{"id":149,"date":"2024-03-17T12:00:00","date_gmt":"2024-03-17T11:00:00","guid":{"rendered":"https:\/\/www.maths-sup.com\/?p=149"},"modified":"2024-03-17T23:36:43","modified_gmt":"2024-03-17T22:36:43","slug":"espace-compact","status":"publish","type":"post","link":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/","title":{"rendered":"Espace compact"},"content":{"rendered":"\n<p>Bien que la notion de compacit\u00e9 puisse sembler abstraite de prime abord, en particulier lorsqu&rsquo;on s&rsquo;attarde sur sa d\u00e9finition, elle se r\u00e9v\u00e8le \u00eatre d&rsquo;une grande richesse en termes d&rsquo;applications pratiques.<\/p>\n\n\n\n<p>Il est important de noter que, dans cet article, notre discussion sur la compacit\u00e9 se limitera aux espaces m\u00e9triques. Cette restriction nous permet de rester dans un cadre coh\u00e9rent avec les exigences de l&rsquo;agr\u00e9gation, bien qu&rsquo;elle puisse \u00e9galement s&rsquo;appliquer dans des contextes plus g\u00e9n\u00e9raux, tels que les espaces topologiques. Dans le contexte des espaces m\u00e9triques, les propri\u00e9t\u00e9s topologiques sont exprimables en termes de convergence de suites, ce qui facilite leur compr\u00e9hension et leur manipulation.<br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Cas de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span><\/h2>\n\n\n\n<p>Pour saisir pleinement l&rsquo;importance de la notion de compacit\u00e9 et ses applications, il est essentiel de bien comprendre son application dans le cas sp\u00e9cifique de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>.<\/p>\n\n\n\n<p>On rappelle les d\u00e9finitions\/propri\u00e9t\u00e9s suivantes:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Soit <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> un sous-ensemble de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>.<br><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> est <strong>ferm\u00e9<\/strong> si et seulement si toute de suite de <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> convergente dans <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span> a sa limite dans <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>.<\/li>\n\n\n\n<li>Soit <span class=\"katex-eq\" data-katex-display=\"false\">(u_n)_{n \\in \\mathbb{N}}<\/span> une suite.<br>Une <strong>extraction<\/strong> de <span class=\"katex-eq\" data-katex-display=\"false\">(u_n)_{n \\in \\mathbb{N}}<\/span> une suite de type <span class=\"katex-eq\" data-katex-display=\"false\">(u_{\\phi(n)})_{n \\in \\mathbb{N}}<\/span>, avec <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> strictement croissante.<br>On dit que <span class=\"katex-eq\" data-katex-display=\"false\">(u_n)_{n \\in \\mathbb{N}}<\/span> admet <strong>une extraction convergente<\/strong> si il existe <br><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> strictement croissante telle que <span class=\"katex-eq\" data-katex-display=\"false\">(u_n)_{n \\in \\mathbb{N}}<\/span> converge.<\/li>\n<\/ul>\n\n\n\n<p><strong><strong>Th\u00e9or\u00e8me fondamental de la compacit\u00e9<\/strong> &#8211; Bolzano-Weierstrass:<\/strong><br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span> est une partie ferm\u00e9e et born\u00e9e de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>.<br>Alors: toute suite de <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span> admet une extraction convergente dans <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span>.<\/p>\n\n\n\n<p><strong>Une mani\u00e8re concr\u00e8te de reformuler le th\u00e9or\u00e8me<\/strong>: si <span class=\"katex-eq\" data-katex-display=\"false\">(u_n)<\/span> est une suite dans une partie ferm\u00e9e et born\u00e9e, alors <span class=\"katex-eq\" data-katex-display=\"false\">(u_n)<\/span> \u00ab\u00a0passe une infinit\u00e9 de fois au m\u00eame endroit\u00a0\u00bb.<br>Plus formellement: il existe un point <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tel que tout voisinage de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> contient une infinit\u00e9 de termes de la suite.<\/p>\n\n\n\n<p>Ceci conduit naturellement \u00e0 la d\u00e9finition d&rsquo;un espace compact:<\/p>\n\n\n\n<p><strong>D\u00e9finition:<\/strong> Soit <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> un espace m\u00e9trique.<br>On dit que <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> est <strong>compact<\/strong> si: toute suite de <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> admet une extraction convergente.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Propri\u00e9t\u00e9 de Borel-Lebesgue<\/h2>\n\n\n\n<p>Les espaces compactes v\u00e9rifie la propri\u00e9t\u00e9 suivant qu&rsquo;on appelle la <strong>propri\u00e9t\u00e9 de Borel-Lebesgue<\/strong>.<\/p>\n\n\n\n<p><strong>Lemme de Borel-Lebesgue<\/strong>:<br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span> un espace m\u00e9trique compact.<br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">(U_i)_{I \\in I}<\/span> une famille d&rsquo;ouverts de <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span> telle que <span class=\"katex-eq\" data-katex-display=\"false\">\\bigcup_{i \\in I} U_i = K<\/span>.<br>Alors il existe une partie finie <span class=\"katex-eq\" data-katex-display=\"false\">J \\subset I<\/span> telle que <span class=\"katex-eq\" data-katex-display=\"false\">\\bigcup_{i \\in J} U_i = K<\/span>.<\/p>\n\n\n\n<p>Dite de mani\u00e8re plus concise: tout recouvrement par des ouverts admet un sous-recouvrement fini.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Propri\u00e9t\u00e9s des espaces compacts<\/h2>\n\n\n\n<p><strong>Th\u00e9or\u00e8me de Heine:<\/strong><br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span> deux espaces m\u00e9triques.<br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">f : X \\longrightarrow Y<\/span> une fonction continue.<br>On suppose que <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> est compacte.<br>Alors <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> est uniform\u00e9ment continue.<\/p>\n\n\n\n<p><strong>Preuve:<\/strong> On raisonne par l&rsquo;absurde.<br>Supposons que <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> n&rsquo;est pas uniform\u00e9ment continue, i.e.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\exists \\varepsilon&gt;0, \\forall \\eta &gt;0, \\exists (x,y) \\in X^2, \\quad |x-y| \\leq \\eta \\text{ et } |f(x)-f(y)| &gt;\\varepsilon.<\/span><\/p>\n\n\n\n<p>Alors il existe <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon&gt;0<\/span> et deux suites <span class=\"katex-eq\" data-katex-display=\"false\">(x_n)_{n \\geq 1}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">(y_n)_{n \\geq 1}<\/span> tels que:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall n \\geq 1, \\quad |x_n - y_n| \\leq \\frac{1}{n} \\text{ et } |f(x_n)-f(y_n)| &gt;\\varepsilon.<\/span><\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> est compact, il existe une extraction <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> telle que <span class=\"katex-eq\" data-katex-display=\"false\">(x_{\\phi(n)})_n<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">(y_{\\phi(n)})_n<\/span> convergent.<br>On note <span class=\"katex-eq\" data-katex-display=\"false\">x := \\lim x_{\\phi(n)}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">y := \\lim y_{\\phi(n)}<\/span>.<br>Comme <span class=\"katex-eq\" data-katex-display=\"false\">\\lim |x_{\\phi(n)} - y_{\\phi(n)}| = 0<\/span>, on a <span class=\"katex-eq\" data-katex-display=\"false\">x=y<\/span>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> est continue, on a:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim |f(x_n)-f(y_n)| = |f(x) - f(y)|<\/span><\/p>\n\n\n\n<p>Or <span class=\"katex-eq\" data-katex-display=\"false\">|f(x_n)-f(y_n)| &gt;\\varepsilon<\/span>,<br>donc <span class=\"katex-eq\" data-katex-display=\"false\">\\lim |f(x_n)-f(y_n)| \\geq \\varepsilon<\/span>.<br>Ce qui contredit <span class=\"katex-eq\" data-katex-display=\"false\">x=y<\/span>. CQFD<\/p>\n\n\n\n<p><strong>Proposition &#8211; Image d&rsquo;un compact:<\/strong><br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span> deux espaces m\u00e9triques.<br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">f : X \\longrightarrow Y<\/span> une fonction continue.<br>On suppose que <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> est compacte.<br>Alors <span class=\"katex-eq\" data-katex-display=\"false\">f(X)<\/span> est compact.<\/p>\n\n\n\n<p>Un <strong>corollaire important<\/strong> est:<br>Si <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> est une fonction continue sur un compact \u00e0 valeurs dans <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>,<br>alors <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> est born\u00e9e et atteint ses bornes.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bien que la notion de compacit\u00e9 puisse sembler abstraite de prime abord, en particulier lorsqu&rsquo;on s&rsquo;attarde sur sa d\u00e9finition, elle se r\u00e9v\u00e8le \u00eatre d&rsquo;une grande richesse en termes d&rsquo;applications pratiques. Il est important de noter que, dans cet article, notre discussion sur la compacit\u00e9 se limitera aux espaces m\u00e9triques. Cette [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":175,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[12,14,13],"tags":[],"class_list":["post-149","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-compacite","category-lecon-203","category-topologie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Espace compact - Maths-Sup<\/title>\n<meta name=\"description\" content=\"Plongez au c\u0153ur de la compacit\u00e9 en math\u00e9matiques \ud83e\udde0\ud83d\udca1! Que vous soyez \u00e9tudiant en math\u00e9matiques, pr\u00e9parant l&#039;agr\u00e9gation, ou simplement curieux des myst\u00e8res math\u00e9matiques, cet article est pour vous.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Espace compact - Maths-Sup\" \/>\n<meta property=\"og:description\" content=\"Plongez au c\u0153ur de la compacit\u00e9 en math\u00e9matiques \ud83e\udde0\ud83d\udca1! Que vous soyez \u00e9tudiant en math\u00e9matiques, pr\u00e9parant l&#039;agr\u00e9gation, ou simplement curieux des myst\u00e8res math\u00e9matiques, cet article est pour vous.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/\" \/>\n<meta property=\"og:site_name\" content=\"Maths-Sup\" \/>\n<meta property=\"article:published_time\" content=\"2024-03-17T11:00:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-03-17T22:36:43+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp\" \/>\n\t<meta property=\"og:image:width\" content=\"1792\" \/>\n\t<meta property=\"og:image:height\" content=\"1024\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/webp\" \/>\n<meta name=\"author\" content=\"Auguste Hoang\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"Auguste Hoang\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/\"},\"author\":{\"name\":\"Auguste Hoang\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25\"},\"headline\":\"Espace compact\",\"datePublished\":\"2024-03-17T11:00:00+00:00\",\"dateModified\":\"2024-03-17T22:36:43+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/\"},\"wordCount\":847,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/www.maths-sup.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp\",\"articleSection\":[\"Compacit\u00e9\",\"Le\u00e7on 203\",\"Topologie\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/\",\"url\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/\",\"name\":\"Espace compact - Maths-Sup\",\"isPartOf\":{\"@id\":\"https:\/\/www.maths-sup.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp\",\"datePublished\":\"2024-03-17T11:00:00+00:00\",\"dateModified\":\"2024-03-17T22:36:43+00:00\",\"description\":\"Plongez au c\u0153ur de la compacit\u00e9 en math\u00e9matiques \ud83e\udde0\ud83d\udca1! Que vous soyez \u00e9tudiant en math\u00e9matiques, pr\u00e9parant l'agr\u00e9gation, ou simplement curieux des myst\u00e8res math\u00e9matiques, cet article est pour vous.\",\"breadcrumb\":{\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage\",\"url\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp\",\"contentUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp\",\"width\":1792,\"height\":1024},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\/\/www.maths-sup.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Espace compact\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.maths-sup.com\/#website\",\"url\":\"https:\/\/www.maths-sup.com\/\",\"name\":\"Maths-Sup\",\"description\":\"Votre Guide Complet pour l&#039;Agr\u00e9gation de Math\u00e9matiques\",\"publisher\":{\"@id\":\"https:\/\/www.maths-sup.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.maths-sup.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"fr-FR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.maths-sup.com\/#organization\",\"name\":\"Maths-Sup\",\"url\":\"https:\/\/www.maths-sup.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png\",\"contentUrl\":\"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png\",\"width\":1024,\"height\":1024,\"caption\":\"Maths-Sup\"},\"image\":{\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25\",\"name\":\"Auguste Hoang\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/www.maths-sup.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g\",\"caption\":\"Auguste Hoang\"},\"sameAs\":[\"https:\/\/www.maths-sup.com\"],\"url\":\"https:\/\/www.maths-sup.com\/index.php\/author\/auguste-hoang\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Espace compact - Maths-Sup","description":"Plongez au c\u0153ur de la compacit\u00e9 en math\u00e9matiques \ud83e\udde0\ud83d\udca1! Que vous soyez \u00e9tudiant en math\u00e9matiques, pr\u00e9parant l'agr\u00e9gation, ou simplement curieux des myst\u00e8res math\u00e9matiques, cet article est pour vous.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/","og_locale":"fr_FR","og_type":"article","og_title":"Espace compact - Maths-Sup","og_description":"Plongez au c\u0153ur de la compacit\u00e9 en math\u00e9matiques \ud83e\udde0\ud83d\udca1! Que vous soyez \u00e9tudiant en math\u00e9matiques, pr\u00e9parant l'agr\u00e9gation, ou simplement curieux des myst\u00e8res math\u00e9matiques, cet article est pour vous.","og_url":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/","og_site_name":"Maths-Sup","article_published_time":"2024-03-17T11:00:00+00:00","article_modified_time":"2024-03-17T22:36:43+00:00","og_image":[{"width":1792,"height":1024,"url":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp","type":"image\/webp"}],"author":"Auguste Hoang","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"Auguste Hoang","Dur\u00e9e de lecture estim\u00e9e":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#article","isPartOf":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/"},"author":{"name":"Auguste Hoang","@id":"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25"},"headline":"Espace compact","datePublished":"2024-03-17T11:00:00+00:00","dateModified":"2024-03-17T22:36:43+00:00","mainEntityOfPage":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/"},"wordCount":847,"commentCount":0,"publisher":{"@id":"https:\/\/www.maths-sup.com\/#organization"},"image":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage"},"thumbnailUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp","articleSection":["Compacit\u00e9","Le\u00e7on 203","Topologie"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/","url":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/","name":"Espace compact - Maths-Sup","isPartOf":{"@id":"https:\/\/www.maths-sup.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage"},"image":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage"},"thumbnailUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp","datePublished":"2024-03-17T11:00:00+00:00","dateModified":"2024-03-17T22:36:43+00:00","description":"Plongez au c\u0153ur de la compacit\u00e9 en math\u00e9matiques \ud83e\udde0\ud83d\udca1! Que vous soyez \u00e9tudiant en math\u00e9matiques, pr\u00e9parant l'agr\u00e9gation, ou simplement curieux des myst\u00e8res math\u00e9matiques, cet article est pour vous.","breadcrumb":{"@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/"]}]},{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#primaryimage","url":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp","contentUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/03\/DALL\u00b7E-2024-03-17-23.31.02-A-classic-school-chalkboard-hanging-on-a-dark-classroom-wall.-The-chalkboard-is-filled-with-mathematical-formulas-written-in-white-chalk-featuring-s.webp","width":1792,"height":1024},{"@type":"BreadcrumbList","@id":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/17\/espace-compact\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/www.maths-sup.com\/"},{"@type":"ListItem","position":2,"name":"Espace compact"}]},{"@type":"WebSite","@id":"https:\/\/www.maths-sup.com\/#website","url":"https:\/\/www.maths-sup.com\/","name":"Maths-Sup","description":"Votre Guide Complet pour l&#039;Agr\u00e9gation de Math\u00e9matiques","publisher":{"@id":"https:\/\/www.maths-sup.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.maths-sup.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"fr-FR"},{"@type":"Organization","@id":"https:\/\/www.maths-sup.com\/#organization","name":"Maths-Sup","url":"https:\/\/www.maths-sup.com\/","logo":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/","url":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png","contentUrl":"https:\/\/www.maths-sup.com\/wp-content\/uploads\/2024\/01\/logo_mahs_sup.png","width":1024,"height":1024,"caption":"Maths-Sup"},"image":{"@id":"https:\/\/www.maths-sup.com\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/www.maths-sup.com\/#\/schema\/person\/3db9aed7da3584090baf6ea59af7ff25","name":"Auguste Hoang","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.maths-sup.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/1d43100e5c3215e0e670d56118dca059dded0701685d6d4a0e2eb80874f150fc?s=96&d=mm&r=g","caption":"Auguste Hoang"},"sameAs":["https:\/\/www.maths-sup.com"],"url":"https:\/\/www.maths-sup.com\/index.php\/author\/auguste-hoang\/"}]}},"_links":{"self":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts\/149","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/comments?post=149"}],"version-history":[{"count":25,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts\/149\/revisions"}],"predecessor-version":[{"id":176,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/posts\/149\/revisions\/176"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/media\/175"}],"wp:attachment":[{"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/media?parent=149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/categories?post=149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.maths-sup.com\/index.php\/wp-json\/wp\/v2\/tags?post=149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}