math.GN 2011-06-28 0

A counterexample in the theory of $D$-spaces

Assuming $\diamondsuit$, we construct a $T_2$ example of a hereditarily Lindel\"of space of size $\omega_1$ which is not a $D$-space. The example has the property that all finite powers are also Lindel\"of.

Authors: Daniel T. Soukup, Paul J. Szeptycki

A COUNTEREXA MPLE IN THE TH EOR Y OF D -SP A CES D ÁNIEL T. SOUKUP AND P AUL J. SZEPTYCKI Abstract. Assuming ♦ , we construct a T 2 example o f a her edi- tarily Lindelöf space of size ω 1 which is not a D - space. The example has the prop erty that all finite pow ers are also Lindelöf. 1. In tr oduction The follow ing notion is due to v an D ou w en, first studied with Pfeffer in [3]. Definition 1.1. A T 1 sp ac e X is said to b e a D -space if for e ac h op en neighb ourho o d as signment { U x : x ∈ X } ther e is a close d and discr ete subset D ⊆ X such that { U x : x ∈ D } c overs the s p ac e. The question whether ev ery regular Lindelöf space is D has b een at- tributed to v an Dou wen [8]. Moreo v er, v an Dou w en and Pfeffer p o in ted out that "No satisfactory example of a space whic h is not a D - space is kno wn, where b y satisfactory example w e mean an example ha ving a co v ering prop ert y at least as strong as metacompactne ss or subparacompactnes s." Indeed, the lac k of satisfactory examples of D -spaces satisfying some in teresting co v ering prop erties con tin ues and there has b een quite a bit of activit y in the area in the last decades (see the surv eys [5] and [8] for other related results and op en pro blems). Whether regular Lindelöf spaces are D -spaces was listed as Problem 14 in Hrušák and Mo ore’s list of 2 0 op en problems in set-theoretic top olog y [10], and there are no consistency results in either direction ev en f or here ditarily Lindelöf spaces. The question whether Lindelöf implies D for the class o f T 1 spaces w as also op en a nd explicitly ask ed in [6] and more recen tly in [1]. 2010 Mathematics S ubje ct Classific ation. 54D20 , 54A3 5 . Key wor ds and phr ases. D -space s , Lindelöf spaces . The second author ackno wledges supp ort from NSERC gr a nt 238944. Corresp onding author: Dániel T. Soukup. 1 2 DÁNIEL T . SOUKU P AND P AUL J . SZEPTYCKI In this note, assuming ♦ , w e construct an example of a hereditarily Lindelöf T 2 space that is not a D - space. The example also has the prop ert y that ev ery finite p o w er is Lindelöf, but w e do not kno w if it can b e made regular. The ar ticle is structured as follow s; in Section 2 we gather a few gen- eral facts and definitions, and in Section 3 w e presen t the construction. In Section 4, w e mak e some remarks and prov e f urther prop erties of our cons truction. Finally , in Section 5 we state a few o p en problems. 2. Preliminaries Delicate use of elemen tary submo dels pla y crucial role in our argu- men ts. W e do not in tend to giv e a precise introduction to this p ow erful to ol since elemen tary submo dels are widely used in top o lo gy no wada ys; let us refer to [4]. How ev er, w e presen t here a few easy facts and a lemma whic h could serv e a s a w arm-up exercise for the readers less in v olve d in the use o f ele men tary submodels. Let H ( ϑ ) denote the sets which hav e transitiv e closure of size less than ϑ for some cardinal ϑ . The following facts will b e used regularly without exp licitly referring to them. F act 2.1. Supp ose that M ≺ H ( ϑ ) for some c ar dinal ϑ and M is c ountable. (a) I f F ∈ M and ther e is F ∈ F \ M then F is unc ountable. (b) I f B ∈ M and B is c ountable then B ⊆ M . The next lemma is w ell-kno wn, nonetheless w e presen t a pro of. Lemma 2.2. L et F ⊆ [ ω 1 ] <ω and s upp ose that M is a c o untable el- ementary submo del o f H ( ϑ ) for some c ar dinal ϑ such that F ∈ M . If ther e is an F ∈ F s uch that F / ∈ M then ther e is an unc ountable ∆ -system G ⊆ F in M with kernel F ∩ M . Mor e over, if ψ ( x, ... ) is any form ula with p ar ameters fr om M and ψ ( F , ... ) holds then G c an b e chosen in such a wa y that ψ ( G, ... ) holds for every G ∈ G , as wel l. Pr o of. Supp o se that F , M , F ∈ F \ M and ψ is as ab ov e. Let D = F ∩ M and let F 0 = { G ∈ F : D ⊆ G and ψ ( G, ... ) } . Clearly , F ∈ F 0 ∈ M and F / ∈ M thu s F 0 is uncoun table. Moreo ver, F ∩ α = D for all α in a tail of M ∩ ω 1 ; that is, ∃ G ∈ F 0 : G ∩ α = D and this holds in M as w ell, b y elemen tary . Th us M | = ∃ β < ω 1 ∀ α ∈ ( β , ω 1 ) ∃ G ∈ F 0 : G ∩ α = D . A COUNTEREXAMPLE IN THE THEOR Y OF D - SP AC ES 3 Th us this holds in H ( ϑ ) as w ell, by eleme n tary . Hence w e can select inductiv ely an uncoun t a ble ∆ -system from F 0 . Using elemen ta ry again, there is suc h a ∆ -sys tem in M to o.  F or an y set-theoretic notion, inc luding bac kground on ♦ , see [11]. There are different conv en t io ns in general topolo gy whether to add regularit y to the definition of a Lindelöf space . In t his article, any top ological space X is said to b e Lindelöf iff ev ery o p en cov er has a coun ta ble subco v er; that is, no separation is assume d. Finally , w e need a f ew other definitions. Definition 2.3. A c ol le ction U of subsets of a sp ac e X is c al le d an ω - co v er if for every finite F ⊆ X ther e is U ∈ U such that F ⊆ U . Definition 2.4. A c ol l e ction N of subsets of a s p ac e X is c al le d a lo cal π -netw ork at the p oin t x if for e ach op en neighb ourho o d U of x in X , ther e is an N ∈ N such that N ⊆ U (it is not r e quir e d that the sets in N b e op en, nor that they c ontain the p oint x ). 3. T he Constr uction W e construct a top ology b y constructing a sequence { U γ : γ < ω 1 } of subsets of ω 1 suc h that γ ∈ U γ for eve ry γ ∈ ω 1 . The example will b e obtained by first taking the family { U γ : γ < ω 1 } as a subbasis for a top o logy o n ω 1 and then refining it with a Hausdorff top ology of coun ta ble w eigh t. The follo wing lemma will b e used to pro v e the Lindelöf prop erty . Lemma 3.1. Consider a top olo gy on ω 1 gener ate d by a family { U γ : γ < ω 1 } as a subb ase ; sets of the form U F = T { U γ : γ ∈ F } for F ∈ [ ω 1 ] <ω form a b ase. If for every unc ountable family B ⊆ [ ω 1 ] <ω of p airwise disjoin t s e ts ther e is a c ountable B ′ ⊆ B such that | ω 1 \ [ { U F : F ∈ B ′ }| ≤ ω then the top olo gy i s her e ditarily Lind elöf. Pr o of. Fix an op en family U ; w e ma y assu me that U = { U F : F ∈ G } for some G ⊆ [ ω 1 ] <ω . Let M b e a countably elemen tary submo del of H ( ϑ ) for some suffi cien tly large ϑ suc h that { U γ : γ ∈ ω 1 } , G ∈ M . It suffices to prov e that S U is co v ered by the coun table f a mily V = { U F : F ∈ G ∩ M } . V clearly co v ers  S U  ∩ M th us we consider an arbitrary α ∈  S U  \ M . Th ere is G 0 ∈ G suc h that α ∈ U G 0 ; let F = G 0 ∩ M . If F = G 0 then V cov ers α thus w e a re done. 4 DÁNIEL T. SOU KUP AND P AUL J. SZEPTYCKI Otherwise there is an uncoun table ∆ -system D ⊆ G in M with k ernel F b y Lemma 2.2. Consider the uncoun table, pairwise disjoint family B = { G \ F : G ∈ D } ; b y o ur h yp othesis there is a coun table B ′ ⊆ B suc h that | ω 1 \ [ { U H : H ∈ B ′ }| ≤ ω . B ′ can b e c hosen in M since B ∈ M ; note that B ′ ⊆ M . Th us the coun ta ble set of p oin ts not co vere d also lie in M . Therefore, there is G \ F ∈ B ′ suc h that α ∈ U G \ F . Hence α ∈ U G since α ∈ U G 0 ⊆ U F and U G = U G \ F ∩ U F . This completes the pro of of the lemma.  Let us defi ne no w the top ology whic h will b e used to ensure the Hausdorff prop ert y . Definition 3.2. De fine a top olo gy on [ R ] <ω as fol lows. L et Q ⊆ R b e a Euclide an op en set and let Q ∗ = { H ∈ [ R ] <ω : H ⊆ Q } . Sets of the form Q ∗ define a ρ top olo gy on [ R ] <ω . The pro of of the follo wing claim is straigh tforw ard. Claim 3.3. (1) ([ R ] <ω , ρ ) is of c ountable weight, (2) an y fam ily X ⊆ [ R ] <ω of p airwise disjo int n o nempty sets forms a Hausdorff subsp ac e of ([ R ] <ω , ρ ) . Let us fix the coun table base W = { Q ∗ : Q is a disjoin t union of finitely many in terv als with rational endp oints } for ([ R ] <ω , ρ ) . F or the remainder o f t his section w e supp ose ♦ ; th us 2 ω = ω 1 and w e can fix an enume ration • { C α } α<ω 1 = [ ω 1 ] ≤ ω suc h that C α ⊆ α for all α < ω 1 . Also, there is a ♦ -seque nce { B γ } γ <ω 1 on [ ω 1 ] <ω ; tha t is, • for ev ery uncoun table B ⊆ [ ω 1 ] <ω there are stationary man y β ∈ ω 1 suc h that B ∩ [ β ] <ω = B β . The next theorem is t he k ey to our main result; w e encourage the reader to first skip the quite tec hnical proo f of Theorem 3.4 and go to Corollary 3.8 to see how our main resu lt is deduced. In particular, IH (3) assure s the space is not a D -space and IH (4) makes the space hereditarily Lindelöf. Theorem 3.4. Th er e e x ist { U α γ } γ ≤ α and ϕ α : ( α + 1) → [ R ] <ω for α < ω 1 with the fo l lowing pr op erties: IH (1) U α γ ⊆ α + 1 and U α α = α + 1 for every γ ≤ α < ω 1 , and the family ϕ α [ α + 1] is p airwise disjoint for every α < ω 1 . IH (2) U α γ = U α 0 γ ∩ ( α + 1) and ϕ α = ϕ α 0 ↾ ( α + 1) f o r al l γ ≤ α ≤ α 0 . A COUNTEREXAMPLE IN THE THEOR Y OF D - SP AC ES 5 L et τ α denote the top olo gy gene r ate d by the sets { U α γ : γ ≤ α } ∪ { ϕ − 1 α ( W ) : W ∈ W } as a subb ase. L et U α F = T { U α γ : γ ∈ F } for F ∈ [ α + 1] <ω . IH (3) I f C α is τ α close d di s cr ete then S { U α γ : γ ∈ C α } 6 = α + 1 . IH (4) L et T α = { β ≤ α : B β is a p airwise disjoint family of fi nite subsets of β and ther e is a c o untable elementary submo del M ≺ H ( ϑ ) for some sufficiently lar ge ϑ such that (i) - ( v) holds fr om b elow } . (i) M ∩ ω 1 = β , (ii) W , { B γ } γ <ω 1 ∈ M , (iii) ther e is a function ϕ ∈ M such that ϕ ↾ β = ϕ β ↾ β , (iv) ther e is an unc ountable B ∈ M such that M ∩ B = B β , and (v) ther e is a { V γ } γ <ω 1 ∈ M such that V γ ∩ β = U β γ ∩ β for al l γ < β . Then (a) if β ∈ T α then B β is a lo c al π - n etwork at β in τ α , (b) if β ∈ T α ∩ α then for every V ∈ τ α with β ∈ V the family { U α F : F ∈ B β , F ⊆ V } is an ω -c over of ( β , α ] . Pr o of. W e prov e by induction on α < ω 1 with inductional hy p othesises IH (1)- IH (4)! Supp ose w e constructed { U β γ } γ ≤ β for β < α . Let U <α γ = ∪{ U β γ : γ ≤ β < α } and ϕ <α = ∪{ ϕ β : β < α } . Let τ − α denote the topolog y on α generated by the sets { U <α γ : γ < ω 1 } ∪ { ϕ − 1 <α ( W ) : W ∈ W } as a sub base. Let U <α F = T { U <α γ : γ ∈ F } for F ∈ [ α ] <ω . Note that if β ∈ T α ∩ α then β ∈ T α ′ for ev ery α ′ ∈ ( β , α ) ; hence N (i) and N (ii) b elo w holds by IH (4): N (i) fo r eve ry V ∈ τ − α with β ∈ V the family { U <α F : F ∈ B β , F ⊆ V } is an ω - cov er of ( β , α ) ; N (ii) B β is a lo cal π -net w ork at β in τ − α . Therefore, it suffices to define U α α = α + 1 and U α γ =  U <α γ or U <α γ ∪ { α } and ϕ α ↾ α = ϕ <α , ϕ α ( α ) = x α ∈ [ R ] <ω suc h that 6 DÁNIEL T. SOU KUP AND P AUL J. SZEPTYCKI D (i) x α is disjoin t from x β = ϕ <α ( β ) for all β < α , D (ii) if β ∈ T α ∩ α then for eve ry β ∈ V ∈ τ α the family { U α F : F ∈ B β , F ⊆ V } is an ω -cov er of ( β , α ] , D (iii) if C α is τ − α closed discrete then α / ∈ U α γ for all γ ∈ C α , D (iv) if α ∈ T α then B α is a lo cal π -netw ork at α in τ α . Case I. T α ∩ α = ∅ Let U α α = α + 1 , U α γ = U <α γ for γ < α . W e pro ceed differen tly according to whether α / ∈ T α or α ∈ T α . Sub case A. α / ∈ T α Pic k an y x α ∈ [ R ] <ω disjoin t from x β for all β < α . Clearly , D ( i)- D (iv) ar e satisfied. Sub case B. α ∈ T α It is clear that D (ii) and D (iii) are satisfied. Let M b e a coun ta ble elemen tary submo del of H ( ϑ ) for some sufficie n tly large ϑ sho wing that α ∈ T α . T o find the appropriate x α ∈ [ R ] <ω w e need that B α is a lo cal π -netw ork at α in τ α . Since { ϕ − 1 α ( W ) : W ∈ W , x α ∈ W } will b e a base at α in τ α w e need that for all W ∈ W suc h that x α ∈ W there is an F ∈ B α suc h that ϕ [ F ] ⊆ W . Since ϕ [ F ] ⊆ W iff ∪ ϕ [ F ] ∈ W , w e need to find an accum ulation p oin t of t he finite sets {∪ ϕ [ F ] : F ∈ B α } . W e pro ve the follo wing whic h will suffice : Claim 3.5. Ther e is an x α ∈ [ R ] <ω such that x α ∩ x β = ∅ for al l β < α and for al l W ∈ W such that x α ∈ W we have M | = |{ F ∈ B : ∪ ϕ [ F ] ∈ W }| > ω . Indeed, D (i) is satisfied. Let us c hec k D (iv); clearly , M | = ∃ F ∈ B : ∪ ϕ [ F ] ∈ W for ev ery W ∈ W suc h that x α ∈ W . Hence there is F ∈ B ∩ M = B α suc h that ∪ ϕ [ F ] = ∪ ϕ α [ F ] ∈ W , that is ϕ α [ F ] ⊆ W . Th us B α is a lo cal π -net w ork at α in τ α ; that is, D (iv) is satisfied. A COUNTEREXAMPLE IN THE THEOR Y OF D - SP AC ES 7 Pr o of of Claim 3.5. Since M | = | B | > ω there is e B ∈ [ B ] ω 1 ∩ M and k ∈ ω , { n i : i < k } ⊆ ω suc h that | F | = k for all F ∈ e B and if F = { γ i : i < k } then | ϕ ( γ i ) | = n i for all i < k . Let s = P i ω for ev ery W ∈ W with x ∈ W  . Hence, there is x α ∈ [ R ] s disjoin t from x β for a ll β < α suc h t hat |{ F ∈ e B : ∪ ϕ [ F ] ∈ W }| > ω for all W ∈ W with x α ∈ W . Th us M | = |{ F ∈ e B : ∪ ϕ [ F ] ∈ W }| > ω whic h we w an ted to pro v e.  Case I I. T α ∩ α 6 = ∅ Let T α ∩ α = { β n : n ∈ ω } and { G n : n ∈ ω } ⊆ [ α ] <ω suc h that fo r all β ∈ T α ∩ α and G ⊆ ( β , α ) there are infinitely man y n ∈ ω suc h that β = β n and G = G n . Let { V k ( β ) : k < ω } denote a decreasing neigh b ourho o d base for the p oint β ∈ T α ∩ α in τ − α . Note that { V n ( β n ) : n ∈ ω , β n = β } is a base for β ∈ T α ∩ α . Sub case A. α / ∈ T α W e need the followin g claim: Claim 3.6. T her e is F n ∈ B β n for n ∈ ω s uch that A (i) F n ⊆ V n ( β n ) , A (ii) G n ⊆ U <α F n , A (iii) F n ∩ C α = ∅ if C α is τ − α close d di s cr ete, Pr o of. There is V ∈ τ − α suc h that β n ∈ V ⊆ V n ( β n ) and if C α is close d discrete, then C α ∩ V ⊆ { β n } . The family { U <α F : F ∈ B β n , F ⊆ V } is an ω -co v er of ( β n , α ) b y N (ii), th us there is F n ∈ B β n suc h that F n ⊆ V and G n ⊆ U <α F n .  8 DÁNIEL T. SOU KUP AND P AUL J. SZEPTYCKI Let U α α = α + 1 and f o r γ < α let U α γ = ( U <α γ if γ / ∈ ∪{ F n : n ∈ ω } , U <α γ ∪ { α } if γ ∈ ∪{ F n : n ∈ ω } . Pic k any x α ∈ [ R ] <ω disjoin t from x β for all β < α . D (i), D (iii), and D (iv) are trivially satisfied. Let us c hec k D (ii); fix β ∈ T α ∩ α , an y neigh b ourho o d V ∈ τ α suc h that β ∈ V , and a finite subset G ⊆ ( β , α ) . W e sho w that there is an F ∈ B β , suc h that U α F co v ers G ∪ { α } and F ⊆ V . There is n ∈ ω suc h that β n = β , G n = G , and V n ( β n ) ⊆ V ; then F n ∈ B β n and F = F n do es the job b y A (i), A (ii) and the fact that α ∈ U α F n . Sub case B. α ∈ T α Let M b e a coun table elemen tary submo del of H ( ϑ ) f o r some suf- ficien tly large ϑ show ing that α ∈ T α . Since M | = | B | > ω there is e B ∈ [ B ] ω 1 ∩ M and k ∈ ω , { n i : i < k } ⊆ ω suc h that | F | = k fo r all F ∈ e B and if F = { γ i : i < k } then | ϕ ( γ i ) | = n i for all i < k . Let s = P i ω . Pr o of. W e construct F n and W n b y induction on n ∈ ω . Supp ose we constructed F k and W k for k < n suc h that the hypothesises B (i)- B (vii) ab ov e are satisfied. Let D =  F ∈ e B : F ⊆ ∩{ V F k : k < n } and ∪ ϕ [ F ] ∈ W n − 1  if n > 0 and D = e B if n = 0 ; then M | = | D | > ω . Just as in Claim 3.5 M | =  there are uncoun tably man y pairwise disjoin t x ∈ R s ∩ W n − 1 suc h that |{ F ∈ D : ∪ ϕ [ F ] ∈ W }| > ω for ev ery W ∈ W with x ∈ W  . A COUNTEREXAMPLE IN THE THEOR Y OF D - SP AC ES 9 Cho ose x ∈ R s ∩ W n − 1 suc h that |{ F ∈ D : ∪ ϕ [ F ] ∈ W }| > ω for ev ery W ∈ W with x ∈ W and x ∩ ϕ ( α n ) = ∅ . Let x = { x i : i < s } and c ho ose W n =  ∪{ Q n,i : i < s }  ∗ suc h that • Q n,i is a rational inte rv al of diameter less then 1 n for ev ery i < s , • x i ∈ Q n,i for ev ery i < s , • Q n,i ⊆ Q n − 1 ,i for ev ery i < s in the Euclidean top ology (if n > 0 ), • ∪{ Q n,i : i < s } ∩ ϕ ( α n ) = ∅ . Let D ′ =  F ∈ e B : F ⊆ ∩{ V F k : k < n } , ∪ ϕ [ F ] ∈ W n and β n < min F  ; clearly , M | = | D ′ | > ω . Let V ∈ τ − α suc h that β n ∈ V ⊆ V n ( β n ) and V ∩ C α ⊆ { β n } if C α is τ − α closed discrete. Applyin g N (i) to F ∪ G n for F ∈ D ′ ∩ M giv es us that there is F n ( F ) ∈ B β n suc h that F n ( F ) ⊆ V and U <α F n ( F ) co v ers F ∪ G n and hence V F n ( F ) co v ers F ∪ G n . Th us M | =  for ev ery F ∈ D ′ there is F n ( F ) ∈ B β n suc h that F n ( F ) ⊆ V and V F n ( F ) co v ers F ∪ G n .  Finally , not e that M | = | B β n | ≤ ω ; th us M | =  there is F n ∈ B β n suc h that F n ⊆ V and V F n co v ers F ∪ G n for uncoun tably man y F ∈ D ′  . It is no w easily c hec k ed that F n and W n satisfies prop erties B (i)- B (vii).  Let U α α = α + 1 and f o r γ < α let U α γ = ( U <α γ if γ / ∈ ∪{ F n : n ∈ ω } , U <α γ ∪ { α } if γ ∈ ∪{ F n : n ∈ ω } . Let x α ∈ [ R ] <ω b e the unique s -elemen t subse t of R in the in tersec- tion ∩{∪{ Q n,i : i < s } : n ∈ ω } ; ex istence and uniqueness follo ws from B (iv) and B (v), and x α is disjoint from x β for all β < α b y B (vi). Note that  ∩{ U α F k : k ≤ n } ∩ ϕ − 1 α ( W n ) : n ∈ ω  is a base for the p oint α in τ α . D (i) is satisfied b y B (vi) and the fact that x α ⊆ ∪{ Q n,i : i < s } . Let us c hec k D (ii); fix β ∈ T α ∩ α , any neighbourho o d V ∈ τ α suc h that β ∈ V , and a finite subset G ⊆ ( β , α ) . W e sho w that there is a n F ∈ B β , suc h that U α F co v ers G ∪ { α } and F ⊆ V . There is n ∈ ω suc h 10 DÁNIEL T. SOU KUP AND P AUL J. SZEPTYCKI that β n = β , G n = G , a nd V n ( β n ) ⊆ V ; F n ∈ B β n do es the job by B (i), B (ii) and the fact that α ∈ U α F n . D (iii) is satisfied b y B (iii) and the definition of U α γ . Finally , let us chec k D (iv); it suffices to sho w that for ev ery n ∈ ω there is F ∈ B α suc h F ⊆ ∩{ U α F k : k ≤ n } ∩ ϕ − 1 α ( W n ) . Condition B (vii) giv es us this, using the observ ation that ϕ α [ F ] ⊆ W iff ∪ ϕ α [ F ] ∈ W for an y F ∈ B α and W ∈ W . By all means, this completes the pro o f of the theorem.  No w w e are ready to deduce our main result. Corollary 3.8. Supp ose that { U α γ } γ ≤ α and ϕ α : ( α + 1) → [ R ] <ω for α < ω 1 ar e as in The or em 3.4 and let U γ = ∪{ U α γ : γ ≤ α < ω 1 } f o r γ < ω 1 and ϕ = ∪{ ϕ α : α < ω 1 } . L et τ denote the top olo g y on ω 1 gener ate d by the sets { U γ : γ < ω 1 } ∪ { ϕ − 1 ( W ) : W ∈ W } as a subb ase. The sp ac e ( ω 1 , τ ) is her e d itarily Lin delöf, Hausdorff but not a D - sp ac e. A lso , ( ω 1 , τ ) has c ountable Ψ -weigh t. Pr o of. First, w e sho w that ( ω 1 , τ ) is hereditarily L indelöf and Haus- dorff. W e need the follow ing observ a t io n. Claim 3.9. A Hausdorff top olo g y of c ountable w eight τ sc r efine d by a her e ditarily Lind elöf top olo gy τ hl on some s e t X is again a her e ditarily Lindelöf, Hausdorff top ol o gy on X . Pr o of. Let τ ref denote the to p ology generated b y τ sc ∪ τ hl as a su bbase; that is, τ ref is the common refinemen t of τ sc and τ hl . τ ref is clearly Haus- dorff, w e pro v e that for an y open family U ⊆ τ ref there is a countable U 0 ⊆ U suc h that ∪U 0 = ∪U . W e can suppose that U = { U i ∩ V i j : i ∈ ω , j ∈ I i } where { U i : i ∈ ω } ⊆ τ sc and { V i j : i ∈ ω , j ∈ I i } ⊆ τ hl for some index sets { I i : i ∈ ω } . F or ev ery i ∈ ω there is a coun t a ble J i ⊆ I i suc h that U i ∩ [ { V i j : j ∈ I i } = U i ∩ [ { V i j : j ∈ J i } b y the hereditarily Linelöfness of τ hl . Th us ∪U = [ { U i ∩ V i j : i ∈ ω , j ∈ J i } A COUNTEREXAMPLE IN THE THEOR Y OF D - SP AC ES 11 whic h completes the pro of.  Therefore, it suffices to pro v e that the top ology generated b y { U γ : γ < ω 1 } as a subbase on ω 1 is hereditarily Lindelöf. L emma 3.1 and the prop osition b elow giv es us this result. Let U F = T { U γ : γ ∈ F } for F ∈ [ ω 1 ] <ω . Prop osition 3.10. F or any unc ountable family of p airwise disjoint sets B ⊆ [ ω 1 ] <ω , ther e is a c ountable B ′ ⊆ B such that { U F : F ∈ B ′ } is a c over, mor e ov er an ω -c over of a tail of ω 1 . Pr o of. Fix some uncoun table family B ⊆ [ ω 1 ] <ω of pairwise disjoint sets. There is an M ≺ H ( ϑ ) for some sufficien tly large ϑ suc h that B , ϕ, { U γ : γ < ω 1 } , { B γ : γ < ω 1 } , W ∈ M and M ∩ ω 1 = β and B ∩ M = B ∩ [ β ] <ω = B β . W e claim that S { U F : F ∈ B ′ } is an ω -cov er of ω 1 \ ( β + 1) for the coun ta ble B ′ = B β . Indeed, fix some finite K ⊆ ω 1 \ ( β + 1) and let α ∈ ω 1 \ ( β + 1) suc h that K ⊆ α . Then β ∈ T α ensured b y the mo del M , and hence there is some F ∈ B β = B ′ suc h that K ⊆ U α F ⊆ U F b y IH (4).  No w w e pro v e that ( ω 1 , τ ) is not a D -space. Consider the neigh b our- ho o d assignmen t γ 7→ U γ ; w e sho w that ∪{ U γ : γ ∈ C } 6 = ω 1 for ev ery closed discrete C ⊆ ω 1 . Since ( ω 1 , τ ) is Lindelöf, | C | ≤ ω and hence there is α < ω 1 suc h that C α = C . It suffices to note that C α is τ α closed discrete if τ closed discrete; indeed, then ∪{ U γ : γ ∈ C α } 6 = α + 1 b y IH (3). Finally , ( ω 1 , τ ) has coun table Ψ -we igh t since τ is a refinemen t of a Hausdorff to p ology whic h is of coun table w eight.  4. Fur t her pr oper ties In [12] the authors ask ed the follo wing: Problem 4.1 ([1 2, Problem 4.6]) . Supp ose that a sp ac e X ha s the pr op erty that for every op e n n e ighb ourho o d assignm e n t { U x : x ∈ X } ther e is a se c on d c ountable s ubs p ac e Y of X such that S { U x : x ∈ Y } = X ( dually second coun table , in short). Is X a D -sp ac e? Our construction answ ers this question in the negativ e. Prop osition 4.2. The s p ac e X c onstructe d in Cor ol lary 3.8 is dual ly se c ond c ountable, howe ver not a D -sp ac e. 12 DÁNIEL T. SOU KUP AND P AUL J. SZEPTYCKI Pr o of. The space X has the prop erty that ev ery coun table subsp ace is second coun table; indeed, the subspace top ology on α ∈ ω 1 is generated b y the sets U β ∩ α for β < α and { ϕ − 1 ( W ) : W ∈ W } , using the nota- tions of the previous section. Therefore, b y the Lindelöf prop ert y , for ev ery op en neigh b ourho o d assignmen t there is a coun ta ble and hence second coun ta ble subs pace whose neighbourho o ds co v er the space.  Our aim no w is to pro ve that the sp ace cons tructed in Corollary 3.8 has the prop ert y that all its finite p ow ers are Lindelöf. Indeed, by a theorem of Gerlits and Nagy [9], a space has all finite p ow ers Lindelöf if and only if t he space is an ( ε ) -space, i.e., ev ery ω -cov er has a coun ta ble ω - sub cov er. Let us call our space from Corollary 3.8 X , and now establish the follo wing theorem: Theorem 4.3. Every subsp ac e of X i s an ( ε ) -sp ac e. Pr o of. First, let us pro v e the follo wing analogue of Lemma 3.1. Lemma 4.4. Consider a top olo gy on ω 1 gener ate d by a family { U γ : γ < ω 1 } a s a subb a se. If for e very unc ountable fa m ily B ⊆ [ ω 1 ] <ω of p airwise disjoin t s e ts ther e is a c ountable B ′ ⊆ B such that { U F : F ∈ B ′ } is an ω -c o ver of a tail of ω 1 then the top olo gy i s a her e d itarily ( ε ) -sp ac e. Pr o of. Fix Y ⊆ X and an ω - co v er U of Y ; w e can supp ose that U = {∪{ U F i : i < m } : { F i : i < m } ∈ F } for some F ⊆  [ ω 1 ] <ω  <ω . Let M b e a countably elemen tary submo del of H ( ϑ ) for some sufficien t ly large ϑ suc h that { U γ : γ ∈ ω 1 } , F ∈ M . It su ffices to pro ve the follo wing. Claim 4.5. M ∩ U is a c ountable ω -c over of Y . Pr o of. Let K ∈ [ Y ] <ω and let L = K ∩ M . Clearly , M ∩ U co v ers K if K = L ⊆ M ; th us, w e can supp ose that K 6 = L a nd hence K / ∈ M . There is some { F i : i < m } ∈ F suc h that K ⊆ ∪{ U F i : i < m } . Let D i = F i ∩ M for i < m and w e can suppo se that there is some n ≤ m suc h that F i 6 = D i for i < n and F i = D i for n ≤ i < m . It follo ws from Lemma 2.2 that there is an uncoun ta ble sequence {{ F α i : i < m } : α < ω 1 } ⊆ F in M suc h that (1) { F α i : α < ω 1 } is an uncoun table ∆ - system with k ernel D i for ev ery i < n , (2) F α i = F i for all α < ω 1 and n ≤ i < m , (3) β ∈ U F i iff β ∈ U F α i for ev ery β ∈ L and α < ω 1 , i < m . A COUNTEREXAMPLE IN THE THEOR Y OF D - SP AC ES 13 The uncoun t a ble family { F α i \ D i : α < ω 1 } is pairwise disjoin t for ev ery i < n . Hence if we let F α = S i

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Leave a Comment