aa r X i v : . [ m a t h . GN ] J a n MADNESS AND (WEAK) NORMALITY
CÉSAR CORRAL
Abstract.
We consider weakenings of normality in Ψ -spaces andprove that the existence of a MAD family whose Ψ -space is almost-normal is independent of ZFC . We also construct a partly-normal notquasi-normal AD family, answering questions from García-Balan andSzeptycki. We finish by showing that the concepts of almost-normaland strongly ℵ -separated AD families are different, even under CH ,answering a question from Oliveira-Rodrigues and Santos-Ronchim Introduction and notation
Two subsets
A, B ⊆ ω are almost disjoint if | A ∩ B | < ω . A family A ⊆ P ( ω ) is almost disjoint (AD for short) if its elements are pairwisealmost disjoint. We say that A is maximal almost disjoint (MAD) if itis AD and maximal with respect to this property (equivalently, for everyinfinite X ⊆ ω , there exists A ∈ A such that | A ∩ X | = ω ).Each AD family A has naturally a topological space associated with it.The Ψ -space or the Mrówka-Isbell space associated to A is denoted by Ψ( A ) and the underlying set is ω ∪ A , where the points in ω are isolated and forevery A ∈ A , sets of the form { A } ∪ ( A \ F ) are basic neighborhoods for A ,with F ranging over all finite subsets of ω .It follows easily from the definition that Ψ( A ) is a separable, locally com-pact, zero dimensional, scattered, first countable Moore space (see [20] and[26]). Despite their simplicity, almost disjoint families and their Ψ -spacesare a very central tool in set theoretic topology, since many important prob-lems have an equivalent reformulation in the realm of Ψ -spaces. In partic-ual, every separable hereditarily locally compact space is homeomorphic toa Ψ -space [15]. Mathematics Subject Classification.
Key words and phrases.
Almost-normal MAD family, almost disjoint family, normal,almost-normal. partly-normal, quasi-normal, strongly ℵ -separated.The author gratefully acknowledges support from CONACyT scholarship 742627. Mroówka-Isbell spaces provide a wide and numerous source of examplesand counterexamples in many areas of topology. Many examples of the useof AD families and their Ψ -spaces can be found in [14]. Normality is noexception. A MAD family is never normal, AD families of size c are notnormal by Jones’ lemma, since A is a discrete subspace of size continuum ofa separable space. One of the first examples of an AD family with specialcombinatorial properties, was the construction of a Luzin family [17]. AnAD family A is a Luzin family, if it can be enumerated as A = { A α : α <ω } in such a way that { β < α : A α ∩ A β ⊆ n } is finite for every α < ω and every n ∈ ω . The key property of Luzin families is that if B , C ⊆ A aretwo uncountable subfamilies, they can not be separated, in consequence,Luzin families are not normal. This suggest that normality is easily notfulfilled for an AD family. One of the first applications of AD familiesto problems related to normality, was the equivalence of the existence ofa normal, separable, non-metrizable Moore space and the existence of anuncountable AD family which is not normal. The later was proved to beindependent of
ZFC [26].In [25], weak normality properties on Ψ -spaces were considered. Recallthat a space X is normal if every two disjoint closed sets C, D ⊆ X canbe separated by two disjoint open sets U, V ⊆ X (that is C ⊆ U , D ⊆ V and U ∩ V = ∅ ). A subset C ⊆ X of a topological space is regularclosed if C = int ( C ) . Thus, the definition of normality becomes weakerif we require one or both of the closed sets to be regular closed or a finiteintersection of regular closed sets (which is called π -closed). Ranging overthese possibilities, several weaknesses of normality arise, and so do someimplications between them. We summarize these implications in the nextdiagram without defining all the concepts involved simply to organize themand have a visual support. We will define each term that we will focus onwhen necessary.( ∗ ) normal = ⇒ almost − normal = ⇒ quasi − normal = ⇒ partly − normal = ⇒ mildly − normal. Counterexamples of some of these implications were provided: A mildly-normal which is not partly-normal and a quasi-normal which is not almost-normal AD families were constructed, whilst counterexamples of the re-maining two implications were left open (Question 4.3 in[25]). In particular,the existence of an almost-normal MAD family, was left open. In Section
ADNESS AND (WEAK) NORMALITY 3
2, we provide an example of an almost-normal MAD family under CH . InSection 3 we show that under PFA , no MAD family can be almost normal,proving that the existence of an almost-normal MAD family is independentof the axioms of
ZFC . In Section 4, we built a partly-normal AD familywhich is not quasi-normal, hence, completing all the counterexamples in ( ∗ ) , at least, consistently. Finally, in Section 5, we will built a strongly ℵ -separated AD family which is not almost-normal under CH , answeringa question from Oliveira-Rodrigues and Santos-Ronchim [6].We will say that an AD family A satisfies a topological property P , ifand only if Ψ( A ) does. Given a set X and a cardinal κ , we denote by [ X ] κ and [ X ] <κ the set of subsets of X of size κ and < κ , respectively. Also, [ X ] ≤ κ = [ X ] κ ∪ [ X ] <κ . For each X ⊆ ω , X will denote X and X willdenote ω \ X . For two infinite subsets A, B of ω , we will say that A meets B if A ∩ B is infinite. We follow [10] for topological notation and [16] for settheoretic notation. Each undefined weakening of normality can be found in[25]. 2. An almost-normal MAD family
As we mentioned above, in [25], several counterexamples for the reverseimplications in ( ∗ ) were given, however, some questions were left open,among them the following two: • Is there an almost-normal not normal AD family? • Is there an almost-normal MAD family?A space X is almost-normal ([24]) if each pair of closed sets C, D ⊆ X ,where one of them is regular closed, can be separated.Of course, a positive answer for the second question provide a negativeanswer for the first one. In [6], a negative answer for the first question wasgiven. For a subset X ⊆ ω , the AD family A X ⊆ P (2 <ω ) is defined asthe family of all sets of the form { f ↾ n : n ∈ ω } with f ∈ X . The resultin [6] is obtained by defining a special class of subsets of ω , called almost Q -sets, such that A X is the desired family whenever X is an almost Q -setand then forcing the set X . This result cannot be improved to get MADsince AD families of the form A X are never MAD.In this section we will prove that under CH there is an almost normalMAD family, consistently answering the second question above and improv-ing a result in [1], where the authors prove that there is a quasi-normal MAD CÉSAR CORRAL family under the same assumption (this result was previously improved in[25], adding the property that the family is Luzin).
Definition 2.1.
Let A be an almost disjoint family. A set D ∈ [ ω ] ω is a partitioner for A , if for every A ∈ A either, A ⊆ ∗ D or A ∩ D is finite.We will say that two disjoint subfamilies B , C ⊆ A can be separated , if thereis a partinioner D for A , such that B ⊆ ∗ D for every B ∈ B and | C ∩ D | < ω for every C ∈ C . In this case, we will say that D is a partitioner for B and C . Notice that if D is a partitioner, ω \ D is a partitioner as well, wherethe properties of “almost contained” and “is almost disjoint” have beenexchanged. Thus, we can always decide which part of our family is almostcontained in the partinioner.It is known that an AD family A is normal, if and only if for each B ⊆ A , B and A\B can be separated [26]. The respective result for almost normalityalso holds. We will need the following easy observation.
Fact . Let A be an AD family. For every regular closed set K ⊆ Ψ( A ) , K = C ∪ { A ∈ A : | A ∩ C | = ω } with C = K ∩ ω . Proposition 2.3.
An AD family A is almost-normal iff for every C ∈ [ ω ] ω ,there exists a partitioner for B = { A ∈ A : | A ∩ C | = ω } and A \ B .Proof.
Assume A is almost-normal and let C ⊆ ω . Let B = { A ∈ A : | A ∩ C | = ω } . Then K = B ∪ C is regular closed and A \ B is closed in Ψ( A ) . Since A is almost-normal, we can find open disjoint subsets U, V ⊆ Ψ( A ) such that K ⊆ U and A \ B ⊆ V . Define D = U ∩ ω and let B ∈ B . Since U containsa basic neighborhood of B , it follows that B ⊆ ∗ D . On the other hand,if A ∈ A \ B , there exists a basic neighborhood of A contained in V , thus A ⊆ ∗ V ∩ ω and therefore | A ∩ D | < ω .Now suppose that each pair B , A \ B as in the proposition can be sepa-rated. Let
F, K ⊆ Ψ( A ) be two disjoint closed sets with K regular closed.There exist C ⊆ ω such that K = C ∪ B with B = { A ∈ A : | A ∩ C | = ω } .Let D be a partitioner for B and A \ B (where the elements of B arethose which are almost contained in D ) and let E = F ∩ ω . Define U = ( B ∪ C ∪ D ) \ E . Claim:
U is clopen.Given that ω is discrete, we only care about the points in A . Let A ∈ A \ B . ADNESS AND (WEAK) NORMALITY 5
Since
A / ∈ B and | A ∩ D | < ω , it follows that { A } ∪ ( A \ ( C ∪ D )) is a basicneighborhood of A disjoint from U . Then U is closed. If B ∈ B , | B ∩ E | < ω (otherwise B ∈ F ) and B ⊆ ∗ D . Then { B } ∪ ( D \ E ) ⊆ U contains a basicneighborhood of B showing that U is open.Finally note that U is a clopen subset disjoint from F and K ⊆ U . Thus A is almost normal. (cid:3) A very related notion on AD families called weakly separation was con-sidered in [5] and [7]. Given B , C ⊆ A , we say that D ∈ [ ω ] ω weaklyseparates B and C , if D meets B for every B ∈ B and D ∩ C is finite forevery C ∈ C . An AD family is weakly separated if for any two disjointsubfamilies B , C ⊆ A , there is a set D ∈ [ ω ] ω that weakly separates B and C . It follows easily that an AD family is normal iff it is almost normal andweakly separated.Before the next definition, let us pointing out that for every finite familyof partinioners { D i : i < n } of an AD family A , each boolean combination T i Let A be an almost disjoint family and D ⊆ [ ω ] ω bea family of partitioners for A . We will say that D is a nice family ofpartitioners , if for every { D i : i < n } ∈ [ D ] <ω and every f : n → |{ A ∈ A : A ⊆ ∗ \ i Let A be a countable AD family, C ⊆ ω and let D ⊂ [ ω ] ω be acountable nice family of partitioners for A . Then, there exists a partinionerfor B = { A ∈ A : | A ∩ C | = ω } and A \ B , such that D ∪ { D } is a nicefamily of partitioners for A .Proof. Enumerate D as { D m : m ∈ ω } and let F n ( ω, be the set of allfinite partial functions s ; ω → . For each s ∈ F n ( ω, define D s = \ i ∈ dom ( s ) D s ( i ) i . and also define A s = { A ∈ A : A ⊆ ∗ D s } with a partition of A s into twopieces A + s and A − s as follows: A + s = { A ∈ A s : | A ∩ C | = ω } and CÉSAR CORRAL A − s = { A ∈ A s : | A ∩ C | < ω } . We can enumerate F n ( ω, , B = { A ∈ A : | A ∩ C | = ω } and A \ B as { s n : n ∈ ω } , { B n : n ∈ ω } and { B ′ n : n ∈ ω } respectively.For each n ∈ ω define X + n , X − n ⊆ ω according to the following cases: Case 1: Either, both A + s n and A − s n are infinite or both A + s n and A − s n arefinite. In this case simply define X + n = ∅ = X − n . Case 2: A + s n is infinite and A − s n is finite. In this case, define X − n = S A − s n and X + n = D s n \ S A − s n . Case 3: A + s n is finite and A − s n is infinite. In this case, define X + n = S A + s n and X − n = D s n \ S A + s n .Notice that the family { D s n \ ∪F : n ∈ ω ∧ F ∈ [ A ] <ω } is also a nicefamily of partitioners. Besides, each finite union of elements of A is clearlya partitioner and it is easy to see that the family( ∗ ) { D s n \ ∪F : n ∈ ω ∧ F ∈ [ A ] <ω } S {∪F : F ∈ [ A ] <ω } is still a nice family of partitioners which contains each X + n and X − n . Wepoint out two properties that follow directly from the definitions of thepartitioners X + n and X − n which will be useful later:(a) B ∩ X − n is finite for every B ∈ B and n ∈ ω .(b) B ′ ∩ X + n is finite for every B ′ ∈ A \ B and n ∈ ω. Define D = [ n ∈ ω (cid:0) B n ∪ X + n (cid:1) \ [ i D∪{ D } is a nice family of partition-ers. Given s ∈ F n ( ω, and i ∈ such that |{ A ∈ A : A ⊆ ∗ D s ∩ D i }| < ω ,if A s is already finite, we have that D s ∩ D i = ∗ { A ∈ A : A ⊆ ∗ D s ∩ D i } since D is a partitioner. Moreover, D is a partitioner for B and A \ B and this implies that either { A ∈ A : A ⊆ ∗ D s ∩ D i } = A + s if i = 0 or { A ∈ A : A ⊆ ∗ D s ∩ D i } = A − s if i = 1 . ADNESS AND (WEAK) NORMALITY 7 Then, we can assume that A s is infinite, and either, A + s is finite if i = 0 or A − s is finite if i = 1 . Let n ∈ ω such that s = s n . Assume i = 0 , hencewe defined X + n and X − n according to case 3 and X + n = S A + s n is a finiteunion of elements of B . Thus X + n is almost disjoint from X − m and B ′ m forevery m ∈ ω . It follows that [ (cid:0) A + s (cid:1) ⊆ ∗ ( B n ∪ X + n ) \ [ i Recall that given ad AD family A , the ideal generated by A∪ ω is denotedby I ( A ) , and I + ( A ) = P ( ω ) \ I ( A ) consists of those subsets of ω that cannot be coveret by a finite union of elements of A and a finite set. Lemma 2.6. Let A be a countable AD family, X ∈ [ ω ] ω almost disjointwith A and let D = { D n : n ∈ ω } be a nice family of partitioners for A . Assume that for each n ∈ ω there exists C n ∈ [ ω ] ω such that D n is apartitioner for B = { A ∈ A : | A ∩ C n | = ω } and A \ B . Then there exists A ∈ [ ω ] ω such that | A ∩ X | = ω , A ∪ { A } is AD, each D n is a partitionerfor A ∪ { A } and A ⊆ ∗ D n iff | A ∩ C n | = ω for every n ∈ ω .Proof. For every s ∈ <ω let D s = \ i< | s | D s ( i ) i . We can recursively construct f ∈ ω such that | D f ↾ n ∩ X | = ω for every n ∈ ω . We have that { A ∈ A : A ⊆ ∗ D f ↾ n } is infinite for every n ∈ ω .Otherwise, since D is a nice family of partitioners, X ∩ D f ↾ n = ∗ X ∩ S i There is an almost-normal MAD family.Proof. Enumerate [ ω ] ω = { X α : α < ω } with X n = ω for every n ∈ ω . Wewill recursively construct a MAD family A = { A α : α < ω } and a familyof partitioners D = { D α : α < ω } such that if A α = { A β : β < α } and D α = { D β : β < α } then: ADNESS AND (WEAK) NORMALITY 9 (1) A α is AD.(2) If X α is AD with A α , then | A α ∩ X α | = ω .(3) D α is a partitioner for B = { A ∈ A α : | A ∩ X α | = ω } and A α \ B .(4) Either, A α ⊆ ∗ D β or | A α ∩ D β | < ω for every β ≤ α .(5) A α ⊆ ∗ D β iff | A α ∩ X β | = ω .(6) { D β : β ≤ α } is a nice family of partitioners for A α .Let { A n : n ∈ ω } ⊆ [ ω ] ω be a partition of ω into infinite pieces and define D n = ω for every n ∈ ω . This family clearly satisfies the above conditions.Assume we have constructed A α and D α as above. We can apply lemma2.5 to the triple ( A α , D α , X α ) to obtain D α .For the construction of A α , let X = X α if X α is AD with A α , otherwiselet X be any infinite subset of ω almost disjoint with A α . By lemma 2.6applied to A α , { D β : β ≤ α } (with their respective C β = X β ) and X , wecan find A α as required.It is clear from point (2) that A is MAD. Also, if C ∈ [ ω ] ω , there exists α < ω such that C = X α . Hence D α is a partitioner for A α as in proposi-tion 2.3. Moreover, point (4) and point (5) ensure that D α is preserved for β ≥ α . Thus D α is a partitioner for A as required in proposition 2.3 with C = X α . We can conclude that A is almost-normal. (cid:3) It was mentioned before that in [25], a quasi-normal Luzin MAD familywas constructed, then it is natural to ask the following question: Question 1. (CH) Is there a Luzin MAD family which is almost-normal? There may be no almost-normal MAD families There are many reasons for which one could think that it is not possibleto obtain Theorem 2.7 without assuming CH . The most obvious reason isthat after ω -many steps, we have already constructed a Luzin family A .Then, we can not get a partitioner as in Proposition 2.3 for a given set C ⊆ [ ω ] ω , whenever it meets uncountable many elements of A and it isalmost disjoint from uncountable many elements of A as well. Indeed, thissituation could be unavoidable as we will see below.Recall that the Proper Forcing Axiom ( PFA ) is the assertion that forevery proper forcing P and every family D of ω -many open dense subsetsof P there exists a D -generic filter for P . If we replace “proper” by “ccc”and “ ω ” by “ < c ” we get the definition of Martin’s Axiom ( MA ). It is well known that PFA implies MA + c = ω . Under PFA we can not avoid theexistence of Luzin subfamilies due to the following result. Theorem 3.1. [8] Each MAD family contains a Luzin subfamily. The existence of a set C as above, that “wants to separate” the Luzinsubfamily is also insured by the next theorem. Theorem 3.2. [18] ( MA ) For every pair of families A , B ⊆ [ ω ] ω of size < c such that for every K ∈ [ A ] <ω and B ∈ B , B \ S K is infinite, thereexists C ∈ [ ω ] ω such that C ∩ A is finite for every A ∈ A and C meets B for every B ∈ B . Now it follows easily that there are no almost normal MAD families inthe presence of PFA . Corollary 3.3. PFA implies that there are no almost normal MAD families.Proof. Let A be a MAD family and let A ′ ⊆ A be a Luzin subfamily.We can split A ′ into two uncountable disjoint subfamilies B , C ⊆ A ′ . ByTheorem 3.2, we can find a set X ⊆ ω that weakly separates B and C ,that is, X ∩ C is finite for every C ∈ C and X meets B for every B ∈ B .Thus, K = { A ∈ A : | X ∩ A | = ω } and A \ K cannot be separated since B ⊆ K , C ⊆ A \ K and B , C are uncountable subfamilies of a Luzin family.Therefore A is not almost-normal. (cid:3) In [9], it is shown that it is consistent with MA that there is a MADfamily which contains no Luzin subfamilies, it could be possible that theonly thing that blocks the existence of almost normal MAD families is theexistence of Luzin subfamilies, so we ask the following: Question 2. Is it consistent with MA that there are almost-normal MADfamilies? Partly-normal not quasi-normal AD families In this section, we will consider the next question stated in [25] and willprovide a positive answer. • Is there a partly-normal not quasi-normal AD family?We will say that a space X is partly-normal if any pair of disjoint closedsets A, B ⊆ X , where A is regular closed and B is π -closed (a finite intersec-tion of regular closed sets), can be separated [1]. A space X is quasi-normalif any two disjoint π -closed sets can be separated [27]. ADNESS AND (WEAK) NORMALITY 11 Most of the examples in [25] were constructed using AD families of truecardinality c . For an AD family A and W ⊆ ω , we will denote by A ↾ W theset of A ∈ A such that A meets W . An AD family is of true cardinality c , iffor every W ⊆ ω , either, A ↾ W is finite or has size c . It is well known thatthe existence of (M)AD families of true cardinality c is equivalent to the ex-istence of completely separable (M)AD families. An AD family is completelyseparable, if for any X ⊆ ω such that |A ↾ X | = ω , there is an A ∈ A suchthat A ⊆ X [13]. While completely separable AD families do exist in ZFC ,the existence of completely separable MAD families in ZFC , asked first byErdös and Shelah [11], is one of the more interesting and central questionsconcerning almost disjoint families. Completely separable MAD familieswere constructed under several assumptions, a = c , b = d , d ≤ a and s = ω (see [2, 3, 23]). Then Shelah showed that they exists under s < a , and alsounder s ≥ a assuming covering type assumptions [22], in particular, theyexists under c < ℵ ω (see [14]). Later work of Mildenberger, Raghavan andSteprans [19], showed that the covering type assumption is not needed inthe case s = a , and then completely separable MAD families exists under s ≤ a . The existence of AD families of true cardinality c is particularlyuseful for constructions of AD families with strong combinatorial proper-ties, since they usually need recursive constructions of length continuum(see, for example, [21]). We will use an AD family of true cardinality c toconstruct a partly-normal not quasi-normal AD family. First, observe thatwe can always assume that an infinite AD family A , contains an infinitepartition of ω into infinite pieces, since we can take { A n : n ∈ ω } ⊆ A andsubstitute A n by A ′ n = ( A n ∪ { n } ) \ S i There is a partly-normal AD family which is not quasi-normal. Proof. Partition ω in four infinite sets W , W , V and V . Further, partitionboth, W and W into infinitely many infinite sets, that is, W = [ n ∈ ω P n and W = [ n ∈ ω Q n . Let A W , A W , A V and A V be AD families of true cardinality c in each ofthe four sets W , W , V and V . We can assume that { P n : n ∈ ω } ⊆ A W and { Q n : n ∈ ω } ⊆ A W . We will recursively construct our family puttingtogether some elements of these AD families of true cardinality c . For easeof notation, let E = A W ∪ A W ∪ A V ∪ A V .For every n ∈ ω , define A n = P n ∪ Q n . Let { f α : α < c } be a dominatingfamily of functions in ω ω . List all pairs ( C, D ) where D ∈ [[ ω ] ω ] <ω and C ∈ [ ω ] ω as { ( C α , D α ) : α < c } . We will built finite sets F α , F α ∈ [ E ] <ω recursively, so that each F α will contain exactly one element of each family A W , A V and A V , and each F α will intersects exactly two of the families A W , A W , A V , A V . In particular, no element of the form A n or ∪F iα willmeet the four sets W , W , V and V Assume we have constructed F β and F β for β < α . For F α , considerthe set X = S n ∈ ω ( P n \ f α ( n )) . Since X meets infinitely many elements of A W , it follows that X meets c -many elements of A W . Choose A ∈ A W \ [ β<α ( F β ∪ F β ) ∪ { P n : n ∈ ω }} such that A meets X . Also pick B ∈ A V \ S β<α ( F β ∪ F β ) and C ∈A V \ S β<α ( F β ∪ F β ) arbitrary and define F α = { A, B, C } .For F α consider the pair ( C α , D α ) and let D α = { D jα : j < n } . Define C = { A ∈ E : A meets C α } and B = { A ∈ E : ∀ j < n ( A meets D jα ) } . If either B or C are finite, simply define F α = ∅ . Otherwise, we havesome cases. Since B and C are infinite, there are Y, Z ∈ { W , W , V , V } such that B ↾ Y and C ↾ Z are infinite. Notice that since B ↾ Y is infinite,in particular, there are infinitely many elements of A Y which meet D jα for ADNESS AND (WEAK) NORMALITY 13 every j < n . Thus, there are c -many of these elements. Pick, for every j < n , an element B j ∈ A Y such that B j meets D jα and B j ∈ A Y \ F α ∪ [ β<α (cid:0) F β ∪ F β (cid:1) . Notice that S j ZFC alone. Hence, it is natural to ask if such a space canalso exists in ZFC . We already know that no counterexample can be MADby Corollary 3.3. In [6], the cardinal an is defined as the least cardinalityof an almost-normal not normal AD family, and it is noted that ap ≤ an ,whenever an is well defined, i.e., whenever there is an almost-normal notnormal AD family. Here ap is defined as the least cardinality of an ADfamily which is not weakly separated [5]. Since it is consistent that ap = c ,the unresolved portion of the question of whether there are almost-normalMAD families, can be stated as follows: Question 3. Does there exist (in ZFC ) an almost-normal AD family whichis not normal? (an almost-normal AD family of size c ?) On the other hand, it was proved in [6], that there is an almost-normalnot normal AD family of size ω < c . Hence, even though the first part ofthe above question might have a “yes” answer, the proof may go by cases(in some models all such families have size < c while in others, all suchfamilies have size c ) and then the second part of the question could have a“no” answer. 5. On strongly ℵ -separated AD families The concept of strongly ℵ -separated AD families was introduced in [25]by the authors. An AD family A is strongly ℵ -separated, if for every twodisjoint countable subfamilies B , C ⊆ A , there is a partitioner D for A , suchthat B ⊆ ∗ D for every B ∈ B and C ∩ D is finite for every C ∈ C . There, itwas shown that almost-normal AD families are strongly ℵ -separated andthat there is a strongly ℵ -separated MAD family under CH .The requirement of one of the subfamilies being countable was modifiedin [6] in order to define a stronger concept: An AD family is strongly ( ℵ , < c ) -separated, if for every two disjoint subfamilies B , C ⊆ A , where B is countable and |C| < c , there is a partitioner D for A such that B ⊆ ∗ D for every B ∈ B and C ∩ D is finite for every C ∈ C . The relation of these ADNESS AND (WEAK) NORMALITY 15 two concepts and almost-normality was studied in [25] and [6], however,the next question remained unanswered [6]: • Does CH imply that strongly ℵ -separated AD families are almost-normal?We will answer this question in the negative. For this purpose, recall thatfor S, X ∈ [ ω ] ω , we say that S splits X , if both S ∩ X and X \ S areinfinite. A family S ⊆ P ( ω ) is a splitting family, if for every X ∈ [ ω ] ω ,there exists S ∈ S such that S splits X , and s is the least size of a splittingfamily. The splitting number s is a cardinal invariant of the continuum,hence ω < s ≤ c . In particular, for every countable subfamily H ⊆ [ ω ] ω ,there exists X ∈ [ ω ] ω which is not split by any element of H , i.e., either, X ∩ H is finite or X ⊆ ∗ H for every H ∈ H (see [4]). Theorem 5.1. (CH) There is a strongly ℵ -separated AD family which isnot almost-normal.Proof. Let { d α : α < ω } be a dominating family of functions and enumer-ate all pairs ( a, b ) ∈ [ ω ] ≤ ω × [ ω ] ≤ ω , such that a ∩ b = ∅ as { ( a α , b α ) : ω ≤ α < ω } . We can assume without loss of generality that a α ∪ b α ⊆ α for ev-ery α < ω . Partition ω = V ∪ W into two infinite sets and let ϕ : V → W be a bijection. Moreover, partition V = S n ∈ ω A n , into infinitely manyinfinite sets.We will recursively construct A α and D α for ω ≤ α < ω such that if A α = { A β : β < α } , then the following holds:(1) A α is an almost disjoint family.(2) | A α ∩ A n | ≤ for every ω ≤ α < ω and every n ∈ ω .(3) A α meets S n ∈ ω ( A n \ d α ( n )) .(4) D α is a partitioner for A α such that A β ⊆ ∗ D α for every β ∈ a α and A γ ∩ D α is finite for every γ ∈ b α .(5) Either, A α ⊆ ∗ D β or A α ∩ D β is finite for every β < α .Suppose we have defined A α = { A β : β < α } and D α = { D β : ω ≤ β <α } with the above properties. We shall define D α and A α .Consider the pair ( a α , b α ) . Let B = { A β : β ∈ a α } and C = { A β : β < α ∧ β / ∈ a α } . Since α is countable we can enumerate both sets as B = { B n : n ∈ ω } and C = { C n : n ∈ ω } . Define D α = [ n ∈ ω B n \ [ i Similarly define D α = D α ∪ ϕ [ D α ] for ω ≤ α < ω . Since ϕ is a bi-jection between two disjoint sets V and W , if A = { A α : α < ω } and A α = { A β : β < α } , properties (1)-(5) also hold replacing A τ by A τ and D τ by D τ . Claim: A is strongly ℵ -separated.Let A , A ∈ [ A ] ≤ ω be disjoint subfamilies. Define a = { δ : A δ ∈ A } and b = { β : A β ∈ A } . There exists α < ω such that ( a, b ) = ( a α , b α ) . Thus D α is a partitioner for A α and was chosen so that D α separates A and A by property (4). Moreover, since A γ is either, almost disjoint or almostcontained in D α for every γ ≥ α , D α is indeed, a partitioner for A whichseparates A and A . Claim: A is not almost-normal.For every n ∈ ω , A n ⊆ V which is disjoint from W . In addition, A α ∩ W = ϕ [ A α ] is an infinite set. It suffices now to prove that { A n : n ∈ ω } and { A α : ω ≤ α < ω } can not be separated. Let D be such that A n ⊆ ∗ D forevery n ∈ ω . There exists f ∈ ω ω such that A n \ f ( n ) ⊆ D . Choose α < ω such that d α > ∗ f . Then A α meets S n ∈ ω (cid:0) A n \ d α ( n ) (cid:1) ⊆ ∗ D . Hence, D isnot a partitioner for { A n : n ∈ ω } and { A α : ω ≤ α < ω } . (cid:3) We have answered Question 7.3 from [6] in the negative, in particu-lar, under CH , there is a strogly ( ℵ , < c ) -separated AD family which isnot almost-normal. This result also follows from PFA , actually, somethingstronger is true. Let P be a given property. We will say that MAD familieswith property P exists generically if all AD families of size less than c canbe extended to a MAD family with property P . Generic existence of MADfamilies was introduced in [12] and it was proved in [6] that under b = c = s ,completely separable MAD families which are strongly ( ℵ , < c ) -separatedexist generically. Since the hypothesis hold under PFA and we have provedthat PFA implies no MAD family is almost-normal, we get the following: Corollary 5.2. 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