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

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

Oct 09, 2023 167 likes •1.3k 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

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SLIDE 1

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 UMO-2014/12/T/ST1/00627

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SLIDE 2

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X

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SLIDE 3

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X F1 ⊆ O1 X X

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SLIDE 4

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X F1 ⊆ O1 X

. . .

X X X

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SLIDE 5

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X F1 ⊆ O1 X

. . .

X X X

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SLIDE 6

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

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SLIDE 7

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

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SLIDE 8

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X Menger ⇒ Lindel¨

  • f

F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

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SLIDE 9

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X σ-compact ⇒ Menger ⇒ Lindel¨

  • f

F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

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SLIDE 10

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X σ-compact ⇒ Menger ⇒ Lindel¨

  • f

Aurichi: Every Menger space is D F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

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SLIDE 11

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X σ-compact ⇒ Menger ⇒ Lindel¨

  • f

Aurichi: Every Menger space is D Chodunsky, Repovˇ s, Zdomskyy: Mengers property characterizes filters whose Mathias forcing notion does not add dominating functions F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

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SLIDE 12

The Menger property

Menger’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that F1 ∪ F2 ∪ . . . covers X σ-compact ⇒ Menger ⇒ Lindel¨

  • f

Aurichi: Every Menger space is D Chodunsky, Repovˇ s, Zdomskyy: Mengers property characterizes filters whose Mathias forcing notion does not add dominating functions Tsaban: The most general class for which a general form of Hindmans Finite Sums Theorem holds F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

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SLIDE 13

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN

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SLIDE 14

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN x ≤ y if x(n) ≤ y(n) for all n y x

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SLIDE 15

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN x ≤ y if x(n) ≤ y(n) for all n x ≤∗ d if x(n) ≤ y(n) for almost all n y x

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SLIDE 16

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN x ≤ y if x(n) ≤ y(n) for all n x ≤∗ d if x(n) ≤ y(n) for almost all n Y is dominating if ∀x∈[N]∞ ∃y∈Y x ≤∗ y y x

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SLIDE 17

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN x ≤ y if x(n) ≤ y(n) for all n x ≤∗ d if x(n) ≤ y(n) for almost all n Y is dominating if ∀x∈[N]∞ ∃y∈Y x ≤∗ y d: minimal cardinality of a dominating set y x

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SLIDE 18

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN x ≤ y if x(n) ≤ y(n) for all n x ≤∗ d if x(n) ≤ y(n) for almost all n Y is dominating if ∀x∈[N]∞ ∃y∈Y x ≤∗ y d: minimal cardinality of a dominating set y x

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  • Theorem (Hurewicz)

Assume that X is Lindel¨

  • f and zero-dimensional

X is Menger ⇔ continuous image of X into [N]∞ is nondominating

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SLIDE 19

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN x ≤ y if x(n) ≤ y(n) for all n x ≤∗ d if x(n) ≤ y(n) for almost all n Y is dominating if ∀x∈[N]∞ ∃y∈Y x ≤∗ y d: minimal cardinality of a dominating set y x

  • •
  • •
  • •
  • •
  • • •
  • •
  • Theorem (Hurewicz)

Assume that X is Lindel¨

  • f and zero-dimensional

X is Menger ⇔ continuous image of X into [N]∞ is nondominating A Lindel¨

  • f X with |X| < d is Menger
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SLIDE 20

Menger meets combinatorics

[N]∞: infinite subsets of N [N]∞ ∋ x = {x(1), x(2), . . .} : increasing enumeration, [N]∞ ⊆ NN x ≤ y if x(n) ≤ y(n) for all n x ≤∗ d if x(n) ≤ y(n) for almost all n Y is dominating if ∀x∈[N]∞ ∃y∈Y x ≤∗ y d: minimal cardinality of a dominating set y x

  • •
  • •
  • •
  • •
  • • •
  • •
  • Theorem (Hurewicz)

Assume that X is Lindel¨

  • f and zero-dimensional

X is Menger ⇔ continuous image of X into [N]∞ is nondominating A Lindel¨

  • f X with |X| < d is Menger

A dominating X ⊆ [N]∞ is not Menger

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SLIDE 21

Products of Menger spaces

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SLIDE 22

Products of Menger spaces

Todorˇ cevi´ c (ZFC): There is a Menger set whose square is not Menger

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SLIDE 23

Products of Menger spaces

Todorˇ cevi´ c (ZFC): There is a Menger set whose square is not Menger Sets of reals

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SLIDE 24

Products of Menger spaces

Todorˇ cevi´ c (ZFC): There is a Menger set whose square is not Menger Sets of reals Just, Miller, Scheepers, Szeptycki (CH): There is a Menger M ⊆ R whose square M × M is not Menger

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SLIDE 25

Products of Menger spaces

Todorˇ cevi´ c (ZFC): There is a Menger set whose square is not Menger Sets of reals Just, Miller, Scheepers, Szeptycki (CH): There is a Menger M ⊆ R whose square M × M is not Menger Scheepers, Tsaban (cov(M) = cof(M)): There is a Menger M ⊆ R whose square M × M is not Menger

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SLIDE 26

Products of Menger spaces

Todorˇ cevi´ c (ZFC): There is a Menger set whose square is not Menger Sets of reals Just, Miller, Scheepers, Szeptycki (CH): There is a Menger M ⊆ R whose square M × M is not Menger Scheepers, Tsaban (cov(M) = cof(M)): There is a Menger M ⊆ R whose square M × M is not Menger Problem (sets of reals) Find the minimal hypotheses that Menger’s property is not productive

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SLIDE 27

Products of Menger spaces

Todorˇ cevi´ c (ZFC): There is a Menger set whose square is not Menger Sets of reals Just, Miller, Scheepers, Szeptycki (CH): There is a Menger M ⊆ R whose square M × M is not Menger Scheepers, Tsaban (cov(M) = cof(M)): There is a Menger M ⊆ R whose square M × M is not Menger Problem (sets of reals) Find the minimal hypotheses that Menger’s property is not productive P(N) ≈ {0, 1}ω: the Cantor space P(N) = [N]∞ ∪ Fin

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SLIDE 28

d-unbounded sets

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

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SLIDE 29

d-unbounded sets

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

A

c

  • a
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SLIDE 30

d-unbounded sets

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

A

c

  • a
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SLIDE 31

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

A

Fin c

  • a
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SLIDE 32

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

F1 ∪ {O1} ⊆ O1 Fin A

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SLIDE 33

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 Fin A Fin A Fin A F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3

. . .

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SLIDE 34

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 F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3

. . .

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SLIDE 35

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

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3 Fin A

. . .

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SLIDE 36

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 ⊆

n On

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3 Fin A

. . .

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SLIDE 37

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3 Fin A

. . .

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SLIDE 38

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact, ∃c∈[N]∞ P(N) \ n On ≤ c

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3 Fin A

. . . c

  • • •
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SLIDE 39

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact, ∃c∈[N]∞ P(N) \ n On ≤ c

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3 Fin A

. . . c

  • • •

a

  • • •
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SLIDE 40

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact, ∃c∈[N]∞ P(N) \ n On ≤ c

|A \

n On| < d

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3 Fin A

. . . c

  • • •

a

  • • •
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SLIDE 41

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact, ∃c∈[N]∞ P(N) \ n On ≤ c

|A \

n On| < d ⇒ A \ n On is Menger

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3 Fin A

. . . c

  • • •

a

  • • •
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SLIDE 42

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact, ∃c∈[N]∞ P(N) \ n On ≤ c

|A \

n On| < d ⇒ A \ n On is Menger

Fin A F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3

. . . c

  • • •

a

  • • •
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SLIDE 43

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact, ∃c∈[N]∞ P(N) \ n On ≤ c

|A \

n On| < d ⇒ A \ n On is Menger

Fin A

a

  • • •

c

  • • •

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3

. . .

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SLIDE 44

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 ⊆

n On

P(N) \

n On ⊆ [N]∞ is compact, ∃c∈[N]∞ P(N) \ n On ≤ c

|A \

n On| < d ⇒ A \ n On is Menger

A ∪ Fin is Menger

Fin A

a

  • • •

c

  • • •

F1 ∪ {O1} ⊆ O1 F2 ∪ {O2} ⊆ O2 F3 ∪ {O3} ⊆ O3

. . .

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SLIDE 45

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

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SLIDE 46

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

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SLIDE 47

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)

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SLIDE 48

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

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SLIDE 49

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

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SLIDE 50

Main results

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

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SLIDE 51

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 { ac : a ∈ A } are d-unbounded

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SLIDE 52

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 { ac : 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

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SLIDE 53

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 { ac : 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

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SLIDE 54

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 { ac : 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 cFin [N]∞, ∞

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SLIDE 55

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 { ac : 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 d ≤ r ⇔ ∃ bi-d-unbounded A ⊆ [N]∞, ∞

P(N) Fin cFin [N]∞, ∞ Fin [N]∞, ∞

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SLIDE 56

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 { ac : 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 d ≤ r ⇔ ∃ bi-d-unbounded A ⊆ [N]∞, ∞ A ∪ Fin is Menger

P(N) Fin cFin [N]∞, ∞ Fin [N]∞, ∞

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SLIDE 57

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 { ac : 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 d ≤ r ⇔ ∃ bi-d-unbounded A ⊆ [N]∞, ∞ A ∪ Fin is Menger τ : P(N) → P(N), τ(a) = ac = a ⊕ N

P(N) Fin cFin [N]∞, ∞ Fin [N]∞, ∞

slide-58
SLIDE 58

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 { ac : 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 d ≤ r ⇔ ∃ bi-d-unbounded A ⊆ [N]∞, ∞ A ∪ Fin is Menger τ : P(N) → P(N), τ(a) = ac = a ⊕ N X = τ[A ∪ Fin] = { ac : a ∈ A } ∪ cFin ⊆ [N]∞

P(N) Fin cFin [N]∞, ∞ Fin [N]∞, ∞

slide-59
SLIDE 59

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 { ac : 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 d ≤ r ⇔ ∃ bi-d-unbounded A ⊆ [N]∞, ∞ A ∪ Fin is Menger τ : P(N) → P(N), τ(a) = ac = a ⊕ N X = τ[A ∪ Fin] = { ac : a ∈ A } ∪ cFin ⊆ [N]∞

P(N) Fin cFin [N]∞, ∞ Fin [N]∞, ∞ Fin [N]∞, ∞ cFin

slide-60
SLIDE 60

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 { ac : 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 d ≤ r ⇔ ∃ bi-d-unbounded A ⊆ [N]∞, ∞ A ∪ Fin is Menger τ : P(N) → P(N), τ(a) = ac = a ⊕ N X = τ[A ∪ Fin] = { ac : a ∈ A } ∪ cFin ⊆ [N]∞ d-unbounded { ac : a ∈ A } ⊆ X

P(N) Fin cFin [N]∞, ∞ Fin [N]∞, ∞ Fin [N]∞, ∞ cFin

slide-61
SLIDE 61

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 { ac : 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 d ≤ r ⇔ ∃ bi-d-unbounded A ⊆ [N]∞, ∞ A ∪ Fin is Menger τ : P(N) → P(N), τ(a) = ac = a ⊕ N X = τ[A ∪ Fin] = { ac : a ∈ A } ∪ cFin ⊆ [N]∞ d-unbounded { ac : a ∈ A } ⊆ X ∃ Menger Y ⊆ P(N), X × Y is not Menger

P(N) Fin cFin [N]∞, ∞ Fin [N]∞, ∞ Fin [N]∞, ∞ cFin

slide-62
SLIDE 62

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 { ac : 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

slide-63
SLIDE 63

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 { ac : 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

slide-64
SLIDE 64

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 { ac : 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 ?

slide-65
SLIDE 65

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 { ac : 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

slide-66
SLIDE 66

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite

slide-67
SLIDE 67

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite F1 ⊆ O1 X X

slide-68
SLIDE 68

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite F1 ⊆ O1 X

. . .

X X X

slide-69
SLIDE 69

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite F1 ⊆ O1 X

. . .

X X X

slide-70
SLIDE 70

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

slide-71
SLIDE 71

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite X X X F1 ⊆ O1 F2 ⊆ O2 F3 ⊆ O3 X

. . .

slide-72
SLIDE 72

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

slide-73
SLIDE 73

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

  • x
slide-74
SLIDE 74

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite Hurewicz ⇒ Menger F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

  • x
slide-75
SLIDE 75

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite σ-compact ⇒ Hurewicz ⇒ Menger F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

  • x
slide-76
SLIDE 76

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite σ-compact ⇒ Hurewicz ⇒ Menger Aurichi, Tall (d = ℵ1): metrizable productively Lindel¨

  • f ⇒ Hurewicz

F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

  • x
slide-77
SLIDE 77

The Hurewicz property

Hurewicz’s property: for every sequence of open covers O1, O2, . . . of X there are finite F1 ⊆ O1, F2 ⊆ O2, . . . such that for each x ∈ X, the set { n ∈ N : x / ∈ Fn } is finite σ-compact ⇒ Hurewicz ⇒ Menger Aurichi, Tall (d = ℵ1): metrizable productively Lindel¨

  • f ⇒ Hurewicz

Sz (ZFC): separable productively paracompact ⇒ Hurewicz F1 ⊆ O1 X F2 ⊆ O2 X F3 ⊆ O3 X X

. . .

  • x
slide-78
SLIDE 78

Hurewicz meets combinatorics

x ≤∗ y if x(n) ≤ y(n) for almost all n y x

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slide-79
SLIDE 79

Hurewicz meets combinatorics

x ≤∗ y if x(n) ≤ y(n) for almost all n y ≤∞ x if x ≤∗ y y x

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slide-80
SLIDE 80

Hurewicz meets combinatorics

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

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slide-81
SLIDE 81

Hurewicz meets combinatorics

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

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slide-82
SLIDE 82

Hurewicz meets combinatorics

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

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  • Theorem (Hurewicz)

Assume that X is Lindel¨

  • f and zero-dimensional

X is Hurewicz ⇔ continuous image of X into [N]∞ is unbounded

slide-83
SLIDE 83

Hurewicz meets combinatorics

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

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  • Theorem (Hurewicz)

Assume that X is Lindel¨

  • f and zero-dimensional

X is Hurewicz ⇔ continuous image of X into [N]∞ is unbounded

slide-84
SLIDE 84

Hurewicz meets combinatorics

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

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  • Theorem (Hurewicz)

Assume that X is Lindel¨

  • f and zero-dimensional

X is Hurewicz ⇔ continuous image of X into [N]∞ is unbounded A Lindel¨

  • f X with |X| < b is Hurewicz

An unbounded X ⊆ [N]∞ is not Hurewicz

slide-85
SLIDE 85

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

slide-86
SLIDE 86

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

A

Fin c

  • • •

a

  • • •
slide-87
SLIDE 87

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

A

Fin c

  • • •

a

  • • •

Tsaban, Zdomskyy: H is Hurewicz and hereditarily Lindel¨

  • f ⇒ H × Y is Menger
slide-88
SLIDE 88

Productivity of Menger and Hurewicz

X is productively Menger if for each Menger M, X × M is Menger

slide-89
SLIDE 89

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz

slide-90
SLIDE 90

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz

slide-91
SLIDE 91

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded

slide-92
SLIDE 92

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα sβ

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slide-93
SLIDE 93

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα ≤∞ yα ∈ Y sα

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slide-94
SLIDE 94

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα ≤∞ yα ∈ Y sα yα

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slide-95
SLIDE 95

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα ≤∞ yα ∈ Y d-unbounded { yα : α < b } ⊆ Y sα yα

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slide-96
SLIDE 96

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα ≤∞ yα ∈ Y d-unbounded { yα : α < b } ⊆ Y ∃ Menger M ⊆ P(N), Y × M not Menger sα yα

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slide-97
SLIDE 97

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα ≤∞ yα ∈ Y d-unbounded { yα : α < b } ⊆ Y ∃ Menger M ⊆ P(N), Y × M not Menger (X ×H)×M → Y ×M, (X ×H)×M not Menger sα yα

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slide-98
SLIDE 98

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα ≤∞ yα ∈ Y d-unbounded { yα : 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 sα yα

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slide-99
SLIDE 99

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz Asm X prod Menger, X × H not Hurewicz X × H → Y ⊆ [N]∞ unbounded (b = d) ∃ dominating { sα : α < b }, sβ ≤∗ sα, β ≤ α sα ≤∞ yα ∈ Y d-unbounded { yα : 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 sα yα

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slide-100
SLIDE 100

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz What about general spaces?

slide-101
SLIDE 101

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz What about general spaces? Miller, Tsaban, Zdomskyy (d = ℵ1): metrizable productively Lindel¨

  • f ⇒ productively Hurewicz

metrizable productively Lindel¨

  • f ⇒ productively Menger
slide-102
SLIDE 102

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¨

  • f spaces

productively Menger ⇒ productively Hurewicz What about general spaces? Miller, Tsaban, Zdomskyy (d = ℵ1): metrizable productively Lindel¨

  • f ⇒ productively Hurewicz

metrizable productively Lindel¨

  • f ⇒ productively Menger

Theorem (Sz, Tsaban) d = ℵ1, general spaces productively Lindel¨

  • f ⇒ productively Menger ⇒ productively Hurewicz
slide-103
SLIDE 103

Open problems

Sets of reals

slide-104
SLIDE 104

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive?

slide-105
SLIDE 105

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive? Any Lusin set is not productively Menger?

slide-106
SLIDE 106

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive? Any Lusin set is not productively Menger? Hereditarily Lindel¨

  • f spaces
slide-107
SLIDE 107

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive? Any Lusin set is not productively Menger? Hereditarily Lindel¨

  • f spaces

(b = d) productively Menger = productively Hurewicz?

slide-108
SLIDE 108

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive? Any Lusin set is not productively Menger? Hereditarily Lindel¨

  • f spaces

(b = d) productively Menger = productively Hurewicz? General spaces

slide-109
SLIDE 109

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive? Any Lusin set is not productively Menger? Hereditarily Lindel¨

  • f spaces

(b = d) productively Menger = productively Hurewicz? General spaces X ⊆ R, |X| < b ⇒ X× Hurewicz is Lindel¨

  • f?
slide-110
SLIDE 110

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive? Any Lusin set is not productively Menger? Hereditarily Lindel¨

  • f spaces

(b = d) productively Menger = productively Hurewicz? General spaces X ⊆ R, |X| < b ⇒ X× Hurewicz is Lindel¨

  • f?

(d = ℵ1) productively Menger = productively Hurewicz

slide-111
SLIDE 111

Open problems

Sets of reals d ≤ r ⇔ Menger is not productive? Any Lusin set is not productively Menger? Hereditarily Lindel¨

  • f spaces

(b = d) productively Menger = productively Hurewicz? General spaces X ⊆ R, |X| < b ⇒ X× Hurewicz is Lindel¨

  • f?

(d = ℵ1) productively Menger = productively Hurewicz Any Sierpi´ nski set is not productively Hurewicz? is not productively Menger? under CH?