Topological applications of long ω 1 -approximation sequences David Milovich Texas A&M International University 2015 Winter School in Abstract Analysis Hejnice, Czech Republic
A long time ago, some authors used “curve” to denote an isometric copy of a graph of a function R → R . (Continuity is not required.) If such a curve is a measurable subset of R 2 , then it is null. However, Sierpi´ nski showed (1933) that, assuming CH, the plane is a countable union of graphs of functions and their converses: • Let ⊳ order R with type ω 1 . • Let f x map ω onto { y : y � x } . • Let g n ( x ) = f x ( n ). n<ω ( g n ∪ g − 1 x ∈ R { ( x, g n ( x )) , ( g n ( x ) , x ) } = R 2 • � n ) = � � n<ω Thus, CH implies that the plane is a countable union of curves. 1
Sierpi´ nski asked (1951) if CH is needed to cover the plane by count- ably many curves. Roy O. Davies answered “no” (1963) with an ingenious ZFC cov- ering. (Never underestimate the axiom of choice!) To cover the plane by countable many curves, it is enough to parti- tion the plane into countably many partial curves. Fix an ω -sequence pairwise non-parallel lines ( L n : n < ω ). (For us, identical lines are considered parallel.) n<ω C n = R 2 such that | L ∩ C n | ≤ 1 Davies constructed a partition � for all n and all lines L || L n . (Davies remarked that an argument of Sierpi´ nski implicitly shows that, given a covering of R 2 by countably many curves, there is a covering of R 2 by countably many pairwise isometric curves.) 2
To a set theorist, the tastiest ingredient of Davies’ proof is his following implicit lemma. Lemma (Davies’ Lemma) . Let L be a countable first order lan- guage. Let A be an uncountable L -structure. Then there is a transfinite sequence M = ( M α ) α<η such that • every M α is a countable substructure of A , • � ran( M ) = A , and • M has the Davies property : for all α ≤ η , M <α = � β<α M β is a finite union of substructures of A . 3
Davies’ partition of the plane applies his lemma to a partial Skolem- ization of ( P , L , ∈ ; L n : n < ω ) where P is the set R 2 of points in the plane and L is the set of lines in the plane. We will simply let A be a complete Skolemization of ( P , L , ∈ ; L n : n < ω ). Therefore, all substructures are elementary substructures. Let M = ( M α ) α<η be as in Davies’ Lemma. Suppose that α < η and we have constructed a partition � n<ω C n = P ∩ M <α such that | L ∩ C n | ≤ 1 for all n and all lines L || L n . n<ω C ′′′ It suffices to show that we can extend C to a partition � n = P ∩ M <α +1 such that | L ∩ C ′′′ n | ≤ 1 for all n and all lines L || L n . 4
Let ν ≤ ω and let p = ( p k ) k<ν biject from ν to P ∩ M α \ M <α . n<ω C ′ Suppose that k < ν and we have extended C to a partition � n = P ∩ M <α ∪ { p j : j < k } such that | L ∩ C ′ n | ≤ 1 for all n and all lines L || L n . It suffices to show that that we can extend C ′ to a partition � n<ω C ′′ n = P ∩ M <α ∪ { p j : j < k + 1 } such that | L ∩ C ′′ n | ≤ 1 for all n and all lines L || L n . Let d < ω and N = ( N i ) i<d be such that M <α = � ran( N ) and each N i is a substructure of A . 5
For each n < ω , let K n be the line through p k that is parallel to L n . It suffices to show that there exists n < ω such that K n is disjoint from M <α ∪ { p j : j < k } . For each j < k , there is at most one n < ω such that p j ∈ K n . For each i < d , there is at most one n < ω such that K n intersects P ∩ N i . Why? If m < n < ω , x ∈ K m ∩ N i , and y ∈ K n ∩ N i , then K m , K n ∈ N i ; then p k ∈ N i because K m ∩ K n = { p k } . But p �∈ N i . Thus, K n is disjoint from M <α ∪ { p j : j < k } for almost all n . � 6
Davies’ Lemma apparently was not used in print again until 2002 by Jackson and Mauldin, and then by Milovich starting in 2008. Jackson and Mauldin constructed (in ZFC) a Steinhaus set, that is, a subset of R 2 that intersects every isometric copy of Z 2 at exactly one point. Without Davies’ Lemma, Jackson and Mauldin’s proof would have needed CH. We do not know if higher-dimensional analogs of Steinhaus sets exist. 7
References R. O. Davies, Covering the plane with denumerably many curves , J. London Math. Soc. 38 (1963), 433–438. S. Jackson and R. D. Mauldin, On a lattice problem of H. Steinhaus , J. Amer. Math. Soc. 15 (2002), no. 4, 817–856. D. Milovich, Noetherian types of homogeneous compacta and dyadic compacta , Topology Appl. 156 (2008), 443–464. D. Milovich, The ( λ, κ ) -Freese-Nation property for Boolean algebras and compacta , Order 29 (2012), 361–379. D. Milovich, On the strong Freese-Nation property (2014), arXiv:1412.7443. 8
How did Davies prove his lemma? Recall: Lemma (Davies’ Lemma) . Let L be a countable first order lan- guage. Let A be an uncountable L -structure. Then there is a transfinite sequence M = ( M α ) α<η such that • every M α is a countable substructures of A , • � ran( M ) = A , and • M has the Davies property : for all α ≤ η , M <α = � β<α M β is a finite union of substructures of A . 9
Proof: The Davies tree. Recursively construct as follows a sequence ( B t : t ∈ T ) with T a subtree of O rd <ω . • B () = A . • If B t is countable, declare t to be a leaf of T . • If | B t | = κ > ℵ 0 , declare t ⌢ ( α ) ∈ T for all α < κ and choose an increasing sequence ( B t ⌢ ( α ) ) α<κ of substructures of B t with union B t such that | B t ⌢ ( α ) | < | B t | for all α . T is well-founded. Therefore, the set L of leaves of T is well ordered by its lexicographic order < lex . Moreover, � t ∈ L B t = A . Finally, if t ∈ L , then � s< lex t B s = � � α<t i B ( t ↾ i ) ⌢ ( α ) . i< dom( t ) 10
Note that if | A | = ℵ n < ℵ ω , then the Davies tree has height n + 1. Therefore: Lemma. Let L be a countable first order language. Let A be an uncountable L -structure of size ℵ n < ℵ ω . Then there is a transfinite sequence M = ( M α ) α<η such that • every M α is a countable substructure of A , • � ran( M ) = A , and • for all α ≤ η , M <α is a union at most n substructures of A . 11
For each cardinal κ , let H ( κ ) denote the set of all sets x with tran- � n x of cardinality less than κ . sitive closure � n<ω For each regular uncountable cardinal θ , ( H ( θ ) , ∈ ) is a model of ZFC except possibly for the power set axiom. We will always implicitly choose θ large enough to include all the sets and power sets we need for the problem at hand. The notation N ≺ H ( θ ) means that N is an elementary {∈} -substructure of H ( θ ). 12
A long ω 1 -approximation sequence is a transfinite sequence M = ( M α ) α<η of countable elementary substructures of ( H ( θ ) , ∈ ) that is retrospective : for each α < η , the sequence ( M β ) β<α is an element of M α . Warning: If α is uncountable, then ( M β ) β<α , { M β : β < α } , and M <α = � β<α M β are not subsets of M α . 13
If M is a long ω 1 -approximation sequence, A ∈ M 0 , and 0 < α < dom( M ), then M 0 and α are definable from ( M β ) β<α , and hence elements of M α . Recall that if X ∈ N ≺ H ( θ ) and | X | ≤ ℵ 0 , then X ⊂ N . Therefore, M 0 ⊂ M α for all α ∈ dom( M ). Also, M β ⊂ M α for all β ≤ α ∈ ω 1 ∩ dom( M ). More generally, for all α, β ∈ dom( M ), we have M β � M α ⇔ M β ∈ M α ⇔ β ∈ α ∩ M α . 14
Recall that if A is a first order structure for a countable language L and A ∈ N ≺ H ( θ ), then A ∩ N ≺ L A . Therefore, assuming A ∈ M 0 , we have A ∩ M α ≺ L A for all α ∈ dom( M ). Moreover, if every M <α is a finite union of elementary substructures of H ( θ ) (and we will show that it is), then every A ∩ M <α is a finite union of L -elementary substructures of A . Choose a surjection f : | A | → A in M 0 . Assuming | A | ≤ dom( M ), we have f ( α ) ∈ M α for all α < | A | . Therefore, � α< | A | ( A ∩ M α ) = A . 15
Long ω 1 -approximation sequences are canonical sequences of count- able structures that are sufficiently rich to encode Davies trees of which they are leaves. A Davies tree is built top-down, starting from a large structure. Long ω 1 -approximation sequences are more flexibly built up from count- able structures, which simplifies the construction of large structures “from scratch.” Long ω 1 -approximation sequences provide a uniformly definable ver- sion of the Davies property and additional coherence properties. 16
The cardinal normal form of an ordinal α is the polynomial ω β 0 · γ 0 + ω β 1 · γ 1 + · · · + ω β m − 1 · γ m − 1 + γ m that equals α and satisfies • β 0 > · · · > β m − 1 ≥ 1, • 1 ≤ γ i < ω + β i for all i < m , and • γ m < ω 1 . An example cardinal normal form: ω ω ω 7 � � + ω 1 · ω 1 + ( ω ω + ω · 2 + 3) 3 ω ω +1 · 4 + ω ω + ω 7 · + ω 6 · ω 7 17
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