# Course Ideas

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.  Convergence.  Examples of convergent, of non-convergent sequences (e.g., based on Figure 3.3).  Read: Section 3.1, Section 3.2 until warning sign on p. 22.
• (Sequentially) closed and open subsets of a metric space.  Read: Section 3.2.
• Compactness.  The Borel-Lebesgue Theorem.  Tychonoff’s Theorem for finite products of compact metric spaces.  Read: Section 3.3.
• Completeness.  The Banach Fixed Point Theorem. The compact metric spaces are the complete, precompact metric spaces.  Read: Section 3.4.
• Continuous maps.  Preservation of limits.  Images of compact subspaces.  Lipschitz maps, uniformly continuous maps.  Read: Section 3.5 until and including Corollary 3.5.6, with its proof (p.40).
• Notions of convergence on spaces of continuous maps.  Pointwise, uniform convergence.  The Arzelà-Ascoli Theorem.  Read: Section 3.6.
• Beyond metric spaces: topological spaces.  Generalizing opens, closed subsets, and continuity.  Bases and subbases.  Separation properties.  Read: Section 4.1, Section 4.3.
• Compactness in the general topological setting.  Read: Section 4.4.
• The product topology, and Tychonoff’s Theorem (general form).  Read: Section 4.5.
• Back to convergence: Moore-Smyth convergence, nets, Kelley’s Theorem.  Read: Section 4.7.
• Advanced topology for domain theory: 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.
• A quick summary of basic topological notions: opens, closed subsets, continuous maps, compact subsets and spaces, products, Tychonoff’s Theorem.  The important example of the Scott topology (Section 4.2), dcpos.  A few warnings: Scott is not Hausdorff, compact subsets need not be closed, limits are not unique (if you decide to talk about limits).  Read: Sections 4.1 through 4.5.
• Continuous dcpos, locally compact spaces.  Why continuous dcpos matter: e.g., observe that products are not the same in the categories Cpo and Top, but this hassle is avoided with continuous dcpos.  Read: Section 5.1.
• 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.  Read: Sections 5.2 through 5.4.
• Topologies on spaces of functions 2: Cartesian-closed categories, relevance to programming language semantics; an important Cartesian-closed category, bc-domains.  Read: Sections 4.12, 5.5 (relevant parts needed to understand categories, products, Cartesian-closedness), Section 5.7.
• Alternative Cartesian-closed categories: C-generated spaces, Kelley spaces, Day’s theorem.  Read: Section 5.6.  (This lecture is optional, depending on time spent on the previous lectures.)
• Home project 1: why the Hausdorff C-generated spaces are not satisfactory in computer science, after K. H. Hofmann and M. Mislove’s paper: explain why and explain the result of the paper.
• Stone Duality 1: how can one recover a topological space from a purported description of its lattice of open subsets?  Frames, spatial lattices, sober spaces; sobrification.  Limits, characterization of sober spaces through limits.  Read: Sections 8.1, 8.2 (and 4.7 for limits).
• Stone Duality 2: The Hofmann-Mislove theorem, the Hofmann-Lawson theorem and various other equivalences between categories of topological spaces and of frames.  Read: Section 8.3.
• Stably compact spaces 1: introduction via Stone duality with fully arithmetic lattices; examples: compact  Hausdorff spaces, bc-domains.  De Groot duality, and Nachbin’s theorem: compact pospaces.  Read: Section 9.1.
• Stably compact spaces 2: products and retracts of stably compact spaces; proper and perfect maps.  Read: Sections 9.3, 9.4.
• Stably compact spaces 3: spectral spaces.  Johnstone’s theorem: the stably compact spaces are the retracts of spectral spaces.  Stone duality in its original form; Priestley spaces.  Read: Section 9.5.
• Stably compact spaces 4: bifinite domains, retracts of bifinite domains, yielding larger Cartesian-closed categories of continuous dcpos than just bc-domains.  Read: Section 9.6.
• Home project 2: study A. Jung’s FS-domains, and do Exercises 9.6.25 through 9.6.32.
• Home project 3: apply the theory of spectral spaces this to understand Samson’s Abramsky Domain Theory in Logical Form.
• Stably compact spaces 5: Noetherian spaces, wqos, the topological Higman and Kruskal theorems.  Read: Section 9.7.
• Home project 3: explain application in verification given in this paper.

Jean Goubault-Larrecq (February 13th, 2013)