{"id":1448,"date":"2022-02-07T13:54:51","date_gmt":"2022-02-07T05:54:51","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1448"},"modified":"2025-02-26T11:05:06","modified_gmt":"2025-02-26T03:05:06","slug":"seifert_van_kampen_theorem","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/07\/seifert_van_kampen_theorem\/","title":{"rendered":"Seifert &#038; Van Kampen\u5b9a\u7406"},"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>\u6625\u8282\u5728\u5bb6\u7684\u65f6\u95f4\u4ec5\u5269\u5927\u7ea6\u4e09\u5929\u4e86, \u65f6\u95f4\u8fc7\u5f97\u771f\u5feb, \u5047\u671f\u524d\u51e0\u5929\u5176\u5b9e\u5e76\u6ca1\u6709\u600e\u4e48\u770b\u4ee3\u6570\u62d3\u6251\u7684\u4e1c\u897f(\u56de\u5bb6\u8def\u4e0a\u770b\u4e86\u4e00\u4e9bSeifert &#038; Van Kampen\u5b9a\u7406\u76f8\u5173\u7684\u5185\u5bb9), \u8fd9\u4e24\u5929\u4e3b\u8981\u8865\u5145\u4e00\u4e0bSeifert &#038; Van Kampen\u5b9a\u7406\u76f8\u5173\u5185\u5bb9\u7684\u5b66\u4e60, \u5e76\u4ee5\u6b64\u6587\u4f5c\u4e3a\u8bb0\u5f55.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.zhihu.com\/question\/263975466\">\u5982\u4f55\u7406\u89e3 Van-Kampen \u5b9a\u7406\uff1f<\/a><br \/>\n2. <a href=\"https:\/\/www.zhihu.com\/question\/299117208\/answer\/521329199\">\u4ec0\u4e48\u662f\u7fa4\u80da(groupoid)\uff1f<\/a><br \/>\n3. <a href=\"https:\/\/www.bananaspace.org\/wiki\/%E7%BE%A4%E8%83%9A\">\u7fa4\u80da<\/a><\/p>\n<p><strong>1. Seifert &#038; Van Kampen\u5b9a\u7406<\/strong> <\/p>\n<p>\u8bbe$X, Y$\u90fd\u662f$Z$\u7684\u5f00\u5b50\u96c6, $Z = X \\cup Y$, $X \\cap Y$\u975e\u7a7a\u5e76\u4e14\u9053\u8def\u8fde\u901a. \u8bbe$i: X \\cap Y $$ \\mapsto X$, $j: X \\cap Y \\mapsto Y$, $p : X \\mapsto X \\cup Y$, $q : Y \\mapsto $$ X \\cup Y$\u662f\u76f8\u5e94\u7684\u542b\u5165\u6620\u5c04.<br \/>\n$\\\\$ \u5728$X \\cap Y$\u4e2d\u53d6\u5b9a\u57fa\u70b9$x_0$, \u5e76\u8bbe$\\pi_1(X, x_0)$\u5177\u6709\u8868\u51fa$\\left \\langle C \\ | \\ R \\right \\rangle$, $\\pi_1(Y, $$ x_0)$\u5177\u6709\u8868\u51fa$\\left \\langle D \\ | \\ S \\right \\rangle$, \u800c\u4ea4\u96c6\u7684\u57fa\u672c\u7fa4$\\pi_1(X \\cap Y, x_0)$\u5177\u6709\u8868\u51fa$\\left \\langle E \\ | \\ T \\right \\rangle$. \u5bf9\u4e8e\u6bcf\u4e2a\u751f\u6210\u5143$e \\in $$ E$, \u53d6\u5b9a\u4e00\u4e2a\u4ee5$C$\u4e3a\u5b57\u7b26\u96c6\u7684\u5b57$e_C$, \u4f7f\u5f97\u5728$\\pi_1(X, x_0)$\u6309$e_C$\u8ba1\u7b97\u5f97$i_\\pi(e)$, \u518d\u53d6\u5b9a\u4e00\u4e2a\u4ee5$D$\u4e3a\u5b57\u7b26\u96c6\u7684\u5b57$e_D$, \u4f7f\u5f97\u5728$\\pi_1(Y, x_0)$\u4e2d\u6309$e_D$\u8ba1\u7b97\u5f97$j_\\pi(e)$. \u5219$\\pi_1( $$ X \\cup Y, x_0)$\u5177\u6709\u8868\u51fa$$\\left \\langle C_{p_\\pi} \\cup D_{q_\\pi} \\ | \\ R_{p_\\pi} \\cup S_{q_\\pi} \\cup \\widetilde{E} \\right \\rangle,$$\u5176\u4e2d$\\widetilde{E} = \\{ p_\\pi(e_C)^{-1}q_\\pi(e_D) \\ | \\ e \\in E \\}$. \u6216\u8005\u7b49\u4ef7\u5730\u8bb2,$$\\pi_1(X \\cup Y, x_0) \\cong (\\pi_1(X, x_0) * \\pi_1(Y, x_0))\/N \\\\ := \\pi_1(X, x_0) *_{N} \\pi_1(Y, x_0),$$\u5176\u4e2d, \u8bba\u8ff0&#8221;\u5728$\\pi_1(X, x_0)$\u6309$e_C$\u8ba1\u7b97\u5f97$i_\\pi(e)$&#8221;\u662f\u6307\u82e5\u5b57\u7b26\u96c6$C$\u751f\u6210\u7684\u81ea\u7531\u7fa4$F(C)$\u5230\u5b57\u7b26\u96c6$C$\u751f\u6210\u7684\u81ea\u7531\u7fa4$\\pi_1(X, x_0)$\u4e4b\u95f4\u7684\u6620\u5c04\u4e3a$\\xi$, \u5219$\\xi(e_C) = i_\\pi(e)$; $N$\u662f\u81ea\u7531\u79ef\u4e2d\u5305\u542b\u6240\u6709\u5f62\u5982$i_\\pi(e)^{-1}j_\\pi(e)$\u7684\u5b57(\u8fd9\u91cc$e \\in \\pi_1(X \\cap Y, x_0)$) \u7684\u6700\u5c0f\u6b63\u89c4\u5b50\u7fa4, \u5373$$N = [\\{ i_\\pi(e)^{-1}j_\\pi(e) \\ | \\ e \\in \\pi_1(X \\cap Y, x_0) \\}],$$\u4e4b\u6240\u4ee5\u8981\u6a21\u53bb\u8fd9\u4e2a\u6b63\u89c4\u5b50\u7fa4, \u662f\u56e0\u4e3a$p_\\pi(e_C)^{-1}q_\\pi(e_D)$\u5728\u4e00\u6761\u9053\u8def\u7684\u5b57$w$\u4e2d\u662f\u591a\u4f59\u7684, \u6b64\u65f6$p_\\pi$\u4e0e$q_\\pi$\u4f5c\u7528\u7684\u5bf9\u8c61$e_C$(\u6216$e_D$) \u672c\u8d28\u4e0a\u662f\u4e00\u81f4\u7684, \u5747\u4e3a$X \\cap Y$\u4e2d\u7684\u7684\u95ed\u9053\u8def\u7c7b$e$.<br \/>\n$\\\\$ \u5b9a\u7406\u5185\u5bb9\u6bd4\u8f83\u957f, \u4f46\u80cc\u540e\u7684\u601d\u60f3\u8fd8\u662f\u6bd4\u8f83\u7b80\u5355\u7684. \u5b9a\u7406\u662f\u6307\u82e5\u8981\u8ba1\u7b97\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4, \u53ef\u4ee5\u628a\u5b83\u62c6\u6210\u4e24\u4e2a\u6709\u8fde\u901a\u4ea4\u96c6\u7684\u8fde\u901a\u5f00\u96c6\u7684\u5e76, \u5219\u5176\u57fa\u672c\u7fa4\u662f\u7531\u4e24\u4e2a\u5f00\u96c6\u7684\u57fa\u672c\u7fa4, \u5176\u4ea4\u96c6\u7684\u57fa\u672c\u7fa4\u4ee5\u53ca\u4ea4\u96c6\u5230\u5f00\u96c6\u3001\u5f00\u96c6\u5230\u7a7a\u95f4\u7684\u5d4c\u5165\u6620\u5c04\u6240\u5b8c\u5168\u51b3\u5b9a\u7684; \u5177\u4f53\u6765\u8bf4, \u5373\u5f00\u96c6\u7684\u57fa\u672c\u7fa4\u7ed9\u51fa\u751f\u6210\u5143, \u800c\u4ea4\u96c6\u7684\u57fa\u672c\u7fa4\u5219\u7ed9\u51fa\u751f\u6210\u5173\u7cfb. \u5728\u4ea4\u96c6\u4e0d\u8fde\u901a\u7684\u60c5\u51b5\u4e0b\u4e5f\u6709\u8868\u8ff0\u66f4\u590d\u6742\u7684\u7248\u672c, \u540c\u65f6\u8fd8\u6709\u5176\u5b83\u4e00\u4e9b\u53d8\u4f53.<br \/>\n$\\\\$ \u6211\u4eec\u53ef\u4ee5\u7c7b\u6bd4\u96c6\u5408\u5e76\u96c6\u5143\u7d20\u6570\u91cf\u7684\u8ba1\u7b97\u516c\u5f0f:$$|A \\cup B| = |A| + |B| &#8211; |A \\cap B|.$$\u4e5f\u53ef\u4ee5\u7c7b\u6bd4\u7ebf\u6027\u7a7a\u95f4\u5e76\u96c6\u7ef4\u6570\u7684\u8ba1\u7b97\u516c\u5f0f:$$dim(A + B) = dim(A) + dim(B) &#8211; dim(A \\cap B).$$\u4e0a\u8ff0\u4e24\u4e2a\u4f8b\u5b50\u90fd\u5f88\u76f4\u89c2, \u96c6\u5408$A, B$\u653e\u5728\u4e00\u8d77\u540e, \u90a3\u4e48\u5728\u6700\u540e\u7684\u7ed3\u679c\u4e2d, \u96c6\u5408$A, B$\u90fd\u4f5c\u51faContribute, \u4f46\u662f\u6709\u4e00\u90e8\u5206$A, B$\u88ab\u89c6\u4e3a\u7b49\u540c\u5730\u4f4d\u4e86(\u4e5f\u5c31\u662f\u4ea4\u96c6), \u4ece\u800c\u8fd9\u4e00\u90e8\u5206\u7684$A, B$\u5206\u522b\u4f5c\u51fa\u7684Contribute\u4e5f\u88ab\u89c6\u4e3a\u7b49\u540c\u5730\u4f4d. \u8fd9\u4e5f\u5c31\u662f\u4e3a\u4ec0\u4e48\u8981\u51cf\u53bb\u4ea4\u96c6\u7684\u539f\u56e0. \u66f4\u76f4\u89c2\u7684\u7406\u89e3\u89d2\u5ea6\u662f\u770b\u96c6\u5408\u91cc\u7684\u5143\u7d20, \u6216\u8005\u7ebf\u6027\u7a7a\u95f4\u7684\u57fa, \u5747\u5305\u542b\u4e09\u90e8\u5206, \u53ea\u5728\u96c6\u5408$A$\u91cc\u7684, \u53ea\u5728\u96c6\u5408$B$\u91cc\u7684, \u4ee5\u53ca\u88ab\u89c6\u4e3a\u7b49\u540c\u5730\u4f4d\u7684\u90a3\u90e8\u5206(\u5373\u5728\u4ea4\u96c6\u91cc\u7684). \u5728\u67d0\u79cd\u7a0b\u5ea6\u4e0a, \u5f53\u628a\u4e24\u4e2a\u7ebf\u6027\u7a7a\u95f4\u653e\u5728\u4e00\u8d77\u540e, \u5c31\u81ea\u7136\u7ed9\u51fa\u4e86\u4e00\u4e2a\u77ed\u6b63\u5408\u5e8f\u5217. \u4e0a\u8ff0\u4e24\u4e2a\u4f8b\u5b50\u90fd\u544a\u8bc9\u6211\u4eec, \u96c6\u5408\u7684\u57fa\u6570\u4e0e\u7ebf\u6027\u7a7a\u95f4\u7684\u7ef4\u5ea6\u90fd\u6ee1\u8db3\u77ed\u6b63\u5408\u5e8f\u5217\u4e2d\u95f4\u9879\u7684\u6027\u8d28, \u800cVan Kampen\u5b9a\u7406\u544a\u8bc9\u6211\u4eec\u57fa\u672c\u7fa4\u4e5f\u662f\u540c\u7406\u4ea6\u7136.<\/p>\n<p><strong>2. Seifert &#038; Van Kampen\u5b9a\u7406\u7fa4\u80da\u7248\u672c<\/strong><\/p>\n<p>Seifert &#038; Van Kampen\u5b9a\u7406\u4e5f\u6709\u4f7f\u7528\u57fa\u672c\u7fa4\u80da\u8bed\u8a00\u63cf\u8ff0\u7684\u7248\u672c, \u5148\u4ecb\u7ecd\u7fa4\u80da\u7684\u5b9a\u4e49. \u4e00\u4e2a\u7fa4\u80da\u662f\u6307\u4e00\u4e2a\u8303\u7574$\\mathcal{C}$, \u5176\u4e2d\u6bcf\u4e2a\u6001\u5c04\u90fd\u662f\u53ef\u9006\u7684: \u5bf9\u4efb\u610f$X, Y \\in \\mathcal{C}$\u53ca$f \\in \\mathcal{C}(X, Y)$, \u5b58\u5728\u552f\u4e00\u7684$f^{-1} \\in \\mathcal{C}(Y, X)$, \u4f7f\u5f97$$f^{-1} \\circ f = 1_X, f \\circ f^{-1} = 1_Y.$$\u5bf9\u8303\u7574$\\mathcal{C}$, \u5b83\u7684\u7fa4\u80da\u5316$\\mathcal{C}^{\\simeq}$\u662f\u5982\u4e0b\u5b9a\u4e49\u7684\u7fa4\u80da:<br \/>\n$\\\\$ $\\cdot$ \u5bf9\u8c61: \u4e0e$\\mathcal{C}$\u4e2d\u76f8\u540c.<br \/>\n$\\\\$ $\\cdot$ \u6001\u5c04: $\\mathcal{C}$\u4e2d\u6240\u6709\u53ef\u9006\u6001\u5c04.<br \/>\n$\\\\$ \u7fa4\u53ef\u4ee5\u770b\u6210\u662f\u53ea\u6709\u4e00\u4e2a\u5bf9\u8c61\u7684\u7fa4\u80da. \u6362\u8a00\u4e4b, \u7fa4\u80da\u662f\u7fa4\u7684\u80da\u5316: \u5bf9\u4e8e\u4efb\u610f\u7fa4$G$, \u5b9a\u4e49\u7fa4\u80da(\u8bb0\u4e3a$G^{\\simeq}$) \u5982\u4e0b:<br \/>\n$\\\\$ $\\cdot$ $\\mathcal{O}b(G^{\\simeq}) = \\{ pt \\}$\u4e3a\u4e00\u4e2a\u5355\u70b9\u96c6.<br \/>\n$\\\\$ $\\cdot$ $Mor(G^{\\simeq}) = \\{ g \\in G\\}$, \u6b64\u5904\u4ea6\u53ef\u7406\u89e3\u4e3a\u7fa4\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20$g$\u5747\u8bf1\u5bfc\u4e86\u4e00\u4e2a\u552f\u4e00\u7684\u6001\u5c04$g^{\\simeq}$.<br \/>\n$\\\\$ $\\cdot$ \u6001\u5c04\u7684\u590d\u5408\u89c4\u5219\u4e3a$g^{\\simeq} \\circ h^{\\simeq} := h \\cdot g$.<br \/>\n$\\\\$ \u8fd9\u6837\u4e00\u6765, \u7fa4\u7684\u4e58\u6cd5\u8fd0\u7b97\u7684\u7ed3\u5408\u5f8b\u4fdd\u8bc1\u4e86\u6001\u5c04\u7684\u590d\u5408\u8fd0\u7b97\u7684\u7ed3\u5408\u5f8b, \u7fa4\u7684\u5355\u4f4d\u5143\u4fdd\u8bc1\u4e86\u6052\u7b49\u6001\u5c04\u7684\u5b58\u5728\u6027, \u8fd9\u4e24\u70b9\u8bf4\u660e$G^{\\simeq}$\u662f\u4e00\u4e2a\u8303\u7574; \u800c\u7fa4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u90fd\u6709\u9006\u5143\u7684\u4e8b\u5b9e\u5219\u4fdd\u8bc1\u4e86\u6bcf\u4e2a\u6001\u5c04\u5747\u4e3a\u540c\u6784\u6620\u5c04\u7684\u4e8b\u5b9e, \u4ece\u800c\u8bf4\u660e$G^{\\simeq}$\u4e3a\u4e00\u4e2a\u7fa4\u80da.<br \/>\n$\\\\$ \u518d\u6765\u770b\u770b\u57fa\u672c\u7fa4\u80da$\\pi_1(X)$, <strong>$\\pi_1(X)$\u4ea6\u4e3a\u4e00\u4e2a\u8303\u7574<\/strong>, \u5bf9\u8c61\u662f$X$\u4e2d\u6240\u6709\u7684\u70b9, \u6001\u5c04\u662f\u8fde\u7ed3\u4e24\u70b9\u7684\u6240\u6709\u8fde\u7eed\u8def\u5f84\u7684\u540c\u4f26\u7c7b, \u5355\u4f4d\u5143\u4e3a\u5e38\u8def\u5f84, \u590d\u5408\u8fd0\u7b97\u4e3a\u8def\u5f84\u7684\u4e32\u63a5. \u7ed9\u5b9a\u4efb\u610f$x \\in X$, \u7531\u5b83\u5f20\u6210\u7684\u6ee1\u5b50\u8303\u7574\u5c31\u662f\u57fa\u672c\u7fa4$\\pi_1(X, x)$. \u4ece\u6734\u7d20\u7684\u89c2\u70b9\u6765\u770b, \u57fa\u672c\u7fa4\u80da\u542b\u6709\u7684\u4fe1\u606f\u76f8\u5f53\u4e8e\u4e00\u65cf\u57fa\u672c\u7fa4$\\{ \\pi_1(X, x) | [x] \\in \\pi_0 $$ (X) \\}$, \u5373\u5bf9$X$\u7684\u6bcf\u4e2a\u8fde\u901a\u5206\u652f(\u8def\u5f84) \u5404\u9009\u4e00\u4e2a\u70b9, \u5e76\u5404\u81ea\u53d6\u8fd9\u4e9b\u70b9\u4e0a\u7684\u57fa\u672c\u7fa4. \u8fd9\u80cc\u540e\u7684\u539f\u56e0\u662f, \u82e5\u4e24\u70b9$x, y$\u5904\u4e8e\u540c\u4e00\u4e2a\u8fde\u901a\u5206\u652f(\u8def\u5f84), \u90a3\u4e48$\\pi_1(X, x)$(\u975e\u5178\u8303) \u540c\u6784\u4e8e$\\pi_1(X, $$ y)$, \u8fd9\u4e2a\u540c\u6784\u53d6\u51b3\u4e8e\u8fde\u7ed3$x, y$\u7684\u8def\u5f84(\u7684\u540c\u4f26\u7c7b), \u5373\u5b58\u5728\u4e00\u4e2a\u5982\u4e0b\u5b9a\u4e49\u7684\u540c\u6784\u6620\u5c04$r_*$,$$r_* : \\pi_1(X, x) \\to \\pi_1(X, y),\\\\\\left \\langle \\alpha \\right \\rangle \\mapsto \\left \\langle r^{-1} \\alpha r \\right \\rangle.$$\u5728\u4ecb\u7ecd\u5b8c\u7fa4\u80da\u7684\u5b9a\u4e49\u4ee5\u540e, \u6211\u4eec\u4fbf\u53ef\u4ee5\u7528\u7fa4\u80da\u7684\u8bed\u8a00\u6765\u63cf\u8ff0Seifert &#038; Van Kampen\u5b9a\u7406: \u8bbe$X$\u4e3a\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4, $X_0, X_1 \\subset X$, $X_{01} = $$ X_0 \\cap X_1$, $int( $$ X_0) \\cup int(X_1) = X$, \u6211\u4eec\u6709\u5982\u4e0b\u4e24\u4e2a\u56fe\u8868, \u5176\u4e2d\u56fe\u8868(#)\u4e3a\u4e00\u4e2aCocartesian\u56fe\u8868.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-1.png\" alt=\"\" width=\"387\" height=\"236\" class=\"aligncenter size-full wp-image-1543\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-1.png 1162w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-1-300x183.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-1-768x467.png 768w\" sizes=\"(max-width: 387px) 100vw, 387px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-2.png\" alt=\"\" width=\"418\" height=\"265\" class=\"aligncenter size-full wp-image-1544\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-2.png 1254w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-2-300x190.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian-2-768x488.png 768w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><\/p>\n<p>\u8fdb\u4e00\u6b65\u5047\u8bbe: $p \\in X_{01}$, $X_0, X_1, X_{01}$\u5747\u9053\u8def\u8fde\u901a, \u5219\u5c06\u5e26\u70b9\u7684\u62d3\u6251\u7a7a\u95f4\u6620\u5c04\u5230\u7fa4\u80da\u7684\u51fd\u5b50$\\pi_1$\u5c06\u56fe\u8868(#\u2019)\u53d8\u4e3a\u7fa4\u80da\u4e2d\u7684Cocartesian\u56fe\u8868, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian_Groupoid1.png\" alt=\"\" width=\"371\" height=\"265\" class=\"aligncenter size-full wp-image-1545\" \/><\/p>\n<p><strong>\u63a8\u8bba<\/strong> $\\pi_1(X, p) \\cong \\pi_1(X_0, p) *_{\\pi_1(X_{01}, p)} \\pi_1(X_1, p)$.<\/p>\n<p>Seifert &#038; Van Kampen\u5b9a\u7406\u66f4\u4e00\u822c\u7684\u5f62\u5f0f\u53c8\u88ab\u79f0\u4e3aR. Brown\u5b9a\u7406: \u8bbe$X$\u4e3a\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4, $X_0, X_1 \\subset X$, $X_{01} = X_0 \\cap X_1$, $int(X_0) \\cup $$ int(X_1) = X$(\u6ce8\u610f\u65e0\u9700\u5047\u8bbe$X_0, X_1, X_{01}$\u5747\u9053\u8def\u8fde\u901a), \u5219\u5c06\u62d3\u6251\u7a7a\u95f4\u6620\u5c04\u5230\u7fa4\u80da\u7684\u51fd\u5b50$\\prod$\u628aCocartesian\u56fe\u8868(#)\u53d8\u4e3a\u7fa4\u80da\u4e2d\u7684Cocartesian\u56fe\u8868, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian_Groupoid2.png\" alt=\"\" width=\"395\" height=\"228\" class=\"aligncenter size-full wp-image-1547\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian_Groupoid2.png 1186w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian_Groupoid2-300x173.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/02\/Cocartesian_Groupoid2-768x443.png 768w\" sizes=\"(max-width: 395px) 100vw, 395px\" \/><\/p>\n<p>\u7531\u4e8e\u5b9a\u7406\u7684\u8bc1\u660e\u8fc7\u7a0b\u8f83\u957f, \u9650\u4e8e\u7bc7\u5e45, \u6b64\u5904\u4e5f\u5c31\u4e0d\u9644\u8bc1\u660e. \u8be6\u7ec6\u7684\u8bc1\u660e\u8fc7\u7a0b\u53ef\u53c2\u8003\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u89c6\u9891:<br \/>\n1. <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW?p=37\">(36)Seifert&#038;van Kampen\u5b9a\u7406groupoid\u7248\u672c<\/a><br \/>\n2. <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW?p=38\">(37)Seifert&#038;van Kampen\u5b9a\u7406<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6625\u8282\u5728\u5bb6\u7684\u65f6\u95f4\u4ec5\u5269\u5927\u7ea6\u4e09\u5929\u4e86, \u65f6\u95f4\u8fc7\u5f97\u771f\u5feb, \u5047\u671f\u524d\u51e0\u5929\u5176\u5b9e\u5e76\u6ca1\u6709\u600e\u4e48\u770b\u4ee3\u6570\u62d3\u6251\u7684\u4e1c\u897f(\u56de\u5bb6\u8def\u4e0a\u770b\u4e86\u4e00\u4e9bSe &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/07\/seifert_van_kampen_theorem\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">Seifert &#038; Van Kampen\u5b9a\u7406<\/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\/1448"}],"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=1448"}],"version-history":[{"count":37,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1448\/revisions"}],"predecessor-version":[{"id":3614,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1448\/revisions\/3614"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1448"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1448"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1448"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}