WEEK 43:

 

·         Priestley space of a bounded distributive lattice

 

·         Definition of the Cantor space

 

·         The Priestley space of  LINDA is homeomorphic to the Cantor space

 

 

Homework:

 

1.* Let B be the set of all those subsets S of N so that S is finite or the complement of S is finite.

(a) Show that B is a Boolean subalgebra of the power set of N.

(b) 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.

 

2.* (a) Let B be the lattice from exercise 1, and  D = B Å 1, the distributive lattice obtained by adding a new top to B. 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.

(d) Show that the dual spaces of D and F are also homeomorphic as topological spaces but that they aren’t order-homeomorphic.

 

3.* A subset of a topological space is clopen if both the set itself and its complement are in the topology.

(a) Show that the clopen subsets of any topological space is a Boolean subalgebra of the power set of the set underlying the space.

(b) Find the Boolean algebra of all clopen subsets of the Cantor space.