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Ccc Indestructible S Spaces

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andrejko

A presentation on Soukup's work on ccc indestructible spaces with strong separability properties.

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INDESTRUCTIBLE SPACES WITH STRONG CCC SEPARABILITY AND SEPARABILITY IN 2ω1 MAℵ1 Erik A. Andrejko University of Wisconsin - Madison Feb 27 2007 ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
CH TO MAℵ1 TO PFA ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
CH TO MAℵ1 TO PFA THFD HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
CH TO MAℵ1 TO PFA THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
CH TO MAℵ1 TO PFA ? THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. If X is hereditarily separable and not Lindelöf then X is called an S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. If X is hereditarily separable and not Lindelöf then X is called an S-space. DEFINITION A is finally dense if for some γ < ω1 A is dense in 2ω1 γ . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. If X is hereditarily separable and not Lindelöf then X is called an S-space. DEFINITION A is finally dense if for some γ < ω1 A is dense in 2ω1 γ . X is a weak HFD iff for all Y ∈ [X ]ω1 there is some A ∈ [Y ]ω such that A is finally dense. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY DEFINITION X is a HFD iff for all A ∈ [X ]ω , A is finally dense. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY DEFINITION X is a HFD iff for all A ∈ [X ]ω , A is finally dense. LEMMA If there is an HFD, there is a weak HFD. If there is a weak HFD, there is an S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SEPARABILITY DEFINITION X is a HFD iff for all A ∈ [X ]ω , A is finally dense. LEMMA If there is an HFD, there is a weak HFD. If there is a weak HFD, there is an S-space. HFD HFDw S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
O-SPACES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. LEMMA If there is a weak HFD there is an O-space. If there is an O-space, there is S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. LEMMA If there is a weak HFD there is an O-space. If there is an O-space, there is S-space. HFD HFDw S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. LEMMA If there is a weak HFD there is an O-space. If there is an O-space, there is S-space. HFD HFDw O-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONG SPACES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONG SPACES DEFINITION A Φ space X is called a strong Φ space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , and hence a strong S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , and hence a strong S-space. COROLLARY (CH) There exists a HFDn space and S-spacen for all n < ω. w ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , and hence a strong S-space. COROLLARY (CH) There exists a HFDn space and S-spacen for all n < ω. w THEOREM (CH) There exists an HFD. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SPACES UNDER CH ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SPACES UNDER CH HFD HFDw O-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
SPACES UNDER CH HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE THEOREM (ROITMAN) Let r be a Cohen real. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE THEOREM (ROITMAN) Let r be a Cohen real. V [r ] |= ∃a strong HFD ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of ω1 . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of ω1 . Then there is an infinite E ⊆ ω such that either An n∈E ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of ω1 . Then there is an infinite E ⊆ ω such that either (ω1 An ) An or n∈E n∈E is uncountable. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE COROLLARY If V |= MAℵ1 , and r is a Cohen real, then V [r ] |= MAℵ1 . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
EXISTENCE AND NONEXISTENCE COROLLARY If V |= MAℵ1 , and r is a Cohen real, then V [r ] |= MAℵ1 . THEOREM (KUNEN) (MAℵ1 ) There are no strong S-spaces. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
UNDER MAℵ1 ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
UNDER MAℵ1 THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
UNDER MAℵ1 ? THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
UNDER MAℵ1 THFD THFD THFD HFD HFDw HFD ω HFD ω ω HFDw HFDw HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
UNDER MAℵ1 THFD THFD THFD HFD HFDw HFD ω HFD ω ω HFDw HFDw ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
UNDER MAℵ1 THFD THFD THFD HFD HFDw HFD ω HFD ω ω HFDw HFDw ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space QUESTION Does there exist an S-space, O-space, or weak HFD under MAℵ1 ? Finite powers? ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space in any ccc extension. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space in any ccc extension. DEFINITION Let X be an S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space in any ccc extension. DEFINITION Let X be an S-space. Then let PX be the natural order to add an uncountable discrete subspace with finite conditions. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A Let A, B we well ordered of type α, β < ω1 limit ordinals. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A Let A, B we well ordered of type α, β < ω1 limit ordinals. Then A is tight in B iff for some n, ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A Let A, B we well ordered of type α, β < ω1 limit ordinals. Then A is tight in B iff for some n, every interval of B of length n meets A. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE S-SPACE CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. THFD THFD THFD HFD HFDw HFD HFDw ω HFD HFDw ω ω ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. THFD THFD THFD HFD HFDw HFD HFDw ω HFD HFDw ω ω HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w DEFINITION A graph G is strongly solid iff G is m-solid for every m < ω. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w DEFINITION A graph G is strongly solid iff G is m-solid for every m < ω. THEOREM (SOUKUP) Let V |= quot;G is strongly solidquot;. For any m there is a P such that ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w DEFINITION A graph G is strongly solid iff G is m-solid for every m < ω. THEOREM (SOUKUP) Let V |= quot;G is strongly solidquot;. For any m there is a P such that V P |= quot;G is ccc-indestructibly m-solidquot;. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC Assume 2ω1 = ω2 . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ |Qη | = ω1 1Pη ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . 1Pη ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
? THFD THFD HFD HFDwω HFD HFDwω HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
THFD THFD THFD HFD HFDw HFD ω HFD HFDw ω ω HFDw ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
THFD THFD THFD HFD HFDw HFD ω HFD HFDw ω ω HFDw HFDw HFDw HFDw ω ω O-space S-space ω O-space S-space O-space S-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
OPEN QUESTIONS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
OPEN QUESTIONS Consistency questions: ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
OPEN QUESTIONS Consistency questions: HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
OPEN QUESTIONS Consistency questions: HFD HFDw HFD HFDw ω ω HFDw HFDw ω ω O-space S-space O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
OPEN QUESTIONS Consistency questions: HFD HFDw HFD HFDw ω ω HFDw HFDw ω ω O-space S-space O-space S-space S-space S-space QUESTION (JUHASZ) Does there exists a (c, →)-HFD? ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
REFERENCES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC

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Ccc Indestructible S Spaces

  • 1. INDESTRUCTIBLE SPACES WITH STRONG CCC SEPARABILITY AND SEPARABILITY IN 2ω1 MAℵ1 Erik A. Andrejko University of Wisconsin - Madison Feb 27 2007 ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 2. CH TO MAℵ1 TO PFA ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 3. CH TO MAℵ1 TO PFA THFD HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 4. CH TO MAℵ1 TO PFA THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 5. CH TO MAℵ1 TO PFA ? THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 6. SEPARABILITY ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 7. SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 8. SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 9. SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 10. SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. If X is hereditarily separable and not Lindelöf then X is called an S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 11. SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. If X is hereditarily separable and not Lindelöf then X is called an S-space. DEFINITION A is finally dense if for some γ < ω1 A is dense in 2ω1 γ . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 12. SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). DEFINITION X is separable if X has a countable dense subset. X is hereditarily separable if every subspace has a countable dense subset. If X is hereditarily separable and not Lindelöf then X is called an S-space. DEFINITION A is finally dense if for some γ < ω1 A is dense in 2ω1 γ . X is a weak HFD iff for all Y ∈ [X ]ω1 there is some A ∈ [Y ]ω such that A is finally dense. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 13. SEPARABILITY DEFINITION X is a HFD iff for all A ∈ [X ]ω , A is finally dense. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 14. SEPARABILITY DEFINITION X is a HFD iff for all A ∈ [X ]ω , A is finally dense. LEMMA If there is an HFD, there is a weak HFD. If there is a weak HFD, there is an S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 15. SEPARABILITY DEFINITION X is a HFD iff for all A ∈ [X ]ω , A is finally dense. LEMMA If there is an HFD, there is a weak HFD. If there is a weak HFD, there is an S-space. HFD HFDw S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 16. O-SPACES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 17. O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 18. O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. LEMMA If there is a weak HFD there is an O-space. If there is an O-space, there is S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 19. O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. LEMMA If there is a weak HFD there is an O-space. If there is an O-space, there is S-space. HFD HFDw S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 20. O-SPACES DEFINITION X is an O-space iff every open set is countable or co-countable. LEMMA If there is a weak HFD there is an O-space. If there is an O-space, there is S-space. HFD HFDw O-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 21. STRONG SPACES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 22. STRONG SPACES DEFINITION A Φ space X is called a strong Φ space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 23. STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 24. STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 25. STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , and hence a strong S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 26. STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , and hence a strong S-space. COROLLARY (CH) There exists a HFDn space and S-spacen for all n < ω. w ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 27. STRONG SPACES DEFINITION A Φ space X is called a strong Φ space if every finite power X n is a Φ space. e.g. HFDw , S-space. THEOREM (CH) There exists a strong HFDw , and hence a strong S-space. COROLLARY (CH) There exists a HFDn space and S-spacen for all n < ω. w THEOREM (CH) There exists an HFD. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 28. SPACES UNDER CH ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 29. SPACES UNDER CH HFD HFDw O-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 30. SPACES UNDER CH HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 31. EXISTENCE AND NONEXISTENCE THEOREM (ROITMAN) Let r be a Cohen real. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 32. EXISTENCE AND NONEXISTENCE THEOREM (ROITMAN) Let r be a Cohen real. V [r ] |= ∃a strong HFD ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 33. EXISTENCE AND NONEXISTENCE ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 34. EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 35. EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 36. EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of ω1 . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 37. EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of ω1 . Then there is an infinite E ⊆ ω such that either An n∈E ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 38. EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. LEMMA (SILVER’S LEMMA) Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of ω1 . Then there is an infinite E ⊆ ω such that either (ω1 An ) An or n∈E n∈E is uncountable. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 39. EXISTENCE AND NONEXISTENCE ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 40. EXISTENCE AND NONEXISTENCE COROLLARY If V |= MAℵ1 , and r is a Cohen real, then V [r ] |= MAℵ1 . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 41. EXISTENCE AND NONEXISTENCE COROLLARY If V |= MAℵ1 , and r is a Cohen real, then V [r ] |= MAℵ1 . THEOREM (KUNEN) (MAℵ1 ) There are no strong S-spaces. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 42. UNDER MAℵ1 ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 43. UNDER MAℵ1 THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 44. UNDER MAℵ1 ? THFD THFD HFD HFDwω HFD ω HFDw HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 45. UNDER MAℵ1 THFD THFD THFD HFD HFDw HFD ω HFD ω ω HFDw HFDw HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 46. UNDER MAℵ1 THFD THFD THFD HFD HFDw HFD ω HFD ω ω HFDw HFDw ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 47. UNDER MAℵ1 THFD THFD THFD HFD HFDw HFD ω HFD ω ω HFDw HFDw ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space QUESTION Does there exist an S-space, O-space, or weak HFD under MAℵ1 ? Finite powers? ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 48. INDESTRUCTIBLE SPACES CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 49. INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 50. INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 51. INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 52. INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 53. INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space in any ccc extension. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 54. INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space in any ccc extension. DEFINITION Let X be an S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 55. INDESTRUCTIBLE SPACES CCC DEFINITION A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and any P-generic filter G over V , V |= ϕ(X ) =⇒ V [G] |= ϕ(X ) e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space in any ccc extension. DEFINITION Let X be an S-space. Then let PX be the natural order to add an uncountable discrete subspace with finite conditions. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 56. INDESTRUCTIBLE SPACES CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 57. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 58. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 59. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 60. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 61. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 62. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 63. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A Let A, B we well ordered of type α, β < ω1 limit ordinals. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 64. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A Let A, B we well ordered of type α, β < ω1 limit ordinals. Then A is tight in B iff for some n, ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 65. INDESTRUCTIBLE SPACES CCC THEOREM (SZENTMIKLÓSSY) If X is a ccc destructible S-space, then some uncountable A ⊆ PX has ccc. DEFINITION X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω : [ε] ∩ A is tight in A Let A, B we well ordered of type α, β < ω1 limit ordinals. Then A is tight in B iff for some n, every interval of B of length n meets A. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 66. INDESTRUCTIBLE S-SPACE CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 67. INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 68. INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 69. INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 70. INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. THFD THFD THFD HFD HFDw HFD HFDw ω HFD HFDw ω ω ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 71. INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. THEOREM (SZENTMIKLÓSSY) If X is a tight HFD, then X is a ccc indestructible S-space. THFD THFD THFD HFD HFDw HFD HFDw ω HFD HFDw ω ω HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 72. STRONGLY SOLID GRAPHS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 73. STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 74. STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 75. STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 76. STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 77. STRONGLY SOLID GRAPHS DEFINITION A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are α < β < ω1 such that [sα , sβ ] ⊆ G ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 78. STRONGLY SOLID GRAPHS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 79. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 80. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 81. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 82. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 83. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w DEFINITION A graph G is strongly solid iff G is m-solid for every m < ω. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 84. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w DEFINITION A graph G is strongly solid iff G is m-solid for every m < ω. THEOREM (SOUKUP) Let V |= quot;G is strongly solidquot;. For any m there is a P such that ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 85. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , there is a graph GX such that GX is m-solid ⇐⇒ X is HFDm w DEFINITION A graph G is strongly solid iff G is m-solid for every m < ω. THEOREM (SOUKUP) Let V |= quot;G is strongly solidquot;. For any m there is a P such that V P |= quot;G is ccc-indestructibly m-solidquot;. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 86. INDESTRICTIBLE m-SOLID CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 87. INDESTRICTIBLE m-SOLID CCC Assume 2ω1 = ω2 . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 88. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 89. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ |Qη | = ω1 1Pη ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 90. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . 1Pη ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 91. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 92. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 93. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 94. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 95. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 96. INDESTRICTIBLE m-SOLID CCC ◦ Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that ◦ 1Pη |Qη | = ω1 and so (2ω1 )V Pω2 = ω2 . Furthermore P satisfies the previous theorem. { { ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 97. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 98. ? THFD THFD HFD HFDwω HFD HFDwω HFDw HFDw ω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 99. THFD THFD THFD HFD HFDw HFD ω HFD HFDw ω ω HFDw ? HFDw HFDw ω S-spaceω O-space S-space ω O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 100. THFD THFD THFD HFD HFDw HFD ω HFD HFDw ω ω HFDw HFDw HFDw HFDw ω ω O-space S-space ω O-space S-space O-space S-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 101. OPEN QUESTIONS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 102. OPEN QUESTIONS Consistency questions: ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 103. OPEN QUESTIONS Consistency questions: HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 104. OPEN QUESTIONS Consistency questions: HFD HFDw HFD HFDw ω ω HFDw HFDw ω ω O-space S-space O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 105. OPEN QUESTIONS Consistency questions: HFD HFDw HFD HFDw ω ω HFDw HFDw ω ω O-space S-space O-space S-space S-space S-space QUESTION (JUHASZ) Does there exists a (c, →)-HFD? ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  • 106. REFERENCES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC