· Stone Prime Filter Theorem
· The Stone space of a bounded distributive lattice and of a Boolean algebra.
Material for this week: 10.15-10.18 (also please read 10.20-10.22), Chapter 11 from beginning through 11.2.
1. Prove Theorem 10.18 using filters instead of ideals.
2. Let B be the set all subsets of N that are either finite or their complement is finite.
(a) Show that B is a sublattice of the power set of N.
(b) Is B a complete lattice?
(c) Find all the prime filters of B [Hint: they are all principal except for one].
(d) Identify the topology on the dual space of B.
3. (a) Let B be the lattice from exercise 2, and D = B Å 1. Identify the poset of prime filters of D as well as the topology on the dual space of D.
(b) Let F be the bounded sublattice of B generated by the singleton sets. Identify the poset of prime filters of F as well as the topology on the dual space of F.
(c) Show that the posets of prime filters of D and F are isomorphic, but that D and F aren't isomorphic.