- Read sections 1, 2, and 4 of Data Types as Lattices Update: I obtained a better typeset version from Scott which is here.
- Look up the definitions of
- topological space,
- open set,
- a basis for a topological space,
- and continuous functions between topological spaces. Give an instance of the above notions for the real number line.
- Prove that P(ω) is a topological space (as defined via open sets), using the basis given by Scott on page 584.
- Prove that a function is continuous (page 584) iff it is continuous in the topological sense, that is, if inverse images of open sets are open sets.
- Restate the proof of Theorem 2.2 in your own words. (The proof is in the Appendix, page 642.)
- Prove that (1), (2), and (3) on page 604 are retracts.
- Given the definition of int in equation (7) on page 606, prove that int(\bot) = \bot int(n) = n (n is an integer) int(\top) = \top
- Work out the semantics of (λx.x+1)(2) using the semantics defined on page 613. Show your work.