc \uc798 \ubcf4\uba74 $ p(t) $ \ub294 \uc77c\ucc28\ud568\uc218\ub85c \uc815\uc758\ub418\uc5b4 \uc788\uc73c\ub2c8\uae4c \uad73\uc774 \uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8\uc744 \uc4f0\uc9c0 \uc54a\uc544\ub3c4 \ud560 \uc218 \uc788\uc744\uac70\uc608\uc694. limit point\ub098 supremum, infimum\uacfc \uac19\uc740 \uac83\ub4e4\uc744 \uc0ac\uc6a9\ud558\uba74 \ub420\uac70\uc608\uc694. \ub450 \uc810 a, b\ub97c \uace0\uc815\uc2dc\ud0a4\uace0 \uadf8\ub9bc\uc744 \uadf8\ub824 \ub193\uace0 \uc0dd\uac01\ud574 \ubcf4\uc138\uc694. \ud480\ub9ac\uba74 \uc5ec\uae30\uc5d0 \ub2f5\uc744 \uc801\uc5b4\uc8fc\uc138\uc694. – \uae40\uc601\uc6b1 <\/p>\n

\uc800\ub294 $ p(t)$ \uac00 \uc5f0\uc18d\uc784\uc744 \uc4f0\ub294 \ud480\uc774 \ubc16\uc5d0 \ubaa8\ub974\uaca0\ub124\uc694. <\/p>\n

(pf) Suppose that Ao and Bo are not separated. <\/p>\n

Then \u2203x s.t. x \u2208 Ao \u2229 (the closure of Bo) or x \u2208 (the closure of Ao) \u2229 Bo. <\/p>\n

WLOG, suppose that x \u2208 Ao \u2229 (the closure of Bo). <\/p>\n

Then x \u2208 Ao & x \u2208 (the closure of Bo). <\/p>\n

Then x \u2208 Ao & x \u2208 Bo or x \u2208 Ao & x \u2208 Bo’.(\u2235(the closure of Bo) = Bo \u222a Bo’) <\/p>\n

In the former case, p(x) \u2208 A & p(x) \u2208 B. <\/p>\n

Then p(x) \u2208 A \u2229 (the closure of B). <\/p>\n

Then A & B are not separated.(contradiction) <\/p>\n

In the latter case, p(x) \u2208 A & p(x) \u2208 B’. <\/p>\n

(If x \u2208 Bo’ but p(x) is not in B’, there exists a deleted nbhd N’ centered at p(x) with radius r s.t. N’ \u2229 B = \uacf5\uc9d1\ud569. <\/p>\n

Since x is a limit point of Bo, there is no y s.t. d(p(x),p(t)) < r for all points t \u2208 Bo for which d(x,t) < y. <\/p>\n

Then p is not a continuous function. And this is a contradiction since p is a continuous function.) <\/p>\n

Thus Ao and Bo are separated. QED (2006.7.10 \uc774\ubcd1\ucc2c) <\/p>\n

A: \ubb38\uc81c\ub294 $ x∈ B_0’$ \uc77c \ub54c $ p(x)∈ B’ $ \uc784\uc744 \ubcf4\uc774\ub294 \uac83\uc774\uaca0\uc9c0\uc694? $ x∈ B_0′ $ \uc774\uba74 \ubb34\ud55c\ud788 \ub9ce\uc740 $ x_n∈ B_0 $ \uac00 \uc788\uc5b4\uc11c $ \\|x-x_n\\| < 1\/n $ \uc774 \ub420 \uc218 \uc788\ub2e4\uace0 \ud574\ub3c4 \ub418\uaca0\uc9c0\uc694? \uc774\uc81c <\/p>\n

$ \\|p(x_n)-p(x)\\|=|x_n-x|⋅\\|\\mathbf{a}-\\mathbf{b}\\| < \\|\\mathbf{a}-\\mathbf{b}\\|\/n $ <\/p>\n

(\uc774 \uc2dd\uc774 $ p $ \uac00 \uc5f0\uc18d\uc774\ub77c\ub294 \uc870\uac74\uc744 \ub300\uc2e0\ud558\ub294 \ubd80\ubd84\uc774\uc9c0\uc694. \uad6c\uccb4\uc801\uc73c\ub85c \ud568\uc218\uc2dd\uc744 \uac00\uc9c0\uace0 \uc788\uae30 \ub54c\ubb38\uc5d0 \uac00\ub2a5\ud55c \uac83\uc774\uc608\uc694. 1\ucc28\ud568\uc218\ub2c8\uae4c.) <\/p>\n

\uc774\uace0 $ p(x_n)∈ B $ \uc774\ubbc0\ub85c, $ p(x)∈ B’ $ \uc774\ub77c\uace0 \ud560 \uc218 \uc788\uc9c0\uc694? \ubb3c\ub860 \ubaa8\ub450 \uc11c\ub85c \ub2e4\ub974\ub2e4\ub294 \uac83\uc744 \uccb4\ud06c\ud558\uace0\uc694. ( $ p(x) $ \ub294 1-1 \uc774\ub2c8\uae4c.) – \uae40\uc601\uc6b1 <\/p>\n

Q: ”’2\uc7a5 24\ubc88”’\ub3c4 \uc9c8\ubb38\ud560\uaed8\uc694. \uac04\ub2e8\ud558\uac8c \uc544\uc774\ub514\uc5b4\ub9cc \uc368\ubcf4\uba74 X\uac00 \uc784\uc758\uc758 $ δ $ \uc5d0 \ub300\ud558\uc5ec \uc720\ud55c\uac1c\uc758 radius\uac00 $ δ $ \uc778 nbhd\ub85c cover\ub418\ubbc0\ub85c. $ δ=1\/n (n=1,2,3,…) $ \uc778 \ubaa8\ub4e0 nbhd\uc758 center\ub9cc \ubaa8\uc544\ub193\uace0 \ubcf4\uba74, \uc815\ud574\uc9c4 $ δ $ \uc5d0\ub300\ud574 center\ub294 \uc720\ud55c\uac1c, \uadf8\ub9ac\uace0 $ δ $ \ub294 countable\ud558\ubbc0\ub85c. \uacb0\uad6d finite \uc9d1\ud569\uc758(\uc815\ud574\uc9c4 $ δ $ \uc5d0\ub300\ud55c center) countable union( $ δ=1\/n, n=1,2,3,… $ ) \uc774\ub418\ubbc0\ub85c \uacb0\uad6d \ubaa8\ub4e0 center\uc758 \uc9d1\ud569\uc740 countable\uc774\ub418\uace0 \uc774\uac8c dense subset\uc774 \ub418\ubbc0\ub85c X\ub294 separable\ud558\ub2e4. \uc774\ub7f0\uc2dd\uc73c\ub85c \ud480\uba74 \ub418\ub294\uac74\uac00\uc694? \ud63c\uc790\ud558\ub2e4\ubcf4\ub2c8\uae50 \ubb50\uac00 \uc798\ubabb\ub418\ub294\uac74\uc9c0 \uc798\ub418\ub294\uac74\uc9c0\ub97c \uc798 \ubaa8\ub974\uaca0\ub124\uc694..(2006. 7. 7. \uae40\uc775\ud658) <\/p>\n

A: \ub9de\ub294 story\uc778 \uac83 \uac19\ub124\uc694. \ubb38\uc81c\ub294 \ub9e8 \uccab step\uc778 \uc720\ud55c\uac1c\uc758 $ δ $ ball nbhd\ub85c cover \ub418\ub294\uac00\ub97c \uc124\uba85\ud558\ub294 \uac83\uc774\uaca0\uc9c0\uc694. \ubb3c\ub860 \ub9cc\uc77c \ubb34\ud55c\uac1c\uac00 \ud544\uc694\ud558\ub2e4\uba74 \uc774 ball\ub4e4\uc758 center \uc810\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubb34\ud55c\uc9d1\ud569\uc740 limit point\ub97c \uac00\uc9c8 \uc218 \uc5c6\ub2e4\ub294 \uac83\uc744 \ubcf4\uc5ec\uc57c\uaca0\uc9c0\uc694. \uac01 \uc2a4\ud15d\uc744 \uc815\ub9ac\ud574\uc11c \uc5ec\uae30 \uc62c\ub824\uc8fc\uba74 \uc5b4\ub5a8\ub978\uc9c0? – \uae40\uc601\uc6b1 <\/p>\n<\/div>\n

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06\/07\/06\uc5d0 \uacf5\ubd80\ud55c \ub0b4\uc6a9<\/h3>\n
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Compactness<\/h4>\n
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    \n
  1. \uc77c\ucc28\uc801\uc73c\ub85c compactness\ub294 $ \\mathbb{R}^n $ \uc5d0\uc11c closed + boundedness \ub77c\uace0 \uc774\ud574\ud558\uba74 \ub41c\ub2e4. \uc774\uac83\uc73c\ub85c \uc6b0\ub9ac\uac00 \ub2e4\ub8e8\ub294 \uac04\ub2e8\ud55c \uc9d1\ud569\uc774 compact\uc778\uac00\ub97c \ub300\ubd80\ubd84 \ud310\ub2e8\ud560 \uc218 \uc788\ub2e4.<\/li>\n
  2. \uadf8\ub7ec\ub098 compact set\uacfc \uad00\ub828\ub41c \uc131\uc9c8\uc744 \uc99d\uba85\ud558\ub294\ub370\ub294 \uc774\uac83\ub9cc\uc73c\ub85c\ub294 \ucda9\ubd84\ud558\uc9c0 \uc54a\ub2e4. \ud2b9\ud788 \uc5f0\uc18d\ud568\uc218(continuous function)\uc640 \uc5f0\uad00\ub41c \uc131\uc9c8\ub4e4\uc744 \ub2e4\ub8f0 \ub54c\ub294 closed + bounded \ub77c\ub294 \uc870\uac74\uc740 \ubcc4\ub85c \uc88b\uc9c0 \uc54a\ub2e4\uace0 \ud560 \uc218 \uc788\ub2e4. \uadf8 \uc774\uc720\ub294 bounded\ub77c\ub294 \uc870\uac74\uc740 open\uc774\ub098 closed\ub77c\ub294 \uc870\uac74\uc5d0 \ube44\ud558\uc5ec \uc5f0\uc18d\ud568\uc218\uc640 \uc798 \ub9de\uc9c0 \uc54a\ub294\ub2e4. \ub530\ub77c\uc11c \uc5f0\uc18d\ud568\uc218\uc640 \uad00\ub828\ub420 \ub54c\ub294 \uc21c\uc804\ud788 open \ub610\ub294 closed \uac19\uc740 \uac1c\ub150\ub9cc\uc744 \uc0ac\uc6a9\ud55c compactness\uc758 \uc815\uc758\uac00 \ub354 \uc88b\uc73c\uba70, \uc774 \uc810\uc5d0\uc11c\ub294 open cover\ub97c \uc0ac\uc6a9\ud55c \uc815\uc758\ub97c \uc798 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub3c4\ub85d \uc5f0\uc2b5\ud560 \ud544\uc694\uac00 \uc788\ub2e4. \uc774\ub7ec\ud55c \ubaa9\uc801\uc73c\ub85c \uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758 \ubc14\ub85c \ub2e4\uc74c\uc5d0 \uc608\ub85c\uc11c \ub098\uc624\ub294 \uc9d1\ud569\ub4e4\uc774 compact\uc778\uac00 \uc544\ub2cc\uac00, \ub610, \uc5b4\uc9f8\uc11c \uadf8\ub7f0\uac00 \ub4f1\uc744 open cover\ub97c \uc368\uc11c \ud655\uc778\ud574 \ubcf4\ub294 \uc5f0\uc2b5\uc744 \ud574 \ubcf4\uc558\ub2e4.<\/li>\n
  3. compactness\ub294 \ub2e8\uc21c\ud55c \uba87 \ub9c8\ub514 \ub9d0\ub85c\uc11c \uadf8 \ud2b9\uc9d5\uc744 \uc124\uba85\ud558\uae30\uc5d0\ub294 \ub9e4\uc6b0 delicate\ud55c \uac1c\ub150\uc774\ub2e4. \uc774\ub97c \uc81c\ub300\ub85c \uc774\ud574\ud558\ub294 \uac83\uc740 \uc801\uc5b4\ub3c4 \uba87 \ub144\ub3d9\uc548 \ud574\uc11d\ud559\uacfc \uc704\uc0c1\uc218\ud559\uc744 \uacf5\ubd80\ud574\uc57c \ud55c\ub2e4\uace0 \ud560 \uc218 \uc788\ub2e4. \uc6b0\uc120 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\ub9cc \uc774\uc57c\uae30\ud55c\ub2e4\uba74 compactness\ub97c \uba87 \uac00\uc9c0 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774 \uac00\uc6b4\ub370 \uac00\uc7a5 \ub9ce\uc774 \uc0ac\uc6a9\ub418\ub294 \uac83\uc774 Bolzano-Weierstrass\uc758 \uc815\ub9ac\uc774\ub2e4. \uc989, ”’compact\uc9d1\ud569 $ K $ \uc758 \ubb34\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 \ud56d\uc0c1 limit point\ub97c \uac00\uc9c4\ub2e4.”’ (\uc774 \ub54c, \uc774 limit point\ub294 \ud56d\uc0c1 $ K $ \uc548\uc5d0 \ub193\uc778\ub2e4. \uc65c\ub0d0 \ud558\uba74 $ K $ \uac00 closed\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.) \ub530\ub77c\uc11c compactness\ub97c \uc774\uc6a9\ud558\ub294 \uac00\uc7a5 \ub9ce\uc774 \uc4f0\uc774\ub294 \ubc29\ubc95\uc740 \uc704\uc758 \uc815\ub9ac \ud615\ud0dc\uc5d0\uc11c\uc774\uace0 \uc774 \ubc16\uc5d0 nested property \ub4f1\uc774 \uc788\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
    \n

    Connectedness<\/h4>\n
    \n

    \uc544\uc9c1 \uc548 \ud588\ub2e4. <\/p>\n<\/div>\n<\/div>\n<\/div>\n

    \n

    06\/06\/29\uc5d0 \uacf5\ubd80\ud55c \ub0b4\uc6a9<\/h3>\n
    \n<\/div>\n
    \n

    2\uc7a5 2\ubc88<\/h4>\n
    \n

    Prove that the set of all algebraic numbers is countable. <\/p>\n

    \n

    \uc5b4\ub5a4 \ub300\uc0c1\uc774 countable\uc784\uc744 \ubcf4\uc774\uae30 \uc704\ud558\uc5ec \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uba74 \uc774 \ub300\uc0c1\uc9d1\ud569\uacfc \uc790\uc5f0\uc218\uc9d1\ud569 $ \\mathbb{N} $ \uc0ac\uc774\uc5d0 1\ub3001 \ub300\uc751\uc774 \uc788\uc74c\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4. \uc989, 1\ub3001 \ub300\uc751\uc744 \ub9cc\ub4e4\uc5b4\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ub098 \uadf8\ub7ec\ud55c \ubc29\ubc95 \uc678\uc5d0\ub3c4 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\uc774 \uc788\ub2e4. <\/p>\n

    ”’\uc815\ub9ac:”’ countable set\ub97c countable\uac1c union \ud558\uba74 \ub2e4\uc2dc countable set\uac00 \ub41c\ub2e4. <\/p>\n

    \uc774\uc81c \ubaa8\ub4e0 algebraic number\ub294 \uc5b4\ub5a4 \uc815\uc218\uacc4\uc218 \ub2e4\ud56d\uc2dd\uc758 \uadfc\uc774\ub2e4. \ud55c\ud3b8 \uac01\uac01\uc758 \uc815\uc218\uacc4\uc218 \ub2e4\ud56d\uc2dd\uc758 \uadfc\uc740 \uc720\ud55c\uac1c \ubfd0\uc774\ub2e4. \ub530\ub77c\uc11c algebraic numbers\uc758 \uc9d1\ud569\uc740 \ubaa8\ub4e0 \uc815\uc218\uacc4\uc218\ub2e4\ud56d\uc2dd\uc758 \uadfc\uc9d1\ud569(the set of roots)\uc758 \ud569\uc9d1\ud569\uc774\ub2e4. <\/p>\n

    \uc774\uc81c \uc815\uc218\uacc4\uc218\ub2e4\ud56d\uc2dd\uc758 \uac1c\uc218\uac00 countable\uc784\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4. \ucc45\uc758 Hint\ub294 \uc774\uac83\uc744 \uc124\uba85\ud55c \uac83\uc774\ub2e4. \uc704\uc640 \ub611\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uac01\uac01\uc758 \uc815\uc218\uacc4\uc218 \ub2e4\ud56d\uc2dd\uc5d0\uc11c <\/p>\n

    \\[ N=n+|a_0|+\\cdots+|a_n| \\] <\/p>\n

    \uc744 \uacc4\uc0b0\ud558\uba74 \uc774 \ub2e4\ud56d\uc2dd\uc740 \ucc28\uc218\uc640 \uacc4\uc218\uc808\ub300\uac12\uc758 \ucd1d\ud569\uc774 $ N $ \uc778 \ub2e4\ud56d\uc2dd\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc774\ub7ec\ud55c \ub2e4\ud56d\uc2dd\uc740 \uc720\ud55c\uac1c \ubc16\uc5d0 \uc5c6\ub2e4.(”’\uc65c \uadf8\ub7f0\uac00?”’) <\/p>\n

    \uadf8\ub7ec\ubbc0\ub85c \uc815\uc218\uacc4\uc218\ub2e4\ud56d\uc2dd\uc758 \uac1c\uc218\ub294 \uc720\ud55c\uc9d1\ud569(\ucc28\uc218\uc640 \uacc4\uc218\uc808\ub300\uac12\uc758 \ucd1d\ud569\uc774 $ N $ \uc778 \ub2e4\ud56d\uc2dd\ub4e4\uc758 \uc9d1\ud569)\uc744 countable\uac1c( $ N $ \uc744 \ub530\ub77c\uc11c) union \ud558\uc5ec \ub9cc\ub4e0 \uac1c\uc218\uc778 countable\uac1c \uc774\ub2e4. <\/p>\n<\/div>\n<\/div>\n

    \n

    2\uc7a5 2\ubc88\uc758 \uc720\uc81c<\/h4>\n
    \n

    \uc774 \ubb38\uc81c\ub294 \uc9d1\ud569\ub860\uc5d0\uc11c \uc911\uc694\ud55c \ubb38\uc81c\uc774\ubbc0\ub85c \uac19\uc740 \ubc29\ubc95\uc744 \uc368\uc11c \ud574\uacb0\ud560 \uc218 \uc788\ub294 \ubb38\uc81c\ub97c \ud55c \ub450\uac1c \uc801\uc5b4\ub461\ub2c8\ub2e4. <\/p>\n

      \n
    1. \uc2e4\uc218 \uc9c1\uc120 \uc704\uc5d0 \uc11c\ub85c \ub9cc\ub098\uc9c0 \uc54a\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc758 \uc9d1\ud569 $ \\{ (a_λ, b_λ) \\} $ \uc744 \uc7a1\uc73c\uba74 \uc774 \uad6c\uac04\uc758 \uac1c\uc218\ub294 at most countable\uac1c\uc774\ub2e4. (\ub610\ub294 \ub2e8\uc870\uc99d\uac00\uc778 \ud568\uc218\uac00 \uac16\ub294 \ubd88\uc5f0\uc18d\uc810\uc758 \uac1c\uc218\ub294 countable\uac1c\ub97c \ub118\uc9c0 \uc54a\ub294\ub2e4.)<\/li>\n
    2. \uc6b0\ub9ac\uac00 \ub9d0\ub85c \uad6c\uccb4\uc801\uc73c\ub85c \ud45c\ud604\ud560(\uc774\ub984\ubd99\uc77c) \uc218 \uc788\ub294 \uac1c\ub150\uc758 \uac1c\uc218\ub294 at most countable\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
      \n

      2\uc7a5 7\ubc88<\/h4>\n
      \n

      Let $ A_1,A_2,A_3,… $ be subsets of a metric space. (a) If $ B_n=\\bigcup_{i=1}^n A_i $ , prove that $ \\overline{B}_n=\\bigcup_{i=1}^n \\overline{A}_i $ . (b) If $ B_n=\\bigcup_{i=1}^∞ A_i $ , prove that $ \\overline{B}_n⊃\\bigcup_{i=1}^∞ \\overline{A}_i $ Show, by an example, that this inclusion can be proper. <\/p>\n

      \n

      \uc774 \ubb38\uc81c\uc758 \ud575\uc2ec\uc774 \ub418\ub294 \ubd80\ubd84\uc740 $ \\overline{B}_n⊂\\bigcup_{k=1}^n \\overline{A}_k $ \uc784\uc744 \ubcf4\uc774\ub294 \uac83\uc774\ub2e4. <\/p>\n

      $ \\overline{B}_n $ \uc758 \uc784\uc758\uc758 \ud55c \uc810 $ p $ \ub97c \uc7a1\ub294\ub2e4. \uc774 \uc810\uc740 $ B_n $ \uc758 \uc810\uc774\uac70\ub098, \uc544\ub2c8\uba74 $ B_n $ \uc758 limit point\uc774\ub2e4. $ B_n $ \uc758 \uc810\uc774\uba74 \uc27d\ub2e4. \ubb38\uc81c\ub294 limit point\uc77c \ub54c \ubfd0\uc774\ub2e4. <\/p>\n

      \uc774 \ub54c\ub294 $ p $ \ub85c \ub2e4\uac00\uac00\ub294 $ B_n $ \uc758 ( $ p $ \uac00 \uc544\ub2cc) \uc810\uc744 \ubb34\ud55c \uac1c \uc7a1\uc744 \uc218 \uc788\ub2e4. (\uc774 \uc0ac\uc2e4\uc744 limit point\uc758 \uc815\uc758\ub97c \uc368\uc11c \uc5c4\ubc00\ud558\uac8c \uc11c\uc220\ud558\uc5ec \ubcf8\ub2e4.) \uc774\uc81c $ A_k $ \uac00\uc6b4\ub370 \ud558\ub098\ub294 \uc774 \ubb34\ud55c\uac1c\uc758 \uc810 \uac00\uc6b4\ub370 \ubb34\ud55c \ud788 \ub9ce\uc740 \uc810\ub4e4\uc744 \ud3ec\ud568\ud558\uace0 \uc788\uc5b4\uc57c \ud55c\ub2e4.(\uc544\ub2c8\ub77c\uba74 \ubaa8\ub4e0 $ A_k $ \uc5d0 \uc720\ud55c\uac1c\uc529 \ubc16\uc5d0 \uc5c6\uc5b4\uc11c \uc804\uccb4\ub3c4 \uc720\ud55c\uac1c\uac00 \ub418\uace0 \ub9cc\ub2e4.) <\/p>\n

      \uc774 \ubb34\ud55c\ud788 \ub9ce\uc740 \uc810\ub4e4 \ub54c\ubb38\uc5d0 $ p $ \ub294 $ A_k $ \uc758 limit point\uc774\ub2e4. \uadf8\ub7ec\ub2c8\uae4c $ p∈ \\overline{A}_k $ \uc774\ub2e4. <\/p>\n

      ”’\uc774 \ubb38\uc81c (b)\uc758 \ubc18\ub840\ub97c \uc801\uc5b4\ubcf4\uc790”’ <\/p>\n

      proper\uc77c\ub54c\uc758 \ubc18\ub840\ub97c \uc801\uc5b4\ubcf4\ub77c\ub294 \ub9d0\uc500\uc774\uc2dc\uc8e0? – \ub9de\uc544\uc694. \uadf8\ub7ec\ub2c8\uae4c ”’\uc774 \ubb38\uc81c (b)\uc5d0\uc11c inclusion\uc774 proper\uc778 \uacbd\uc6b0\uac00 \ub418\ub294 \uc608\ub97c \uc801\uc5b4\ubcf4\uc790.”’\uac00 \ub9de\uaca0\uc8e0? <\/p>\n

      $ A_k=(1\/k,2) $ \uadf8\ub7ec\uba74 $ B=(0,2) $ \uac00 \ub418\uaca0\uc8e0. \uadf8\ub824\uba74 $ \\overline{B}=[0,2] $ \uac00 \ub418\uace0 $ \\bigcup_{k=1}^∞ \\overline{A}_k =(0,2] $ \uac00 \ub418\uaca0\uc8e0. (2006.7.8.\uae40\uc775\ud658) <\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

      = Basic Topology = \ub0b4\uc6a9 \uc694\uc57d: http:\/\/math.korea.ac.kr\/~ywkim\/courses\/2k6rudin\/rudin_ch2.pdf http:\/\/math.korea.ac.kr\/~ywkim\/courses\/2k6rudin\/rudin_ch2_compactness.pdf 06\/07\/20\uc5d0 \uacf5\ubd80\ud55c \ub0b4\uc6a9 ”’2\uc7a5 26\ubc88”’: \uc774 \ubb38\uc81c\ub294 \uc55e\uc758 23\ubc88, 24\ubc88\uc744 \uc54c\uc544\uc57c \ud55c\ub2e4\uace0 \ub418\uc5b4 \uc788\uc5b4\uc694. \uc774\ub7f4 \ub54c \uba87 \uc2a4\ud15d\uc73c\ub85c \ub098\ub204\uc5b4\uc11c \uc0dd\uac01\ud574 \ubd10\uc57c \ud574\uc694. \uc6b0\uc120 23\ubc88, 24\ubc88\uc758 \ub0b4\uc6a9\uc744 \ubcf4\uba74 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc774\ub7ec\uc774\ub7ec \ud558\uba74, \uc774\ub7ec\uc774\ub7ec\ud55c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4\uace0 \ud588\uc9c0\uc694. \uc774\ub7f0 \uacb0\uacfc\uac00 \uc131\ub9bd\ub418\ub294 \uac70\ub9ac\uacf5\uac04\uc758 \uc608\ub97c \ucc3e\uc544\ubcfc \ud544\uc694\uac00 \uc788\uc5b4\uc694. \uc801\uc5b4\ub3c4 22~24\ubc88\uc740 $ \\mathbb{R}$ \uc774\ub098 … \ub354 \uc77d\uae30<\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-3760","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"distributor_meta":false,"distributor_terms":false,"distributor_media":false,"distributor_original_site_name":"\uae40\uc601\uc6b1","distributor_original_site_url":"https:\/\/mathematicians.korea.ac.kr\/ywkim","push-errors":false,"_links":{"self":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3760","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/comments?post=3760"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3760\/revisions"}],"predecessor-version":[{"id":3761,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3760\/revisions\/3761"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3760"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3760"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3760"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}