A separable non-remainder of H

A SEP ARABLE NON-REMAINDER OF H ALAN DOW † AND KLAAS PIETER HAR T Abstract. W e pro v e that there is a compact s eparable contin uum that (con- sisten tly) is not a remainder of the real line. Introduction Much is known ab out the co n tin uous images of N ∗ , the ˇ Cech-Stone remainder of the discrete space N . It is nigh on trivial to prov e that every separable compa ct Hausdorff spa ce is a co n tin uous image of N ∗ (w e a bbreviate this as ‘ N ∗ -image’), it is a ma jor result of P arovi ˇ cenko, from [99, 9], that ev ery compact Hausdorff space o f w eight ℵ 1 is an N ∗ -image and in [10 10, 10] P rzym usi ´ nski used the latter result to pr o v e that all p erfectly normal co mpact spaces are N ∗ -images. Under the ass umption of the Contin uum Hyp othesis Parovi ˇ cenko’s result encompasses all three r e sults: a compact Ha usdorff space is an N ∗ -image if and only if it has weigh t c or less. In [77, 7 ] the authors fo r m ulated a nd proved a v ersion o f Paroviˇ cenk o’s theorem in the class o f contin ua: every co ntin uum o f weigh t ℵ 1 is a contin uous image of H ∗ (an ‘ H ∗ -image’), the ˇ Cech-Stone remainder of the subspace H = [0 , ∞ ) of the real line. This result built o n a nd extended the cor respo nding res ult for metric contin ua from [1 1, 1]. Thu s the Contin uum Hyp othesis ( CH ) allows one to characterize the H ∗ -images as the contin ua of weigh t c or less . The pap er [77, 7] contains further results on H ∗ -images that parallel older results ab out N ∗ -images: Martin’s Axiom ( MA ) implies all contin ua of w eight less than c are H ∗ -images, in the Cohen model the long segment of leng th ω 2 is no t an H ∗ -image, and it is consistent with MA that not ev ery contin uum of weigh t c is an H ∗ -image. The na tural question whether the ‘trivial’ result on separ able compa ct s pa ces has its par allel version for contin ua proved harder to answer than expected. W e show that in this case the parallelism actually breaks do wn. Ther e is a well-defined separable contin uum K that is not an H ∗ -image if the Op en C o louring Axiom ( OCA ) is as s umed. This also answers a mor e general que s tion ra is ed b y G. D. F aulkner ([77; 7, Q uestion 7.3 ]): if a contin uum is an N ∗ -image m ust it be an H ∗ -image? Indeed, K is s eparable a nd hence an N ∗ -image. It is readily s een that β H itself is an H ∗ -image: b y moving back and for th in ever larger sweeps one constructs a map fro m H onto itself whose ˇ Cech-Stone extension maps H ∗ onto β H . Indeed the same argument applies to an y space that is the union of a connected collectio n of Peano co n tin ua: its ˇ Cech-Stone compa ctification Date : T uesda y 25-09-2007 at 22:40:23 (cest) . 2000 Mathematics Subject Classific ation. Primary: 54F15. Secondary: 03E50, 03E65, 54A35, 54D15, 54D40, 54D65. Key wor ds and phr ases. separable cont inuum, cont inuo us image, H ∗ , β X , OCA . † Supported by NSF grant DMS-0554896. 1 2 ALAN DOW AND KLAAS P IETER HAR T is an H ∗ -image. Th us, e.g., for every n the space β R n is an H ∗ -image. Our example is one step up from these e x amples: it is the ˇ Cech-Stone compactifica tion of a string of sin 1 x -curves. Our result also shows that the pr oof in [77, 7] ca nno t b e extended beyond ℵ 1 , as OCA is comp ossible with Martin’s Axio m ( MA ). The adage that M A makes all cardinals b elow c behave as if they a re co un ta ble w ould suggest that the aforemen- tioned pro of, an in verse-limit cons truction, could b e ma de c lo ng, at least if MA holds. W e see that this is not pos sible, ev en if the con tin uum is s eparable. The pa per is orga nized as follo ws. Section 1 contains a few preliminaries, in- cluding the consequences of OCA that w e shall use. In Section 2 we construct the contin uum K and show ho w OCA implies that it is not an H ∗ -image. Finally , in Section 4 we give a few more details on the lack o f effica cy of MA in this and we discuss and ask whether other potential H ∗ -images are indeed H ∗ -images. W e thank the r eferee for p oin ting out that there was muc h ro om for improvemen t in our present ation. 1. Preliminaries Closed and op en sets in β X . Since we will b e working with subsets of the pla ne we ca n economize a bit o n notation and write β F fo r the c lo sure-in- β X of a closed subset of the s pace X itself; we a lso wr ite F ∗ = β F \ F . If O is an o pen subse t of X then E x O = β X \ β ( X \ O ) is the lar g est open subse t of β X whose intersection with X is O . In dealing with closed subsets of H ∗ the following, which is P ropo sition 3.2 from [88, 8], is very useful. Prop osition 1.1. Le t F and G b e disjoint close d set s in H ∗ . Ther e is an incr e asing and c ofinal se quenc e se quenc e h a k : k ∈ ω i in H such that F ⊆ Ex S k ( a 2 k +1 , a 2 k +2 ) and G ⊆ E x S k ( a 2 k , a 2 k +1 ) .  W e shall b e working with closed subsets of the plane (or H ) that can be written as the union of a dis crete sequence h F n : n ∈ ω i of compact sets. The ex tension of the natural map π from F = S n F n to ω , that sends the p oin ts of F n to n , partitions β F into sets indexed by β ω : for u ∈ β ω we write F u = β π ← ( u ). If the F n are all connected then so is every F u and, indeed, the F u are the comp onents of β F , see [88; 8, Corolla ry 2.2 ]. F or use b elow we note the following. Lemma 1.2. If e ach F n is an irr e ducible c ontinuum, b etwe en the p oints a n and b n say, then so is e ach F u , b etwe en t he p oints a u and b u .  The Op en C olouring Axio m. The Op en Coloring Axiom ( OCA ) was formulated by T odorˇ cevi´ c in [1111, 11]. It reads as follows: if X is s e pa rable and metr izable and if [ X ] 2 = K 0 ∪ K 1 , where K 0 is op en in the pro duct top ology of [ X ] 2 , then either X has an uncoun table K 0 -homogeneo us subset Y or X is the union of countably many K 1 -homogeneo us subsets. One can deduce the conjunction OCA and MA from the Pr op er F or cing Ax iom or prov e it c o nsisten t in an ω 2 -length countable suppor t proper itera ted for cing construction, using ♦ on ω 2 to predict all p ossible subsets o f the Hilb ert cube and all po ssible op en co lourings of these, as well as all pos sible ccc pos ets of cardinality ℵ 1 . A SEP ARABLE NON-REMAINDER OF H 3 W e shall mak e use of OCA only but w e noted the co mpossibility with MA in order to substa n tia te the claim that the la tter pr inciple do es not imply tha t all separable contin ua are H ∗ -images. T riviality of m ap s. W e shall use tw o consequences of OCA . The first says that contin uous surjections fro m ω ∗ onto β ω are ‘trivial’ on la rge pieces of ω ∗ . If ϕ : ω ∗ → β ω is a contin uo us surjection then it induces, by Stone duality , an embedding of Φ : P ( ω ) → P ( ω ) / fin by Φ( A ) = ϕ ← [ β A ]. The following is a consequence of [6 6; 6, Theo rem 3.1], wher e for a subset M of ω w e write ˜ M = ( M − 1 ) ∪ M ∪ ( M + 1 ). Prop osition 1.3 ( OCA ) . With the notation as ab ove ther e ar e infinite subset D and M of ω and a map ψ : D → ˜ M such t ha t fo r every subset A of ˜ M one ha s Φ( A ) = ψ ← [ A ] ∗ .  Thu s, on the s e t D ∗ the map ϕ is determined by the map ψ : D → ˜ M ; this is the sense in which ϕ ↾ D ∗ might b e called trivial. It is also impor tan t to no te that D = ∗ ψ ← [ ˜ M ], which follo ws fro m D ∗ = ϕ ← [ β ˜ M ]; this will be used in our pro of. Non-images of N ∗ . The final nail in the coffin of a purp orted map fro m H ∗ onto the contin uum K will b e the follo wing result from [55, 5], where D = ω × ( ω + 1). Prop osition 1.4 ( OCA ) . The ˇ Ce ch-Stone r emainder D ∗ is n ot an N ∗ -image.  2. The non -ima ge The example . W e start by replicating the sin 1 x -curve along the x -axis in the plane: for n ∈ ω we set K n =  { n } × [ − 1 , 1]  ∪  h n + t, sin π t i : 0 < t ≤ 1  . The union K = S n K n is connected and its ˇ Cech-Stone compactificatio n β K is separable cont in uum. W e sha ll show that OCA implies that β K is not a co n tinuous image of H ∗ . W e define four closed sets that play a n impo r tan t part in the proo f. F or n ∈ ω we define: • S n = { h x, y i ∈ X : n + 1 3 ≤ x ≤ n + 2 3 } , • S + n = { h x, y i ∈ X : n + 1 4 ≤ x ≤ n + 3 4 } , • T n = { h x, y i ∈ X : n − 1 4 ≤ x ≤ n + 1 4 } , • T + n = {h x, y i ∈ X : n − 1 3 ≤ x ≤ n + 1 3 } ; we put S = S n ∈ ω S n , S + = S n ∈ ω S + n , T = S n ∈ ω T n and T + = S n ∈ ω T + n . Note that S ∩ T = ∅ and hence β S ∩ β T = ∅ in β X . W e also no te that the four sets S n , S + n , T n and T + n are a ll connected and that w e therefore know exactly what the compo nen ts of β S , β S + , β T and β T + are. Note that, b y Lemma 1.2, each co n tinuum T + u (as well a s S u , S + u and T u ) is irreducible, as each T + n is irreducible (b et ween its end points h n − 1 3 , − 1 i and h n + 1 3 , 0 i ). Finally w e note that S n meets the sets T + n and T + n +1 only , that T n meets S + n − 1 and S + n only , etcetera. This b eha viour p ersists when we mov e to the co ntin ua S u , T + u , S + u and T + u , when we define u + 1 and u − 1, for u ∈ ω ∗ , in the o b v ious w ay: u + 1 is gener ated by { A + 1 : A ∈ u } and u − 1 is generated b y { A : A + 1 ∈ u } . Prop erties of a p otential surjection. Assume h : H ∗ → β K is a contin uous surjection a nd apply Prop osition 1.1 to the closed subs e ts h ← [ β S ] and h ← [ β T ] of H ∗ to get a sequence h a k : k ∈ ω i . After comp osing h with a piecewise linea r map w e may assume, without loss of gener a lit y , that a k = k for all k . W e obtain 4 ALAN DOW AND KLAAS P IETER HAR T h ← [ β S ] ∩ β S k ∈ ω I 2 k +1 = ∅ and h ← [ β T ] ∩ β S k ∈ ω I 2 k = ∅ , where I k = [ k , k + 1]. W e wr ite 2 ω and 2 ω + 1 for the sets of even and o dd natural num ber s resp ectiv ely . The ma p h induces maps from (2 ω ) ∗ and (2 ω + 1) ∗ onto β ω , as follows. If u ∈ (2 ω ) ∗ then h [ I u ] is a connected set tha t is disjoint fro m β T , hence it must be contained in a compo nent of β S + . Likewise, if v ∈ (2 ω + 1) ∗ then h [ I v ] is contained in a compone nt of β T + . Thu s we g et maps ϕ 0 : (2 ω ) ∗ → β ω and ϕ 1 : (2 ω + 1) ∗ → β ω defined by • ϕ 0 ( u ) = x iff h [ I u ] ⊆ S + x , • ϕ 1 ( v ) = y iff h [ I v ] ⊆ T + y . Lemma 2.1. The maps ϕ 0 and ϕ 1 ar e c ontinuous. Pr o of. F or k ∈ ω put r k = k + 1 2 . Obser v e that, by connectivity , h ( r u ) ∈ S + x iff h [ I u ] ⊆ S + x , so that ϕ 0 can be decomp osed as u 7→ r u 7→ h ( r u ) 7→ π 0 ( h ( r u )), where π 0 : β S + → β ω is the natural map. The argument for ϕ 1 is similar.  The maps ϕ 0 and ϕ 1 are not unrelated. Let u ∈ (2 ω ) ∗ and put x = ϕ 0 ( u ). Then h [ I u ] ⊆ S + x , so that h [ I u − 1 ] and h [ I u +1 ] both intersect S + x . How ever, S + x int ersects only the con tinua T + x and T + x +1 , s o tha t ϕ 1 ( u − 1) , ϕ 1 ( u + 1) ∈ { x, x + 1 } . By symmetry a simila r statement can be made if y = ϕ 1 ( v ): then ϕ 0 ( v − 1) , ϕ 0 ( v + 1) ∈ { y − 1 , y } . Using these relationships we can deduce so me extra pro perties of ϕ 0 and ϕ 1 . Lemma 2.2. If u ∈ (2 ω ) ∗ then ϕ 0 ( u − 2) and ϕ 0 ( u + 2) b oth ar e in { x − 1 , x, x + 1 } , wher e x = ϕ 0 ( x ) .  3. An applica tion of OCA W e apply Pr opositio n 1.3 to the embedding Φ 0 of P ( ω ) into P (2 ω ) / fin defined by Φ 0 [ A ] = ϕ ← 0 [ β A ]. W e find infinite sets D ⊆ 2 ω a nd M ⊆ ω together with a map ψ : D → ˜ M that induces Φ 0 on its r a nge: for every subset A of ˜ M we have Φ 0 [ A ] = ψ ← 0 [ A ] ∗ . As noted above this implies that ϕ 0 ↾ D ∗ = β ψ 0 ↾ D ∗ . F or m ∈ ˜ M and u ∈ (2 ω ) ∗ we have the equiv alence ϕ 0 ( u ) = m iff ψ ← 0 [ { m } ] ∈ u . Using the pr operties of ϕ 0 stated in Le mma 2.2 we deduce the followin g inc lus ion- mo d-finite  ψ ← 0  { m }  − 2  ∪ ψ ← 0  { m }  ∪  ψ ← 0  { m }  + 2  ⊆ ∗ ψ ← 0  { m − 1 , m, m + 1 }  ⊆ D Therefore we get for every m ∈ M a j m such that: if n ≥ j m and ψ 0 ( n ) = m then n − 2 , n + 2 ∈ D . Lemma 3.1. F or every m ∈ M ther e ar e infinitely many n ∈ D such that ψ 0 ( n ) = m and ψ 0 ( n + 2) 6 = m . Pr o of. Let m ∈ M and take m ′ ∈ M \ { m } . Let n ∈ D b e a rbitrary such that ψ 0 ( n ) = m and n ≥ j m ; cho ose n ′ > n suc h that ψ 0 ( n ′ ) = m ′ . There m ust be a first index i such tha t ψ 0 ( n + 2 i ) 6 = ψ 0 ( n + 2 i + 2 ) a s otherwise we c o uld s ho w inductively that n + 2 i ∈ D and ψ 0 ( n + 2 i ) = m for all i , whic h would imply that ψ 0 ( n ′ ) = m . F or this minimal i we hav e ψ 0 ( n + 2 i ) = m and ψ 0 ( n + 2 i + 2) 6 = m .  W e use this le mma to find an infinite subset L o f D whe r e ϕ 0 and ϕ 1 are very well-behav ed. A SEP ARABLE NON-REMAINDER OF H 5 Let m 0 = min M and cho ose l 0 ≥ j m 0 such that ψ 0 ( l 0 ) = m 0 and ψ 0 ( l 0 + 2) 6 = m 0 . Pro ceed recursively: c ho ose m i +1 ∈ M large r than m i + 3 and ψ 0 ( l i + 2) + 3, and then pick l i +1 larger than l i and j m i +1 such that ψ 0 ( l i +1 ) = m i +1 and ψ 0 ( l i +1 + 2) 6 = m i +1 . Consider the set L = { l i : i ∈ ω } and thin out M so that it will be equal to { m i : i ∈ ω } . L et u ∈ L ∗ and let x = ϕ 0 ( u ) = ψ 0 ( u ); we assume, without loss of gener alit y , that { l ∈ L : ψ 0 ( l + 2) = ψ 0 ( l ) + 1 } b elongs to u . It follows that ϕ 0 ( u + 2 ) = x + 1 and this means that ϕ 1 ( u + 1 ) = x . W e find that h [ I u ] ⊆ S + x , h [ I u +1 ] ⊆ T + x and h [ I u +2 ] ⊆ S + x +1 . Therefore the image h [ I u ∪ I u +1 ∪ I u +2 ] is a sub cont in uum of S + x ∪ T + x ∪ S + x +1 that meets S + x and S + x +1 . Because T + x is irr educible w e find that T + x ⊆ h [ I u ∪ I u +1 ∪ I u +2 ] and hence T x ⊆ h [ I u +1 ], be c ause the o ther tw o parts of this contin uum are disjoint from β T . W e now have infinite sets L a nd M where the ma ps ϕ 0 and ϕ 1 behave very nicely indeed. B ecause ψ 0 maps L onto M the map ϕ 0 maps L ∗ onto M ∗ . F urthermore, if u ∈ L ∗ and x = ϕ 0 ( u ) then also x = ϕ 1 ( u + 1) and T x ⊆ h [ I u +1 ] ⊆ T + x . W e put L = S { I u +1 : u ∈ L ∗ } and we observe that, by the inclusions above, [ { T x : x ∈ M ∗ } ⊆ h [ L ] ⊆ [ { T + x : x ∈ M ∗ } . ( ∗ ) W e put h L = h ↾ L . A map from N ∗ on to D ∗ . W e no w use h L to c reate a ma p from N ∗ onto D ∗ , which will yield the contradiction tha t finis he s the pro of. Let F = S m ∈ M T m ∩  [0 , ∞ ) × [ 1 2 , 1]  and G = S m ∈ M T m ∩  [0 , ∞ ) × [0 , 1]  . Observe that the inclusion map fr om F to G induces the iden tit y map betw e e n their resp ectiv e component spa ces and hence also the iden tit y map betw een the comp onen t space s of β F and β G . W e work with the closed subsets F ∗ and G ∗ of K ∗ ; the former is con tained in the interior of the latter, hence the same holds for h ← L [ F ∗ ] and h ← L [ G ∗ ]. W e apply P ropo sition 1 .1 a nd obtain, for ev ery l ∈ L , a finite family I l of subin terv als of I l such that for the closed set H = S l ∈ L S I l we hav e h ← L [ F ∗ ] ⊆ int H ∗ ⊆ H ∗ ⊆ int h ← L [ G ∗ ] ( † ) Endow the countable set o f in terv als I = S l ∈ L I l with the discrete top ology and let p ∈ I ∗ ; the corr e s ponding comp onen t of H ∗ is mapp ed by h L int o a compo nen t of G ∗ . Thus we obtain a map from I ∗ int o the comp onen t space of G ∗ . This map is onto: let C G be a co mponent of G ∗ and let C F be the unique compo nen t of F ∗ contained in C G . Because o f ( † ) and ( ∗ ) there is a family of compo nen ts of H ∗ that cov e r s C F ; all these components are mapped into C G . W e obtain a map from I ∗ onto the comp onent space of G ∗ . This map is contin- uous; this ca n b e shown as for the maps ϕ 0 and ϕ 1 using midpo in ts of the in terv als and the q uotien t map from G ∗ onto its comp onent s pace. The comp onent s pa ce of G its e lf is D , so that G ∗ has D ∗ as its comp onen t space. Thus the assumption that H ∗ maps onto β K leads, assuming OCA , to a contin uo us surjection from ω ∗ onto D ∗ , whic h, b y Prop osition 1.4 is imp ossible. 4. Fur ther remarks 4.1. Comments on the construction. The pro ofs in [55, 566, 6] tha t cer ta in spaces are not N ∗ -images follow the same t w o-step pa ttern: first show that no ‘trivial’ map exists a nd then show that OCA implies that if there is a map a t all then there m ust a lso b e a ‘trivial’ one. In the context of our exa mple it should 6 ALAN DOW AND KLAAS P IETER HAR T be clear that there is no ma p from H to the plane that induces a ma p fro m H ∗ onto β K ; it would hav e been nic e to ha v e found a map from S l ∈ L I l +1 to the plane that w ould hav e induced h L but w e did not see how to construct one. 4.2. MA is not strong enough. As mentioned in the in tro duction the principa l result of [77, 7] states that every con tin uum of weight ℵ 1 is an H ∗ -image. In that pap er the authors a lso prov e that under MA every contin uum of weight less than c is a n H ∗ -image; the sta rting p oin t of that pro of was the result o f V an Douw en and Przymusi´ nski [44, 4] that, under M A , every co mpact Hausdorff space of w e ig h t less than c is an N ∗ -image. Given such a contin uum X , of weigh t κ < c , one assumes it is embedded in the Tyc ho noff cub e I κ and takes a contin uous map f : β N → I κ such that f [ N ∗ ] = X . What the pro of then establishes, using MA , is that f has an extension F : β H → I κ such that F [ H ∗ ] = X . Thus, in a very real sens e, one can simply connec t the do ts of N to pr oduce a map from H ∗ onto X that extends the given map from N ∗ onto X . Since MA and OCA are c ompossible our ex ample shows that MA do es not im- ply that all s e parable con tin ua are H ∗ -images a nd, a fortio ri, that the tw o pr oofs from [77, 7] cannot be amalg amated to show that the answer to F aulkner’s question is pos itive under MA , not ev en for separable spaces. 4.3. Other images. As no ted in the int ro duction there are many para lle ls betw een the res ults o n N ∗ -images and those on H ∗ -images. The example in this pap er shows that there is no co mplete parallelism. There are s ome results on N ∗ -images wher e no parallel has been found or disproved to exist. W e mentioned Przymusi´ nski’s theorem from [1010, 10 ] that every p erfectly nor- mal compact spac e is an N ∗ -image. By co mpactness every p erfectly normal co mpact space is first-countable and by Arhangel ′ ski ˘ ı’s theorem ([22, 2]) every first-countable compact space has w eight c and is therefore an N ∗ -image if CH is assumed. Thu s we get tw o questions on H ∗ -images. Question 4.1 . Is every p erfectly normal compact cont inu um an H ∗ -image? Question 4.2 . Is every first-co un ta ble contin uum an H ∗ -image? The questions are r elated of course but the q ue s tion on first-co un ta ble con tin ua might get a consistent negative answer so oner than the one on perfectly normal contin ua in the light of Bell’s consistent example, from [33, 3], of a first-countable compact space that is not an N ∗ -image. References [1] ]cite.MR 0214 0331J. M. Aarts and P . v an Emde Boas, Continua as r emainders in c omp act extensions , Nieu w Archief voor Wiskunde (3) 15 (1967), 34–37. MR 0214033 (35 #4885) [2] ]cite.MR 0251 6952A. V. Arhangel ′ ski ˘ ı, The p ower of bic omp acta with first axiom of co unt- ability , Doklady Ak ademii Nauk SSSR 1 87 (1969) , 967–970 (Russian); Engli sh transl., Soviet Mathematics Doklady 10 (1969), 951–955. MR 0251695 (40 #4922) [3] ]cite.MR 1058 7953Murra y G. Bell , A first c ountable c omp act sp ac e that is not an N ∗ image , T op ology and its Applications 35 (1990), no. 2-3, 153–156. MR 1058795 ( 91m: 54028) [4] ]cite.MR 5615 884Eric K. v an Dou wen and T eo dor C. Przymusi´ nski, Sep ar able extensions of first c ountable sp ac es , F undament a M athe maticae 105 (19 79/80), no. 2, 147–158. MR 561588 (82j: 54051) [5] ]cite.MR 1679 5865Alan Dow and Kl aas Pi ete r Har t, ω ∗ has (almost) no c ontinuous images , Israel Journal of Mathematics 1 09 (1999), 29–39. MR 1679586 (2000d: 54031) A SEP ARABLE NON-REMAINDER OF H 7 [6] ]cite.MR 1752 1026 , The me asur e algebr a do es not always emb e d , F undamen ta Mathe- maticae 16 3 (2000), no. 2, 163–176. MR 175210 2 (2001g: 0308 9) [7] ]cite.MR 1707 4897 , A universal c ontinuum of weight ℵ , T ransactions of the Ameri can Mathematical So ciet y 353 (2001 ), no. 5, 1819–1838. MR 1707489 (2001g: 54037) [8] ]cite.Hart8Klaas Pieter Hart, The ˇ Ce ch-Stone c omp actific ation of the R e al Line , Recent progress i n general topology , 1992, pp. 317–352. MR 95g: 54004 [9] ]cite.Paro vicenk o639I. I. P aro viˇ cenko, A unive rsal bic omp act of weight ℵ , Soviet Mathematics Doklady 4 (1963), 592–5 95. Russian original: Ob o dnom universal ′ nom bikomp akte vesa ℵ , Doklady A k ademi ˘ ı Nauk SSSR 150 (1963) 36–39. MR 27#719 [10] ] cite.MR671232 10T eo dor C. Przymusi´ nski, Perfe ctly normal c omp act sp ac es ar e c ontinuous images of β N \ N , Pr o ceedings of the American Mathematical So ciet y 86 (1982), no. 3, 541–544. MR 671232 (85c: 54014) [11] ] cite.MR980949 11Stev o T o dor ˇ cev i´ c, Partition pr oblems in top olo gy , Con temp orary Math- ematics, vol. 84, Amer ican Mathemat ical Society , Pro vidence, RI, 1989. MR 9809 49 (90d: 04001) [12] ] cite.MR932983 12Stephe n W atson and Will iam W eiss, A top olo gy on the union of the double arr ow sp ac e and the inte gers , T op ology and its Appli cat ions 28 (1988), no. 2, 177–179. Special issue on set-theoretic topology. MR 932983 ( 89d: 54019) Dep ar tment of Ma thema tics, UNC-Charlotte, 9201 University City Bl v d., Char- lotte, NC 2822 3-0001 E-mail addr ess : adow@uncc. edu URL : http:/ /www.mat h.uncc.e du/~adow F acul ty of Electrical Engineerin g, Ma thema tics and Computer Science, TU Delft, Postbus 5031, 26 00 G A Delft, the Netherlands E-mail addr ess : k.p.hart@t udelft.n l URL : http:/ /fa.its. tudelft. nl/~hart