{"id":1695,"date":"2022-05-05T22:24:12","date_gmt":"2022-05-05T14:24:12","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1695"},"modified":"2025-02-26T11:03:32","modified_gmt":"2025-02-26T03:03:32","slug":"jordan_curve_theorem_proof_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/05\/jordan_curve_theorem_proof_mark\/","title":{"rendered":"Jordan\u66f2\u7ebf\u5b9a\u7406\u8bc1\u660e\u6ce8\u8bb0"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u4eca\u5929\u662f\u4e94\u4e00\u5047\u671f\u7684\u7b2c\u4e00\u5929, \u7531\u4e8e\u6ca1\u80fd\u4e70\u5230\u4ece\u6df1\u5733\u56de\u5bb6\u7684\u7968, \u6545\u53ea\u80fd\u628a\u4e94\u4e00\u56de\u5bb6\u7684\u65e5\u5b50\u5ef6\u8fdf\u5230\u660e\u5929\u4e86. \u8fd9\u4e2a\u4e94\u4e00\u5047\u671f\u56e0\u4e3a\u540e\u9762\u81ea\u5df1\u8fd8\u4f1a\u5f3a\u884c\u8bf7\u4e09\u5929\u5047, \u8fde\u8d77\u6765\u5927\u6982\u603b\u5171\u4f1a\u67099\u5929\u7684\u5047\u671f, \u8fd8\u662f\u9700\u8981\u597d\u597d\u5229\u7528\u7684~ \u6700\u8fd1\u770b\u5b8c\u4e86Jordan\u66f2\u7ebf\u5b9a\u7406\u7684\u8bc1\u660e, \u8bc1\u660e\u8fc7\u7a0b\u4e2d\u6709\u4e9b\u5e76\u4e0d\u662f\u5341\u5206\u663e\u7136\u7684\u8bba\u8ff0\u8fd8\u662f\u9700\u8981\u53e6\u5916\u505a\u4e00\u4e0b\u6ce8\u8bb0\u7684~<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/313906\/compactness-in-subsets-of-mathbbr2-example\">Compactness in subsets of R2 example<\/a><br \/>\n2. <a href=\"https:\/\/mathhelpboards.com\/threads\/if-the-induced-homomorphsim-is-trivial-then-the-map-is-nullhomotopic.15578\/\">If the Induced Homomorphsim is Trivial then the map is Nullhomotopic<\/a><br \/>\n3. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/77235997\">\u7d27\u81f4\u7a7a\u95f4\u4e0e\u5355\u70b9\u7d27\u5316<\/a><br \/>\n4. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/04\/Jordan\u66f2\u7ebf\u5b9a\u7406\u8bfb\u4e66\u7b14\u8bb0.pdf\">Jordan\u66f2\u7ebf\u5b9a\u7406\u8bfb\u4e66\u7b14\u8bb0<\/a><br \/>\n5. <a href=\"https:\/\/math.stackexchange.com\/questions\/639606\/locally-path-connected-implies-that-the-components-are-open\">Locally path-connected implies that the components are open<\/a><\/p>\n<p>1. \u4e66\u4e0aP218\u7b2c\u4e8c\u6bb5\u5904$i_\\pi$\u4e0e$j_\\pi$\u90fd\u662f\u96f6\u540c\u6001.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u9996\u5148\u9700\u8981\u8bc1\u660e$X = S^2 \\backslash \\gamma(A)$\u662f\u7d27\u81f4\u7684, \u90a3\u4e48\u6211\u4eec\u5c31\u9700\u8981\u8bc1\u660e\u8fd9\u6837\u4e00\u4e2a\u547d\u9898: \u7d27\u6027\u662f\u4e00\u79cd\u4e0d\u4f9d\u8d56\u4e8e\u8d85\u96c6\u7684\u62d3\u6251\u6027\u8d28. \u82e5$X$\u662f\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4, $Y \\subseteq X$, \u5219$Y$\u662f$X$\u4e2d\u7684\u7d27\u96c6\u5f53\u4e14\u4ec5\u5f53$Y$\u5728\u5b83\u7684\u5b50\u7a7a\u95f4\u62d3\u6251\u4e0a\u662f\u7d27\u7684.<br \/>\n$\\\\$ \u4e3a\u4e86\u8bc1\u660e\u4e0a\u8ff0\u547d\u9898, \u5047\u8bbe$Y$\u662f$X$\u7684\u4e00\u4e2a\u7d27\u96c6, $\\{ U_i \\}_{i \\in I}$\u662f$Y$\u5728\u5b83\u7684\u5b50\u7a7a\u95f4\u62d3\u6251\u4e0a\u7684\u4e00\u4e2a\u5f00\u8986\u76d6. \u6839\u636e\u5b50\u7a7a\u95f4\u62d3\u6251\u7684\u5b9a\u4e49, \u5bf9\u4e8e\u4efb\u610f$i \\in I$, \u5b58\u5728\u5f00\u96c6$V_i \\subseteq X$\u4f7f\u5f97$U_i $$ = V_i \\cap Y$, \u5219$\\{ V_i \\}_{i \\in I}$\u662f$Y$\u5728$X$\u4e2d\u7684\u5f00\u8986\u76d6. \u6839\u636e$Y$\u7684\u7d27\u6027, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u6709\u9650\u5b50\u8986\u76d6$\\{ V_{ik} \\}_{k = 1}^n$, \u5219$\\{ U_{ik} $$ \\}_{k = 1}^n$\u662f$\\{ U_i \\}_{i \\in I}$\u662f\u5b50\u7a7a\u95f4\u62d3\u6251\u4e0a\u4e0a\u7684\u6709\u9650\u5b50\u8986\u76d6. \u53cd\u8fc7\u6765, \u5047\u8bbe$Y \\subseteq $$ X$\u662f\u4e00\u4e2a\u7d27\u81f4\u7684\u62d3\u6251\u7a7a\u95f4, $Y$\u4e0a\u7684\u62d3\u6251\u662f\u5728$X$\u4e2d\u7684\u5b50\u7a7a\u95f4\u62d3\u6251. \u53d6$Y$\u5728$X$\u4e2d\u7684\u4e00\u4e2a\u5f00\u8986\u76d6$\\{ U_i \\}_{i \\in I}$, \u5219\u5bf9\u4e8e\u4efb\u610f$i \\in I$, \u5b9a\u4e49$Y$\u4e0a\u7684\u5f00\u96c6$V_i $$ := U_i $$ \\cap Y$, $\\{ V_i \\}_{i \\in I}$\u662f$Y$\u5728\u5b83\u7684\u5b50\u7a7a\u95f4\u62d3\u6251\u4e0a\u7684\u4e00\u4e2a\u5f00\u8986\u76d6. \u6839\u636e$Y$\u7684\u7d27\u6027, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u6709\u9650\u5b50\u8986\u76d6$\\{ V_{ik} \\}_{k = 1}^n$, \u5219$\\{ U_{ik} \\}_{k = 1}^n$\u4e3a$Y$\u5728$X$\u4e2d\u7684\u6709\u9650\u5b50\u8986\u76d6.<br \/>\n$\\\\$ \u56de\u5230\u539f\u547d\u9898\u7684\u8bc1\u660e\u4e0a\u6765: $X = S^2 \\backslash \\gamma(A)$\u662f\u7d27\u81f4\u7684. \u53d6$X$\u7684\u4efb\u610f\u4e00\u4e2a\u5f00\u8986\u76d6$\\{ U_i $$ \\}_{i \\in I_U}$, $\\gamma(A)$\u7684\u4efb\u610f\u4e00\u4e2a\u5f00\u8986\u76d6$\\{ V_i \\}_{i \\in I_V}$, \u5219$\\{ U_i \\}_{i \\in I_U} \\cup $$ \\{ V_i \\}_{i \\in I_V}$\u6784\u6210$S^2$\u7684\u4e00\u4e2a\u5f00\u8986\u76d6. \u7531\u4e8e$S^2$\u662f$E^3$\u4e2d\u7684\u7d27\u96c6\u5f53\u4e14\u4ec5\u5f53$S^2$\u5728\u5b83\u7684\u5b50\u7a7a\u95f4\u62d3\u6251\u4e0a\u662f\u7d27\u7684, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230$S^2$\u7684\u4e00\u4e2a\u6709\u9650\u5b50\u8986\u76d6$\\{ $$ U_{ik} \\}_{k = 1}^{n_U} \\cup \\{ V_{ik} \\}_{k = 1}^{n_V}$, \u4ece\u800c$\\{ U_{ik} \\}_{k = 1}^{n_U}$\u662f$X$\u7684\u4e00\u4e2a\u6709\u9650\u5b50\u8986\u76d6($X \\cap $$ \\gamma(A) = \\emptyset$), $X$\u662f$S^2$\u4e2d\u7684\u4e00\u4e2a\u7d27\u96c6. \u7531\u521a\u521a\u8bc1\u660e\u7684\u547d\u9898\u53ef\u77e5, $X$\u4e5f\u5728\u5b83\u7684\u5b50\u7a7a\u95f4\u62d3\u6251\u4e0a\u662f\u7d27\u7684.<br \/>\n$\\\\$ \u5728\u8bc1\u660e\u4e86$X$\u662f\u7d27\u81f4\u7684\u4e8b\u5b9e\u4ee5\u540e, \u4fbf\u53ef\u4ee5\u5229\u7528\u96f6\u4f26\u5f15\u7406\u77e5$i$\u96f6\u4f26. \u4efb\u53d6\u57fa\u70b9$x_0 \\in X$, \u5219$i: X \\mapsto X \\cup Y$\u8bf1\u5bfc\u57fa\u672c\u7fa4\u7684\u540c\u6001$$i_\\pi: \\pi_1(X, x_0) \\to \\pi_1(X \\cup Y, i(x_0)), \\left \\langle a \\right \\rangle \\mapsto \\left \\langle i \\circ a \\right \\rangle.$$\u7531\u4e8e$i$\u96f6\u4f26, \u5bf9\u4e8e\u4efb\u610f$\\left \\langle a \\right \\rangle \\in \\pi_1(X, x_0)$, \u6211\u4eec\u6709$$i_\\pi(\\left \\langle a \\right \\rangle) = \\left \\langle i \\circ a \\right \\rangle = \\left \\langle e_{i(x_0)} \\circ a \\right \\rangle = \\left \\langle e_{i(x_0)} \\right \\rangle = 0.$$\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>2. \u4e66\u4e0aP219\u7b2c\u4e8c\u6bb5\u5904\u9053\u8def$H \\circ a: [0, 1] \\to S^1, t \\mapsto \\frac{u(t) &#8211; v(0)}{\\| u(t) &#8211; v(0) \\|}$\u59cb\u7ec8\u5728\u4e0a\u534a\u5706\u5468\u4e2d\u8d70.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u6b64\u5904\u7684&#8221;\u59cb\u7ec8\u5728\u4e0a\u534a\u5706\u5468\u4e2d\u8d70&#8221;\u7684\u63cf\u8ff0\u6307\u7684\u662f$H \\circ a$\u7684\u50cf\u70b9\u9664\u4e86\u9700\u8981\u6ee1\u8db3\u5706\u5fc3\u4f4d\u4e8e\u539f\u70b9\u7684\u5355\u4f4d\u5706\u65b9\u7a0b\u4ee5\u5916, \u8fd8\u9700\u8981\u6ee1\u8db3$y$\u5206\u91cf\u4e0d\u5c0f\u4e8e0\u7684\u7ea6\u675f. \u56e0\u6b64\u6211\u4eec\u53ea\u9700\u8981\u8003\u8651$H \\circ a$\u7684\u50cf\u70b9\u7684$y$\u5206\u91cf\u5373\u53ef, \u7531\u4e8e$v(0) $$ = (0, -1)$, \u56e0\u6b64$u(t) &#8211; v(0)$\u7684$y$\u5206\u91cf\u4e3a$u(t)$\u7684$y$\u5206\u91cf\u52a0\u4e0a1. \u800c$u(t)$\u7684$y$\u5206\u91cf\u662f\u5728$[-1, 1]$\u5185\u7684, \u4ece\u800c$\\frac{u(t) &#8211; v(0)}{\\| u(t) &#8211; v(0) \\|}$\u7684$y$\u5206\u91cf\u603b\u662f\u4e0d\u5c0f\u4e8e0. \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>3. \u8bbe$\\gamma: S^1 \\to E^2$\u662f\u4e00\u4e2a\u5d4c\u5165, \u5219$E^2 \\backslash \\gamma(S^1)$\u5b58\u5728\u552f\u4e00\u4e00\u4e2a\u6709\u754c\u5206\u652f$U$, \u5e76\u4e14$U$\u662f\u4ee5$\\gamma(S^1)$\u4e3a\u8fb9\u754c, \u5373$\\overline{U} \\backslash U^{\\circ} = \\gamma(S^1)$, \u8fd9\u4e2a\u7ed3\u8bba\u4e0eJordan\u66f2\u7ebf\u5b9a\u7406\u662f\u5b8c\u5168\u7b49\u4ef7\u7684.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u5e73\u9762\u6dfb\u52a0\u4e00\u4e2a\u65e0\u7a77\u8fdc\u70b9(\u4e00\u70b9\u7d27\u5316) \u540e\u5c31\u662f\u7403\u9762. \u5148\u4ecb\u7ecd\u4e00\u4e0b\u4e00\u70b9\u7d27\u5316(\u672c\u6587\u53ea\u4ecb\u7ecd\u4e9a\u5386\u5c71\u5fb7\u7f57\u592b\u5355\u70b9\u7d27\u5316): &#8220;\u7d27&#8221;\u662f\u4e00\u4e2a\u5f88\u91cd\u8981\u7684\u6982\u5ff5, \u5f88\u591a\u65f6\u5019\u6211\u4eec\u9700\u8981\u8ba9\u4e00\u4e2a\u975e\u7d27\u7684\u62d3\u6251\u7a7a\u95f4\u53d8\u6210\u7d27\u7684, \u8fd9\u53ea\u9700\u8981\u5728\u539f\u6765\u7684\u7a7a\u95f4\u4e0a\u6dfb\u52a0\u4e00\u4e2a\u70b9\u5c31\u53ef\u4ee5\u505a\u5230. \u6211\u4eec\u4e4b\u524d\u4fbf\u5b66\u8fc7\u5546\u96c6\u7684\u6982\u5ff5, \u5546\u96c6\u5c31\u50cf\u662f\u80f6\u6c34\u4e00\u6837, \u53ef\u4ee5\u628a\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4\u7c98\u5408\u8d77\u6765; \u5355\u70b9\u7d27\u5316\u4e5f\u662f\u5982\u6b64, \u800c\u5b83\u6240\u6dfb\u52a0\u7684\u90a3\u4e2a\u62bd\u8c61\u7684\u65e0\u7a77\u8fdc\u70b9\u5c31\u662f\u80f6\u6c34\u7684\u7c98\u5408\u70b9. \u5728\u8ba8\u8bba\u5355\u70b9\u7d27\u5316\u65f6, \u4e00\u822c\u6709\u5927\u524d\u63d0\u2014\u2014\u8bbe\u62d3\u6251\u7a7a\u95f4$X$\u662fHausdorff\u7684\u4e14\u5c40\u90e8\u7d27\u7684.<br \/>\n$\\\\$ <strong>\u5b9a\u4e49<\/strong> \u8bbe$X$\u662f\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4, $\\widetilde{X} = X \\cup \\{ \\infty \\}$, \u5176\u4e2d$\\infty$\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u70b9. \u540c\u65f6, \u5b9a\u4e49$\\widetilde{X}$\u4e0a\u7684\u62d3\u6251$\\widetilde{\\tau}$:<br \/>\n$\\\\$ (1) \u82e5$\\infty \\notin U$, \u5219$U$\u5728$\\widetilde{X}$\u4e2d\u5f00\u5f53\u4e14\u4ec5\u5f53$U$\u5728$X$\u4e2d\u5f00.<br \/>\n$\\\\$ (2) \u82e5$\\infty \\in U$, \u5219$U$\u5728$\\widetilde{X}$\u4e2d\u5f00\u5f53\u4e14\u4ec5\u5f53$U$\u7684\u8865\u96c6\u5728$X$\u4e2d\u662f\u7d27\u7684.<br \/>\n$\\\\$ \u5355\u70b9\u7d27\u5316\u7684\u60f3\u6cd5\u662f\u5f15\u5165\u4e86\u4e00\u4e2a\u62bd\u8c61\u7684\u65e0\u7a77\u8fdc\u70b9, \u7136\u540e\u5c06\u975e\u7d27\u96c6\u7684\u8fb9\u7f18\u90e8\u5206\u548c\u8fd9\u4e2a\u65e0\u7a77\u8fdc\u70b9\u8fde\u5728\u4e00\u8d77, \u6700\u7ec8\u4f7f\u5f97\u53d8\u5f62\u540e\u7684\u7a7a\u95f4\u6210\u4e3a\u7d27\u7684. \u63a5\u4e0b\u6765\u4ecb\u7ecd\u4e00\u4e2a\u5f88\u91cd\u8981\u7684\u7684\u4e0e\u4e00\u70b9\u7d27\u5316\u76f8\u5173\u7684\u5f15\u7406.<br \/>\n$\\\\$ <strong>\u5f15\u7406<\/strong> \u8bbe$C$\u662f\u4e00\u4e2a$S^2$\u7684\u4e00\u4e2a\u7d27\u81f4\u5b50\u7a7a\u95f4, $b$\u662f$S^2 \\backslash C$\u7684\u4e00\u4e2a\u70b9, $h: S^2 $$ \\backslash \\{ b \\} \\to E^2$\u662f\u4e00\u4e2a\u540c\u80da, \u4e14$U$\u662f$S^2 \\backslash C$\u7684\u4e00\u4e2a\u5206\u652f. \u5982\u679c$b \\notin U$, \u90a3\u4e48$h(U)$\u662f$E^2 \\backslash h(C)$\u7684\u4e00\u4e2a\u6709\u754c\u5206\u652f; \u5982\u679c$b \\in U$, \u90a3\u4e48$h(U \\backslash \\{ b \\})$\u662f$E^2 \\backslash h(C)$\u7684\u4e00\u4e2a\u65e0\u754c\u5206\u652f. \u7279\u522b\u5730, \u5982\u679c$S^2 \\backslash C$\u6709$n$\u4e2a\u5206\u652f, \u90a3\u4e48$E^2 \\backslash $$ h(C)$\u5c31\u6709$n$\u4e2a\u8fde\u901a\u5206\u652f.<br \/>\n$\\\\$ \u5f15\u7406\u7684\u8bc1\u660e\u8fc7\u7a0b\u8be6\u89c1\u53c2\u8003\u6750\u65994. \u6709\u4e86\u4e0a\u8ff0\u5f15\u7406, \u6211\u4eec\u4fbf\u53ef\u4ee5\u628aJordan\u66f2\u7ebf\u5b9a\u7406\u7684\u4e0e$S^2$\u76f8\u5173\u7684\u8bc1\u660e\u8fc7\u7a0b&#8221;\u65e0\u7f1d&#8221;\u5207\u6362\u81f3\u4e0e$E^2$\u76f8\u5173\u7684\u8bc1\u660e\u8fc7\u7a0b.<\/p>\n<p>4. \u4e66\u4e0aP219\u5012\u6570\u7b2c\u4e8c\u6bb5\u5904$E^2 \\backslash \\gamma(S^1)$\u7684\u6bcf\u4e2a\u5206\u652f\u90fd\u662f\u5f00\u96c6, \u6240\u4ee5$U^\\circ $$ = U$, \u5e76\u4e14$\\overline{U}$\u4e0d\u4e0e\u5176\u5b83\u5206\u652f\u76f8\u4ea4, \u53ea\u80fd\u5305\u542b\u4e8e$\\gamma(S^1) \\cup U$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u9996\u5148\u8bc1\u660e$E^2 \\backslash \\gamma(S^1)$\u662f\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684. \u7531\u4e8e$S^2$\u662f\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684, \u6545\u6211\u4eec\u53ea\u9700\u8bc1\u660e\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7a7a\u95f4\u7684\u5f00\u5b50\u7a7a\u95f4\u662f\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684. (\u7136\u800c\u6b64\u5904\u53ea\u8bc1\u660e\u4e86\u5c40\u90e8\u8fde\u901a\u7a7a\u95f4\u7684\u5f00\u5b50\u7a7a\u95f4\u662f\u5c40\u90e8\u8fde\u901a\u7684=.= \u540e\u9762\u770b\u770b\u80fd\u4e0d\u80fd\u60f3\u51fa\u6765) \u5047\u8bbe\u62d3\u6251\u7a7a\u95f4$X$\u662f\u5c40\u90e8\u8fde\u901a\u7684, $O$\u662f$X$\u7684\u4e00\u4e2a\u5f00\u5b50\u96c6, \u4efb\u53d6$p \\in O$, \u5bf9\u4e8e\u4efb\u610f\u5305\u542b$p$\u7684\u5f00\u96c6$U $$ \\subset O$\u4e0e\u8fde\u901a\u5f00\u96c6$V \\subset $$ X$, \u5219$p \\in W = U \\cap V$, \u4ece\u800c$W$\u662f$p$\u7684\u4e00\u4e2a\u5f00\u90bb\u57df, $W \\subset $$ U \\subset O$. \u82e5$W$\u4e0d\u662f\u8fde\u901a\u7684, \u5219$W$\u53ef\u4ee5\u5199\u4e3a\u4e24\u4e2a\u65e2\u5f00\u53c8\u95ed\u7684\u96c6\u5408$A, B$\u7684\u65e0\u4ea4\u5e76, \u4ece\u800c$A \\cup B = W = U \\cap V \\subset V$. \u7531\u4e8e$V$\u662f\u8fde\u901a\u7684, $V$\u4e2d\u65e2\u5f00\u53c8\u95ed\u7684\u96c6\u5408\u53ea\u6709$\\emptyset$\u4e0e$V$\u672c\u8eab. \u6211\u4eec\u8ba8\u8bba\u5982\u4e0b\u4e24\u79cd\u60c5\u5f62:<br \/>\n$\\\\$ $\\cdot$ $A = B = \\emptyset$, \u8fd9\u4e0e$p \\in W = A \\cup B$\u77db\u76fe.<br \/>\n$\\\\$ $\\cdot$ $A = V$\u6216\u8005$B = V$, \u5219$V = A \\cup B = W = U \\cap V \\subset U \\subset O$. \u4f46$V$\u4e0e$W$\u7684\u8fde\u901a\u6027\u4e0d\u540c, \u77db\u76fe.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, \u5c40\u90e8\u8fde\u901a\u7a7a\u95f4\u7684\u5f00\u5b50\u7a7a\u95f4\u662f\u5c40\u90e8\u8fde\u901a\u7684. \u800c$S^2 \\backslash \\gamma(S^1)$\u662f\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7a7a\u95f4$S^2$\u7684\u4e00\u4e2a\u5f00\u5b50\u7a7a\u95f4, \u6545\u4e5f\u662f\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684, \u4ece\u800c$E^2 \\backslash \\gamma(S^1)$\u662f\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684.<br \/>\n$\\\\$ \u518d\u6765\u8bc1\u660e\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7a7a\u95f4\u7684\u6bcf\u4e2a\u9053\u8def\u8fde\u901a\u5206\u652f\u90fd\u662f\u5f00\u96c6. \u5047\u8bbe$U$\u4e3a\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7a7a\u95f4$X$\u7684\u4e00\u4e2a\u9053\u8def\u8fde\u901a\u5206\u652f, \u4e14$U$\u4e0d\u662f\u5f00\u96c6. \u5219\u5b58\u5728\u8fb9\u754c\u70b9$b \\in U$, \u6839\u636e\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684\u5b9a\u4e49, \u6211\u4eec\u53ef\u4ee5\u627e\u5230$b$\u7684\u4e00\u4e2a\u9053\u8def\u8fde\u901a\u90bb\u57df$V$.<br \/>\n$\\\\$ \u7531\u4e8e\u9053\u8def\u8fde\u901a\u5b50\u96c6\u603b\u662f\u8fde\u901a\u7684, \u4e14$b \\in U \\cap V$, \u6545$U \\cup V$\u662f\u4e00\u4e2a\u4e25\u683c\u5927\u4e8e$U$\u7684\u8fde\u901a\u5b50\u96c6, \u4e14$V \\cap (X &#8211; U) \\ne \\emptyset$. \u8fd9\u4e0e$U$\u662f\u4e00\u4e2a\u6781\u5927\u7684\u9053\u8def\u8fde\u901a\u5b50\u96c6\u7684\u4e8b\u5b9e\u77db\u76fe, \u6545$U$\u4e3a\u4e00\u4e2a\u5f00\u96c6.<br \/>\n$\\\\$ \u6700\u540e\u6765\u8bc1\u660e$\\overline{U}$\u4e0d\u4e0e\u5176\u5b83\u5206\u652f\u76f8\u4ea4, \u53ea\u80fd\u5305\u542b\u4e8e$\\gamma(S^1) \\cup U$. \u503c\u5f97\u4e00\u63d0\u7684\u662f, \u4e0d\u5e94\u8be5\u4ece\u95ed\u5305\u7684\u9053\u8def\u8fde\u901a\u6027\u53bb\u5165\u624b, \u56e0\u4e3a\u9053\u8def\u8fde\u901a\u96c6\u5408\u7684\u95ed\u5305\u4e0d\u4e00\u5b9a\u662f\u9053\u8def\u8fde\u901a\u7684, \u53cd\u4f8b\u8be6\u89c1<a href=\"https:\/\/mathstrek.blog\/2013\/03\/07\/topology-path-connected-spaces\/\">Topology: Path-Connected Spaces<\/a>. \u5047\u8bbe$\\overline{U}$\u4e0e\u5176\u5b83\u5206\u652f$U&#8217;$\u76f8\u4ea4, \u5219$\\partial U$\u4e0e\u5176\u5b83\u5206\u652f$U&#8217;$\u76f8\u4ea4, \u4e0d\u59a8\u8bbe$b$\u5c5e\u4e8e$\\partial U$\u4e0e\u5176\u5b83\u5206\u652f$U&#8217;$\u7684\u4ea4\u96c6. \u7531\u4e8e$b \\in \\partial U$, \u7531\u8fb9\u754c\u70b9\u7684\u5b9a\u4e49\u77e5, $b$\u7684\u6bcf\u4e2a\u90bb\u57df\u90fd\u542b$U$\u4e2d\u7684\u70b9; \u6b64\u5916, $b \\in S^2$, $S^2$\u662f\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684, \u7531\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684\u5b9a\u4e49\u77e5$b$\u5b58\u5728\u9053\u8def\u8fde\u901a\u7684\u90bb\u57df, \u4e14\u542b$U$\u4e2d\u7684\u70b9; \u53c8$b \\in U&#8217;$, \u5219\u4efb\u53d6$x \\in U$, $y \\in U&#8217;$, $S^2$\u4e2d\u5b58\u5728\u4ece$x$\u5230$y$\u7684\u9053\u8def, \u8fd9\u4e0e\u9053\u8def\u8fde\u901a\u5206\u652f\u7684\u5b9a\u4e49\u77db\u76fe, \u6545$\\overline{U}$\u4e0d\u4e0e\u5176\u5b83\u5206\u652f\u76f8\u4ea4, \u53ea\u80fd\u5305\u542b\u4e8e$\\gamma(S^1) \\cup U$.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>5. \u4e66\u4e0aP220\u7b2c\u4e8c\u6bb5\u5904\u5229\u7528\u4e86\u524d\u9762\u7ae0\u8282\u7684\u4e60\u9898\u7ed3\u8bba, \u9700\u8981\u8bc1\u660e$U^c$\u662f$E^2$\u7684\u9053\u8def\u8fde\u901a\u771f\u5b50\u96c6.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4efb\u53d6$x, y \\in U^c$, \u5728$E^2$\u4e2d\u4f5c\u4ece$x$\u5230$y$\u7684\u7ebf\u6bb5$l$, \u82e5$l \\cap U = \\emptyset$, \u5219\u5728$U^c$\u4e2d\u5b58\u5728\u4ece$x$\u5230$y$\u7684\u9053\u8def; \u5426\u5219, \u6211\u4eec\u53ef\u4ee5\u4f5c$l$\u7684\u4e2d\u5782\u7ebf, \u7531\u4e8e$U$\u662f\u6709\u754c\u7684, \u6545\u4e2d\u5782\u7ebf\u4e0a\u603b\u5b58\u5728\u4e00\u70b9$p$, \u4f7f\u5f97$x$\u4e0e$p$\u7684\u8fde\u7ebf, $y$\u4e0e$p$\u7684\u8fde\u7ebf\u5747\u5728$U$\u4e4b\u5916, \u4ece\u800c\u5728$U^c$\u4e2d\u5b58\u5728\u4ece$x$\u5230$y$\u7684\u9053\u8def, \u547d\u9898\u5f97\u8bc1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u5929\u662f\u4e94\u4e00\u5047\u671f\u7684\u7b2c\u4e00\u5929, \u7531\u4e8e\u6ca1\u80fd\u4e70\u5230\u4ece\u6df1\u5733\u56de\u5bb6\u7684\u7968, \u6545\u53ea\u80fd\u628a\u4e94\u4e00\u56de\u5bb6\u7684\u65e5\u5b50\u5ef6\u8fdf\u5230\u660e\u5929\u4e86. \u8fd9\u4e2a\u4e94\u4e00\u5047\u671f\u56e0\u4e3a &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/05\/jordan_curve_theorem_proof_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">Jordan\u66f2\u7ebf\u5b9a\u7406\u8bc1\u660e\u6ce8\u8bb0<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1695"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1695"}],"version-history":[{"count":54,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1695\/revisions"}],"predecessor-version":[{"id":3608,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1695\/revisions\/3608"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1695"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1695"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1695"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}