Productively (and non-productively) Menger spaces Piotr Szewczak - PowerPoint PPT Presentation

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin Fin Fin Fin . . . A A A A F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin . . . A F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin . . . A F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin . . . A F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin . . . A F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact P ( N ) \ �

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin c • • • . . . A F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact, ∃ c ∈ [ N ] ∞ P ( N ) \ � P ( N ) \ � n O n ≤ c

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin c • • • . . . • • • A a F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact, ∃ c ∈ [ N ] ∞ P ( N ) \ � P ( N ) \ � n O n ≤ c

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin c • • • . . . • • • A a F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact, ∃ c ∈ [ N ] ∞ P ( N ) \ � P ( N ) \ � n O n ≤ c | A \ � n O n | < d

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin c • • • . . . • • • A a F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact, ∃ c ∈ [ N ] ∞ P ( N ) \ � P ( N ) \ � n O n ≤ c | A \ � n O n | < d ⇒ A \ � n O n is Menger

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin c • • • . . . • • • A a F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact, ∃ c ∈ [ N ] ∞ P ( N ) \ � P ( N ) \ � n O n ≤ c | A \ � n O n | < d ⇒ A \ � n O n is Menger

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin c • • • . . . • • • A a F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact, ∃ c ∈ [ N ] ∞ P ( N ) \ � P ( N ) \ � n O n ≤ c | A \ � n O n | < d ⇒ A \ � n O n is Menger

d -unbounded sets A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ is d -unbounded ⇒ A ∪ Fin is Menger Fin c • • • . . . • • • A a F 1 ∪ { O 1 } ⊆ O 1 F 2 ∪ { O 2 } ⊆ O 2 F 3 ∪ { O 3 } ⊆ O 3 Fin ⊆ � n O n n O n ⊆ [ N ] ∞ is compact, ∃ c ∈ [ N ] ∞ P ( N ) \ � P ( N ) \ � n O n ≤ c | A \ � n O n | < d ⇒ A \ � n O n is Menger A ∪ Fin is Menger

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger Corollary cf( d ) < d ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger Corollary cf( d ) < d ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger ∃ cf( d )-unbounded X ⊆ [ N ] ∞ , | X | = cf( d )

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger Corollary cf( d ) < d ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger ∃ cf( d )-unbounded X ⊆ [ N ] ∞ , | X | = cf( d ) | X | = cf( d ) < d ⇒ X is Menger

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger Corollary cf( d ) < d ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger ∃ cf( d )-unbounded X ⊆ [ N ] ∞ , | X | = cf( d ) | X | = cf( d ) < d ⇒ X is Menger ∃ Menger Y ⊆ [ N ] ∞ , X × Y is not Menger

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) Fin [ N ] ∞ , ∞ cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) d ≤ r ⇔ ∃ bi- d -unbounded A ⊆ [ N ] ∞ , ∞ Fin Fin [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) d ≤ r ⇔ ∃ bi- d -unbounded A ⊆ [ N ] ∞ , ∞ Fin Fin A ∪ Fin is Menger [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) d ≤ r ⇔ ∃ bi- d -unbounded A ⊆ [ N ] ∞ , ∞ Fin Fin A ∪ Fin is Menger τ : P ( N ) → P ( N ), τ ( a ) = a c = a ⊕ N [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) d ≤ r ⇔ ∃ bi- d -unbounded A ⊆ [ N ] ∞ , ∞ Fin Fin A ∪ Fin is Menger τ : P ( N ) → P ( N ), τ ( a ) = a c = a ⊕ N X = τ [ A ∪ Fin ] = { a c : a ∈ A } ∪ cFin ⊆ [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) d ≤ r ⇔ ∃ bi- d -unbounded A ⊆ [ N ] ∞ , ∞ Fin Fin Fin A ∪ Fin is Menger τ : P ( N ) → P ( N ), τ ( a ) = a c = a ⊕ N X = τ [ A ∪ Fin ] = { a c : a ∈ A } ∪ cFin ⊆ [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) d ≤ r ⇔ ∃ bi- d -unbounded A ⊆ [ N ] ∞ , ∞ Fin Fin Fin A ∪ Fin is Menger τ : P ( N ) → P ( N ), τ ( a ) = a c = a ⊕ N X = τ [ A ∪ Fin ] = { a c : a ∈ A } ∪ cFin ⊆ [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ d -unbounded { a c : a ∈ A } ⊆ X cFin cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger P ( N ) d ≤ r ⇔ ∃ bi- d -unbounded A ⊆ [ N ] ∞ , ∞ Fin Fin Fin A ∪ Fin is Menger τ : P ( N ) → P ( N ), τ ( a ) = a c = a ⊕ N X = τ [ A ∪ Fin ] = { a c : a ∈ A } ∪ cFin ⊆ [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ d -unbounded { a c : a ∈ A } ⊆ X ∃ Menger Y ⊆ P ( N ), X × Y is not Menger cFin cFin

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller ?

Main results A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d A ⊆ [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A ⊆ [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≤ r ⇒ ∃ Menger X , Y ⊆ P ( N ) , X × Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller ? Theorem ? (Zdomskyy) In the Miller model Menger is productive

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x /

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X F 1 ⊆ O 1

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . F 1 ⊆ O 1

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . F 1 ⊆ O 1

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . • x F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . • x F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3 Hurewicz ⇒ Menger

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . • x F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3 σ -compact ⇒ Hurewicz ⇒ Menger

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . • x F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3 σ -compact ⇒ Hurewicz ⇒ Menger Aurichi, Tall ( d = ℵ 1 ): metrizable productively Lindel¨ of ⇒ Hurewicz

The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 ⊆ O 1 , F 2 ⊆ O 2 , . . . such that for each x ∈ X , the set ∈ � F n } is finite { n ∈ N : x / X X X X . . . • x F 1 ⊆ O 1 F 2 ⊆ O 2 F 3 ⊆ O 3 σ -compact ⇒ Hurewicz ⇒ Menger Aurichi, Tall ( d = ℵ 1 ): metrizable productively Lindel¨ of ⇒ Hurewicz Sz (ZFC): separable productively paracompact ⇒ Hurewicz

Hurewicz meets combinatorics • y • x ≤ ∗ y if x ( n ) ≤ y ( n ) for almost all n • • • • • • • • • x • • • • •

Hurewicz meets combinatorics • • • y x ≤ ∗ y if x ( n ) ≤ y ( n ) for almost all n • • x • • y ≤ ∞ x if x �≤ ∗ y • • • • • • • • •

Hurewicz meets combinatorics • c • x ≤ ∗ y if x ( n ) ≤ y ( n ) for almost all n • • • y ≤ ∞ x if x �≤ ∗ y • • • y Y is bounded if ∃ c ∈ [ N ] ∞ ∀ y ∈ Y y ≤ ∗ c • • • • • • • •

Hurewicz meets combinatorics • c • x ≤ ∗ y if x ( n ) ≤ y ( n ) for almost all n • • • y ≤ ∞ x if x �≤ ∗ y • • • y Y is bounded if ∃ c ∈ [ N ] ∞ ∀ y ∈ Y y ≤ ∗ c • • • • • b : minimal cardinality of an unbounded set • • •

Hurewicz meets combinatorics • c • x ≤ ∗ y if x ( n ) ≤ y ( n ) for almost all n • • • y ≤ ∞ x if x �≤ ∗ y • • • y Y is bounded if ∃ c ∈ [ N ] ∞ ∀ y ∈ Y y ≤ ∗ c • • • • • b : minimal cardinality of an unbounded set • • • Theorem (Hurewicz) Assume that X is Lindel¨ of and zero-dimensional X is Hurewicz ⇔ continuous image of X into [ N ] ∞ is unbounded

Hurewicz meets combinatorics • c • x ≤ ∗ y if x ( n ) ≤ y ( n ) for almost all n • • • y ≤ ∞ x if x �≤ ∗ y • • • y Y is bounded if ∃ c ∈ [ N ] ∞ ∀ y ∈ Y y ≤ ∗ c • • • • • b : minimal cardinality of an unbounded set • • • Theorem (Hurewicz) Assume that X is Lindel¨ of and zero-dimensional X is Hurewicz ⇔ continuous image of X into [ N ] ∞ is unbounded

Hurewicz meets combinatorics • c • x ≤ ∗ y if x ( n ) ≤ y ( n ) for almost all n • • • y ≤ ∞ x if x �≤ ∗ y • • • y Y is bounded if ∃ c ∈ [ N ] ∞ ∀ y ∈ Y y ≤ ∗ c • • • • • b : minimal cardinality of an unbounded set • • • Theorem (Hurewicz) Assume that X is Lindel¨ of and zero-dimensional X is Hurewicz ⇔ continuous image of X into [ N ] ∞ is unbounded A Lindel¨ of X with | X | < b is Hurewicz An unbounded X ⊆ [ N ] ∞ is not Hurewicz

Main theorem again A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger

Main theorem again A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger Y = A ∪ Fin , A is d -unbounded Fin c • • • • • • A a

Main theorem again A ⊆ [ N ] ∞ is d -unbounded if | A | ≥ d and ∀ c ∈ [ N ] ∞ |{ a ∈ A : a ≤ c }| < d Theorem (Sz, Tsaban) If X ⊆ [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y ⊆ P ( N ), X × Y is not Menger Y = A ∪ Fin , A is d -unbounded Fin c • • • • • • A a Tsaban, Zdomskyy: H is Hurewicz and hereditarily Lindel¨ of ⇒ H × Y is Menger

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [ N ] ∞ unbounded

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded • s α ( b = d ) • • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α • • • • s β • • • • • •

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded s α ( b = d ) • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α s α ≤ ∞ y α ∈ Y • • • •

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz • Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded • • s α ( b = d ) • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α • • s α ≤ ∞ y α ∈ Y y α • • • • • • •

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz • Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded • • s α ( b = d ) • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α • • s α ≤ ∞ y α ∈ Y y α • • • • d -unbounded { y α : α < b } ⊆ Y • • •

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz • Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded • • s α ( b = d ) • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α • • s α ≤ ∞ y α ∈ Y y α • • • • d -unbounded { y α : α < b } ⊆ Y • • ∃ Menger M ⊆ P ( N ), Y × M not Menger •

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz • Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded • • s α ( b = d ) • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α • • s α ≤ ∞ y α ∈ Y y α • • • • d -unbounded { y α : α < b } ⊆ Y • • ∃ Menger M ⊆ P ( N ), Y × M not Menger • ( X × H ) × M → Y × M , ( X × H ) × M not Menger

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz • Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded • • s α ( b = d ) • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α • • s α ≤ ∞ y α ∈ Y y α • • • • d -unbounded { y α : α < b } ⊆ Y • • ∃ Menger M ⊆ P ( N ), Y × M not Menger • ( X × H ) × M → Y × M , ( X × H ) × M not Menger H × M is Menger, X × ( H × M ) is Menger

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz • Asm X prod Menger, X × H not Hurewicz • • X × H → Y ⊆ [ N ] ∞ unbounded • • s α ( b = d ) • • ∃ dominating { s α : α < b } , s β ≤ ∗ s α , β ≤ α • • s α ≤ ∞ y α ∈ Y y α • • • • d -unbounded { y α : α < b } ⊆ Y • • ∃ Menger M ⊆ P ( N ), Y × M not Menger • ( X × H ) × M → Y × M , ( X × H ) × M not Menger H × M is Menger, X × ( H × M ) is Menger

Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X × M is Menger Theorem (Sz, Tsaban) b = d , hereditarily Lindel¨ of spaces productively Menger ⇒ productively Hurewicz What about general spaces?