{"id":1912,"date":"2022-07-10T23:01:40","date_gmt":"2022-07-10T15:01:40","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1912"},"modified":"2025-02-26T11:16:57","modified_gmt":"2025-02-26T03:16:57","slug":"fibration_covering_map_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/07\/10\/fibration_covering_map_mark\/","title":{"rendered":"\u7ea4\u7ef4\u5316\u4e0e\u590d\u8fed\u6620\u5c04\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>\u6700\u8fd1\u6bcf\u5929\u7684\u5929\u6c14\u90fd\u662f\u706b\u7089\u5929\u6c14, \u5728\u5bb6\u91cc\u5b9e\u5728\u662f\u6709\u70b9\u96be\u53d7, \u4e2a\u4eba\u611f\u89c9\u53bb\u5e74\u540c\u4e00\u65f6\u5019\u7684\u6e29\u5ea6\u8981\u4f4e\u4e00\u4e9b\u2026\u2026 \u672c\u6587\u4e3b\u8981\u662f\u5173\u4e8e\u7ea4\u7ef4\u5316\u4e0e\u590d\u8fed\u6620\u5c04\u4e00\u8282\u5185\u5bb9\u7684\u6ce8\u8bb0, \u9664\u4e86\u4e66\u4e0a\u7684\u5185\u5bb9\u4e4b\u5916, \u81ea\u5df1\u8fd8\u4f1a\u5b66\u4e60\u5e84\u6653\u6ce2\u8001\u5e08\u5173\u4e8e\u8986\u76d6\u6620\u5c04\u7684\u76f8\u5173\u89c6\u9891, \u4ee5\u6b64\u52a0\u6df1\u5bf9\u8fd9\u4e00\u8282\u5185\u5bb9\u7684\u7406\u89e3.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/ncatlab.org\/nlab\/show\/Lebesgue+number+lemma\">Lebesgue number lemma<\/a><br \/>\n2. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/solutions_09.pdf\">solutions_09<\/a><br \/>\n3. <a href=\"https:\/\/www.zhihu.com\/question\/63131822\">\u4e3a\u4ec0\u4e48\u8bf4\u79bb\u6563\u7a7a\u95f4\u4e2d\uff0cX\u7684\u6bcf\u4e00\u4e2a\u5b50\u96c6\u90fd\u662f\u5f00\u96c6\uff1f<\/a><br \/>\n4. <a href=\"https:\/\/math.stackexchange.com\/questions\/40315\/the-fiber-of-a-covering-space-over-a-connected-space-has-constant-cardinality\">The fiber of a covering space over a connected space has constant cardinality [duplicate]<\/a><br \/>\n5. <a href=\"https:\/\/math.stackexchange.com\/questions\/260578\/4-sheet-covering-of-the-wedge-sum-of-two-circles\">4-sheet covering of the wedge sum of two circles<\/a><br \/>\n6. <a href=\"https:\/\/www.researchgate.net\/post\/How_do_I_see_that_the_fundamental_group_of_M_is_finite\">How do I see that the fundamental group of M is finite?<\/a><\/p>\n<p>1. \u4e66\u4e0aP233\u5b9a\u74065.2.1\u7684\u8bc1\u660e\u4e2d\u4f7f\u7528\u4e86Lebesgue\u5f15\u7406, \u5176\u5f15\u7406\u5185\u5bb9\u8be6\u89c1\u53c2\u8003\u6750\u65991. \u7531\u4e8e$[0, 1]$\u662f\u4e00\u4e2a\u5e8f\u5217\u7d27\u7684\u5ea6\u91cf\u7a7a\u95f4, \u6545Lebsgue\u5f15\u7406\u7684\u4f7f\u7528\u662f\u5408\u7406\u7684. \u7c7b\u4f3c\u5730, \u7531\u4e8e\u8bc1\u660e\u4e2d\u6784\u9020\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df\u7684\u539f\u50cf\u662f\u5f00\u96c6, \u4e14$[\\frac{i}{n}, \\frac{i + 1}{n}]$\u662f\u7d27\u81f4\u7684, \u6545\u7ba1\u72b6\u5f15\u7406\u7684\u4f7f\u7528\u4ea6\u662f\u5408\u7406\u7684.<br \/>\n$\\\\$ \u6b64\u5916, \u4e66\u4e0a\u5e76\u672a\u5bf9\u8fde\u7eed\u6269\u5f20\u7684\u552f\u4e00\u6027\u8fdb\u884c\u8be6\u7ec6\u8bba\u8ff0, \u540e\u7eed\u9700\u8981\u8865\u5145\u6b64\u5904\u7ec6\u8282.<\/p>\n<p>2. \u628a\u5706\u5468\u7ed5\u81ea\u5df1$n$\u5708\u7684\u6620\u5c04$$f_\\pi : S^1 \\to S^1, (cos(2\\pi t), sin(2\\pi t)) \\mapsto (cos(2n\\pi t), sin(2n\\pi t))$$\u662f\u4e00\u4e2a$n$\u5c42\u590d\u8fed.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u6211\u4eec\u4ece\u590d\u6570\u7684\u89d2\u5ea6\u51fa\u53d1, \u5c06\u95ee\u9898\u8f6c\u5316\u4e3a\u5982\u4e0b\u5f62\u5f0f:$$p: S^1 \\to S^1, p(z) := z^n$$\u662f\u4e00\u4e2a$n$\u5c42\u590d\u8fed.<br \/>\n$\\\\$ \u9996\u5148\u8981\u6ce8\u610f1\u7684$n$\u6b21\u65b9\u6839, \u5373$z^n = 1$\u7684\u89e3\u4e3a$$\\xi_k := e^{2\\pi ik\/n} = cos(2\\pi k \/ n) + i sin(2\\pi k \/ n), k = 1, \\cdots, n.$$\u56e0\u6b64, \u7ed9\u5b9a\u4efb\u610f$z = e^{i\\theta} \\in S^1$, \u6211\u4eec\u6709$$p^{-1}(z) = \\{ e^{i\\theta \/ n} \\xi_1, e^{i\\theta \/ n} \\xi_2, \\cdots, e^{i\\theta \/ n} \\xi_n \\}.$$\u53d6\u5b9a$z_0 = e^{i\\theta_0} \\in S^1$, \u4ee4$U := \\{ e^{i\\theta} : |\\theta &#8211; \\theta_0 | < \\pi \/ 2\\} \\subseteq S^1$\u4e3a$z_0$\u7684\u4e00\u4e2a\u5747\u5300\u590d\u8fed\u90bb\u57df. \u6839\u636e\u53c2\u8003\u6750\u65991\u4e2d\u7684\u547d\u98989.1, \u53ef\u5f97$p^{-1}(U) = \\cup_{k = 1}^n V_k$, \u5176\u4e2d$$V_k := \\{ e^{i\\theta \/n} \\xi_k : |\\theta - \\theta_0| < \\pi \/ 2 \\}.$$\u6ce8\u610f\u5230$V_k \\cap V_h = \\emptyset, \\forall k \\ne h$. \u4e8b\u5b9e\u4e0a, \u5047\u8bbe$\\exists \\theta, \\theta' \\in (\\theta_0 - \\pi \/2, $$ \\theta_0 + \\pi \/ 2)$, $e^{i\\theta \/ n} \\xi_k = e^{i\\theta' \/ n} \\xi_h, k \\ne h$, \u5219$$\\frac{\\theta}{n} + \\frac{2\\pi k}{n} \\equiv \\frac{\\theta'}{n} + \\frac{2\\pi h}{n} (mod \\ 2\\pi) \\\\ \\Longleftrightarrow \\theta - \\theta' \\equiv 2\\pi (h - k) (mod \\ 2\\pi n).$$\u4f46$|\\theta - \\theta'| < \\pi < 2\\pi |h - k| < 2\\pi n$, \u8fd9\u4e0e\u4e0a\u8ff0\u7b49\u5f0f\u76f8\u77db\u76fe. \u6545$\\{ $$ V_k \\}_{k = 1, \\cdots, n}$\u4e24\u4e24\u65e0\u4ea4. \u66f4\u8fdb\u4e00\u6b65\u5730, \u5bf9\u4e8e$k = 1, \\cdots, n$, $p | V_k : V_k $$ \\to U$\u662f\u4e00\u4e2a\u8fde\u7eed\u53cc\u5c04. \u6b64\u5916, \u5bf9\u4e8e\u4efb\u610f$e^{i\\theta} \\in U$, \u5373$|\\theta - \\theta_0| < \\pi \/ 2$, $p | V_k$\u7684\u9006\u6620\u5c04$(p | V_k)^{-1} : U \\to V_k$\u53ef\u5b9a\u4e49\u5982\u4e0b$$(p | V_k)^{-1}(e^{i\\theta}) = e^{i\\theta \/ n} \\xi_k.$$\u6545$(p | V_k)^{-1}$\u4ea6\u662f\u8fde\u7eed\u7684, \u4ece\u800c\u8bc1\u660e\u4e86\u5bf9\u4e8e$\\forall k = 1, \\cdots, n$, $p | V_k : $$ V_k $$ \\to U$\u662f\u4e00\u4e2a\u540c\u80da. \u547d\u9898\u5f97\u8bc1.\n\n$\\\\$ $\\\\$ 3. \u8bbe$f: E \\to B \\times F$\u662f\u540c\u80da, $p: B \\times F \\to B, (x, y) \\mapsto x$\u662f\u6295\u5c04. \u8bc1\u660e$p \\circ $$ f: E \\to B$\u662f\u4e00\u4e2a\u7ea4\u7ef4\u5316.\n$\\\\$ <strong>\u8bc1:<\/strong> \u8fd9\u4e2a\u547d\u9898\u7684\u8bc1\u660e\u5176\u5b9e\u4e0d\u96be, \u4f46\u81ea\u5df1\u4e00\u5f00\u59cb\u4e0d\u7406\u89e3\u8be5\u547d\u9898\u7684\u8bc1\u660e\u7684\u539f\u56e0\u662f\u641e\u6df7\u4e86\u7ea4\u7ef4\u5316\u4e0e\u590d\u8fed\u6620\u5c04\u7684\u6982\u5ff5.<\/p>\n<p>$\\\\$ $\\\\$ 4. \u8bbe$p: E \\to B$\u662f\u590d\u8fed\u6620\u5c04, \u4efb\u53d6$x \\in B$, \u8bc1\u660e\u7ea4\u7ef4$p^{-1}(x) \\subseteq E$\u662f\u79bb\u6563\u62d3\u6251\u7a7a\u95f4(\u5373\u6bcf\u4e2a\u5355\u70b9\u96c6\u662f\u5f00\u96c6).<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u5bf9\u4e8e\u4efb\u610f$x \\in B$, \u6211\u4eec\u9996\u5148\u8bc1\u660e: $p^{-1}(x)$\u4e3a\u4e00\u4e2a\u79bb\u6563\u5b50\u96c6, \u5373\u5bf9\u4e8e\u4efb\u610f$y \\in E$, \u5b58\u5728\u4e00\u4e2a$y$\u7684\u5f00\u90bb\u57df$U$\u4f7f\u5f97$(U \\backslash \\{ y \\}) \\cap p^{-1}(x) = \\emptyset$. \u6211\u4eec\u9700\u8981\u5bf9\u4e0b\u8ff0\u4e24\u79cd\u60c5\u51b5\u5206\u522b\u8fdb\u884c\u8ba8\u8bba:<br \/>\n$\\\\$ $\\cdot$ \u82e5$y \\in p^{-1}(x)$, \u53d6$x$\u7684\u4e00\u4e2a\u5747\u5300\u590d\u8fed\u90bb\u57df$V$, \u5219$p^{-1}(V) = \\cup_{i \\in I}U_i$, \u5176\u4e2d$U_i \\subseteq $$ E$\u662f\u4e24\u4e24\u65e0\u4ea4\u7684, \u4e14\u5747\u540c\u80da\u4e8e$V$. \u7279\u522b\u5730, $U_i$\u5747\u4e3a\u5f00\u96c6, \u4e14\u5bf9\u4e8e$\\forall i \\in I$, \u5b58\u5728\u552f\u4e00\u7684$y_i \\in p^{-1}(x) \\cap U_i$. \u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4ee4$U := $$ U_i$, \u5176\u4e2d$y \\in U_i$, \u8fd9\u4ea6\u662f$y$\u7684\u4e00\u4e2a\u5f00\u90bb\u57df, \u4f7f\u5f97$(U \\backslash \\{ y \\}) \\cap p^{-1}(x) $$ = \\emptyset$.<br \/>\n$\\\\$ $\\cdot$ \u82e5$p(y) \\ne x$, \u7531\u4e8e$B$\u662fHausdorff\u7684(\u4e8b\u5b9e\u4e0a, $B$\u4ec5\u9700\u6ee1\u8db3T1\u516c\u7406\u5373\u53ef, \u8fd9\u662f\u989d\u5916\u5047\u8bbe???), \u6545\u6211\u4eec\u53ef\u53d6$p(y)$\u7684\u4e00\u4e2a\u4e0d\u5305\u542b$x$\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df$V$, \u5176\u539f\u50cf$U := $$ p^{-1}(V)$\u662f$y$\u7684\u4e00\u4e2a\u4e0d\u5305\u542b$p^{-1}(x)$\u7684\u4efb\u610f\u5143\u7d20\u7684\u5f00\u90bb\u57df(\u56e0\u4e3a$U$\u540c\u80da\u4e8e$V$\u4e14$x \\notin $$ V$), \u4ece\u800c\u6211\u4eec\u53ef\u5f97$(U \\backslash \\{ y \\}) \\cap $$ p^{-1}(x) = \\emptyset$.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8bc1\u660e$p^{-1}(x)$\u4e3a\u4e00\u4e2a\u6709\u9650\u5b50\u96c6, \u4e0d\u59a8\u5047\u8bbe$E$\u662f\u7d27\u81f4\u7684, \u5219\u4e00\u4e2a\u7d27\u81f4\u62d3\u6251\u7a7a\u95f4\u7684\u4efb\u610f\u79bb\u6563\u5b50\u96c6\u5747\u662f\u79bb\u6563\u7684; \u5426\u5219, \u6211\u4eec\u5047\u8bbe\u5728\u7d27\u81f4\u62d3\u6251\u7a7a\u95f4$E$\u4e2d\u5b58\u5728\u4e00\u4e2a\u65e0\u9650\u79bb\u6563\u5b50\u96c6$A$. \u7531\u79bb\u6563\u5b50\u96c6\u7684\u5b9a\u4e49, \u5bf9\u4e8e\u4efb\u610f$y \\in E$, \u5b58\u5728$y$\u7684\u4e00\u4e2a\u5f00\u90bb\u57df$U_y$, \u4f7f\u5f97$(U_y \\backslash \\{ y \\}) \\cap A = \\emptyset$. \u6839\u636e$E$\u7684\u7d27\u81f4\u6027, $E$\u7684\u5f00\u8986\u76d6$\\{ U_y \\}_{y \\in E}$\u6709\u4e00\u4e2a\u6709\u9650\u5b50\u8986\u76d6$\\{ U_{y_1}, \\cdots, $$ U_{y_n} \\}$, \u5219\u6211\u4eec\u6709$$\\emptyset = ((U_{y_1} \\backslash \\{ y_1 \\}) \\cup \\cdots \\cup (U_{y_n} \\backslash \\{ y_n \\})) \\cap A \\\\ \\supseteq ((U_{y_1} \\cup \\cdots \\cup U_{y_n}) \\backslash \\{ y_1, \\cdots, y_n \\}) \\cap A,$$\u4ece\u800c$A \\subseteq \\{ y_1, \\cdots, y_n \\}$\u662f\u6709\u9650\u7684.<br \/>\n$\\\\$ \u6700\u540e\u6211\u4eec\u8bc1\u660e\u5728\u79bb\u6563\u5b50\u96c6$p^{-1}(x)$\u53ef\u5b9a\u4e49\u79bb\u6563\u62d3\u6251. \u6211\u4eec\u4e0d\u59a8\u5728\u79bb\u6563\u5b50\u96c6$p^{-1}(x)$\u4e0a\u5b9a\u4e49\u79bb\u6563\u8ddd\u79bb$d_d$:$$d_d(x, y) = \\left\\{\\begin{matrix}<br \/>\n 1 &#038; if \\ x \\ne y, \\\\<br \/>\n 0 &#038; if \\ x = y.<br \/>\n\\end{matrix}\\right.$$\u8fd9\u4e2a\u8ddd\u79bb\u4e0b$\\{ x \\} = Ball(x, 1)$, \u4e5f\u5c31\u662f\u8bf4\u6bcf\u4e2a\u5706\u5fc3$x$\u7684&#8221;\u5355\u4f4d\u5f00\u7403&#8221; \u5c31\u662f\u8fd9\u4e2a\u5706\u5fc3\u672c\u8eab. \u56e0\u4e3a\u5728\u8ddd\u79bb\u7a7a\u95f4\u4e2d\u5ea6\u91cf\u8bf1\u5bfc\u51fa\u6765\u7684\u5f00\u96c6\u5b9a\u4e49\u4e3a: $U$\u4e3a\u4e00\u4e2a\u5f00\u96c6, \u5f53\u4e14\u4ec5\u5f53\u5bf9\u4e8e\u4efb\u610f$x \\in U$, \u5b58\u5728\u4e00\u4e2a\u5f00\u7403$Ball(x, d) $$ := \\{ x; d(x, d) \\} \\subset U$. \u4e8b\u5b9e\u4e0a, \u7531\u53c2\u8003\u6750\u65993\u4e2d\u63d0\u53ca\u7684\u5b9a\u7406, \u79bb\u6563\u8ddd\u79bb\u8bf1\u5bfc\u51fa\u6765\u7684\u62d3\u6251\u548c\u79bb\u6563\u62d3\u6251\u7b49\u4ef7, \u5373\u6bcf\u4e2a\u70b9\u90fd\u662f\u5f00\u96c6(\u56e0\u4e3a\u662f\u4e00\u4e2a\u5f00\u7403), \u7531\u4e8e\u6bcf\u4e2a\u96c6\u5408\u90fd\u662f\u5176\u4e2d\u7684\u5355\u70b9\u7684\u5f00\u96c6, \u81ea\u7136\u6bcf\u4e2a\u96c6\u5408\u90fd\u662f\u5f00\u96c6. \u4e5f\u5c31\u662f\u8bf4$$A = \\cup_{x \\in A} Ball(x, 1) = \\cup_{x \\in A} \\{ x \\}.$$\u82e5\u6211\u4eec\u76f4\u63a5\u5bf9\u4e00\u4e2a\u7a7a\u95f4\u8d4b\u4e88\u79bb\u6563\u62d3\u6251$X = R, \\tau = 2^R$, \u4e5f\u5c31\u662f\u8bf4\u4efb\u4f55\u5b50\u96c6\u90fd\u662f\u5728\u8fd9\u4e2a\u62d3\u6251$\\tau = 2^R$\u4e0b\u7684\u5f00\u96c6, \u5219\u8fd9\u4e2a\u62d3\u6251\u548c\u7528\u8ddd\u79bb\u7a7a\u95f4\u8bf1\u5bfc\u51fa\u7684\u62d3\u6251\u662f\u7b49\u4ef7\u7684. \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>5. \u8bbe$p: E \\to B$\u662f\u590d\u8fed\u6620\u5c04, \u5982\u679c$B$\u9053\u8def\u8fde\u901a, \u5219\u4efb\u53d6$x, y \\in B$, $p^{-1}(x) \\cong $$ p^{-1}(y)$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4e0d\u59a8\u5c06\u6761\u4ef6\u5f31\u5316, \u8bc1\u660e\u66f4\u4e00\u822c\u7684\u7ed3\u8bba: \u8bbe$p: E \\to B$\u662f\u590d\u8fed\u6620\u5c04, \u5982\u679c$B$\u8fde\u901a, \u5219\u4efb\u53d6$x, y \\in B$, $p^{-1}(x) \\cong $$ p^{-1}(y)$. \u7531\u4e0a\u8ff0\u7ed3\u8bba, \u5373\u987b\u8bc1\u660e: \u82e5\u5bf9\u4e8e\u4e00\u4e9b$b_0 \\in $$ B$, $p^{-1}(b_0)$\u542b\u6709$k$\u4e2a\u5143\u7d20, \u5219\u5bf9\u4e8e\u4efb\u610f$b \\in B$, $p^{-1}(b)$\u542b\u6709$k$\u4e2a\u5143\u7d20.<br \/>\n$\\\\$ \u5047\u8bbe\u5b58\u5728$b_0 \\in B$, $p^{-1}(b_0)$\u542b\u6709$k$\u4e2a\u5143\u7d20. \u4ee4$A := \\{ b \\in B: |p^{-1}(b $$ )| = k \\}$, \u6ce8\u610f\u5230<br \/>\n$\\\\$ 1) $\\because b_0 \\in A$, $A \\ne \\emptyset$.<br \/>\n$\\\\$ 2) $A \\cap (B \\backslash A) = \\emptyset$.<br \/>\n$\\\\$ 3) $A \\cup (B \\backslash A) = B$.<br \/>\n$\\\\$ 4) $A$\u4e3a\u5f00\u96c6.<br \/>\n$\\\\$ 5) $B \\backslash A$\u4e3a\u5f00\u96c6.<br \/>\n$\\\\$ \u82e5\u4ee5\u4e0a\u4e94\u70b9\u5747\u6210\u7acb, \u5219\u53ef\u63a8\u51fa$B \\backslash A = \\emptyset$, \u5426\u5219\u6211\u4eec\u53ef\u4ee5\u5f97\u5230$B$\u7684\u4e00\u4e2a\u5206\u5272, \u8fd9\u4e0e$B$\u7684\u8fde\u901a\u6027\u77db\u76fe. \u6545$B \\backslash A = \\emptyset \\Rightarrow A = B$, \u4ece\u800c\u4efb\u610f$p$\u7684\u7ea4\u7ef4(\u5373$p^{-1}(b)$) \u542b\u6709$k$\u4e2a\u5143\u7d20.<br \/>\n$\\\\$ \u524d\u9762\u4e09\u70b9\u662f\u663e\u7136\u7684, \u63a5\u4e0b\u6765\u6211\u4eec\u8bc1\u660e\u540e\u9762\u4e24\u70b9. \u9996\u5148\u8bc1\u660e4): \u4ee4$a \\in A$, \u56e0\u4e3a$p$\u4e3a\u4e00\u4e2a\u590d\u8fed\u6620\u5c04, \u6545\u5b58\u5728\u5f00\u96c6$U \\ni a$\u4f7f\u5f97$p^{-1}(U) = $$ \\cup_{\\alpha \\in \\Lambda} V_\\alpha$, \u5176\u4e2d\u6bcf\u4e2a$V_\\alpha$\u5747\u901a\u8fc7$p | V_\\alpha$\u540c\u80da\u4e8e$U$. \u7531\u4e8e$p^{-1}(a)$\u4e0e\u6bcf\u4e2a$V_\\alpha$\u4ec5\u76f8\u4ea4\u4e8e\u4e00\u70b9, \u6545$\\Lambda$\u542b\u6709$k$\u4e2a\u5143\u7d20, \u6211\u4eec\u53ef\u5c06$p^{-1}(U)$\u5199\u4e3a$p^{-1}(U) = \\cup_{n = 1}^k V_n$. \u4efb\u53d6$x \\in U$, \u5219$p^{-1}(x) \\subseteq p^{-1}(U) = $$ \\cup_{n = 1}^k V_n$. \u53c8$p | V_n$\u662f\u4e00\u4e2a\u540c\u80da, \u5219$p^{-1}(x)$\u4ea6\u542b\u6709$k$\u4e2a\u5143\u7d20, \u4ece\u800c$x \\in A$. \u56e0\u4e3a$\\forall a \\in $$ A$, $a \\in U \\subseteq A$, \u6545$A$\u662f\u4e00\u4e2a\u5f00\u96c6. \u540c\u7406\u53ef\u8bc1\u660e5).<br \/>\n\u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>6. \u8bf7\u753b\u56fe\u63cf\u8ff0\u51fa\u4e24\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76$S^1 \\vee S^1$\u7684\u4e24\u79cd\u5168\u7a7a\u95f4\u76f8\u4e92\u4e0d\u540c\u80da\u76844\u5c42\u590d\u8fed.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u7531\u4e0a\u8ff0\u7ed3\u8bba, \u7531\u4e8e\u4e24\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76$S^1 \\vee S^1$\u662f\u9053\u8def\u8fde\u901a\u7684, \u6545$p^{-1}(x)$\u542b\u6709\u7684\u5143\u7d20\u6570\u91cf\u662f\u590d\u8fed\u6620\u5c04$p: E \\to S^1 \\vee S^1$\u7684Number of Sheets.<br \/>\n$\\\\$ Hatcher\u7684Algebraic Topology\u7684P58\u4e0a\u7684\u4f8b7, 8, 9\u5747\u7ed9\u51fa\u4e86\u4e24\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76$S^1 \\vee S^1$\u76844\u5c42\u590d\u8fed. \u4ee5\u4f8b7\u4e3a\u4f8b, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/07\/Number-of-Sheets.png\" alt=\"\" width=\"651\" height=\"246\" class=\"aligncenter size-full wp-image-1992\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/07\/Number-of-Sheets.png 651w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/07\/Number-of-Sheets-300x113.png 300w\" sizes=\"(max-width: 651px) 100vw, 651px\" \/><\/p>\n<p>\u5728\u5706\u7684\u5185\u90e8\u5305\u542b\u4e00\u4e2a\u53d8\u5f62\u7684\u6b63\u65b9\u5f62, \u8fd9\u4e2a\u53d8\u5f62\u7684\u6b63\u65b9\u5f62\u4e0e\u5706\u76f8\u4ea4\u4e8e4\u4e2a\u70b9. \u8fd94\u4e2a\u70b9\u662f\u4e24\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76$S^1 \\vee S^1$\u7684\u57fa\u70b9\u5728\u590d\u8fed\u6620\u5c04\u4e0b\u7684\u539f\u50cf, \u8be5\u590d\u8fed\u6620\u5c04\u540c\u65f6\u4e5f\u5c06$E$\u4e2d\u7684\u8fb9$b$\u6620\u5c04\u81f3\u4e24\u4e2a\u5706\u5468\u7684\u4e00\u70b9\u5e76$S^1 \\vee $$ S^1$\u4e2d\u7684\u5bf9\u5e94\u8fb9(\u6b64\u5904\u4ea6\u8bb0\u4e3a$b$), \u8fb9$a$\u4ea6\u7136. \u6545\u590d\u8fed\u6620\u5c04$p: E \\to S^1 \\vee $$ S^1$\u7684Number of Sheets\u7b49\u4e8e\u53d8\u5f62\u7684\u6b63\u65b9\u5f62\u4e0e\u5706\u7684\u76f8\u4ea4\u7684\u9876\u70b9\u6570, 4.<\/p>\n<p>7. \u8bbe$p: E \\to B$\u662f\u6709\u9650\u590d\u8fed, $q: \\widetilde{E} \\to E$\u4e5f\u662f\u590d\u8fed\u6620\u5c04, \u8bc1\u660e$p \\circ q $$ : $$ \\widetilde{E} \\to B$\u662f\u590d\u8fed\u6620\u5c04.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u53d6$b \\in B$, \u4ee4$y_1, \\cdots, y_n$\u4e3a$b$\u5728\u6620\u5c04$p$\u4e0b\u7684\u539f\u50cf, $U$\u4e3a$b$\u5728\u590d\u8fed\u6620\u5c04$p: $$ E \\to B$\u4e0b\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df. \u5bf9\u4e8e$\\forall i = 1, \\cdots, n$, \u4ee4$U_i$\u4e3a$y_i$\u5728\u590d\u8fed\u6620\u5c04$q: \\widetilde{E} \\to E$\u4e0b\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df, $V := U_i \\cap q^{-1}(U)$, \u5219\u6211\u4eec\u6709<br \/>\n$\\\\$ $\\cdot$ $V_i$\u4e3a$y_i$\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df(\u67d0\u70b9\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df\u7684\u5305\u542b\u8be5\u70b9\u7684\u5f00\u5b50\u96c6\u4ecd\u4e3a\u8be5\u70b9\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df);<br \/>\n$\\\\$ $\\cdot$ $p | _{V_i}$\u4e3a$V_i$\u4e0e$p(V_i)$\u4e4b\u95f4\u7684\u540c\u80da;<br \/>\n$\\\\$ $\\cdot$ $V_i$\u5176\u4e3a\u590d\u8fed\u6620\u5c04$q: \\widetilde{E} \\to E$\u4e0b\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765, \u6211\u4eec\u5b9a\u4e49$W \\subseteq B$\u4e3a$$W := \\bigcap_{i = 1}^{n} p(V_i).$$\u7531\u4e8e$n$\u662f\u6709\u9650\u7684, $W$\u4e3a$B$\u7684\u4e00\u4e2a\u5f00\u5b50\u96c6. \u6b64\u5916, \u7531\u4e8e$W$\u88ab\u5305\u542b\u5728$b$\u7684\u4e00\u4e2a\u5728\u590d\u8fed\u6620\u5c04$p: E \\to B$\u4e0b\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df\u4e2d, \u6545$W$\u4ea6\u4e3a$b$\u7684\u4e00\u4e2a\u5747\u5300\u590d\u8fed\u90bb\u57df. \u53e6\u4e00\u65b9\u9762, \u7531\u4e8e$p^{-1}(W) \\subseteq \\cup^n_{i = 1}V_i$, \u6211\u4eec\u6709$p^{-1}(W) $$ = $$ \\cup^n_{i = 1}W_i$, \u5176\u4e2d, \u5bf9\u4e8e$\\forall i = 1, \\cdots, $$ n$, $W_i$\u88ab\u5305\u542b\u4e8e$V_i$\u4e2d, \u6545$W_i$\u4ea6\u4e3a$y_i$\u7684\u5728\u590d\u8fed\u6620\u5c04$q: \\widetilde{E} \\to E$\u4e0b\u7684\u4e24\u4e24\u4e0d\u76f8\u4ea4\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df.<br \/>\n$\\\\$ \u6700\u540e\u6211\u4eec\u6765\u8bc1\u660e$W$\u662f$b$\u7684\u4e00\u4e2a\u5728\u590d\u8fed\u6620\u5c04$p \\circ q$\u4e0b\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df. \u6ce8\u610f\u5230$(p \\circ $$ q)^{-1}(W) = \\cup^n_{i = 1}q^{-1}(W_i)$, \u5176\u4e2d$q^{-1}(W_i)$\u662f\u4e24\u4e24\u4e0d\u76f8\u4ea4\u7684(\u56e0\u4e3a$W_i$\u662f\u4e24\u4e24\u4e0d\u76f8\u4ea4\u7684). \u6b64\u5916, \u7531\u4e8e\u4efb\u4e00$W_i$\u5747\u4e3a\u590d\u8fed\u6620\u5c04$q$\u4e0b\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df, \u6211\u4eec\u6709$q^{-1}(W_i) $$ = \\cup_{j \\in J_i}T_i^j$, \u5176\u4e2d, $T_j^i \\subseteq \\widetilde{E}$\u662f\u4e24\u4e24\u4e0d\u76f8\u4ea4\u7684, $q | _{T_i^j} : T_i^j \\to W_i$\u4e3a\u4e00\u4e2a\u540c\u80da. \u56e0\u6b64, \u6211\u4eec\u53ef\u5f97$(p \\circ q $$ ) | _{T_i^j} = p | _{W_i} $$ \\circ q | _{T_i^j} : T_i^j \\to W$\u4e3a\u4e00\u4e2a\u540c\u80da, \u56e0\u4e3a\u5b83\u4e3a\u4e24\u4e2a\u540c\u80da\u7684\u590d\u5408. \u7efc\u4e0a\u6240\u8ff0, $W$\u4e3a$b \\in B$\u7684\u4e00\u4e2a\u5728\u590d\u8fed\u6620\u5c04$p \\circ q$\u4e0b\u7684\u5747\u5300\u590d\u8fed\u90bb\u57df. \u6839\u636e$b $$ \\in B$\u7684\u9009\u53d6\u7684\u4efb\u610f\u6027, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p>8. \u8bbe$X$\u662f\u4e00\u4e2a\u4ee5$S^n$\u4e3a\u590d\u8fed\u7a7a\u95f4\u7684$n$\u7ef4\u6d41\u5f62, \u8bc1\u660e\u8fd9\u4e2a\u590d\u8fed\u4e00\u5b9a\u662f\u6709\u9650\u590d\u8fed.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u53d6\u5b9a$x \\in X$, $p^{-1}(x)$\u662f$S^n$\u7684\u95ed\u5b50\u96c6, \u56e0\u6b64\u7d27\u81f4. \u4e8e\u662f\u53d6$x$\u7684\u4e00\u4e2a\u5747\u5300\u590d\u8fed\u90bb\u57df$U$, \u5219$p^{-1}(U) = \\cup_{i = 1}^m U_i$\u7684\u6709\u9650\u591a\u5c42\u5c31\u53ef\u4ee5\u8986\u76d6$p^{-1}( $$ x)$\u4e86. \u82e5$m$\u662f\u65e0\u9650\u7684, \u5219\u7531\u4e8e$U_i$\u662f\u4e24\u4e24\u4e0d\u76f8\u4ea4\u7684, $p^{-1}(U) = $$ \\cup_{i = 1}^m U_i$\u9664\u53bb\u4e00\u4e2a\u6709\u9650\u5b50\u8986\u76d6\u4ee5\u540e, \u4e0d\u5305\u542b$p^{-1}(x)$\u4e2d\u7684\u4efb\u4f55\u70b9, \u8fd9\u4e0e\u590d\u8fed\u6620\u5c04\u7684\u5b9a\u4e49\u77db\u76fe. \u7efc\u4e0a\u6240\u8ff0, $m$\u662f\u6709\u9650\u7684, \u547d\u9898\u5f97\u8bc1.<br \/>\n$\\\\$ PS: \u8be5\u547d\u9898\u4ea6\u53ef\u4ece\u540c\u8c03\u7fa4\u7684\u89d2\u5ea6\u51fa\u53d1\u53bb\u8bc1\u660e, \u8be6\u89c1\u53c2\u8003\u6750\u65996. <\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u6bcf\u5929\u7684\u5929\u6c14\u90fd\u662f\u706b\u7089\u5929\u6c14, \u5728\u5bb6\u91cc\u5b9e\u5728\u662f\u6709\u70b9\u96be\u53d7, \u4e2a\u4eba\u611f\u89c9\u53bb\u5e74\u540c\u4e00\u65f6\u5019\u7684\u6e29\u5ea6\u8981\u4f4e\u4e00\u4e9b\u2026\u2026 \u672c\u6587\u4e3b\u8981\u662f\u5173\u4e8e\u7ea4 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/07\/10\/fibration_covering_map_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u7ea4\u7ef4\u5316\u4e0e\u590d\u8fed\u6620\u5c04\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\/1912"}],"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=1912"}],"version-history":[{"count":89,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1912\/revisions"}],"predecessor-version":[{"id":3626,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1912\/revisions\/3626"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1912"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1912"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1912"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}