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

compact spaces with a p diagonal
SMART_READER_LITE
LIVE PREVIEW

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

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


slide-1
SLIDE 1

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

slide-2
SLIDE 2

P-domination

  • K. P. Hart

Compact spaces with a P-diagonal 2 / 16

slide-3
SLIDE 3

P-domination

Definition A space, X, is P-dominated

  • K. P. Hart

Compact spaces with a P-diagonal 2 / 16

slide-4
SLIDE 4

P-domination

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

  • K. P. Hart

Compact spaces with a P-diagonal 2 / 16

slide-5
SLIDE 5

P-domination

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

  • K. P. Hart

Compact spaces with a P-diagonal 2 / 16

slide-6
SLIDE 6

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.

  • K. P. Hart

Compact spaces with a P-diagonal 2 / 16

slide-7
SLIDE 7

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.

  • K. P. Hart

Compact spaces with a P-diagonal 2 / 16

slide-8
SLIDE 8

P-diagonal

  • K. P. Hart

Compact spaces with a P-diagonal 3 / 16

slide-9
SLIDE 9

P-diagonal

Definition A space, X, has a P-diagonal

  • K. P. Hart

Compact spaces with a P-diagonal 3 / 16

slide-10
SLIDE 10

P-diagonal

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

  • K. P. Hart

Compact spaces with a P-diagonal 3 / 16

slide-11
SLIDE 11

Origins

Geometry of topological vector spaces (Cascales, Orihuela)

  • K. P. Hart

Compact spaces with a P-diagonal 4 / 16

slide-12
SLIDE 12

Origins

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

  • K. P. Hart

Compact spaces with a P-diagonal 4 / 16

slide-13
SLIDE 13

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

slide-14
SLIDE 14

Question

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

  • K. P. Hart

Compact spaces with a P-diagonal 5 / 16

slide-15
SLIDE 15

An answer

Yes if CH (Dow, Guerrero S´ anchez).

  • K. P. Hart

Compact spaces with a P-diagonal 6 / 16

slide-16
SLIDE 16

An answer

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

  • K. P. Hart

Compact spaces with a P-diagonal 6 / 16

slide-17
SLIDE 17

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

  • K. P. Hart

Compact spaces with a P-diagonal 6 / 16

slide-18
SLIDE 18

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

  • K. P. Hart

Compact spaces with a P-diagonal 6 / 16

slide-19
SLIDE 19

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

  • K. P. Hart

Compact spaces with a P-diagonal 6 / 16

slide-20
SLIDE 20

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

  • K. P. Hart

Compact spaces with a P-diagonal 6 / 16

slide-21
SLIDE 21

The answer

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

  • K. P. Hart

Compact spaces with a P-diagonal 7 / 16

slide-22
SLIDE 22

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.

  • K. P. Hart

Compact spaces with a P-diagonal 7 / 16

slide-23
SLIDE 23

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?

  • K. P. Hart

Compact spaces with a P-diagonal 7 / 16

slide-24
SLIDE 24

BIG sets

We work with the Cantor cube 2ω1.

  • K. P. Hart

Compact spaces with a P-diagonal 8 / 16

slide-25
SLIDE 25

BIG sets

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

  • K. P. Hart

Compact spaces with a P-diagonal 8 / 16

slide-26
SLIDE 26

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.

  • K. P. Hart

Compact spaces with a P-diagonal 8 / 16

slide-27
SLIDE 27

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)

  • K. P. Hart

Compact spaces with a P-diagonal 8 / 16

slide-28
SLIDE 28

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

  • K. P. Hart

Compact spaces with a P-diagonal 8 / 16

slide-29
SLIDE 29

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.

  • K. P. Hart

Compact spaces with a P-diagonal 8 / 16

slide-30
SLIDE 30

BIG sets

A nice property of BIG sets.

  • K. P. Hart

Compact spaces with a P-diagonal 9 / 16

slide-31
SLIDE 31

BIG sets

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

  • K. P. Hart

Compact spaces with a P-diagonal 9 / 16

slide-32
SLIDE 32

BIG sets

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

  • K. P. Hart

Compact spaces with a P-diagonal 9 / 16

slide-33
SLIDE 33

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

  • K. P. Hart

Compact spaces with a P-diagonal 9 / 16

slide-34
SLIDE 34

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 .

  • K. P. Hart

Compact spaces with a P-diagonal 9 / 16

slide-35
SLIDE 35

P-domination in 2ω1

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

  • K. P. Hart

Compact spaces with a P-diagonal 10 / 16

slide-36
SLIDE 36

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.

  • K. P. Hart

Compact spaces with a P-diagonal 10 / 16

slide-37
SLIDE 37

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

  • K. P. Hart

Compact spaces with a P-diagonal 10 / 16

slide-38
SLIDE 38

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.

  • K. P. Hart

Compact spaces with a P-diagonal 10 / 16

slide-39
SLIDE 39

The proof, case 1

d = ℵ1

  • K. P. Hart

Compact spaces with a P-diagonal 11 / 16

slide-40
SLIDE 40

The proof, case 1

d = ℵ1: straightforward construction of a point not in

f Kf if we

assume no Kf is BIG

  • K. P. Hart

Compact spaces with a P-diagonal 11 / 16

slide-41
SLIDE 41

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.

  • K. P. Hart

Compact spaces with a P-diagonal 11 / 16

slide-42
SLIDE 42

The proof, case 3

b > ℵ1

  • K. P. Hart

Compact spaces with a P-diagonal 12 / 16

slide-43
SLIDE 43

The proof, case 3

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

  • K. P. Hart

Compact spaces with a P-diagonal 12 / 16

slide-44
SLIDE 44

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.

  • K. P. Hart

Compact spaces with a P-diagonal 12 / 16

slide-45
SLIDE 45

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 .

  • K. P. Hart

Compact spaces with a P-diagonal 12 / 16

slide-46
SLIDE 46

The proof, case 2

d > b = ℵ1

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-47
SLIDE 47

The proof, case 2

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

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-48
SLIDE 48

The proof, case 2

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

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-49
SLIDE 49

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

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-50
SLIDE 50

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

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-51
SLIDE 51

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

slide-52
SLIDE 52

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.

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-53
SLIDE 53

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

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-54
SLIDE 54

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

slide-55
SLIDE 55

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.

  • K. P. Hart

Compact spaces with a P-diagonal 13 / 16

slide-56
SLIDE 56

Finishing up

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

  • nto [0, 1]ω1.
  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

slide-57
SLIDE 57

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

slide-58
SLIDE 58

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

slide-59
SLIDE 59

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

slide-60
SLIDE 60

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

slide-61
SLIDE 61

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 .
  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

slide-62
SLIDE 62

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

  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

slide-63
SLIDE 63

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.

  • K. P. Hart

Compact spaces with a P-diagonal 14 / 16

slide-64
SLIDE 64

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.

  • K. P. Hart

Compact spaces with a P-diagonal 15 / 16

slide-65
SLIDE 65

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?

  • K. P. Hart

Compact spaces with a P-diagonal 15 / 16

slide-66
SLIDE 66

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

slide-67
SLIDE 67

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