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´ 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

P -domination K. P. Hart Compact spaces with a P -diagonal 2 / 16

P -domination Definition A space, X , is P -dominated K. P. Hart Compact spaces with a P -diagonal 2 / 16

P -domination Definition A space, X , is P -dominated (stop giggling) K. P. Hart Compact spaces with a P -diagonal 2 / 16

P -domination Definition A space, X , is P -dominated if there is a cover { K f : f ∈ P } of X by compact sets K. P. Hart Compact spaces with a P -diagonal 2 / 16

P -domination Definition A space, X , is P -dominated (stop giggling) if there is a cover { K f : f ∈ P } of X by compact sets such that f � g (pointwise) implies K f ⊆ K g . K. P. Hart Compact spaces with a P -diagonal 2 / 16

P -domination Definition A space, X , is P -dominated if there is a cover { K f : f ∈ P } of X by compact sets such that f � g (pointwise) implies K f ⊆ K g . We call { K f : f ∈ P } a P -dominating cover. K. P. Hart Compact spaces with a P -diagonal 2 / 16

P -diagonal K. P. Hart Compact spaces with a P -diagonal 3 / 16

P -diagonal Definition A space, X , has a P -diagonal K. P. Hart Compact spaces with a P -diagonal 3 / 16

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

Origins Geometry of topological vector spaces (Cascales, Orihuela) K. P. Hart Compact spaces with a P -diagonal 4 / 16

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

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( ℵ 1 ) holds). (Cascales, Orihuela, Tkachuk). K. P. Hart Compact spaces with a P -diagonal 4 / 16

Question So, question: are compact spaces with P -diagonals metrizable? K. P. Hart Compact spaces with a P -diagonal 5 / 16

An answer Yes if CH (Dow, Guerrero S´ anchez). K. P. Hart Compact spaces with a P -diagonal 6 / 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

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

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

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] ω 1 K. P. Hart Compact spaces with a P -diagonal 6 / 16

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] ω 1 , when it has uncountable tightness K. P. Hart Compact spaces with a P -diagonal 6 / 16

The answer Theorem Every compact space with a P -diagonal is metrizable. K. P. Hart Compact spaces with a P -diagonal 7 / 16

The answer Theorem Every compact space with a P -diagonal is metrizable. Proof. No compact space with a P -diagonal maps onto [0 , 1] ω 1 . K. P. Hart Compact spaces with a P -diagonal 7 / 16

The answer Theorem Every compact space with a P -diagonal is metrizable. Proof. No compact space with a P -diagonal maps onto [0 , 1] ω 1 . How does that work? K. P. Hart Compact spaces with a P -diagonal 7 / 16

BIG sets We work with the Cantor cube 2 ω 1 . K. P. Hart Compact spaces with a P -diagonal 8 / 16

BIG sets We work with the Cantor cube 2 ω 1 . We call a closed subset, Y , of 2 ω 1 BIG K. P. Hart Compact spaces with a P -diagonal 8 / 16

BIG sets We work with the Cantor cube 2 ω 1 . We call a closed subset, Y , of 2 ω 1 BIG if there is a δ in ω 1 such that π δ [ Y ] = 2 ω 1 \ δ . K. P. Hart Compact spaces with a P -diagonal 8 / 16

BIG sets We work with the Cantor cube 2 ω 1 . We call a closed subset, Y , of 2 ω 1 BIG if there is a δ in ω 1 such that π δ [ Y ] = 2 ω 1 \ δ . ( π δ projects onto 2 ω 1 \ δ ) K. P. Hart Compact spaces with a P -diagonal 8 / 16

BIG sets We work with the Cantor cube 2 ω 1 . We call a closed subset, Y , of 2 ω 1 BIG if there is a δ in ω 1 such that π δ [ Y ] = 2 ω 1 \ δ . ( π δ projects onto 2 ω 1 \ δ ) Combinatorially K. P. Hart Compact spaces with a P -diagonal 8 / 16

BIG sets We work with the Cantor cube 2 ω 1 . We call a closed subset, Y , of 2 ω 1 BIG if there is a δ in ω 1 such that π δ [ Y ] = 2 ω 1 \ δ . ( π δ projects onto 2 ω 1 \ δ ) Combinatorially: a closed set Y is BIG if there is a δ such that for every s ∈ Fn( ω 1 \ δ, 2) there is y ∈ Y such that s ⊆ y . K. P. Hart Compact spaces with a P -diagonal 8 / 16

BIG sets A nice property of BIG sets. K. P. Hart Compact spaces with a P -diagonal 9 / 16

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

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

BIG sets A nice property of BIG sets. Proposition A closed set is big if and only if there are a δ ∈ ω 1 and ρ ∈ 2 δ K. P. Hart Compact spaces with a P -diagonal 9 / 16

BIG sets A nice property of BIG sets. Proposition A closed set is big if and only if there are a δ ∈ ω 1 and ρ ∈ 2 δ such that { x ∈ 2 ω 1 : ρ ⊆ x } ⊆ Y . K. P. Hart Compact spaces with a P -diagonal 9 / 16

P -domination in 2 ω 1 Of course 2 ω 1 is P -dominated: take K f = 2 ω 1 for all f ∈ P . K. P. Hart Compact spaces with a P -diagonal 10 / 16

P -domination in 2 ω 1 Of course 2 ω 1 is P -dominated: take K f = 2 ω 1 for all f ∈ P . Here is a Baire category-like result for 2 ω 1 . K. P. Hart Compact spaces with a P -diagonal 10 / 16

P -domination in 2 ω 1 Of course 2 ω 1 is P -dominated: take K f = 2 ω 1 for all f ∈ P . Here is a Baire category-like result for 2 ω 1 . Theorem If { K f : f ∈ P } is a P -dominating cover of 2 ω 1 then K. P. Hart Compact spaces with a P -diagonal 10 / 16

P -domination in 2 ω 1 Of course 2 ω 1 is P -dominated: take K f = 2 ω 1 for all f ∈ P . Here is a Baire category-like result for 2 ω 1 . Theorem If { K f : f ∈ P } is a P -dominating cover of 2 ω 1 then some K f is BIG. K. P. Hart Compact spaces with a P -diagonal 10 / 16

The proof, case 1 d = ℵ 1 K. P. Hart Compact spaces with a P -diagonal 11 / 16

The proof, case 1 d = ℵ 1 : straightforward construction of a point not in � f K f if we assume no K f is BIG K. P. Hart Compact spaces with a P -diagonal 11 / 16

The proof, case 1 d = ℵ 1 : straightforward construction of a point not in � f K f if we assume no K f is BIG, using a cofinal family of ℵ 1 many K f ’s. K. P. Hart Compact spaces with a P -diagonal 11 / 16

The proof, case 3 b > ℵ 1 K. P. Hart Compact spaces with a P -diagonal 12 / 16

The proof, case 3 b > ℵ 1 : find there are ℵ 1 many s ∈ Fn( ω 1 , 2) and K. P. Hart Compact spaces with a P -diagonal 12 / 16

The proof, case 3 b > ℵ 1 : find there are ℵ 1 many s ∈ Fn( ω 1 , 2) and for each there are many h ∈ P such that s ⊆ y for some y ∈ K h . K. P. Hart Compact spaces with a P -diagonal 12 / 16

The proof, case 3 b > ℵ 1 : find there are ℵ 1 many s ∈ Fn( ω 1 , 2) and for each there are many h ∈ P such that s ⊆ y for some y ∈ K h . We cleverly found ℵ 1 many h ’s such that each � ∗ -upper bound, f , for this family has a BIG K f . K. P. Hart Compact spaces with a P -diagonal 12 / 16

The proof, case 2 d > b = ℵ 1 K. P. Hart Compact spaces with a P -diagonal 13 / 16

The proof, case 2 d > b = ℵ 1 : this is the trickiest one. K. P. Hart Compact spaces with a P -diagonal 13 / 16

The proof, case 2 d > b = ℵ 1 : this is the trickiest one. We borrow K. P. Hart Compact spaces with a P -diagonal 13 / 16

The proof, case 2 d > b = ℵ 1 : this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ 1 then 2 ω 1 has a subset X of cardinality ℵ 1 K. P. Hart Compact spaces with a P -diagonal 13 / 16

The proof, case 2 d > b = ℵ 1 : this is the trickiest one. We borrow Theorem (Todorˇ cevi´ c) If b = ℵ 1 then 2 ω 1 has a subset X of cardinality ℵ 1 and such that every uncountable A ⊆ X K. P. Hart Compact spaces with a P -diagonal 13 / 16