Compact spaces with a P -diagonal T a sc eil n agam K. P. Hart - - PowerPoint PPT Presentation

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Compact spaces with a P -diagonal T a sc eil n agam K. P. Hart - - PowerPoint PPT Presentation

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Compact spaces with a P -diagonal T a sc eil n agam K. P. Hart Faculty EEMCS TU Delft Praha, 25. cervence, 2016: 12:00 12:30 K. P. Hart Compact spaces with a P -diagonal 1 / 16 P -domination K. P. Hart Compact spaces with

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Compact spaces with a P-diagonal

T´ a sc´ eil´ ın agam

  • K. P. Hart

Faculty EEMCS TU Delft

Praha, 25. ˇ cervence, 2016: 12:00 – 12:30

  • K. P. Hart

Compact spaces with a P-diagonal 1 / 16

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P-domination

  • K. P. Hart

Compact spaces with a P-diagonal 2 / 16

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P-domination

Definition A space, X, is P-dominated

  • K. P. Hart

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P-domination

Definition A space, X, is P-dominated (stop giggling)

  • K. P. Hart

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P-domination

Definition A space, X, is P-dominated if there is a cover {Kf : f ∈ P} of X by compact sets

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P-domination

Definition A space, X, is P-dominated (stop giggling) if there is a cover {Kf : f ∈ P} of X by compact sets such that f g (pointwise) implies Kf ⊆ Kg.

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Compact spaces with a P-diagonal 2 / 16

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P-domination

Definition A space, X, is P-dominated if there is a cover {Kf : f ∈ P} of X by compact sets such that f g (pointwise) implies Kf ⊆ Kg. We call {Kf : f ∈ P} a P-dominating cover.

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P-diagonal

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P-diagonal

Definition A space, X, has a P-diagonal

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P-diagonal

Definition A space, X, has a P-diagonal if the complement of the diagonal in X 2 is P-dominated.

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Origins

Geometry of topological vector spaces (Cascales, Orihuela)

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Origins

Geometry of topological vector spaces (Cascales, Orihuela); P-domination yields metrizability for compact subsets.

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Compact spaces with a P-diagonal 4 / 16

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Origins

Geometry of topological vector spaces (Cascales, Orihuela); P-domination yields metrizability for compact subsets. A compact space with a P-diagonal is metrizable if it has countable tightness (no extra conditions if MA(ℵ1) holds). (Cascales, Orihuela, Tkachuk).

  • K. P. Hart

Compact spaces with a P-diagonal 4 / 16

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Question

So, question: are compact spaces with P-diagonals metrizable?

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An answer

Yes if CH (Dow, Guerrero S´ anchez).

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An answer

Yes if CH (Dow, Guerrero S´ anchez). Two important steps in that result: a compact space with a P-diagonal

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An answer

Yes if CH (Dow, Guerrero S´ anchez). Two important steps in that result: a compact space with a P-diagonal does not map onto [0, 1]c

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Compact spaces with a P-diagonal 6 / 16

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An answer

Yes if CH (Dow, Guerrero S´ anchez). Two important steps in that result: a compact space with a P-diagonal does not map onto [0, 1]c, ever

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An answer

Yes if CH (Dow, Guerrero S´ anchez). Two important steps in that result: a compact space with a P-diagonal does not map onto [0, 1]c, ever does map onto [0, 1]ω1

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Compact spaces with a P-diagonal 6 / 16

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An answer

Yes if CH (Dow, Guerrero S´ anchez). Two important steps in that result: a compact space with a P-diagonal does not map onto [0, 1]c, ever does map onto [0, 1]ω1, when it has uncountable tightness

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Compact spaces with a P-diagonal 6 / 16

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The answer

Theorem Every compact space with a P-diagonal is metrizable.

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The answer

Theorem Every compact space with a P-diagonal is metrizable. Proof. No compact space with a P-diagonal maps onto [0, 1]ω1.

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The answer

Theorem Every compact space with a P-diagonal is metrizable. Proof. No compact space with a P-diagonal maps onto [0, 1]ω1. How does that work?

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BIG sets

We work with the Cantor cube 2ω1.

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BIG sets

We work with the Cantor cube 2ω1. We call a closed subset, Y , of 2ω1 BIG

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BIG sets

We work with the Cantor cube 2ω1. We call a closed subset, Y , of 2ω1 BIG if there is a δ in ω1 such that πδ[Y ] = 2ω1\δ.

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BIG sets

We work with the Cantor cube 2ω1. We call a closed subset, Y , of 2ω1 BIG if there is a δ in ω1 such that πδ[Y ] = 2ω1\δ. (πδ projects onto 2ω1\δ)

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BIG sets

We work with the Cantor cube 2ω1. We call a closed subset, Y , of 2ω1 BIG if there is a δ in ω1 such that πδ[Y ] = 2ω1\δ. (πδ projects onto 2ω1\δ) Combinatorially

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BIG sets

We work with the Cantor cube 2ω1. We call a closed subset, Y , of 2ω1 BIG if there is a δ in ω1 such that πδ[Y ] = 2ω1\δ. (πδ projects onto 2ω1\δ) Combinatorially: a closed set Y is BIG if there is a δ such that for every s ∈ Fn(ω1 \ δ, 2) there is y ∈ Y such that s ⊆ y.

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BIG sets

A nice property of BIG sets.

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BIG sets

A nice property of BIG sets. Proposition A closed set is big if

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BIG sets

A nice property of BIG sets. Proposition A closed set is big if and only if

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BIG sets

A nice property of BIG sets. Proposition A closed set is big if and only if there are a δ ∈ ω1 and ρ ∈ 2δ

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BIG sets

A nice property of BIG sets. Proposition A closed set is big if and only if there are a δ ∈ ω1 and ρ ∈ 2δ such that {x ∈ 2ω1 : ρ ⊆ x} ⊆ Y .

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P-domination in 2ω1

Of course 2ω1 is P-dominated: take Kf = 2ω1 for all f ∈ P.

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P-domination in 2ω1

Of course 2ω1 is P-dominated: take Kf = 2ω1 for all f ∈ P. Here is a Baire category-like result for 2ω1.

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P-domination in 2ω1

Of course 2ω1 is P-dominated: take Kf = 2ω1 for all f ∈ P. Here is a Baire category-like result for 2ω1. Theorem If {Kf : f ∈ P} is a P-dominating cover of 2ω1 then

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Compact spaces with a P-diagonal 10 / 16

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P-domination in 2ω1

Of course 2ω1 is P-dominated: take Kf = 2ω1 for all f ∈ P. Here is a Baire category-like result for 2ω1. Theorem If {Kf : f ∈ P} is a P-dominating cover of 2ω1 then some Kf is BIG.

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The proof, case 1

d = ℵ1

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The proof, case 1

d = ℵ1: straightforward construction of a point not in

f Kf if we

assume no Kf is BIG

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The proof, case 1

d = ℵ1: straightforward construction of a point not in

f Kf if we

assume no Kf is BIG, using a cofinal family of ℵ1 many Kf ’s.

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The proof, case 3

b > ℵ1

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The proof, case 3

b > ℵ1: find there are ℵ1 many s ∈ Fn(ω1, 2) and

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The proof, case 3

b > ℵ1: find there are ℵ1 many s ∈ Fn(ω1, 2) and for each there are many h ∈ P such that s ⊆ y for some y ∈ Kh.

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The proof, case 3

b > ℵ1: find there are ℵ1 many s ∈ Fn(ω1, 2) and for each there are many h ∈ P such that s ⊆ y for some y ∈ Kh. We cleverly found ℵ1 many h’s such that each ∗-upper bound, f , for this family has a BIG Kf .

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The proof, case 2

d > b = ℵ1

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The proof, case 2

d > b = ℵ1: this is the trickiest one.

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ1 then 2ω1 has a subset X of cardinality ℵ1

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ1 then 2ω1 has a subset X of cardinality ℵ1 and such that every uncountable A ⊆ X

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Compact spaces with a P-diagonal 13 / 16

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ1 then 2ω1 has a subset X of cardinality ℵ1 and such that every uncountable A ⊆ X has a countable subset D such that

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ1 then 2ω1 has a subset X of cardinality ℵ1 and such that every uncountable A ⊆ X has a countable subset D such that πδ[D] is dense in 2ω1\δ for some δ.

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Compact spaces with a P-diagonal 13 / 16

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ1 then 2ω1 has a subset X of cardinality ℵ1 and such that every uncountable A ⊆ X has a countable subset D such that πδ[D] is dense in 2ω1\δ for some δ. This yields another set of ℵ1 many h’s

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Compact spaces with a P-diagonal 13 / 16

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ1 then 2ω1 has a subset X of cardinality ℵ1 and such that every uncountable A ⊆ X has a countable subset D such that πδ[D] is dense in 2ω1\δ for some δ. This yields another set of ℵ1 many h’s; the special properties of X ensure

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

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The proof, case 2

d > b = ℵ1: this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ1 then 2ω1 has a subset X of cardinality ℵ1 and such that every uncountable A ⊆ X has a countable subset D such that πδ[D] is dense in 2ω1\δ for some δ. This yields another set of ℵ1 many h’s; the special properties of X ensure: if f is not dominated by any one of the h’s then Kf is BIG.

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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.
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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.

Then we have a closed subset Y with a P-diagonal and a continuous map ϕ of Y onto 2ω1.

  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.

Then we have a closed subset Y with a P-diagonal and a continuous map ϕ of Y onto 2ω1. Then we find closed sets Y0 ⊃ Y1 ⊃ · · · and

  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.

Then we have a closed subset Y with a P-diagonal and a continuous map ϕ of Y onto 2ω1. Then we find closed sets Y0 ⊃ Y1 ⊃ · · · and points yn ∈ Yn \ Yn+1 such that

  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.

Then we have a closed subset Y with a P-diagonal and a continuous map ϕ of Y onto 2ω1. Then we find closed sets Y0 ⊃ Y1 ⊃ · · · and points yn ∈ Yn \ Yn+1 such that ϕ[Yn] is always BIG and

  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.

Then we have a closed subset Y with a P-diagonal and a continuous map ϕ of Y onto 2ω1. Then we find closed sets Y0 ⊃ Y1 ⊃ · · · and points yn ∈ Yn \ Yn+1 such that ϕ[Yn] is always BIG and (ultimately) one f such that

  • n({yn} × Yn+1) ⊆ Kf .
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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.

Then we have a closed subset Y with a P-diagonal and a continuous map ϕ of Y onto 2ω1. Then we find closed sets Y0 ⊃ Y1 ⊃ · · · and points yn ∈ Yn \ Yn+1 such that ϕ[Yn] is always BIG and (ultimately) one f such that

  • n({yn} × Yn+1) ⊆ Kf .

For every accumulation point, y, of yn : n ∈ ω we’ll have

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Finishing up

The final step: assume X has a P-diagonal and a continuous map

  • nto [0, 1]ω1.

Then we have a closed subset Y with a P-diagonal and a continuous map ϕ of Y onto 2ω1. Then we find closed sets Y0 ⊃ Y1 ⊃ · · · and points yn ∈ Yn \ Yn+1 such that ϕ[Yn] is always BIG and (ultimately) one f such that

  • n({yn} × Yn+1) ⊆ Kf .

For every accumulation point, y, of yn : n ∈ ω we’ll have y, y ∈ Kf , a contradiction.

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Compact spaces with a P-diagonal 14 / 16

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Question

Cascales, Orihuela and Tkachuk also asked if a compact space with a P-diagonal would have a small diagonal (answer: yes); this would imply metrizability.

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Question

Cascales, Orihuela and Tkachuk also asked if a compact space with a P-diagonal would have a small diagonal (answer: yes); this would imply metrizability. I’d like to turn that around: does a space with a small diagonal have a P-diagonal?

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Compact spaces with a P-diagonal 15 / 16

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Question

Cascales, Orihuela and Tkachuk also asked if a compact space with a P-diagonal would have a small diagonal (answer: yes); this would imply metrizability. I’d like to turn that around: does a space with a small diagonal have a P-diagonal? This would settle the metrizability question for spaces with a small diagonal.

  • K. P. Hart

Compact spaces with a P-diagonal 15 / 16

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Light reading

Website: fa.its.tudelft.nl/~hart Alan Dow and Klaas Pieter Hart, Compact spaces with a P-diagonal, Indagationes Mathematicae, 27 (2016), 721–726.

  • K. P. Hart

Compact spaces with a P-diagonal 16 / 16