Guessing clubs for aD, non D-spaces

GUESSING CLUBS FOR aD, NON D-SP A CES DÁNIEL SOUKUP Abstract. W e pro ve tha t there exists a 0-dimensional, scatter ed T 2 space X suc h that X is aD but not linearly D, answ ering a question of Arhangel’skii. The construc tions are based on Shelah’s club guessing principles. 1. Introduction The notion o f a D -sp ac e was probably first in tro duced by v an Douw en and since than, many work ha d b een done in this topic. In vestigating the prop erties o f D - spaces and the connections b etw ee n other cov ering prop erties led to the definition of aD-sp ac es , defined by Ar hangel’skii in [2 ]. As it turned out, pro pe r t y aD is muc h more docile then prop erty D. In [3] Arhangel’skii asked the following: Problem 4. 6. Is there a Tyc honoff aD-space whic h is not a D-space? A neg ative answ er to this question would settle almo s t all o f the questions a b out the relationship of classical covering proper ties to pr op e rty D. Quite similarly , Guo and Junnila in [6] asked the following ab out a weakening of prop erty D: Problem 2. 12. Is every a D-space linea rly D? In G. Gruenha g e’s survey on D-spaces [5], another version of this question is stated (besides the original Arhangel’skii), namely: Question 3.6(2) Is every scattered, aD-spa c e a D-space? The main result of this pap er is the following a ns w er to the questions ab ov e. Theorem 1.1. Ther e exists a 0-dimensional T 2 sp ac e X such that X is sc atter e d, aD and non line arly D. In [9] the author show ed that the existence of a locally countable, lo ca lly compact space X o f size ω 1 which is a D and non linear ly D is independent of ZF C. Here we refine tho s e methods and using Shelah’s club g ues sing theo r y w e answ er the above questions in ZF C. The pap er has the follo wing structure. Sections 2, 3 and 4 ga ther all the necessar y facts ab out D-spa ces, MAD families and c lub guessing. In Sec tion 5 we define spaces X [ λ, µ, M , C ] , where λ a nd µ = cf ( µ ) are ca rdinals, M is a MAD family o n µ a nd C is a guessing sequence. I t is sho wn in Claim 5.2 that 2000 Mathematics Subje ct Classific ation. 54D20, 03E75. Key wor ds and phr ases. D-spaces, aD-spaces, club guessing. 1 2 D. SOUKUP (0) X [ λ, µ, M , C ] is alwa ys T 2 , 0-dimensional and scattered. Section 6 cont ains tw o impor tant results: (1) X [ λ, µ, M , C ] is not linearly D if cf ( λ ) ≥ µ (see Corollary 6.3), (2) X [ λ, µ, M , C ] is aD under certain assumptions (see Corolla ry 6.9). Finally in Sectio n 7 we show how to pro duce suc h spaces X [ λ, µ, M , C ] depending on the cardinal arithmetic and using Shelah’s club guessing. The author would lik e to thank Assaf Rino t for his ideas a nd advices to lo o k deepe r into the theory of club guessing in ZFC. 2. Definitions An op en neighb orho o d assignment (ONA, in short) on a s pace ( X , τ ) is a ma p U : X → τ such that x ∈ U ( x ) for ev ery x ∈ X . X is said to b e a D-sp ac e if for e very neighbor ho o d assig nmen t U , one can find a closed discr ete D ⊆ X s uch that X = S d ∈ D U ( d ) = S U [ D ] (such a set D is called a kernel for U ). In [2] the authors in tro duced prop erty aD : Definition 2.1 . A sp ac e ( X , τ ) is said to b e aD iff for e ach close d F ⊆ X and for e ach op en c over U of X t her e is a close d discr ete A ⊆ F and φ : A → U with a ∈ φ ( a ) for al l a ∈ A such that F ⊆ ∪ φ [ A ] . It is clear that D-spaces are aD. Proving that a spac e is aD, the notion of an irr e ducible sp ac e will play a key ro le. A spa ce X is irr e ducible iff every open cover U has a minimal op en r efi nement U 0 ; meaning that no prop er subfamily of U 0 cov ers X . In [3] Arhangel’skii show ed the following equiv alence. Theorem 2.2 ([3, Theorem 1.8]) . A T 1 -sp ac e X is an aD-sp ac e if and only if every close d subsp ac e of X is irr e ducible. Another genera lization of prop erty D is due to Guo and Junnila [6]. F or a space X a cover U is monotone iff it is linearly or dered by inclusion. Definition 2. 3. A sp ac e ( X , τ ) is said to b e linearly D iff for any ONA U : X → τ for which { U ( x ) : x ∈ X } is monotone, one c an find a close d discr ete set D ⊆ X such that X = S U [ D ] . W e will use the following characterization o f linear D prop erty . A set D ⊆ X is said to be U -big for a cov er U iff there is no U ∈ U such that D ⊆ U . Theorem 2. 4 ([6, Theorem 2.2]) . The fol lowing ar e e quivalent for a T 1 -sp ac e X: (1) X is line arly D. (2) F or every non- trivial monotone op en c over U of X , t her e exists a close d discr ete U -bi g set in X . 3. Notes on MAD f amilies As MAD families will play an essen tial part in o ur constructions w e observe some easy facts about them. Let µ b e any infinite cardinal. W e call M ⊆ [ µ ] µ an almost disjoi nt family if | M ∩ N | < µ for all distinct M , N ∈ M . M is a maximal almost disjoi nt family (in shor t, a MAD family ) if for all A ∈ [ µ ] µ there is some M ∈ M such that | A ∩ M | = µ . W e will use the following rather trivial com binatoria l fact. GUESSING CLUBS FOR aD, NON D-SP ACES 3 Claim 3. 1. L et M ⊆ [ µ ] µ b e a MAD family and M = { M ϕ : ϕ < κ } . Supp ose that N ∈ [ µ ] µ and | N \ ∪M ′ | = µ fo r a l l M ′ ∈ [ M ] <µ . Then | Φ | > µ fo r Φ = { ϕ < κ : | N ∩ M ϕ | = µ } . Pr o of. If | Φ | < µ then with e N = N \ S { M ϕ : ϕ ∈ Φ } ∈ [ µ ] µ we can extend the MAD family , which is a contradiction. If | Φ | = µ then let Φ = { ϕ ζ : ζ < µ } . By transfinite induction, construct e N = { n ξ : ξ < µ } s uch that n ξ ∈ N \ ( S { M ϕ ζ : ζ < ξ } ∪ { n ζ : ζ < ξ } ) for ξ < µ . It is straightforw ard that e N / ∈ M and M ∪ { e N } is almost disjoin t, which is a cont radiction.  F rom our p oint of view the size s of MAD fa milies ar e impo rtant. Clearly there is a MAD family o n ω of size 2 ω . The analo gue of this do es not always hold for ω 1 . Baumgartner in [4 ] proves that it is consistent with ZF C that there is no a lmost disjoint family on ω 1 of size 2 ω 1 . How ever, we have the following fact. Claim 3.2. If 2 ω = ω 1 then ther e is a MAD family M on ω 1 of size 2 ω 1 . In Section 7 we use nonstationary MAD families M N S ⊆ [ µ ] µ meaning that M N S is a MAD family such tha t every M ∈ M N S is nons tationary in µ . Obser ve, that using Z orn’s lemma to almos t disjoint families of no ns ta tionary sets of µ w e can get nonstationary MAD families. 4. Fragments of S helah’s club guessing The constructions o f the upco ming sections will use the follo wing a mazing results of Shelah. F or a ca rdinal λ and a r egular cardinal µ let S λ µ denote the ordina ls in λ with cofinality µ . F o r an S ⊆ S λ µ an S-club se quenc e is a se q uence C = h C δ : δ ∈ S i such that C δ ⊆ δ is a club in δ of order type µ . Theorem 4.1 ([7, Claim 2.3]) . L et λ b e a c ar dinal such that cf ( λ ) ≥ µ ++ for some r e gular µ and let S ⊆ S λ µ stationary. Then t her e is an S -club se quenc e C = h C δ : δ ∈ S i su ch t hat for every club E ⊆ λ ther e is δ ∈ S (e quivalently, stationary many) such that C δ ⊆ E . A detailed pro of of Theorem 4.1 can b e found in [1, Theore m 2.1 7]. Theorem 4.2 ([8, Claim 3.5]) . L et λ b e a c ar dinal such that λ = µ + for some unc oun table, r e gular µ and S ⊆ S λ µ stationary. Then ther e is an S -club se quenc e C = h C δ : δ ∈ S i such t hat C δ = { α δ ζ : ζ < µ } ⊆ δ and for every club E ⊆ λ t her e is δ ∈ S (e qu ivalently, stationary many) such that: { ζ < µ : α δ ζ +1 ∈ E } is stationary . F or a detailed pro o f, s ee [10]. 5. The general construction Definition 5.1. L et λ > µ = cf ( µ ) b e infinite c ar dinals. L et M ⊆ [ µ ] µ b e a MAD family, M = { M ϕ : ϕ < κ } and let C = { C α : α ∈ S λ µ } denote an S λ µ -club se quenc e. W e define a t op olo gic al sp ac e X = X [ λ, µ, M , C ] as fol lows. The u n derlying set of our top olo gy wil l b e a subset of t he pr o duct λ × κ . L et • X α = { ( α, 0) } for α ∈ λ \ S λ µ , • X α = { α } × κ for α ∈ S λ µ , • X = S { X α : α < λ } . 4 D. SOUKUP L et C α = { a ξ α : ξ < µ } denote t he incr e asing enumer ation for α ∈ S λ µ . F or e ach α ∈ S λ µ let • I ξ α = ( a ξ α , a ξ +1 α ] for ξ ∈ suc c ( µ ) ∪ { 0 } , • I ξ α = [ a ξ α , a ξ +1 α ] for ξ ∈ lim( µ ) . Note that S { I ξ α : ξ < µ } = ( a 0 α , α ) is a disjoi nt union. Define t he t op olo gy on X by neighb orho o d b ases as fol lows; (i) for α ∈ S λ µ and ϕ < κ let U ( ( α, ϕ ) , η ) = { ( α, ϕ ) } ∪ [ { X γ : γ ∈ ∪{ I ξ α : ξ ∈ M ϕ \ η }} for η < µ and let B ( α, ϕ ) = { U (( α, ϕ ) , η ) : η < µ } b e a b ase for the p oint ( α, ϕ ) . P S f r a g r e p l a c e m e n t s α ( α, ϕ ) λ X α a ξ α a ξ +1 α (ii) for α ∈ S λ <µ ∪ suc c ( λ ) ∪ { 0 } let ( α, 0 ) b e an isolate d p oint, (iii) for α ∈ S λ µ ′ wher e µ ′ > µ let U ( α, β ) = [ { X γ : β < γ ≤ α } for β < α and let B ( α ) = { U ( α, β ) : β < α } b e a b ase for the p oint ( α, 0) . It is straightforward to c heck that these b asic op en sets form neighbo rho o d bases. ⋆ Fix some cardinals λ > µ = cf ( µ ) , a MAD family M = { M ϕ : ϕ < κ } ⊆ [ µ ] µ and S λ µ -club sequence C . In the following X = X [ λ, µ, M , C ] . Claim 5.2 . The sp ac e X [ λ, µ, M , C ] is 0-dimensional, T 2 and sc atter e d. Observe that (a) X α is close d discr ete for al l α < λ , mor e over (b) S { X α : α ∈ A } is close d discr ete for al l A ∈ [ λ ] <µ , (c) X ≤ α = S { X β : β ≤ α } is clop en for al l α < λ . Pr o of. First we prove that X [ λ, µ, M , C ] is T 2 . Note that ( ∗ ) S { X γ : β < γ ≤ α } is clo pen for all β < α < λ . Thu s ( α, ϕ ) , ( α ′ , ϕ ′ ) ∈ X c a n b e sepa rated trivially if α 6 = α ′ . Suppose that α = α ′ ∈ S λ µ and ϕ 6 = ϕ ′ < κ . There is η < µ such that ( M ϕ ∩ M ϕ ′ ) \ η = ∅ since | M ϕ ∩ M ϕ ′ | < µ . Thus U (( α, ϕ ) , η ) ∩ U (( α, ϕ ′ ) , η ) = ∅ . Next we show tha t X [ λ, µ, M , C ] is 0-dimensio na l. By ( ∗ ) it is enough to prov e that U (( α, ϕ ) , η ) is closed for all α ∈ S λ µ , ϕ < κ and η < µ . Supp ose x = ( α ′ , ϕ ′ ) ∈ GUESSING CLUBS FOR aD, NON D-SP ACES 5 X \ U (( α, ϕ ) , η ) , w e w ant to s eparate x from U (( α, ϕ ) , η ) by an open set. Let α = α ′ . There is η ′ < µ such that ( M ϕ ∩ M ϕ ′ ) \ η ′ = ∅ , thu s U (( α, ϕ ) , η ) ∩ U (( α, ϕ ′ ) , η ′ ) = ∅ . Let α 6 = α ′ . If α ′ ∈ S λ <µ ∪ succ ( λ ) ∪ { 0 } then x is isolated, thus we are do ne. Supp ose α ∈ S λ µ ′ where µ ′ ≥ µ . Then β = sup( C α \ α ′ ) < α ′ th us U ( α ′ , β ) ∩ U (( α, ϕ ) , η ) = ∅ . X [ λ, µ, M , C ] is scattered since X [ λ, µ, M , C ] is right separ ated by the lexico - graphical ordering on λ × κ . (a) and (c) is trivial, we prove (b). Suppose x = ( α ′ , ϕ ′ ) ∈ X , we prove that there is a neighbo rho o d U of x such that | U ∩ S { X α : α ∈ A }| ≤ 1 . If α ′ ∈ S λ <µ ∪ succ ( λ ) ∪ { 0 } then x is isolated, thus we are done. Supp ose α ∈ S λ µ ′ where µ ′ ≥ µ . Then β = sup( A \ α ′ ) < α ′ th us the op en set U = { x } ∪ S { X γ : β < γ < α } will do the job.  6. Focusing on pr oper ty D and aD Again fix some ca r dinals λ > µ = cf ( µ ) , a MAD family M = { M ϕ : ϕ < κ } ⊆ [ µ ] µ and S λ µ -club sequence C . Our next aim is to in vestigate the spaces X = X [ λ, µ, M , C ] concerning prop erty D and aD. Definition 6.1. L et π ( F ) = { α < λ : F ∩ X α 6 = ∅} for F ⊆ X . F is s aid to b e (un)bounded if π ( F ) is (u n )b ounde d in λ . Claim 6.2. If F ⊆ X and π ( F ) ac cu mulates to α ∈ S λ η such that µ ≤ η < λ then F ′ ∩ X α 6 = ∅ . Pr o of. If η > µ then X α = { ( α, 0) } and each neig h b orho o d U ( α, β ) o f ( α, 0) inter- sects F . Thus F ′ ∩ X α 6 = ∅ . Let us supp ose tha t π ( F ) accumulates to α ∈ S λ µ . Since S { I ξ α : ξ < µ } = ( a 0 α , α ) , the set N = { ξ < µ : I ξ α ∩ π ( F ) 6 = ∅} has ca r dinalit y µ . Thu s there is s ome ϕ < κ such that | N ∩ M ϕ | = µ , since M is MAD family . It is straightforw ard that ( α, ϕ ) ∈ F ′ since U (( α, ϕ ) , η ) ∩ F 6 = ∅ for all η < µ .  Corollary 6.3 . If cf ( λ ) ≥ µ then a close d unb ounde d subsp ac e F ⊆ X is not a line arly D-subsp ac e of X . Henc e X [ λ, µ, M , C ] is not a line arly D-sp ac e. Pr o of. Let F ⊆ X b e closed unbounded. | π ( D ) | < µ for every closed discre te D ⊆ X by Claim 6 .2. Thus there is no big closed discrete set for the op en cover { X ≤ α : α < λ } which shows that F is no t linea rly D b y Theorem 2.4.  Our aim now is to prov e that in ce r tain cases the space X [ λ, µ, M , C ] is an aD-space, equiv alently every clo sed subspace of it is irreducible. Claim 6. 4. Every close d, b oun de d subsp ac e F ⊆ X is a D-subsp ac e of X ; henc e F is irr e ducible. Pr o of. W e prov e that F ⊆ X is a D-subspace of X by induction o n α = sup π ( F ) < λ . Let U : F → τ b e an ONA. If α is a s ucc e ssor (or α = 0 ), then F 0 = F \ U (( α, 0)) is closed and sup( F 0 ) < α th us we are easily do ne by induction. Let α ∈ S λ µ ′ where µ ≤ µ ′ < λ . Then sup π ( F 0 ) < α where F 0 = F \ ∪ U [ X α ∩ F ] b y Claim 6.2. Thus we are easily do ne b y induction a nd the fact that X α is closed discrete. Now let ν = cf ( α ) < µ , let sup { α ξ : ξ < ν } = α such tha t α 0 = 0 and { α ξ : ξ < ν } is strictly increasing. Let J ξ = S { X γ : α ξ ≤ γ ≤ α ξ +1 } if ξ < ν is limit or ξ = 0 a nd J ξ = S { X γ : α ξ < γ ≤ α ξ +1 } if ξ < ν is a succe s sor. Let J ν = X α . Clearly { J ξ : ξ ≤ ν } is a discrete family of disjoint clop en sets such that 6 D. SOUKUP S { J ξ : ξ ≤ ν } = X ≤ α . F = S { F ξ : ξ ≤ ν } where F ξ = F ∩ J ξ is clo sed for ξ ≤ ν . By induction, for a ll ξ < ν there is some c losed discrete kernel D ξ ⊆ F ξ for the restriction o f U to F ξ . Let D ν = F ν . Then D = S { D ξ : ξ ≤ ν } is closed disc r ete and F ⊆ ∪ U [ D ] .  T o handle the unbounded clo sed subsets w e need the following definition. Definition 6.5. L et F α = F ∩ X α for F ⊆ X and α < λ . A subset F ⊆ X is high enough if |{ α < λ : | F α | = | F |}| ≥ µ. W e say that a subse t F ⊆ X is high if every close d u nb ounde d subset of F is high enough. The following rather technical claim will be useful. Claim 6.6 . F or any F ⊆ X and ONA U : F → τ such that U ( x ) is a b asic op en neighb orho o d of x ∈ F , let Y F = { x ∈ F : ∃ α < λ : F α ⊆ U ( x ) , | F α | = | F |} , Γ F = { α < λ : | F α | = | F | , ∃ x ∈ F : F α ⊆ U ( x ) } . If F is close d and high enough then Y F , Γ F 6 = ∅ . Pr o of. Since Y F 6 = ∅ iff Γ F 6 = ∅ , it is enough to show that there is some x ∈ Y F . Since F is high enough, | Z | ≥ µ for Z = { α ′ < λ : | F | = | F α ′ |} . Let D = S { F α ′ : α ′ ∈ Z } ⊆ F . Let β ∈ S λ µ be an a c cum ulation p oint of Z = π ( D ) . Then by Claim 6.2 there is some x ∈ D ′ ∩ X β th us x ∈ F . Clea rly x ∈ Y F .  Theorem 6. 7 . If the close d u nb ounde d F ⊆ X is high then F is irr e ducible. Pr o of. Supp ose that U is an o p en cover of F . W e can supp ose that w e re fined it to the form { U ( x ) : x ∈ F } wher e each U ( x ) is basic op en. F rom Claim 6 .6 w e know that Y F , Γ F 6 = ∅ . W e define Y ξ ⊆ F b y induction. • Let α 0 ∈ Γ F and Y 0 = { x ∈ Y F : F α 0 ⊆ U ( x ) } . Fix some h 0 : Y 0 → F α 0 injection; this exists b e c ause | F α 0 | = | F | ≥ | Y F | ≥ | Y 0 | . • Supp ose we defined α ζ < λ and Y ζ for ζ < ξ . Let F ξ = F \  [  U ( x ) : x ∈ ∪{ Y ζ : ζ < ξ }  ∪ X ≤ α  where α = sup { α ζ : ζ < ξ } . • If F ξ is bounded then sto p. Notice that F ξ is b ounded iff F \ S  U ( x ) : x ∈ ∪{ Y ζ : ζ < ξ }  is bounded. • Supp ose F ξ is un b ounded. F ξ ⊆ F is c lo sed either thus F ξ is high enough since F is high. Hence Y F ξ , Γ F ξ 6 = ∅ . • Let α ξ ∈ Γ F ξ ; thus | F ξ α ξ | = | F ξ | and F ξ α ξ is cov ered by so me U ( x ) fo r x ∈ F ξ . Let Y ξ = { x ∈ Y F ξ : F ξ α ξ ⊆ U ( x ) } . Fix s o me h ξ : Y ξ → F ξ α ξ injection; this exists b e c ause | F ξ α ξ | = | F ξ | ≥ | Y F ξ | ≥ | Y ξ | . Lemma 6.8 . The induction stops b efor e µ many st eps. Pr o of. Supp ose we defined this w ay { α ξ : ξ < µ } and let α = sup { α ξ : ξ < µ } ∈ S λ µ . Let D = S { F α ξ : ξ < µ } . By Claim 6 .2 there is some x ∈ D ′ ∩ X α , th us x ∈ F either. Clearly F α ξ ⊆ U ( x ) for µ man y ξ < µ . By the definition of the induction ( ∗ ) for every ζ < ξ < µ a nd every y ∈ Y ζ : F ξ α ξ ∩ U ( y ) = ∅ GUESSING CLUBS FOR aD, NON D-SP ACES 7 Clearly by ( ∗ ) , x / ∈ Y ζ for all ζ < µ since there is ζ < ξ < µ such tha t F ξ α ξ ⊆ U ( x ) . Moreov er x / ∈ U ( y ) for every y ∈ Y ζ and ζ < µ ; if x ∈ U ( y ) then since x 6 = y there is some β < α suc h that S { X γ : β < γ ≤ α } ⊆ U ( y ) . This contradicts ( ∗ ) since there is ζ < ξ < µ such that β < α ξ , thu s F ξ α ξ ⊆ U ( y ) . Thus x ∈ F ξ for all ξ < µ . Then x ∈ Y ξ for all ξ < µ such tha t F α ξ ⊆ U ( x ) . This is a contradiction.  Thu s let us suppo se that the induction stopp e d a t s tep ξ < µ , meaning that e F = F \ S { U ( x ) : x ∈ Y } is b ounded wher e Y = ∪{ Y ζ : ζ < ξ } . Let h = S { h ζ : ζ < ξ } , h : Y → F is a 1-1 function since the sets dom ( h ζ ) = Y ζ and ran ( h ζ ) ⊆ F ζ α ζ are pairwise disjoint for ζ < ξ . Note that ran ( h ) ⊆ S { F α ζ : ζ < ξ } is closed discrete b y Claim 5.2. F or x ∈ Y let U 0 ( x ) = ( U ( x ) \ ran ( h )) ∪ { h ( x ) } , note that U 0 ( x ) is op en. Then [ { U 0 ( x ) : x ∈ Y } = [ { U ( x ) : x ∈ Y } is a minimal op en refinement, since h ( x ) is only covered b y U 0 ( x ) for all x ∈ Y . Let U 0 = { U 0 ( x ) : x ∈ Y } Let V ( x ) = U ( x ) \ S { F α ζ : ζ < ξ } . Then V = { V ( x ) : x ∈ e F } is a n op en cover o f e F , refining U ; F α ζ ∩ e F = ∅ b y co ns tr uction for all ζ < ξ . e F is closed and b ounded th us irreducible by Claim 6.4, hence there is an ir reducible op en refinement V 0 of V . It is straightforward that V 0 ∪ U 0 is a minimal op en refinemen t of U covering F .  Corollary 6.9. Supp ose that λ > µ = cf ( µ ) ar e infinite c ar dinals su ch that cf ( λ ) ≥ µ . L et M = { M ϕ : ϕ < κ } ⊆ [ µ ] µ b e a MAD family and C an S λ µ -club se quenc e. If X [ λ, µ, M , C ] is high then X [ λ, µ, M , C ] is a 0-dimensional, Hausdorff, sc att er e d sp ac e which is aD however n ot line arly D. Pr o of. X [ λ, µ, M , C ] is 0-dimensional, Ha us dorff a nd scattered b y Claim 5.2 and not linearly D by Coro llary 6.3. It suffices to show that ev ery closed F ⊆ X is irreducible. If F is b ounded then F is a D-spa ce by Claim 6.4 hence irreducible. If F is unbounded, then F is high since X is high. Hence F is irr educible by Theorem 6.7.  7. Examples of aD, non linearl y D-sp a ces In this section we give examples of aD, non linea r ly D-spa ces of the form X = X [ λ, µ, M , C ] . First let us make an observ atio n. Claim 7.1. If C α ⊆ π ( F ) ′ for a close d F ⊆ X and α ∈ S λ µ , t hen F α = X α . Pr o of. Clearly S { X γ : γ ∈ I ξ α } ∩ F 6 = ∅ for all ξ < µ . Thus every po in t in X α is a n accumulation p oint of F , thus F α = X α since F is closed.  Corollar ies 7.3 and 7.5 below give certain ex amples of high X [ λ, µ, M , C ] spaces . Prop osition 7.2. Su pp ose that µ is a r e gu lar c ar dinal, cf ( λ ) ≥ µ ++ . L et C b e an S λ µ -club guessing se quenc e fr om The or em 4.1 . If M ⊆ [ µ ] µ is a MAD family of size at le ast λ then X [ λ, µ, M , C ] is high. 8 D. SOUKUP Pr o of. Let F ⊆ X closed, un b ounded. Then π ( F ) ′ is a club in λ , hence there exists a stationary S ⊆ S λ µ such that C α ⊆ π ( F ) ′ for all α ∈ S . Thus F α = X α b y Cla im 7.1 hence | F α | = |M| = | X | for all α ∈ S .  Corollary 7.3. (1) Supp ose that 2 ω ≥ ω 2 . L et M b e a MAD family on ω of size 2 ω and let C b e an S ω 2 ω -club guessing se quenc e fr om The or em 4.1. Then X [ ω 2 , ω , M , C ] is high. (2) Supp ose t hat 2 ω = ω 1 and 2 ω 1 ≥ ω 3 . L et M b e a MAD family on ω 1 of size 2 ω 1 (exists by Claim 3.2) and let C b e an S ω 3 ω 1 -club guessing se quenc e fr om The or em 4.1. Then X [ ω 3 , ω 1 , M , C ] is high. Prop osition 7.4 . Su pp ose that λ = µ + > µ = cf ( µ ) > ω and let C b e an S µ + µ - club guessing se quenc e fr om The or em 4.2. If ther e is a nonst ationary MAD family M N S ⊆ [ µ ] µ such that |M N S | = µ + then X = X [ µ + , µ, M N S , C ] is high. Pr o of. Let M N S = { M ϕ : ϕ < µ + } and C = h C α : α ∈ S µ + µ i such that C α = { a ξ α : ξ < µ } ⊆ α . Suppo s e that the clo s ed F ⊆ X is unbounded. Then π ( F ) ′ is a c lub in µ + , hence there exists a stationary S ⊆ S µ + µ such that N α = { ξ < µ : a ξ +1 α ∈ π ( F ) ′ } is stationary in µ for all α ∈ S . Fix any α ∈ S , we prov e that | F α | = | F | . N α is stationary so b y applying Claim 3 .1 we g e t that | Φ α | = µ + for Φ α = { ϕ < µ + : | N α ∩ M ϕ | = µ } . Note that F ∩ S { X γ : γ ∈ I ξ α } 6 = ∅ for ξ ∈ N α . Thus ( α, ϕ ) is an accumulation po in t of F for ϕ ∈ Φ α , hence { α } × Φ α ⊆ F α . Thus | F α | = µ + = | X | .  Corollary 7.5. Su pp ose that 2 ω 1 = ω 2 . L et C b e an S ω 2 ω 1 -club guessing se quenc e fr om The or em 4.2 and let M N S b e a n onstationary MAD family on ω 1 . Then X [ ω 2 , ω 1 , M N S , C ] is high. Thu s, by all mea ns w e can deduce the pro of of Theor em 1.1. Pr o of of The or em 1.1. Note that in any mo del of ZFC, either (2 ω ≥ ω 2 ) or (2 ω = ω 1 ∧ 2 ω 1 ≥ ω 3 ) or (2 ω 1 = ω 2 ) . Using Co r ollaries 7.3 a nd 7.5 ab ov e, depe nding on the size s o f 2 ω and 2 ω 1 , we see that there exists a high X [ λ, µ, M , C ] s pace. W e are done b y Corolla ry 6.9.  References [1] U. Abraham, M. Magidor, Cardinal Arithmetic – Handbo ok of Set Theory (Eds. F oreman, Kanamori), V olume 2, 1149-1229 (ht tp://www.cs.bgu.ac.il/ ∼ abraham/papers/math/Pcf.dvi ) [2] A. V .Arhangel’skii and R. Buzy ak ov a, Addition theorems and D-spaces, Commen t. Mat. Univ. Car. 43(2002), 653-663. [3] A. V . Arhangel’skii, D-spaces and cov ering prop erties, T op ology and Appl. 146-147(2005), 437- 449. [4] J. E. Baumgartne r, Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9 (1976), no. 4, 401-439. [5] G. Gruenhage, A survey of D-spaces, to app ear. [6] H. Guo and H.J.K. Junn ila, On space s which are linearly D, T op ology and Appl., V olume 157, Issue 1, 1 Jan uary 2010, Pages 102-107. [7] S. Shelah, Cardinal Arithmetic – Oxf ord Logic Guides, vol. 29, Oxford Uni v. Press (1994) [8] S. Shelah, Colouring and non-productivity of ℵ 2 -c.c. – Annals Pure and Applied Logic 84 (1997) 153-174 [9] D. Soukup, Properties D and aD are different, submitted to T op. Proc. [10] D. Soukup, L. Soukup, Club guessing for dummies, Arxiv note GUESSING CLUBS FOR aD, NON D-SP ACES 9 Eötvös Lóránd Univer sity E-mail addr ess : daniel.t.souku p@gmail.com