Compact spaces with a -diagonal - Tá scéilín agam

Compact spaces with a P-diagonal
Tá scéilı́n agam
K. P. Hart
Faculty EEMCS
TU Delft
Praha, 25. července, 2016: 12:00 – 12:30
K. P. Hart
Compact spaces with a P-diagonal
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P-domination
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P-domination
Definition
A space, X , is P-dominated
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P-domination
Definition
A space, X , is P-dominated (stop giggling)
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P-domination
Definition
A space, X , is P-dominated
{Kf : f ∈ P} of X by compact sets
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if there is a cover
Compact spaces with a P-diagonal
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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 6 g (pointwise)
implies Kf ⊆ Kg .
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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 6 g (pointwise)
implies Kf ⊆ Kg .
We call {Kf : f ∈ P} a P-dominating cover.
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Compact spaces with a P-diagonal
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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
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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).
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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 Sánchez).
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An answer
Yes if CH (Dow, Guerrero Sánchez).
Two important steps in that result: a compact space with a
P-diagonal
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An answer
Yes if CH (Dow, Guerrero Sánchez).
Two important steps in that result: a compact space with a
P-diagonal
does not map onto [0, 1]c
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An answer
Yes if CH (Dow, Guerrero Sánchez).
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 Sánchez).
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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An answer
Yes if CH (Dow, Guerrero Sánchez).
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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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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Compact spaces with a P-diagonal
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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
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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
assume no Kf is BIG
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S
Compact spaces with a P-diagonal
f
Kf if we
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The proof, case 1
S
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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Compact spaces with a P-diagonal
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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 6∗ -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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Compact spaces with a P-diagonal
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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 (Todorčević)
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 (Todorčević)
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
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The proof, case 2
d > b = ℵ1 : this is the trickiest one.
We borrow
Theorem (Todorčević)
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
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Compact spaces with a P-diagonal
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The proof, case 2
d > b = ℵ1 : this is the trickiest one.
We borrow
Theorem (Todorčević)
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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The proof, case 2
d > b = ℵ1 : this is the trickiest one.
We borrow
Theorem (Todorčević)
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
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The proof, case 2
d > b = ℵ1 : this is the trickiest one.
We borrow
Theorem (Todorčević)
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
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Compact spaces with a P-diagonal
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The proof, case 2
d > b = ℵ1 : this is the trickiest one.
We borrow
Theorem (Todorčević)
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
onto [0, 1]ω1 .
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Finishing up
The final step: assume X has a P-diagonal and a continuous map
onto [0, 1]ω1 .
Then we have a closed subset Y with a P-diagonal and a
continuous map ϕ of Y onto 2ω1 .
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Finishing up
The final step: assume X has a P-diagonal and a continuous map
onto [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
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Finishing up
The final step: assume X has a P-diagonal and a continuous map
onto [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
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Compact spaces with a P-diagonal
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Finishing up
The final step: assume X has a P-diagonal and a continuous map
onto [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
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Compact spaces with a P-diagonal
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Finishing up
The final step: assume X has a P-diagonal and a continuous map
onto [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
S
n ({yn } × Yn+1 ) ⊆ Kf .
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Compact spaces with a P-diagonal
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Finishing up
The final step: assume X has a P-diagonal and a continuous map
onto [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
S
n ({yn } × Yn+1 ) ⊆ Kf .
For every accumulation point, y , of hyn : n ∈ ωi we’ll have
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Compact spaces with a P-diagonal
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Finishing up
The final step: assume X has a P-diagonal and a continuous map
onto [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
S
n ({yn } × Yn+1 ) ⊆ Kf .
For every accumulation point, y , of hyn : n ∈ ωi we’ll have
hy , y i ∈ Kf , a contradiction.
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Compact spaces with a P-diagonal
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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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Compact spaces with a P-diagonal
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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?
K. P. Hart
Compact spaces with a P-diagonal
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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
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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.
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Compact spaces with a P-diagonal
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