Distributivity of the algebra of regular open subsets of βr \ R

Topology and its Applications 149 (2005) 1 7 www.elsevier.com/locate/topol Distributivity of the algebra of regular open subsets of βr \ R Bohuslav Balcar a,, Michael Hrušák b a Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic b Instituto de matemáticas, UNAM, Unidad Morelia, A.P. 61-3, Xangari, C.P. 58089, Morelia, Mich., México Received 12 February 2004; accepted 24 August 2004 Abstract We compare the structure of the algebras P(ω)/fin and A ω /Fin, where A denotes the algebra of clopen subsets of the Cantor set. We show that the distributivity number of the algebra A ω /Fin is bounded by the distributivity number of the algebra P(ω)/fin and by the additivity of the meager ideal on the reals. As a corollary we obtain a result of A. Dow, who showed that in the iterated Mathias model the spaces βω \ ω and βr \ R are not co-absolute. We also show that under the assumption t = h the spaces βω \ ω and βr \ R are co-absolute, improving on a result of E. van Douwen. 2004 Published by Elsevier B.V. MSC: 54G05; 06E15; 03E17 Keywords: Distributivity of Boolean algebras; Cardinal invariants of the continuum; Čech Stone compactification The authors research was partially supported by a grant GAČR 201/03/0933. First author s research was also partially supported by grant GAČR 201/02/0857 and the research of the second author was supported by PAPIIT grant IN108802-2 and CONACYT grant 40057-F. * Corresponding author. E-mail addresses: balcar@cts.cuni.cz (B. Balcar), michael@matmor.unam.mx (M. Hrušák). 0166-8641/$ see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.topol.2004.08.009

2 B. Balcar, M. Hrušák / Topology and its Applications 149 (2005) 1 7 0. Introduction The object studied here is the algebra RO(βR \ R) of regular open subsets of the Čech Stone remainder of the real line. This algebra is naturally isomorphic to the completion of the Boolean algebra A ω /Fin, an object combinatorially easier to grasp. It is easily seen that A ω /Fin is σ -closed (i.e., every countable decreasing chain of non-zero elements of A ω /Fin has a non-zero lower bound), homogeneous Boolean algebra of cardinality c satisfying c + - c.c. The algebra A ω /Fin is closely related to the algebra P(ω)/fin = Clop(βω \ ω). It is a consequence of the classical theorem of Parovičenko that, assuming the Continuum Hypothesis, the two algebras are, in fact, isomorphic. Recall that topological spaces X and Y are coabsolute if the Boolean algebras RO(X) and RO(Y ) are isomorphic. Answering a longstanding question of E. van Douwen, Dow in [7] showed that the spaces βω \ ω and βr \ R are not co-absolute in the iterated Mathias model. His method for proving this was to show that the distributivity number h of P(ω)/fin is equal to ω 2 while the distributivity number h(a ω /Fin) of A ω /Fin equals ω 1 in this model. We present a ZFC theorem, from which Dow s result easily follows. We also study the tower number t(a ω /Fin), a natural lower bound for h(a ω /Fin), and show that t(a ω /Fin) = t. Thiswe use to show that under the assumption t = h the spaces βω\ ω and βr \ R are co-absolute, improving on a result of E. van Douwen. 1. Preliminaries In this section we review basic notions concerning Boolean algebras and their cardinal invariants as well as standard cardinal invariants of the continuum. The terminology used here is standard and follows [12,13,10]. Throughout this paper A denotes the Boolean algebra of closed and open subsets of the Cantor set 2 ω. It is well known that it is up to isomorphism the unique countable atomless Boolean algebra. Given a Boolean algebra B the product B ω is also a Boolean algebra with the operations defined coordinatewise. For f B ω the support of f is the set {n ω: f(n) 0}. By Fin we denote the ideal {f B ω : support(f ) <ω}. Algebras studied in the paper are the quotient algebras B ω /Fin. Note that P(ω)/fin is naturally isomorphic to B ω /Fin, where B is the trivial algebra {0, 1}. It easily follows that P(ω)/fin can be regularly embedded into B ω /Fin for any Boolean algebra B. It is therefore natural to compare cardinal invariants of the algebra B ω /Fin with those of P(ω)/fin. We will often treat B + as a set partially ordered by the canonical ordering on B. The distributivity number of a Boolean algebra B, denoted by h(b), is defined as the minimal size of a family of maximal antichains in B without common refinement. For homogeneous B, h(b) is equal to the minimal size of a collection of dense open subsets of B whose intersection is empty. By a tower in a Boolean algebra B we mean a well ordered decreasing chain in B + without a lower bound in B +.Thetower number t(b) equals the minimal size of a tower in B. For atomless Boolean algebras B, t(b) is a regular infinite cardinal and t(b) h(b). Furthermore, t(b) is uncountable if and only if B + is σ -closed. All algebras of the type B ω /Fin have this property. A fundamental difference between h(b) and t(b) is that h(b) = h(ro(b)) whereas t(b) = ω for every complete atomless Boolean algebra B.

B. Balcar, M. Hrušák / Topology and its Applications 149 (2005) 1 7 3 Next we recall the definitions and basic facts about the relevant cardinal invariants of the continuum (see, e.g., [4]). The symbol b denotes the unbounding number of (ω ω, ) and d denotes the dominating number of (ω ω, ),cov(m) is the minimal size of a family of meager subsets of 2 ω that cover 2 ω and add(m) stands for the additivity of the meager ideal, i.e., the minimal size of a family of meager subsets of 2 ω whose union is not meager. Using standard notation the tower number of P(ω)/fin is written simply as t and the distributivity number of P(ω)/fin as h. The following proposition sums up provable relationships between theses cardinal invariants: Proposition 1.1. (i) t h b d, (ii) (Piotrowski Szymański) t add(m), (iii) (Bartoszyński Miller) add(m) = min{b, cov(m)}. The invariants h and add(m) as well as b and cov(m) are in ZFC not provably comparable. For proofs and additional information consult [4]. In the proof of our main result we will use the following reformulation of a result of Keremedis [11], for an alternative proof see [1]. The symbol Q denotes the set of rational numbers equipped with its usual topology. Theorem 1.2 (Keremedis). cov(m) is equal to the minimal size of a family F of nowhere dense subsets of Q such that for every infinite set Y Q there is an F F intersecting Y in an infinite set. 2. Main results As mentioned above P(ω)/fin can be regularly embedded into B ω /Fin for any Boolean algebra B. It is well known, that the algebra P(ω)/fin generically adds a selective ultrafilter U on ω. Next, we will show that B ω /Fin can be written as an iteration of P(ω)/fin and an ultra-power of B modulo U. In particular, P(ω)/fin is a regular subalgebra of A ω /Fin. The relation denotes forcing equivalence or, in other words, the fact that the completions of the algebras are isomorphic. Proposition 2.1. B ω /Fin P(ω)/fin B ω / U, where U is the P(ω)/fin-name for the selective ultrafilter added by P(ω)/fin. Proof. Define a function Φ : B ω /Fin P(ω)/fin B ω / U by putting Φ(f) = (support(f ), [ f ] U ), where [ f ] U is a P(ω)/fin-name for {g B ω : {n ω: f(n)= g(n)} U}. It is easy to verify that Φ is a dense embedding. We use the proposition to show that h(a ω /Fin) h, a fact known already to van Douwen (see [6]).

4 B. Balcar, M. Hrušák / Topology and its Applications 149 (2005) 1 7 Note that the Boolean algebra A ω /Fin is naturally isomorphic to the algebra Clop(βX \ X), where the space X = 2 ω \ 0 or, equivalently, X is a disjoint sum of countably many copies of 2 ω. In particular, the spaces βx \ X and βr \ R are co-absolute. Now we are ready to state and prove our main result: Theorem 2.2. h(a ω /Fin) min{h, add(m)}. Proof. In this proof we identify the rationals Q with {q 2 ω : ( n ω) q(n) = 0}. Recall that if f A ω and n support(f ) then f(n)is a nonempty clopen subset of 2 ω. By Proposition 2.1, P(ω)/fin can be regularly embedded into A ω /Fin. Consequently, h(a ω /Fin) h. Since h b and add(m) = min{b, cov(m)} all that needs to be shown is that h(a ω /Fin) cov(m). To that end fix a family {K α : α<cov(m)} of nowhere dense subsets of Q such that for every infinite set Y Q there is an α<cov(m) such that K α intersects Y in an infinite set. For α<cov(m) put H α = { f A ω : ( n ω) f (n) K α = }. Note that: (a) H α is closed under finite changes, i.e., if f H α and g = Fin f then g H α. (b) H α is dense in the algebra A ω /Fin. To see this, let f A ω have infinite support. As K α is nowhere dense, it is easy to find g A ω with the same support as f such that g(n) f(n)and g(n) K α =, for all n ω. (c) H α is open (downward closed). To finish the proof it suffices to check that: (d) {H α : α<cov(m)}=fin. Obviously, Fin H α for all α<cov(m). Now,takef A ω with infinite support. For every n in the support of f recursively pick q n Q f(n)\{q m : m<n} and set Y ={q n : n support(f )}. There is an α<cov(m) such that Y K α =ω. Consequently, there are infinitely many n ω such that f(n) K α and hence f/ H α. As a corollary we now get Dow s result. Theorem 2.3 (Dow). h(a ω /Fin)<h in the iterated Mathias model. Proof. It is a folklore fact, first observed in [5], that h = c = ω 2 in the Mathias model. Equally standard is the fact that cov(m) = ω 1 in the Mathias model (see, e.g., [4]). By Theorem 2.2 the result follows. It actually follows from Theorem 2.2, that h(b ω /Fin) min{h, add(m)} for any B which contains A as a regular subalgebra, i.e., for any B adding a Cohen real.

B. Balcar, M. Hrušák / Topology and its Applications 149 (2005) 1 7 5 Theorem 2.4. t = t(a ω /Fin). Proof. Let {T α : α<t} beatowerinp(ω)/fin. Let χ α A ω be the characteristic function of T α, i.e., χ α (n) = 1ifn T α and χ α (n) = 0ifn/ T α. Then {χ α : α<t} forms a tower in A ω /Fin, for if f A ω were its lower bound then the support of f would be a lower bound for {T α : α<t}. Hence t t(a ω /Fin). In order to prove t t(a ω /Fin) let {f α : α<κ} be a decreasing chain in A ω /Fin, where κ<t. We will show that {f α : α<κ} has a lower bound. To that end we first prove the following claim. Claim 1. There is a one-to-one sequence c n : n ω (2 ω ) ω such that for every α<κ the set A α ={n ω: c n f α (n)} is infinite. To prove the claim, note that the set X = { cn : n ω (2 ω ) ω : ( n m)c n / f α (n) } α<κ m ω { cn : n ω (2 ω ) ω } : c n = c m m n ω is a union of κ many meager subsets of (2 ω ) ω.asκ<t cov(m), (2 ω ) ω \ X is not empty and any c n : n ω (2 ω ) ω \ X has the required property. Claim 2. If α<β<κthen A β A α. As f β Fin f α, f β (n) f α (n) for all but finitely many n ω. Hence, A β A α. So, {A α : α<κ} form a decreasing chain in P(ω)/fin and as κ<tthe chain has a lower bound A [ω] ω.forα<κand n A set { min{k ω: cn k f g α (n) = α (k)} if n A α A, 0 otherwise, where c n k denotes the clopen set {h 2 ω : h k = c n k}. Asκ<t b there is a function g : A ω which -dominates all g α,α<κ. Set { cn g(n) if n A, f(n)= 0 otherwise. Note that support(f ) = A is an infinite set. To finish the proof it suffices to check that f Fin f α for every α<κ. This follows as for all but finitely many n A, g(n) g α (n) and hence f α (n) c n g α (n) c n g(n) =f(n). The following corollary is a strengthening of a theorem of van Douwen [6] who proved that, assuming p = c, βω \ ω and βr \ R are co-absolute. Corollary 2.5. (t = h) βω\ ω and βr \ R are co-absolute.

6 B. Balcar, M. Hrušák / Topology and its Applications 149 (2005) 1 7 Proof. If t = h = κ then by Theorems 2.2 and 2.4 also t(a ω /Fin) = h(a ω /Fin) = κ. Hence by the base tree theorem of [2] (see also [3,8]) both algebras P(ω)/fin and A ω /Fin have dense subsets isomorphic to c-branching tree of height κ.ast = t(a ω /Fin) = κ these trees have no short branches and hence are isomorphic to the tree c <κ. So the algebras RO(P(ω)/fin) and RO(A ω /Fin) are isomorphic and the result follows. We have shown that t h(a ω /Fin) min{h, add(m)}. It is a natural question whether one of the inequalities is, in fact, an equality. Proposition 2.6. It is relatively consistent with ZFC that t < h(a ω /Fin). Proof. Let V be a model of p = c >ω 1 in which there is a Suslin tree T. To obtain such model it suffices to add a single Cohen real to a model of Martin s Axiom (see, e.g., [4]). Treat T as a forcing notion ordered by reversing the tree order. T is then a c.c.c. forcing of size ω 1 which does not add any new reals. Let G be a T-generic filter over V.InV[G], t = ω 1 as T can be isomorphically embedded into P(ω)/fin and the generic branch produces a tower. This fact was probably first explicitly stated in [9]. To see that V [G] =h(a ω /Fin) = c fix some κ<c and assume that, Ȧ α, α<κ are T-names for maximal antichains in A ω /Fin. For α<κand t T let A t α = { f A ω } /Fin: t f A α and extend each A t α to a maximal antichain Bt α in Aω /Fin. As V is a model of p = h(a ω /Fin) = c there is a maximal antichain A in A ω /Fin which refines Bα t for every α<κ and t T. Then, in V [G], A refines A α for all α<κ. Another model for t < h(a ω /Fin) can be found in [2]; Take the product forcing P Q where P is the Solovay Tennenbaum c.c.c. poset forcing MA plus c = ω 2, and Q is the forcing for adding more than ω 2 subsets of ω 1 with countable conditions. A completely analogous reasoning as in [2] shows that t < h(a ω /Fin) holds in this model. We conjecture that Dow s method for proving the consistency of h(a ω /Fin)<h can be used to prove the following consistency result. Question 2.7. Is it consistent with ZFC that h(a ω /Fin)<min{h, add(m)}? Acknowledgements The work on this paper was initiated during the second author s visit at the Center for Theoretical Study of the Charles University and completed during the subsequent stay of the first mentioned author at the Instituto de matemáticas de la UNAM, Unidad Morelia. The authors wish to thank both institutions for their hospitality.

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