{"id":2433,"date":"2019-03-05T03:11:33","date_gmt":"2019-03-04T18:11:33","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2433"},"modified":"2019-07-25T22:36:37","modified_gmt":"2019-07-25T13:36:37","slug":"baire-category-theorem-basic","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/baire-category-theorem-basic\/","title":{"rendered":"\ubc30\uc5b4\uc758 \ubc94\uc8fc \uc815\ub9ac"},"content":{"rendered":"

\uac00\uc0b0\uc9d1\ud569\uacfc \ube44\uac00\uc0b0\uc9d1\ud569\uc740 \uc6d0\uc18c\uc758 \uac1c\uc218\uc5d0 \ub530\ub77c \uc9d1\ud569\uc744 \ubd84\ub958\ud55c \uac83\uc774\ub2e4. \ud574\uc11d\ud559\uacfc \uc704\uc0c1\uc218\ud559\uc5d0\uc11c\ub294 \uc6d0\uc18c\uc758 \uac1c\uc218\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc6d0\uc18c\uc758 \ubd84\ud3ec \ud615\ud0dc\uae4c\uc9c0 \uace0\ub824\ud558\uc5ec \uc9d1\ud569\uc744 \ubd84\ub958\ud558\ub294\ub370 \uadf8\ub7ec\ud55c \ubd84\ub958\ubc95 \uc911 \ud558\ub098\uac00 \uc9d1\ud569\uc758 \ubc94\uc8fc\uc774\ub2e4.<\/p>\n

\n

\uc815\uc758 1.<\/span>
\n\\(X\\)\uac00 \uc704\uc0c1\uacf5\uac04\uc774\uace0 \\(E\\subseteq X\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\left(\\overline{E}\\right)^o = \\varnothing\\)\uc774\uba74 \\(E\\)\ub294 \\(X\\)\uc758 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\ub2e4<\/span>(nowhere dense)\uace0 \ub9d0\ud55c\ub2e4.\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 1.<\/span>
\n\\(\\mathbb{R}\\)\uac00 \ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\mathbb{N}\\)\uc740 \\(\\mathbb{R}\\)\uc758 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\ubcf4\uae30 2.<\/span>
\n\\(\\mathbb{Z}\\)\uac00 \ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\[\\left(\\overline{\\mathbb{N}}\\right)^o = \\mathbb{N}^o = \\mathbb{N}\\]\uc774\ubbc0\ub85c \\(\\mathbb{N}\\)\uc740 \\(\\mathbb{Z}\\)\uc758 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\ub2e4\uace0 \ud560 \uc218 \uc5c6\ub2e4<\/span>.
\n<\/span><\/p>\n<\/div>\n

\uc704 \ub450 \ubcf4\uae30\ub97c \ud1b5\ud574 \ub3d9\uc77c\ud55c \uc9d1\ud569\uc774\ub77c\ub3c4 \uc804\uccb4\uacf5\uac04\uc774 \ub2ec\ub77c\uc9c0\uba74 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc131\uc9c8\uc740 \ub2ec\ub77c\uc9c8 \uc218 \uc788\uc74c\uc744 \uc54c \uc218 \uc788\ub2e4. \ub610\ud55c \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\ub2e4\ub294 \uc131\uc9c8\uc774 \uc870\ubc00\ud558\ub2e4\ub294 \uc131\uc9c8\uc758 \ubc18\ub300\uac00 \uc544\ub2d8\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc815\uc758 2. (\uc9d1\ud569\uc758 \ubc94\uc8fc)<\/span><\/p>\n

\\(X\\)\uac00 \uc704\uc0c1\uacf5\uac04\uc774\uace0 \\(E\\subseteq X\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

    \n
  1. \\(E\\)\uac00 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0\ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub420 \uc218 \uc788\uc73c\uba74 \u2018\\(E\\)\ub294 \\(X\\)\uc5d0\uc11c \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n
  2. \\(E\\)\uac00 \\(X\\)\uc5d0\uc11c \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774 \uc544\ub2c8\uba74 \u2018\\(E\\)\ub294 \\(X\\)\uc5d0\uc11c \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n

    \ub9cc\uc57d \uc804\uccb4\uacf5\uac04\uc774 \uba85\ud655\ud558\uc5ec \uc5b8\uae09\ud560 \ud544\uc694\uac00 \uc5c6\uc744 \ub54c\uc5d0\ub294 \u2018\\(X\\)\uc5d0\uc11c \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\u2019\uacfc \u2018\\(X\\)\uc5d0\uc11c \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\u2019\uc744 \uac04\ub2e8\ud788 \u2018\uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\u2019\uacfc \u2018\uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\u2019\uc774\ub77c\uace0 \ud45c\ud604\ud558\uae30\ub3c4 \ud55c\ub2e4.\n<\/p><\/div>\n

    \n

    \ubcf4\uae30 3.<\/span>
    \n\ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04 \\(\\mathbb{Q}\\)\ub294 \\(\\mathbb{Q}\\)\uc5d0\uc11c \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74
    \n\\[\\mathbb{Q} = \\bigcup_{q\\in\\mathbb{Q}}\\left\\{q\\right\\}\\]
    \n\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.
    \n<\/span>\n<\/p>\n<\/div>\n

    \n

    \ubcf4\uae30 4.<\/span>
    \n\ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04 \\(\\mathbb{R}\\)\uc5d0\uc11c \\(\\left\\{0 \\right\\}\\)\uc740 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4. \uadf8\ub7ec\ub098 \ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04 \\(\\left\\{0\\right\\}\\)\uc5d0\uc11c \\(\\left\\{0\\right\\}\\)\uc740 \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\left\\{0\\right\\}\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \uc911 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uac83\uc740 \\(\\varnothing\\) \ubfd0\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.
    \n<\/span>\n<\/p>\n<\/div>\n

    \n

    \ubcf4\uae30 5.<\/span>
    \n\ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04 \\(\\mathbb{N}\\)\uc740 \\(\\mathbb{N}\\)\uc5d0\uc11c \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\mathbb{N}\\)\uc5d0\uc11c\uc758 \ubcf4\ud1b5\uc704\uc0c1\uc740 \uc774\uc0b0\uc704\uc0c1\uc774\ubbc0\ub85c \\(\\mathbb{N}\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \uc911 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uac83\uc740 \\(\\varnothing\\) \ubfd0\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.
    \n<\/span>\n<\/p>\n<\/div>\n

    \n

    \ucc38\uace0 1.<\/span>
    \n\\(A\\subseteq B\\)\uc774\uace0 \\(B\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\uba74 \\(A\\)\ub3c4 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(B\\)\uac00 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0\ud569\uc9d1\ud569 \\(B = \\bigcup E_i\\)\ub85c \ud45c\ud604\ub418\uba74 \\(A\\)\ub3c4 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0\ud569\uc9d1\ud569 \\(A = \\bigcup (E_i \\cap A)\\)\ub85c \ud45c\ud604\ub41c\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \ucc38\uace0 2.<\/span>
    \n\uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(E_i\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\uba74
    \n\\[E = \\bigcup_{i\\in\\mathbb{N}}E_i\\]
    \n\ub3c4 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(E_i\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ubbc0\ub85c \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569 \\(E_{ij}\\)\uc640 \\(\\mathbb{N}\\)\uc758 \uac00\uc0b0\ubd80\ubd84\uc9d1\ud569 \\(I_i\\)\uac00 \uc874\uc7ac\ud558\uc5ec
    \n\\[E_i = \\bigcup_{j\\in I_i} E_{ij}\\]
    \n\uc774\ub2e4. \uc774\ub54c
    \n\\[I = \\left\\{ (i,\\,j) \\,\\vert\\, i\\in\\mathbb{N},\\,j\\in I_i\\right\\} \\subseteq \\mathbb{N} \\times \\mathbb{N}\\]
    \n\uc774\ubbc0\ub85c \\(I\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\uace0
    \n\\[E= \\bigcup_{i\\in\\mathbb{N}} E_i = \\bigcup_{i\\in\\mathbb{N}} \\bigcup_{j\\in I_i} E_{ij} = \\bigcup_{(i,\\,j)\\in I}E_{ij}\\]
    \n\uc774\ubbc0\ub85c \\(E\\)\ub294 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \ucc38\uace0 3.<\/span>
    \n\uc81c 1 \ubc94\uc8fc\uc640 \uc81c 2 \ubc94\uc8fc\ub294 \uc704\uc0c1\uc801 \uc131\uc9c8\uc774\ub2e4. \uc989 \\(X,\\) \\(Y\\)\uac00 \uc704\uc0c1\uacf5\uac04\uc774\uace0 \\(\\phi : X \\,\\to\\,Y\\)\uac00 \uc704\uc0c1\ub3d9\ud615\uc0ac\uc0c1\uc77c \ub54c, \\(X\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(Y\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc778 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(X\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(X\\)\ub294 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0\ud569\uc9d1\ud569
    \n\\[X = \\bigcup_{i\\in I}E_i\\]
    \n\ub85c \ud45c\ud604\ub41c\ub2e4. \\(\\phi\\)\uc640 \\(\\phi^{-1}\\)\uac00 \ubaa8\ub450 \uc5f0\uc18d\uc774\ubbc0\ub85c \uac01 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(\\overline{\\phi (E_i )} = \\phi \\left(\\overline{E_i}\\right)\\)\uc774\uace0
    \n\\[\\left(\\overline{\\phi (E_i )} \\right)^o = \\phi\\left( \\left( \\overline{E_i} \\right)^o \\right) = \\phi (\\varnothing ) = \\varnothing\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370
    \n\\[Y = \\phi (X) = \\bigcup_{i\\in I} \\phi (E_i )\\]
    \n\uc774\ubbc0\ub85c \\(Y\\)\ub3c4 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0\ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \ub530\ub77c\uc11c \\(Y\\)\ub3c4 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

    \ud55c\ud3b8 \\(\\phi^{-1}\\)\ub3c4 \uc77c\ub300\uc77c \ub300\uc751\uc774\uace0 \uc5f0\uc18d\uc774\ubbc0\ub85c \uc704 \uacfc\uc815\uc5d0\uc11c \\(X\\)\uc640 \\(Y\\)\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74, \\(Y\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc77c \ub54c \\(X\\)\ub3c4 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \ucc38\uace0 4.<\/span>
    \n\\(E\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\uace0 \\(E^o = \\varnothing\\)\uc774\uba74 \\(E\\)\ub294 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \ub2eb\ud78c \uc9d1\ud569\uc758 \ud3d0\ud3ec\ub294 \uc790\uae30 \uc790\uc2e0\uc774\ubbc0\ub85c
    \n\\[\\left( \\overline{E} \\right)^o = E^o =\\varnothing\\]
    \n\uc774\ub2e4. \ub530\ub77c\uc11c \\(E\\)\ub294 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \uc774\uc81c \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774 \uc5b4\ub5a0\ud55c \ubc94\uc8fc\uc5d0 \uc18d\ud558\ub294\uc9c0 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

    \n

    \ubcf4\uc870\uc815\ub9ac 1.<\/span>
    \n\\((X,\\,\\rho)\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(\\left\\{ U_n \\right\\}\\)\uc774 \\(X\\)\uc758 \uc870\ubc00\ud55c \uc5f4\ub9b0\uc9d1\ud569\uc5f4\uc774\uba74 \\(U_n\\)\ub4e4\uc758 \uad50\uc9d1\ud569\uc740 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(U = \\bigcap U_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(x\\in X\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(x\\in U ‘ \\)\uc784\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<\/p>\n

    \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(U_1\\)\uc774 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\ubbc0\ub85c \\(U_1 \\cap B_{\\epsilon} (x)\\)\uc758 \uc6d0\uc18c \\(y_1\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(U_1 \\cap B_{\\epsilon} (x)\\)\uac00 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\epsilon_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec
    \n\\[B_{\\epsilon_1} (y_1 ) \\subseteq U_1 \\cap B_{\\epsilon} (x)\\]
    \n\uc774\ub2e4. \\(\\delta_1 := \\min \\left\\{ \\epsilon_1 \/2 ,\\,1\\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \\(U_2\\)\uac00 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\ubbc0\ub85c \\(U_2 \\cap B_{\\delta_1} (y_1 )\\)\uc758 \uc6d0\uc18c \\(y_2\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(U_2 \\cap B_{\\delta_1} (y_1 )\\)\uc774 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ubbc0\ub85c \\(\\epsilon_2 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec
    \n\\[B_{\\epsilon_2} (y_2 ) \\subseteq U_2 \\cap B_{\\delta_1} (y_1 )\\]
    \n\uc774\ub2e4. \\(\\delta_2 := \\min\\left\\{ \\epsilon_2 \/2 ,\\, 1\/2 \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

    \uc774\ub7ec\ud55c \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\ub294 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[B_{\\epsilon_{n+1}} (y_{n+1} ) \\subseteq U_{n+1} \\cap B_{\\delta_n} (y_n ) ,\\\\[6pt]
    \n\\delta_n = \\min\\left\\{ \\frac{\\epsilon _n}{2} ,\\, \\frac{1}{n} \\right\\}\\]
    \n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218\uc5f4 \\(\\left\\{ y_n \\right\\},\\) \\(\\left\\{ \\epsilon_n \\right\\},\\) \\(\\left\\{\\delta _n \\right\\}\\)\uc744 \uc5bb\ub294\ub2e4.
    \n\\[B_{\\delta_{n+1}}(y_{n+1} ) \\subseteq B_{\\epsilon_{n+1}}(y_{n+1})\\subseteq B_{\\delta_n} (y_n )\\]
    \n\uc774\ubbc0\ub85c \\(m > n\\)\uc77c \ub54c\ub9c8\ub2e4 \\(y_m \\in B_{\\delta_n} (y_n)\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\delta_n \\le 1\/n\\)\uc774\ubbc0\ub85c \\(m > n\\)\uc774\uba74 \\(\\rho(y_m ,\\,y_n )<1\/n\\)\uc774\ub2e4. \uc989 \\(\\left\\{y_n \\right\\}\\)\uc740 \ucf54\uc2dc \uc218\uc5f4\uc774\ub2e4. \\(X\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\ubbc0\ub85c \\(\\left\\{y_n \\right\\}\\)\uc740 \\(X\\)\uc758 \uc810 \\(y\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/p>\n

    \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(y\\in\\overline{B_{\\delta_n} (y_n )}\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\delta_n \\le \\epsilon_n \/2\\)\uc774\ubbc0\ub85c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[y \\in \\overline{B_{\\delta_n} (y_n )} \\subseteq \\overline{B_{\\epsilon_n \/2} (y_n )} \\subseteq B_{\\epsilon _n} (y_n ) \\subseteq U_n\\]
    \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \\(y \\in \\bigcap U_n\\)\uc774\ub2e4. \ub610\ud55c
    \n\\[y\\in B_{\\epsilon_1} (y_1 ) \\subseteq B_{\\epsilon} (x)\\]
    \n\uc774\ubbc0\ub85c \\(\\rho (x,\\,y) < \\epsilon\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\in U '\\)\uc774\ub2e4. \\(x\\)\ub294 \\(X\\)\uc758 \uc784\uc758\uc758 \uc810\uc774\ubbc0\ub85c \\(U\\)\ub294 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \n

    \uc815\ub9ac 1. (\ubc30\uc5b4\uc758 \ubc94\uc8fc \uc815\ub9ac; Baire’s category theorem)<\/span><\/p>\n

    \\(X\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uba74 \\(X\\)\ub294 \\(X\\)\uc5d0\uc11c \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(X\\)\uac00 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(X\\)\uc758 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569 \\(A_n\\)\ub4e4\uc774 \uc874\uc7ac\ud558\uc5ec
    \n\\[X = \\bigcup_{n=1}^{\\infty} A_n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[\\bigcup_{n=1}^{\\infty} \\left( \\overline{A_n} \\right)^c
    \n= \\left( \\bigcup_{n=1}^{\\infty} \\overline{A_n} \\right)^c
    \n= X^c = \\varnothing\\tag{1}\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370 \uac01 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left( \\overline{A_n} \\right)^c\\)\uc740 \\(X\\)\uc758 \uc870\ubc00\ud55c \uc5f4\ub9b0\ubd80\ubd84\uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \ubcf4\uc870\uc815\ub9ac 1\uacfc (1)\uc5d0 \uc758\ud558\uc5ec \\(\\varnothing\\)\uc740 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(X\\)\ub294 \\(X\\)\uc5d0\uc11c \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \ub530\ub984\uc815\ub9ac.<\/span>
    \n\\((X,\\,T)\\)\uac00 \uc704\uc0c1\uacf5\uac04\uc774\uace0 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc73c\ub85c \uac70\ub9ac\ud654 \uac00\ub2a5\ud558\uba74 \\(X\\)\ub294 \\(X\\)\uc5d0\uc11c \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\((X,\\,\\rho )\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(\\rho\\)\uc5d0 \uc758\ud558\uc5ec \uc720\ub3c4\ub41c \uc704\uc0c1\uc774 \\(T\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(G\\)\uac00 \\((x,\\,\\rho )\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\((X,\\,T)\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc778 \uac83\uc774\ub2e4. \ub530\ub77c\uc11c \ubc30\uc5b4\uc758 \ubc94\uc8fc \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(X\\)\ub294 \\((X,\\,T)\\)\uc5d0\uc11c \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \ubc94\uc8fc \uc815\ub9ac\ub97c \ud65c\uc6a9\ud558\uc5ec \ud574\uacb0\ud558\ub294 \ubb38\uc81c\ub4e4\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

    \\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(x\\in X\\)\uc774\uc9c0\ub9cc \\(x\\in X ‘ \\)\uc774\uba74 \\(x\\)\ub97c \\(X\\)\uc758 \uace0\ub9bd\uc810<\/span>(isolated point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(x\\)\uac00 \\(X\\)\uc758 \uace0\ub9bd\uc810\uc774\ub77c\ub294 \uac83\uc740 \\(x\\in X\\)\uc774\uace0 \\(B_{\\epsilon} ‘ (x) \\cap X = \\varnothing\\)\uc778 \\(\\epsilon > 0\\)\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n

    \n

    \uc608\uc81c 6.<\/span>
    \n\\(X\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uace0 \uace0\ub9bd\uc810\uc744 \uac16\uc9c0 \uc54a\uc73c\uba74 \\(X\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ud480\uc774<\/p>\n

    \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(X\\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uba85\ubc31\ud788 \\(X\\)\ub294 \uc720\ud55c\uc9d1\ud569\uc774 \uc544\ub2c8\ubbc0\ub85c \\(X = \\left\\{ x_n \\,\\vert\\, n\\in\\mathbb{N}\\right\\}\\)\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \\(X\\)\uac00 \uace0\ub9bd\uc810\uc744 \uac16\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\{ x_n \\right\\}\\)\uc740 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\ub2e4. \ub530\ub77c\uc11c \\(X\\)\ub294 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\uc740 \uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0\ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub418\ubbc0\ub85c \\(X\\)\ub294 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4. \ubc30\uc5b4\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(X\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.
    \n<\/span>\n<\/p>\n<\/div>\n

    \n

    \uc608\uc81c 7.<\/span>
    \n\ubc94\uc8fc \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \ubb34\ub9ac\uc218\uac00 \uc874\uc7ac\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ud480\uc774<\/p>\n

    \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \ubb34\ub9ac\uc218\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\mathbb{R}\\)\uc5d0\ub294 \uc720\ub9ac\uc218\ub9cc \uc874\uc7ac\ud558\ubbc0\ub85c \\(\\mathbb{R}\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\mathbb{R}\\)\ub294 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uace0 \uace0\ub9bd\uc810\uc744 \uac16\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(\\mathbb{R}\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(\\mathbb{R}\\)\uc5d0\ub294 \uc720\ub9ac\uc218\uac00 \uc544\ub2cc \uc6d0\uc18c\uac00 \uc874\uc7ac\ud55c\ub2e4.
    \n<\/span>\n<\/p>\n<\/div>\n

    \n

    \uc608\uc81c 8.<\/span>
    \n\uc644\ube44\uac70\ub9ac\uacf5\uac04 \\((X,\\,\\rho )\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(E\\)\uac00 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \ud568\uc218 \\(f:X \\,\\to\\,\\mathbb{R}\\)\uac00 \\(E\\) \uc704\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \\(f\\)\ub294 \\(X\\)\uc758 \uc81c 2 \ubc94\uc8fc\uc778 \ubd80\ubd84\uc9d1\ud569 \uc704\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ud480\uc774<\/p>\n

    \uac01 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[\\begin{align}
    \nU(x) &:= \\inf \\left\\{ \\sup \\left\\{ f(t) \\,\\vert\\, t\\in B_{\\epsilon} (x) \\right\\} \\,\\vert\\, \\epsilon > 0 \\right\\} ,\\\\[6pt]
    \nL(x) &:= \\sup \\left\\{ \\inf \\left\\{ f(t) \\,\\vert\\, t\\in B_{\\epsilon} (x) \\right\\} \\,\\vert\\, \\epsilon > 0 \\right\\}
    \n\\end{align}\\]
    \n\uc774\ub77c\uace0 \uc815\uc758\ud558\uc790. \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[U_n := \\left\\{ x\\in X \\,\\left\\vert\\, U(x) – L(x) < \\frac{1}{n} \\right. \\right\\}\\]\n\uc774\ub77c\uace0 \ud558\uba74 \\(U_n\\)\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \uc784\uc758\uc758 \\(x\\in U_n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\delta := \\frac{1}{n} - (U(x) - L(x))\\]\n\ub77c\uace0 \ub450\uba74, \\(\\epsilon_1 > 0\\)\uacfc \\(\\epsilon_2 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec
    \n\\[\\begin{align}
    \n\\sup \\left\\{ f(t) \\,\\vert\\, t\\in B_{\\epsilon_1} (x) \\right\\} < U(x) + \\frac{\\delta}{3} ,\\\\[6pt]\n\\inf \\left\\{ f(t) \\,\\vert\\, t\\in B_{\\epsilon_2} (x) \\right\\} > L(x) – \\frac{\\delta}{3}
    \n\\end{align}\\]
    \n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(\\epsilon := \\min \\left\\{ \\epsilon_1 ,\\, \\epsilon_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(u\\in B_{\\epsilon \/2} (x)\\)\uc5d0 \ub300\ud558\uc5ec \\(B_{\\epsilon \/2}(u) \\subseteq B_{\\epsilon} (x)\\)\uc774\ubbc0\ub85c
    \n\\[\\begin{align}
    \n\\sup \\left\\{ f(t) \\,\\vert\\, t\\in B_{\\epsilon\/2} (u) \\right\\} &\\le \\sup \\left\\{ f(t)\\,\\vert\\, t\\in B_{\\epsilon} (x) \\right\\} < U(x) + \\frac{\\delta}{3},\\\\[6pt]\n\\inf\\left\\{ f(t) \\,\\vert\\, t\\in B_{\\epsilon \/2} (u) \\right\\} &\\ge \\inf \\left\\{ f(t) \\,\\vert\\, t\\in B_{\\epsilon} (x) \\right\\} > L(x) – \\frac{\\delta}{3}
    \n\\end{align}\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub450 \ubd80\ub4f1\uc2dd\uc744 \uacb0\ud569\ud558\uba74
    \n\\[U(u) – L(u) < U(x) -L(x) + \\frac{2}{3} \\delta < \\frac{1}{n}\\]\n\uc774\ubbc0\ub85c \\(u\\in U_n\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\uac83\uc740 \\(B_{\\epsilon \/2} (x) \\subseteq U_n\\)\uc744 \uc758\ubbf8\ud558\ubbc0\ub85c \\(U_n\\)\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \ud568\uc218 \\(f\\)\uac00 \\(E\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(E\\)\ub294 \\(X\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(E \\subseteq U_n\\)\uc774\ub2e4. \uc774\uac83\uc740 \\(U_n\\)\uc774 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud55c \uc9d1\ud569\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4. \ub530\ub77c\uc11c \ubcf4\uc870\uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(\\bigcap U_n\\)\uc740 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4. \\(\\left( U_n \\right)^c\\)\ub4e4\uc740 \uc30d\ub9c8\ub2e4 \uc11c\ub85c\uc18c\uc778 \uc9d1\ud569\uc774\ubbc0\ub85c\n\\[\\left( \\bigcap U_n \\right)^c = \\bigcup \\left( U_n \\right)^c\\]\n\uc740 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

    \ub530\ub77c\uc11c \\(\\bigcap U_n\\)\uc740 \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\uace0 \\(U_n\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(f\\)\uac00 \uc5f0\uc18d\uc778 \ubaa8\ub4e0 \uc810\uc744 \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4.
    \n<\/span>\n<\/p>\n<\/div>\n

    \n

    \uc608\uc81c 9.<\/span>
    \n\ud568\uc218 \\(f : \\mathbb{R} \\,\\to\\,\\mathbb{R}\\)\uac00 \ubb34\ub9ac\uc218 \uc9d1\ud569\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uba74\uc11c \uc720\ub9ac\uc218 \uc9d1\ud569\uc5d0\uc11c\ub294 \uc5f0\uc18d\uc77c \uc218 \uc5c6\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ud480\uc774<\/p>\n

    \\(f\\)\uac00 \\(\\mathbb{Q}\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc \\(\\mathbb{R} \\setminus \\mathbb{Q}\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c\ub294 \ubd88\uc5f0\uc18d\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(\\mathbb{Q}\\)\ub294 \uc81c 1 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc774\ubbc0\ub85c \\(f\\)\uac00 \\(\\mathbb{Q}\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub824\uba74 \\(f\\)\ub294 \uc81c 2 \ubc94\uc8fc\uc758 \uc9d1\ud569\uc778 \\(\\mathbb{R} \\setminus \\mathbb{Q}\\)\uc5d0\uc11c\ub3c4 \uc5f0\uc18d\uc774 \ub418\uc5b4\uc57c \ud55c\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ub2e4.
    \n<\/span>\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

    \uac00\uc0b0\uc9d1\ud569\uacfc \ube44\uac00\uc0b0\uc9d1\ud569\uc740 \uc6d0\uc18c\uc758 \uac1c\uc218\uc5d0 \ub530\ub77c \uc9d1\ud569\uc744 \ubd84\ub958\ud55c \uac83\uc774\ub2e4. \ud574\uc11d\ud559\uacfc \uc704\uc0c1\uc218\ud559\uc5d0\uc11c\ub294 \uc6d0\uc18c\uc758 \uac1c\uc218\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc6d0\uc18c\uc758 \ubd84\ud3ec \ud615\ud0dc\uae4c\uc9c0 \uace0\ub824\ud558\uc5ec \uc9d1\ud569\uc744 \ubd84\ub958\ud558\ub294\ub370 \uadf8\ub7ec\ud55c \ubd84\ub958\ubc95 \uc911 \ud558\ub098\uac00 \uc9d1\ud569\uc758 \ubc94\uc8fc\uc774\ub2e4. \uc815\uc758 1. \\(X\\)\uac00 \uc704\uc0c1\uacf5\uac04\uc774\uace0 \\(E\\subseteq X\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\left(\\overline{E}\\right)^o = \\varnothing\\)\uc774\uba74 \\(E\\)\ub294 \\(X\\)\uc758 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\ub2e4(nowhere dense)\uace0 \ub9d0\ud55c\ub2e4. \ubcf4\uae30 1. \\(\\mathbb{R}\\)\uac00 \ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\mathbb{N}\\)\uc740 \\(\\mathbb{R}\\)\uc758 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \uc870\ubc00\ud558\uc9c0 \uc54a\ub2e4. \ubcf4\uae30 2. \\(\\mathbb{Z}\\)\uac00 \ubcf4\ud1b5\uc704\uc0c1\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\left(\\overline{\\mathbb{N}}\\right)^o =…<\/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":[53],"tags":[],"class_list":["post-2433","post","type-post","status-publish","format-standard","hentry","category-general-topology"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2433","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=2433"}],"version-history":[{"count":26,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2433\/revisions"}],"predecessor-version":[{"id":2459,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2433\/revisions\/2459"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}