(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.)

**Summary**

- 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 ?

Question

“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.

Note

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.”

**Example 4**.

**Note**. In examples 3 and 4, the space contains no uncountable metric space.

## The Theorem

**Theorem**. Every compact set of Baire-class-1 functions has a dense metrizable subspace.

## Strategy

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

So .

**What if you take **?

So let with each continuous. Then is a sequence of continuous functions.

**Is convergent?**

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.“

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