Covering an uncountable square by countably many continuous - - PowerPoint PPT Presentation

Covering an uncountable square by countably many continuous functions

Wiesław Kubi´ s

Instytut Matematyki Akademia ´ Swie ¸tokrzyska Kielce, POLAND http://www.pu.kielce.pl/˜wkubis/

Measure Theory Edward Marczewski Centennial Conference Be ¸dlewo, 9–15 September 2007

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 1 / 14

Motivations

Theorem (Sierpi´ nski)

Let S be a set of cardinality ℵ1. Then there exists a sequence of functions {fn : S → S}n∈ω, such that S × S =

• n∈ω

(fn ∪ f −1

n ).

Proof.

We assume that S = ω1. For each β ∈ S fix a surjection gβ : ω → β + 1. Define fn(β) = gβ(n).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 2 / 14

Motivations

Theorem (Sierpi´ nski)

Let S be a set of cardinality ℵ1. Then there exists a sequence of functions {fn : S → S}n∈ω, such that S × S =

• n∈ω

(fn ∪ f −1

n ).

Proof.

We assume that S = ω1. For each β ∈ S fix a surjection gβ : ω → β + 1. Define fn(β) = gβ(n).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 2 / 14

Motivations

Theorem (Sierpi´ nski)

Let S be a set of cardinality ℵ1. Then there exists a sequence of functions {fn : S → S}n∈ω, such that S × S =

• n∈ω

(fn ∪ f −1

n ).

Proof.

We assume that S = ω1. For each β ∈ S fix a surjection gβ : ω → β + 1. Define fn(β) = gβ(n).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 2 / 14

Motivations

Theorem (Sierpi´ nski)

Let S be a set of cardinality ℵ1. Then there exists a sequence of functions {fn : S → S}n∈ω, such that S × S =

• n∈ω

(fn ∪ f −1

n ).

Proof.

We assume that S = ω1. For each β ∈ S fix a surjection gβ : ω → β + 1. Define fn(β) = gβ(n).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 2 / 14

Motivations

Theorem (Sierpi´ nski)

Let S be a set of cardinality ℵ1. Then there exists a sequence of functions {fn : S → S}n∈ω, such that S × S =

• n∈ω

(fn ∪ f −1

n ).

Proof.

We assume that S = ω1. For each β ∈ S fix a surjection gβ : ω → β + 1. Define fn(β) = gβ(n).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 2 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Remark (Sierpi´ nski)

If S has the above property then |S| ℵ1.

Proof.

Fix A ∈ [S]ℵ1. For each x ∈ A let Fx = {fn(x): n ∈ ω}. The set

x∈A Fx has cardinality ℵ1.

Suppose p ∈ S is such that p / ∈ Fx for x ∈ A. For each a ∈ A there is n(a) ∈ ω such that a = fn(a)(p). The map a → n(a) must be one-to-one. A contradiction.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 3 / 14

Question

Is it possible that the square of some uncountable subset of R is covered by countably many continuous real functions and their inverses? In other words:

Question

Does there exist a family {fn : R → R}n∈ω consisting of continuous functions such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some S ∈ [R]ℵ1?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 4 / 14

Question

Is it possible that the square of some uncountable subset of R is covered by countably many continuous real functions and their inverses? In other words:

Question

Does there exist a family {fn : R → R}n∈ω consisting of continuous functions such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some S ∈ [R]ℵ1?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 4 / 14

Question

Is it possible that the square of some uncountable subset of R is covered by countably many continuous real functions and their inverses? In other words:

Question

Does there exist a family {fn : R → R}n∈ω consisting of continuous functions such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some S ∈ [R]ℵ1?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 4 / 14

Question

How about covering by (continuous) non-decreasing functions? Suppose S × S ⊆

n∈ω(fn ∪ f −1 n ), where each fn : S → S is a

non-decreasing function. Then both fn and f −1

n

are chains in S × S. Thus, if |S| > ℵ0 then S is a Countryman type!

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 5 / 14

Question

How about covering by (continuous) non-decreasing functions? Suppose S × S ⊆

n∈ω(fn ∪ f −1 n ), where each fn : S → S is a

non-decreasing function. Then both fn and f −1

n

are chains in S × S. Thus, if |S| > ℵ0 then S is a Countryman type!

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 5 / 14

Question

How about covering by (continuous) non-decreasing functions? Suppose S × S ⊆

n∈ω(fn ∪ f −1 n ), where each fn : S → S is a

non-decreasing function. Then both fn and f −1

n

are chains in S × S. Thus, if |S| > ℵ0 then S is a Countryman type!

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 5 / 14

Question

How about covering by (continuous) non-decreasing functions? Suppose S × S ⊆

n∈ω(fn ∪ f −1 n ), where each fn : S → S is a

non-decreasing function. Then both fn and f −1

n

are chains in S × S. Thus, if |S| > ℵ0 then S is a Countryman type!

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 5 / 14

Proposition

There exists a compact line K and a family {fn : K → K}n∈ω consisting

• f continuous non-decreasing functions such that

S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ K.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 6 / 14

Proposition

There exists a compact line K and a family {fn : K → K}n∈ω consisting

• f continuous non-decreasing functions such that

S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ K.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 6 / 14

Another motivation

Proposition (Shelah [5])

There exists an Fσ set A ⊆ R2 with the following properties. S × S ⊆ A for some uncountable set S. X × Y ⊆ A whenever X, Y ∈ [R]ℵ2. X × Y ⊆ A whenever X, Y are perfect subsets of R.

Question

Is it possible that A =

n∈ω(fn ∪ f −1 n ), where each fn is a continuous

real function?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 7 / 14

Another motivation

Proposition (Shelah [5])

There exists an Fσ set A ⊆ R2 with the following properties. S × S ⊆ A for some uncountable set S. X × Y ⊆ A whenever X, Y ∈ [R]ℵ2. X × Y ⊆ A whenever X, Y are perfect subsets of R.

Question

Is it possible that A =

n∈ω(fn ∪ f −1 n ), where each fn is a continuous

real function?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 7 / 14

Another motivation

Proposition (Shelah [5])

There exists an Fσ set A ⊆ R2 with the following properties. S × S ⊆ A for some uncountable set S. X × Y ⊆ A whenever X, Y ∈ [R]ℵ2. X × Y ⊆ A whenever X, Y are perfect subsets of R.

Question

Is it possible that A =

n∈ω(fn ∪ f −1 n ), where each fn is a continuous

real function?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 7 / 14

Another motivation

Proposition (Shelah [5])

There exists an Fσ set A ⊆ R2 with the following properties. S × S ⊆ A for some uncountable set S. X × Y ⊆ A whenever X, Y ∈ [R]ℵ2. X × Y ⊆ A whenever X, Y are perfect subsets of R.

Question

Is it possible that A =

n∈ω(fn ∪ f −1 n ), where each fn is a continuous

real function?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 7 / 14

Another motivation

Proposition (Shelah [5])

There exists an Fσ set A ⊆ R2 with the following properties. S × S ⊆ A for some uncountable set S. X × Y ⊆ A whenever X, Y ∈ [R]ℵ2. X × Y ⊆ A whenever X, Y are perfect subsets of R.

Question

Is it possible that A =

n∈ω(fn ∪ f −1 n ), where each fn is a continuous

real function?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 7 / 14

Another motivation

Proposition (Shelah [5])

There exists an Fσ set A ⊆ R2 with the following properties. S × S ⊆ A for some uncountable set S. X × Y ⊆ A whenever X, Y ∈ [R]ℵ2. X × Y ⊆ A whenever X, Y are perfect subsets of R.

Question

Is it possible that A =

n∈ω(fn ∪ f −1 n ), where each fn is a continuous

real function?

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 7 / 14

Proposition

Assume {fn : S → S}n∈ω and A, B are uncountable sets such that A × B ⊆

• n∈ω

(fn ∪ f −1

n ).

Then |A| = |B| = ℵ1.

Proposition

Let {fn : R → R}n∈ω be a family of continuous functions. Then there are no perfect sets P, Q such that P × Q ⊆

• n∈ω

(fn ∪ f −1

n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 8 / 14

Proposition

Assume {fn : S → S}n∈ω and A, B are uncountable sets such that A × B ⊆

• n∈ω

(fn ∪ f −1

n ).

Then |A| = |B| = ℵ1.

Proposition

Let {fn : R → R}n∈ω be a family of continuous functions. Then there are no perfect sets P, Q such that P × Q ⊆

• n∈ω

(fn ∪ f −1

n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 8 / 14

Proposition

Assume {fn : S → S}n∈ω and A, B are uncountable sets such that A × B ⊆

• n∈ω

(fn ∪ f −1

n ).

Then |A| = |B| = ℵ1.

Proposition

Let {fn : R → R}n∈ω be a family of continuous functions. Then there are no perfect sets P, Q such that P × Q ⊆

• n∈ω

(fn ∪ f −1

n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 8 / 14

Proposition

Assume {fn : S → S}n∈ω and A, B are uncountable sets such that A × B ⊆

• n∈ω

(fn ∪ f −1

n ).

Then |A| = |B| = ℵ1.

Proposition

Let {fn : R → R}n∈ω be a family of continuous functions. Then there are no perfect sets P, Q such that P × Q ⊆

• n∈ω

(fn ∪ f −1

n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 8 / 14

Proposition

Assume {fn : S → S}n∈ω and A, B are uncountable sets such that A × B ⊆

• n∈ω

(fn ∪ f −1

n ).

Then |A| = |B| = ℵ1.

Proposition

Let {fn : R → R}n∈ω be a family of continuous functions. Then there are no perfect sets P, Q such that P × Q ⊆

• n∈ω

(fn ∪ f −1

n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 8 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Main result

Theorem

There exists a ccc forcing which introduces a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n )

for some uncountable set S ⊆ 2ω.

The forcing:

p ∈ P iff p = np, sp, vp, f p, γp, ̺p, where (1) np ∈ ω, sp ∈ [ω]<ω and vp ∈ [ω1]<ω; (2) f p = {f p

i }i∈sp ⊆ Lip1(2np, 2np) and ̺p : [vp]2 → sp;

(3) γp : vp → 2np is one-to-one; (4) γp(α) = f p

̺p(α,β)(γp(β)) whenever α < β and α, β ∈ vp.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 9 / 14

Corollaries

Theorem (ZFC)

There exist a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω and an uncountable set S ⊆ 2ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n ).

Proof.

By Keisler’s absoluteness theorem [2] for the language Lω1,ω(Q).

Theorem (ZFC)

There exist an ℵ1-dense set X ⊆ R and a family of continuous functions {fn : R → R}n∈ω such that X × X ⊆

n∈ω(fn ∪ f −1 n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 10 / 14

Corollaries

Theorem (ZFC)

There exist a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω and an uncountable set S ⊆ 2ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n ).

Proof.

By Keisler’s absoluteness theorem [2] for the language Lω1,ω(Q).

Theorem (ZFC)

There exist an ℵ1-dense set X ⊆ R and a family of continuous functions {fn : R → R}n∈ω such that X × X ⊆

n∈ω(fn ∪ f −1 n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 10 / 14

Corollaries

Theorem (ZFC)

There exist a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω and an uncountable set S ⊆ 2ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n ).

Proof.

By Keisler’s absoluteness theorem [2] for the language Lω1,ω(Q).

Theorem (ZFC)

There exist an ℵ1-dense set X ⊆ R and a family of continuous functions {fn : R → R}n∈ω such that X × X ⊆

n∈ω(fn ∪ f −1 n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 10 / 14

Corollaries

Theorem (ZFC)

There exist a family of 1-Lipschitz functions {fn : 2ω → 2ω}n∈ω and an uncountable set S ⊆ 2ω such that S × S ⊆

• n∈ω

(fn ∪ f −1

n ).

Proof.

By Keisler’s absoluteness theorem [2] for the language Lω1,ω(Q).

Theorem (ZFC)

There exist an ℵ1-dense set X ⊆ R and a family of continuous functions {fn : R → R}n∈ω such that X × X ⊆

n∈ω(fn ∪ f −1 n ).

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 10 / 14

Corollary

It is relatively consistent with ZFC that for every set X ∈ [R]ℵ1 there exists a sequence of continuous functions fn : R → R with X × X ⊆

n∈ω(fn ∪ f −1 n ).

Proof.

This holds in Baumgartner’s model [1] in which every two ℵ1-dense subsets of R are order isomorphic.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 11 / 14

Corollary

It is relatively consistent with ZFC that for every set X ∈ [R]ℵ1 there exists a sequence of continuous functions fn : R → R with X × X ⊆

n∈ω(fn ∪ f −1 n ).

Proof.

This holds in Baumgartner’s model [1] in which every two ℵ1-dense subsets of R are order isomorphic.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 11 / 14

Corollary

It is relatively consistent with ZFC that for every set X ∈ [R]ℵ1 there exists a sequence of continuous functions fn : R → R with X × X ⊆

n∈ω(fn ∪ f −1 n ).

Proof.

This holds in Baumgartner’s model [1] in which every two ℵ1-dense subsets of R are order isomorphic.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 11 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

There exists a family of continuous functions {un : 2ω → 2ω}n∈ω with the following properties:

1

For every family {gn : 2ω → 2ω}n∈ω consisting of continuous functions, there exist quotient maps k : 2ω → 2ω, ℓ: 2ω → 2ω and an injection ψ: ω → ω such that the diagram 2ω

uψ(n) k

• 2ω

ℓ

• 2ω

gn

2ω

commutes for every n ∈ ω.

2

Some sort of homogeneity. The above properties describe the family {un}n∈ω uniquely.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 12 / 14

Theorem

Let {un}n∈ω be the universal homogeneous family of functions from the previous theorem. Then X 2 ⊆

• n∈ω

(un ∪ u−1

n )

for some uncountable set X ⊆ 2ω.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 13 / 14

Theorem

Let {un}n∈ω be the universal homogeneous family of functions from the previous theorem. Then X 2 ⊆

• n∈ω

(un ∪ u−1

n )

for some uncountable set X ⊆ 2ω.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 13 / 14

References

• J. BAUMGARTNER, All ℵ1-dense sets of reals can be isomorphic,
• Fund. Math. 79 (1973) 101–106.
• J. KEISLER, Logic with quantifier “there exists uncountably many”,

Annals of Mathematical Logic 1 (1970) 1–93.

• W. KUBI´

S, S. SHELAH Analytic colorings, Ann. Pure Appl. Logic

121 (2003) 145–161.

• K. KURATOWSKI, Sur une caract´

erisation des alephs, Fund. Math. 38 (1951) 14–17.

• S. SHELAH, Borel sets with large squares, Fund. Math. 159 (1999)

1–50.

W.Kubi´ s (http://www.pu.kielce.pl/∼wkubis/) Covering ℵ1 × ℵ1 by ℵ0 functions Be ¸dlewo, 11 September 2007 14 / 14