{"id":3494,"date":"2019-03-08T12:53:34","date_gmt":"2019-03-08T03:53:34","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=3494"},"modified":"2019-10-16T09:25:51","modified_gmt":"2019-10-16T00:25:51","slug":"calculus-open-and-closed-sets","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-open-and-closed-sets\/","title":{"rendered":"\uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569"},"content":{"rendered":"

\uc9c1\uad00\uc801\uc73c\ub85c \uc5f4\ub9b0 \uc9d1\ud569\uc740 \uacbd\uacc4\uc810\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc774\uba70 \ub2eb\ud78c \uc9d1\ud569\uc740 \ubaa8\ub4e0 \uacbd\uacc4\uc810\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \uc9d1\ud569\uc774\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569\uc744 \ub17c\ub9ac\uc801\uc73c\ub85c \uc815\uc758\ud558\uace0 \uadf8\uc640 \uad00\ub828\ub41c \uc131\uc9c8\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

<\/p>\n

\uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569<\/h3>\n

\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec, \ub2eb\ud78c \uad6c\uac04 \\(I\\)\uc5d0 \uc18d\ud55c \uc218\uc5f4\uc774 \uc218\ub834\ud558\uba74 \uadf8 \uadf9\ud55c\uac12\uc740 \ub2eb\ud78c \uad6c\uac04 \\(I\\)\uc5d0 \uc18d\ud55c\ub2e4. \uc774\uac83\uc744 \uc77c\ubc18\ud654\ud558\uc5ec \ub2eb\ud78c \uc9d1\ud569\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. (\uad00\ub828 \ud3ec\uc2a4\ud2b8: \uc218\uc5f4\uc758 \uadf9\ud55c<\/a> \uc815\ub9ac 8.)<\/p>\n

\n

\uc815\uc758 1. (\ub2eb\ud78c \uc9d1\ud569)<\/span><\/p>\n

\\(F\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(F\\)\uc758 \uc6d0\uc18c\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774 \uc218\ub834\ud560 \ub54c\ub9c8\ub2e4 \uadf8 \uadf9\ud55c\uc774 \\(F\\)\uc5d0 \uc18d\ud558\uba74 \\(F\\)\ub97c \ub2eb\ud78c \uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. [\uc989 \\(F\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\ub294 \uac83\uc740 \ubaa8\ub4e0 \ud56d\uc774 \\(F\\)\uc5d0 \uc18d\ud558\uc9c0\ub9cc \uadf8 \uadf9\ud55c\uc740 \\(F\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uc218\uc5f4\uc774 \ud558\ub098 \uc774\uc0c1 \uc874\uc7ac\ud558\ub294 \uac83\uc774\ub2e4.]<\/p>\n<\/div>\n

\u2018\ub2eb\ud78c \uc9d1\ud569\u2019\uc774\ub77c\ub294 \uc6a9\uc5b4\uc5d0\uc11c \u2018\ub2eb\ud600 \uc788\ub2e4\u2019\ub77c\ub294 \ud45c\ud604\uc740 \u2018\uadf9\ud55c\uc758 \uacb0\uacfc\uac00 \uadf8 \uc9d1\ud569\uc5d0 \uc788\ub2e4\u2019\ub77c\ub294 \ub73b\uc73c\ub85c \uc0dd\uac01\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \uc5f4\ub9b0 \uc9d1\ud569\uc744 \uc815\uc758\ud558\uc790. \uc5f4\ub9b0 \uad6c\uac04\uc740 \ub05d\uc810\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uad6c\uac04\uc774\ub2e4. \ub530\ub77c\uc11c \uc11c\ub85c\uc18c\uc778 \uc720\ud55c \uac1c\uc758 \uc5f4\ub9b0 \uad6c\uac04\uc744 \ud569\uc9d1\ud569\ud558\uc5ec\ub3c4 \uadf8 \uc9d1\ud569\uc740 \ub05d\uc810\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294\ub2e4. \uc774\uac83\uc744 \uc77c\ubc18\ud654\ud558\uc5ec \uc5f4\ub9b0 \uc9d1\ud569\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc815\uc758 2. (\uc5f4\ub9b0 \uc9d1\ud569)<\/span><\/p>\n

\uc5f4\ub9b0 \uad6c\uac04\uc744 \ud569\uc9d1\ud569\ud558\uc5ec \ub9cc\ub4e4 \uc218 \uc788\ub294 \uc9d1\ud569\uc744 \uc5f4\ub9b0 \uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(G\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub77c\ub294 \uac83\uc740 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ubaa8\uc784 \\(\\left\\{ G_j \\,\\vert\\, j\\in J \\right\\}\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[G = \\bigcup_{j\\in J} G_j\\]
\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. [\uc5ec\uae30\uc11c \\(G_j\\)\ub4e4\uc740 \uc11c\ub85c\uc18c\uc77c \ud544\uc694\uac00 \uc5c6\uc73c\uba70, \\(J\\)\ub294 \uc720\ud55c\uc9d1\ud569\uc77c \ud544\uc694\uac00 \uc5c6\ub2e4.]\n<\/p>\n<\/div>\n

\\(G\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(x\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(x\\in I \\subseteq G\\)\uc778 \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uac00 \uc874\uc7ac\ud558\uba74 \\(x\\)\ub97c \\(G\\)\uc758 \ub0b4\uc810<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub0b4\uc810\uc744 \uc774\uc6a9\ud558\uba74 \uc815\uc758 2\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

\n

\\(G\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub77c\ub294 \uac83\uc740 \\(G\\)\uc758 \ubaa8\ub4e0 \uc810\uc774 \\(G\\)\uc758 \ub0b4\uc810\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<\/div>\n

\uc815\uc758\uc5d0 \uc758\ud558\uba74 \uc2e4\uc218 \uc804\uccb4 \uc9d1\ud569\uc744 \uae30\uc900\uc774 \ub418\ub294 \uc9d1\ud569\uc73c\ub85c \ub450\uc5c8\uc744 \ub54c, \\(\\varnothing\\)\uacfc \\(\\mathbb{R}\\)\ub294 \uc5f4\ub9b0 \uc9d1\ud569\uc778 \ub3d9\uc2dc\uc5d0 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \ub610\ud55c \ub450 \uc9d1\ud569
\n\\[(1,\\,4),\\,\\, (1,\\,2) \\cup (3,\\,5)\\]
\n\uc640 \uac19\uc740 \uc9d1\ud569\uc740 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0, \ub450 \uc9d1\ud569
\n\\[[2,\\,7] ,\\,\\, \\left\\{ 1,\\,3,\\,5 \\right\\}\\]
\n\uc640 \uac19\uc740 \uc9d1\ud569\uc740 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \ubc18\uba74 \ub450 \uc9d1\ud569
\n\\[(1,\\,4] ,\\,\\, \\mathbb{Q}\\]
\n\ub294 \uc5f4\ub9b0 \uc9d1\ud569\ub3c4 \uc544\ub2c8\uace0 \ub2eb\ud78c \uc9d1\ud569\ub3c4 \uc544\ub2c8\ub2e4. \uc694\ucee8\ub300 \uc5f4\ub9b0 \uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc774 \ud56d\uc0c1 \ub2eb\ud78c \uc9d1\ud569\uc778 \uac83\uc774 \uc544\ub2c8\uba70, \ub2eb\ud78c \uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc774 \ud56d\uc0c1 \uc5f4\ub9b0 \uc9d1\ud569\uc778 \uac83\ub3c4 \uc544\ub2c8\ub2e4.<\/p>\n

\ud558\uc9c0\ub9cc \uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uad00\uacc4\uac00 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569\uc758 \uad00\uacc4)<\/span><\/p>\n

\\(G\\)\uc640 \\(F\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

    \n
  1. \\(G\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\mathbb{R} \\setminus G \\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc778 \uac83\uc774\ub2e4.<\/li>\n
  2. \\(F\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\mathbb{R} \\setminus F \\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uc99d\uba85<\/p>\n

    \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc77c \ub54c \\(\\mathbb{R} \\setminus ( \\mathbb{R} \\setminus E ) = E\\)\uc774\ubbc0\ub85c, [1]\uacfc [2]\ub294 \ud45c\ud604\ub9cc \ub2e4\ub97c \ubfd0 \uc11c\ub85c \ub3d9\uce58\uc778 \uc9c4\uc220\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 [1]\ub9cc \uc99d\uba85\ud558\uc790.<\/p>\n

    \uba3c\uc800 \ud544\uc694\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(G\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uace0 \\(F = \\mathbb{R} \\setminus G\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(G = \\mathbb{R}\\)\uc774\uac70\ub098 \\(G = \\varnothing\\)\uc774\uba74 \\(F\\)\ub294 \uba85\ubc31\ud788 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \ub450 \uacbd\uc6b0 \ubaa8\ub450 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \\(\\left\\{ x_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774\uace0 \ubaa8\ub4e0 \ud56d\uc774 \\(F\\)\uc5d0 \uc18d\ud55c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(L \\notin F\\)\ub77c\uba74 \\(L \\in G\\)\uc774\ub2e4. \\(G\\)\ub294 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ud569\uc9d1\ud569\uc774\ubbc0\ub85c \\(L \\in (a,\\,b)\\)\uc778 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\epsilon =\\,\\)\\( \\min\\left\\{ L-a ,\\, b-L \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \\(\\left\\lvert x_n – L \\right\\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub54c \\(x_n \\in G\\)\uc774\ubbc0\ub85c \\(x_n \\notin F\\)\uc774\ub2e4. \uc774\uac83\uc740 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(F\\)\uc5d0 \uc18d\ud55c\ub2e4\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(L\\in F\\)\uc774\ub2e4. \uc989 \\(F\\)\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.<\/p>\n

    \ub2e4\uc74c\uc73c\ub85c \ucda9\ubd84\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(F\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(x\\in G\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[B_{1\/n}(x) = \\left( x-\\frac{1}{n} ,\\, x+ \\frac{1}{n} \\right)\\]
    \n\uc774\ub77c\uace0 \uc815\uc758\ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(F \\cap B_{1\/n}(x) \\ne \\varnothing\\)\uc774\ub77c\uba74 \\(x_n \\in F\\cap B_{1\/n} (x)\\)\ub97c \ud0dd\ud558\uc5ec \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uc774\ub54c \\(\\left\\{ x_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc740 \\(F\\)\uc5d0 \uc18d\ud558\uc9c0\ub9cc \uadf8 \uadf9\ud55c\uac12\uc740 \\(x\\in G\\)\uc774\ubbc0\ub85c \\(F\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ub77c\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
    \n\\[B_{1\/N_x} (x)= \\left( x-\\frac{1}{N_x} ,\\, x+\\frac{1}{N_x} \\right) \\subseteq G\\]
    \n\uc778 \uc790\uc5f0\uc218 \\(N_x\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(G_x = B_{1\/N_x}(x)\\)\ub85c \ub450\uba74, \uc9c0\uae08\uae4c\uc9c0 \ub17c\uc758\ud55c \uacb0\uacfc\ub294 \uc784\uc758\uc758 \\(x\\in G\\)\uc5d0 \ub300\ud558\uc5ec \\(x \\in G_x \\subseteq G\\)\uc778 \uc5f4\ub9b0 \uad6c\uac04 \\(G_x\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc774\uc81c
    \n\\[G = \\bigcup_{x\\in G} G_x\\]
    \n\uc774\ubbc0\ub85c \\(G\\)\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \uc989 \\(G\\)\ub294 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569\uc740 \ud569\uc9d1\ud569, \uad50\uc9d1\ud569 \uc5f0\uc0b0\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 2. (\uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569\uc758 \uc131\uc9c8)<\/span><\/p>\n

      \n
    1. \uc5f4\ub9b0 \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uc740 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4. \uc989 \\(\\left\\{ G_j \\,\\vert\\, j\\in J \\right\\}\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uba74
      \n\\[G = \\bigcup_{j\\in J} G_j\\]
      \n\ub3c4 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4.<\/li>\n
    2. \uc720\ud55c \uac1c\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc758 \uad50\uc9d1\ud569\uc740 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4. \uc989 \\(G_1 ,\\) \\(G_2 ,\\) \\(\\cdots ,\\) \\(G_n\\)\uc774 \ubaa8\ub450 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uba74
      \n\\[G = G_1 \\cap G_2 \\cap \\cdots \\cap G_n\\]
      \n\ub3c4 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4.<\/li>\n
    3. \ub2eb\ud78c \uc9d1\ud569\uc758 \uad50\uc9d1\ud569\uc740 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \uc989 \\(\\left\\{ F_j \\,\\vert\\, j\\in J \\right\\}\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\uba74
      \n\\[F = \\bigcap_{j\\in J} F_j\\]
      \n\ub3c4 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.<\/li>\n
    4. \uc720\ud55c \uac1c\uc758 \ub2eb\ud78c \uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uc740 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \uc989 \\(F_1 ,\\) \\(F_2 ,\\) \\(\\cdots ,\\) \\(F_n\\)\uc774 \ubaa8\ub450 \ub2eb\ud78c \uc9d1\ud569\uc774\uba74
      \n\\[F = F_1 \\cup F_2 \\cup \\cdots \\cup F_n\\]
      \n\ub3c4 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc99d\uba85<\/p>\n

      [1]\uc758 \uc99d\uba85.<\/span> \uac01 \\(G_j\\)\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc774\ubbc0\ub85c \\(G_j\\)\ub97c \ud569\uc9d1\ud569\ud558\uc5ec \uc5bb\uc740 \\(G\\) \ub610\ud55c \uc5f4\ub9b0 \uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc774\ub2e4.<\/p>\n

      [2]\uc758 \uc99d\uba85.<\/span> \\(G = \\varnothing\\)\uc774\ub77c\uba74 \uc790\uba85\ud558\ubbc0\ub85c \\(G \\ne \\varnothing\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(x\\in G\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x\\)\ub294 \\(G_1 ,\\) \\(G_2 ,\\) \\(\\cdots ,\\) \\(G_n\\) \ubaa8\ub450\uc5d0 \uc18d\ud558\ubbc0\ub85c \\(j=1,\\) \\(2,\\) \\(\\cdots ,\\) \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\[x\\in I_j \\subseteq G_j\\]\uc778 \uc5f4\ub9b0 \uad6c\uac04 \\(I_j\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(I_x = I_1 \\cap I_2 \\cap \\cdots \\cap I_n\\)\uc774\ub77c\uace0 \ud558\uba74 \\(I_x\\) \ub610\ud55c \uc5f4\ub9b0 \uad6c\uac04\uc774\uace0 \\[x\\in I_x \\subseteq G_1 \\cap G_2 \\cap \\cdots \\cap G_n = G\\]\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc989 \uc784\uc758\uc758 \\(x\\in G\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\in I_x \\subseteq G\\)\uc778 \uc5f4\ub9b0 \uad6c\uac04 \\(I_x\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub85c\uc368
      \n\\[G = \\bigcup_{x\\in G} I_x\\]
      \n\uc774\ubbc0\ub85c \\(G\\)\ub294 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

      [3]\uc758 \uc99d\uba85.<\/span> \\(F = \\varnothing\\)\uc774\ub77c\uba74 \uc790\uba85\ud558\ubbc0\ub85c \\(F \\ne \\varnothing\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(\\left\\{ x_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\uace0 \ubaa8\ub4e0 \ud56d\uc774 \\(F\\)\uc5d0 \uc18d\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\{ x_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(F_j\\)\uc5d0 \uc18d\ud558\uace0 \\(F_j\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \\(L\\in F_j\\)\uc774\ub2e4. \uc989 \\(L\\)\uc740 \ubaa8\ub4e0 \\(F_j\\)\uc5d0 \uc18d\ud558\ubbc0\ub85c \\(L \\in F\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(F\\)\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.<\/p>\n

      [4]\uc758 \uc99d\uba85.<\/span> \ub4dc\ubaa8\ub974\uac04\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\begin{align}
      \n\\mathbb{R}\\setminus F
      \n&= \\mathbb{R} \\setminus \\left( F_1 \\cup F_2 \\cup \\cdots \\cup F_n \\right) \\\\[8pt]
      \n&= \\left( \\mathbb{R} \\setminus F_1 \\right) \\cap \\left( \\mathbb{R} \\setminus F_2 \\right) \\cap \\cdots \\cap \\left( \\mathbb{R} \\setminus F_n \\right)
      \n\\end{align}\\]
      \n\uc774\uace0 \uac01 \\(\\left(\\mathbb{R} \\setminus F_j\\right)\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\mathbb{R} \\setminus F\\)\ub294 \uc720\ud55c \uac1c\uc758 \uc5f4\ub9b0 \uc9d1\ud569\uc758 \uad50\uc9d1\ud569, \uc989 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(F\\)\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n

      \ub2eb\ud78c \uc9d1\ud569\uc758 \uc131\uc9c8<\/h3>\n

      \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4 \uc911 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774 \uc874\uc7ac\ud560 \ub54c \\(\\lambda\\)\ub97c \\(\\left\\{ x_n \\right\\}\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uac1c\ub150\uc744 \uc9d1\ud569\uc5d0 \uc801\uc6a9\ud558\uc5ec \uc9d1\ud569\uc758 \uc9d1\uc801\uc810\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

      \n

      \uc815\uc758 3. (\uc9d1\uc801\uc810\uacfc \ub3c4\uc9d1\ud569)<\/span><\/p>\n

      \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

        \n
      1. \\(\\lambda\\in\\mathbb{R}\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\in E\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < \\left\\lvert \\lambda-x \\right\\rvert < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(\\lambda\\)\ub97c \\(E\\)\uc758 \uc9d1\uc801\uc810<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n
      2. \\(E\\)\uc758 \uc9d1\uc801\uc810\ub4e4\uc758 \ubaa8\uc784\uc744 \\(E\\)\uc758 \ub3c4\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70 \\(E ‘ \\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

        \uc218\uc5f4\uc758 \uc9d1\uc801\uc810\uacfc \uc9d1\ud569\uc758 \uc9d1\uc801\uc810\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uad00\uacc4\ub97c \uac00\uc9c4\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 3. (\uc218\uc5f4\uc758 \uc9d1\uc801\uc810\uacfc \uc9d1\ud569\uc758 \uc9d1\uc801\uc810\uc758 \uad00\uacc4)<\/span><\/p>\n

        \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(\\lambda\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\lambda\\)\uac00 \\(E\\)\uc758 \uc9d1\uc801\uc810\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubaa8\ub4e0 \ud56d\uc774 \\(E \\setminus \\left\\{ \\lambda \\right\\}\\)\uc5d0 \uc18d\ud558\uace0 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \uba3c\uc800 \ud544\uc694\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(\\lambda\\)\uac00 \\(E\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c \\(\\frac{1}{n} > 0\\)\uc774\ubbc0\ub85c
        \n\\[0 < \\left\\lvert \\lambda - x_n \\right\\rvert < \\frac{1}{n} \\]\n\uc778 \\(x_n \\in E\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(\\left\\{ x_n \\right\\}\\)\uc740 \ubaa8\ub4e0 \ud56d\uc774 \\(E \\setminus \\left\\{ \\lambda \\right\\}\\)\uc5d0 \uc18d\ud558\uace0 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774\ub2e4.<\/p>\n

        \ub2e4\uc74c\uc73c\ub85c \ucda9\ubd84\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(\\left\\{ x_n \\right\\}\\)\uc774 \ubaa8\ub4e0 \ud56d\uc774 \\(E \\setminus \\left\\{ \\lambda \\right\\}\\)\uc5d0 \uc18d\ud558\uace0 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\left\\lvert x_n – \\lambda \\right\\rvert < \\epsilon\\)\uc778 \ud56d \\(x_n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(x_n \\ne \\lambda\\)\uc774\ubbc0\ub85c \\(0 < \\,\\)\\(\\left\\lvert x_n - \\lambda \\right\\rvert < \\epsilon\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\lambda\\)\ub294 \\(E\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4.\n<\/span><\/p>\n<\/div>\n

        \n

        \uc815\ub9ac 4. (\ub2eb\ud78c \uc9d1\ud569\uacfc \ub3c4\uc9d1\ud569\uc758 \uad00\uacc4)<\/span><\/p>\n

        \\(F\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(F\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc790\uc2e0\uc758 \uc9d1\uc801\uc810\uc744 \ubaa8\ub450 \uc6d0\uc18c\ub85c \uac16\ub294 \uac83, \uc989 \\(F ‘ \\subseteq F\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \uba3c\uc800 \ud544\uc694\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(F\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\lambda\\)\uac00 \\(F\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubaa8\ub4e0 \ud56d\uc774 \\(F \\setminus \\left\\{ \\lambda \\right\\}\\)\uc5d0 \uc18d\ud558\uace0 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(F\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \ubaa8\ub4e0 \ud56d\uc774 \\(F\\)\uc5d0 \uc18d\ud558\ub294 \uc218\uc5f4\uc774 \uc218\ub834\ud558\uba74 \uadf8 \uadf9\ud55c\uc740 \\(F\\)\uc5d0 \uc18d\ud55c\ub2e4. \uc989 \\(\\lambda \\in F\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(F\\)\ub294 \ubaa8\ub4e0 \uc9d1\uc801\uc810\uc744 \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4.<\/p>\n

        \ub2e4\uc74c\uc73c\ub85c \ucda9\ubd84\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(F\\)\uac00 \uc790\uc2e0\uc758 \uc9d1\uc801\uc810\uc744 \ubaa8\ub450 \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\left\\{ x_n \\right\\}\\)\uc774 \ubaa8\ub4e0 \ud56d\uc774 \\(F\\)\uc5d0 \uc18d\ud558\uace0 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\lambda\\)\uac00 \\(F\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\ne \\lambda\\)\uc774\ubbc0\ub85c \uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec \\(\\lambda\\)\ub294 \\(F\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4. \ub530\ub77c\uc11c \uc815\ub9ac\uc758 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(\\lambda\\)\ub294 \\(F\\)\uc5d0 \uc18d\ud55c\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(\\left\\{ x_n \\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc740 \\(F\\)\uc5d0 \uc18d\ud55c\ub2e4. \uc989 \\(F\\)\uc758 \uc810\uc73c\ub85c \ub9cc\ub4e4\uc5b4\uc9c4 \uc218\uc5f4\uc774 \uc218\ub834\ud558\uba74 \uadf8 \uadf9\ud55c\uc774 \\(F\\)\uc5d0 \uc18d\ud558\ubbc0\ub85c \\(F\\)\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \n

        \uc815\uc758 4. (\ud3d0\ud3ec)<\/span><\/p>\n

        \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \ub2eb\ud78c \uc9d1\ud569 \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc744 \\(E\\)\uc758 \ud3d0\ud3ec<\/span>(closure) \ub610\ub294 \ub2eb\uac1c<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(\\overline{E}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \ubaa8\ub4e0 \ub2eb\ud78c \uc9d1\ud569\ub4e4\uc758 \ubaa8\uc784\uc744 \\(\\left\\{ F_i \\,\\vert\\, i\\in I \\right\\}\\)\ub77c\uace0 \ud558\uba74
        \n\\[\\overline{E} = \\bigcap _{i\\in I} F_i\\]
        \n\uc774\ub2e4.<\/p>\n<\/div>\n

        \ud3d0\ud3ec\ub294 \ub2eb\ud78c \uc9d1\ud569\ub4e4\uc758 \uad50\uc9d1\ud569\uc774\ubbc0\ub85c \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \ub610\ud55c \\(E\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\uba74 \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \ub2eb\ud78c \uc9d1\ud569 \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc740 \\(E\\) \uc790\uc2e0\uc774\ubbc0\ub85c \\(\\overline{E} = E\\)\uc774\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 5. (\ud3d0\ud3ec\uc640 \ub3c4\uc9d1\ud569\uc758 \uad00\uacc4)<\/span><\/p>\n

        \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc77c \ub54c, \\(\\overline{E} = E \\cup E ‘ \\)\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \\(E = \\varnothing\\)\uc774\uac70\ub098 \\(E = \\mathbb{R}\\)\uc778 \uacbd\uc6b0\ub294 \uc790\uba85\ud558\ubbc0\ub85c, \\(E\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \\(\\mathbb{R}\\)\ub3c4 \uc544\ub2c8\ub77c\uace0 \ud558\uc790.<\/p>\n

        \uba3c\uc800 \\(\\overline{E} \\subseteq E\\cup E ‘ \\)\uc744 \uc99d\uba85\ud558\uc790. \\(\\lambda\\in \\overline{E}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\lambda\\in E\\)\uc774\uac70\ub098 \\(\\lambda\\notin E\\)\uc774\ub2e4. \\(\\lambda\\in E\\)\uc774\uba74 \ub2f9\uc5f0\ud788 \\(\\lambda\\in E\\cup E ‘ \\)\uc774\ubbc0\ub85c, \\(\\lambda\\notin E\\)\uc778 \uacbd\uc6b0\ub9cc \uc0dd\uac01\ud558\uba74 \ub41c\ub2e4. \ub9cc\uc57d \\(\\lambda\\notin E ‘ \\)\uc774\ub77c\uba74 \\(\\lambda\\)\ub294 \\(E\\)\uc758 \uc9d1\uc801\uc810\uc774 \uc544\ub2c8\ub2e4. \ub530\ub77c\uc11c \uc9d1\uc801\uc810\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\epsilon > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\lvert \\lambda -x \\right\\rvert \\ge \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
        \n\\[B_{\\epsilon}(\\lambda) = (\\lambda – \\epsilon ,\\, \\lambda + \\epsilon )\\]
        \n\uc740 \uc5f4\ub9b0 \uc9d1\ud569\uc774\uba70 \\(E\\)\uc758 \uc6d0\uc18c\ub97c \ud558\ub098\ub3c4 \uac16\uc9c0 \uc54a\ub294\ub2e4. \\(F_\\lambda = \\mathbb{R} \\setminus B_{\\epsilon}(\\lambda )\\)\ub77c\uace0 \ud558\uba74 \uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(F_\\lambda\\)\ub294 \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\uba70 \\(\\lambda\\)\ub97c \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294\ub2e4. \\(\\overline{E}\\)\ub294 \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \ub2eb\ud78c \uc9d1\ud569 \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc774\ubbc0\ub85c \\(\\overline{E} \\subseteq F_{\\lambda}\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\lambda\\)\ub294 \\(\\overline{E}\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(\\lambda \\in E ‘\\)\uc77c \uc218\ubc16\uc5d0 \uc5c6\ub2e4. \uc774\ub85c\uc368
        \n\\[\\lambda \\in \\overline{E} \\quad \\Rightarrow \\quad \\lambda \\in E \\cup E ‘\\]
        \n\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n

        \ub2e4\uc74c\uc73c\ub85c \\(\\overline{E} \\supseteq E\\cup E ‘ \\)\uc744 \uc99d\uba85\ud558\uc790. \\(\\lambda \\in E \\cup E ‘ \\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\lambda \\in E\\)\uc774\uba74 \ub2f9\uc5f0\ud788 \\(\\lambda \\in \\overline{E}\\)\uc774\ubbc0\ub85c, \\(\\lambda \\in E ‘ \\)\uc778 \uacbd\uc6b0\ub9cc \uc0dd\uac01\ud558\uba74 \ub41c\ub2e4. \ub9cc\uc57d \\(\\lambda \\notin \\overline{E}\\)\ub77c\uba74, \\(\\mathbb{R} \\setminus \\overline{E}\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ubbc0\ub85c, \\((\\lambda – \\epsilon ,\\, \\lambda + \\epsilon ) \\subseteq (\\mathbb{R} \\setminus \\overline{E})\\)\uc778 \\(\\epsilon > 0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(\\lambda \\in E ‘ \\)\uc774\ubbc0\ub85c \ubaa8\ub4e0 \ud56d\uc774 \\(E\\)\uc5d0 \uc18d\ud558\uace0 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(x_n \\in (\\lambda – \\epsilon ,\\, \\lambda + \\epsilon )\\)\uc778 \ud56d \\(x_n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774 \ud56d \\(x_n\\)\uc740 \\(\\overline{F}\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(F\\)\uc5d0\ub3c4 \uc18d\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(\\lambda\\in\\overline{E}\\)\uc77c \uc218\ubc16\uc5d0 \uc5c6\ub2e4. \uc774\ub85c\uc368
        \n\\[\\lambda \\in E\\cup E ‘ \\quad \\Rightarrow \\quad \\lambda \\in \\overline{E}\\]
        \n\uac00 \uc99d\uba85\ub418\uc5c8\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \uc720\uacc4\uc778 \uc218\uc5f4\uc740 \uc9d1\uc801\uc810\uc744 \uac00\uc9c4\ub2e4. \uc774\uc640 \uac19\uc740 \uc131\uc9c8\uc744 \uc9d1\ud569\uc5d0\uc11c\ub3c4 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 6. (\uc9d1\ud569\uc5d0 \ub300\ud55c \ubcfc\ucc28\ub178-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac)<\/span><\/p>\n

        \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(E\\)\uac00 \uc720\uacc4\uc774\uace0 \ubb34\ud55c\uc9d1\ud569\uc774\uba74 \\(E\\)\uc758 \uc9d1\uc801\uc810\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \\(E\\)\uac00 \ubb34\ud55c\uc9d1\ud569\uc774\ubbc0\ub85c \\(E\\)\ub294 \uac00\ubd80\ubc88\uc778 \ubd80\ubd84\uc9d1\ud569\uc744 \ud3ec\ud568\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(E\\)\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uace0 \uc11c\ub85c \uacb9\uce58\ub294 \ud56d\uc774 \uc5c6\ub294 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc774\ub7ec\ud55c \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 \uc720\uacc4\uc774\ubbc0\ub85c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{ x_{n_k} \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\left\\{ x_{n_k} \\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc744 \\(\\lambda\\)\ub77c\uace0 \ud558\uc790. \\(\\left\\{ x_{n_k} \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc740 \\(E\\)\uc5d0 \uc18d\ud558\uba70, \\(\\left\\{ x_{n_k} \\right\\}\\)\uc758 \ud56d \uc911\uc5d0\uc11c \\(\\lambda\\)\uc640 \uc77c\uce58\ud558\ub294 \uac83\uc740 \ub9ce\uc544\uc57c \ud558\ub098 \uc874\uc7ac\ud558\ubbc0\ub85c \uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ x_{n_k} \\right\\}\\)\uc758 \uadf9\ud55c\uac12 \\(\\lambda\\)\ub294 \\(E\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        <\/p>\n

        \ucef4\ud329\ud2b8 \uc9d1\ud569<\/h3>\n

        \\(K\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(\\mathcal{C} = \\left\\{ \\mathcal{O}_i \\,\\vert\\, i \\in I \\right\\}\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d
        \n\\[K \\subseteq \\bigcup_{i\\in I} \\mathcal{O}_i\\]
        \n\uc774\uba74 \\(\\mathcal{C}\\)\ub97c \\(K\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c<\/span>(open covering)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\mathcal{C} ‘\\)\uc774 \\(\\mathcal{C}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(K\\)\uc758 \ub36e\uac1c\uc774\uba74 \\(\\mathcal{C} ‘\\)\uc744 \u2018\\(K\\)\ub97c \ub36e\ub294 \\(\\mathcal{C}\\)\uc758 \ubd80\ubd84\ub36e\uac1c<\/span>\u2019\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

        \n

        \uc815\uc758 5. (\ucef4\ud329\ud2b8 \uc9d1\ud569)<\/span><\/p>\n

        \\(K\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(K\\)\ub97c \ub36e\ub294 \u2018\uc784\uc758\uc758\u2019 \uc5f4\ub9b0 \ub36e\uac1c\uac00 \\(K\\)\ub97c \ub36e\ub294 \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\ub97c \uac00\uc9c0\uba74 \\(K\\)\ub97c \ucef4\ud329\ud2b8 \uc9d1\ud569<\/span> \ub610\ub294 \uc639\uace8 \uc9d1\ud569<\/span> \ub610\ub294 \uae34\ubc00 \uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

        \\(K\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\ub824\uba74 \\(K\\)\ub97c \ub36e\ub294 \ubaa8\ub4e0 \uc5f4\ub9b0 \ub36e\uac1c\uac00 \uc720\ud55c\uc778 \ubd80\ubd84\ub36e\uac1c\ub97c \uac00\uc9d0\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4. \ubc18\uba74\uc5d0 \\(K\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774 \uc544\ub2d8\uc744 \ubcf4\uc774\ub824\uba74 \uc790\uc2e0\uc740 \\(K\\)\ub97c \ub36e\uc9c0\ub9cc \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\ub294 \\(K\\)\ub97c \ub36e\uc9c0 \ubabb\ud558\ub294 \uc5f4\ub9b0 \ub36e\uac1c\uac00 \ud558\ub098 \uc774\uc0c1 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4.<\/p>\n

        \n

        \ubcf4\uae30 1.<\/span><\/p>\n

          \n
        1. \uc720\ud55c\uc9d1\ud569\uc740 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ub2e4. \\[K = \\left\\{ x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]\uc774\uace0 \\(\\mathcal{C} = \\left\\{ \\mathcal{O}_i \\,\\vert\\, i\\in I\\right\\}\\)\uac00 \\(K\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uac01 \\(x_j\\)\uc5d0 \ub300\ud558\uc5ec \\(x_j \\in \\mathcal{O}_{i_j}\\)\uc778 \\(\\mathcal{O}_{i_j}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\[\\mathcal{C} ‘ = \\left\\{ \\mathcal{O}_{j_1} ,\\, \\mathcal{O}_{j_2} ,\\, \\cdots ,\\, \\mathcal{O}_{j_n}\\right\\}\\]\uc740 \\(K\\)\ub97c \ub36e\ub294 \uc720\ud55c\ub36e\uac1c\uc774\ub2e4.<\/li>\n
        2. \\(\\mathbb{Z}\\)\ub294 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4. \\(\\mathcal{O}_z = (z-1,\\, z+1)\\)\uc774\ub77c\uace0 \ud558\uba74
          \n\\[\\mathcal{C} = \\left\\{ \\mathcal{O}_z \\,\\vert\\, z\\in \\mathbb{Z}\\right\\}\\]
          \n\ub294 \\(\\mathbb{Z}\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c\uc774\uc9c0\ub9cc \\(\\mathbb{Z}\\)\ub97c \ub36e\ub294 \uc720\ud55c\uc778 \ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n
        3. \\(E = (0,\\,1) \\cap \\mathbb{Q}\\)\ub294 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4. \uc65c\ub0d0\ud558\uba74
          \n\\[\\mathcal{O}_i = \\left( \\frac{\\pi}{i+1} ,\\, \\frac{\\pi}{i}\\right)\\]
          \n\uc774\ub77c\uace0 \ud558\uba74 \\(\\mathcal{C} = \\left\\{ \\mathcal{O}_i \\,\\vert\\, i\\in \\mathbb{N} \\right\\}\\)\uc740 \\(E\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c\uc774\uc9c0\ub9cc \\(E\\)\ub97c \ub36e\ub294 \uc720\ud55c\uc778 \ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

          \ub2e4\uc74c \uc815\ub9ac\ub294 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774 \ucef4\ud329\ud2b8\uc778\uc9c0 \uc5ec\ubd80\ub97c \uc27d\uac8c \ud310\ubcc4\ud558\ub294 \ubc29\ubc95\uc744 \uc54c\ub824\uc900\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 7. (\ud558\uc774\ub124-\ubcf4\ub810 \uc815\ub9ac)<\/span><\/p>\n

          \\(K\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(K\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(K\\)\uac00 \uc720\uacc4\uc774\uace0 \ub2eb\ud78c \uc9d1\ud569\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/p>\n

          \uba3c\uc800 \ud544\uc694\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(K\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(K\\)\uac00 \uc720\uacc4\ub77c\ub294 \uac83\uacfc \\(K\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ub77c\ub294 \uac83\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<\/p>\n

          \\(\\mathcal{C} = \\left\\{ (-n ,\\, n) \\,\\vert\\, n\\in\\mathbb{N} \\right\\}\\)\uc740 \\(\\mathbb{R}\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c\uc774\ubbc0\ub85c \\(K\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c\uc774\uae30\ub3c4 \ud558\ub2e4. \\(K\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\mathcal{C}\\)\uc758 \uc6d0\uc18c \uc911 \uc720\ud55c \uac1c\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(K\\)\ub97c \ub36e\uc744 \uc218 \uc788\ub2e4. \\(\\mathcal{C}\\)\uc758 \uc6d0\uc18c\ub294 \\((-n,\\,n)\\) \uaf34\uc774\ubbc0\ub85c, \\(\\mathcal{C}\\)\uc758 \uc6d0\uc18c \uc911 \uc720\ud55c \uac1c\ub97c \ud0dd\ud55c\ub2e4\uba74 \uadf8 \uc6d0\uc18c \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac83\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. \uadf8\uac83\uc744 \\((-m,\\,m)\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(K \\subseteq (-m,\\,m)\\)\uc774\uace0 \\(m\\)\uc740 \uc790\uc5f0\uc218\uc774\ubbc0\ub85c \\(K\\)\ub294 \uc720\uacc4\uc774\ub2e4.<\/p>\n

          \ub2e4\uc74c\uc73c\ub85c \\(K\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc790. \ub9cc\uc57d \\(K\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uba74 \\(K\\)\uc758 \uc9d1\uc801\uc810 \uc911\uc5d0\uc11c \\(K\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8 \uc9d1\uc801\uc810\uc744 \\(\\lambda\\)\ub77c\uace0 \ud558\uc790. \uc790\uc5f0\uc218 \\(i\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[\\mathcal{O}_i = \\mathbb{R} \\setminus \\left[ \\lambda – \\frac{1}{i} ,\\, \\lambda + \\frac{1}{i} \\right]\\]
          \n\uc774\ub77c\uace0 \ud558\uba74 \\(\\mathcal{C} = \\left\\{ \\mathcal{O}_i \\,\\vert\\, i\\in\\mathbb{N} \\right\\}\\)\uc740 \\(K\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c\uc774\ub2e4. \\(K\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\mathcal{C}\\)\uc758 \uc720\ud55c \uac1c\uc758 \uc6d0\uc18c
          \n\\[\\mathcal{O}_{i_1} ,\\,\\mathcal{O}_{i_2} ,\\, \\cdots ,\\, \\mathcal{O}_{i_k}\\]
          \n\uac00 \uc874\uc7ac\ud558\uc5ec \uc774 \uc6d0\uc18c\ub4e4\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \\(K\\)\ub97c \ub36e\uc744 \uc218 \uc788\ub2e4. \\(i_1,\\) \\(i_2,\\) \\(\\cdots,\\) \\(i_k\\) \uc911 \uac00\uc7a5 \ud070 \uac12\uc744 \\(i_M\\)\uc774\ub77c\uace0 \ud558\uba74 \\(K \\subseteq \\mathcal{O}_{i_M}\\) \uc989
          \n\\[K \\cap \\left[ \\lambda – \\frac{1}{i_M} ,\\, \\lambda+ \\frac{1}{i_M} \\right] = \\varnothing\\]
          \n\uc774\ub2e4. \uc774\uac83\uc740 \\(\\lambda\\)\uac00 \\(K\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(K\\)\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.<\/p>\n

          \uc774\uc81c \ucda9\ubd84\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(K\\)\uac00 \uc720\uacc4\uc774\uace0 \ub2eb\ud78c \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(K\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uba74 \uc790\uae30 \uc790\uc2e0\uc740 \\(K\\)\ub97c \ub36e\uc9c0\ub9cc \uc720\ud55c\uc778 \ubd80\ubd84\uc9d1\ud569\uc740 \\(K\\)\ub97c \ub36e\uc744 \uc218 \uc5c6\ub294 \uc5f4\ub9b0 \ub36e\uac1c \\(\\mathcal{C} = \\left\\{ \\mathcal{O}_i \\,\\vert\\, i\\in I \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(K\\)\uac00 \uc720\uacc4\uc774\ubbc0\ub85c \\(K \\subseteq [-B ,\\, B]\\)\uc778 \\(B > 0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(I_0 = [-B ,\\,B]\\)\ub97c \uc798\ub77c\uc11c \uae38\uc774\uac00 \uac19\uc740 \ub450 \uac1c\uc758 \ub2eb\ud78c \uad6c\uac04 \\([ -B ,\\, 0],\\) \\([0,\\,B]\\)\ub97c \ub9cc\ub4e4\uc5c8\uc744 \ub54c, \ub450 \ub2eb\ud78c \uad6c\uac04 \uc911 \ud558\ub098 \uc774\uc0c1\uc740 \\(K\\)\uc640 \uad50\uc9d1\ud569\ud588\uc744 \ub54c \\(\\mathcal{C}\\)\uc758 \uc720\ud55c \uac1c\uc758 \uc6d0\uc18c\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ud55c \ub2eb\ud78c \uad6c\uac04\uc744 \ud0dd\ud558\uc5ec \\(I_1 = \\left[ \\alpha_1 ,\\, \\beta_1 \\right]\\)\uc774\ub77c\uace0 \ud558\uc790. \\(I_1\\)\uc744 \uc798\ub77c\uc11c \uae38\uc774\uac00 \uac19\uc740 \ub450 \uac1c\uc758 \ub2eb\ud78c \uad6c\uac04
          \n\\[\\left[ \\alpha_1 ,\\, \\frac{\\alpha_1 + \\beta_1}{2} \\right] ,\\,\\, \\left[ \\frac{\\alpha_1 + \\beta_1}{2} ,\\, \\beta_1 \\right] \\]
          \n\uc744 \ub9cc\ub4e4\uc5c8\uc744 \ub54c, \ub450 \ub2eb\ud78c \uad6c\uac04 \uc911 \ud558\ub098 \uc774\uc0c1\uc740 \\(K\\)\uc640 \uad50\uc9d1\ud569\ud588\uc744 \ub54c \\(\\mathcal{C}\\)\uc758 \uc720\ud55c \uac1c\uc758 \uc6d0\uc18c\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ud55c \ub2eb\ud78c \uad6c\uac04\uc744 \ud0dd\ud558\uc5ec \\(I_2 = \\left[ \\alpha_2 ,\\, \\beta_2 \\right]\\)\ub77c\uace0 \ud558\uc790. \uc774\uc640 \uac19\uc740 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec[\uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec] \ucd95\uc18c\ud558\ub294 \ub2eb\ud78c \uad6c\uac04 \\(I_k\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

          \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(I_k \\cap K\\)\ub294 \\(\\mathcal{C}\\)\uc758 \uc720\ud55c \uac1c\uc758 \uc6d0\uc18c\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(x_k \\in I_k \\cap K\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\left\\{ \\alpha_n \\right\\}\\)\uacfc \\(\\left\\{ \\beta_n \\right\\}\\)\uc740 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud558\uace0, \ub450 \uc218\uc5f4\uc758 \uadf9\ud55c\uac12\uc774 \uac19\uc73c\uba70 \uc784\uc758\uc758 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha_k \\le x_k \\le \\beta_k\\)\uc774\ubbc0\ub85c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{x_n \\right\\}\\)\ub3c4 \uc218\ub834\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uac12\uc744 \\(\\lambda\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

          \\(\\left\\{x_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(K\\)\uc5d0 \uc18d\ud558\uace0 \\(K\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\lambda\\in K\\)\uc774\ub2e4. \\(\\mathcal{C}\\)\uac00 \\(K\\)\uc758 \uc5f4\ub9b0 \ub36e\uac1c\uc774\uace0 \\(\\lambda\\)\ub294 \\(K\\)\uc5d0 \uc18d\ud558\ub294 \ud55c \uc810\uc774\ubbc0\ub85c \\(\\lambda\\in\\mathcal{O}_j\\)\uc778 \\(\\mathcal{O}_j\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\mathcal{O}_j\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\lambda \\in (\\lambda – \\epsilon ,\\, \\lambda+ \\epsilon ) \\subseteq \\mathcal{O}_j\\)\uc778 \\(\\epsilon > 0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(\\left\\{ \\alpha_n \\right\\}\\)\uacfc \\(\\left\\{ \\beta_n \\right\\}\\)\uc774 \ubaa8\ub450 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c
          \n\\[\\left[\\alpha_m ,\\, \\beta_m \\right] \\subseteq (\\lambda – \\epsilon ,\\, \\lambda+\\epsilon ) \\subseteq \\mathcal{O}_j\\]
          \n\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \\(m\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc989 \\(I_m\\)\uc740 \\(\\mathcal{C}\\)\uc758 \ub2e8 \ud558\ub098\uc758 \uc6d0\uc18c\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\ubbc0\ub85c \\(I_m \\cap K\\) \ub610\ud55c \\(\\mathcal{C}\\)\uc758 \ub2e8 \ud558\ub098\uc758 \uc6d0\uc18c\uc5d0 \uc758\ud558\uc5ec \ub36e\uc778\ub2e4. \uc774\uac83\uc740 \uc784\uc758\uc758 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(I_k \\cap K\\)\uac00 \\(\\mathcal{C}\\)\uc758 \uc720\ud55c \uac1c\uc758 \uc6d0\uc18c\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(K\\)\ub97c \ub36e\ub294 \uc720\ud55c\uc778 \ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uc5f4\ub9b0 \ub36e\uac1c \\(\\mathcal{C}\\)\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc989 \\(K\\)\ub294 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          <\/p>\n

          \uade0\ub4f1\uc5f0\uc18d<\/h3>\n

          \ud568\uc218 \\(f : D \\to \\mathbb{R}\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\ub294 \uac83\uc740
          \n\\[(\\forall \\epsilon > 0)(\\exists \\delta > 0)(\\forall x \\in D )( \\lvert x-c \\rvert < \\delta \\,\\,\\rightarrow\\,\\, \\lvert f(x) - f(c) \\rvert < \\epsilon)\\]\n\uc774 \ucc38\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(\\delta\\)\ub294 \\(\\epsilon\\)\uacfc \\(c\\)\uc758 \uac12\uc5d0 \ub530\ub77c \uacb0\uc815\ub418\ub294 \uc591\uc218\uc774\ub2e4.<\/p>\n

          \uc608\ucee8\ub300 \ud568\uc218
          \n\\[f(x) = \\frac{1}{x}\\]
          \n\uc744 \uc0dd\uac01\ud558\uc790. \uc774 \ud568\uc218\ub294 \\((0,\\, \\infty )\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \\(\\epsilon = 1,\\) \\(c=10\\)\uc77c \ub54c \\(\\delta = 2\\)\ub85c \ub450\uba74 \ucda9\ubd84\ud558\ub2e4. \uadf8\ub7ec\ub098 \ub3d9\uc77c\ud55c \\(\\epsilon = 1\\)\uc5d0 \ub300\ud558\uc5ec \\(c = 1\\)\uc77c \ub54c\uc5d0\ub294 \\(\\delta = 2\\)\ub85c \ub450\uba74 \ucda9\ubd84\ud558\uc9c0 \uc54a\ub2e4. \uc774\ub54c\uc5d0\ub294 \\(\\delta = 0.5\\)\ub85c \ub450\uc5b4\uc57c \ud55c\ub2e4.<\/p>\n

          \ud558\uc9c0\ub9cc \ud568\uc218\uc5d0 \ub530\ub77c\uc11c\ub294 \uc8fc\uc5b4\uc9c4 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta\\)\uc758 \uac12\uc774 \uc815\ud574\uc9c0\uace0 \ub098\uba74 \uc810 \\(c\\)\uac00 \uc815\uc758\uc5ed \\(E\\)\uc758 \uc5b4\ub290 \uacf3\uc5d0 \uc788\ub4e0 \uc0c1\uad00 \uc5c6\uc774 \ubaa8\ub4e0 \\(c\\)\uc640 \\(x\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[ ( \\lvert x-c \\rvert < \\delta) \\,\\to\\, ( \\lvert f(x) - f(c) \\rvert < \\epsilon )\\]\n\uc774 \uc131\ub9bd\ud558\ub294 \uacbd\uc6b0\uac00 \uc788\ub2e4. \uc774\ub7ec\ud55c \ud568\uc218\ub97c \\(E\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc778 \ud568\uc218\ub77c\uace0 \ubd80\ub978\ub2e4. \uc815\ud655\ud55c \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

          \n

          \uc815\uc758 6. (\uade0\ub4f1\uc5f0\uc18d)<\/span><\/p>\n

          \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc9d1\ud569\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\left\\lvert x_1 – x_2 \\right\\rvert < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x_1 ,\\) \\(x_2 \\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\lvert f \\left( x_1 \\right) - f\\left( x_2 \\right) \\right\\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(E\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<\/div>\n

          \uc815\uc758 6\uc744 \uae30\ud638\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
          \n\\[\\left(\\forall \\epsilon > 0\\right)\\left(\\exists \\delta > 0\\right)\\left(\\forall x_1 \\in E\\right)\\left(\\forall x_2 \\in E\\right)\\left( \\left\\lvert x_1 – x_2 \\right\\rvert < \\delta \\,\\,\\rightarrow\\,\\, \\left\\lvert f\\left( x_1 \\right) - f\\left( x_2 \\right) \\right\\rvert < \\epsilon \\right)\\]\n\n\ud568\uc218\uc758 \uc2dd\uc774 \uac19\ub354\ub77c\ub3c4 \uc5b4\ub290 \uc9d1\ud569 \uc704\uc5d0\uc11c \ub530\uc9c0\ub290\ub0d0\uc5d0 \ub530\ub77c \uade0\ub4f1\uc5f0\uc18d\uc77c \uc218\ub3c4 \uc788\uace0 \uc544\ub2d0 \uc218\ub3c4 \uc788\ub2e4. \ub2e4\uc74c \uc608\ub97c \ubcf4\uc790.<\/p>\n

          \n

          \ubcf4\uae30 2.<\/span><\/p>\n

            \n
          1. \\(f\\)\uac00 \uc77c\ucc28\ud568\uc218\uc774\uace0 \\(f(x) = ax+b\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\)\ub294 \\(\\mathbb{R}\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta = \\frac{\\epsilon}{\\lvert a \\rvert}\\)\uc774\ub77c\uace0 \ub450\uba74 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n
          2. \\(f(x) = x^2\\)\uc774\uba74 \\(f\\)\ub294 \\(\\mathbb{R}\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4. \\(\\epsilon = 1\\)\uc77c \ub54c \\(\\delta > 0\\)\uac00 \ubb34\uc5c7\uc774 \uc8fc\uc5b4\uc9c0\ub4e0 \\(x_1 ,\\) \\(x_2 = x_1 + \\frac{\\delta}{2}\\)\ub97c \ucda9\ubd84\ud788 \ud06c\uac8c \ud558\uc5ec \\(\\left\\lvert f\\left( x_1 \\right) – f\\left( x_2 \\right) \\right\\rvert \\ge 1\\)\uc774 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n
          3. \\(f(x)=x^2\\)\uc774\uba74 \\(f\\)\ub294 \\([0,\\,2]\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta = \\frac{\\epsilon}{4}\\)\uc73c\ub85c \ub450\uba74 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

            \uc704 \ubcf4\uae30\uc5d0\uc11c \ubcf4\ub2e4\uc2dc\ud53c \uc5f0\uc18d\uc778 \ud568\uc218\uac00 \ubaa8\ub450 \uade0\ub4f1\uc5f0\uc18d\uc778 \uac83\uc740 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ub098 \ucef4\ud329\ud2b8 \uc9d1\ud569 \uc704\uc5d0\uc11c\ub294 \uc774\uc57c\uae30\uac00 \ub2e4\ub974\ub2e4.<\/p>\n

            \n

            \uc815\ub9ac 8. (\ucef4\ud329\ud2b8 \uc9d1\ud569 \uc704\uc5d0\uc11c\uc758 \uade0\ub4f1\uc5f0\uc18d\uc131)<\/span><\/p>\n

            \\(K\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\uace0 \\(f\\)\uac00 \\(K\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\)\ub294 \\(K\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n

            \n

            \uc99d\uba85<\/p>\n

            \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(f\\)\uac00 \\(K\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta_n = \\frac{1}{n}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\epsilon > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(K\\)\uc758 \ub450 \uc810 \\(s_n\\)\uacfc \\(t_n\\)\uc774 \uc874\uc7ac\ud558\uc5ec
            \n\\[\\left\\lvert s_n – t_n \\right\\rvert < \\delta_n \\,\\,\\,\\text{and}\\,\\,\\, \\left\\lvert f\\left( s_n \\right) - f\\left( t_n \\right) \\right\\rvert \\ge \\epsilon\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(\\left\\{ s_n \\right\\}\\)\uc774 \uc720\uacc4\uc778 \uc218\uc5f4\uc774\ubbc0\ub85c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{ s_{n_k} \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\left\\{ s_{n_k} \\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc744 \\(\\lambda\\)\ub77c\uace0 \ud558\uc790. \\(\\left\\lvert s_{n_k} - t_{n_k} \\right\\rvert < \\,\\)\\( \\delta_{n_k}\\)\uc774\uace0 \\(k\\to\\infty\\)\uc77c \ub54c \\(\\delta_{n_k} \\to 0\\)\uc774\ubbc0\ub85c \\(\\left\\{ t_{n_k}\\right\\}\\) \ub610\ud55c \\(\\lambda\\)\ub85c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc774\ub2e4. \\(K\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\lambda\\in K\\)\uc774\ub2e4.<\/p>\n

            \\(f\\)\uac00 \\(K\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(\\lambda \\in K\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(\\lambda\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \ub610\ud55c \\(k\\to\\infty\\)\uc77c \ub54c \\(s_{n_k} \\to \\lambda ,\\) \\(t_{n_k} \\to \\lambda\\)\uc774\ubbc0\ub85c
            \n\\[\\lim_{k\\to\\infty} f\\left( s_{n_k}\\right) = \\lim_{k\\to\\infty} f\\left( t_{n_k} \\right) = f(\\lambda )\\]
            \n\uc774\ub2e4. \uc774\uac83\uc740 \uc784\uc758\uc758 \\(k\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[\\left\\lvert f\\left( s_{n_k} \\right) – f\\left( t_{n_k} \\right) \\right\\rvert \\ge \\epsilon\\]
            \n\uc774\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uac00 \\(K\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.
            \n<\/span><\/p>\n<\/div>\n

            \ub2e4\uc74c \uc815\ub9ac\ub294 \uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc99d\uba85\ud560 \ub54c \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n

            \n

            \ub530\ub984\uc815\ub9ac 9.<\/span> \ud568\uc218 \\(f\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.\n<\/div>\n

            \n

            \uc99d\uba85<\/p>\n

            \ud558\uc774\ub124-\ubcf4\ub810 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\([a,\\,b]\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ubbc0\ub85c \uc815\ub9ac 8\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.
            \n<\/span><\/p>\n<\/div>\n

            <\/p>\n

            <\/p>\n

            <\/p>\n","protected":false},"excerpt":{"rendered":"

            \uc9c1\uad00\uc801\uc73c\ub85c \uc5f4\ub9b0 \uc9d1\ud569\uc740 \uacbd\uacc4\uc810\uc744 \uc6d0\uc18c\ub85c \uac16\uc9c0 \uc54a\ub294 \uc9d1\ud569\uc774\uba70 \ub2eb\ud78c \uc9d1\ud569\uc740 \ubaa8\ub4e0 \uacbd\uacc4\uc810\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \uc9d1\ud569\uc774\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569\uc744 \ub17c\ub9ac\uc801\uc73c\ub85c \uc815\uc758\ud558\uace0 \uadf8\uc640 \uad00\ub828\ub41c \uc131\uc9c8\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569 \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec, \ub2eb\ud78c \uad6c\uac04 \\(I\\)\uc5d0 \uc18d\ud55c \uc218\uc5f4\uc774 \uc218\ub834\ud558\uba74 \uadf8 \uadf9\ud55c\uac12\uc740 \ub2eb\ud78c \uad6c\uac04 \\(I\\)\uc5d0 \uc18d\ud55c\ub2e4. \uc774\uac83\uc744 \uc77c\ubc18\ud654\ud558\uc5ec \ub2eb\ud78c \uc9d1\ud569\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. (\uad00\ub828 \ud3ec\uc2a4\ud2b8: \uc218\uc5f4\uc758 \uadf9\ud55c \uc815\ub9ac 8.) \uc815\uc758 1. (\ub2eb\ud78c \uc9d1\ud569)…<\/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],"tags":[135,170,178,173,180,175,182,172,169,184,183,171,177,168,174,134,167,136,179,176,181],"class_list":["post-3494","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-bolzano-weierstrass","tag-closed-set","tag-closure","tag-cluster-point","tag-compact","tag-derived-set","tag-heine-borel","tag-inner-point","tag-open-set","tag-uniform-continuity","tag-183","tag-171","tag-177","tag-168","tag-174","tag-134","tag-167","tag-136","tag-179","tag-176","tag-181"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3494","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=3494"}],"version-history":[{"count":35,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3494\/revisions"}],"predecessor-version":[{"id":3924,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3494\/revisions\/3924"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=3494"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=3494"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=3494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}