(This is the thirteenth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the twelfth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)
- We restate the strong lemma.
- A handful of spaces are presented to illustrate the following theorem.
- A theorem about Baire-Class-1 functions is stated. (Which is proved by forcing!)
- The proof strategy is discussed.
The Strong Lemma
Strong Lemma (restated). Assume
THEN for and such that
Question. Can we get for all ?
“You would like to call this lemma trivial, but somehow it always escapes.”
“The proof is by induction on dimension.”
“Try to strengthen the statement and find counterexamples.“
A Topology Apology
We will use forcing to analyze compact subsets of – the collection of Baire-class-1 functions on some Polish space with topology of pointwise convergence. I.e. .
“I need to apologize for this topology stuff.“
Definition. A Baire-class-1 function is a countable limit of continuous functions.
Example 1 (Stone-Cech Compactification of ). For and projection onto the ^th coordinate, .
This is a counterexample to Helley’s Theorem.
Helley’s Theorem. Every bounded sequence of monotone functions has a convergent subsequence.
Example 2 (Helley’s Space). We define .
Example 1.5: Take a separable Banach space with . Look at , where is the dual of , and is the unit ball of a space . By the Banach-Alaoglu Theroem is compact.
and the middle family is a compact set of Baire-Class-1 functions.
Example 3 (Subspace of Helley’s Space). Define , by , where the functions are monotone and only take 2 values.
“This object is important and obviously separable. It is part of a dichotomy. Either you contain a compact, separable collection of Baire-class-1 functions or … [doesn’t want to say.] We could spend a lot of time here, but we should move on.”
Note. In examples 3 and 4, the space contains no uncountable metric space.
Theorem. Every compact set of Baire-class-1 functions has a dense metrizable subspace.
We will outline a general strategy, and sketch the interesting parts.
(1) Take .
The key here is bounded (it is not important that it is compact) and relatively compact. i.e. .
“That is what is wrong with example 1, it is not relatively comapct.“
What is the forcing notion? The regular open algebra .
(2) Force with .
(3) Analyze in the forcing extension.
“Then the miracle happens: It is the right thing!“
To each we associate which is in . Define which is relatively compact in .
“Extending continuous to continuous is a classical (~60s) move. We do for Baire-class-1.“
(4) The generic filter of is countably generated.
“Why you want to prove that? Because you must. (4) is a corollary of the theorem.”
“ collapses something, always.”
“Now we are in business.”
(5) The set of -points of is dense in .
(Claim. This follows from (2) and (3))
Collapse to by countable conditions, then , so it has a point.
[If not, build a complete subtree of height (???)]
(6) [Corollary of (4)] has a -disjoint -base.
(7) [Corollary of (5) and (6)] There is a dense metrizable subspace.
Sketch of Proof
“The two most interesting are (3) and (4).“
(3) “What is the continuous function? What is the code? A sequence of pre-images. You want to use something absolute (), not .”
For and define , a monotone collection of open sets. Thus define .
What is the code? Let
What if you take ?
So let with each continuous. Then is a sequence of continuous functions.
Yes! We say , which is well-defined.
“So this is how you extend.”
“Now comes the surprising part. Pointwise accumulation points are Baire-class-1. This is a first order? (no), second order? (no), third order? (yes!) statement.“