{"id":2126,"date":"2022-09-17T14:07:04","date_gmt":"2022-09-17T06:07:04","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=2126"},"modified":"2025-02-26T11:15:01","modified_gmt":"2025-02-26T03:15:01","slug":"existence_universal_covering_space_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/09\/17\/existence_universal_covering_space_mark\/","title":{"rendered":"\u6cdb\u590d\u8fed\u7a7a\u95f4\u7684\u5b58\u5728\u6027\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>\u7ed3\u675f\u4e86\u4e2d\u79cb\u5047\u671f\u4e0e\u957f\u6c99\u4e4b\u65c5, \u53c8\u8fce\u6765\u4e86\u4e00\u4e2a\u5b85\u5bb6\u7684\u5468\u672b~ \u8fd9\u6b21\u4e2d\u79cb\u5047\u671f\u5e26\u5973\u670b\u53cb\u56de\u5bb6\u89c1\u4e86\u89c1\u7238\u5988, \u611f\u89c9\u8fd8\u4e0d\u9519\u563f\u563f~ \u800c\u540e\u4fbf\u6765\u4e86\u4e00\u573a\u957f\u6c99\u4e4b\u65c5, \u8bf4\u5b9e\u8bdd, \u65c5\u6e38\u4e00\u5929\u6bd4&#8221;\u642c\u7816&#8221; \u4e00\u5929\u8f9b\u82e6\u591a\u4e86QAQ\u2026\u2026 \u8bdd\u4e0d\u591a\u8bf4, \u8fdb\u5165\u6b63\u9898, \u672c\u6587\u7528\u4ee5\u7ed3\u675f\u6cdb\u590d\u8fed\u7a7a\u95f4\u7684\u5b58\u5728\u6027\u4e00\u8282\u5185\u5bb9\u7684\u5b66\u4e60.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/2364579\/locally-simply-connected-vs-semilocally-simply-connected\">locally simply-connected vs. semilocally simply-connected<\/a><br \/>\n2. <a href=\"https:\/\/math.stackexchange.com\/questions\/69698\/wedge-sum-of-circles-and-the-hawaiian-earring?rq=1\">Wedge sum of circles and the Hawaiian Earring<\/a><br \/>\n3. <a href=\"https:\/\/wildtopology.com\/2013\/11\/23\/the-hawaiian-earring\/\">The fundamental group of the earring space<\/a><br \/>\n4. <a href=\"https:\/\/ncatlab.org\/nlab\/show\/Hawaiian+earring+space\">Hawaiian earring space<\/a><br \/>\n5. <a href=\"https:\/\/math.tianpeng.org\/page\/2\/\">\u8fde\u7eed\u7684\u4e00\u4e00\u6620\u5c04\uff0c\u5176\u9006\u6620\u5c04\u600e\u6837\u624d\u8fde\u7eed\uff1f<\/a><br \/>\n6. <a href=\"https:\/\/math.stackexchange.com\/questions\/1138151\/show-that-a-set-that-is-open-in-the-subspace-topology-is-open-in-the-full-space\">Show that a set that is open in the subspace topology is open in the full space topology.<\/a><br \/>\n7. <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW?p=50&#038;vd_source=71042933bac98e100d097571a3b97b3a\">P50 (49)\u4e07\u6709\u8986\u76d6\u4e4b\u6784\u9020<\/a><\/p>\n<p>1. \u82e5\u62d3\u6251\u7a7a\u95f4$X$\u534a\u5c40\u90e8\u5355\u8fde\u901a, \u5219\u4e0b\u8ff0\u5b9a\u4e49\u7b49\u4ef7:<br \/>\n(1) \u5982\u679c\u62d3\u6251\u7a7a\u95f4$X$\u4e2d\u7684\u6bcf\u4e2a\u70b9\u5b58\u5728\u9053\u8def\u8fde\u901a\u90bb\u57df$U$, \u4f7f\u5f97\u5305\u542b\u6620\u5c04$i: U \\hookrightarrow X$\u8bf1\u5bfc\u7684\u57fa\u672c\u7fa4\u7684\u540c\u6001$i_\\pi : \\pi_1(U, x) \\to \\pi_1(X, x)$\u662f\u5e73\u51e1\u540c\u6001, \u5219\u79f0$X$\u534a\u5c40\u90e8\u5355\u8fde\u901a.<br \/>\n(2) \u5982\u679c\u62d3\u6251\u7a7a\u95f4$X$\u4e2d\u7684\u6bcf\u4e2a\u70b9\u5b58\u5728\u9053\u8def\u8fde\u901a\u90bb\u57df$U$, \u4f7f\u5f97\u9053\u8def\u8fde\u901a\u90bb\u57df$U$\u4e2d\u7684\u95ed\u9053\u8def\u90fd<strong>\u5728$X$\u4e2d<\/strong>(\u6ce8\u610f, \u6b64\u5904\u9ed1\u4f53\u5f3a\u8c03\u7684\u90e8\u5206\u4ea6\u662f\u4e0e\u5c40\u90e8\u5355\u8fde\u901a\u6027\u8d28\u6700\u5927\u7684\u533a\u522b\u6240\u5728) \u96f6\u4f26, \u5219\u79f0$X$\u534a\u5c40\u90e8\u5355\u8fde\u901a.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> (1) $\\to$ (2): \u7531\u5305\u542b\u6620\u5c04\u7684\u5b9a\u4e49\u53ef\u77e5, \u5b58\u5728\u6536\u7f29\u6620\u5c04$r : X \\to U$\u4f7f\u5f97$r_\\pi \\circ i_\\pi $$ = id_\\pi$\u6210\u7acb. \u7531\u4e8e$id_\\pi$\u4e3a\u4e00\u4e2a\u540c\u6784, \u6545$i_\\pi$\u4e3a\u4e00\u4e2a\u5355\u540c\u6001, \u5373\u5f53\u95ed\u9053\u8def$a$\u4e0e\u95ed\u9053\u8def$b$\u4e0d\u540c\u4f26\u65f6, $i_\\pi(\\left \\langle a \\right \\rangle) \\ne i_\\pi(\\left \\langle b \\right \\rangle)$. \u53c8$i_\\pi$\u4e3a\u4e00\u4e2a\u5e73\u51e1\u540c\u6001, \u6545\u6240\u6709\u7684\u95ed\u9053\u8def\u7c7b\u5728$i_\\pi$\u7684\u4f5c\u7528\u4e0b\u5747\u53d8\u6362\u4e3a\u4e00\u4e2a\u70b9\u9053\u8def\u7c7b$\\left \\langle e_x \\right \\rangle$, \u4e14$i_\\pi(\\left \\langle e_x \\right \\rangle) $$ = \\left \\langle e_x \\right \\rangle$, \u4ece\u800c\u6240\u6709\u7684\u95ed\u9053\u8def\u90fd\u4e0e\u70b9\u9053\u8def$e_x$\u540c\u4f26.<br \/>\n$\\\\$ (2) $\\to$ (1): \u7531\u4e8e\u9053\u8def\u8fde\u901a\u90bb\u57df$U$\u4e2d\u7684\u95ed\u9053\u8def$a$\u90fd\u96f6\u4f26, \u4e14$i_\\pi(\\left \\langle e_x \\right \\rangle) = $$ \\left \\langle e_x \\right \\rangle$, \u5219$i_\\pi( $$ \\left \\langle a \\right \\rangle) = \\left \\langle e_x \\right \\rangle$, \u4ece\u800c\u8bf4\u660e$i_\\pi$\u4e3a\u4e00\u4e2a\u5e73\u51e1\u540c\u6001.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>2. \u6d41\u5f62\u4ee5\u53ca\u6709\u9650\u5355\u7eaf\u590d\u5f62\u90fd\u662f\u534a\u5c40\u90e8\u5355\u8fde\u901a\u7684.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u7531\u6d41\u5f62\u7684\u5b9a\u4e49\u53ef\u77e5, \u6d41\u5f62$X$\u4e2d\u7684\u6bcf\u4e00\u70b9$x$\u90fd\u6709\u4e00\u4e2a\u5f00\u90bb\u57df$U$\u4e0e\u6b27\u6c0f\u7a7a\u95f4\u540c\u80da, \u800c\u6b27\u6c0f\u7a7a\u95f4\u662f\u4e00\u4e2a\u5355\u8fde\u901a\u7a7a\u95f4, \u90a3\u4e48\u8fd9\u4e2a\u90bb\u57df\u4e00\u5b9a\u9053\u8def\u8fde\u901a\u5e76\u4e14\u57fa\u672c\u7fa4\u5e73\u51e1. \u56e0\u6b64, \u6d41\u5f62\u90fd\u662f\u534a\u5c40\u90e8\u5355\u8fde\u901a\u7684.<br \/>\n$\\\\$ \u540c\u7406, \u7531\u6709\u9650\u5355\u7eaf\u590d\u5f62\u7684\u5b9a\u4e49\u53ef\u77e5, \u5176\u4e2d\u6bcf\u4e2a\u5355\u5f62\u5747\u4e3a\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5b50\u96c6, \u7531\u4e0a\u8ff0\u8bba\u8ff0\u53ef\u77e5, \u6709\u9650\u5355\u7eaf\u590d\u5f62\u4ea6\u90fd\u662f\u534a\u5c40\u90e8\u5355\u8fde\u901a\u7684.<\/p>\n<p>3. \u590f\u5a01\u5937\u8033\u73af$X$\u4e0d\u534a\u5c40\u90e8\u5355\u8fde\u901a, \u800c\u53ef\u6570\u65e0\u7a77\u591a\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76$Y $$ = \\bigvee_{n = 1}^{\\infty} S^1$\u534a\u5c40\u90e8\u5355\u8fde\u901a.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u867d\u7136\u4e24\u4e2a\u7a7a\u95f4\u770b\u4e0a\u53bb\u5f88\u50cf, \u4f46\u7531\u4e8e\u4e24\u8005\u62d3\u6251\u7ed3\u6784\u7684\u4e0d\u540c, \u5bfc\u81f4\u4e24\u8005\u7684\u534a\u5c40\u90e8\u5355\u8fde\u901a\u6027\u8d28\u4e0d\u4e00\u81f4, \u800c\u8fd9\u4e5f\u662f\u8bc1\u660e\u4e2d\u6700\u4e3a\u5173\u952e\u7684\u4e00\u70b9.<br \/>\n$\\\\$ \u590f\u5a01\u5937\u8033\u73af$X$\u7684\u62d3\u6251\u662f\u5173\u4e8e\u4e8c\u7ef4\u6b27\u6c0f\u7a7a\u95f4$E^2$\u7684\u5b50\u7a7a\u95f4\u62d3\u6251, \u5219\u5728$X$\u4e2d\u5750\u6807\u539f\u70b9$O = (0, 0)$\u7684\u4efb\u4f55\u90bb\u57df\u5185\u90fd\u5305\u542b\u4e00\u4e2a\u4e0d\u96f6\u4f26\u7684\u5c0f\u5706(\u5706\u5468$S^1$\u7684\u57fa\u672c\u7fa4\u975e\u5e73\u51e1), \u6240\u4ee5\u5b83\u4e0d\u534a\u5c40\u90e8\u5355\u8fde\u901a.<br \/>\n$\\\\$ \u53ef\u6570\u65e0\u7a77\u591a\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76$Y$\u7684\u62d3\u6251\u662f\u5728\u5546\u96c6\u5408$\\bigcup_{n =1 }^{\\infty} S^1 \/ \\sim$\u4e0a\u7684\u5546\u62d3\u6251, \u5176\u4e2d, $\\sim$\u4e3a$\\bigcup_{n =1 }^{\\infty} S^1$\u4e0a\u7684\u7b49\u4ef7\u5173\u7cfb, \u8868\u793a\u53ef\u6570\u65e0\u7a77\u591a\u4e2a\u5706\u5468\u7684\u4ea4\u70b9\u5747\u4e3a\u7b49\u4ef7\u7684.\u5728\u5546\u62d3\u6251\u4e0b, $Y$\u4e2d\u7684\u5f00\u96c6\u5373\u4e3a$\\bigcup_{n =1 }^{\\infty} S^1$\u4e2d\u7684\u5f00\u96c6, \u6545\u5728$Y$\u4e2d\u4efb\u53d6\u4e00\u70b9$y$, \u5176\u5f00\u90bb\u57df\u603b\u662f\u7531\u5176\u6240\u5728\u7684\u5706\u5468\u4e2d\u7684\u5f00\u96c6\u6784\u6210\u7684(\u5706\u5468$S^1$\u662f\u4e0e\u4e00\u7ef4\u6b27\u6c0f\u7a7a\u95f4$E^1$\u7684\u4e00\u70b9\u7d27\u5316\u540c\u80da\u7684, \u6545\u6b64\u5904\u5706\u5468$S^1$\u4e0a\u7684\u62d3\u6251\u662f\u7531\u4e00\u7ef4\u6b27\u6c0f\u7a7a\u95f4$E^1$\u7684\u62d3\u6251\u5f97\u5230\u7684), \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/09\/wedge_sum_s1_open_neighborhood-e1662300083142-300x123.png\" alt=\"\" width=\"300\" height=\"123\" class=\"aligncenter size-medium wp-image-2181\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/09\/wedge_sum_s1_open_neighborhood-e1662300083142-300x123.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/09\/wedge_sum_s1_open_neighborhood-e1662300083142.png 759w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>4. \u6ce8\u610f, \u5982\u679c$X$\u5c40\u90e8\u9053\u8def\u8fde\u901a, \u5219\u6211\u4eec\u53ef\u4ee5\u5728$x$\u7684\u4efb\u4f55\u4e00\u4e2a\u90bb\u57df$U$\u5185\u53d6\u4e00\u4e2a\u66f4\u5c0f\u7684\u9053\u8def\u8fde\u901a\u5f00\u90bb\u57df$V$. \u4e5f\u5c31\u662f\u8bf4, \u5c40\u90e8\u9053\u8def\u8fde\u901a\u5e76\u4e14\u534a\u5c40\u90e8\u5355\u8fde\u901a\u7a7a\u95f4$X$\u4e2d\u7684\u6bcf\u4e2a\u70b9$x$\u5b58\u5728\u9053\u8def\u8fde\u901a\u7684\u5f00\u90bb\u57df$U$<strong>(\u6ce8\u610f, \u91cd\u70b9\u662f&#8221;\u5f00&#8221;)<\/strong>, \u4f7f\u5f97\u5305\u542b\u6620\u5c04$i: A \\hookrightarrow X$\u8bf1\u5bfc\u7684\u57fa\u672c\u7fa4\u7684\u540c\u6001$i_\\pi : \\pi_1(A, $$ x) \\to \\pi_1(X, x)$\u662f\u5e73\u51e1\u540c\u6001.<\/p>\n<p>5. \u4e66\u4e0aP245\u7684\u8bc1\u660e\u4e2d\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4e00\u4e00\u5bf9\u5e94$p_\\alpha : U_\\alpha \\to V_x$, $\\gamma \\mapsto p( $$ \\gamma)$, \u56e0\u6b64\u53ef\u4ee5\u5728$U_\\alpha$\u4e0a\u5b9a\u4e49\u4e00\u4e2a\u62d3\u6251$\\tau_\\alpha$\u5982\u4e0b:$$\\tau_\\alpha = \\{ A \\subseteq U_\\alpha | p(A) \\subseteq B, p(A) \\  is \\  open. \\},$$\u4f7f\u5f97$p_\\alpha$\u53d8\u6210\u540c\u80da.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u9996\u5148\u8bc1\u660e$p_\\alpha = p | _{U_\\alpha}$\u4e3a\u4e00\u4e2a\u5355\u5c04, \u5bf9\u4e8e$\\forall \\alpha w_1, \\alpha w_2 \\in U_\\alpha$, \u5176\u4e2d, $w_1$, $w_2$\u5747\u4e3a$V_x$\u4e2d\u8d77\u70b9\u4e3a$x$\u7684\u9053\u8def\u5728$B$\u4e2d\u7684\u9053\u8def\u7c7b. \u82e5$p_\\alpha(\\alpha w_1) = $$ p_\\alpha( $$ \\alpha w_2)$, \u5219\u7531$p_\\alpha$\u5b9a\u4e49\u6709$w_1$\u4e0e $w_2$\u7684\u7ec8\u70b9\u76f8\u540c. \u53c8\u7531\u534a\u5c40\u90e8\u5355\u8fde\u901a\u7684\u5b9a\u4e49\u53ef\u77e5$w_1(w_2)^{-1}$\u5728$B$\u4e2d\u96f6\u4f26, \u5373$w_1$\u4e0e$w_2$\u5728$B$\u4e2d\u5b9a\u7aef\u540c\u4f26, \u4ece\u800c$\\alpha w_1 = $$ \\alpha w_2$.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u8bc1\u660e$p_\\alpha$\u4e3a\u4e00\u4e2a\u6ee1\u5c04, \u8fd9\u662f\u663e\u7136\u7684, \u56e0\u4e3a\u5bf9\u4e8e$\\forall y \\in V_x$, \u5728$B$\u4e2d\u603b\u5b58\u5728\u4ece$x$\u5230$y$\u7684\u9053\u8def, \u8fd9\u4ea6\u662f\u7531$B$\u7684\u9053\u8def\u8fde\u901a\u6027\u4fdd\u8bc1\u7684.<br \/>\n$\\\\$ \u6700\u540e, \u6211\u4eec\u6765\u8bc1\u660e$p_\\alpha$\u4e3a\u4e00\u4e2a\u540c\u80da, \u6211\u4eec\u9700\u8981\u4ece\u4e24\u65b9\u9762\u8fdb\u884c\u8bc1\u660e. \u4efb\u53d6$U_\\alpha$\u4e2d\u7684\u5f00\u96c6$U$, \u5219$U$\u5728$p_\\alpha^{-1}$\u4e0b\u7684\u539f\u50cf$p(U)$\u4ea6\u4e3a\u4e00\u4e2a\u5f00\u96c6, \u8fd9\u662f\u7531$\\tau_\\alpha$\u7684\u5b9a\u4e49\u6240\u51b3\u5b9a\u7684.<br \/>\n$\\\\$ \u53e6\u5916\u4e00\u65b9\u9762, \u4efb\u53d6$V_x$\u4e2d\u7684\u5f00\u96c6$V$, \u5219\u8be5\u5f00\u96c6\u4ea6\u4e3a\u5e95\u7a7a\u95f4$B$\u4e2d\u7684\u5f00\u96c6, \u8fd9\u662f\u56e0\u4e3a\u7531\u5b50\u7a7a\u95f4\u62d3\u6251\u7684\u5b9a\u4e49, $V = V_x \\cap U$, \u5176\u4e2d$U$\u4e3a$B$\u4e2d\u7684\u5f00\u96c6. \u56e0\u4e3a$U$\u4e0e$V_x$\u5728\u5e95\u7a7a\u95f4$B$\u4e0a\u90fd\u662f\u5f00\u7684, \u5e76\u4e14\u6709\u9650\u4e2a\u5f00\u96c6\u7684\u4ea4\u96c6\u4f9d\u65e7\u4e3a\u4e00\u4e2a\u5f00\u96c6, \u6545$V$\u5728\u5e95\u7a7a\u95f4$B$\u4e2d\u4ea6\u4e3a\u4e00\u4e2a\u5f00\u96c6. \u8fd9\u6837\u4e00\u6765, $V$\u5728$p_\\alpha$\u4e0b\u7684\u539f\u50cf$p_\\alpha^{-1}( $$ V) \\subseteq U_\\alpha$\u4ea6\u4e3a\u4e00\u4e2a\u5f00\u96c6, \u5426\u5219\u7531$\\tau_\\alpha$\u7684\u5b9a\u4e49\u77e5$p_\\alpha^{-1}(V) \\notin $$ \\tau_\\alpha$($p(p_\\alpha^{-1}(V) $$ ) = $$ V$\u4e3a\u5e95\u7a7a\u95f4$B$\u4e2d\u7684\u5f00\u96c6), \u8fd9\u662f\u77db\u76fe\u7684.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, $p_\\alpha$\u4e3a\u4e00\u4e2a\u8fde\u7eed\u7684\u4e00\u4e00\u5bf9\u5e94, \u4e14\u5176\u9006\u6620\u5c04\u4ea6\u4e3a\u8fde\u7eed\u7684, \u6545$p_\\alpha$\u4e3a\u4e00\u4e2a\u540c\u80da.<\/p>\n<p>6. \u4e66\u4e0aP245\u5728\u5bf9\u4e8e$\\Omega_b$\u9053\u8def\u8fde\u901a\u7684\u8bc1\u660e\u4e2d\u4efb\u53d6\u4e00\u6761\u4ece$b$\u51fa\u53d1\u7684\u9053\u8def$w: $$ [0, 1] \\to $$ B$, \u5b9a\u4e49\u9053\u8def$w_s : [0, 1] \\to B$, $w_s(t) = w(st)$, \u5219$$w^\\uparrow : [0, 1] \\to \\Omega_b, t \\mapsto \\left \\langle w_t \\right \\rangle$$\u5c31\u662f\u4e00\u6761\u4ece$\\left \\langle w \\right \\rangle$\u5230\u4ee5$b$\u4e3a\u57fa\u70b9\u7684\u70b9\u9053\u8def\u7c7b$\\left \\langle e_b \\right \\rangle$\u7684\u9053\u8def.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4ec5\u9700\u8bc1\u660e$w^\\uparrow$\u4e3a\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04\u5373\u53ef. $\\Leftrightarrow$ \u4efb\u53d6\u9053\u8def\u7c7b$\\alpha \\in \\Omega_b$, $(w^\\uparrow $$ )^{-1} (U_\\alpha)$\u4e3a$[0, 1]$\u4e2d\u4e00\u4e2a\u5f00\u96c6. $\\Leftrightarrow$ \u4efb\u53d6$s_0 \\in (w^\\uparrow)^{-1}(U_\\alpha)$, \u5b58\u5728$s_0$\u7684\u4e00\u4e2a\u5f00\u90bb\u57df\u5305\u542b\u4e8e$(w^\\uparrow $$ )^{-1}(U_\\alpha)$\u4e2d.<br \/>\n$\\\\$ \u7531\u4e8e$s_0 \\in (w^\\uparrow)^{-1}(U_\\alpha)$, i.e. $\\left \\langle w_{s_0} \\right \\rangle =  w^\\uparrow(s_0) \\in U_\\alpha$, i.e. \u5b58\u5728\u4e00\u4e2a$V_x$\u4e2d\u8d77\u70b9\u4e3a$x$\u7684\u9053\u8def\u5728$B$\u4e2d\u7684\u9053\u8def\u7c7b$\\beta$, s.t. $\\left \\langle w_{s_0} \\right \\rangle = \\left \\langle \\alpha \\beta \\right \\rangle$, $\\Rightarrow$ $w( $$ s_0) = \\beta(1) \\in V_x$, $\\Rightarrow$ $s_0 $$ \\in w^{-1}(V_x) \\subseteq [0, 1]$. \u53c8$V_x$\u4e3a\u4e00\u4e2a\u5f00\u96c6, $w$\u4e3a\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04, \u6545$w^{-1}(V_x)$\u4ea6\u4e3a\u4e00\u4e2a\u5f00\u96c6. \u63a5\u4e0b\u6765\u6211\u4eec\u4ec5\u9700\u8bc1\u660e$w^{-1}(V_x) $$ \\subseteq (w^\\uparrow)^{-1}(U_\\alpha)$\u5373\u53ef.<br \/>\n$\\\\$ \u5bf9\u4e8e$\\forall s \\in w^{-1}(V_x)$, $\\left \\langle w_s \\right \\rangle = w^\\uparrow(s) \\in U_\\alpha$. \u5df2\u77e5$\\left \\langle w_{s_0} \\right \\rangle = \\left \\langle \\alpha \\beta \\right \\rangle$, \u5219\u5b58\u5728\u4e00\u4e2a$V_x$\u4e2d\u8d77\u70b9\u4e3a$x$\u7684\u9053\u8def\u5728$B$\u4e2d\u7684\u9053\u8def\u7c7b$\\delta_s$, s.t.$$\\left \\langle w_s \\right \\rangle = \\left \\langle w_{s_0} \\delta_s \\right \\rangle = \\left \\langle \\alpha \\beta \\delta_s \\right \\rangle = \\left \\langle \\alpha (\\beta \\delta_s) \\right \\rangle \\in U_\\alpha.$$\u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>7. \u4e66\u4e0aP244\u5b9a\u74065.4.1\u4e4b\u6240\u4ee5\u8981\u6c42\u5e95\u7a7a\u95f4$B$\u9053\u8def\u8fde\u901a\u5e76\u4e14\u5c40\u90e8\u9053\u8def\u8fde\u901a, \u662f\u56e0\u4e3a\u9053\u8def\u8fde\u901a\u4e0e\u5c40\u90e8\u9053\u8def\u8fde\u901a\u8fd9\u4e24\u4e2a\u6027\u8d28\u5e76\u4e0d\u5177\u5907\u76f8\u4e92\u5305\u542b\u7684\u5173\u7cfb, \u5b58\u5728\u9053\u8def\u8fde\u901a\u4f46\u4e0d\u5c40\u90e8\u9053\u8def\u8fde\u901a\u7684\u62d3\u6251\u7a7a\u95f4: <a href=\"https:\/\/en.wikipedia.org\/wiki\/Topologist%27s_sine_curve\">Topologist&#8217;s sine curve<\/a>.<\/p>\n<p>8. \u4e66\u4e0aP244\u5b9a\u74065.4.1\u7684\u8bc1\u660e\u8fc7\u7a0b\u53ef\u63d0\u70bc\u4e3a\u4e09\u6b65:<br \/>\n$\\\\$ (1) \u8003\u8651\u4ece\u57fa\u70b9$b \\in B$\u51fa\u53d1\u7684\u6240\u6709\u9053\u8def\u7c7b\u6784\u6210\u7684\u96c6\u5408$\\Omega_b$, \u4e3a\u5176\u5b9a\u4e49\u4e00\u4e2a\u62d3\u6251. (\u79c1\u4ee5\u4e3a\u8fd9\u4ea6\u662f\u6574\u4e2a\u8bc1\u660e\u8fc7\u7a0b\u4e2d\u6700\u6f02\u4eae\u7684\u4e00\u6b65~)<br \/>\n$\\\\$ (2) \u8bc1\u660e$p: \\Omega_b \\to B$\u662f\u4e2a\u590d\u8fed\u6620\u5c04.<br \/>\n$\\\\$ (3) \u8bc1\u660e$\\Omega_b$\u5355\u8fde\u901a.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u7ed3\u675f\u4e86\u4e2d\u79cb\u5047\u671f\u4e0e\u957f\u6c99\u4e4b\u65c5, \u53c8\u8fce\u6765\u4e86\u4e00\u4e2a\u5b85\u5bb6\u7684\u5468\u672b~ \u8fd9\u6b21\u4e2d\u79cb\u5047\u671f\u5e26\u5973\u670b\u53cb\u56de\u5bb6\u89c1\u4e86\u89c1\u7238\u5988, \u611f\u89c9\u8fd8\u4e0d\u9519\u563f\u563f~  &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/09\/17\/existence_universal_covering_space_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u6cdb\u590d\u8fed\u7a7a\u95f4\u7684\u5b58\u5728\u6027\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\/2126"}],"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=2126"}],"version-history":[{"count":85,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2126\/revisions"}],"predecessor-version":[{"id":3623,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2126\/revisions\/3623"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}