Weak normality properties in Ψ -spaces
aa r X i v : . [ m a t h . GN ] J u l Weak normality properties in Ψ -spaces SERGIO A. GARCIA-BALAN AND PAUL J. SZEPTYCKI
Abstract
Almost disjoint families of true cardinality c are used to produce an exam-ple of a mildly-normal not partly-normal Ψ-space and a quasi-normal notalmost-normal Ψ-space. This is related with a problem posed by Kalantanin [9] where he asks whether there exists a mad family so that the relatedMr´owka-Isbell space is partly-normal. In addition, a consistent exampleof a Luzin mad family such that its associated Ψ-space is quasi-normal isprovided. Mr´owka-Isbell Ψ-spaces give a number of interesting counterexamples in manyareas of topology including normality and related covering properties ([11],[4]). Ψ-spaces associated to maximal almost disjoint families are never nor-mal. Weakenings of normality have been considered in the literaure since thelate 60’s and early 70’s . For instance, quasi-normal [15], almost-normal [1],mildly-normal [13], [14], and more recently π -normal [8] and partly-normal [9].In [10] L. Kalantan and the second author prove that any product of ordinalsis mildly-normal. Kalantan builds a Ψ-space which is not mildly-normal in [7]and, in [9], using CH constructs a mad family so that the associated Ψ-spaceis quasi-normal.Standard notation is followed and any undefined term can be found in [2]. Asubset A of a topological space X is called regularly closed (also called closeddomain), if A = int ( A ) ( cl X ( A ) or simply cl ( A ) will denote the closure of A inthe space X as well). A set A will be called π -closed , if A is a finite intersectionof regularly closed sets. Two subsets A and B of a topological space X are saidto be separated if there exist two disjoint open sets U and V of X such that A ⊆ U and B ⊆ V . July 14, 20202020
Mathematics Subject Classification . Primary 54D15; Secondary 54G20.
Key words and phrases.
Almost disjoint family, Mr´owka-Isbell Psi-spaces, mildly-normal,partly-normal, quasi-normal, almost-normal. efinition 1.1. A regular space X is called:1. π -normal [8] if any two nonintersecting sets A and B , where A is closedand B is π -closed, are separated.2. almost-normal [1] if any two nonintersecting sets A and B , where A isclosed and B is regularly closed, are separated.3. quasi-normal [15] if any two nonintersecting π -closed sets A and B areseparated.4. partly-normal [9] if any two nonintersecting sets A and B , where A isregular closed and B is π -closed, are separated.5. mildly-normal (also called κ -normal), [13] [14] if any two nonintersectingregular closed sets A and B are separated. Since “regular closed → π -closed → closed” holds, it follows that normal spacesare π -normal and: ր quasi-normal ց π -normal partly-normal → mildly-normal. ց almost-normal ր Proposition 1.2.
Almost-normal spaces are π -normal.Proof. Assume X is an almost-normal space. For a positive integer n , call a set n - π -closed, if it is the intersection of n many regular closed sets. We will showby induction on n , that in X every n - π -closed set can be separated from a closedset, provided they are disjoint. This is enough to show that X is π -normal. Base case: n = 1. Since X is almost normal, every closed H and 1- π -closedset K in X such that H ∩ K = ∅ can be separated ( K is a regular closed set). Inductive step:
Assume that for all 1 ≤ i ≤ n if H is closed, K is i - π -closedin X and, H ∩ K = ∅ , then H and K can be separated. Let H ⊂ X be aclosed set and let K be an ( n + 1)- π -closed set such that H ∩ K = ∅ . Thus, K = T ≤ j ≤ n K j , where each K j is a regular closed set in X . We show that H and K can be separated. Case 1: H ∩ ( T j Given an almost disjoint family A , • If B ⊆ ω , let A ↾ B = { a ∈ A : | a ∩ B | = ω } . • I + ( A ) = { B ⊆ ω : |A ↾ B | ≥ ω } is the family of big sets (the sets thathave infinite intersection with infinite many members of the family). • I ( A ) = { B ⊆ ω : |A ↾ B | < ω } , the family of small sets. This familyforms an ideal. • A will be called completely separable [3] if for each B ∈ I + ( A ) , there issome a ∈ A with a ⊆ B . • A will be called of true cardinality c [6] if for every B ⊆ ω either A ↾ B isfinite, or it has size c . • If Ψ( A ) is a normal space (almost-normal, quasi-normal, partly-normal,mildly-normal), it will be said that A is normal (almost-normal, quasi-normal, partly-normal, mildly-normal, respectively). The existence in ZFC of a completely separable mad family is an important openquestion that has many interesting consequences (see [6]). Completely separablealmost disjoint (not maximal) families do exist in ZFC and also have interestingconsequences (see [3]). It is not hard to show that if A is a completely separablealmost disjoint family and B ∈ I + ( A ), then |{ a ∈ A : a ⊆ B }| = c . This facthas the following consequence: if A is completely separable, then for any B ⊆ ω ,the set A ↾ B is either finite or it has size c . That is, every completely separablealmost disjoint family is of true cardinality c , and therefore, almost disjointfamilies of true cardinality c exist in ZFC. Furthermore, every infinite almost3isjoint family A of true cardinality c , has size c and thefore A is not normal (asa consequence of Jones’ Lemma). Actually, something slightly stronger holds: Observation 1.4. If A is an almost disjoint family of true cardinality c , thenfor all C ∈ [ A ] ℵ , C and A r C cannot be separated in Ψ( A ) .Proof. Let U , V be any open sets in Ψ( A ) so that C ⊆ U , A \ C ⊆ V . Let W = U ∩ ω , then for all c ∈ C , c ⊆ ∗ W . Hence, |A ↾ W | ≥ ω . Thus, |A ↾ W | = c .Pick a ∈ A \ C such that | W ∩ a | = ω . Since a ⊆ ∗ V ∩ ω , U ∩ V = ∅ .The following observations are not hard to show and they will be used in variousoccasions in the next section. Observation 1.5. Given any almost disjoint family A , if W ⊆ ω , then cl Ψ( A ) ( W ) is a regular closed subset of Ψ( A ) . Observation 1.6. Given any almost disjoint family A , if H ⊂ Ψ( A ) is aregular closed set, then for each a ∈ A , a ∈ H if and only if | a ∩ H | = ω . Observation 1.7. Given any almost disjoint family A and H, K ⊂ Ψ( A ) suchthat H and K are closed sets, H ∩ K = ∅ and | H ∩ A| < ω , then H and K can be separated. In particular, for each closed set H ⊂ Ψ( A ) that has finiteintersection with A , H and A r H can be separated. Example 2.3 provides a quasi-normal not almost-normal almost disjoint family F which is constructed from a particular non almost-normal almost disjointfamily A of true cardinality c . Each element of F will be a finite union of ele-ments of A . In order to make F quasi-normal, all pairs of disjoint π -closed setsin Ψ( F ) have to be separated. By Observation 1.7, the only pairs of π -closedsets ( A, B ) that might be difficult to separate are the ones where A ∩ F and B ∩ F are infinite. Using that A is of true cardinality c it will be possible tobuild F so that all such pairs have a point in common. Thus, all pairs of disjoint π -closed sets in Ψ( F ) will be trivial, i.e. one of them will have finite intersectionwith F . Hence, F will be quasi-normal. In addition, it won’t be hard to carrythis construction out so that the non almost-normality of A is preserved in F .That is, a closed set C and a regular closed set E with empty intersection thatcannot be separated in Ψ( A ) will be transformed into a pair of witnesses of nonalmost-normality in Ψ( F ). Now, let us obtain the required non almost-normalalmost disjoint family of true cardinality c .The following example is an instance of a machine for converting two almostdisjoint families of the same cardinality, into a single almost disjoint family A with a countable set C ⊂ A and a set E ⊂ Ψ( A ) such that C is closed and E isregular closed in Ψ( A ), C ∩ E = ∅ and A ⊂ C ∪ E .4 xample 2.1. There is an almost disjoint family A of true cardinality c on ω so that there is C ∈ [ A ] ω and W ∈ [ ω ] ω , such that cl Ψ( A ) ( W ) ∩ A = A r C . In particular, there is a non almost-normal almost disjoint family of truecardinality c .Proof. Partition ω into two infinite disjoint sets V, W . Let A , A be almostdisjoint families of true cardinality c on V and W , respectively, and let C ∈ [ A ] ω .Now, a new family is built as follows, let α : A r C ↔ A be a bijective function.Let A = { a ∪ α ( a ) : a ∈ A r C} ∪ C .Let us check that A is the desired family. Clearly, it is almost disjoint. To seethat it has true cardinality c let M ⊆ ω such that |A ↾ M | ≥ ω . Then, either |C ↾ M | ≥ ω or | ( A \ C ) ↾ M | ≥ ω . Hence, |A ↾ M | ≥ ω or |A ↾ M | ≥ ω .Therefore, |A ↾ M | = c or |A ↾ M | = c . In any case, |A ↾ M | = c . Thus, A isof true cardinality c .Now, a ∈ cl Ψ( A ) ( W ) ∩ A ↔ a ∈ A ∧| a ∩ W | = ω ↔ a ∈ A ∧ (cid:0) ∃ a ∈ A [ a = a ∪ α ( a )] (cid:1) ↔ a ∈ A r C . By Observation 1.4, A is not almost-normal.If in the previous example we assume, in addition, that A , A are mad familiesof the same cardinality, the resulting family A is mad as well: If M ∈ [ ω ] ω , then M has infinite intersection either with V or with W , since A , A are both mad,there is a ∈ A such | a ∩ M | = ω . Hence, the following holds: Corollary 2.2. The existence of a mad family of true cardinality c implies theexistence of a mad family A of true cardinality c on ω so that there is C ∈ [ A ] ω and W ∈ [ ω ] ω , such that cl Ψ( A ) ( W ) ∩ A = A r C . In particular, the existence ofa mad family of true cardinality c implies the existence of a non almost-normalmad family of true cardinality c . Example 2.3. There is a quasi-normal not almost-normal almost disjoint fam-ily of true cardinality c .Proof. Let A be a not almost-normal almost disjoint family of true cardinality c as in Example 2.1. Hence, let C ∈ [ A ] ω and W ∈ [ ω ] ω , with | ω r W | = ω ,such that cl Ψ( A ) ( W ) ∩ A = A r C . Consider the family of finite subsets of [ ω ] ω , E = (cid:2) [ ω ] ω (cid:3) <ω and let B = {{ C, D } ∈ [ E ] : ( T C ) ∩ ( T D ) = ∅} . Since |B| = c ,we can list it as B = {{ C α , D α } : α < c } . A sequence of finite sets F α ∈ [ A ] <ω will be built recursively in c many steps.For α = 0, consider { C , D } ∈ B . If for each C ∈ C and D ∈ D , A ↾ C and A ↾ D all have size c , then for each C ∈ C and D ∈ D pick a C , b D ∈ A \ C suchthat | a C ∩ C | = ω = | b D ∩ D | and all the a C ’s and b D ’s are distinct ( |{ a C , b D : C ∈ C , D ∈ D }| = | C | + | D | ). Let F = { a C , b D : C ∈ C , D ∈ D } . If thereis C ∈ C (or D ∈ D ) such that A ↾ C is finite ( A ↾ D is finite), let F = ∅ .Observe that these are the only two possibilities as A is of true cardinality c .Now assume 0 < α < c and that for each β < α , F β is either empty of a finitesubset of A \ ( C ∪ S γ<β F γ ). Consider the pair { C α , D α } . If for each C ∈ C α and D ∈ D α , A ↾ C and A ↾ D all have size c , then for each C ∈ C α and D ∈ D α a C , b D ∈ A \ ( C ∪ S β<α F β ) such that | a C ∩ C | = ω = | b D ∩ D | and all the a C ’s and b D ’s are distinct ( |{ a C , b D : C ∈ C α , D ∈ D α }| = | C α | + | D α | ). Let F α = { a C , b D : C ∈ C α , D ∈ D α } . If there is C ∈ C α (or D ∈ D α ) such that A ↾ C is finite ( A ↾ D is finite), let F α = ∅ . Let F = (cid:8) [ F α : α < c (cid:9) ∪ (cid:0) A \ [ α< c F α (cid:1) . Since each a ∈ F is either an element of A or a finite union of elements of A , itis clear that F is an almost disjoint family of true cardinality c . Claim: Ψ( F ) is quasi-normal. Let A = ∅ 6 = B be disjoint π -closed subsetsof Ψ( F ). A = T ni =1 A i , B = T mj =1 B j , where each A i and B j are regularclosed sets. It can be assumed that for each i ≤ n and for each j ≤ m , | A i ∩ ω | = ω = | B j ∩ ω | . Let α < c be minimal such that C α = { A i ∩ ω : i ≤ n } and D α = { B j ∩ ω : j ≤ m } .At stage α , either F α = ∅ or F α = { a C , b D : C ∈ C α , D ∈ D α } . The latter isnot possible since for each C ∈ C α and each D ∈ D α the a C ’s and b D ’s werechosen so that | a C ∩ C | = ω = | b D ∩ D | and this implies S F α is in the closureof each C ∈ C α and each D ∈ D α (see Observation 1.5 and Observation 1.6).Hence S F α ∈ A ∩ B , but it is assumed that A and B are disjoint.Thus, F α = ∅ . This means that there exists C ∈ C α , such that A ↾ C = H forsome finite set H (or there exists D ∈ D α , such that A ↾ D = H for some finiteset H ). Without loss of generality assume there exists such C ∈ C α . Hence, A ↾ C = H for some finite set H . Observe that since for each a ∈ F , either a ∈ A or a is a finite union of elements of A , then F ↾ C = H for some finite H so that | H | ≤ | H | . Now fix i ≤ n such that A i ∩ ω = C . Since A i is regularclosed, by 1.6 A i ∩ F = H . Thus, A ∩ F ⊆ H and by Observation 1.7, A and B can be separated. Therefore Ψ( F ) is quasi-normal. Claim: Ψ( F ) is not almost-normal.Fix a ∈ F r C , then a ∈ A r C or a is a finite union of elements of A r C .Since cl Ψ( A ) ( W ) ∩ A = A r C , | W ∩ a | = ω . Hence, a ∈ cl Ψ( F ) ( W ), i.e., F r C ⊆ cl Ψ( F ) ( W ). On the other hand, if c ∈ C , c / ∈ cl Ψ( A ) ( W ), thus | c ∩ W | < ω and therefore c / ∈ cl Ψ( F ) ( W ).Hence, C is a closed set, cl Ψ( F ) ( W ) is a regular closed set, they do not intersectand by Observation 1.4 they cannot be separated.If in the construction of Example 2.3, a mad family as in Corollary 2.2 is chosen,then the resulting family F is mad, quasi-normal and not almost-normal. Thus: Corollary 2.4. The existence of a mad family of true cardinality c implies theexistence of a quasi-normal, non almost-normal mad family of true cardinality c . The following example provides a mildly-normal not partly-normal almost dis-joint family F of true cardinality c which is constructed using three almost6isjoint families of true cardinality c . In order to make F mildly-normal allpairs of disjoint regular closed sets in Ψ( F ) have to be separated. A similarapproach as in Example 2.3 is followed. It will be possible to build F so thatall pairs of disjoint regular closed sets in Ψ( F ) will be trivial, i.e., one of themwill have finite intersection with F (Observation 1.7 guarantees they can beseparated). To make F not quasi-normal, there will be a regular closed set A disjoint from a π -closed set B that cannot be separated. The basic idea is topartition ω into three infinite sets, W , V , V , take an almost disjoint family oftrue cardinality c on each one of them (we use the property of true cardinality c to make F mildly-normal), and build F so that in Ψ( F ), A = cl Ψ( F ) ( W ) and B = cl Ψ( F ) ( V ) ∩ cl Ψ( F ) ( V ) are disjoint but cannot be separated. Example 2.5. There exists a mildly-normal not partly-normal almost disjointfamily of true cardinality c .Proof. Partition ω into three disjoint infinite pieces, that is W, V , V ∈ [ ω ] ω and W ∪ V ∪ V = ω . If Y ∈ { W, V , V } let A Y be an almost disjoint family of truecardinality c on Y . List all pairs of infinite subsets of ω with empty intersectionas {{ C α , D α } : α < c } . A sequence of finite sets F α ⊂ A W ∪ A V ∪ A V will bebuilt recursively in c many steps.Fix α < c , assume that for each β < α , F β has been defined such that F β is a possibly empty finite set F β ⊂ ( A W ∪ A V ∪ A V ) \ S γ<β F γ such thateither F β ⊂ A W or F β has nonempty intersection with exactly two elements of {A W , A V , A V } . Consider { C α , D α } . Case 1: Either all three sets A W ↾ C α , A V ↾ C α , A V ↾ C α are finite, or all threesets A W ↾ D α , A V ↾ D α , A V ↾ D α are finite. In this case, let F α = ∅ . Case 2: Case 1 is false. That is (given that A W , A V , A V are of true cardi-nality c ): at least one of the three sets A W ↾ C α , A V ↾ C α , A V ↾ C α has size c and at least one of the three sets A W ↾ D α , A V ↾ D α , A V ↾ D α has size c . Choosethe smallest i such that Subcase 2 .i (below) holds, define F α accordingly, andignore the other subcases. Subcase 2.1: |A W ↾ C α | = c = |A W ↾ D α | . Pick c α , d α ∈ A W \ S β<α F β suchthat c α = d α and | c α ∩ C α | = ω = | d α ∩ D α | . Let F α = { c α , d α } . Subcase 2.2: There exists i ∈ { , } so that |A V i ↾ C α | = c = |A V i ↾ D α | . Pick c α , d α ∈ A V i \ S β<α F β , such that c α = d α and | c α ∩ C α | = ω = | d α ∩ D α | . Inaddition, pick e α ∈ A V − i \ S β<α F β . Let F α = { c α , d α , e α } . Subcase 2.3: |A V ↾ C α | = c = |A V ↾ D α | . Pick c α ∈ A V \ S β<α F β and d α ∈ A V \ S β<α F β such that | c α ∩ C α | = ω = | d α ∩ D α | and let F α = { c α , d α } . Subcase 2.4: |A W ↾ C α | = c and there exists i ∈ { , } so that |A V i ↾ D α | = c .Pick c α ∈ A W \ S β<α F β and d α ∈ A V i \ S β<α F β such that | c α ∩ C α | = ω = | d α ∩ D α | and let F α = { c α , d α } . 7his finishes Case 2 and the construction of F α for α < c . Let F = (cid:8) [ F α : α < c (cid:9) ∪ (cid:0) ( A W ∪ A V ∪ A V ) \ [ α< c F α (cid:1) . It will be shown that F is the desired almost disjoint family. Given that eachof A W , A V and A V is of true cardinality c and if we let a ∈ F , then either a is an element or a finite union of elements of A W ∪ A V ∪ A V , then F is analmost disjoint family of true cardinality c .Ψ( F ) is not partly-normal: Let A = cl Ψ( F ) ( W ) and B = cl Ψ( F ) ( V ) ∩ cl Ψ( F ) ( V ). By Observation 1.5, A isregular closed and B is a π -closed set. Observe that since A V and A V are oftrue cardinality c , there are infinite many pairs { C α , D α } such that C α ⊂ V , D α ⊂ V , and |A V ↾ C α | = c = |A V ↾ D α | . For such pairs Subcase 2.3 appliesand therefore | B ∩ F| ≥ ω . In addition, A ∩ B = ∅ : assume there is a ∈ A ∩ B .Since V ∩ V = ∅ , B ∩ ω = ∅ , hence a ∈ F ∩ A ∩ B . By Observation 1.6, | a ∩ W | = | a ∩ V | = | a ∩ V | = ω . This implies that a / ∈ A W ∪ A V ∪ A V .There is α < c such that a = S F α , but by the construction, F α ⊂ A W or F α intersects exactly two elements of {A W , A V , A V } which contradicts that a hasinfinite intersection with W , V and V . Whence, A ∩ B = ∅ .It remains to show that A and B cannot be separated. Assume, on the contrary,that there are S, T ⊆ Ψ( F ) open such that A ⊆ S , B ⊆ T and S ∩ T = ∅ . Let α < c such that C α = ω ∩ S and D α = ω ∩ T . For the pair { C α , D α } , eitherCase 1 or Case 2 of the construction holds. If Case 1 holds: since W ⊆ C α , A W ↾ C α is not finite. Hence, A W ↾ D α , A V ↾ D α , A V ↾ D α are finite. Thus, F ↾ D α is finite. Since cl Ψ( F ) ( D α ) is regular closedand F ↾ D α is finite, by Observation 1.6, F ∩ cl Ψ( F ) ( D α ) is finite. Now, T isopen and D α = ω ∩ T , therefore T ⊆ cl Ψ( F ) ( D α ). Hence, F ∩ T is finite. Giventhat | B ∩ F| ≥ ω , B T , which is a contradiction. If Case 2 holds: Either F α ⊂ A W or F α intersects exactly two elements of {A W , A V , A V } . In any case S F α is an element of A or B . In addition, thereexist c α , d α ∈ F α such that | c α ∩ C α | = ω = | d α ∩ D α | . If S F α ∈ A , then foreach open neighbourhood U of S F α , U ∩ T = ∅ (which implies U S ), andthis contradicts that S is open. We reach a similar contradiction if S F α ∈ B .Hence, A and B cannot be separated.Ψ( F ) is mildly-normal: Let C = ∅ 6 = D be disjoint regular closed subsets of Ψ( F ). It can be assumedthat | C ∩ ω | = ω = | D ∩ ω | . Fix α < c such that C ∩ ω = C α and D ∩ ω = D α .For the pair { C α , D α } , either Case 1 or Case 2 holds. If Case 2 holds, there exist c α , d α ∈ F α such that | c α ∩ C α | = ω = | d α ∩ D α | . Thus, S F α ∈ cl Ψ( F ) ( C α ) ∩ cl Ψ( F ) ( D α ) ⊆ cl Ψ( F ) ( C ) ∩ cl Ψ( F ) ( D ) = C ∩ D . This contradicts C ∩ D = ∅ .Thus, Case 1 holds. This means that all three sets A W ↾ C α , A V ↾ C α , A V ↾ C α are finite, or all three sets A W ↾ D α , A V ↾ D α , A V ↾ D α are finite.Without loss of generality, assume the former. This implies that F ↾ C α is finite.8iven that C is a regular closed set and C α = C ∩ ω , by Observation 1.6 C ∩ F is finite and by Observation 1.7, C and D can be separated. Therefore Ψ( F ) ismildly-normal.Observe that if in the construction of Example 2.5, the families A W , A V and A V are mad of true cardinality c , then the family F is mad as well. Therefore: Corollary 2.6. If there exists a mad family of true cardinality c , then there isa mildly-normal, not partly-normal mad family of true cardinality c . Definition 2.7. For a positive n ∈ ω , a regular space will be called n -partly-normal if any two nonintersecting sets A and B , where A is regularly closed and B is the intersection of at most n regularly closed sets, are separated. Observe that 1-partly-normal coincides with mildly-normal, and for each posi-tive n ∈ ω , partly-normal → ( n +1)-partly-normal → n -partly-normal → mildly-normal. It is possible to extend the idea in Example 2.5 (partition ω into n + 2pairwise disjoint infinite pieces, take an almost disjoint family of true cardi-nality c on each piece and let { C α : α < c } list all sets C ⊂ [ ω ] ω such that2 ≤ | C | ≤ n + 1), to show the following: Theorem 2.8. For each positive n ∈ ω , there exists a n -partly-normal not ( n + 1) -partly-normal almost disjoint family of true cardinality c . Similarly as Corollary 2.6, it also holds true: Corollary 2.9. If there exists a mad family of true cardinality c , then for eachpositive n ∈ ω , there is a n -partly-normal not ( n + 1) -partly-normal mad familyof true cardinality c . Corollary 2.4 says, in particular, that there is a quasi-normal mad family, pro-vided there is a completely separable mad family. Our next example shows that,assuming CH , not only a quasi-normal mad family exists, but one that it is alsoLuzin. Recall that an almost disjoint family is Luzin if it can be enumerated as { a α : α < ω } so that for each α < ω and each n ∈ ω , { β < α : a α ∩ a β ⊆ n } is finite. Luzin introduced this kind almost disjoint family in [12] to providean example of an almost disjoint family A such that every pair of uncountablesubfamilies of A have no separation (it will be said that two subfamiles B and C of A , have a separation if there is X ⊆ ω such that for each b ∈ B , b ⊆ ∗ X andfor each c ∈ C , c ∩ X = ∗ ∅ ). Thus, Luzin families are far from being normal. Nomad family is normal, no Luzin family is normal, and yet, there is, consistently,a quasi-normal Luzin mad family. Example 2.10 (CH) . There is a Luzin mad family A which is quasi-normal.Proof. The standard construction of a Luzin family is modified to build a family A with the extra following property: for each X ⊆ ω , either X is covered byfinitely many elements of A or the set of elements of A that has finite intersectionwith X is countable.The idea is to use CH to list all infinite subsets X α ⊆ ω , with α < ω and,9t stage α < ω of the construction of the family, X α will be covered by the α -th element of the family, together with finitely many elements of the familypreviously constructed or, if X α has infinite intersection with infinitely manyelements of the family constructed so far, it will be guaranteed that, from thatstage until the end, all elements of the family will have infinite intersection with X α .Partition ω into infinite pairwise disjoint subsets a i , with i ∈ ω , that is ω = S i ∈ ω a i , and i = j implies a i ∩ a j = ∅ . List all infinite subsets of ω as [ ω ] ω = { X α : α < ω } such that for each n ∈ ω , X n = a n . If α is such that ω ≤ α < ω ,recursively assume we have constructed a β for β < α such that { a β : β < α } isan almost disjoint family and for each β < α , X β is covered by finitely manyelements of { a γ : γ ≤ β } or for each β ≤ γ < α , | X β ∩ a γ | = ω .The α -th element of the family will be constructed. Reenumerate the sets A α = { a β : β < α } and J α = { X β : β ≤ α } as A α = { a αn : n ∈ ω } and J α = { X αn : n ∈ ω } . Let I α = { n ∈ ω : X αn ∈ I + ( A α ) } .There are two cases, either X α ∈ I + ( A α ) or X α / ∈ I + ( A α ). We will construct a α depending on whether at this stage, I α is still empty or not.If I α = ∅ (observe that in particular X α / ∈ I + ( A α )), let p αn ⊆ a αn \ S i Definition 3.1. An almost disjoint family A will be called strongly ℵ -separated ,if and only if for each pair of disjoint countable subfamilies there is a clopen par-tition of A that separates them. That is, for each A, B ∈ [ A ] ω , with A ∩ B = ∅ ,there is X ⊂ ω such that1. For each a ∈ A , a ⊆ ∗ X or a ∩ X = ∗ ∅ ,2. For each a ∈ A , a ⊆ ∗ X ,3. For each a ∈ B , a ∩ X = ∗ ∅ . Lemma 3.2. Almost-normal almost disjoint families are strongly ℵ -separated.Proof. Let A be an almost-normal almost disjoint family. First, let us recallthat each pair of disjoint countable closed subsets of a regular space can beseparated. Hence, given that Ψ( A ) is regular and A is a closed discrete subsetof Ψ( A ), if we consider A, B ∈ [ A ] ω so that A ∩ B = ∅ , then A and B can beseparated. Thus, there exist U A , U B open subsets of A such that U A ∩ U B = ∅ and A ⊆ U A , B ⊆ U B . Let C = cl Ψ( A ) ( U A ∩ ω ). By Observation 1.5, C is aregular closed set. Then C and A \ C is a pair of a regular closed set and aclosed set with empty intersection. Since A is almost-normal, there exist V , W open subsets of Ψ( A ) such that V ∩ W = ∅ and C ⊆ V , A \ C ⊆ W .Let us check that X = V ∩ ω has the desired properties. Indeed, let a ∈ A , if a ∈ C , then a ⊆ ∗ V ∩ ω = X . If a ∈ A \ C , then a ⊆ ∗ W ∩ ω , thus a ∩ X = ∗ ∅ .Now, if a ∈ A , a ⊆ ∗ U A ∩ ω ⊆ C ∩ ω ⊆ V ∩ ω = X . If b ∈ B , | b ∩ U A | < ω thus, b ∈ A \ C . Hence, b ⊆ ∗ W , i.e. b ∩ X = ∗ ∅ . Hence, A is strongly ℵ -separated. Proposition 3.3 (CH) . There is a strongly ℵ -separated mad family.Proof. Let { ( A β , B β ) ∈ [ ω ] ω × [ ω ] ω : ω ≤ β < ω } list all disjoint pairs ofcountable subsets of ω in such a way that for each ω ≤ β < ω , A β ∪ B β ⊆ β .In addition, list [ ω ] ω as { Y α : ω ≤ α < ω } .Let ω ≤ α < ω and assume that for each ω ≤ β < α , the sets X β , a β ⊂ ω havebeen defined such that:1. For each γ ∈ A β : a γ ⊆ ∗ X β , 11. For each γ ∈ B β : a γ ∩ X β = ∗ ∅ ,3. For each γ < α : a γ ⊆ ∗ X β or a γ ∩ X β = ∗ ∅ ,4. If there is γ < β such that | a γ ∩ Y β | = ω , then a β = ∅ . Otherwise, | a β ∩ Y β | = ω ,5. For each η, γ < α , a η ∩ a γ = ∗ ∅ ,Let us construct X α . List α r B α and B α as α r B α = { γ n : n ∈ ω } , B α = { β n : n ∈ ω } . Since A α ∪ B α ⊆ α , then A α ⊆ α r B α and for each n ∈ ω , γ n , β n < α . That is, a γ n , a β n have been defined. In addition, for n ∈ ω , W n = a γ n r [ S j ≤ n a β j ] is either empty of infinite. Define X α = S n ∈ ω W n .Observe that ( A α , B α ) and X α satisfy properties 1. and 2. of the recursiveconstruction.Now let us build a α . Reenumerate { X β : β < α }∪{ X α } as { X n : n ∈ ω } . For n ∈ ω , let X n = X n , X n = ω r X n . If there is γ < α such that | a γ ∩ Y α | = ω ,then let a α = ∅ . On the other hand, if for each γ < α , | a γ ∩ Y α | < ω , for n ∈ ω , pick i ( n ) ∈ { , } so that Y α ∩ T j ≤ n X ji ( j ) is infinite. For each n ∈ ω ,pick p n ∈ (cid:2) Y α ∩ T j ≤ n X ji ( j ) (cid:3) r { p j : j < n } . In this case, let a α = { p n : n ∈ ω } .Since a α ⊆ Y α , then for ech β < α , a β ∩ a α is finite.This finishes the recursive construction of X α and a α . Regardless of whether a α is empty or not, it satisfies properties 4. and 5. In addition, it holds true thatfor each γ, β ≤ α : a γ ⊆ ∗ X β or a γ ∩ X β = ∗ ∅ . Thus, property 3. is satisfied.Let A = { a α : ω ≤ α < ω and a α = ∅} . Observe that properties 4. and 5.guarantee that A is a mad family. Properties 1., 2. and 3. guarantee that A isstrongly ℵ -separated. Hence, A is the desired family. We don’t even have consistent examples to answer the following questions: Question 4.1. Is there a partly-normal not quasi-normal almost disjoint fam-ily? Question 4.2. Is there an almost-normal not normal almost disjoint family? Question 4.3. Is there an almost-normal mad family? If A is mad, Ψ( A ) is a pseudocompact and not countably compact space. Recallthat normal pseudocompact spaces are countably compact and so it is naturalto ask the following more general question Question 4.4. Are almost-normal pseudocompact spaces countably compact? Since Ψ-spaces are always Tychonoff and not countably compact, the existenceof an almost-normal mad family would answer this question in the negative.Finally, we have not considered the relationship between these weakenings ofnormality and countable paracompactness: Question 4.5. Is there a relationship between countably paracompact and anyof these weakenings of normality? cknowledgements The first author was partly supported for this research by the Consejo Nacionalde Ciencia y Tecnolog´ıa CONACYT, M´exico, Scholarship 411689. References [1] S. Arya, M. Singal Almost normal and almost completely regular spaces , Kyung-pook Math. J., Volume 25 ,1 (1970), 141-152.[2] R. Engelking, General Topology , Heldermann Verlag, Berlin, Sigma Series in PureMathematics 6, 1989.[3] F. Galvin, P. Simon, A ˇCech function in ZFC , Fund. Math. 193 (2007), 181-188.[4] L. Gillman, M. Jerison, Rings of Continuous Functions , Van Nostrand, Princeton,NJ. 1960.[5] F. Hern´andez-Hern´andez, M. 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Zurnal, 13(1972), 1182-1196.[14] A.R. Singal, M. K .Singal, Mildly normal spaces , Kyungpook Math. J., 13 (1973),27-31.[15] V. Zaitsev, On certain classes of topological spaces and their bicompactifications ,Dokl. Akad. Nauk SSSR, 178 (1968), 778-779. Department of Mathematics and Statistics, York University, 4700 KeeleSt. Toronto, ON M3J 1P3 Canada Email address : S. A. Garcia-Balan: [email protected] Paul J. Szeptycki: [email protected]@yorku.ca