{"id":639,"date":"2017-11-16T00:16:25","date_gmt":"2017-11-16T08:16:25","guid":{"rendered":"http:\/\/wonghoi.humgar.com\/blog\/?p=639"},"modified":"2017-11-16T00:45:08","modified_gmt":"2017-11-16T08:45:08","slug":"super-simplified-what-is-a-topology","status":"publish","type":"post","link":"https:\/\/wonghoi.humgar.com\/blog\/2017\/11\/16\/super-simplified-what-is-a-topology\/","title":{"rendered":"Super-simplified: What is a topology"},"content":{"rendered":"<p>&#8216;Super-simplified&#8217; is my series of brief notes that summarizes what I have learned so I can pick it up at no time. That means summarizing an hour of lecture into a few takeaway points.<\/p>\n<p><iframe loading=\"lazy\" title=\"General Topology Introduction Part 1\" width=\"584\" height=\"329\" src=\"https:\/\/www.youtube.com\/embed\/tWIoUZNYj1g?list=PLa8Zx5-x7OuTYTsEXaX7bJIYBmyff6n1U\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<p>These lectures complemented my gap in understanding open sets in undergrad real analysis, which I understood it under the narrow world-view of the real line.<\/p>\n<hr \/>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-996ff7036e644e89f8ac379fa58d0cf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/>: Universal set<\/p>\n<p>Topology\u00a0\u2261 open + <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-336673f3b8795927d8741cb1bff8666e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#118;&#97;&#114;&#110;&#111;&#116;&#104;&#105;&#110;&#103;&#44;&#32;&#88;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"54\" style=\"vertical-align: -5px;\"\/><\/p>\n<p>Open\u00a0\u2261 preserved under unions, and <span style=\"text-decoration: underline;\">finite<\/span> intersections.<\/p>\n<p>Why <span style=\"text-decoration: underline;\">finite<\/span>\u00a0needed for intersections only? Infinite intersections can squeeze open edge points to limit points, e.g. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-d971da09c246fc7e188df35301660ee1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#105;&#103;&#99;&#97;&#112;&#94;&#123;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#95;&#123;&#110;&#125;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#41;&#32;&#61;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"139\" style=\"vertical-align: -6px;\"\/>.<\/p>\n<p>Never forget that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-336673f3b8795927d8741cb1bff8666e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#118;&#97;&#114;&#110;&#111;&#116;&#104;&#105;&#110;&#103;&#44;&#32;&#88;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"54\" style=\"vertical-align: -5px;\"\/> is always there because it might not have properties that the meat open set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-c74288aabc0e2ca280d25d92bf1a1ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> doesn&#8217;t have. e.g. a discrete topology of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-c2d15a912e0dab6620f72e426e48c8b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#81;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\"\/> on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-52ab2930ccb722d9f3576c62c49a6a6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#48;&#44;&#49;&#41;&#32;&#61;&#32;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-9b674e6e8e2e122ac6073ac810f234fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#98;&#115;&#101;&#116;&#101;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"12\" style=\"vertical-align: -3px;\"\/> universal set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-dcf20f7993ebe2426b1421f7692135e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\"\/> means for any irrational point, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-245bc9fa4fce4d5d0cd20d26f5bce0aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\"\/> is the only open-neighborhood (despite it looks far away) because they cannot be &#8216;synthesized*&#8217; from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-c2d15a912e0dab6620f72e426e48c8b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#81;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\"\/> using operation that preserves openness.<\/p>\n<p>* &#8216;synthesized&#8217; in here means constructed from union and\/or <span style=\"text-decoration: underline;\">finite<\/span> intersections.<\/p>\n<hr \/>\n<p>[Bonus] What I learned from real line topology in real analysis 101:<\/p>\n<ol>\n<li>Normal intuitive cases<\/li>\n<li>Null and universal set are clopen<\/li>\n<li>Look into rationals (countably infinite) and irrationals (uncountable)<\/li>\n<li>Blame Cantor (sets)!<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_639\" class=\"pvc_stats all  \" data-element-id=\"639\" style=\"\"><i class=\"pvc-stats-icon medium\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p>\n<div class=\"pvc_clear\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#8216;Super-simplified&#8217; is my series of brief notes that summarizes what I have learned so I can pick it up at no time. That means summarizing an hour of lecture into a few takeaway points. These lectures complemented my gap in &hellip; <a href=\"https:\/\/wonghoi.humgar.com\/blog\/2017\/11\/16\/super-simplified-what-is-a-topology\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_639\" class=\"pvc_stats all  \" data-element-id=\"639\" style=\"\"><i class=\"pvc-stats-icon medium\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p>\n<div class=\"pvc_clear\"><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[16,6,32],"tags":[],"class_list":["post-639","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-note-to-self","category-topology"],"_links":{"self":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/posts\/639","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/comments?post=639"}],"version-history":[{"count":12,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/posts\/639\/revisions"}],"predecessor-version":[{"id":642,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/posts\/639\/revisions\/642"}],"wp:attachment":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/media?parent=639"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/categories?post=639"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/tags?post=639"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}