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Countable Dense Homogeneous Filters 2012 Winterschool, Hejnice Rodrigo Hernández-Gutiérrez (joint work with Michael Hrušák) [email protected] Universidad Nacional Autónoma de México Countable Dense Homogeneous Filters – p. 1 Countable Dense Homogeneous Spaces A Hausdorff separable space X is countable dense homogeneous (CDH) if every time D, E are countable dense subsets there exists a homeomorphism h : X → X such that h[D] = E . Countable Dense Homogeneous Filters – p. 2 Countable Dense Homogeneous Spaces A Hausdorff separable space X is countable dense homogeneous (CDH) if every time D, E are countable dense subsets there exists a homeomorphism h : X → X such that h[D] = E . That is, any two countable dense subsets of X are placed “in the same way” inside X . Countable Dense Homogeneous Filters – p. 2 Countable Dense Homogeneous Spaces A Hausdorff separable space X is countable dense homogeneous (CDH) if every time D, E are countable dense subsets there exists a homeomorphism h : X → X such that h[D] = E . That is, any two countable dense subsets of X are placed “in the same way” inside X . Examples: Countable Dense Homogeneous Filters – p. 2 Countable Dense Homogeneous Spaces A Hausdorff separable space X is countable dense homogeneous (CDH) if every time D, E are countable dense subsets there exists a homeomorphism h : X → X such that h[D] = E . That is, any two countable dense subsets of X are placed “in the same way” inside X . Examples: the real line is CDH Countable Dense Homogeneous Filters – p. 2 Countable Dense Homogeneous Spaces A Hausdorff separable space X is countable dense homogeneous (CDH) if every time D, E are countable dense subsets there exists a homeomorphism h : X → X such that h[D] = E . That is, any two countable dense subsets of X are placed “in the same way” inside X . Examples: the real line is CDH (use back and forth), Countable Dense Homogeneous Filters – p. 2 Countable Dense Homogeneous Spaces A Hausdorff separable space X is countable dense homogeneous (CDH) if every time D, E are countable dense subsets there exists a homeomorphism h : X → X such that h[D] = E . That is, any two countable dense subsets of X are placed “in the same way” inside X . Examples: the real line is CDH (use back and forth), the rationals are NOT CDH (remove one point), Countable Dense Homogeneous Filters – p. 2 Countable Dense Homogeneous Spaces A Hausdorff separable space X is countable dense homogeneous (CDH) if every time D, E are countable dense subsets there exists a homeomorphism h : X → X such that h[D] = E . That is, any two countable dense subsets of X are placed “in the same way” inside X . Examples: the real line is CDH (use back and forth), the rationals are NOT CDH (remove one point), other spaces known to be CDH: Euclidean spaces, the Cantor set, the Hilbert cube, Hilbert space... Countable Dense Homogeneous Filters – p. 2 Separable Metrizable Spaces Let us restrict to Separable Metrizable Spaces from now on. Countable Dense Homogeneous Filters – p. 3 Separable Metrizable Spaces Let us restrict to Separable Metrizable Spaces from now on. So Euclidean spaces, the Cantor set, the Hilbert cube are CDH. Countable Dense Homogeneous Filters – p. 3 Separable Metrizable Spaces Let us restrict to Separable Metrizable Spaces from now on. So Euclidean spaces, the Cantor set, the Hilbert cube are CDH. Question (Fitzpatrick and Zhou). Is there a CDH metrizable space X that is not completely metrizable? Countable Dense Homogeneous Filters – p. 3 Separable Metrizable Spaces Let us restrict to Separable Metrizable Spaces from now on. So Euclidean spaces, the Cantor set, the Hilbert cube are CDH. Question (Fitzpatrick and Zhou). Is there a CDH metrizable space X that is not completely metrizable? It is not difficult to construct a CDH Bernstein set... Countable Dense Homogeneous Filters – p. 3 Separable Metrizable Spaces Let us restrict to Separable Metrizable Spaces from now on. So Euclidean spaces, the Cantor set, the Hilbert cube are CDH. Question (Fitzpatrick and Zhou). Is there a CDH metrizable space X that is not completely metrizable? It is not difficult to construct a CDH Bernstein set... using some additional hypothesis like CH. Countable Dense Homogeneous Filters – p. 3 Separable Metrizable Spaces Let us restrict to Separable Metrizable Spaces from now on. So Euclidean spaces, the Cantor set, the Hilbert cube are CDH. Question (Fitzpatrick and Zhou). Is there a CDH metrizable space X that is not completely metrizable? It is not difficult to construct a CDH Bernstein set... using some additional hypothesis like CH. Question. For which 0-dimensional subsets X of R is ω X CDH? Countable Dense Homogeneous Filters – p. 3 Some Answers Theorem 1 [Hrušák and Zamora Áviles] Let X be a separable metrizable space. Countable Dense Homogeneous Filters – p. 4 Some Answers Theorem 1 [Hrušák and Zamora Áviles] Let X be a separable metrizable space. (1) If X is CDH and Borel, then X is completely metrizable. Countable Dense Homogeneous Filters – p. 4 Some Answers Theorem 1 [Hrušák and Zamora Áviles] Let X be a separable metrizable space. (1) If X is CDH and Borel, then X is completely metrizable. (2) If ω X is CDH, the X is a Baire space. Countable Dense Homogeneous Filters – p. 4 Some Answers Theorem 1 [Hrušák and Zamora Áviles] Let X be a separable metrizable space. (1) If X is CDH and Borel, then X is completely metrizable. (2) If ω X is CDH, the X is a Baire space. Theorem 2 [Farah, Hrušák and Martinez-Ranero] There is a CDH set of reals X of size ω1 that is a λ-set (all countable subsets are relative Gδ ). Countable Dense Homogeneous Filters – p. 4 Some Answers Theorem 1 [Hrušák and Zamora Áviles] Let X be a separable metrizable space. (1) If X is CDH and Borel, then X is completely metrizable. (2) If ω X is CDH, the X is a Baire space. Theorem 2 [Farah, Hrušák and Martinez-Ranero] There is a CDH set of reals X of size ω1 that is a λ-set (all countable subsets are relative Gδ ). Notice that by (2) in Theorem 1, it is not possible to extend the result of Theorem 2. Countable Dense Homogeneous Filters – p. 4 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. In particular, let’s consider non-principal ultrafilters U ⊂ P(ω). Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. In particular, let’s consider non-principal ultrafilters U ⊂ P(ω). Theorem 3 [Medini and Milovich] Assume MA(countable). Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. In particular, let’s consider non-principal ultrafilters U ⊂ P(ω). Theorem 3 [Medini and Milovich] Assume MA(countable). Then there exists a non-principal ultrafilter U ⊂ P(ω) with one of the following properties: Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. In particular, let’s consider non-principal ultrafilters U ⊂ P(ω). Theorem 3 [Medini and Milovich] Assume MA(countable). Then there exists a non-principal ultrafilter U ⊂ P(ω) with one of the following properties: U is CDH and a P -point, Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. In particular, let’s consider non-principal ultrafilters U ⊂ P(ω). Theorem 3 [Medini and Milovich] Assume MA(countable). Then there exists a non-principal ultrafilter U ⊂ P(ω) with one of the following properties: U is CDH and a P -point, U is CDH and not a P -point, Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. In particular, let’s consider non-principal ultrafilters U ⊂ P(ω). Theorem 3 [Medini and Milovich] Assume MA(countable). Then there exists a non-principal ultrafilter U ⊂ P(ω) with one of the following properties: U is CDH and a P -point, U is CDH and not a P -point, U is not CDH and not a P -point, Countable Dense Homogeneous Filters – p. 5 Medini and Milovich paper The Cantor set ω 2 can be identified to P(ω) using characteristic functions. Thus, any subset X ⊂ P(ω) can be considered with the Cantor set topology. In particular, let’s consider non-principal ultrafilters U ⊂ P(ω). Theorem 3 [Medini and Milovich] Assume MA(countable). Then there exists a non-principal ultrafilter U ⊂ P(ω) with one of the following properties: U is CDH and a P -point, U is CDH and not a P -point, U is not CDH and not a P -point, ω U is CDH. Countable Dense Homogeneous Filters – p. 5 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Countable Dense Homogeneous Filters – p. 6 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Not really!! Countable Dense Homogeneous Filters – p. 6 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Not really!! Theorem (Hernández-Gutiérrez and Hrušák) Let F ⊂ P(ω) be a filter that extends the Fréchet filter. If F is a non-meager P -filter, then both F and ω F are CDH. Countable Dense Homogeneous Filters – p. 6 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Not really!! Theorem (Hernández-Gutiérrez and Hrušák) Let F ⊂ P(ω) be a filter that extends the Fréchet filter. If F is a non-meager P -filter, then both F and ω F are CDH. Remark: Countable Dense Homogeneous Filters – p. 6 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Not really!! Theorem (Hernández-Gutiérrez and Hrušák) Let F ⊂ P(ω) be a filter that extends the Fréchet filter. If F is a non-meager P -filter, then both F and ω F are CDH. Remark: It is not known that non-meager P -filters exist in ZFC. Countable Dense Homogeneous Filters – p. 6 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Not really!! Theorem (Hernández-Gutiérrez and Hrušák) Let F ⊂ P(ω) be a filter that extends the Fréchet filter. If F is a non-meager P -filter, then both F and ω F are CDH. Remark: It is not known that non-meager P -filters exist in ZFC. However, their existence follows from cof [d]ω = d. Countable Dense Homogeneous Filters – p. 6 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Not really!! Theorem (Hernández-Gutiérrez and Hrušák) Let F ⊂ P(ω) be a filter that extends the Fréchet filter. If F is a non-meager P -filter, then both F and ω F are CDH. Remark: It is not known that non-meager P -filters exist in ZFC. However, their existence follows from cof [d]ω = d. Thus, if all P -filters are meager, then there are inner models with large cardinals. Countable Dense Homogeneous Filters – p. 6 Our results Is it really necessary to use MA(countable) in order to construct a CDH filter? Not really!! Theorem (Hernández-Gutiérrez and Hrušák) Let F ⊂ P(ω) be a filter that extends the Fréchet filter. If F is a non-meager P -filter, then both F and ω F are CDH. Remark: It is not known that non-meager P -filters exist in ZFC. However, their existence follows from cof [d]ω = d. Thus, if all P -filters are meager, then there are inner models with large cardinals. Proposition. Any CDH filter must be non-meager. Countable Dense Homogeneous Filters – p. 6 Proof of the Proposition Proposition. Any CDH filter F must be non-meager. Countable Dense Homogeneous Filters – p. 7 Proof of the Proposition Proposition. Any CDH filter F must be non-meager. Proof. The Frechet filter is not CDH because it is countable. Countable Dense Homogeneous Filters – p. 7 Proof of the Proposition Proposition. Any CDH filter F must be non-meager. Proof. The Frechet filter is not CDH because it is countable. If x ∈ F is coinfinite, {y ∈ P(ω) : x ⊂ y} ⊂ F is a Cantor set. Countable Dense Homogeneous Filters – p. 7 Proof of the Proposition Proposition. Any CDH filter F must be non-meager. Proof. The Frechet filter is not CDH because it is countable. If x ∈ F is coinfinite, {y ∈ P(ω) : x ⊂ y} ⊂ F is a Cantor set. Let D be a countable dense set of {y ∈ P(ω) : x ⊂ y} ⊂ F . Countable Dense Homogeneous Filters – p. 7 Proof of the Proposition Proposition. Any CDH filter F must be non-meager. Proof. The Frechet filter is not CDH because it is countable. If x ∈ F is coinfinite, {y ∈ P(ω) : x ⊂ y} ⊂ F is a Cantor set. Let D be a countable dense set of {y ∈ P(ω) : x ⊂ y} ⊂ F . If F is meager, there exists a countable dense subset E of F that is a relative Gδ . Countable Dense Homogeneous Filters – p. 7 Proof of the Proposition Proposition. Any CDH filter F must be non-meager. Proof. The Frechet filter is not CDH because it is countable. If x ∈ F is coinfinite, {y ∈ P(ω) : x ⊂ y} ⊂ F is a Cantor set. Let D be a countable dense set of {y ∈ P(ω) : x ⊂ y} ⊂ F . If F is meager, there exists a countable dense subset E of F that is a relative Gδ . No homeomorphism takes D inside E . Countable Dense Homogeneous Filters – p. 7 Proof of the Proposition Proposition. Any CDH filter F must be non-meager. Proof. The Frechet filter is not CDH because it is countable. If x ∈ F is coinfinite, {y ∈ P(ω) : x ⊂ y} ⊂ F is a Cantor set. Let D be a countable dense set of {y ∈ P(ω) : x ⊂ y} ⊂ F . If F is meager, there exists a countable dense subset E of F that is a relative Gδ . No homeomorphism takes D inside E . Notice that in this proof we really found two different countable dense subsets. Countable Dense Homogeneous Filters – p. 7 ω From F to F Countable Dense Homogeneous Filters – p. 8 ω From F to F If F is a non-meager P -filter, then ω F is homeomorphic to Countable Dense Homogeneous Filters – p. 8 ω From F to F If F is a non-meager P -filter, then ω F is homeomorphic to {A ⊂ ω × ω : ∀n < ω {x : (x, n) ∈ A} ∈ F}, Countable Dense Homogeneous Filters – p. 8 ω From F to F If F is a non-meager P -filter, then ω F is homeomorphic to {A ⊂ ω × ω : ∀n < ω {x : (x, n) ∈ A} ∈ F}, that is also a non-meager P -filter. Countable Dense Homogeneous Filters – p. 8 ω From F to F If F is a non-meager P -filter, then ω F is homeomorphic to {A ⊂ ω × ω : ∀n < ω {x : (x, n) ∈ A} ∈ F}, that is also a non-meager P -filter. Thus, we only have to prove the Theorem for F . Countable Dense Homogeneous Filters – p. 8 The topological idea Take a non-meager P -filter F and two countable dense subsets D0 , D1 . Countable Dense Homogeneous Filters – p. 9 The topological idea Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Countable Dense Homogeneous Filters – p. 9 The topological idea Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Each hk+1 will “extend” hk and the homeomorphism h : P(ω) → P(ω) is the limit of the hk . Countable Dense Homogeneous Filters – p. 9 The topological idea Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Each hk+1 will “extend” hk and the homeomorphism h : P(ω) → P(ω) is the limit of the hk . Intuitively, in each step we decide what open set goes where, depending on D0 . Countable Dense Homogeneous Filters – p. 9 The topological idea Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Each hk+1 will “extend” hk and the homeomorphism h : P(ω) → P(ω) is the limit of the hk . Intuitively, in each step we decide what open set goes where, depending on D0 . Lemma (Medini and Milovich). Let I be an ideal, D a countable dense subset and h : P(ω) → P(ω) a homeomorphism. If there is a x ∈ I such that d △ h(d) ⊂ x for all d ∈ D , then h[I] = I . Countable Dense Homogeneous Filters – p. 9 The topological idea Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Each hk+1 will “extend” hk and the homeomorphism h : P(ω) → P(ω) is the limit of the hk . Intuitively, in each step we decide what open set goes where, depending on D0 . Lemma (Medini and Milovich). Let I be an ideal, D a countable dense subset and h : P(ω) → P(ω) a homeomorphism. If there is a x ∈ I such that d △ h(d) ⊂ x for all d ∈ D , then h[I] = I . F is homeomorphic to its dual ideal I so we may use the Lemma to obtain the homeomorphism we want. Countable Dense Homogeneous Filters – p. 9 The topological idea Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Each hk+1 will “extend” hk and the homeomorphism h : P(ω) → P(ω) is the limit of the hk . Intuitively, in each step we decide what open set goes where, depending on D0 . Lemma (Medini and Milovich). Let I be an ideal, D a countable dense subset and h : P(ω) → P(ω) a homeomorphism. If there is a x ∈ I such that d △ h(d) ⊂ x for all d ∈ D , then h[I] = I . F is homeomorphic to its dual ideal I so we may use the Lemma to obtain the homeomorphism we want. Thus, we need to construct such an x. Countable Dense Homogeneous Filters – p. 9 Forget about the topology So we want to meet the conditions of the Lemma. Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed) Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed) , then there is some e ∈ D1−i such that Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed) , then there is some e ∈ D1−i such that d − x = e − x Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed) , then there is some e ∈ D1−i such that d − x = e − x (that is, d and e have the same misses) Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed) , then there is some e ∈ D1−i such that d − x = e − x (that is, d and e have the same misses) and e restricted to n ∩ x is as t says Countable Dense Homogeneous Filters – p. 10 Forget about the topology So we want to meet the conditions of the Lemma. Forget about the condition x ∈ I for a while. Thus, we want the following conditions (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed) , then there is some e ∈ D1−i such that d − x = e − x (that is, d and e have the same misses) and e restricted to n ∩ x is as t says (so e is in the open set given by t). Countable Dense Homogeneous Filters – p. 10 Forget about the topology (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed), then there is some e ∈ D1−i such that d − x = e − x (that is, d and e have the same misses) and e restricted to n ∩ x is as t says (so e is in the open set given by t.) Countable Dense Homogeneous Filters – p. 11 Forget about the topology (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed), then there is some e ∈ D1−i such that d − x = e − x (that is, d and e have the same misses) and e restricted to n ∩ x is as t says (so e is in the open set given by t.) In this way, in the steps of the topological construction we can make h(d) = e for some appropriate t (given by the construction). Countable Dense Homogeneous Filters – p. 11 Forget about the topology (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed), then there is some e ∈ D1−i such that d − x = e − x (that is, d and e have the same misses) and e restricted to n ∩ x is as t says (so e is in the open set given by t.) In this way, in the steps of the topological construction we can make h(d) = e for some appropriate t (given by the construction). It is easy to achieve these conditions with an induction, construct such x by finite steps. Countable Dense Homogeneous Filters – p. 11 Forget about the topology (1) for all d ∈ D0 ∪ D1 , d ⊂∗ x (intuitively, d finitely misses x), (2) if i ∈ {0, 1}, d ∈ Di and we have a partial function t : n ∩ x → 2 for n < ω (that is, some basic open set that is compatible with the x constructed), then there is some e ∈ D1−i such that d − x = e − x (that is, d and e have the same misses) and e restricted to n ∩ x is as t says (so e is in the open set given by t.) In this way, in the steps of the topological construction we can make h(d) = e for some appropriate t (given by the construction). It is easy to achieve these conditions with an induction, construct such x by finite steps. But what about x ∈ I ? That’s the tricky part. Countable Dense Homogeneous Filters – p. 11 How we constructed x ∈ I At first, I did not believe Theorem 3 Countable Dense Homogeneous Filters – p. 12 How we constructed x ∈ I At first, I did not believe Theorem 3 until I saw the following. Countable Dense Homogeneous Filters – p. 12 How we constructed x ∈ I At first, I did not believe Theorem 3 until I saw the following. Let X ⊂ [ω]ω . A tree T ⊂ <ω ([ω]<ω ) is called a X -tree of finite subsets if for each s ∈ T there is Xs ∈ X such that for every a ∈ [Xs ]<ω we have s⌢ a ∈ T . Countable Dense Homogeneous Filters – p. 12 How we constructed x ∈ I At first, I did not believe Theorem 3 until I saw the following. Let X ⊂ [ω]ω . A tree T ⊂ <ω ([ω]<ω ) is called a X -tree of finite subsets if for each s ∈ T there is Xs ∈ X such that for every a ∈ [Xs ]<ω we have s⌢ a ∈ T . Lemma. Let F be a filter on P(ω) that extends the Fréchet filter. Then F is a non-meager P -filter if and only if every F -tree of finite subsets has a branch whose union is in F . Countable Dense Homogeneous Filters – p. 12 How we constructed x ∈ I At first, I did not believe Theorem 3 until I saw the following. Let X ⊂ [ω]ω . A tree T ⊂ <ω ([ω]<ω ) is called a X -tree of finite subsets if for each s ∈ T there is Xs ∈ X such that for every a ∈ [Xs ]<ω we have s⌢ a ∈ T . Lemma. Let F be a filter on P(ω) that extends the Fréchet filter. Then F is a non-meager P -filter if and only if every F -tree of finite subsets has a branch whose union is in F . This means that if we do our construction all the ways possible, there will be some one of those that gives x ∈ I . Countable Dense Homogeneous Filters – p. 12 How we constructed x ∈ I At first, I did not believe Theorem 3 until I saw the following. Let X ⊂ [ω]ω . A tree T ⊂ <ω ([ω]<ω ) is called a X -tree of finite subsets if for each s ∈ T there is Xs ∈ X such that for every a ∈ [Xs ]<ω we have s⌢ a ∈ T . Lemma. Let F be a filter on P(ω) that extends the Fréchet filter. Then F is a non-meager P -filter if and only if every F -tree of finite subsets has a branch whose union is in F . This means that if we do our construction all the ways possible, there will be some one of those that gives x ∈ I . And we’re done!! Countable Dense Homogeneous Filters – p. 12 Summary Countable Dense Homogeneous Filters – p. 13 Summary Any non-meager P -filter F is CDH and ω F is CDH. (HG-H) Countable Dense Homogeneous Filters – p. 13 Summary Any non-meager P -filter F is CDH and ω F is CDH. (HG-H) It is consistent that there exist ultrafilters non P -filters that are CDH. (M-M) Countable Dense Homogeneous Filters – p. 13 Summary Any non-meager P -filter F is CDH and ω F is CDH. (HG-H) It is consistent that there exist ultrafilters non P -filters that are CDH. (M-M) CDH filters are non-meager. (HG-H) Countable Dense Homogeneous Filters – p. 13 Summary Any non-meager P -filter F is CDH and ω F is CDH. (HG-H) It is consistent that there exist ultrafilters non P -filters that are CDH. (M-M) CDH filters are non-meager. (HG-H) It is not known in ZFC if there are non-meager filters that are CDH or non-CDH. (one of them should be true!!) Countable Dense Homogeneous Filters – p. 13 Summary Any non-meager P -filter F is CDH and ω F is CDH. (HG-H) It is consistent that there exist ultrafilters non P -filters that are CDH. (M-M) CDH filters are non-meager. (HG-H) It is not known in ZFC if there are non-meager filters that are CDH or non-CDH. (one of them should be true!!) Thank you. Countable Dense Homogeneous Filters – p. 13