aa r X i v : . [ m a t h . L O ] M a r SPLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω DAVID MILOVICH
Abstract.
Extending some results of Malykhin, we prove several indepen-dence results about base properties of βω \ ω and its powers, especially theNoetherian type Nt ( βω \ ω ), the least κ for which βω \ ω has a base that is κ -like with respect to containment. For example, Nt ( βω \ ω ) is at least s , butcan consistently be ω , c , c + , or strictly between ω and c . Nt ( βω \ ω ) is alsoconsistently less than the additivity of the meager ideal. Nt ( βω \ ω ) is closelyrelated to the existence of special kinds of splitting families. Introduction
Definition 1.1.
Given a cardinal κ , define a poset to be κ - like ( κ op - like ) if noelement is above (below) κ -many elements. Define a poset to be almost κ op - like ifit has a κ op -like dense subset.In the context of families of subsets of a topological space, we will always im-plicitly order by inclusion. We are particularly interested in κ op -like bases, π -bases,local bases, and local π -bases of the space ω ∗ of nonprincipal ultrafilters on ω .Recall that a local base (local π -base) at a point in a space is a family of openneighborhoods of that point (family of nonempty open subsets) such that everyneighborhood of the point contains an element of the family; a base ( π -base) ofa space is family of open sets that contains local bases (local π -bases) at everypoint. See Engelking [9] for the more background on bases and their cousins. Alsorecall the following basic cardinal functions. For more about these functions, seeJuh´asz [12]. Definition 1.2.
Given a space X , let the weight of X , or w ( X ), be the least κ ≥ ω such that X has a base of size at most κ . Given p ∈ X , let the character of p ,or χ ( p, X ), be the least κ ≥ ω such that there is a local base at p of size at most κ . Let the character of X , or χ ( X ), be the supremum of the characters of itspoints. Analogously define π -weight and local π -character , respectively denotingthem using π and πχ .Now consider the following order-theoretic parallels. Definition 1.3.
Given a space X , let the Noetherian type of X , or N t ( X ), be theleast κ ≥ ω such that X has a base that is κ op -like. Given p ∈ X , let the localNoetherian type of p , or χN t ( p, X ), be the least κ ≥ ω such that there is a κ op -likelocal base at p . Let the local Noetherian type of X , or χN t ( X ), be the supremumof the local Noetherian types of its points. Analogously define Noetherian π -type and local Noetherian π -type , respectively denoting them using πN t and πχN t . Date : March 26, 2008.Support provided by an NSF graduate fellowship.
Noetherian type and Noetherian π -type were introduced by Peregudov [16]. Let ω ∗ denote the space of nonprincipal ultrafilters on ω . Malykhin [15] proved thatMA implies πN t ( ω ∗ ) = c and CH implies N t ( ω ∗ ) = c . We extend these results byinvestigating N t ( ω ∗ ), πN t ( ω ∗ ), χN t ( ω ∗ ), and πχN t ( ω ∗ ) as cardinal characteristicsof the continuum. For background on such cardinals, see Blass [7]. We also examinethe sequence h N t (( ω ∗ ) α ) i α ∈ O n . Definition 1.4.
Let b denote the minimum of |F| where F ranges over the subsetsof ω ω that have no upper bound in ω ω with respect to eventual domination. Definition 1.5. A tree π -base of a space X is a π -base that is a tree when orderedby containment. Let h be the minimum of the set of heights of tree π -bases of ω ∗ .Balcar, Pelant, and Simon [1] proved that tree π -bases of ω ∗ exist, and that h ≤ min { b , cf c } . They also proved that the above definition of h is equivalent tothe more common definition of h as the distributivity number of [ ω ] ω ordered by ⊆ ∗ . Definition 1.6.
Given x, y ∈ [ ω ] ω , we say that x splits y if | y ∩ x | = | y \ x | = ω .Let r be the minimum value of | A | where A ranges over the subsets of [ ω ] ω suchthat no x ∈ [ ω ] ω splits every y ∈ A . Let s be the minimum value of | A | where A ranges over the subsets of [ ω ] ω such that every x ∈ [ ω ] ω is split by some y ∈ A .It is known that b ≤ r and h ≤ s . (See Theorems 3.8 and 6.9 of [7].)Clearly, N t ( ω ∗ ) ≤ w ( ω ∗ ) + = c + . We will show that also πχN t ( ω ∗ ) = ω and πN t ( ω ∗ ) = h and s ≤ N t ( ω ∗ ). Furthermore, N t ( ω ∗ ) can consistently be c , c + , orany regular κ satisfying 2 <κ = c . Also, N t ( ω ∗ ) = ω is relatively consistent withany values of b and c . The relations ω < b = s = N t ( ω ∗ ) < c and ω = b = s < N t ( ω ∗ ) < c are also each consistent. We also prove some relations between r and N t ( ω ∗ ), as well as some consistency results about the local Noetherian type ofpoints in ω ∗ . 2. Basic results
The following proposition is essentially due to Peregudov (see Lemma 1 of [16]).
Proposition 2.1.
Suppose a point p in a space X satisfies πχ ( p, X ) < cf κ ≤ κ ≤ χ ( p, X ) . Then N t ( X ) > κ .Proof. Let A be a base of X . Let U and V be, respectively, a local π -base at p of size at most πχ ( p, X ) and a local base at p of size χ ( p, X ). For each element of U , choose a subset in A , thereby producing local π -base U at p that is a subsetof A of size at most πχ ( p, X ). Similarly, for each element of V , choose a smallerneighborhood of p in A , thereby producing a local base V at p that is a subset of A of size χ ( p, X ). Every element of V contains an element of U . Hence, some elementof U is contained in κ -many elements of V ; hence, A is not κ op -like. (cid:3) Definition 2.2.
For all x ∈ [ ω ] ω , set x ∗ = { p ∈ ω ∗ : p ∈ x } . Theorem 2.3.
It is relatively consistent with any value of c satisfying cf c > ω that N t ( ω ∗ ) = c + .Proof. We may assume cf c > ω . By Exercise A10 on p. 289 of Kunen [14], thereis a ccc generic extension V [ G ] such that ˇ c = c V [ G ] and, in V [ G ], there exists p ∈ ω ∗ PLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω such that χ ( p, ω ∗ ) = ω . Henceforth work in V [ G ]. Let ϕ be a bijection from ω to ω . Define ψ : ω ∗ → ω ∗ by x
7→ { E ⊆ ω : { m < ω : { n < ω : ϕ ( m, n ) ∈ E } ∈ p } ∈ x } . Since πχ ( p, ω ∗ ) ≤ χ ( p, ω ∗ ) = ω , there exists h E α i α<ω ∈ ([ ω ] ω ) ω such that everyneighborhood of p contains E ∗ α for some α < ω . Hence, for all x ∈ ω ∗ , everyneighborhood of ψ ( x ) contains ( ϕ “( { m } × E α )) ∗ for some m < ω and α < ω ;whence, πχ ( ψ ( x ) , ω ∗ ) = ω . Since ψ is easily verified to be a topological embedding, χ ( x, ω ∗ ) ≤ χ ( ψ ( x ) , ω ∗ ) for all x ∈ ω ∗ . By a result of Pospiˇsil [17], there exists q ∈ ω ∗ such that χ ( q, ω ∗ ) = c . Hence, πχ ( ψ ( q ) , ω ∗ ) = ω and χ ( ψ ( q ) , ω ∗ ) = c . ByProposition 2.1, N t ( ω ∗ ) > χ ( ψ ( q ) , ω ∗ ) = c . (cid:3) Definition 2.4.
Given n < ω , let ss n ( ss ω ) denote the least cardinal κ for whichthere exists a sequence h f α i α< c of functions on ω each with range contained in n (each with finite range) such that for all I ∈ [ c ] κ and x ∈ [ ω ] ω there exists α ∈ I such that f α is not eventually constant on x . (The notation ss was chosen with thephrase “supersplitting number” in mind.) Note that if such an h f α i α< c does notexist for any κ ≤ c , then ss n ( ss ω ) is by definition equal to c + .Clearly ss n ≥ ss n +1 ≥ ss ω for all n < ω . Moreover, since cf c > ω , we have ss ω = ss n for some n < ω . However, for any particular n ∈ ω \
2, it is not clearwhether ZFC proves ss ω = ss n . Definition 2.5.
Given λ ≥ κ ≥ ω and a space X , a h λ, κ i - splitter of X is a sequence hF α i α<λ of finite open covers of X such that, for all I ∈ [ λ ] κ and h U α i α ∈ I ∈ Q α ∈ I F α , the interior of T α ∈ I U α is empty. Lemma 2.6.
Suppose X is a compact space with a base A of size at most w ( X ) suchthat U ∩ V ∈ A ∪ {∅} for all U, V ∈ A . If κ ≤ w ( X ) and X has a h w ( X ) , κ i -splitter,then A contains a κ op -like base of X . Hence, N t ( ω ∗ ) ≤ ss ω .Proof. Set λ = w ( X ) and let hF α i α<λ be a h λ, κ i -splitter of X . For each α < λ ,the cover F α is refined by a finite subcover of A ; hence, we may assume F α ⊆ A .Let A = { U α : α < λ } . For each α < λ , set B α = { U α ∩ V : V ∈ F α } . Set B = S α<λ B α \ {∅} . Then B is easily seen to be a base of X and a κ op -like subsetof A . (cid:3) Lemma 2.7.
Let X be a compact space without isolated points and let ω ≤ κ ≤ λ ≤ min p ∈ X χ ( p, X ) . If X has no h λ, κ i -splitter, then N t ( X ) > κ .Proof. Let A be a base of X . Construct a sequence hF α i α<λ of finite subcovers of A as follows. Suppose we have α < λ and hF β i β<α . For each p ∈ X , choose V p ∈ A such that p ∈ V p S β<α F β . Let F α be a finite subcover of { V p : p ∈ X } . Then F α ∩ F β = ∅ for all α < β < λ . Suppose X has no h λ, κ i -splitter. Then choose I ∈ [ λ ] κ and h U α i α ∈ I ∈ Q α ∈ I F α such that T α ∈ I U α has nonempty interior. Thenthere exists W ∈ A such that W ⊆ T α ∈ I U α . Thus, A is not κ op -like. (cid:3) Definition 2.8.
Let u denote the minimum of the set of characters of points in ω ∗ . Let π u denote the minimum of the set of π -characters of points in ω ∗ .By a theorem of Balcar and Simon [2], π u = r . Theorem 2.9.
Suppose u = c . Then N t ( ω ∗ ) = ss ω . DAVID MILOVICH
Proof.
By Lemma 2.6,
N t ( ω ∗ ) ≤ ss ω . Suppose κ ≤ c . Since every finite open coverof ω ∗ is refined by a finite, pairwise disjoint, clopen cover, ω ∗ has a h c , κ i -splitter ifand only if ss ω ≤ κ . Hence, N t ( ω ∗ ) ≥ ss ω by Lemma 2.7. (cid:3) Lemma 2.10.
Suppose r = c . Then ss ≤ c .Proof. Let h x α i α< c enumerate [ ω ] ω . Construct h y α i α< c ∈ ([ ω ] ω ) c as follows. Given α < c and h y β i β<α , choose y α such that y α splits every element of { x α } ∪ { y β : β <α } . Suppose I ∈ [ c ] c and α < c . Then x α is split by y β for all β ∈ I \ α . Thus, h{ y α , ω \ y α }i α< c witnesses ss ≤ c . (cid:3) Theorem 2.11.
The cardinals r and N t ( ω ∗ ) are related as follows. (1) If r = c , then N t ( ω ∗ ) = ss ω ≤ c . (2) If r < c , then N t ( ω ∗ ) ≥ c . (3) If r < cf c , then N t ( ω ∗ ) = c + .Proof. Statement (1) follows from Lemma 2.10, Theorem 2.9, and π u = r . Theproof of Theorem 2.3 shows how to construct p ∈ ω ∗ such that πχ ( p, ω ∗ ) = π u = r and χ ( p, ω ∗ ) = c . Hence, (2) and (3) follow from Proposition 2.1. (cid:3) Definition 2.12.
A subset A of [ ω ] ω has the strong finite intersection property (SFIP) if the intersection of every finite subset of A is infinite. Given A ⊆ [ ω ] ω with the SFIP, define the Booth forcing for A to be [ ω ] <ω × [ A ] <ω ordered by h σ , F i ≤ h σ , F i if and only if F ⊇ F and σ ⊆ σ ⊆ σ ∪ T F . Define a generic pseudointersection of A to be S h σ,F i∈ G σ where G is a generic filter of[ ω ] <ω × [ A ] <ω . Theorem 2.13.
For all cardinals κ satisfying κ > cf κ > ω , it is consistent that r = u = cf κ and N t ( ω ∗ ) = ss = c = κ .Proof. Assuming GCH in the ground model, construct a finite support iteration h P α i α ≤ κ as follows. First choose some U ∈ ω ∗ . Then suppose we have α < κ and P α and (cid:13) α U α ∈ ω ∗ . Let P α +1 ∼ = P α ∗ Q α where Q α is a P α -name for the Boothforcing for U α . Let x α be a P α +1 -name for a generic pseudointersection of U α addedby Q α ; let U α +1 be a P α +1 -name for an element of ω ∗ containing U α ∪ { x α } . Forlimit α < κ , let U α = S β<α U β .Let h η α i α< cf κ be an increasing sequence of ordinals with supremum κ . Then { x η α : α < cf κ } is forced to generate an ultrafilter in V P κ . Hence, (cid:13) κ r ≤ u ≤ cf κ < κ = c . Therefore, by Lemma 2.6 and Theorem 2.11, it suffices to showthat (cid:13) κ ss ≤ κ . Every nontrivial finite support iteration of infinite length addsa Cohen real. Hence, we may choose for each α < κ a P ω ( α +1) -name y α for anelement of [ ω ] ω that is Cohen over V P ωα . Then every name S for the range of acofinal subsequence of h y α i α<κ is such that (cid:13) κ ∀ z ∈ [ ω ] ω ∃ w ∈ S w splits z. Hence, h y α i α<κ witnesses that (cid:13) κ ss ≤ κ . (cid:3) Theorem 2.14.
N t ( ω ∗ ) ≥ s .Proof. Suppose
N t ( ω ∗ ) = κ < s . Since N t ( ω ∗ ) < c , we have r = c by Theorem 2.11.Hence, u = c . By Theorem 2.9, it suffices to show that ss ω > κ . Suppose h f α i α< c isa sequence of functions on ω with finite range and I ∈ [ c ] κ . Since κ < s , there exists x ∈ [ ω ] ω such that f α is eventually constant on x for all α ∈ I . Thus, ss ω > κ . (cid:3) PLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω Lemma 2.15.
Let κ be a cardinal and let P and Q be mutually dense subsets of acommon poset. Then P is almost κ op -like if and only if Q is.Proof. Suppose D is a κ op -like dense subset of P . Then it suffices to construct a κ op -like dense subset of Q . Define a partial map f from | D | + to Q as follows. Set f = ∅ . Suppose α < | D | + and we have constructed a partial map f α from α to Q . Set E = { d ∈ D : d q for all q ∈ ran f α } . If E = ∅ , then set f α +1 = f α .Otherwise, choose q ∈ Q such that q ≤ e for some e ∈ E and let f α +1 be thesmallest function extending f α such that f α +1 ( α ) = q . For limit ordinals γ ≤ | D | + ,set f γ = S α<γ f α . Set f = f | D | + .Let us show that ran f is a κ op -like. Suppose otherwise. Then there exists q ∈ ran f and an increasing sequence h ξ α i α<κ in dom f such that q ≤ f ( ξ α ) forall α < κ . By the way we constructed f , there exists h d α i α<κ ∈ D κ such that f ( ξ β ) ≤ d β = d α for all α < β < κ . Choose p ∈ P such that p ≤ q . Then choose d ∈ D such that d ≤ p . Then d ≤ d β = d α for all α < β < κ , which contradictsthat D is κ op -like. Therefore, ran f is κ op -like.Finally, let us show that ran f is a dense subset of Q . Suppose q ∈ Q . Choose p ∈ P such that p ≤ q . Then choose d ∈ D such that d ≤ p . By the way weconstructed f , there exists r ∈ ran f such that r ≤ d ; hence, r ≤ q . (cid:3) Theorem 2.16. πN t ( ω ∗ ) = h .Proof. First, we show that πN t ( ω ∗ ) ≤ h . Let A be a tree π -base of ω ∗ such that A has height h with respect to containment. Then A is clearly h op -like. To show that h ≤ πN t ( ω ∗ ), let A be as above and let B be a πN t ( ω ∗ ) op -like π -base of ω ∗ . Then A and B are mutually dense; hence, by Lemma 2.15, A contains a πN t ( ω ∗ ) op -like π -base C of ω ∗ . Since C is also a tree π -base, it has height at most πN t ( ω ∗ ). Hence, h ≤ πN t ( ω ∗ ). (cid:3) Corollary 2.17. If h = c , then πN t ( ω ∗ ) = N t ( ω ∗ ) = ss = c .Proof. Suppose h = c . Then r = c because h ≤ b ≤ r ≤ c . Hence, by Theorem 2.16,Theorem 2.11, and Lemma 2.10, c ≤ πN t ( ω ∗ ) ≤ N t ( ω ∗ ) = ss ω ≤ ss ≤ c . (cid:3) Models of
N t ( ω ∗ ) = ω Adding c -many Cohen reals collapses ss to ω . By Lemma 2.6, it therefore alsocollapses N t ( ω ∗ ) to ω . The same result holds for random reals and Hechler reals. Theorem 3.1.
Suppose κ ω = κ and P = B (2 κ ) / I where B (2 κ ) is the Borel alegebraof the product space κ and I is either the meager ideal or the null ideal (with respectto the product measure). (In other words, P adds κ -many Cohen reals or κ -manyrandom reals in the usual way.) Then P (cid:13) ω = ss .Proof. Working in the generic extension V [ G ], we have κ = c and a sequence h x α i α<κ in [ ω ] ω such that V [ G ] = V [ h x α i α<κ ] and, if E ∈ P ( κ ) ∩ V and α ∈ κ \ E ,then x α is Cohen or random over V [ h x β i β ∈ E ]. (See [13] for a proof.) Suppose I ∈ [ κ ] ω and y ∈ [ ω ] ω . Then y ∈ V [ h x α i α ∈ J ] for some J ∈ [ κ ] ω ∩ V ; hence, x α splits y for all α ∈ I \ J . Thus, h{ x α , ω \ x α }i α<κ witnesses ss = ω . (cid:3) Definition 3.2.
Let d denote the minimum of the cardinalities of subsets of ω ω that are cofinal with respect to eventual domination. DAVID MILOVICH
Corollary 3.3.
Every transitive model of ZFC has a ccc forcing extension thatpreserves b , d , and c , and collapses ss to ω .Proof. Add c -many random reals to the ground model. Then every element of ω ω in the extension is eventually dominated by an element of ω ω in the ground model;hence, b , d , and c are preserved by this forcing, while ss becomes ω . (cid:3) Definition 3.4.
We say that a transfinite sequence h x α i α<η of subsets of ω is eventually splitting if for all y ∈ [ ω ] ω there exists α < η such that for all β ∈ η \ α the set x β splits y . Theorem 3.5.
Let κ = κ ω . Then ss = ω is forced by the κ -long finite supportiteration of Hechler forcing.Proof. Let P be the κ -long finite support iteration of Hechler forcing. Let G bea generic filter of P . For each α < κ , let g α be the generic dominating functionadded at stage α ; set x α = { n < ω : g α ( n ) is even } . Suppose p ∈ G and I and y are names such that p forces I ∈ [ κ ] ω and y ∈ [ ω ] ω . Choose q ∈ G and a name h such that q ≤ p and q forces h to be an increasing map from ω to I . For each α < ω , set E α = { β < κ : q (cid:13) h ( α ) = ˇ β } ; let k α be a surjection from ω to E α . Let q ≥ r ∈ G and n < ω and γ ≤ κ and J be a name such that r forces J ∈ [ ω ] ω and sup ran h = ˇ γ and h ( α ) = k α ( n ) ˇ for all α ∈ J . Set F = { k α ( n ) : α < ω } ∩ γ ;let j be the order isomorphism from some ordinal η to F . Then cf η = cf γ = ω .For all α < κ , the set x α is Cohen over V [ h g β i β<α ]; hence, h x j ( α ) i α<η is eventuallysplitting in V [ h g α i α<γ ]. By a result of Baumgartner and Dordal [5], h x j ( α ) i α<η isalso eventually splitting in V [ G ]. Choose β < η such that x j ( α ) splits y G for all α ∈ η \ β . Then there exist s ∈ G and α ∈ γ \ j ( β ) such that r ≥ s (cid:13) ˇ α ∈ h “ J .Hence, α ∈ I G and x α splits y G . Thus, h{ x α , ω \ x α }i α<κ witnesses ss = ω in V [ G ]. (cid:3) Definition 3.6.
Let add( B ) denote the additivity of the ideal of meager sets ofreals.It is known that add( B ) ≤ b and that it is consistent that add( B ) < b . (See 5.4and 11.7 of [7] and 7.3.D of [4]). Corollary 3.7. If κ = cf κ > ω , then it is consistent that ss = ω and add( B ) = c = κ .Proof. Starting with GCH in the ground model, perform a κ -long finite supportiteration of Hechler forcing. This forces add( B ) = c = κ (see 11.6 of [7]). ByTheorem 3.5, this also forces ss = ω . (cid:3) Models of ω < N t ( ω ∗ ) < c To prove the consistency of ω < N t ( ω ∗ ) < c , we employ generalized iterationof forcing along posets as defined by Groszek and Jech [10]. We will only use finitesupport iterations along well-founded posets. For simplicity, we limit our definitionof generalized iterations to this special case. Definition 4.1.
Suppose X is a well-founded poset and P a forcing order consistingof functions on X . Given any x ∈ X , partial map f on X , and down-set Y of X ,set P ↾ Y = { p ↾ Y : p ∈ P } , X ↾ x = { y ∈ X : y < x } , X ↾ ≤ x = { y ∈ X : y ≤ x } , P ↾ x = P ↾ ( X ↾ x ), P ↾ ≤ x = P ↾ ( X ↾ ≤ x ), f ↾ x = f ↾ ( X ↾ x ), and f ↾ ≤ x = f ↾ PLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω ( X ↾ ≤ x ). Then P is a finite support iteration along X if there exists a sequence h Q x i x ∈ X satisfying the following conditions for all x ∈ X and all p, q ∈ P .(1) P ↾ x is a finite support iteration along X ↾ x .(2) Q x is a ( P ↾ x )-name for a forcing order.(3) P ↾ ≤ x = { p ∪ {h x, q i} : h p, q i ∈ ( P ↾ x ) ∗ Q x } .(4) P ↾ x (cid:13) P ( x ) = Q x .(5) P is the set of functions r on X for which r ↾ ≤ y ∈ P ↾ ≤ y for all y ∈ X and P ↾ z (cid:13) r ( z ) = Q z for all but finitely many z ∈ X .(6) p ≤ q if and only if p ↾ y ≤ q ↾ y and p ↾ y (cid:13) p ( y ) ≤ q ( y ) for all y ∈ X .Given a finite support iteration P along X and x ∈ X and a filter G of P , set G x = { p ( x ) : p ∈ G } , G ↾ x = { p ↾ x : p ∈ G } , and G ↾ ≤ x = { p ↾ ≤ x : p ∈ G } .Given any down-set Y of X , set G ↾ Y = { p ↾ Y : p ∈ G } . Remark. If P is a finite support iteration along a well-founded poset X withdown-set Y , then P ↾ Y is an iteration along Y , and P ↾ Y = P ↾ Y . Definition 4.2.
Suppose P is a finite support iteration along a well-founded poset X with down-sets Y and Z such that Y ⊆ Z . Then there is a complete embedding j ZY : P ↾ Y → P ↾ Z given by j ZY ( p ) = p ∪ ( P ↾ Z \ Y ) for all p ∈ P ↾ Y . Thisembedding naturally induces an embedding of the class of ( P ↾ Y )-names, whichin turn naturally induces an embedding of the class of atomic forumlae in the( P ↾ Y )-forcing language. Let j ZY also denote these embeddings. Proposition 4.3.
Suppose P , Y , and Z are as in the above definition, and ϕ is anatomic formula in the ( P ↾ Y ) -forcing language. Then, for all p ∈ P ↾ Z , we have p (cid:13) j ZY ( ϕ ) if and only if p ↾ Y (cid:13) ϕ .Proof. If p ↾ Y (cid:13) ϕ , then p ≤ j ZY ( p ↾ Y ) (cid:13) j ZY ( ϕ ). Conversely, suppose p ↾ Y (cid:13) ϕ .Then we may choose q ≤ p ↾ Y such that q (cid:13) ¬ ϕ . Hence, j ZY ( q ) (cid:13) ¬ j ZY ( ϕ ). Set r = q ∪ ( p ↾ Z \ Y ). Then j ZY ( q ) ≥ r ≤ p ; hence, p (cid:13) j ZY ( ϕ ). (cid:3) Lemma 4.4.
Suppose P is a finite support iteration along a well-founded poset X and x is a maximal element of X . Set Y = X \ { x } . Then there is a denseembedding φ : P → ( P ↾ Y ) ∗ j YX ↾ x ( Q x ) given by φ ( p ) = h p ↾ Y, j YX ↾ x ( p ( x )) i . Hence,if G is a P -generic filter, then G x is ( Q x ) G ↾ x -generic over V [ G ↾ Y ] .Proof. First, let us show that φ is an order embedding. Suppose r, s ∈ P . Then r ≤ s if and only if r ↾ Y ≤ s ↾ Y and r ↾ x (cid:13) r ( x ) ≤ s ( x ). Also, φ ( r ) ≤ φ ( s ) ifand only if r ↾ Y ≤ s ↾ Y and r ↾ Y (cid:13) j YX ↾ x ( r ( x ) ≤ s ( x )). By Proposition 4.3, r ↾ Y (cid:13) j YX ↾ x ( r ( x ) ≤ s ( x )) if and only if r ↾ x (cid:13) r ( x ) ≤ s ( x ); hence, r ≤ s if andonly if φ ( r ) ≤ φ ( s ).Finally, let us show that ran φ is dense. Suppose h p, q i ∈ ( P ↾ Y ) ∗ j YX ↾ x ( Q x ). Thenthere exist r ≤ p and s ∈ dom (cid:0) j YX ↾ x ( Q x ) (cid:1) such that r (cid:13) s = q ∈ j YX ↾ x ( Q x ). Hence, h r, s i ≤ h p, q i . Also, s is a ( j YX ↾ x “( P ↾ x ))-name; hence, there exists a ( P ↾ x )-name t such that j YX ↾ x ( t ) = s . Hence, r (cid:13) j YX ↾ x ( t ∈ Q x ); hence, r ↾ x (cid:13) t ∈ Q x . Hence, r ∪ {h x, t i} ∈ P and φ ( r ∪ {h x, t i} ) = h r, s i . Thus, ran φ is dense. (cid:3) Remark.
Proposition 4.3 and Lemma 4.4 and their proofs remain valid for arbitraryiterations along posets as defined in [10].
Lemma 4.5.
Let P be a forcing order, A a subset of [ ω ] ω with the SFIP, Q theBooth forcing for A , x a Q -name for a generic pseudointersection of A , and B a DAVID MILOVICH P -name such that P forces ˇ A ⊆ B ⊆ [ ω ] ω and forces B to have the SFIP. Let i and j be the canonical embeddings, respectivly, of P -names and Q -names into ( P ∗ ˇ Q ) -names. Then P ∗ ˇ Q forces i ( B ) ∪ { j ( x ) } to have the SFIP.Proof. Seeking a contradiction, suppose r = h p , h σ, F i ˇ i ∈ P ∗ ˇ Q and n < ω and p (cid:13) H ∈ [ B ] <ω and r (cid:13) j ( x ) ∩ T i ( H ) ⊆ ˇ n . Then p forces ˇ F ∪ H ⊆ B , whichis forced to have the SFIP; hence, there exist p ≤ p and m ∈ ω \ n such that p (cid:13) ˇ m ∈ T ( ˇ F ∪ H ). Set r = h p , h σ ∪{ m } , F i ˇ i . Then r ≥ r (cid:13) ˇ m ∈ j ( x ) ∩ T i ( H ),contradicting how we chose r . (cid:3) Lemma 4.6.
Suppose P and Q are forcing orders such that P is ccc and Q hasproperty (K). Then P forces ˇ Q to have property (K).Proof. Suppose the lemma fails. Then there exist p ∈ P and f such that p (cid:13) f ∈ ˇ Q ω and p (cid:13) ∀ J ∈ [ ω ] ω ∃ α, β ∈ J f ( α ) ⊥ f ( β ). For each α < ω , choose p α ≤ p and q α ∈ Q such that p α (cid:13) f ( α ) = ˇ q α . Then there exists I ∈ [ ω ] ω such that q α q β for all α, β ∈ I . Let J be the P -name {h ˇ α, p α i : α ∈ I } . Then p (cid:13) ∀ α, β ∈ J f ( α ) = ˇ q α ˇ q β = f ( β ). Hence, p (cid:13) | J | ≤ ω . Since P is ccc, thereexists α ∈ I such that p (cid:13) J ⊆ ˇ α . But this contradicts p ≥ p α (cid:13) ˇ α ∈ J . (cid:3) Lemma 4.7.
Suppose P is a finite support iteration along a well-founded poset X and P ↾ x forces Q x to have property (K) for all x ∈ X . Then P has property (K).Proof. We may assume the lemma holds whenever X is replaced by a poset of lesserheight. Let I ∈ [ P ] ω . We may assume { supp( p ) : p ∈ I } is a ∆-system; let σ beits root. Set Y = S x ∈ σ X ↾ x . Then P ↾ Y has property (K). Let n = | σ \ Y | and h x i i i Suppose cf κ = κ ≤ λ = λ <κ . Then there exists a κ -like, κ -directed,well-founded poset Ξ with cofinality and cardinality λ .Proof. Let { x α : α < λ } biject from λ to [ λ ] <κ . Construct h y α i α<λ ∈ ([ λ ] <κ ) λ asfollows. Given α < λ and h y β i β<α , choose ξ α ∈ λ \ S β<α y β and set y α = x α ∪ { ξ α } .Let Ξ be { y α : α < λ } ordered by inclusion. Then Ξ is cofinal with [ λ ] <κ ; hence,Ξ is κ -directed and has cofinality λ . Also, Ξ is well-founded because h y α i α<λ isnondecreasing. Finally, Ξ is κ -like because for all I ∈ [ λ ] κ we have | S α ∈ I y α | ≥|{ ξ α : α ∈ I }| = κ ; whence, { y α : α ∈ I } has no upper bound in [ λ ] <κ . (cid:3) Definition 4.9. A point q in a space X is a P κ -point if every intersection of fewerthan κ -many neighborhoods of q contains a neighborhood of q . Definition 4.10. For all x, y ⊆ ω , define x ⊆ ∗ y as | x \ y | < ω . Let p denote theminimum value of | A | where A ranges over the subsets of [ ω ] ω that have SFIP yethave no pseudointersection. Remark. It easily seen that ω ≤ p ≤ h . PLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω Theorem 4.11. Suppose ω ≤ cf κ = κ ≤ λ = λ <κ . Then there is a property (K)forcing extension in which p = πN t ( ω ∗ ) = N t ( ω ∗ ) = ss = b = κ ≤ λ = c . Moreover, in this extension ω ∗ has P κ -points; whence, max q ∈ ω ∗ χN t ( q, ω ∗ ) = κ .Proof. Let Ξ be as in Lemma 4.8. Let h σ α i α<λ biject from λ to Ξ. Let hh ζ α , η α ii α<λ biject from λ to λ . Given α < λ and h τ ζ β ,η β i β<α ∈ Ξ α , choose τ ζ α ,η α ∈ Ξ suchthat σ ζ α < τ ζ α ,η α τ ζ β ,η β for all β < α . We may so choose τ ζ α ,η α because Ξ isdirected and has cofinality λ .Let us construct a finite support iteration P along Ξ. Since Ξ is well-founded,we may define Q σ in terms of P ↾ σ for each σ ∈ Ξ. Suppose σ ∈ Ξ and, for all τ < σ , we have | P ↾ ≤ τ | < κ and P ↾ τ forces Q τ to have property (K). Then P ↾ σ has property (K) by Lemma 4.7, and hence is ccc. Moreover, | P ↾ σ | < κ because P ↾ σ is a finite support iteration along Ξ ↾ σ and | Ξ ↾ σ | < κ . Hence, P ↾ σ (cid:13) | c <κ | ≤ (( κ ω ) <κ ) ˇ ≤ λ . Let E σ be a ( P ↾ σ )-name for the set of all E in the( P ↾ σ )-generic extension for which E ∈ [[ ω ] ω ] <κ and E has the SFIP. Then we maychoose a ( P ↾ σ )-name f σ such that P ↾ σ forces f σ to be a surjection from λ to E σ .We may assume we are given corresponding f τ for all τ < σ . If there exist α, β < λ such that σ = τ α,β , then let Q σ be a ( P ↾ σ )-name for Q ′ σ × Fn( ω, 2) where Q ′ σ is a ( P ↾ σ )-name for the Booth forcing for f σ α ( β ). If there are no such α and β ,then let Q σ be a ( P ↾ σ )-name for a singleton poset. Then P ↾ σ forces Q σ to haveproperty (K). Also, we may assume | Q σ | < κ . Hence, | P ↾ ≤ σ | < κ .By induction, | P ↾ ≤ σ | < κ and P ↾ σ forces Q σ to have property (K) for all σ ∈ Ξ. Hence, P has property (K) by Lemma 4.7, and hence is ccc. Also, since | Ξ | ≤ λ and P is a finite support iteration, | P | ≤ λ . Let G be a P -generic filter.Then c V [ G ] ≤ λ ω = λ . Moreover, c V [ G ] ≥ λ because P adds λ -many Cohen reals.By Theorem 2.16 and Lemma 2.6, it suffices to show that b V [ G ] ≤ κ ≤ p V [ G ] , that ss V [ G ]2 ≤ κ , and that some q ∈ ( ω ∗ ) V [ G ] is a P κ -point. First, we prove κ ≤ p V [ G ] .Suppose E ∈ ([[ ω ] ω ] <κ ) V [ G ] and E has the SFIP. Then there exists α < λ suchthat E ∈ V [ G ↾ σ α ] because Ξ is κ -directed. Hence, there exists β < λ such that( f σ α ) G ↾ σ α ( β ) = E . Hence, E has a pseudointersection in V [ G ↾ ≤ τ α,β ]. Thus, κ ≤ p V [ G ] .Second, let us show that b V [ G ] ≤ κ . For each α < κ , let u α be the increasingenumeration of the Cohen real added by the Fn( ω, 2) factor of Q τ ,α . Then itsuffices to show that { u α : α < κ } is unbounded in ( ω ω ) V [ G ] . Suppose v ∈ ( ω ω ) V [ G ] .Then there exists σ ∈ Ξ such that v ∈ V [ G ↾ σ ]. Since Ξ is κ -like, there exists α < κ such that τ ,α σ . By Lemma 4.4, u α enumerates a real Cohen generic over V [ G ↾ σ ]; hence, u α is not eventually dominated by v .Third, let us prove ss V [ G ]2 ≤ κ . For each α < λ , let x α be the Cohen real addedby the Fn( ω, 2) factor of Q τ ,α . Suppose I ∈ ([ λ ] κ ) V [ G ] and y ∈ ([ ω ] ω ) V [ G ] . Thenthere exists σ ∈ Ξ such that y ∈ V [ G ↾ σ ]. Since Ξ is κ -like, there exists α ∈ I suchthat τ ,α σ . By Lemma 4.4, x α is Cohen generic over V [ G ↾ σ ], and thereforesplits y . Thus, h{ x α , ω \ x α }i α<λ witnesses ss V [ G ]2 ≤ κ .Finally, let us construct a P κ -point q ∈ ( ω ∗ ) V [ G ] . Let ⊑ be an extension of theordering of Ξ to a well-ordering of Ξ. For each σ ∈ Ξ, set Y σ = { τ ∈ Ξ : τ ⊏ σ } .Set ρ = min ⊑ Ξ and choose U ρ ∈ ( ω ∗ ) V . Suppose τ ∈ Ξ and σ is a final predecessorof τ with respect to ⊑ and U σ ∈ ( ω ∗ ) V [ G ↾ Y σ ] . If there are no α, β < λ such that σ = τ α,β and ( f σ α ) G ↾ σ α ( β ) ⊆ U σ , then choose U τ ∈ ( ω ∗ ) V [ G ↾ Y τ ] such that U τ ⊇ U σ .Now suppose such α and β exist. Let v σ be the pseudointersection of ( f σ α ) G ↾ σ α ( β )added by Q ′ σ .By Lemmas 4.4 and 4.5, U σ ∪ { v σ } has the SFIP; hence, we may choose U τ ∈ ( ω ∗ ) V [ G ↾ Y τ ] such that U τ ⊇ U σ ∪ { v σ } . For τ ∈ Ξ that are limit points with respectto ⊑ , choose U τ ∈ ( ω ∗ ) V [ G ↾ Y τ ] such that U τ ⊇ S σ ⊏ τ U σ ; set q = S τ ∈ Ξ U τ . Then,arguing as in the proof of κ ≤ p V [ G ] , we have that q is a P κ -point in ( ω ∗ ) V [ G ] . (cid:3) The forcing extension of Theorem 4.11 can be modified to satisfy b = s Given a class J of posets and a cardinal κ , let M A ( κ ; J ) denotethe statement that, given any P ∈ J and fewer than κ -many dense subsets of P ,there is a filter of P intersecting each of these dense sets. We may replace J witha descriptive term for J when there is no ambiguity. For example, M A ( c ; ccc) isMartin’s axiom. Theorem 4.13. Suppose ω < cf κ = κ ≤ λ = λ <κ . Then there is a property (K)forcing extension in which ω = πN t ( ω ∗ ) = b = s < N t ( ω ∗ ) = ss = κ ≤ λ = c . Proof. Let P be as in the proof of Theorem 4.11. Set R = P × Fn( ω , P does. Let K be a generic filter of R . Let π and π bethe natural coordinate projections on R ; let π and π also denote their respectivenatural extensions to the class of R -names. Set G = π “ K and H = π “ K . Then c V [ K ] = λ clearly holds. Adding ω -many Cohen reals to any model of ZFC forces b = s = ω , and πN t ( ω ∗ ) = h ≤ b , so πN t ( ω ∗ ) V [ K ] = b V [ K ] = s V [ K ] = ω .For each α < λ , let x α be the Cohen real added by the Fn( ω, 2) factor of Q τ ,α .Suppose I ∈ ([ λ ] κ ) V [ K ] and y ∈ ([ ω ] ω ) V [ K ] . Then there exists σ ∈ Ξ such that y ∈ V [( G ↾ σ ) × H ]. Since Ξ is κ -like, there exists α ∈ I such that τ ,α σ .By Lemma 4.4, x α is Cohen generic over V [ G ↾ σ ]; hence, x α is Cohen generic over V [( G ↾ σ ) × H ] and therefore splits y . Thus, h{ x α , ω \ x α }i α<λ witnesses ss V [ K ]2 ≤ κ .Therefore, it suffices to show that N t ( ω ∗ ) V [ K ] ≥ κ . Suppose µ < κ and A is an R -name for a base of ω ∗ . Choose an R -name q for an element of ω ∗ with character λ . Let f be a name for an injection from λ into A such that q ∈ T ran f . Let g be a name for an element of ([ ω ] ω ) λ such that q ∈ g ( α ) ∗ ⊆ f ( α ) for all α < λ . Foreach α < λ , let u α be a name for g ( α ) such that u α = {{ ˇ n } × A α,n : n < ω } whereeach A α,n is a countable antichain of R . Since max { ω , µ } < λ , there exist ξ < ω and J ∈ [ λ ] µ such that ran π ( u α ) ⊆ Fn( ξ, 2) for all α ∈ J . It suffices to show that { ( u α ) K : α ∈ J } has a pseudointersection in V [ K ].For each α ∈ J , set v α = {h ˇ n, r i : h ˇ n, h p, r ii ∈ u α and p ∈ G } . Set H = H ∩ Fn( ξ, A ( p ; σ -centered) is a theorem of ZFC. Hence, V [ G ] satisfies M A ( κ ; σ -centered). By an argument of Baumgartner and Tall com-municated by Roitman [18], adding a single Cohen real preserves M A ( κ ; σ -centered).Since Booth forcing for { ( v α ) H : α ∈ J } is σ -centered, { ( v α ) H : α ∈ J } , which isequal to { ( u α ) K : α ∈ J } , has a pseudointersection in V [ G × H ]. (cid:3) Local Noetherian type and π -type Definition 5.1. For every infinite cardinal κ , let u ( κ ) denote the space of uniformultrafilters on κ . PLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω Dow and Zhou [8] proved that there is a point in ω ∗ that (along with satisfyingsome additional properties) has an ω op -like local base. We present a simpler con-struction of an ω op -like local base which also naturally generalizes to every u ( κ ).This construction is essentially due to Isbell [11], who was interested in actualintersections as opposed to pseudointersections. Definition 5.2. Given cardinals λ ≥ κ ≥ ω and a point p in a space X , a local h λ, κ i - splitter is a set U of λ -many open neighborhoods of p such that p is not inthe interior of T V for any V ∈ [ U ] κ . Lemma 5.3. Every poset P is almost | P | op -like.Proof. Let κ = | P | and let h p α i α<κ biject from κ to P . Define a partial map f : κ → P as follows. Suppose α < κ and we have a partial map f α : α → P . Ifran f α is dense in P , then set f α +1 = f α . Otherwise, set β = min { δ < κ : p δ q for all q ∈ ran f α } and set f α +1 = f α ∪ {h α, p β i} . For limit ordinals γ ≤ κ , set f γ = S α<γ f α . Set f = f κ . Then f is nonincreasing; hence, ran f is κ op -like.Moreover, ran f is dense in P . (cid:3) Lemma 5.4. Suppose X is a space with a point p at which there is no finite localbase. Then χN t ( p, X ) is the least κ ≥ ω for which there is a local h χ ( p, X ) , κ i -splitterat p . Moreover, if λ > χ ( p, X ) , then p does not have a local h λ, κ i -splitter at p forany κ < λ or κ ≤ cf λ .Proof. By Lemma 5.3, χ ( p, X ) ≥ χN t ( p, X ); hence, a χN t ( p, X ) op -like local baseat p (which necessarily has size χ ( p, X )) is a local h χ ( p, X ) , χN t ( p, X ) i -splitter at p . To show the converse, let λ = χ ( p, X ) and let h U α i α<λ be a sequence of openneighborhoods of p . Let { V α : α < λ } be a local base at p . For each α < λ , choose W α ∈ { V β : β < λ } such that W α ⊆ U α ∩ V α . Then { W α : α < λ } is a localbase at p . Let κ < χN t ( p, X ). Then there exist α < λ and I ∈ [ λ ] κ such that W α ⊆ T β ∈ I W β . Hence, p is in the interior of T β ∈ I U β . Hence, { U α : α < λ } is nota local h λ, κ i -splitter at p .To prove the second half of the lemma, suppose λ > χ ( p, X ) and A is a set of λ -many open neighborhoods of p . Let B be a local base at p of size χ ( p, X ). Then,for all κ < λ and κ ≤ cf λ , there exist U ∈ B and C ∈ [ A ] κ such that U ⊆ T C .Hence, A is not a local h λ, κ i -splitter at p . (cid:3) Theorem 5.5. For each κ ≥ ω , there exists p ∈ u ( κ ) such that χN t ( p, u ( κ )) = ω and χ ( p, u ( κ )) = 2 κ .Proof. Let A be an independent family of subsets of κ of size 2 κ . Set B = S F ∈ [ A ] ω { x ⊆ κ : ∀ y ∈ F | x \ y | < κ } . Since A is independent, we may extend A toan ultrafilter p on κ such that p ∩ B = ∅ . For each x ⊆ κ , set x ∗ = { q ∈ u ( κ ) : x ∈ q } .Then { x ∗ : x ∈ A } is a local h κ , ω i -splitter at p . Since χ ( p, u ( κ )) ≤ κ , it followsfrom Lemma 5.4 that χN t ( p, u ( κ )) = ω and χ ( p, u ( κ )) = 2 κ . (cid:3) Definition 5.6. Let a denote the minimum of the cardinalities of infinite, maximalalmost disjoint subfamilies of [ ω ] ω . Let i denote the minimum of the cardinalitiesof infinite, maximal independent subfamilies of [ ω ] ω .It is known that b ≤ a and r ≤ i ≥ d ≥ s . (See 8.4, 8.12, 8.13 and 3.3 of [7].)Because of Kunen’s result that a = ℵ in the Cohen model (see VIII.2.3 of [14]), itis consistent that a < r . Also, Shelah [20] has constructed a model of r ≤ u < a . In ZFC, the best upper bound of χN t ( ω ∗ ) of which we know is c by Lemma 5.3.We will next prove Theorem 5.10, which implies that, except for c and possiblycf c , all of the cardinal characteristics of the continuum with definitions included inBlass [7] can consistently be simultaneously strictly less than χN t ( ω ∗ ). Lemma 5.7. Suppose κ , λ , and µ are regular cardinals and κ ≤ λ > µ . Then ( κ × λ ) op is not almost µ op -like.Proof. Let I be a cofinal subset of κ × λ . Then it suffices to show that I is not µ -like. If κ = λ , then I is not µ -like because it is λ -directed. Suppose κ < λ .Then there exists α < κ such that | I ∩ ( { α } × λ ) | = λ ; hence, I has an increasing λ -sequence; hence, I is not µ -like. (cid:3) Lemma 5.8. Given any infinite independent subfamily I of [ ω ] ω , there exists J ⊆ [ ω ] ω such that if x is a generic pseudointersection of J then I ∪ { x } is independent,but I ∪ { x, y } is not independent for any y ∈ [ ω ] ω ∩ V \ I .Proof. See Exercise A12 on page 289 of Kunen [14]. (cid:3) Definition 5.9. We say a P κ -point in a space is simple if it has a local base oforder type κ op . Theorem 5.10. Suppose ω ≤ cf κ = κ ≤ cf λ = λ = λ <κ . Then there is aproperty (K) forcing extension satisfying p = a = i = u = κ ≤ λ = χN t ( ω ∗ ) = c .Proof. We will construct a finite support iteration h P α i α ≤ λκ where λκ denotes theordinal product of λ and κ . It suffices to ensure that the iteration is at every stageproperty (K) and of size at most λ , and that V P λκ satisfies max { a , i , u } ≤ κ ≤ p and λ ≤ χN t ( ω ∗ ). Our strategy is to interleave an iteration of length λκ and threeiterations of length κ . At every stage below λκ , add another piece of what will bean ultrafilter base that, ordered by ⊇ ∗ , will be isomorphic to a cofinal subset of κ × λ . Also, at every stage we will add a pseudointersection, such that the finalmodel satisfies p ≥ κ . After each limit stage of cofinality λ , add an element to eachof three objects that, when completed, will be a maximal almost disjoint family ofsize κ , a maximal independent family of size κ , and a base of a simple P κ -point in ω ∗ .Let ϕ : λ → λ be a bijection such that ϕ ( α, β ) ≥ α for all α, β < λ . Foreach h α, β i ∈ κ × λ , set E α,β = {h γ, δ i ∈ κ × λ : λγ + δ < λα + β } . Suppose h α, β i ∈ κ × λ and we have constructed h P γ i γ ≤ λα + β to have property (K) and sizeat most λ at all of its stages, and a sequence h x γ,δ i h γ,δ i∈ E α,β of P λα + β -names eachforced to be in [ ω ] ω . Set B = { x γ,δ : h γ, δ i ∈ E α,β } . Let h S γ i γ<κ be a partition of λ into κ -many stationary sets such that S contains all successor ordinals. Supposewe have constructed a sequence h ρ γ,δ i h γ,δ i∈ E α,β ∈ λ E α,β such that we always have ρ γ,δ ∈ S γ and ρ γ,δ < ρ γ,δ whenever δ < δ . Set D α,β = {h γ, ρ γ,δ i : h γ, δ i ∈ E α,β } .Further suppose that {hh γ, ρ γ,δ i , x γ,δ i : h γ, δ i ∈ E α,β } is forced to be an orderembedding of D α,β into h [ ω ] ω , ⊇ ∗ i and that its range B is forced to have the SFIP.Also suppose that we have the following if α > (cid:13) λα + β ∀ σ ∈ [ B ] <ω ∃ δ < λ \ σ ∗ x ,δ For each ε < λ , set A ε = { x γ,δ : h γ, δ i ∈ E α,β and h γ, ρ γ,δ i < h α, ε i} .Let y β be a P λα + β -name for a surjection from λ to [ ω ] ω . We may assume thatcorresponding y γ have already been constructed for all γ < β . Let ϕ ( ζ, η ) = β . PLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω Claim. If α > , then we may choose z ∈ { y ζ ( η ) , ω \ y ζ ( η ) } such that (cid:13) λα + β ∀ σ ∈ [ B ] <ω ∃ δ < λ z ∩ \ σ ∗ x ,δ . Proof. Suppose not. Let { z , z } = { y ζ ( η ) , ω \ y ζ ( η ) } . Then, working in a genericextension by P λα + β , there exist σ , σ ∈ [ B ] <ω and such that z i ∩ T σ i ⊆ ∗ x ,δ forall i < δ < λ . Hence, T S i< σ i ⊆ ∗ x ,δ for all δ < λ , in contradiction with(5.1). (cid:3) If α > 0, then choose z as in the above claim; otherwise, choose z arbitrarily.If α = 0, then set ρ α,β = β + 1. Otherwise, we may choose ρ α,β ∈ S α such that ρ α,β > ρ α,γ for all γ < β and (cid:13) λα + β ∀ σ ∈ [ A ρ α,β ] <ω ∃ δ < ρ α,β z ∩ \ σ ∗ x ,δ . Set D α,β +1 = D α,β ∪ {h α, ρ α,β i} . Let A ′ be a P λα + β -name forced to satisfy A ′ = A ρ α,β ∪ { z } if z splits B and A ′ = A ρ α,β otherwise. Let Q be a name for the Boothforcing for A ′ ∪ { ω \ n : n < ω } ; let x α,β be a name for a generic pseudointersectionof A ′ ∪ { ω \ n : n < ω } . (The purpose of { ω \ n : n < ω } is to ensure that x α,β doesnot almost contain any element of [ ω ] ω ∩ V P λα + β .)Let F λα + β to be a P λα + β -name for a surjection from λ to the elements of [[ ω ] ω ] <κ that have the SFIP. We may assume that corresponding F γ have already beenconstructed for all γ < λα + β . Let Q be a name for the Booth forcing for F λα + ζ ( η ).Further suppose we have constructed sequences h w γ i γ<α and h U γ i γ<α of P λα -namessuch that (cid:13) λγ U δ ∪ { w δ } ⊆ U γ ∈ ω ∗ for all δ < γ < α , and such that w γ is forcedto be a pseudointersection U γ for all γ < α . If β = 0, then let Q be a name for thetrivial forcing. If β = 0, then choose U α such that (cid:13) λα S γ<α U γ ∪ { w γ } ⊆ U α ∈ ω ∗ ,let Q be a name for the Booth forcing for U α , and let w α be a name for a genericpseudointersection of U α .Further suppose we have constructed a sequence h a γ i γ<α of P λα -names whoserange is forced to be an almost disjoint subfamily of [ ω ] ω . If β = 0, then let Q be a name for the trivial forcing. If β = 0, then let Q be a name for the Boothforcing for { ω \ a γ : γ < α } , and let a α be a name for a generic pseudointersectionof { ω \ a γ : γ < α } .Further suppose we have constructed a sequence h i γ i γ<α of P λα -names whoserange is forced to be an independent subfamily of [ ω ] ω . If β = 0, then let Q be aname for the trivial forcing. If β = 0, then set I = { i γ : γ < α } and let J and x beas in Lemma 5.8; let Q be a name for the Booth forcing for J ; let i α be a namefor x .Set P λα + β +1 = P λα + β ∗ Q n< Q n . We may assume | Q n< Q n | ≤ λ ; hence, P λα + β +1 has property (K) and size at most λ . Also, B ∪ { x α,β } is forced to havethe SFIP by Q -genericity because for every b ∈ B we have that { b }∪ A ′ is forced tohave the SFIP because { b } ∪ A ′ ⊆ B ∪ { z } if z splits B and { b } ∪ A ′ ⊆ B otherwise.Let us also show that (5.1) holds if we replace β with β + 1. We may assume α > σ ∈ [ B ] <ω . Then there exists δ < λ such that (cid:13) λα + β z ∩ T ( σ ∪ τ ) ∗ x ,δ for all τ ∈ [ A ρ α,β ] <ω ; hence, (cid:8)(cid:0)T σ (cid:1) \ x ,δ (cid:9) ∪ A ′ is forced to have the SFIP; hence, (cid:13) λα + β +1 x α,β ∩ T σ ∗ x ,δ by Q -genericity. Thus, (5.1) holds as desired.To complete our inductive construction of h P γ i γ ≤ λκ , it suffices to show that {hh γ, ρ γ,δ i , x γ,δ i : h γ, δ i ∈ E α,β +1 } is forced to be an order embedding of D α,β +1 into h [ ω ] ω , ⊇ ∗ i . Suppose h γ, δ i ∈ E α,β . Then h α, ρ α,β i 6≤ h γ, ρ γ,δ i and (cid:13) λα + β +14 DAVID MILOVICH x α,β ∗ x γ,δ by Q -genericity. If h γ, ρ γ,δ i < h α, ρ α,β i , then x γ,δ ∈ A ′ ; whence, (cid:13) λα + β +1 x γ,δ ) ∗ x α,β . Suppose h γ, ρ γ,δ i 6 < h α, ρ α,β i . Then ρ α,β < ρ γ,δ ; hence, ρ γ,δ ≥ ρ α,β +1 = ρ ,ρ α,β ; hence, x γ,δ ⊆ ∗ x ,ρ α,β . By construction, A ′ ∪{ ω \ x ,ρ α,β } isforced to have the SFIP; hence, (cid:13) λα + β +1 x γ,δ ⊆ ∗ x ,ρ α,β ∗ x α,β by Q -genericity.Thus, {hh γ, ρ γ,δ i , x γ,δ i : h γ, δ i ∈ E α,β +1 } is forced to be an embedding as desired.Let us show that V P λκ satisfies λ ≤ χN t ( ω ∗ ). Let G be a generic filter of P λκ and set B = { ( x α,β ) ∗ G : h α, β i ∈ κ × λ } . Then B is a local base at some p ∈ ( ω ∗ ) V [ G ] because every element of ([ ω ] ω ) V [ G ] is handled by an appropriate Q .By Lemma 2.15, B contains a χN t ( p, ω ∗ ) op -like local base { ( x α,β ) ∗ G : h α, β i ∈ I } at p for some I ⊆ κ × λ . Set J = {h α, ρ α,β i : h α, β i ∈ I } . Then J is cofinal in κ × λ ;hence, by Lemma 5.7, J is not ν -like for any ν < λ . Hence, χN t ( ω ∗ ) V [ G ] ≥ λ .Finally, let us show that V P λκ satisfies max { a , i , u } ≤ κ ≤ p . Working in V [ G ],notice that u ≤ κ because S α<κ ( U α ) G ∈ ω ∗ and { ( w α ) ∗ G : α < κ } is a local baseat S α<κ ( U α ) G . Moreover, { ( a α ) G : α < κ } and { ( i α ) G : α < κ } witness that a ≤ κ and i ≤ κ . For p ≥ κ , note that very element of [[ ω ] ω ] <κ with the SFIP is( F λα + ζ ( η )) G for some α < κ and ζ, η < λ . By Q -genericity, a pseudointersectionof ( F λα + ζ ( η )) G is added at stage λα + ϕ ( ζ, η ). (cid:3) Theorem 5.11. πχN t ( ω ∗ ) = ω .Proof. Fix p ∈ ω ∗ . By a result of Balcar and Vojt´aˇs [3], there exists h y x i x ∈ p suchthat y x ∈ [ x ] ω for all x ∈ p and { y x } x ∈ p is an almost disjoint family. Clearly, { y ∗ x } x ∈ p is a pairwise disjoint—and therefore ω op -like—local π -base at p . (cid:3) Powers of ω ∗ Definition 6.1. A box is a subset E of a product space Q i ∈ I X i such that thereexist σ ∈ [ I ] <ω and h E i i i ∈ σ such that E = T i ∈ σ π − i E i . Let N t box ( Q i ∈ I X i ) denotethe least infinite κ such that Q i ∈ I X i has a κ op -like base of open boxes. Lemma 6.2 (Peregudov [16]) . In any product space X = Q i ∈ I X i , we have N t ( X ) ≤ N t box ( X ) ≤ sup i ∈ I N t ( X i ) . Lemma 6.3 (Malykhin [15]) . Let X = Q i ∈ I X i where each X i is a nonsingleton T space. If w ( X ) ≤ | I | , then N t ( X ) = N t box ( X ) = ω .Remark. In Lemma 6.3, the hypothesis that the factor spaces be nonsingleton and T can be weakened to merely require that each factor space is the union of twonontrivial open sets. Also, the conclusion of Lemma 6.3 may be amended withthe statement that X has a h| I | , ω i -splitter: use h{ π − i U i , π − i V i }i i ∈ I where each { U i , V i } is a nontrivial open cover of X i . Theorem 6.4. The sequence h N t (( ω ∗ ) ω + α ) i α ∈ O n is nonincreasing. Moreover, N t (( ω ∗ ) c ) = ω .Proof. Note that if ω ≤ α ≤ β , then ( ω ∗ ) β ∼ = (( ω ∗ ) α ) β . Then apply Lemmas 6.2and 6.3. (cid:3) Lemma 6.5. Let < n < ω and X be a space. Then N t box ( X n ) = N t ( X ) .Proof. Set κ = N t box ( X n ). By Lemma 6.2, κ ≤ N t ( X ). Let us show that N t ( X ) ≤ κ . Let A be a κ op -like base of X n consisting only of boxes. Let B denote the set of allnonempty open V ⊆ X for which there exists Q i N t box ( X n ) ≤ N t ( X n ). By Lemma 2.7,either X n has a h w ( X n ) , N t ( X n ) i -splitter, or N t ( X n ) = w ( X n ) + . Hence, byLemma 2.6, N t box ( X n ) ≤ N t ( X n ). (cid:3) Theorem 6.7. If < n < ω , then N t ( ω ∗ ) ≥ N t (( ω ∗ ) n ) ≥ min { N t ( ω ∗ ) , c } . More-over, max { u , cf c } = c implies N t ( ω ∗ ) = N t (( ω ∗ ) n ) .Proof. Lemma 6.2 implies N t ( ω ∗ ) ≥ N t (( ω ∗ ) n ). To prove the rest of the theorem,first consider the case r < c . As in the proof of Theorem 2.3, construct a point p ∈ ω ∗ such that πχ ( p, ω ∗ ) = r and χ ( p, ω ∗ ) = c . Then πχ ( h p i i Suppose max { u , cf c } = c . Then h N t (( ω ∗ ) α ) i α ∈ O n is nonincreas-ing.Proof. By Theorem 6.7 and Lemma 6.2, N t (( ω ∗ ) n ) = N t ( ω ∗ ) ≥ N t (( ω ∗ ) α ) when-ever 0 < n < ω ≤ α . The rest follows from Theorem 6.4. (cid:3) Theorem 6.9. Suppose u = c . Then N t (( ω ∗ ) α ) = N t ( ω ∗ ) for all α < cf c .Proof. Let λ be an arbitrary infinite cardinal less than N t ( ω ∗ ). By Lemma 2.7, itsuffices to show that ( ω ∗ ) α does not have a h c , λ i -splitter. Seeking a contradiction,suppose hF β i β< c is such a h c , λ i -splitter. We may assume S β< c F β consists onlyof open boxes because we can replace each F β with a suitable refinement. Since α < cf c , there exist σ ∈ [1 + α ] <ω and I ∈ [ c ] c such that, for every U ∈ S β ∈ I F β ,there exists ϕ ( U ) ⊆ ( ω ∗ ) σ such that U = π − σ ϕ ( U ). Let j be a bijection from c to I . Then h ϕ “ F j ( β ) i β< c is a h c , λ i -splitter of ( ω ∗ ) σ . Hence, N t (( ω ∗ ) σ ) ≤ λ < N t ( ω ∗ )by Lemma 2.6. But N t (( ω ∗ ) σ ) < N t ( ω ∗ ) contradicts Theorem 6.7. (cid:3) Lemma 6.10. Suppose a space X has a h cf w ( X ) , cf w ( X ) i -splitter. Then N t ( X ) ≤ w ( X ) .Proof. Set κ = cf w ( X ) and λ = w ( X ). Let hF α i α<κ be a h κ, κ i -splitter of X . Let h : λ → κ satisfy | h − { α }| < λ for all α < κ . Then hF h ( α ) i α<λ is a h λ, λ i -splitterbecause if I ∈ [ λ ] λ , then h “ I ∈ [ κ ] κ . By Lemma 2.6, N t ( X ) ≤ λ . (cid:3) Remark. The proof of the above lemma shows that for any infinite cardinal κ , aspace with a h cf κ, cf κ i -splitter also has a h κ, κ i -splitter. Theorem 6.11. N t (( ω ∗ ) cf c ) ≤ c .Proof. The sequence h{ π − α ( { n : n < ω } ∗ ) , π − α ( { n + 1 : n < ω } ∗ ) }i α< cf c is a h cf c , ω i -splitter of ( ω ∗ ) cf c . Apply Lemma 6.10. (cid:3) Theorem 6.12. For all cardinals κ satisfying κ > cf κ > ω , it is consistent that c = κ and r < cf c . The last inequality implies N t (( ω ∗ ) α ) = c + for all α < cf c and N t (( ω ∗ ) β ) = c = κ for all β ∈ c \ cf c .Proof. Starting with c = κ in the ground model, the proof of Theorem 2.3 showshow to force r = u = ω while preserving c . Now suppose r < cf c . Fix α < cf c and β ∈ c \ cf c . By Theorems 6.11 and 6.4, N t (( ω ∗ ) β ) ≤ c . To see that N t (( ω ∗ ) β ) ≥ c ,proceed as in the proof of Theorem 6.7, constructing a point with character c and π -character | β | . Similarly prove N t (( ω ∗ ) α ) = c + by constructing a point withcharacter c and π -character | r + α | . (cid:3) Lemma 6.13. Suppose κ , λ , and µ are cardinals and p is a point in a productspace X = Q α<κ X α satisfying the following for all α < κ . (1) 0 < κ < w ( X ) and ω ≤ λ ≤ w ( X ) . (2) κ < cf w ( X ) or λ < w ( X ) . (3) µ < λ or µ = cf λ . (4) χ ( p ( α ) , X α ) < λ or the intersection of any µ -many neighborhoods of p ( α ) has nonempty interior.Then χ ( p, X ) < w ( X ) or N t ( X ) > µ .Proof. Let A be a base of X . Set B = { U ∈ A : p ∈ U } . For each α < κ , let C α be a local base at p ( α ) of size χ ( p ( α ) , X α ). Set F = S r ∈ [ κ ] <ω Q α ∈ r C α . Foreach σ ∈ F , set U σ = T α ∈ dom σ π − α σ ( α ). For each V ∈ B , choose σ ( V ) ∈ F suchthat p ∈ U σ ( V ) ⊆ V . We may assume χ ( p, X ) = w ( X ); hence, by (1) and (2),there exist r ∈ [ κ ] <ω and D ∈ [ B ] λ such that dom σ ( V ) = r for all V ∈ D . Set s = { α ∈ r : χ ( p ( α ) , X α ) < λ } and t = r \ s . By (3), there exist τ ∈ Q α ∈ s C α and E ∈ [ D ] µ such that σ ( V ) ↾ s = τ for all V ∈ E . By (4), T V ∈E σ ( V )( α )has nonempty interior for all α ∈ t . Hence, T E has nonempty interior because itcontains U τ ∩ T α ∈ t π − α T V ∈E σ ( V )( α ). Thus, N t ( X ) > µ . (cid:3) Theorem 6.14. Suppose < α < c and h X β i β<α is a sequence of spaces each withweight at most c . Then N t ( Q β<α ( X β ⊕ ω ∗ )) > ν for all regular ν < p .Proof. Let ν be an arbitrary infinite regular cardinal less than p . Set κ = | α | and λ = µ = ν . Choose q ∈ ω ∗ such that χ ( q, ω ∗ ) = c ; set p = h q i β<α . ApplyingLemma 6.13, we have N t ( Q β<α ( X β ⊕ ω ∗ )) > ν . (cid:3) Corollary 6.15. Suppose p = c . Then N t (( ω ∗ ) α ) = c for all α < c .Proof. By Theorem 2.11, N t ( ω ∗ ) ≤ c . Hence, by Corollary 6.8, N t (( ω ∗ ) α ) ≤ c for all α ∈ O n . By Theorem 6.14, N t (( ω ∗ ) α ) ≥ c for all α < c . (cid:3) Corollary 6.16. Suppose α < c and h X β i β<α is a sequence of spaces each withweight at most c . Then Q β<α ( X β ⊕ ω ∗ ) is not homeomorphic to a product of c -manynonsingleton spaces.Proof. Combine Theorem 6.14 and Lemma 6.3. (cid:3) PLITTING FAMILIES AND THE NOETHERIAN TYPE OF βω \ ω Questions Question . Is it consistent that N t ( ω ∗ ) = c + and r ≥ cf c ? Question . Is N t ( ω ∗ ) < ss ω consistent? This inequality implies u < c . Hence, byTheorem 2.11, the inequality further impliescf c ≤ r ≤ u < c = N t ( ω ∗ ) < ss ω = c + . More generally, does any space X have a base that does not contain an N t ( X ) op -likebase? Question . Is ss ω < ss consistent? Question . Letting g denote the groupwise density number (see 6.26 of [7]), is N t ( ω ∗ ) < g consistent? χN t ( ω ∗ ) < g ? In particular, what are N t ( ω ∗ ) and χN t ( ω ∗ )in the Laver model (see 11.7 of [7])? Question . Is cf N t ( ω ∗ ) < N t ( ω ∗ ) < c consistent? cf N t ( ω ∗ ) = ω ? Question . Is cf c < N t ( ω ∗ ) < c consistent? Question . What is χN t ( ω ∗ ) in the forcing extension of the proof of Theorem 4.13?More generally, is it consistent that χN t ( ω ∗ ) < N t ( ω ∗ ) ≤ c ? Question . Is χN t ( ω ∗ ) = ω consistent? An affirmative answer would be a strength-ening of Shelah’s result [19] that ω ∗ consistently has no P-points. If the answer isnegative, then which, if any, of p , h , s , and g are lower bounds of χN t ( ω ∗ ) in ZFC? Question . Is cf c < χN t ( ω ∗ ) consistent? cf c < χN t ( ω ∗ ) < c ? Question . Does any Hausdorff space have uncountable local Noetherian π -type?(It is easy to construct such T spaces: give ω + 1 the topology { ( ω + 1) \ ( α ∪ σ ) : α < ω and σ ∈ [ ω + 1] <ω } ∪ {∅} .) Question . Is it consistent that N t (( ω ∗ ) α ) < min { N t ( ω ∗ ) , c } for some α < c ?Is it consistent that N t (( ω ∗ ) α ) < N t ( ω ∗ ) for some α < cf c ? References [1] B. Balcar, J. Pelant, and P. Simon, The space of ultrafilters on N covered by nowhere densesets , Fund. Math. (1980), no. 1, 11–24.[2] B. Balcar and P. Simon, Reaping number and π -character of Boolean algebras , Topological,algebraical and combinatorial structures. Discrete Math. 108 (1992), no. 1-3, 5–12.[3] B. Balcar and P Vojt´aˇs, Almost disjoint refinement of families of subsets of N , Proc. Amer.Math. Soc., , no. 3, 1980.[4] T. Bartoszy´nski and H. Judah, Set theory. On the structure of the real line , A K Peters, Ltd.,Wellesley, MA, 1995.[5] J. E. Baumgartner and P. Dordal, Adjoining Dominating Functions , J. Symbolic Logic (1985), no. 1, 94–101.[6] M. G. Bell, On the combinatorial principle P ( c ), Fund. Math. (1981), no. 2, 149–157.[7] A. Blass, Combinatorial Cardinal Characteristics of the Continuum. In M. Foreman, A.Kanamori, and M. Magidor, eds., Handbook of Set Theory . Kluwer, to appear.[8] A. Dow and J. Zhou, Two real ultrafilters on ω , Topology Appl. (1999), no. 1-2, 149–154.[9] R. Engelking, General Topology , Heldermann Verlag, Berlin, 2nd ed., 1989.[10] M. Groszek and T. Jech, Generalized iteration of forcing , Trans. Amer. Math. Soc. (1991), no. 1, 1–26.[11] J. Isbell, The category of cofinal types. II , Trans. Amer. Math. Soc. (1965), 394–416.[12] I. Juh´asz, Cardinal functions in topology—ten years later , Mathematical Centre Tracts, 123,Mathematisch Centrum, Amsterdam, 1980. [13] K. Kunen, Random and Cohen reals , Handbook of set-theoretic topology, 887–911,North-Holland, Amsterdam, 1984.[14] K. Kunen, Set theory. An introduction to independence proofs , Studies in Logic and theFoundations of Mathematics, 102. North-Holland Publishing Co., Amsterdam-New York,1980.[15] V. I. Malykhin, On Noetherian Spaces , Amer. Math. Soc. Transl. (1987), no. 2, 83–91.[16] S. A. Peregudov, On the Noetherian type of topological spaces , Comment. Math. Univ. Car-olin. (1997), no. 3, 581–586.[17] B. Pospiˇsil, On bicompact spaces , Publ. Fac. Sci. Univ. Masaryk (1939), no. 270.[18] J. Roitman, Correction to: Adding a random or a Cohen real: topological consequences andthe effect on Martin’s axiom , Fund. Math. (1988), no. 2, 141.[19] S. Shelah, Proper forcing , Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982.[20] S. Shelah, Two cardinal invariants of the continuum ( d < a ) and FS linearly ordered iteratedforcing , Acta Math. (2004), no. 2, 187-223 University of Wisconsin-Madison Mathematics Dept. E-mail address ::