{"id":97,"date":"2013-02-13T18:45:57","date_gmt":"2013-02-13T17:45:57","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=97"},"modified":"2023-03-31T14:09:03","modified_gmt":"2023-03-31T12:09:03","slug":"course-ideas","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=97","title":{"rendered":"Course Ideas"},"content":{"rendered":"<p>Here are few ideas of courses that can be given, based on the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>.<\/p>\n<ul>\n<li><strong>Introduction to topology<\/strong>: level L3 (European)\/bachelor level, 10 x 2h.\n<ul>\n<li>Metric spaces, the example of the Euclidean place. \u00a0Convergence. \u00a0Examples of convergent, of non-convergent sequences (e.g., based on Figure 3.3). \u00a0Read: Section 3.1, Section 3.2 until warning sign on p. 22.<\/li>\n<li>(Sequentially) closed and open subsets of a metric space. \u00a0Read: Section 3.2.<\/li>\n<li>Compactness. \u00a0The Borel-Lebesgue Theorem. \u00a0Tychonoff&#8217;s Theorem for finite products of compact metric spaces. \u00a0Read: Section 3.3.<\/li>\n<li>Completeness. \u00a0The Banach Fixed Point Theorem. The compact metric spaces are the complete, precompact metric spaces. \u00a0Read: Section 3.4.<\/li>\n<li>Continuous maps. \u00a0Preservation of limits. \u00a0Images of compact subspaces. \u00a0Lipschitz maps, uniformly continuous maps. \u00a0Read: Section 3.5 until and including Corollary 3.5.6, with its proof (p.40).<\/li>\n<li>Notions of convergence on spaces of continuous maps. \u00a0Pointwise, uniform convergence. \u00a0The Arzel\u00e0-Ascoli Theorem. \u00a0Read: Section 3.6.<\/li>\n<li>Beyond metric spaces: topological spaces. \u00a0Generalizing opens, closed subsets, and continuity. \u00a0Bases and subbases. \u00a0Separation properties. \u00a0Read: Section 4.1, Section 4.3.<\/li>\n<li>Compactness in the general topological setting. \u00a0Read: Section 4.4.<\/li>\n<li>The product topology, and Tychonoff&#8217;s Theorem (general form). \u00a0Read: Section 4.5.<\/li>\n<li>Back to convergence: Moore-Smyth convergence, nets, Kelley&#8217;s Theorem. \u00a0Read: Section 4.7.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Advanced topology for domain theory<\/strong>: level M2 (European)\/first year of PhD in theoretical computer science, 12 x 2h; ideally, paired with or following a course on semantics of programming languages.\n<ul>\n<li>A quick summary of basic topological notions: opens, closed subsets, continuous maps, compact subsets and spaces, products, Tychonoff&#8217;s Theorem. \u00a0The important example of the Scott topology (Section 4.2), dcpos. \u00a0A few warnings: Scott is not Hausdorff, compact subsets need not be closed, limits are not unique (if you decide to talk about limits). \u00a0Read: Sections 4.1 through 4.5.<\/li>\n<li>Continuous dcpos, locally compact spaces. \u00a0Why continuous dcpos matter: e.g., observe that products are not the same in the categories <strong>Cpo<\/strong> and <strong>Top<\/strong>, but this hassle is avoided with continuous dcpos. \u00a0Read: Section 5.1.<\/li>\n<li>Topologies on spaces of functions 1: core-compactness, as a refinement of local compactness; relevance to the lattice of open subsets; the exponentiable spaces are the core-compact spaces; uniqueness of the exponential topology. \u00a0Read: Sections 5.2 through 5.4.<\/li>\n<li>Topologies on spaces of functions 2: Cartesian-closed categories, relevance to programming language semantics; an important Cartesian-closed category, bc-domains. \u00a0Read: Sections 4.12, 5.5 (relevant parts needed to understand categories, products, Cartesian-closedness), Section 5.7.<\/li>\n<li>Alternative Cartesian-closed categories: C-generated spaces, Kelley spaces, Day&#8217;s theorem. \u00a0Read: Section 5.6. \u00a0(This lecture is optional, depending on time spent on the previous lectures.)\n<ul>\n<li>Home project 1: why the Hausdorff C-generated spaces are not satisfactory in computer science, after K. H. Hofmann\u00a0and M. Mislove&#8217;s\u00a0<a href=\"https:\/\/scholar.google.fr\/scholar_url?hl=fr&amp;q=https:\/\/citeseerx.ist.psu.edu\/viewdoc\/download%3Fdoi%3D10.1.1.206.819%26rep%3Drep1%26type%3Dpdf&amp;sa=X&amp;scisig=AAGBfm11bNbvB12WjuwPa7mf1ebRX-8fJg&amp;oi=scholarr&amp;ei=Ks4bUd3CHuuW0QWcy4DADA&amp;ved=0CCwQgAMoADAA\">paper<\/a>: explain why and explain the result of the paper.<\/li>\n<\/ul>\n<\/li>\n<li>Stone Duality 1: how can one recover a topological space from a purported description of its lattice of open subsets? \u00a0Frames, spatial lattices, sober spaces; sobrification. \u00a0Limits, characterization of sober spaces through limits. \u00a0Read: Sections 8.1, 8.2 (and 4.7 for limits).<\/li>\n<li>Stone Duality 2: The Hofmann-Mislove theorem, the Hofmann-Lawson theorem and various other equivalences between categories of topological spaces and of frames. \u00a0Read: Section 8.3.<\/li>\n<li>Stably compact spaces 1: introduction via Stone duality with fully arithmetic lattices; examples: compact\u00a0\u00a0Hausdorff spaces, bc-domains. \u00a0De Groot duality, and Nachbin&#8217;s theorem: compact pospaces. \u00a0Read: Section 9.1.<\/li>\n<li>Stably compact spaces 2: products and retracts of stably compact spaces; proper and perfect maps. \u00a0Read: Sections 9.3, 9.4.<\/li>\n<li>Stably compact spaces 3: spectral spaces. \u00a0Johnstone&#8217;s theorem: the stably compact spaces are the retracts of spectral spaces. \u00a0Stone duality in its original form; Priestley spaces. \u00a0Read: Section 9.5.<\/li>\n<li>Stably compact spaces 4: bifinite domains, retracts of bifinite domains, yielding larger Cartesian-closed categories of continuous dcpos than just bc-domains. \u00a0Read: Section 9.6.\n<ul>\n<li>Home project 2: study A. Jung&#8217;s FS-domains, and do Exercises 9.6.25 through 9.6.32.<\/li>\n<li>Home project 3: apply the theory of spectral spaces this to understand Samson&#8217;s Abramsky\u00a0<a href=\"https:www.cs.ox.ac.uk\/files\/295\/dtlf.pdf\">Domain Theory in Logical Form<\/a>.<\/li>\n<\/ul>\n<\/li>\n<li>Stably compact spaces 5: Noetherian spaces, wqos, the topological Higman and Kruskal theorems. \u00a0Read: Section 9.7.\n<ul>\n<li>Home project 3: explain application in verification given in <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/mes_publis.php?onlykey=JGL-icalp10\">this paper<\/a>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right;\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\" rel=\"attachment wp-att-993\">Jean Goubault-Larrecq<\/a>\u00a0(February 13th, 2013)<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-993 alignright\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" width=\"32\" height=\"44\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here are few ideas of courses that can be given, based on the book. Introduction to topology: level L3 (European)\/bachelor level, 10 x 2h. Metric spaces, the example of the Euclidean place. \u00a0Convergence. \u00a0Examples of convergent, of non-convergent sequences (e.g., &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=97\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-97","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/97","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=97"}],"version-history":[{"count":9,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/97\/revisions"}],"predecessor-version":[{"id":5973,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/97\/revisions\/5973"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}