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)