The weak Freeze Nation FN property and the Noetherian types of - PowerPoint PPT Presentation

wFN � ω s.t. � P has wFN iff ∃ f : P → P if p , q ∈ P and p ≤ q then ∃ r ∈ f ( p ) ∩ f ( q ) with p ≤ r ≤ q . Q ≤ σ P iff cf { q ∈ Q : q ≤ p } = ci { q ∈ Q : q ≥ p } = ω . � ω 1 : Q ≤ σ P } contains a club P has wFN iff { Q ∈ � P Can we weaken the assumption � ω 1 : Q ≤ σ P } contains a club”? � “ { Q ∈ P Let P be a poset, and χ be a large enough regular cardinal . (Fuchino-Koppelberg-Shelah) TFAE: (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Soukup, L (HAS) RIMS 2010 3 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization Let χ = cf ( χ ) be a large enough. TFAE for a poset P : (1) P has the wFN property (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with | M | = ω 1 and P ∈ M . Hard to verify P ∩ M ≤ σ P for an arbitrary submodel M . Consider nicer submodels Definition M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elementary submodels of H ( χ ) . � ω , ⊆ �� � (CH) If P = κ , then P ∩ M ≤ σ P for each V ω 1 -like M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . Soukup, L (HAS) RIMS 2010 4 / 23

Consistent counterexample TFAE: (1) P has the wFN property, (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with M | = ω 1 and P ∈ M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . � ω ⊆ �� � If CH holds then for each cardinal κ , the poset κ satisfies (w2). Theorem � ω , ⊆ �� � If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then the poset ℵ ω does not have the wFN property. Levinsky-Magidor-Shelah: The assumption is consistent modulo a huge cardinal. Soukup, L (HAS) RIMS 2010 5 / 23

Consistent counterexample TFAE: (1) P has the wFN property, (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with M | = ω 1 and P ∈ M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . � ω ⊆ �� � If CH holds then for each cardinal κ , the poset κ satisfies (w2). Theorem � ω , ⊆ �� � If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then the poset ℵ ω does not have the wFN property. Levinsky-Magidor-Shelah: The assumption is consistent modulo a huge cardinal. Soukup, L (HAS) RIMS 2010 5 / 23

Consistent counterexample TFAE: (1) P has the wFN property, (2) P ∩ M ≤ σ P for each M ≺ H ( χ ) with M | = ω 1 and P ∈ M . Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M . � ω ⊆ �� � If CH holds then for each cardinal κ , the poset κ satisfies (w2). Theorem � ω , ⊆ �� � If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then the poset ℵ ω does not have the wFN property. Levinsky-Magidor-Shelah: The assumption is consistent modulo a huge cardinal. Soukup, L (HAS) RIMS 2010 5 / 23

A positive theorem Let χ = cf ( χ ) be a large enough regular cardinal . M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elemen- tary submodels of H ( χ ) . Theorem (Fuchino, Soukup) Assume that λ is a cardinal with (i) cf ([ µ ] ω , ⊆ ) = µ if ω 1 < µ < λ and cf ( µ ) ≥ ω 1 (ii) � ∗∗∗ holds for each singular µ < λ with cofinality ω µ Then for each poset P of cardinality ≤ λ the following are equivalent: (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M. Soukup, L (HAS) RIMS 2010 6 / 23

A positive theorem Let χ = cf ( χ ) be a large enough regular cardinal . M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elemen- tary submodels of H ( χ ) . Theorem (Fuchino, Soukup) Assume that λ is a cardinal with (i) cf ([ µ ] ω , ⊆ ) = µ if ω 1 < µ < λ and cf ( µ ) ≥ ω 1 (ii) � ∗∗∗ holds for each singular µ < λ with cofinality ω µ Then for each poset P of cardinality ≤ λ the following are equivalent: (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M. Soukup, L (HAS) RIMS 2010 6 / 23

A positive theorem Let χ = cf ( χ ) be a large enough regular cardinal . M ≺ H ( χ ) is V ω 1 -like iff M is the union of an ω 1 chain of countable elemen- tary submodels of H ( χ ) . Theorem (Fuchino, Soukup) Assume that λ is a cardinal with (i) cf ([ µ ] ω , ⊆ ) = µ if ω 1 < µ < λ and cf ( µ ) ≥ ω 1 (ii) � ∗∗∗ holds for each singular µ < λ with cofinality ω µ Then for each poset P of cardinality ≤ λ the following are equivalent: (w1) P has the wFN property, (w2) P ∩ M ≤ σ P for each V ω 1 -like M ≺ H ( χ ) with P ∈ M. Soukup, L (HAS) RIMS 2010 6 / 23

The wFN property of P ( ω ) Theorem (Fuchino, Soukup) Assume that λ is a cardinal with (i) cf ([ µ ] ω , ⊆ ) = µ if ω 1 < µ < λ and cf ( µ ) ≥ ω 1 (ii) � ∗∗∗ holds for each singular µ < λ with cofinality ω µ Then V Fn ( λ, 2 ; ω ) | = P ( ω ) has the wFN property. Theorem (Fuchino, Geschke, Shelah, Soukup) If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then V D∗ Fn ( ℵ ω , 2 ; ω ) | = P ( ω ) does not have the wFN property . Soukup, L (HAS) RIMS 2010 7 / 23

The wFN property of P ( ω ) Theorem (Fuchino, Soukup) Assume that λ is a cardinal with (i) cf ([ µ ] ω , ⊆ ) = µ if ω 1 < µ < λ and cf ( µ ) ≥ ω 1 (ii) � ∗∗∗ holds for each singular µ < λ with cofinality ω µ Then V Fn ( λ, 2 ; ω ) | = P ( ω ) has the wFN property. Theorem (Fuchino, Geschke, Shelah, Soukup) If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then V D∗ Fn ( ℵ ω , 2 ; ω ) | = P ( ω ) does not have the wFN property . Soukup, L (HAS) RIMS 2010 7 / 23

The wFN property of P ( ω ) Theorem (Fuchino, Soukup) Assume that λ is a cardinal with (i) cf ([ µ ] ω , ⊆ ) = µ if ω 1 < µ < λ and cf ( µ ) ≥ ω 1 (ii) � ∗∗∗ holds for each singular µ < λ with cofinality ω µ Then V Fn ( λ, 2 ; ω ) | = P ( ω ) has the wFN property. Theorem (Fuchino, Geschke, Shelah, Soukup) If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then V D∗ Fn ( ℵ ω , 2 ; ω ) | = P ( ω ) does not have the wFN property . Soukup, L (HAS) RIMS 2010 7 / 23

The wFN property of P ( ω ) Theorem (Fuchino, Soukup) Assume that λ is a cardinal with (i) cf ([ µ ] ω , ⊆ ) = µ if ω 1 < µ < λ and cf ( µ ) ≥ ω 1 (ii) � ∗∗∗ holds for each singular µ < λ with cofinality ω µ Then V Fn ( λ, 2 ; ω ) | = P ( ω ) has the wFN property. Theorem (Fuchino, Geschke, Shelah, Soukup) If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then V D∗ Fn ( ℵ ω , 2 ; ω ) | = P ( ω ) does not have the wFN property . Soukup, L (HAS) RIMS 2010 7 / 23

Noetherian type Basic notions Peregudov, Malykhin, Shapirovskii Definition Let X be a topological space. The Noetherian type of X , Nt ( X ) , is the least cardinal κ such that X has a base B such that |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B . Noetherian type ≈ cellularity 2 κ has countable Noetherian type and cellularity For a topological space X , let X ( δ ) denote the space obtained by declaring the G δ -sets to be open. Theorem (Spadaro) (GCH) Let X be a compact space such that Nt ( X ) has uncountable cofinality . Then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) . Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type Basic notions Peregudov, Malykhin, Shapirovskii Definition Let X be a topological space. The Noetherian type of X , Nt ( X ) , is the least cardinal κ such that X has a base B such that |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B . Noetherian type ≈ cellularity 2 κ has countable Noetherian type and cellularity For a topological space X , let X ( δ ) denote the space obtained by declaring the G δ -sets to be open. Theorem (Spadaro) (GCH) Let X be a compact space such that Nt ( X ) has uncountable cofinality . Then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) . Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type Basic notions Peregudov, Malykhin, Shapirovskii Definition Let X be a topological space. The Noetherian type of X , Nt ( X ) , is the least cardinal κ such that X has a base B such that |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B . Noetherian type ≈ cellularity 2 κ has countable Noetherian type and cellularity For a topological space X , let X ( δ ) denote the space obtained by declaring the G δ -sets to be open. Theorem (Spadaro) (GCH) Let X be a compact space such that Nt ( X ) has uncountable cofinality . Then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) . Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type Basic notions Peregudov, Malykhin, Shapirovskii Definition Let X be a topological space. The Noetherian type of X , Nt ( X ) , is the least cardinal κ such that X has a base B such that |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B . Noetherian type ≈ cellularity 2 κ has countable Noetherian type and cellularity For a topological space X , let X ( δ ) denote the space obtained by declaring the G δ -sets to be open. Theorem (Spadaro) (GCH) Let X be a compact space such that Nt ( X ) has uncountable cofinality . Then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) . Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type Basic notions Peregudov, Malykhin, Shapirovskii Definition Let X be a topological space. The Noetherian type of X , Nt ( X ) , is the least cardinal κ such that X has a base B such that |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B . Noetherian type ≈ cellularity 2 κ has countable Noetherian type and cellularity For a topological space X , let X ( δ ) denote the space obtained by declaring the G δ -sets to be open. Theorem (Spadaro) (GCH) Let X be a compact space such that Nt ( X ) has uncountable cofinality . Then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) . Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type Basic notions Peregudov, Malykhin, Shapirovskii Definition Let X be a topological space. The Noetherian type of X , Nt ( X ) , is the least cardinal κ such that X has a base B such that |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B . Noetherian type ≈ cellularity 2 κ has countable Noetherian type and cellularity For a topological space X , let X ( δ ) denote the space obtained by declaring the G δ -sets to be open. Theorem (Spadaro) (GCH) Let X be a compact space such that Nt ( X ) has uncountable cofinality . Then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) . Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type Basic notions Peregudov, Malykhin, Shapirovskii Definition Let X be a topological space. The Noetherian type of X , Nt ( X ) , is the least cardinal κ such that X has a base B such that |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B . Noetherian type ≈ cellularity 2 κ has countable Noetherian type and cellularity For a topological space X , let X ( δ ) denote the space obtained by declaring the G δ -sets to be open. Theorem (Spadaro) (GCH) Let X be a compact space such that Nt ( X ) has uncountable cofinality . Then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) . Soukup, L (HAS) RIMS 2010 8 / 23

Question Def: Nt ( X ) ≤ κ iff |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B Def: X ( δ ) denotes the G δ -topology Spadaro: If GCH holds, X is compact, cf ( Nt ( X )) > ω then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) Spadaro’s question What happens if Nt ( X ) has countable cofinality? Can we drop GCH? Nt ( 2 κ ) = ω . What about Nt ( 2 κ ( δ ) ) ? Theorem (Milovich) ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Soukup, L (HAS) RIMS 2010 9 / 23

Question Def: Nt ( X ) ≤ κ iff |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B Def: X ( δ ) denotes the G δ -topology Spadaro: If GCH holds, X is compact, cf ( Nt ( X )) > ω then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) Spadaro’s question What happens if Nt ( X ) has countable cofinality? Can we drop GCH? Nt ( 2 κ ) = ω . What about Nt ( 2 κ ( δ ) ) ? Theorem (Milovich) ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Soukup, L (HAS) RIMS 2010 9 / 23

Question Def: Nt ( X ) ≤ κ iff |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B Def: X ( δ ) denotes the G δ -topology Spadaro: If GCH holds, X is compact, cf ( Nt ( X )) > ω then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) Spadaro’s question What happens if Nt ( X ) has countable cofinality? Can we drop GCH? Nt ( 2 κ ) = ω . What about Nt ( 2 κ ( δ ) ) ? Theorem (Milovich) ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Soukup, L (HAS) RIMS 2010 9 / 23

Question Def: Nt ( X ) ≤ κ iff |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B Def: X ( δ ) denotes the G δ -topology Spadaro: If GCH holds, X is compact, cf ( Nt ( X )) > ω then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) Spadaro’s question What happens if Nt ( X ) has countable cofinality? Can we drop GCH? Nt ( 2 κ ) = ω . What about Nt ( 2 κ ( δ ) ) ? Theorem (Milovich) ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Soukup, L (HAS) RIMS 2010 9 / 23

Question Def: Nt ( X ) ≤ κ iff |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B Def: X ( δ ) denotes the G δ -topology Spadaro: If GCH holds, X is compact, cf ( Nt ( X )) > ω then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) Spadaro’s question What happens if Nt ( X ) has countable cofinality? Can we drop GCH? Nt ( 2 κ ) = ω . What about Nt ( 2 κ ( δ ) ) ? Theorem (Milovich) ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Soukup, L (HAS) RIMS 2010 9 / 23

Question Def: Nt ( X ) ≤ κ iff |{ B ′ ∈ B : B ⊂ B ′ }| < κ for each B ∈ B Def: X ( δ ) denotes the G δ -topology Spadaro: If GCH holds, X is compact, cf ( Nt ( X )) > ω then Nt ( X ( δ ) ) ≤ 2 Nt ( X ) Spadaro’s question What happens if Nt ( X ) has countable cofinality? Can we drop GCH? Nt ( 2 κ ) = ω . What about Nt ( 2 κ ( δ ) ) ? Theorem (Milovich) ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Soukup, L (HAS) RIMS 2010 9 / 23

Answer ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Thm: Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Theorem If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then Nt ( 2 ℵ ω ( δ ) ) ≥ ω 2 . Theorem Assume that ( ℵ ω ) ω = ℵ ω + 1 and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) . If D is cofinal in � ω such that |D ∩ P ( A ) | > ω . � ω , ⊆ �� � � ℵ ω , then there is A ∈ ℵ ω What is the relationship between these properties and the wFN � ω , ⊂ �� � property of ℵ ω ? Soukup, L (HAS) RIMS 2010 10 / 23

Answer ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Thm: Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Theorem If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then Nt ( 2 ℵ ω ( δ ) ) ≥ ω 2 . Theorem Assume that ( ℵ ω ) ω = ℵ ω + 1 and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) . If D is cofinal in � ω such that |D ∩ P ( A ) | > ω . � ω , ⊆ �� � � ℵ ω , then there is A ∈ ℵ ω What is the relationship between these properties and the wFN � ω , ⊂ �� � property of ℵ ω ? Soukup, L (HAS) RIMS 2010 10 / 23

Answer ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Thm: Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Theorem If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then Nt ( 2 ℵ ω ( δ ) ) ≥ ω 2 . Theorem Assume that ( ℵ ω ) ω = ℵ ω + 1 and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) . If D is cofinal in � ω such that |D ∩ P ( A ) | > ω . � ω , ⊆ �� � � ℵ ω , then there is A ∈ ℵ ω What is the relationship between these properties and the wFN � ω , ⊂ �� � property of ℵ ω ? Soukup, L (HAS) RIMS 2010 10 / 23

Answer ( δ ) ) = ω 1 for κ < ℵ ω . If � ℵ ω + ( ℵ ω ) ω = ℵ ω + 1 then Nt ( 2 κ Thm: Nt ( 2 κ ( δ ) ) = ω 1 ( δ ) ) ≤ 2 ω in ZFC? Question: Nt ( 2 ℵ ω Theorem If GCH holds and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) then Nt ( 2 ℵ ω ( δ ) ) ≥ ω 2 . Theorem Assume that ( ℵ ω ) ω = ℵ ω + 1 and ( ℵ ω + 1 , ℵ ω ) ։ ( ℵ 1 , ℵ 0 ) . If D is cofinal in � ω such that |D ∩ P ( A ) | > ω . � ω , ⊆ �� � � ℵ ω , then there is A ∈ ℵ ω What is the relationship between these properties and the wFN � ω , ⊂ �� � property of ℵ ω ? Soukup, L (HAS) RIMS 2010 10 / 23

Resolvability of monotonically normal spaces Soukup, L (HAS) RIMS 2010 11 / 23

The beginnings Basic notions E. Hewitt, 1943 Definition A topological space X is κ -resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2 -resolvable irresolvable it is not resolvable Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings Basic notions E. Hewitt, 1943 Definition A topological space X is κ -resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2 -resolvable irresolvable it is not resolvable Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings Basic notions E. Hewitt, 1943 Definition A topological space X is κ -resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2 -resolvable irresolvable it is not resolvable Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings Basic notions E. Hewitt, 1943 Definition A topological space X is κ -resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2 -resolvable irresolvable it is not resolvable Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings Basic notions E. Hewitt, 1943 Definition A topological space X is κ -resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2 -resolvable irresolvable it is not resolvable Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings. Basic notions X is κ -resolvable iff X contains κ disjoint dense subsets. If D is dense and U is a non-empty open set, then U ∩ D � = ∅ . So if X is κ -resolvable then κ ≤ min {| U | : U ∈ τ X \ {∅}} =∆( X ) . ∆( X ) is the dispersion character of X . Definition (Ceder, Pearson, 1967) X is maximally resolvable iff it is ∆( X ) -resolvable. Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings. Basic notions X is κ -resolvable iff X contains κ disjoint dense subsets. If D is dense and U is a non-empty open set, then U ∩ D � = ∅ . So if X is κ -resolvable then κ ≤ min {| U | : U ∈ τ X \ {∅}} =∆( X ) . ∆( X ) is the dispersion character of X . Definition (Ceder, Pearson, 1967) X is maximally resolvable iff it is ∆( X ) -resolvable. Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings. Basic notions X is κ -resolvable iff X contains κ disjoint dense subsets. If D is dense and U is a non-empty open set, then U ∩ D � = ∅ . So if X is κ -resolvable then κ ≤ min {| U | : U ∈ τ X \ {∅}} =∆( X ) . ∆( X ) is the dispersion character of X . Definition (Ceder, Pearson, 1967) X is maximally resolvable iff it is ∆( X ) -resolvable. Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings. Basic notions X is κ -resolvable iff X contains κ disjoint dense subsets. If D is dense and U is a non-empty open set, then U ∩ D � = ∅ . So if X is κ -resolvable then κ ≤ min {| U | : U ∈ τ X \ {∅}} =∆( X ) . ∆( X ) is the dispersion character of X . Definition (Ceder, Pearson, 1967) X is maximally resolvable iff it is ∆( X ) -resolvable. Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings. Basic notions X is κ -resolvable iff X contains κ disjoint dense subsets. If D is dense and U is a non-empty open set, then U ∩ D � = ∅ . So if X is κ -resolvable then κ ≤ min {| U | : U ∈ τ X \ {∅}} =∆( X ) . ∆( X ) is the dispersion character of X . Definition (Ceder, Pearson, 1967) X is maximally resolvable iff it is ∆( X ) -resolvable. Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings X is κ -resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆( X ) -resolvable . irresolvable ≡ not 2-resolvable There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces? Soukup, L (HAS) RIMS 2010 14 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Given a space X , define the family of marked open sets as follows: � � M ( X ) = � x , U � ∈ X × τ ( X ) : x ∈ U X is monotonically normal ( MN ) if (1) X is T 1 , (2) X admits a monotone normality operator i.e. there is function H : M ( X ) → τ ( X ) such that (1) x ∈ H ( x , U ) ⊂ U for each � x , U � ∈ M ( X ) , (2) if ( x , U ) , ( y , V ) ∈ M ( X ) , x / ∈ V and y / ∈ U then H ( x , U ) ∩ H ( y , V ) = ∅ . Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable? Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces Are the monotonically normal spaces maximally resolvable? Theorem (Juhász,S, Szentmiklóssy) A dense-in-itself monotonically normal space is ω -resolvable Problem (Ceder, Pearson 1967) Does ω -resolvable imply maximally resolvable ? Theorem (Juhász, S, Szentmiklóssy) For each infinite κ there is a 0-dimensional T 2 space X = � κ, τ � , s.t. ∆( X ) = κ , X is ω -resolvable , but not ω 1 -resolvable . Soukup, L (HAS) RIMS 2010 16 / 23

Monotonically normal spaces Are the monotonically normal spaces maximally resolvable? Theorem (Juhász,S, Szentmiklóssy) A dense-in-itself monotonically normal space is ω -resolvable Problem (Ceder, Pearson 1967) Does ω -resolvable imply maximally resolvable ? Theorem (Juhász, S, Szentmiklóssy) For each infinite κ there is a 0-dimensional T 2 space X = � κ, τ � , s.t. ∆( X ) = κ , X is ω -resolvable , but not ω 1 -resolvable . Soukup, L (HAS) RIMS 2010 16 / 23

Monotonically normal spaces Are the monotonically normal spaces maximally resolvable? Theorem (Juhász,S, Szentmiklóssy) A dense-in-itself monotonically normal space is ω -resolvable Problem (Ceder, Pearson 1967) Does ω -resolvable imply maximally resolvable ? Theorem (Juhász, S, Szentmiklóssy) For each infinite κ there is a 0-dimensional T 2 space X = � κ, τ � , s.t. ∆( X ) = κ , X is ω -resolvable , but not ω 1 -resolvable . Soukup, L (HAS) RIMS 2010 16 / 23

Monotonically normal spaces Are the monotonically normal spaces maximally resolvable? Theorem (Juhász,S, Szentmiklóssy) A dense-in-itself monotonically normal space is ω -resolvable Problem (Ceder, Pearson 1967) Does ω -resolvable imply maximally resolvable ? Theorem (Juhász, S, Szentmiklóssy) For each infinite κ there is a 0-dimensional T 2 space X = � κ, τ � , s.t. ∆( X ) = κ , X is ω -resolvable , but not ω 1 -resolvable . Soukup, L (HAS) RIMS 2010 16 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction ? X dense-in-itself, monotonically normal = ⇒ X maximally resolvable? If T is a trees, t ∈ T , then succ ( t ) = the succerrors of t in T T is everywhere infinitely branching iff succ T ( t ) is infinite a filtration on T is a map F s.t. dom ( F ) = T and F ( t ) is a filter on succ T ( t ) . Define the topological space X F = � T , τ F � : U ⊂ T is open iff for each t ∈ U the set succ T ( t ) ∩ U ∈ F ( t ) . X F is monotonically normal: H ( t , V )= { u ∈ V : [ t , u ] ⊂ V } We conjectured: If T = ω <ω and F ( t ) is an uniform ultrafilter on “ ω 1 ” 1 then X F is not ω 1 -resolvable Proposition If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 Soukup, L (HAS) RIMS 2010 17 / 23

A consistent counterexample If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . Theorem (J-S-Sz) Let F be an ultrafiltration on T. If every F ( t ) is ω -decomposable , then X F is ω 1 -resolvable . Theorem (J-S-Sz) If κ is a measurable cardinal , T = κ <ω , F ( t ) is a measure on succ T ( t ) ≈ κ , then X F is ω 1 -irresolvable . Soukup, L (HAS) RIMS 2010 18 / 23

A consistent counterexample If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . Theorem (J-S-Sz) Let F be an ultrafiltration on T. If every F ( t ) is ω -decomposable , then X F is ω 1 -resolvable . Theorem (J-S-Sz) If κ is a measurable cardinal , T = κ <ω , F ( t ) is a measure on succ T ( t ) ≈ κ , then X F is ω 1 -irresolvable . Soukup, L (HAS) RIMS 2010 18 / 23

A consistent counterexample If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . Theorem (J-S-Sz) Let F be an ultrafiltration on T. If every F ( t ) is ω -decomposable , then X F is ω 1 -resolvable . Theorem (J-S-Sz) If κ is a measurable cardinal , T = κ <ω , F ( t ) is a measure on succ T ( t ) ≈ κ , then X F is ω 1 -irresolvable . Soukup, L (HAS) RIMS 2010 18 / 23

A consistent counterexample If F is a uniform ultrafiltration on T = ω <ω then X F is ω 1 -resolvable . 1 An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . Theorem (J-S-Sz) Let F be an ultrafiltration on T. If every F ( t ) is ω -decomposable , then X F is ω 1 -resolvable . Theorem (J-S-Sz) If κ is a measurable cardinal , T = κ <ω , F ( t ) is a measure on succ T ( t ) ≈ κ , then X F is ω 1 -irresolvable . Soukup, L (HAS) RIMS 2010 18 / 23

Smaller counterexamples An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . An ultrafilter U is λ -descendingly complete iff � { X ξ : ξ < λ } ∈ U for each decreasing sequence { X ξ : ξ < λ } ⊂ U . Theorems: (J-S-Sz) If λ = cf ( λ ) , F is an ultrafiltration on T , F ( t ) is λ -descendingly complete for all t ∈ T , then X F is λ + -irresolvable . (Magidor): It is consistent from a supercompact that there is an ω 1 -descendingly complete uniform ultrafilter on ℵ ω Con ( ∃ supercompact) implies Con( there is a monotonically normal space X with | X | = ∆( X ) = ℵ ω that is no ω 2 -resolvable ). Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . An ultrafilter U is λ -descendingly complete iff � { X ξ : ξ < λ } ∈ U for each decreasing sequence { X ξ : ξ < λ } ⊂ U . Theorems: (J-S-Sz) If λ = cf ( λ ) , F is an ultrafiltration on T , F ( t ) is λ -descendingly complete for all t ∈ T , then X F is λ + -irresolvable . (Magidor): It is consistent from a supercompact that there is an ω 1 -descendingly complete uniform ultrafilter on ℵ ω Con ( ∃ supercompact) implies Con( there is a monotonically normal space X with | X | = ∆( X ) = ℵ ω that is no ω 2 -resolvable ). Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . An ultrafilter U is λ -descendingly complete iff � { X ξ : ξ < λ } ∈ U for each decreasing sequence { X ξ : ξ < λ } ⊂ U . Theorems: (J-S-Sz) If λ = cf ( λ ) , F is an ultrafiltration on T , F ( t ) is λ -descendingly complete for all t ∈ T , then X F is λ + -irresolvable . (Magidor): It is consistent from a supercompact that there is an ω 1 -descendingly complete uniform ultrafilter on ℵ ω Con ( ∃ supercompact) implies Con( there is a monotonically normal space X with | X | = ∆( X ) = ℵ ω that is no ω 2 -resolvable ). Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . An ultrafilter U is λ -descendingly complete iff � { X ξ : ξ < λ } ∈ U for each decreasing sequence { X ξ : ξ < λ } ⊂ U . Theorems: (J-S-Sz) If λ = cf ( λ ) , F is an ultrafiltration on T , F ( t ) is λ -descendingly complete for all t ∈ T , then X F is λ + -irresolvable . (Magidor): It is consistent from a supercompact that there is an ω 1 -descendingly complete uniform ultrafilter on ℵ ω Con ( ∃ supercompact) implies Con( there is a monotonically normal space X with | X | = ∆( X ) = ℵ ω that is no ω 2 -resolvable ). Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . An ultrafilter U is λ -descendingly complete iff � { X ξ : ξ < λ } ∈ U for each decreasing sequence { X ξ : ξ < λ } ⊂ U . Theorems: (J-S-Sz) If λ = cf ( λ ) , F is an ultrafiltration on T , F ( t ) is λ -descendingly complete for all t ∈ T , then X F is λ + -irresolvable . (Magidor): It is consistent from a supercompact that there is an ω 1 -descendingly complete uniform ultrafilter on ℵ ω Con ( ∃ supercompact) implies Con( there is a monotonically normal space X with | X | = ∆( X ) = ℵ ω that is no ω 2 -resolvable ). Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples An ultrafilter U is λ -decomposable iff there is a decreasing sequence { X ξ : ξ < λ } ⊂ U with � { X ξ : ξ < λ } = ∅ . An ultrafilter U is λ -descendingly complete iff � { X ξ : ξ < λ } ∈ U for each decreasing sequence { X ξ : ξ < λ } ⊂ U . Theorems: (J-S-Sz) If λ = cf ( λ ) , F is an ultrafiltration on T , F ( t ) is λ -descendingly complete for all t ∈ T , then X F is λ + -irresolvable . (Magidor): It is consistent from a supercompact that there is an ω 1 -descendingly complete uniform ultrafilter on ℵ ω Con ( ∃ supercompact) implies Con( there is a monotonically normal space X with | X | = ∆( X ) = ℵ ω that is no ω 2 -resolvable ). Soukup, L (HAS) RIMS 2010 19 / 23