Stevo’s Forcing Class Fall 2012 – Class 13

(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.)

The Strong Lemma

Strong Lemma (restated). Assume

  • <p_s : s \in \omega^d> \subseteq \textrm{Fn}(I, \omega) ;
  • \vert \textrm{dom}(p_s) \vert \leq m ;
  • \vert \textrm{ran}(p_s) \vert \leq n, \forall s \in \omega^d ;

THEN \forall k, \exists M_i \subseteq \omega for i<d and \vert M_i \vert \geq k such that
\displaystyle  \bigcup_{s \in \prod_{i < d} M_i} p_s \in \textrm{Fn}(I, \omega)

. Can we get \vert M_i \vert = \aleph_0 for all i < d ?

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 \mathcal{B}_1 (X) – the collection of Baire-class-1 functions on some Polish space X with topology of pointwise convergence. I.e. \mathcal{B}_1 (X) \subseteq \mathbb{R}^X .

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 \mathbb{N} ). For X = 2^\N and f_n :=  projection onto the n ^th coordinate, \beta(\N) = \overline{\{f_n : n <\omega\}} \subseteq \{0,1\}^{2^\N} .

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 H := \{f \in [0,1]^{[0,1]} : f \textrm{ is monotone}\} .

Example 1.5: Take X a separable Banach space with \ell_1 \not \subseteq X . Look at \mathcal{B}_1 (B_{X^*}) , where X^* is the dual of X , and B_Y is the unit ball of a space Y . By the Banach-Alaoglu Theroem B_{X^*} is compact.

\displaystyle  \mathcal{B}_1 (B_{X^*}) \supseteq B_{X^{X^*}} \supseteq B_X
and the middle family is a compact set of Baire-Class-1 functions.

Example 3 (Subspace of Helley’s Space). Define S(I) \subseteq H , by S(I) := [0,1]\times \{0,1\} , 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. \{\mathbb{1}_{\{x\}} : x \in 2^{\N}\} \cup \{\overline{0}\} \subseteq \mathcal{B}_1 (2^\N)

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.


We will outline a general strategy, and sketch the interesting parts.

(1) Take F \subseteq \mathcal{B}_1 (\N^\N) .

The key here is bounded (it is not important that it is compact) and relatively compact. i.e. \overline{F} \subseteq \mathcal{B}_1 (\N^\N) .

That is what is wrong with example 1, it is not relatively comapct.

What is the forcing notion? The regular open algebra \mathcal{B} := \textrm{ro}(F) .

(2) Force with \mathcal{B} := \textrm{ro}(F) .
(3) Analyze \overline{F} in the forcing extension.

Then the miracle happens: It is the right thing!

To each f \in F we associate \hat{f} : (\N^\N)^{V^\mathcal{B}} \rightarrow \mathbb{R}^{V^\mathcal{B}} which is in \mathcal{B}_1 ((\N^\N)^{V^\mathcal{B}}) . Define \hat{F} := \{\hat{f} : f \in F\} which is relatively compact in \mathcal{B}_1 ((\N^\N)^{V^\mathcal{B}}) .

Extending continuous to continuous is a classical (~60s) move. We do for Baire-class-1.

(4) The generic filter of \B is countably generated.

Why you want to prove that? Because you must. (4) is a corollary of the theorem.

\B collapses something, always.”

Now we are in business.”

(5) The set of G_\delta -points of \overline{F} = K is dense in K .

(Claim. This follows from (2) and (3))

Collapse 2^{\aleph_0} to \aleph_1 by countable conditions, then \vert K \vert = \aleph_1 , so it has a G_\delta point.

[If not, build a complete subtree of height \aleph_1 (???)]

(6) [Corollary of (4)] K has a \sigma -disjoint \pi -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 (\mathbb{Q} ), not \mathbb{R} .”

For f : \mathbb{N}^\N \rightarrow \mathbb{R} and q \in \mathbb{Q} define O_q^f := f^{-1} (-\infty, q) , a monotone collection of open sets. Thus define f(x) := \inf\{q: x \in O_q^f\} .

What is the code? Let \lceil O_q^f \rceil := \{t \in \mathbb{N}^{<\N} : [t] \subseteq O_q^f\}

So \hat{f} \in V^{\mathcal{B}} .

What if you take g \in \mathcal{B}_1 (\N^\N) ?

So let f_n \rightarrow g with f_n each continuous. Then (\hat{f_n}) is a sequence of continuous functions.

Is (\hat{f_n}) convergent?

Yes! We say \hat{g} := \lim \hat{f_n} , 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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