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# Domains

We will begin by reading the first five chapters of the Pisa Notes by Gordon Plotkin. The weekly meetings will consist of discussion about our different solutions/attempts to solve the exercises, in the fashion of Moore.

### Errata

• Page 1, one should have C:Stat→C.
• Page 9 ex 8 is lacking an assumption: the dcpo should be pointed.
• Page 22 ex 12: the given definition of V-sequentiality is wrong. It should be read:
“[…] iff either f is a constant or, for each x, there is an integer i (with 0≤in) such that […]”

### Hints for some exercises

• 1.8. Pataria found a proof (in 1997!) that doesn't invoke transfinite induction (link, exercise O-2.21).
• 1.19. There is a tricky proof in the book Domains and lambda-calculi by R. Amadio and P.L. Curien. It proceeds in two steps (proposition 7.1.27 and corollary 7.1.28):
• There is no monotone surjection from D to Dop → 𝕆, hence there is no monotone surjection from D to (D→𝕆)→𝕆,
• There is no retraction from DD to D since otherwise that would imply the above (using the retraction from D to 𝕆).
• The function space construction does not preserve ω-algebraicity. The counterexamples are lacking some explanation: the idea is to show that approx(idD) is not directed, hence D → D cannot be ω-algebraic.