TOPOSYM 2016 Weak diamonds and topology Michael Hruˇ s´ ak CCM UNAM michael@matmor.unam.mx Praha July 2016 M. Hruˇ s´ ak Weak diamonds and topology
Typical problem to consider... There are two powerful paradigms in set theory for solving (e.g. topological) problems The Axiom of constructibility V=L, and forcing axioms (MA, PFA, MM,. . . ) We shall look at problems not settled by these (or rather ”settled in the same way”). Usually problems of the form: Is there a topological space (or a family of spaces, or a combinatorial object) with property P? M. Hruˇ s´ ak Weak diamonds and topology
Typical problem to consider... There are two powerful paradigms in set theory for solving (e.g. topological) problems The Axiom of constructibility V=L, and forcing axioms (MA, PFA, MM,. . . ) We shall look at problems not settled by these (or rather ”settled in the same way”). Usually problems of the form: Is there a topological space (or a family of spaces, or a combinatorial object) with property P? M. Hruˇ s´ ak Weak diamonds and topology
Typical analysis of such a problem CH ⇒ Yes MA ⇒ Yes ”Optimize” the above proofs to get inv = c ⇒ YES. Cardinal invariants of the continuum serve primarily as a scale against which we measure the complexity (or strength) of our LONG ( c -many tasks in c -many steps) recursive constructions. There is typically a companion SHORT ( c -many tasks in ω 1 -many steps) recursive construction using a parametrized (weak) ♦ -principle. The intention being to EITHER split the problem into manageable cases to produce a ZFC result, OR to obtain more information for the search of a suitable forcing model to prove a consistency result. M. Hruˇ s´ ak Weak diamonds and topology
Typical analysis of such a problem CH ⇒ Yes MA ⇒ Yes ”Optimize” the above proofs to get inv = c ⇒ YES. Cardinal invariants of the continuum serve primarily as a scale against which we measure the complexity (or strength) of our LONG ( c -many tasks in c -many steps) recursive constructions. There is typically a companion SHORT ( c -many tasks in ω 1 -many steps) recursive construction using a parametrized (weak) ♦ -principle. The intention being to EITHER split the problem into manageable cases to produce a ZFC result, OR to obtain more information for the search of a suitable forcing model to prove a consistency result. M. Hruˇ s´ ak Weak diamonds and topology
Cardinal invariants of the continuum ”...The cardinal characteristics are simply the smallest cardinals for which various results true for ℵ 0 become false....” ——– A. Blass: Combinatorial Cardinal Characteristics of the Continuum b = min {|F| : F ⊆ ω ω ∀ g ∈ ω ω ∃ f ∈ F |{ n : f ( n ) > g ( n ) }| = ω } s = min {|S| : S ⊆ [ ω ] ω ∀ A ∈ [ ω ] ω ∃ S ∈ S | S ∩ A | = | A \ S | = ω } M. Hruˇ s´ ak Weak diamonds and topology
Cardinal invariants of the continuum ”...The cardinal characteristics are simply the smallest cardinals for which various results true for ℵ 0 become false....” ——– A. Blass: Combinatorial Cardinal Characteristics of the Continuum b = min {|F| : F ⊆ ω ω ∀ g ∈ ω ω ∃ f ∈ F |{ n : f ( n ) > g ( n ) }| = ω } s = min {|S| : S ⊆ [ ω ] ω ∀ A ∈ [ ω ] ω ∃ S ∈ S | S ∩ A | = | A \ S | = ω } M. Hruˇ s´ ak Weak diamonds and topology
� � � � � � � � Cardinal invariants of the continuum � non( M ) � cof( M ) � cof( N ) cov( N ) � d b � add( M ) add( N ) cov( M ) non( N ) Cicho´ n’s diagram M. Hruˇ s´ ak Weak diamonds and topology
Weak diamond Definition (Devlin-Shelah 1978) The weak diamond principle Φ is the following assertion: ∀ F : 2 <ω 1 → 2 ∃ g : ω 1 → 2 ∀ f ∈ 2 ω 1 { α < ω 1 : F ( f ↾ α ) = g ( α ) } is stationary. Theorem (Devlin-Shelah 1978) Φ is equivalent to 2 ω < 2 ω 1 . M. Hruˇ s´ ak Weak diamonds and topology
Malykhin’s problem Problem (Malykhin 1978) Is there a separable (equvalently, countable) Fr´ echet group which is not metrizable? Partial positive solutions: p > ω 1 . . . Yes (Gerlits-Nagy 1982 ) There is an uncountable γ -set . . . Yes (Nyikos 1989) p = b . . . Yes Theorem (H.-Ramos Garc´ ıa 2014) It is consistent with ZFC that every separable Fr´ echet group is metrizable. M. Hruˇ s´ ak Weak diamonds and topology
Malykhin’s problem Problem (Malykhin 1978) Is there a separable (equvalently, countable) Fr´ echet group which is not metrizable? Partial positive solutions: p > ω 1 . . . Yes (Gerlits-Nagy 1982 ) There is an uncountable γ -set . . . Yes (Nyikos 1989) p = b . . . Yes Theorem (H.-Ramos Garc´ ıa 2014) It is consistent with ZFC that every separable Fr´ echet group is metrizable. M. Hruˇ s´ ak Weak diamonds and topology
Weak diamond and Fr´ echet groups Theorem (H.–Ramos-Garc´ ıa 2014) Assuming Φ, there is a countable non-metrizable Fr´ echet group (of weight ℵ 1 ). Given a filter F on ω let F <ω = { A ⊆ [ ω ] <ω : ( ∃ F ∈ F )[ F ] <ω ⊆ A } . Declaring F <ω the filter of neighbourhoods of the ∅ induces a group topology τ F on the Boolean group [ ω ] <ω with the symmetric difference as the group operation. M. Hruˇ s´ ak Weak diamonds and topology
Φ ⇒ ∃ countable non-metrizable Fr´ echet group We shall use Φ to show that there is a pair of mutually orthogonal, ⊆ ∗ -increasing sequences of infinite subsets of ω (in fact, a Hausdorff gap ) � A α : α < ω 1 � , � B α : α < ω 1 � so that for every X ⊆ [ ω ] <ω \ {∅} there exists an α < ω 1 such that either there is an n ∈ ω such that a ∩ ( A α ∪ n ) � = ∅ for every a ∈ X , or 1 for every n ∈ ω there is an a ∈ X such that min a � n and a ⊂ B α . 2 Having done that, let F be the filter generated by the complements of the A α ’s and the co-finite sets. Then τ F is Fr´ echet group which is not metrizable. M. Hruˇ s´ ak Weak diamonds and topology
Φ ⇒ ∃ countable non-metrizable Fr´ echet group Recall - Φ: ∀ F : 2 <ω 1 → 2 Borel ∃ g : ω 1 → 2 ∀ f ∈ 2 ω 1 { α < ω 1 : F ( f ↾ α ) = g ( α ) } is stationary . Want � A α : α < ω 1 � , � B α : α < ω 1 � , ⊆ ∗ -increasing mutually orthogonal so that for every X ⊆ [ ω ] <ω \ {∅} there exists an α < ω 1 such that either there is an n ∈ ω such that a ∩ ( A α ∪ n ) � = ∅ for every a ∈ X , or 1 for every n ∈ ω there is an a ∈ X such that min a � n and a ⊂ B α . 2 M. Hruˇ s´ ak Weak diamonds and topology
Φ ⇒ ∃ countable non-metrizable Fr´ echet group The domain of F (using a suitable coding) is the set of all triples � X , � A β : β < α � , � B β : β < α �� such that: X ⊆ [ ω ] <ω \ {∅} . 1 α is an infinite countable ordinal. 2 � A β : β < α � , � B β : β < α � is a pair of mutually orthogonal, 3 ⊆ ∗ -increasing sequences of infinite subsets of ω . Given a pair � A β : β < α � , � B β : β < α � as above, fix disjoint sets A and B such that A almost contains all A β , β < α , while B almost contains all B β , β < α , and ω = A ∩ B . 1 � if ∃ n ∈ ω ∀ a ∈ X ( a ∩ ( A ∪ n ) � = ∅ ); 0 F ( t ) = 1 if ∀ n ∈ ω ∃ a ∈ X ( a ∩ ( A ∪ n ) = ∅ ) . 1 Let α = { α n : n ∈ ω } be an enumeration of α . For each n ∈ ω , let A n +1 = A n ∪ � k � n B k � and B n +1 = B n ∪ � k � n +1 A k � A α n +1 \ � B α n +1 \ � , where A 0 = A α 0 and B 0 = B α 0 \ A α 0 . Then, A = � n ∈ ω A n and B = ω \ A are as required. M. Hruˇ s´ ak Weak diamonds and topology
Φ ⇒ ∃ countable non-metrizable Fr´ echet group Now suppose that g : ω 1 → 2 is a ♦ -sequence for F . Construct � A α : α < ω 1 � , � B α : α < ω 1 � as follows: Let � A n : n < ω � , � B n : n < ω � be any pair of mutually orthogonal, ⊆ ∗ -increasing sequences of infinite subsets of ω . If � A β : β < α � , � B β : β < α � have been defined, consider the corresponding partition ω = A ∪ B such that A almost contains all A β , β < α , while B almost contains all B β , β < α constructed by the algorithm described above. If g ( α ) = 0, then let A α = A , and let B α be a co-infinite subset of B still almost containing all B β , β < α . If g ( α ) = 1, then let B α = B , and let A α be a co-infinite subset of A almost containing all A β , β < α . M. Hruˇ s´ ak Weak diamonds and topology
Weakest weak diamond Definition (Devlin-Shelah 1978) The weak diamond principle Φ is the following assertion: ∀ F : 2 <ω 1 → 2 ∃ g : ω 1 → 2 ∀ f ∈ 2 ω 1 { α < ω 1 : F ( f ↾ α ) = g ( α ) } is stationary (unbounded). Definition (Moore-H.-Dˇ zamonja 2004) The weakest (or Borel) weak diamond principle ♦ (2 , =) is the following assertion: ∀ F : 2 <ω 1 → 2 Borel ∃ g : ω 1 → 2 ∀ f ∈ 2 ω 1 { α < ω 1 : F ( f ↾ α ) = g ( α ) } is stationary (unbounded). a a F is Borel if F ↾ 2 α is Borel for every α < ω 1 . Borel .... F ↾ 2 α is Borel for every α < ω 1 . M. Hruˇ s´ ak Weak diamonds and topology
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