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productively and non productively menger spaces

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

Oct 09, 2023 β€’167 likes β€’1.29k views

Productively (and non-productively) Menger spaces Piotr Szewczak Cardinal Stefan Wyszy nski University, Poland, and Bar-Ilan University, Israel joint work with Boaz Tsaban Toposym 2016 Supported by National Science Center Poland

  1. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin Fin Fin Fin . . . A A A A F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3

  2. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin . . . A F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3

  3. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin . . . A F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3

  4. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin . . . A F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n

  5. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin . . . A F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact P ( N ) \ οΏ½

  6. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin c β€’ β€’ β€’ . . . A F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact, βˆƒ c ∈ [ N ] ∞ P ( N ) \ οΏ½ P ( N ) \ οΏ½ n O n ≀ c

  7. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin c β€’ β€’ β€’ . . . β€’ β€’ β€’ A a F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact, βˆƒ c ∈ [ N ] ∞ P ( N ) \ οΏ½ P ( N ) \ οΏ½ n O n ≀ c

  8. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin c β€’ β€’ β€’ . . . β€’ β€’ β€’ A a F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact, βˆƒ c ∈ [ N ] ∞ P ( N ) \ οΏ½ P ( N ) \ οΏ½ n O n ≀ c | A \ οΏ½ n O n | < d

  9. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin c β€’ β€’ β€’ . . . β€’ β€’ β€’ A a F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact, βˆƒ c ∈ [ N ] ∞ P ( N ) \ οΏ½ P ( N ) \ οΏ½ n O n ≀ c | A \ οΏ½ n O n | < d β‡’ A \ οΏ½ n O n is Menger

  10. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin c β€’ β€’ β€’ . . . β€’ β€’ β€’ A a F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact, βˆƒ c ∈ [ N ] ∞ P ( N ) \ οΏ½ P ( N ) \ οΏ½ n O n ≀ c | A \ οΏ½ n O n | < d β‡’ A \ οΏ½ n O n is Menger

  11. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin c β€’ β€’ β€’ . . . β€’ β€’ β€’ A a F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact, βˆƒ c ∈ [ N ] ∞ P ( N ) \ οΏ½ P ( N ) \ οΏ½ n O n ≀ c | A \ οΏ½ n O n | < d β‡’ A \ οΏ½ n O n is Menger

  12. d -unbounded sets A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ is d -unbounded β‡’ A βˆͺ Fin is Menger Fin c β€’ β€’ β€’ . . . β€’ β€’ β€’ A a F 1 βˆͺ { O 1 } βŠ† O 1 F 2 βˆͺ { O 2 } βŠ† O 2 F 3 βˆͺ { O 3 } βŠ† O 3 Fin βŠ† οΏ½ n O n n O n βŠ† [ N ] ∞ is compact, βˆƒ c ∈ [ N ] ∞ P ( N ) \ οΏ½ P ( N ) \ οΏ½ n O n ≀ c | A \ οΏ½ n O n | < d β‡’ A \ οΏ½ n O n is Menger A βˆͺ Fin is Menger

  13. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger

  14. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger Corollary cf( d ) < d β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger

  15. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger Corollary cf( d ) < d β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger βˆƒ cf( d )-unbounded X βŠ† [ N ] ∞ , | X | = cf( d )

  16. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger Corollary cf( d ) < d β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger βˆƒ cf( d )-unbounded X βŠ† [ N ] ∞ , | X | = cf( d ) | X | = cf( d ) < d β‡’ X is Menger

  17. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger Corollary cf( d ) < d β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger βˆƒ cf( d )-unbounded X βŠ† [ N ] ∞ , | X | = cf( d ) | X | = cf( d ) < d β‡’ X is Menger βˆƒ Menger Y βŠ† [ N ] ∞ , X Γ— Y is not Menger

  18. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d

  19. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded

  20. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite

  21. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger

  22. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) Fin [ N ] ∞ , ∞ cFin

  23. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) d ≀ r ⇔ βˆƒ bi- d -unbounded A βŠ† [ N ] ∞ , ∞ Fin Fin [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

  24. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) d ≀ r ⇔ βˆƒ bi- d -unbounded A βŠ† [ N ] ∞ , ∞ Fin Fin A βˆͺ Fin is Menger [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

  25. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) d ≀ r ⇔ βˆƒ bi- d -unbounded A βŠ† [ N ] ∞ , ∞ Fin Fin A βˆͺ Fin is Menger Ο„ : P ( N ) β†’ P ( N ), Ο„ ( a ) = a c = a βŠ• N [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

  26. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) d ≀ r ⇔ βˆƒ bi- d -unbounded A βŠ† [ N ] ∞ , ∞ Fin Fin A βˆͺ Fin is Menger Ο„ : P ( N ) β†’ P ( N ), Ο„ ( a ) = a c = a βŠ• N X = Ο„ [ A βˆͺ Fin ] = { a c : a ∈ A } βˆͺ cFin βŠ† [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin

  27. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) d ≀ r ⇔ βˆƒ bi- d -unbounded A βŠ† [ N ] ∞ , ∞ Fin Fin Fin A βˆͺ Fin is Menger Ο„ : P ( N ) β†’ P ( N ), Ο„ ( a ) = a c = a βŠ• N X = Ο„ [ A βˆͺ Fin ] = { a c : a ∈ A } βˆͺ cFin βŠ† [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ cFin cFin

  28. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) d ≀ r ⇔ βˆƒ bi- d -unbounded A βŠ† [ N ] ∞ , ∞ Fin Fin Fin A βˆͺ Fin is Menger Ο„ : P ( N ) β†’ P ( N ), Ο„ ( a ) = a c = a βŠ• N X = Ο„ [ A βˆͺ Fin ] = { a c : a ∈ A } βˆͺ cFin βŠ† [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ d -unbounded { a c : a ∈ A } βŠ† X cFin cFin

  29. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger P ( N ) d ≀ r ⇔ βˆƒ bi- d -unbounded A βŠ† [ N ] ∞ , ∞ Fin Fin Fin A βˆͺ Fin is Menger Ο„ : P ( N ) β†’ P ( N ), Ο„ ( a ) = a c = a βŠ• N X = Ο„ [ A βˆͺ Fin ] = { a c : a ∈ A } βˆͺ cFin βŠ† [ N ] ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ [ N ] ∞ , ∞ d -unbounded { a c : a ∈ A } βŠ† X βˆƒ Menger Y βŠ† P ( N ), X Γ— Y is not Menger cFin cFin

  30. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller

  31. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller

  32. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller ?

  33. Main results A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d A βŠ† [ N ] ∞ , ∞ is bi- d -unbounded if A and { a c : a ∈ A } are d -unbounded r : min card of A βŠ† [ N ] ∞ , there is no r ∈ [ N ] ∞ s.t. for all a ∈ A r ∩ a and r \ a are infinite Corollary d ≀ r β‡’ βˆƒ Menger X , Y βŠ† P ( N ) , X Γ— Y is not Menger Productivity of Menger MA Cohen Random Sacks Hechler Laver Mathias Miller ? Theorem ? (Zdomskyy) In the Miller model Menger is productive

  34. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x /

  35. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X F 1 βŠ† O 1

  36. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . F 1 βŠ† O 1

  37. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . F 1 βŠ† O 1

  38. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3

  39. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3

  40. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3

  41. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . β€’ x F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3

  42. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . β€’ x F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3 Hurewicz β‡’ Menger

  43. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . β€’ x F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3 Οƒ -compact β‡’ Hurewicz β‡’ Menger

  44. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . β€’ x F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3 Οƒ -compact β‡’ Hurewicz β‡’ Menger Aurichi, Tall ( d = β„΅ 1 ): metrizable productively LindelΒ¨ of β‡’ Hurewicz

  45. The Hurewicz property Hurewicz’s property: for every sequence of open covers O 1 , O 2 , . . . of X there are finite F 1 βŠ† O 1 , F 2 βŠ† O 2 , . . . such that for each x ∈ X , the set ∈ οΏ½ F n } is finite { n ∈ N : x / X X X X . . . β€’ x F 1 βŠ† O 1 F 2 βŠ† O 2 F 3 βŠ† O 3 Οƒ -compact β‡’ Hurewicz β‡’ Menger Aurichi, Tall ( d = β„΅ 1 ): metrizable productively LindelΒ¨ of β‡’ Hurewicz Sz (ZFC): separable productively paracompact β‡’ Hurewicz

  46. Hurewicz meets combinatorics β€’ y β€’ x ≀ βˆ— y if x ( n ) ≀ y ( n ) for almost all n β€’ β€’ β€’ β€’ β€’ β€’ β€’ β€’ β€’ x β€’ β€’ β€’ β€’ β€’

  47. Hurewicz meets combinatorics β€’ β€’ β€’ y x ≀ βˆ— y if x ( n ) ≀ y ( n ) for almost all n β€’ β€’ x β€’ β€’ y ≀ ∞ x if x �≀ βˆ— y β€’ β€’ β€’ β€’ β€’ β€’ β€’ β€’ β€’

  48. Hurewicz meets combinatorics β€’ c β€’ x ≀ βˆ— y if x ( n ) ≀ y ( n ) for almost all n β€’ β€’ β€’ y ≀ ∞ x if x �≀ βˆ— y β€’ β€’ β€’ y Y is bounded if βˆƒ c ∈ [ N ] ∞ βˆ€ y ∈ Y y ≀ βˆ— c β€’ β€’ β€’ β€’ β€’ β€’ β€’ β€’

  49. Hurewicz meets combinatorics β€’ c β€’ x ≀ βˆ— y if x ( n ) ≀ y ( n ) for almost all n β€’ β€’ β€’ y ≀ ∞ x if x �≀ βˆ— y β€’ β€’ β€’ y Y is bounded if βˆƒ c ∈ [ N ] ∞ βˆ€ y ∈ Y y ≀ βˆ— c β€’ β€’ β€’ β€’ β€’ b : minimal cardinality of an unbounded set β€’ β€’ β€’

  50. Hurewicz meets combinatorics β€’ c β€’ x ≀ βˆ— y if x ( n ) ≀ y ( n ) for almost all n β€’ β€’ β€’ y ≀ ∞ x if x �≀ βˆ— y β€’ β€’ β€’ y Y is bounded if βˆƒ c ∈ [ N ] ∞ βˆ€ y ∈ Y y ≀ βˆ— c β€’ β€’ β€’ β€’ β€’ b : minimal cardinality of an unbounded set β€’ β€’ β€’ Theorem (Hurewicz) Assume that X is LindelΒ¨ of and zero-dimensional X is Hurewicz ⇔ continuous image of X into [ N ] ∞ is unbounded

  51. Hurewicz meets combinatorics β€’ c β€’ x ≀ βˆ— y if x ( n ) ≀ y ( n ) for almost all n β€’ β€’ β€’ y ≀ ∞ x if x �≀ βˆ— y β€’ β€’ β€’ y Y is bounded if βˆƒ c ∈ [ N ] ∞ βˆ€ y ∈ Y y ≀ βˆ— c β€’ β€’ β€’ β€’ β€’ b : minimal cardinality of an unbounded set β€’ β€’ β€’ Theorem (Hurewicz) Assume that X is LindelΒ¨ of and zero-dimensional X is Hurewicz ⇔ continuous image of X into [ N ] ∞ is unbounded

  52. Hurewicz meets combinatorics β€’ c β€’ x ≀ βˆ— y if x ( n ) ≀ y ( n ) for almost all n β€’ β€’ β€’ y ≀ ∞ x if x �≀ βˆ— y β€’ β€’ β€’ y Y is bounded if βˆƒ c ∈ [ N ] ∞ βˆ€ y ∈ Y y ≀ βˆ— c β€’ β€’ β€’ β€’ β€’ b : minimal cardinality of an unbounded set β€’ β€’ β€’ Theorem (Hurewicz) Assume that X is LindelΒ¨ of and zero-dimensional X is Hurewicz ⇔ continuous image of X into [ N ] ∞ is unbounded A LindelΒ¨ of X with | X | < b is Hurewicz An unbounded X βŠ† [ N ] ∞ is not Hurewicz

  53. Main theorem again A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger

  54. Main theorem again A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger Y = A βˆͺ Fin , A is d -unbounded Fin c β€’ β€’ β€’ β€’ β€’ β€’ A a

  55. Main theorem again A βŠ† [ N ] ∞ is d -unbounded if | A | β‰₯ d and βˆ€ c ∈ [ N ] ∞ |{ a ∈ A : a ≀ c }| < d Theorem (Sz, Tsaban) If X βŠ† [ N ] ∞ contains a d -unbounded set or a cf( d )-unbounded set, then there is a Menger Y βŠ† P ( N ), X Γ— Y is not Menger Y = A βˆͺ Fin , A is d -unbounded Fin c β€’ β€’ β€’ β€’ β€’ β€’ A a Tsaban, Zdomskyy: H is Hurewicz and hereditarily LindelΒ¨ of β‡’ H Γ— Y is Menger

  56. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger

  57. 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

  58. 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

  59. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz Asm X prod Menger, X Γ— H not Hurewicz X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded

  60. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded β€’ s Ξ± ( b = d ) β€’ β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± β€’ β€’ β€’ β€’ s Ξ² β€’ β€’ β€’ β€’ β€’ β€’

  61. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded s Ξ± ( b = d ) β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± s Ξ± ≀ ∞ y Ξ± ∈ Y β€’ β€’ β€’ β€’

  62. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz β€’ Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded β€’ β€’ s Ξ± ( b = d ) β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± β€’ β€’ s Ξ± ≀ ∞ y Ξ± ∈ Y y Ξ± β€’ β€’ β€’ β€’ β€’ β€’ β€’

  63. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz β€’ Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded β€’ β€’ s Ξ± ( b = d ) β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± β€’ β€’ s Ξ± ≀ ∞ y Ξ± ∈ Y y Ξ± β€’ β€’ β€’ β€’ d -unbounded { y Ξ± : Ξ± < b } βŠ† Y β€’ β€’ β€’

  64. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz β€’ Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded β€’ β€’ s Ξ± ( b = d ) β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± β€’ β€’ s Ξ± ≀ ∞ y Ξ± ∈ Y y Ξ± β€’ β€’ β€’ β€’ d -unbounded { y Ξ± : Ξ± < b } βŠ† Y β€’ β€’ βˆƒ Menger M βŠ† P ( N ), Y Γ— M not Menger β€’

  65. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz β€’ Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded β€’ β€’ s Ξ± ( b = d ) β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± β€’ β€’ s Ξ± ≀ ∞ y Ξ± ∈ Y y Ξ± β€’ β€’ β€’ β€’ d -unbounded { y Ξ± : Ξ± < b } βŠ† Y β€’ β€’ βˆƒ Menger M βŠ† P ( N ), Y Γ— M not Menger β€’ ( X Γ— H ) Γ— M β†’ Y Γ— M , ( X Γ— H ) Γ— M not Menger

  66. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz β€’ Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded β€’ β€’ s Ξ± ( b = d ) β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± β€’ β€’ s Ξ± ≀ ∞ y Ξ± ∈ Y y Ξ± β€’ β€’ β€’ β€’ d -unbounded { y Ξ± : Ξ± < b } βŠ† Y β€’ β€’ βˆƒ Menger M βŠ† P ( N ), Y Γ— M not Menger β€’ ( X Γ— H ) Γ— M β†’ Y Γ— M , ( X Γ— H ) Γ— M not Menger H Γ— M is Menger, X Γ— ( H Γ— M ) is Menger

  67. Productivity of Menger and Hurewicz X is productively Menger if for each Menger M , X Γ— M is Menger Theorem (Sz, Tsaban) b = d , hereditarily LindelΒ¨ of spaces productively Menger β‡’ productively Hurewicz β€’ Asm X prod Menger, X Γ— H not Hurewicz β€’ β€’ X Γ— H β†’ Y βŠ† [ N ] ∞ unbounded β€’ β€’ s Ξ± ( b = d ) β€’ β€’ βˆƒ dominating { s Ξ± : Ξ± < b } , s Ξ² ≀ βˆ— s Ξ± , Ξ² ≀ Ξ± β€’ β€’ s Ξ± ≀ ∞ y Ξ± ∈ Y y Ξ± β€’ β€’ β€’ β€’ d -unbounded { y Ξ± : Ξ± < b } βŠ† Y β€’ β€’ βˆƒ Menger M βŠ† P ( N ), Y Γ— M not Menger β€’ ( X Γ— H ) Γ— M β†’ Y Γ— M , ( X Γ— H ) Γ— M not Menger H Γ— M is Menger, X Γ— ( H Γ— M ) is Menger

  68. 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?

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