Almost-normality of Isbell-Mrówka spaces
aa r X i v : . [ m a t h . GN ] D ec Almost-normality of Isbell-Mr´owka spaces
Vinicius de Oliveira Rodrigues [email protected]
Victor dos Santos Ronchim [email protected]
Institute of Mathematics and StatisticsUniversity of S˜ao PauloDecember 4, 2020
Abstract
We explore almost-normality in Isbell-Mr´owka spaces and some related concepts. We use forcingto provide an example of an almost-normal not normal almost disjoint family, explore the concept ofsemi-normality in Isbell-Mr´owka spaces, define the concept of strongly ( ℵ , < c )-separated almost disjointfamilies and prove the generic existence of completely separable strongly ( ℵ , < c )-separated almostdisjoint families assuming s = c and b = c . We also provide an example of a Tychonoff almost-normal notnormal pseudocompact space which is not countably compact, answering a question from P. Szeptyckiand S. Garcia-Balan. Primary 54D15, 54D80, 54G20; Secondary 54A35
Keywords:
Isbell-Mr´owka spaces, almost disjoint families, almost-normal, semi-normal
Isbell-Mr´owka spaces are topological spaces associated to almost disjoint families. This class of spacesis used to provide examples and counter examples to numerous questions in General Topology, includingquestions that are initially not related to them. The topological properties of such spaces often depend onthe combinatorial properties of the associated almost disjoint family. We cite the surveys [12] and [10] asreferences for this field of study.If N is a countable infinite set such that N ∩ [ N ] ω = ∅ , an almost disjoint family (over N ) is an infinitecollection A of infinite subsets of N such that for all distinct a, b ∈ A , a ∩ b is finite. A MAD family (maximalalmost disjoint family) is an almost disjoint family which is not properly contained in any other almostdisjoint family. By Zorn’s Lemma, every almost disjoint family can be extended to a MAD family and it iswell known that there exist almost disjoint families of size c [12]. The least cardinality of a MAD family iscalled a , and it is well known that a ≥ ω .Given an almost disjoint family A over N , the Isbell-Mr´owka space associated to A , also known as Ψ-spaceof A , and denoted by Ψ( A ) is the set N ∪ A with the topology generated by {{ n } : n ∈ N } ∪ {{{ a } ∪ ( a \ F ) : a ∈ A , F ∈ [ N ] <ω } . It is immediate that A is a Hausdorff, locally compact (therefore Tychonoff) notcountably compact zero dimensional separable topological space.In general, Ψ( A ) does not need to be normal (e.g., if |A| = c , |A| is a closed discrete subspace of size c of the separable space Ψ( A ), so it is not normal by Jones’s Lemma) but it may be normal, since Ψ( A ) is1etrizable iff A is countable. The existence of a uncountable normal Isbell-Mr´owka space is independent ofthe Axioms of ZFC, and is equivalent to the existence of a normal separable non-metrizable Moore space [21][11], [20].In this paper we study weakenings of normality on Isbell-Mr´owka spaces. We say a topological space isnormal iff every two closed disjoint subsets can be separated by open disjoint subsets. Various weakeningsof normality have been proposed and studied, such as quasi-normality [22], almost-normality [18], mildly-normality [17] and semi-normality [18]. In this paper we will focus on the study of almost-normality andsemi-normality on Isbell-Mr´owka spaces. Recent results regarding the study of some weakenings of normalityand Isbell-Mr´owka spaces include [8] and [1].Given a topological space X , a regular closed set of X is a closed set F such that F = cl(int( F )), and anopen set U is said to be regular open iff U = int(cl( U )). We say a topological space X is almost-normal iffwhenever F is a closed set and K is a regular closed set disjoint from F , there exist disjoint open sets U, V such that F ⊆ U , K ⊆ V . We say that X is semi-normal iff for every closed set F and every open set U containing F there exists a regular open set V such that F ⊆ V ⊆ U . The following proposition is from [18]and can be easily verified: Proposition 1.1 ([19]) . A topological space is normal iff it is almost-normal and semi-normal.
In this paper, we say that an almost disjoint family A is [semi, almost]-normal iff Ψ( A ) is [semi, almost]-normal.In [8], P. Szeptycki, S. Garcia-Balan provided, among several other examples, an example in ZFC, ofan almost disjoint family of true cardinality c which is not almost-normal but satisfies weaker separationproperties, such as quasi-normality. They also showed that this example could be made MAD in case thereexists a MAD family of true cardinality c . They asked the following question (Question 4.2 of [8]): Question 1.2.
Is there an almost-normal not normal almost disjoint family?
Recall Ψ( A ) is pseudocompact iff A is MAD [12]. Thus, if A is MAD, it cannot be normal since, as aconsequence of Tietze’s theorem, pseudocompact normal spaces are countably compact. The authors of [8]also asked the following (Question 4.3 and 4.4 of [8]): Question 1.3.
Is there an almost-normal MAD family?
Question 1.4.
Are almost-normal pseudocompact spaces countably compact?
In this paper, we use iterated forcing and a generalization of the notion of Q -set to provide a partial answerto Question 1.2 (consistently, yes) and answer negatively Question 1.4 in ZFC by providing a subspace of βω which serves as a counter example. Question 1.3 remains open.In [8], they also define the concept of strongly ℵ -separated almost disjoint family, which is related toalmost-normal almost disjoint families, as follows: an almost disjoint family A (over N ) is said to be strongly ℵ -separated iff for every two countable disjoint subsets B , C of A there exists X ⊆ N such that:(1) For every a ∈ A , a ⊆ ∗ X or A ∩ X = ∗ ∅ ;(2) for every a ∈ B , a ⊆ ∗ X ;(3) for every a ∈ C , a ∩ X = ∗ ∅ . 2hey showed that every almost-normal almost disjoint family is strongly ℵ -separated and showed that,under CH , there exist MAD families which are strongly ℵ -separated. In the last section of this paper wedefine a stronger concept we call strongly ( ℵ , < c )-separated almost disjoint family and prove that b = c plus s = c implies the generic existence of ( ℵ , < c )-separated completely separable MAD families.Regarding notation, we define some of the set theoretical topological and cardinal characteristcs conceptsas we need them, for undefined concepts we refer (resp.) to [16], [5] and [2].It is worth mentioning a stronger version of almost-normality, called π -normality, was proposed [14]: asubset of a topological space X is said to be π -closed if it is a finite intersection of regular closed sets, and X is said to be π -normal iff whenever F ⊆ X is π -closed, K ⊆ X is closed and F ∩ K = ∅ , there exists disjointopen sets separating F from K . However, in [8] it was proven that almost-normality and π -normality areequivalent. In this section we give, in ZFC, a negative answer for Question 1.4 by constructing a suitable subspace of βω .As noted by Kalantan in [13], extremely disconnected spaces are almost-normal since every regular closedset is a clopen set, so it can be separated from any set disjoint from it. This fact will be useful to obtain ourcounterexample.The following lemma is well known and can be easily proved by the reader. We refer [5]. Lemma 2.1. If X is extremally disconnected and D ⊆ X is a dense subset, then D is also extremallydisconnected. The following Lemma is also known. We prove it for the sake of completeness.
Lemma 2.2. If D ⊆ X is dense and every sequence in D has an accumulation point in X , then X ispseudocompact.Proof. Suppose, by contradiction, that X is not pseudocompact. There exists an unbounded continuousfunction h : X → [0 , ∞ ) ⊆ R . For each n ∈ ω , let d ( n ) ∈ D ∩ h − [( n, ∞ )]. Then d : ω → D has no limitpoint x , for if it had, we would have x ∈ cl( { d ( n ) : n ≥ m } ) for every m ∈ ω , thus, by continuity, f ( x ) ≥ m for every m ∈ ω , a contradiction.Now we present our example. For the construction, we identify βω with the space of ultrafilters over ω ,where U n is the principal ultrafilter generated by { n } for each n ∈ ω (and n is identified with U n ). We write N = {U n : n ∈ ω } . ω ∗ ⊆ βω is the set of free ultrafilters over ω . Given A ⊆ ω , ˆ A is the basic clopen set { p ∈ βω : A ∈ p } . Example 2.3.
There exists a Tychonoff extremely disconnected (thus, almost normal) pseudocompact spacewhich is not countably compact.
Construction.
Let ( P n : n ∈ ω ) be a partition of ω into pairwise disjoint infinite sets. For each n ∈ ω , let p n be a free ultrafilter such that P n ∈ p n . Let F = { p n : n ∈ ω } . F is infinite and discrete since given n , { p n } = F ∩ ˆ P n .Given A ∈ [ ω ] ω , let q A ∈ ω ∗ be defined as follows:31) If there exists n ∈ ω such that A ∈ p n , let q A = p n , for any such n (e.g. the least such n ), or(2) if for all n ∈ ω A / ∈ p n , let q A ∈ ω ∗ be any free ultrafilter such that A ∈ q A .In any case, A ∈ q A . Let X = N ∪ { q A : A ∈ [ ω ] ω } and notice that, for each n ∈ ω , q P n = p n by (1).Hence, F ⊆ X . X is a dense subspace of βω (since it contains N ) and by Lemma 2.1, X is extremely disconnected. Inparticular, X is also almost normal. X is pseudocompact: since N is dense in X , it suffices to see that every sequence f : ω → N has anaccumulation point. By passing to a subsequence, we can suppose f is either constant or injective. Constantsequences converge, so suppose f is injective. Let g : ω → ω be such that f ( n ) = U g ( n ) . Let A = ran( g ). Weclaim q A is an accumulation point of f . Given a basic nhood ˆ B ∋ q A , we know B ∩ A ∈ q A is infinite, so itfollows that g − [ A ∩ B ] ⊆ { n ∈ ω : f ( n ) ∈ ˆ B } is also infinite. Since B is arbitrary, the proof is complete. X is not countably compact: we know F is an infinite discrete subspace of X (since it is in βω ). Thus,it suffices to show that F is closed in X . We show X \ F is open in X . Clearly, every point of N is in theinterior of X \ F since N is open. If A ∈ [ ω ] ω and q A / ∈ F , then (2) holds, so q A ∈ ˆ A and F ∩ ˆ A = ∅ , that is, q A ∈ X ∩ ˆ A ⊆ X \ F .The space constructed in Example 2.3 answers negatively the Question 4.4 from [8]. It is worth mentioningthat Isbell-Mr´owka spaces are never extremally disconnected, so a similar strategy cannot be employed whentrying to address Questions 4.2 and 4.3. Ψ( A ) In this section we start to explore the notion of almost-normality in the realm of Isbell-Mr´owka spaces.In particular, we aim to provide some characterizations for “ A is almost-normal”. In order to do so, we willuse the well known notion of a partitioner of an almost disjoint family. As in the introduction, N denotes aninfinite countable set for which N ∩ [ N ] ω = ∅ . Definition 3.1.
Let A be an almost disjoint family (over N ). We say that X ⊆ N is a partitioner for A iffor each a ∈ A , a ⊆ ∗ X or a ∩ X = ∗ ∅ .We say that a partitioner X for A is a partitioner for B , C ⊆ A if b ⊆ ∗ X and c ∩ X = ∗ ∅ for each b ∈ B and c ∈ C .The main motivation for our equivalences is the following classical result. We give [12] and [11] asreferences. Proposition 3.2.
Let A be an almost disjoint family. Then A is normal if, and only if, for all B ⊆ A , B and A \ B can be separated by disjoint open sets of ψ ( A ) . Recall there is a one to one correspondence between the clopen subsets of Ψ( A ) and the partitioners of A which can be defined as follows: for each X ⊆ N consider B X = { a ∈ A : a ⊆ ∗ X } and C X = { a ∈ A : a ∩ X = ∗ ∅} . It follows that: • B X ∪ X and C X ∪ ( N \ X ) are disjoint open subsets of Ψ ( A ); • If X is a partitioner for Ψ ( A ), then A = B X ∪· C X and Ψ ( A ) = ( B X ∪ X ) S · ( C X ∪ ( N \ X )) is unionof clopen subsets. 4hen it easily follows that: Lemma 3.3. If A is an almost disjoint family, then F : Clop(Ψ ( A )) −→ { X ⊆ N : X is a partitioner for A} ,defined by F ( W ) = W ∩ N , is a bijective function, with inverse given by F − ( X ) = B X ∪ X . The regular closed subsets of Ψ( A ) are easily characterized by the following proposition: Lemma 3.4.
Let A be an almost disjoint family. Then F ⊆ Ψ( A ) is a regular closed set iff there exists W ⊆ N such that F = cl( W ) = W ∪ { a ∈ A : | a ∩ W | = ω } .Proof. First, notice that given a subset W of N , W ⊆ int(cl( W )), therefore cl( W ) ⊆ cl(int(cl( W ))), conclud-ing that cl(int(cl( W ))) = cl( W ) since cl( W ) is closed. Also, it is easy to see that cl( W ) = W ∪ { a ∈ A : | a ∩ W | = ω } . This proves the “if” clause.To prove the “only if”, suppose F is a regular closed set. Let W = F ∩ N . It is straightforward to verifythat F = cl( W ). Lemma 3.5. If Ψ ( A ) is almost-normal, then for all B, C ⊆ A , B ∩ C = ∅ , the following holds: B and C are separated by open sets ⇐⇒ B and C are separated by clopen sets.Proof. If B and C are separated by disjoint open sets U B and U C , respectively, then F . = cl Ψ( A ) ( U B ∩ N ) isa regular closed set and A \ F is closed. Since A is almost-normal, there exists V, W disjoint open subsetsof Ψ ( A ) such that F ⊆ V and A \ F ⊆ W . One can verify that X . = V ∩ N is a partitioner for A , then byLemma 3.3, B X ∪ X and its complement are the desired clopen sets.From this lemma, it easily follows that Ψ ( A ) is normal iff every two disjoint subsets B, C ⊆ A areseparated by clopen sets. Now we are ready to characterize the almost-normality of Isbell-Mr´owka space byusing partitioners and clopen sets:
Theorem 3.6. If A is an almost disjoint family then the following are equivalent:(1) Ψ ( A ) is almost-normal;(2) For each F regular closed set, there exists a partitioner X for F ∩ A and A \ F ;(3) For each F regular closed set, there exists a clopen set C such that F ∩ A ⊆ C and A \ F ⊆ Ψ ( A ) \ C ;(4) For each F regular closed set, there exists a clopen set C such that F ⊆ C and A \ F ⊆ Ψ ( A ) \ C ;(5) Closed sets are separated from regular closed sets by clopen sets.Proof. (1) = ⇒ (3) : If F is a regular closed, there exist disjoint open sets U, V such that F ⊆ U and A \ F ⊆ V . By Lemma 3.5, F ∩ A and A \ F are separated by clopen sets.(2) ⇐⇒ (3) : This is clear by Lemma 3.3.(3) = ⇒ (4) : If F is regular closed set of Ψ ( A ), let C be a clopen set such that F ∩ A ⊆ C and A \ F ⊆ Ψ ( A ) \ C , it follows that: F ⊆ C ∪ (( F ∩ N ) \ C | {z } clopen set ) = C ∪ ( F ∩ N )It is straightforward to verify that Y . = ( F ∩ N ) \ C is a clopen set.54) = ⇒ (5) : Let F, K ⊆ Ψ( A ) be disjoint closed sets, where F is regular closed. By (4), there exists aclopen set C such that F ⊆ C and A \ F ⊆ Ψ( A ) \ C .Let C ′ = C \ ( K ∩ N ). Clearly, C ′ is a closed set containing F . K is disjoint from C ′ since K ∩ A ⊆ A \ F is disjoint from C . It is straightforward to verify that C ′ is also open.(5) = ⇒ (1) : Trivial.This characterization will be useful in the next section to provide an example of an almost disjoint familywhich is almost-normal but not normal (consistently). In this section we partially answer Question 1.2 by using iterated forcing to create a model for ZFC+CHwhich has an almost-normal almost disjoint family which is not normal. We will use the equivalence between(1) and (2) of Theorem 3.6 and a generalization of the notion of Q -set.Given X ⊆ ω , the almost disjoint family over N = 2 <ω induced by X is the family A X = { A x : x ∈ X } ,were A x = { x | n : n ∈ ω } for each x ∈ X and for F ⊆ ω , we denote ˆ F = { x | n : n ∈ ω, x ∈ F } . As in [11], wesay that an uncountable X ⊆ ω is a Q -set iff every subset of X is an F σ of X . The following folklore resultholds (a proof can be found in Proposition 2.2 of [11]): Proposition 4.1 ([11]) . Given an uncountable X ⊆ ω , Ψ ( A X ) is normal iff X is a Q -set. In what follows next we give a similar characterization for almost-normal almost disjoint families. Forthis purpose, we need the following:
Definition 4.2.
An almost Q -set in 2 ω is an uncountable subset X ⊆ ω such that for every W ⊆ <ω ,[ W ] X = { x ∈ X : ∀ m ∈ ω ∃ n ≥ m ( x | n ∈ W ) } (which is { x ∈ X : | A x ∩ W | = ω } ) is an F σ in X .We note that the definition of [ W ] X is absolute for transitive models of ZFC.The next proof can be extrated from the proof of Proposition 4.1 of [11]. We write it here for the sake ofcompleteness. Lemma 4.3.
Given an uncountable X ⊆ ω and disjoint subsets B , C ⊆ A X such that e B = { x ∈ X : A x ∈ B} and e C = { x ∈ X : A x ∈ C} are F σ sets in X , then there exists a partitioner J ⊆ <ω which separates them.Proof. Write e B = S n ∈ ω F n and e C = S n ∈ ω G n , where F n and G n are closed in X . We proceed with a standardshoelace argument, defining J = b F , K = b G \ b F , J n = b F n \ (cid:0) S i Given an uncountable X ⊆ ω , A X is almost-normal iff X is an almost Q -set.Proof. (= ⇒ ) For W ⊆ <ω fixed, consider the regular closed set F = cl A X ( W ). Since Ψ ( A X ) is almost-normal, by Theorem 3.6, there exists a partitioner J ⊆ <ω for A X ∩ F and = A X \ F . It follows that:[ W ] X = { x ∈ X : | A x ∩ W | = ω } = { x ∈ X : A x ∈ F } = { x ∈ X : A x ⊆ ∗ J } = [ m ∈ ω \ n ≥ m { x ∈ X : x | n ∈ J } | {z } closed in X . Hence, X is an almost Q -set in 2 ω . 6 ⇐ =) By Theorem 3.6 it suffices to show that for every regular closed set F , there exists a partitioner for B = A X ∩ F and C = A X \ F .If F is a regular closed set in A X , there exists W ⊆ <ω such that F = cl A X ( W ). Notice that [ W ] X is a G δ since: [ W ] X = \ m ∈ ω [ n ≥ m { x ∈ X : x | n ∈ W. } | {z } open in X Since X is almost Q -set, it follows that both e B = [ W ] X and e C = X \ [ W ] X are F σ in X , so by Lemma4.3 there exists a partitioner for B and C .Before providing the forcing example, notice that if M, N are countable transitive models for ZFC and M ⊆ N , then for every X, Y ⊆ ω in M with Y ⊆ X , Y ( Y is an F σ of X ) M → ( Y is an F σ of X ) N sincecountable sets of M are countable sets of N , and since closed/open subsets of X in M are closed/open subsetsof X in N .Now we are ready for the main result of this section. Proposition 4.5. Suppose that (in the ground model) X ⊆ ω is infinite, and let κ = c . Then there exists ac.c.c. forcing notion P of size κ such that in every forcing extension by P , κ = c and X is an almost Q -set(thus, by the previous corollary, Ψ( A X ) is almost-normal in the extension).Proof. We will proceed by iterated forcing. For the forcing notation, we adopt the countable transitiveapproach, where M is a fixed ctm for ZFC. Let λ = | X | .First we study the basic step of the iteration which may be found in [6]. Given A ⊆ X in M , let P ( A, X )be the sets of all finite r ∈ [ ω × (2 <ω ∪ A )] <ω such that for all n ∈ ω , x ∈ A and s ∈ <ω , if ( n, x ) ∈ r and( n, s ) ∈ r , then s x . We order P ( A, X ) by r ≤ r ′ ( r is stronger than r ′ ) iff r ′ ⊆ r . P ( A, X ) is σ -centered(thus, c.c.c.) since for all r, r ′ , if r ∩ ( ω × <ω ) = r ′ ∩ ( ω × <ω ), r ∪ r ′ ∈ P ( A, X ) is a common extension.Also, notice that in M , | P ( A, X ) | ≤ max {| X | , ω } = µ .If G is P ( A, X ) generic over M , consider, for each n , the set U n = { x ∈ X : ∃ r ∈ G ∃ s ∈ <ω ( n, s ) ∈ r and s ⊆ x } ∈ M [ G ]. Clearly, U n is an open subset of X . Then A = S n ∈ ω X \ U n since the sets D y = { r ∈ P ( A, X ) : ∃ n ∈ ω ( n, y ) ∈ r } and E nx = { r ∈ P ( A, X ) : ∃ s ∈ <ω s ⊆ x and ( n, s ) ∈ r } for x ∈ X \ A , y ∈ A and n ∈ ω are all dense. Thus, A is a F σ of X in M [ G ].Now we recursively construct, working in M a finite support κ -stage iterated forcing construction ( h ( P ξ , ≤ ξ , ξ ) : ξ ≤ ω i , h (˚ Q ξ , ˚ ≤ ξ , ˚ ξ ) : ξ < κ i ). As in [16], if ζ, ξ ≤ ω , i ξζ is the usual complete embedding from P ζ to P ξ . Moreover, if i : P → Q is a complete embedding between forcing posets and τ is a P -name, i ∗ ( τ ) is the Q -name recursively defined as { ( i ∗ ( σ ) , i ( p )) : ( σ, p ) ∈ τ } .Fix a function f from κ onto κ × κ such that if f ( ξ ) ≤ ( ζ, µ ), then ζ ≤ ξ . We will use f as a bookkeepingdevice. Each ˚ Q ξ will have size µ and will be forced by P ξ to have the ccc, therefore for each ξ , P ξ will havecardinality at most κ and will have the ccc as well.Suppose we have constructed ( h ( P ζ , ≤ ζ , ζ ) : ζ ≤ ξ i , h (˚ Q ζ , ˚ ≤ ζ , ˚ ζ ) : ζ < ξ i ) for some ξ < κ . We mustdetermine (˚ Q ξ , ˚ ≤ ξ , ˚ ξ ). Suppose that for each stage ζ < ξ we have also listed all P ζ -nice names for subsetsof ˇ ω as ( τ µζ : µ < κ ). This is possible since | P ζ | ≤ κ = κ ω and has since P ζ has the countable chain condition.List all P ξ -nice names for subsets of ˇ ω as ( τ µξ : µ < κ ) as well.Let f ( ξ ) = ( ζ, µ ). Since ζ ≤ ξ , the name ( i ξζ ) ∗ ( τ µζ ) is a nice P ξ -name for a subset of ˇ ω . Let (˚ Q ξ , ˚ ≤ ξ , ˚ ξ )be such that α (cid:13) α (˚ Q ξ , ˚ ≤ ξ , ˚ ξ ) ≈ P (cid:16)h i ξζ ∗ ( τ µζ ) i ˇ X , ˇ X (cid:17) and | ˚ Q ξ | ≤ µ (which is possible since | P ( A, X ) | = λ .For instance, we may take ˚ Q ξ to be ˇ λ ). 7et P = P κ . P has the ccc and | P | = κ , c ≤ κ in any extension by P (by counting nice names of subsets of ˇ ω ), so c = κ must hold since P preserves cardinals since it has the countable chain condition.Let G be P -generic over M . We claim X is an almost Q -set in M [ G ]. It is uncountable since P preservescardinals. Now let W be a subset of ω in M [ G ]. Since cf( κ ) M > ω , There exists ζ < κ such that W ∈ M [ G ζ ],where G ζ = ( i κζ ) − [ G ]. There exists µ < κ such that W = val( τ µζ , G ζ ). Let ξ be such that f ( ξ ) = ( ζ, µ ).Then, since W = val( i ξζ ∗ ( τ µζ ) , G ξ ). Hence, by the choice of ˚ Q ξ , M [ G ξ +1 ] contains a P ([ W ] X , X )-generic filterover M [ G ξ ], so, in M [ G ξ +1 ], [ W ] X is an F σ -subset of X , hence, the same happens in M [ G ]. Corollary 4.6. The following are relatively consistent with ZFC:1. There exists an almost-normal almost disjoint family which is not normal plus CH.2. There exists an almost-normal almost disjoint family of size ω < c .Proof. For 1., apply the previous proposition assuming c = κ = ω = | X | . For 2. assume, for concreteness,that in the ground model, | X | = ω < c = ω < ω = ω .In both examples Ψ( A X ) is not normal, because if it was normal, then by Jones’s lemma we wouldhave (in the extension) that 2 | Ψ( A X ) | = 2 | X | = 2 ω ≤ c , contradicting 2 ω = 2 c > c in the first case, and2 ω ≥ ω > c in the second case. This last ≥ inequality holds since P preserves cardinals due to the countablechain condition.Thus, it is consistent that there exists an almost-normal almost disjoint family of cardinality c (therefore,not normal), which gives a partial answer to Question 1.2. The almost disjoint families we constructed arenot MAD since no A X is a MAD family (since it can be extended by an infinite antichain of 2 <ω ), so Question1.3 remains fully open. In the previous section we have constructed an almost disjoint family which is almost-normal but is notnormal by using iterated forcing. We do not know if such an almost disjoint family exists in ZFC. Due toProposition 1.1, a semi-normal almost disjoint family A is normal iff A is almost-normal. Thus, the studyof semi-normality may come in handy when looking for an almost-normal a.d. family which is not normal inZFC.Semi-normality can be translated in combinatorial terms for Isbell-Mrowka spaces. In the end, it followsthat semi-normality is equivalent to a weaker form of separation, which was considered by Dow in [4] andBrendle in [3] and we state next: Definition 5.1. Let A an almost disjoint family (over N ) and two subfamilies B , C ⊆ A , we say that a set X ⊆ N weakly separates B and C if for all b ∈ B and c ∈ C , | X ∩ b | < ω and | X ∩ c | = ω .We say that A is weakly separated if for every B ⊆ A , the pair B and A \ B can be weakly separated.Now we are ready to present the combinatorial characterization of semi-normality in Isbell-Mr´owka spaces.(2) is a combinatorial property that looks like semi-normality. Proposition 5.2. Let A be an almost disjoint family. The following are equivalent:(1) Ψ ( A ) is semi-normal; 2) For each B ⊆ A and each W ⊆ N such that b ⊆ ∗ W for all b ∈ B , there exists W ⊆ W satisfying thefollowing: for all a ∈ A : a ∈ B ⇐⇒ a ⊆ ∗ W . (5.1) (3) A is weakly separated.Proof. (1) = ⇒ (2): Fix B ⊆ A and W ⊆ N such that each b ∈ B , b ⊆ ∗ W . Since B is closed and B ∪ W isopen, there exists a regular open set such that B ⊆ V ⊆ B ∪ W . We claim that (5.1) holds for W . = V ∩ N :If b ∈ B , then b ∈ V , so b ⊆ ∗ W . On the other hand, if b ∈ A is such that b ⊆ ∗ W , then b ∈ cl( W ) =cl( V ). Since b ⊆ ∗ V , it follows that b ∈ int(cl( V )) = V ⊆ B ∪ W , hence b ∈ B .(2) = ⇒ (3): Fix B ⊆ A . By hypothesis, there exists W ⊆ N satisfying (5.1). We claim that X = N \ W weakly separates B and A \ B . Indeed, if b ∈ B then b ⊆ ∗ W , so b ∩ ( N \ W ) = b ∩ X is finite. On the otherhand, if a ∈ A \ B , then a \ W = a ∩ X is infinite since a * ∗ W .(3) = ⇒ (1): Let F closed and U open such that F ⊆ U , then F = B ∪· K where B = F ∩ A and K = F ∩ N . By hypothesis, there exists X ⊆ N such that for each b ∈ B , | b ∩ X | < ω and for each a ∈ A \ B , | a ∩ X | = ω . Consider W = U ∩ N and let V = B ∪ K ∪ ( W \ X ), then F ⊆ V ⊆ U . We claim that V is aregular open set.Clearly, V is open. For the regularity, let x ∈ int(cl( V )). If x ∈ N , then x ∈ K ∪ ( W \ X ) ⊆ V . If x ∈ A ,then x ⊆ ∗ cl( V ) and it follows that x ⊆ ∗ K ∪ ( W \ X ). In the case of x ∩ K is infinite, x ∈ F ⊆ V since F isclosed, otherwise x ⊆ ∗ ( W \ X ) and it follows that x ∩ X is finite, thus x ∈ B ⊆ V .Recall a subset A of [ ω ] ω is centered iff every finite subset of A has infinite intersection, and a pseudoin-tersection of A is an infinite set X such that X ⊆ ∗ A whenever A ∈ A . The pseudointersection number p is defined as the least size of a centered subset of [ ω ] ω which does not admit a pseudointersection. It is wellknown that ω ≤ p ≤ a [2].In [3], Brendle observes that p ≤ ap , where ap is defined as the smallest cardinal κ for which there existsan almost disjoint family A of size |A| = κ that is not weakly separated. The reader can verify this inequalitydirectly by applying the following famous classical result : Proposition 5.3. Given C , D ⊆ P ( N ) such that max {|C| , |D|} < p and for all x ∈ D and F ∈ [ C ] <ω , | x \ S F | = ω . Then, there exists d ⊆ N , such that for each x ∈ C , | d ∩ x | < ω and for each y ∈ D , | d ∩ y | = ω . Corollary 5.4. If A is an almost disjoint family with |A| < p , then A is semi-normal.Proof. This follows from p ≤ ap and from Proposition 5.2. Corollary 5.5. If A is an almost disjoint family with |A| < p , then A is normal iff A is almost-normal. This gives us (consistently) a family of uncountable almost disjoint families for which normality andalmost-normality are the same. In particular, if p > ω , then no Luzin family is almost-normal. One may askif, consistently, every almost disjoint family is semi-normal, since if this was the case, every almost-normalalmost disjoint family would be normal. However, this is false. Proposition 5.6. If an almost disjoint family A is semi-normal, then |A| = c . In particular, almost disjointfamilies of cardinality c are not semi-normal. The proof can be found in [15, Theorem 2.15] were it is defined a σ -centered order, so the hypothesis about MA ( κ ) can bereplaced by limiting the size of C and D by p . roof. If A is semi-normal, then A is weakly separated by Proposition 5.2. But then we may inject P ( A )into P ( N ) by letting, for each B ⊆ A , X B be a subset of N such that, for all a ∈ A , | a ∩ X B | = ω iff a ∈ B .By Proposition 5.2, ap is the least cardinality of a non semi-normal almost disjoint family. In [3], Brendleshowed that ap ≤ min { add( M ) , q } , where q is defined as the least cardinality of a subset of 2 ω which is not a Q -set and add( M ) is the least cardinality of a collection of meager subsets of R whose union is not meager.Thus, there exists a non semi-normal almost disjoint family of size ≤ min { add( M ) , q } . In particular, thisdiscussion wields the following: Corollary 5.7. If add ( M ) = ω , there exists a non semi-normal almost disjoint family of size ω . ( ℵ , < c ) - separated MAD families In [8], it is defined the concept of strongly ℵ -separated almost disjoint family, which is related to almost-normality. They show that every almost-normal almost disjoint family is strongly ℵ -separated and thatan strongly ℵ -separated exists under CH. Their paper does not say anything about the converse, whichwe are going to argue to be consistently false. In this section we modify their technique to weaken the CHhypothesis. First, we define a suitable separation concept. Definition 6.1. We say that an almost disjoint family A is strongly ( ℵ , < c )-separated iff for every twodisjoint B , C ⊆ A , with B countable and |C| < c , there exists a partitioner X ⊆ ω for A and B .Clearly, every strongly ( ℵ , < c )-separated almost disjoint family is strongly ℵ -separated and these con-cepts are equivalent under CH.Now we recall the definitions of b and s . If f, g ∈ ω <ω , we say that f < ∗ g iff the set { n ∈ ω : f ( n ) ≥ g ( n ) } is finite. An unbounded family in ω ω is a set B ⊆ ω ω such that for every f ∈ ω ω there exists g ∈ B such that g < ∗ f . The bounding number b is the smallest cardinality of an unbounded family.We say S ⊆ P ( ω ) is a splitting family iff for every X ∈ [ ω ] ω there exists A ∈ S such that both X \ A and X ∩ A are infinite. The splitting s is the least size of a splitting family.It is well known that p ≤ s ≤ c and that p ≤ b ≤ a , and all inequalities are consistent to be strict [2]. Lemma 6.2. Let A an almost disjoint family with |A| < b . If B ⊆ A is a countable set, then there exists apartitioner for B and A \ B . In particular, A is strongly ( ℵ , < c ) -separated.Proof. Let B = { b n : n ∈ ω } list all elements of B . For each a ∈ A \ B , consider the function f a ∈ ω ω definedby f a ( n ) = sup( a ∩ b n ). Since F = { f a ∈ ω ω : a ∈ A \ B} is family of functions with |F| < b , there exists g ∈ ω ω such that f a < ∗ g , for all a ∈ A \ B .Let X = S n ∈ ω ( b n \ g ( n )). We claim that X is a separator for B and A \ B .Clearly, we have that b n ⊆ ∗ X , for all n ∈ ω . Given a ∈ A \ B , since g > ∗ f a , there exists k ∈ ω such that b n \ g ( n ) = ∅ for all n ≥ k , thus X ∩ a = ∗ ∅ . Corollary 6.3. If p > ω , there exists a strongly ( ℵ , < c ) -separated almost disjoint family which is notalmost-normal.Proof. By Corollary 5.5 and Lemma 6.2, every non normal almost disjoint family of size ω is strongly( ℵ , < c )-separated and is not almost-normal. So, assuming p > ω , any non-normal almost disjoint family ofsize ω suffices. E.g., a Luzin family. 10iven an almost disjoint family A , In what follows, J + ( A ) = { X ∈ ω : |{ a ∈ A : | a ∩ X | = ω }| ≥ ω } .An almost disjoint family is said to be completely separable iff for every A ∈ J + ( A ) there exists a ∈ A such that a ⊆ A . Completely separable almost disjoint families exist in ZFC [7], however, we don’t know ifcompletely separable MAD families exist in ZFC even thought we know they exist in most models [12]. Aconcept related to completely separability is the true cardinality c .In [9, Definition 1.2], the authors introduce the definition of generic existence of a MAD family in termsof a given property P . More precisely, we say that MAD families with a property P exist generically iff allalmost disjoint families of size less than c can be extended to a MAD family with the property P . In thissense, we have the following result: Theorem 6.4 ( b = s = c ) . Completely separable MAD families which are strongly ( ℵ , < c ) -separated existgenerically.Proof. Let A ′ be an infinite almost disjoint family of size κ < c and write A ′ = { a γ : γ < κ } so that a γ = a ν whenever γ = ν .Let { B β ∈ [ c ] ω : κ ≤ β < c } list all countable subsets of c such that for each β , B β ⊆ β and, for all B ∈ [ c ] ω , |{ β : B β = B }| = c and list [ ω ] ω = { Y α : κ ≤ α < c } .We will define recursively almost disjoint families A α , a α ∈ [ ω ] ω and X α ⊆ ω , for κ ≤ α < c such that:(1) A β = { a γ : γ < β } ;(2) A κ = A ′ .(3) ∀ β < c : ∀ γ ∈ B β , a γ ⊆ ∗ X β ;(4) ∀ β < c : ∀ γ ∈ β \ B β , a γ ∩ X β = ∗ ∅ ;(5) ∀ γ < c : ∀ β ≤ γ , a γ ⊆ ∗ X β or a γ ∩ X β = ∗ ∅ ;(6) if Y β ∈ J + ( A β ), a β ⊆ Y β is an infinite subset.(7) ∀ η < γ < c : a η ∩ a γ is finite.Fix α < c and suppose that X β and a β are defined for κ ≤ β < α . Since B α is countable and | α | < c = b ,using Lemma 6.2, let X α ⊆ ω be a partitioner for { a ξ : ξ ∈ B α } and { a ξ : ξ ∈ α \ B α } .To define a α , notice that since | α | < a there exists an infinite Y ⊆ ω almost disjoint from a β , for all β < α . In addition, if Y α ∈ J + ( A α ), we can take Y ⊆ Y α :Indeed, if { γ < α : | a γ ∩ Y α | = ω } is finite, take Y . = Y α \ S { a γ : γ < α ∧ | a γ ∩ Y α | = ω } . Otherwise, notethat B = { a γ ∩ Y α : γ < α ∧ | a γ ∩ Y α | = ω } is an almost disjoint family in Y α . Since |B| < a , there exists Y ⊆ Y α almost disjoint from each element of B .Since s = c , { X γ ∩ Y : γ ≤ α } is not a splitting family in Y . Thus, there exists a α ⊆ Y such that for all γ ≤ α , a α ∩ X γ = ∗ ∅ or a α ⊆ ∗ X γ .Notice that A is an almost disjoint family extending A ′ by (2) and (7)We show that for every infinite Y ⊆ ω , either Y / ∈ J + ( A ) or there exists α < c such that a α ⊆ Y , thusproving A is MAD and completely separable. If Y ∈ J + ( A ), let α be such that Y = Y α . Then Y ∈ J + ( A α ),thus, by 6., a α ⊆ Y α .Finally, we prove that A is ( ℵ , < c )-separated. Given an infinite countable set B ⊆ A and C ∈ [ A ] < c suchthat B ∩ C = ∅ , let B = { α < c : a α ∈ B} and C = { α < c : a α ∈ C} . Notice that B is infinite and countable,11 C | < c , and B ∩ C = ∅ . Let α = sup C , which is less than c since c = b is regular, and let α > α be suchthat B α = B . In particular, B = B α ⊆ α and C ⊆ α \ B α . By (3), for all b ∈ B , b ⊆ ∗ X α .By (4), for all c ∈ C , c ∩ X α = ∗ ∅ By (3) and (4) together by using α in the place of β , we see that for every γ < α , a γ ⊆ ∗ X α or a γ ∩ X α = ∗ ∅ .If γ > α , we apply (5) for this γ and α in the place of β to conclude that a γ ⊆ ∗ X α or a γ ∩ X α = ∗ ∅ .It is well know that the cardinal characteristic par , as defined in [2], equals the minimum of b , s . Thus,the previous theorem could have its hypothesis replaced by par = c . We have answered Question 4.4 of [8] by providing a counter example in βω and partially answeredQuestion 4.2 of [8] by providing an example by using forcing. We have shown that an almost disjointfamily is semi-normal iff it is weakly separated, thus, for weakly separated almost disjoint families normalityand almost-normality are the same. However, Question 4.2 remains open. We may define an as the leastcardinality of an almost-normal almost disjoint family which is not normal. We don’t know if this numberis well defined in ZFC, however, if there is such an almost disjoint family, it follows that ap ≤ an . We mayrefine Question 4.2. as follows: Question 7.1. Is an well defined in ZFC? If there is an almost-normal not normal almost disjoint family,does an = ap hold? Recall that in [8] it was proven that almost-normal almost disjoint families are strongly ℵ -separated.Here we have defined the concept of strongly ( ℵ , < c )-almost disjoint families and we have proved thatstrongly ( ℵ , < c )-separation property does not hold for all almost-normal almost disjoint families, at leastconsistently. However, the relation between these concepts is not fully understood. Thus, we ask: Question 7.2. Are almost-normal almost disjoint families strongly ( ℵ , < c ) -separated? Question 7.3. Does CH imply that strongly ℵ -separated almost disjoint families are almost-normal? The first author was funded by FAPESP (Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo, processnumber 2017/15502-2). The second author was funded by CNPq (Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico, process number 141881/2017-8).The authors would like to thank Sergio A. 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