(In the fall of 2012 I will be taking Stevo Todorcevic’s class in Forcing at the University of Toronto. I will try to publish my notes here, although that won’t always be possible.)
Summary of class 1.
- Discuss examples of ccc posets whose product is not ccc.
- Prove a theorem of Baumgartner’s that relates the branches in a tree to its antichain structure.
- Display the differences in chain conditions.
This class is going to be held Wed 1-2 and Friday 3:30-5:30 in BA 6183.
The theme for today is “Things that are ccc, but whose products are not”. So first some examples:
Example 2. Let finite antichains of , where is a Suslin Tree. Then is powerfully ccc (its finite products are ccc), but is not ccc.
“A Suslin tree is the skeleton of a Suslin continuum. If you take x-rays, that’s what you get.” – Stevo
This example was first studied by Baumgartner in his thesis:
Theorem (Baumgartner, 1969): TFAE for a tree :
- is ccc;
- has no uncountable branches.
It is clear that NOT (2) imples NOT (1), as the singletons of a branch form an antichain.
“There are many proofs of this fact, but Baumgartner’s original proof is still the most useful.” – Stevo
Proof. Like most proofs that a poset is ccc, we start with an uncountable subset of the poset and refine, refine, refine.
Let , we want to find such that . We refine 3 times to an uncountable set so that:
- (All conditions are the same length) for all ,
- ( is a -sytem) There is an such that for in we have .
- (Heights are increasing) The for .
We want to assume that is actually empty, and we can do this by considering . Note that implies .
WLOG, we get .
We suppose that for all in (and look for a contradiction). This means that if then there is an and a such that .
Now take a uniform ultrafilter on such that .
Fix . Write .
There is an such that .
Question: Is there a large antichain in X?
Now for each there is a so that . Thus and we can choose an in this intersection so that for all .
Now if does not depend on then we are done. This follows from exercise 2.
Exercise 1: Suppose that is a ccc poset and let . Show that there is an infinite set such that are pairwise compatible.
Exercise 2: Let be a tree with an uncountable subset that has no antichains of size . Show that has an uncountable chain.
Some chain conditions, listed from easiest to satisfy to hardest to satisfy:
- powerfully ccc
- productively ccc
- -n-linked ()
Note that Borel posets can distinguish all of the blue and dark blue properties. By the following exercise, the dark blue posets are small.
Exercise 3. If is an atomless -(2)-linked poset then .
Hint. Use the Erdös-Rado theorem.
Question. Can a boolean algebra supporting a continuous submeasure algebra distinguish -finite-cc from -bounded-cc?