{"id":2438,"date":"2022-10-23T14:22:20","date_gmt":"2022-10-23T06:22:20","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=2438"},"modified":"2022-10-23T16:26:54","modified_gmt":"2022-10-23T08:26:54","slug":"summary_study_notes_introduction_point_set_topology_algebraic_topology","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/23\/summary_study_notes_introduction_point_set_topology_algebraic_topology\/","title":{"rendered":"\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u5b66\u4e60\u7b14\u8bb0\u6c47\u603b"},"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>\u524d\u540e\u5386\u65f6\u5c06\u8fd1\u4e24\u5e74, \u7ec8\u4e8e\u5229\u7528\u5404\u79cd\u4e1a\u4f59\u65f6\u95f4\u5b66\u5b8c\u4e86\u5305\u5fd7\u5f3a\u8001\u5e08\u7684\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u4e00\u4e66. \u5b66\u5b8c\u672c\u4e66\u5e76\u4e0d\u610f\u5473\u7740\u4ee3\u6570\u62d3\u6251\u5b66\u4e60\u7684\u7ed3\u675f, \u76f8\u53cd\u5730, \u8fd9\u6070\u6070\u662f\u81ea\u5df1\u4ee3\u6570\u62d3\u6251\u5b66\u4e60\u7684\u5f00\u59cb. \u5e73\u5fc3\u800c\u8bba, \u5b66\u5b8c\u672c\u4e66, \u81ea\u5df1\u53ea\u80fd\u8bf4\u5bf9\u5404\u79cd\u4ee3\u6570\u62d3\u6251\u7684\u6982\u5ff5\u6709\u4e86\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u7684\u4e86\u89e3, \u4f46\u79bb\u771f\u6b63\u610f\u4e49\u4e0a\u7684\u878d\u4f1a\u8d2f\u901a\u8fd8\u6709\u7740\u975e\u5e38\u8fdc\u7684\u8ddd\u79bb; \u6b64\u5916, \u7531\u4e8e\u6ca1\u6709\u5b9a\u65f6\u5730\u590d\u4e60\u4e4b\u524d\u6240\u5b66\u7684\u5185\u5bb9, \u4e5f\u5bfc\u81f4\u81ea\u5df1\u5bf9\u4e4b\u524d\u5b66\u4e60\u7684\u5185\u5bb9\u751f\u758f\u4e86\u4e0d\u5c11. \u63a5\u4e0b\u6765, \u81ea\u5df1\u4f1a\u7ed9\u81ea\u5df1\u5728\u4e1a\u4f59\u65f6\u95f4\u653e\u4e2a\u5c0f\u5047\u671f, \u901a\u5173\u4e00\u4e0b\u4e4b\u524d\u4e00\u76f4\u60f3\u901a\u5173\u7684\u6e38\u620f(\u4f8b\u5982\u5deb\u5e083\u7b49\u2026\u2026), \u800c\u540e\u4f1a\u5f00\u59cb\u7814\u7a76Dey T K, Fan F, Wang Y. An efficient computation of handle and tunnel loops via Reeb graphs[J]. ACM Transactions on Graphics (TOG), 2013, 32(4): 1-10.\u8fd9\u7bc7\u8bba\u6587, \u6bd5\u7adf\u81ea\u5df1\u5e76\u4e0d\u6ee1\u8db3\u4e8e\u4ec5\u4ec5\u662f\u5b66\u4e60, \u8fd8\u671f\u671b\u81ea\u5df1\u80fd\u591f\u7814\u7a76\u4e00\u4e9b\u5185\u5bb9. \u672a\u6765, \u81ea\u5df1\u8fd8\u6253\u7b97\u5b66\u4e60Hatcher\u7684Algebraic Topology\u4e00\u4e66, \u4e3b\u8981\u662f\u56e0\u4e3a\u5305\u5fd7\u5f3a\u8001\u5e08\u7684\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u4e00\u4e66\u5927\u7bc7\u5e45\u4ecb\u7ecd\u4e86\u540c\u4f26\u76f8\u5173\u7684\u5185\u5bb9, \u4f46\u5bf9\u540c\u8c03\u76f8\u5173\u7684\u5185\u5bb9\u4ec5\u4ec5\u662f\u5c0f\u7bc7\u5e45\u5e26\u8fc7, \u8fd9\u5728\u4ee3\u6570\u62d3\u6251\u7684\u5b66\u4e60\u4e2d\u662f\u8fdc\u8fdc\u4e0d\u591f\u7684. \u5b66\u65e0\u6b62\u5883, \u8c28\u4ee5\u672c\u6587\u6c47\u603b\u4e4b\u524d\u5b66\u4e60\u5305\u5fd7\u5f3a\u8001\u5e08\u7684\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u4e00\u4e66\u7684\u6240\u6709\u7b14\u8bb0~<\/p>\n<p><!--more--><\/p>\n<p><strong>\u7b2c\u4e00\u7ae0 \u62d3\u6251\u7a7a\u95f4\u4e0e\u8fde\u7eed\u6027<\/strong><br \/>\n$\\\\$ <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2020\/11\/02\/order_topology\/\">\u5173\u4e8e\u5728\u5177\u5907\u5b57\u5178\u5e8f\u5b9a\u4e49\u7684\u96c6\u5408\u4e0a\u5b9a\u4e49\u5e8f\u62d3\u6251\u7684\u95ee\u9898<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/05\/03\/open_connected_subspace_e2_path_connected\/\">\u8bc1\u660eE^2\u7684\u8fde\u901a\u5f00\u5b50\u96c6\u4e00\u5b9a\u9053\u8def\u8fde\u901a<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/05\/04\/discret_topology_r_compact_set\/\">\u5e26\u4e0a\u79bb\u6563\u62d3\u6251\u7684\u5b9e\u6570\u96c6R\u4e0d\u662f\u7d27\u81f4\u96c6<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/06\/08\/interior_points_relativity\/\">\u5185\u70b9\u7684\u76f8\u5bf9\u6027<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/25\/gluing_map\/\">\u7c98\u5408\u6620\u5c04<\/a><\/p>\n<p><strong>\u7b2c\u4e8c\u7ae0 \u5e38\u7528\u70b9\u96c6\u62d3\u6251\u6027\u8d28<\/strong><br \/>\n$\\\\$ <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/05\/23\/lebesgue_cover_theorem\/\">Lebesgue\u8986\u76d6\u5b9a\u7406\u8bc1\u660e\u7ec6\u8282\u6ce8\u89e3<\/a><\/p>\n<p><strong>\u7b2c\u4e09\u7ae0 \u95ed\u66f2\u9762\u7684\u62d3\u6251\u5206\u7c7b<\/strong><br \/>\n$\\\\$ <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/06\/17\/manifold_regular_space\/\">\u6d41\u5f62\u662f\u6b63\u5219\u7a7a\u95f4\u7684\u8bc1\u660e<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/06\/27\/abstract_simplex_and_geometric_realization\/#more-896\">\u62bd\u8c61\u5355\u5f62\u53ca\u5176\u51e0\u4f55\u5b9e\u73b0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/01\/real_projective_plane_and_homogeneous_coordinates1\/\">\u5b9e\u5c04\u5f71\u5e73\u9762\u53ca\u56fe\u5f62\u5b66\u4e2d\u7684\u9f50\u6b21\u5750\u6807(\u4e00)<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/11\/real_projective_plane_and_homogeneous_coordinates2\/\">\u5b9e\u5c04\u5f71\u5e73\u9762\u53ca\u56fe\u5f62\u5b66\u4e2d\u7684\u9f50\u6b21\u5750\u6807(\u4e8c)<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/17\/homogeneous_space_clipping\/\">\u9f50\u6b21\u7a7a\u95f4\u88c1\u526a<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/13\/closed_surface_connected_sum\/\">\u95ed\u66f2\u9762\u7684\u8fde\u901a\u548c\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/22\/orientation\/\">\u53ef\u5b9a\u5411\u6027\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/29\/homology\/\">\u540c\u8c03\u7684\u76f8\u5173\u7406\u89e3\u53ca\u8bc1\u660e<\/a><\/p>\n<p><strong>\u7b2c\u56db\u7ae0 \u57fa\u672c\u7fa4\u53ca\u5176\u5e94\u7528<\/strong><br \/>\n$\\\\$ <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/09\/25\/homotopy_fixed_point\/\">\u5173\u4e8e\u4e0d\u52a8\u70b9\u7684\u540c\u4f26\u95ee\u9898<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/10\/03\/deformation_retraction_mark\/\">\u5f62\u53d8\u6536\u7f29\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/10\/09\/group_common_knowledge_review\/\">\u7fa4\u7684\u5e38\u7528\u77e5\u8bc6\u590d\u4e60<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/10\/24\/fundamental_group_mark\/\">\u57fa\u672c\u7fa4\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/11\/03\/s1_multiply_connected\/\">S^1\u662f\u591a\u8fde\u901a\u7684<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/11\/07\/fundamental_group_related_exercises\/\">\u57fa\u672c\u7fa4\u76f8\u5173\u4e60\u9898<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/12\/05\/free_group_mark\/\">\u81ea\u7531\u7fa4\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/12\/12\/direct_sum_and_direct_product\/\">\u76f4\u548c\u4e0e\u76f4\u79ef<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/01\/09\/coefficient_matrix_abelianized_relations_mark\/\">\u4ea4\u6362\u5316\u7cfb\u6570\u77e9\u9635\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/07\/seifert_van_kampen_theorem\/\">Seifert &#038; Van Kampen\u5b9a\u7406<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/12\/generator_related_terminology_description\/\">\u751f\u6210\u5143\u76f8\u5173\u672f\u8bed\u8bf4\u660e<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/02\/13\/examples_seifert_van_kampens_theorem_mark\/\">Seifert &#038; Van Kampen\u5b9a\u7406\u76f8\u5173\u4f8b\u5b50\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/05\/seifert_van_kampen_theorem_related_exercises\/\">Seifert &#038; Van Kampen\u5b9a\u7406\u76f8\u5173\u4e60\u9898<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/20\/fundamental_groups_applications_mark\/\">\u57fa\u672c\u7fa4\u53ca\u5176\u5e94\u7528\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/27\/degree_mapping\/\">\u6620\u5c04\u5ea6<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/04\/10\/nulhomotopy_lemma\/\">\u96f6\u4f26\u5f15\u7406<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/05\/jordan_curve_theorem_proof_mark\/\">Jordan\u66f2\u7ebf\u5b9a\u7406\u8bc1\u660e\u6ce8\u8bb0<\/a><\/p>\n<p><strong>\u7b2c\u4e94\u7ae0 \u590d\u8fed\u7a7a\u95f4<\/strong><br \/>\n$\\\\$ <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/29\/group_action\/\">\u7fa4\u4f5c\u7528<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/29\/free_group_action_proper_maps_and_proper_discontinuity\/\">\u81ea\u7531\u7fa4\u4f5c\u7528, Proper Maps\u4e0e\u7eaf\u4e0d\u8fde\u7eed<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/06\/12\/hyperbolic_tree\/\">\u53cc\u66f2\u6811<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/06\/25\/cayley_klein_metric\/\">Cayley-Klein\u5ea6\u91cf<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/07\/10\/fibration_covering_map_mark\/\">\u7ea4\u7ef4\u5316\u4e0e\u590d\u8fed\u6620\u5c04\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/08\/20\/fundamental_groups_covering_spaces_mark\/\">\u590d\u8fed\u7a7a\u95f4\u7684\u57fa\u672c\u7fa4\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/08\/21\/winding_number_non-closed_curve\/\">\u975e\u95ed\u66f2\u7ebf\u4e0a\u7684\u5708\u6570<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/09\/17\/existence_universal_covering_space_mark\/\">\u6cdb\u590d\u8fed\u7a7a\u95f4\u7684\u5b58\u5728\u6027\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/03\/lifting_criterion_mark\/\">\u6620\u5c04\u63d0\u5347\u5b9a\u7406\u6ce8\u8bb0<\/a><br \/>\n<a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/14\/deck_transformation_mark\/\">\u590d\u8fed\u53d8\u6362\u6ce8\u8bb0<\/a><\/p>\n<p>\u6700\u540e, \u63a8\u8350\u4e00\u4e2a\u81ea\u5df1\u770b\u4e86\u89c9\u5f97\u8fd8\u7b97\u4e0d\u9519\u7684\u4e00\u4e2a\u4ee3\u6570\u62d3\u6251\u7684\u590d\u4e60\u89c6\u9891: <a href=\"https:\/\/www.bilibili.com\/video\/BV1eJ411e7oA\/\">\u7b80\u5355\u7684\u4ee3\u6570\u62d3\u6251\u603b\u590d\u4e60\uff08\u4ec5\u4f9b\u8003\u524d\u590d\u4e60\u7528\uff09<\/a>. \u53ef\u60dc\u7684\u662f, \u8be5\u89c6\u9891\u4ec5\u4ec5\u5927\u81f4\u590d\u4e60\u4e86\u7b2c\u56db\u7ae0\u7684\u5185\u5bb9, \u5176\u5b83\u7ae0\u8282\u7684\u5185\u5bb9\u5219\u57fa\u672c\u672a\u63d0\u53ca; \u6b64\u5916, \u89c6\u9891\u6700\u540e\u4e00\u5c0f\u90e8\u5206\u4e5f\u6bd4\u8f83\u6c34, Up\u4e3b\u8c8c\u4f3c\u5bf9\u5f62\u53d8\u6536\u7f29\u7684\u6982\u5ff5\u638c\u63e1\u5f97\u4e0d\u662f\u5f88\u597d(\u5927\u96fe\u2026\u2026 \u9003\u03b5=\u03b5=\u03b5=\u250f(\u309c\u30ed\u309c;)\u251b).<\/p>\n<p>PS: \u4e0a\u8ff0\u6240\u6709\u7b14\u8bb0\u7684\u683c\u5f0f\u5747\u5728\u516c\u53f8\u7535\u8111\u4e0a\u7ecf\u8fc7\u591a\u6b21\u8c03\u6574, \u4f46\u65e0\u6cd5\u4fdd\u8bc1\u5728\u6240\u6709\u8bbe\u5907\u4e0a\u5747\u80fd\u591f\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u6548\u679c.<\/p>\n<p>\u5b8c\u7ed3, \u6492\u82b1*\u2605,\u00b0*:.\u2606(\uffe3\u25bd\uffe3)\/$:*.\u00b0\u2605* \u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u524d\u540e\u5386\u65f6\u5c06\u8fd1\u4e24\u5e74, \u7ec8\u4e8e\u5229\u7528\u5404\u79cd\u4e1a\u4f59\u65f6\u95f4\u5b66\u5b8c\u4e86\u5305\u5fd7\u5f3a\u8001\u5e08\u7684\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u4e00\u4e66. \u5b66\u5b8c\u672c\u4e66\u5e76\u4e0d\u610f\u5473\u7740 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/10\/23\/summary_study_notes_introduction_point_set_topology_algebraic_topology\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u300a\u70b9\u96c6\u62d3\u6251\u4e0e\u4ee3\u6570\u62d3\u6251\u5f15\u8bba\u300b\u5b66\u4e60\u7b14\u8bb0\u6c47\u603b<\/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\/2438"}],"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=2438"}],"version-history":[{"count":20,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2438\/revisions"}],"predecessor-version":[{"id":2583,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2438\/revisions\/2583"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2438"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2438"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2438"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}