{"id":4471,"date":"2020-05-11T15:57:03","date_gmt":"2020-05-11T06:57:03","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4471"},"modified":"2020-05-12T00:01:50","modified_gmt":"2020-05-11T15:01:50","slug":"counterexample-for-an-intersection-of-open-sets","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/counterexample-for-an-intersection-of-open-sets\/","title":{"rendered":"\uc5f4\ub9b0\uc9d1\ud569\uc758 \ubb34\ud55c \uad50\uc9d1\ud569 \ubc18\ub840"},"content":{"rendered":"<p>\uc5f4\ub9b0\uc9d1\ud569\ub4e4\uc758 \ud569\uc9d1\ud569\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \uc5f4\ub9b0\uc9d1\ud569\uc758 \uac1c\uc218\uac00 \ubb34\ud55c\uc77c\uc9c0\ub77c\ub3c4 \uadf8\ub4e4\uc744 \ud569\uc9d1\ud569\ud558\uc5ec \uc5bb\uc740 \uacb0\uacfc\ub294 \ud56d\uc0c1 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. (\ucc38\uace0: <a href=\"https:\/\/sasamath.com\/blog\/articles\/calculus-open-and-closed-sets\/\">\uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569<\/a>) \uadf8\ub7ec\ub098 \uc5f4\ub9b0\uc9d1\ud569\uc744 \uad50\uc9d1\ud569\ud55c \uacb0\uacfc\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2d0 \uc218\ub3c4 \uc788\ub2e4. \uadf8 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\uc804\uccb4\uc9d1\ud569\uc744 \\(\\mathbb{R}\\)\ub77c\uace0 \ud558\uace0, \uc790\uc5f0\uc218 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \uc9d1\ud569 \\(A_j\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[A_j = \\left( &#8211; \\frac{1}{j} ,\\, 1+ \\frac{1}{j} \\right).\\tag{1}\\]<br \/>\n\uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(A_j\\)\ub294 \uc5f4\ub9b0\uad6c\uac04\uc774\ubbc0\ub85c, \\(\\mathbb{R}\\)\uc5d0\uc11c\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c<br \/>\n\\[A = \\bigcap_{j=1}^{\\infty} A_j\\tag{2}\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774\uc81c<br \/>\n\\[A = [0, 1]\\tag{3}\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc790. \uc6b0\uc120 \\(x\\in[0,\\,1]\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(j\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[x\\in \\left( &#8211; \\frac{1}{j} ,\\, 1+ \\frac{1}{j} \\right) \\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[[0,\\,1] \\subseteq A\\tag{4}\\]<br \/>\n\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c (4)\uc758 \ud3ec\ud568\uad00\uacc4\uac00 \ubc18\ub300\ub85c \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790. \\(x\\notin [0,\\,1]\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x < 0\\) \ub610\ub294 \\(x > 1\\)\uc774\ub2e4. \uba3c\uc800 \\(x < 0\\)\uc778 \uacbd\uc6b0\ubd80\ud130 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<ul>\n<li>\ub9cc\uc57d \\(x < 0\\)\uc774\uba74 \\(j > -1\/x\\)\uc778 \uc790\uc5f0\uc218 \\(j\\)\ub97c \ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \\(x < -1\/j\\)\uc774\ubbc0\ub85c \\(x \\notin A_j\\)\uc774\ub2e4. \uc989 \\(A_j\\) \uc911\uc5d0\uc11c \\(x\\)\ub97c \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uac83\uc774 \uc874\uc7ac\ud558\ubbc0\ub85c, \\(A_j\\)\ub4e4\uc758 \uad50\uc9d1\ud569\uc778 \\(A\\) \ub610\ud55c \\(x\\)\ub97c \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(x > 1\\)\uc774\uba74 \\(j > 1\/(x-1)\\)\uc778 \uc790\uc5f0\uc218 \\(j\\)\ub97c \ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \\(x > 1+ (1\/j)\\)\uc774\ubbc0\ub85c \\(x\\notin A_j\\)\uc774\ub2e4. \uc989 \\(A_j\\) \uc911\uc5d0\uc11c \\(x\\)\ub97c \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uac83\uc774 \uc874\uc7ac\ud558\ubbc0\ub85c, \\(A_j\\)\ub4e4\uc758 \uad50\uc9d1\ud569\uc778 \\(A\\) \ub610\ud55c \\(x\\)\ub97c \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ul>\n<p>\uc694\ucee8\ub300 \\([0,\\,1]\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uc6d0\uc18c\ub294 \\(A\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \uadf8 \ub300\uc6b0\ub85c\uc11c \\(A\\)\uc5d0 \uc18d\ud558\ub294 \ubaa8\ub4e0 \uc6d0\uc18c\ub294 \\([0,\\,1]\\)\uc5d0 \uc18d\ud55c\ub2e4. \uc989 \\(A \\subseteq [0,\\,1]\\)\uc774\ub2e4. \uc774 \ud3ec\ud568\uad00\uacc4\uc640 (4)\ub97c \uacb0\ud569\ud558\uba74 \ub4f1\uc2dd \\(A = [0,\\,1]\\)\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\uadf8\ub807\ub2e4\uba74 \\([0,\\,1]\\)\uc740 \\(\\mathbb{R}\\)\uc5d0\uc11c \uc65c \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2d0\uae4c? \u201c\ub2eb\ud78c\uad6c\uac04\uc774\ubbc0\ub85c \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.\u201d\ub77c\uace0 \ub2f5\ud55c\ub2e4\uba74 \uc798\ubabb\ub41c \ub2f5\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \ub2eb\ud78c\uc9d1\ud569\uc774\uba74\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc77c \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4. \ub300\uc2e0 \\([0,\\,1]\\)\uc774 \uc5f4\ub9b0\uc9d1\ud569\uc758 \uc815\uc758\ub97c \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc774\uc790.<\/p>\n<p>\\([0,\\,1]\\)\uc774 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub824\uba74 \\([0,\\,1]\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \\([0,\\,1]\\)\uc758 \ub0b4\ubd80\uc810\uc774\uc5b4\uc57c \ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(1\\)\uc740 \\([0,\\,1]\\)\uc758 \uc6d0\uc18c\uc774\uc9c0\ub9cc \\([0,\\,1]\\)\uc758 \ub0b4\ubd80\uc810\uc774 \uc544\ub2c8\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(1\\)\uc744 \uc911\uc2ec\uc73c\ub85c \ud558\uace0 \ubc18\uc9c0\ub984\uc774 \uc591\uc218\uc778 \uc6d0\uc740 \uadf8 \ub0b4\ubd80\uc5d0 \ud56d\uc0c1 \\([0,\\,1]\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uc6d0\uc18c\ub97c \uac00\uc9c0\uae30 \ub54c\ubb38\uc774\ub2e4. \uc989 \\([0,\\,1]\\)\uc740 \ub0b4\ubd80\uc810\uc774 \uc544\ub2cc \uc6d0\uc18c\ub97c \uac00\uc9c0\ubbc0\ub85c \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc5f4\ub9b0\uc9d1\ud569\ub4e4\uc758 \ud569\uc9d1\ud569\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \uc5f4\ub9b0\uc9d1\ud569\uc758 \uac1c\uc218\uac00 \ubb34\ud55c\uc77c\uc9c0\ub77c\ub3c4 \uadf8\ub4e4\uc744 \ud569\uc9d1\ud569\ud558\uc5ec \uc5bb\uc740 \uacb0\uacfc\ub294 \ud56d\uc0c1 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. (\ucc38\uace0: \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569) \uadf8\ub7ec\ub098 \uc5f4\ub9b0\uc9d1\ud569\uc744 \uad50\uc9d1\ud569\ud55c \uacb0\uacfc\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2d0 \uc218\ub3c4 \uc788\ub2e4. \uadf8 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc804\uccb4\uc9d1\ud569\uc744 \\(\\mathbb{R}\\)\ub77c\uace0 \ud558\uace0, \uc790\uc5f0\uc218 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \uc9d1\ud569 \\(A_j\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790. \\(A_j = \\left( &#8211; \\frac{1}{j} ,\\, 1+ \\frac{1}{j} \\right).\\) \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(A_j\\)\ub294 \uc5f4\ub9b0\uad6c\uac04\uc774\ubbc0\ub85c, \\(\\mathbb{R}\\)\uc5d0\uc11c\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(A = \\bigcap_{j=1}^{\\infty} A_j\\) \ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(A =&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47,53],"tags":[],"class_list":["post-4471","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","category-general-topology"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4471","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=4471"}],"version-history":[{"count":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4471\/revisions"}],"predecessor-version":[{"id":4481,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4471\/revisions\/4481"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4471"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4471"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4471"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}