A first-countable non-remainder of H

A FIRST-COUNT ABLE NON-RE MAINDER OF H ALAN DOW † AND KLAAS PIETER HAR T Abstract. W e give a (consistent ) example of a first-counta ble co n tin uum th at is not a remainder of the real line. Introduction The purp o s e o f this note is to confir m a suspicion ra ised in [55; 5 , Question 4 .2]: we show that Bell’s example, from [33, 3 ], of a firs t-countable compact space that is no t an N ∗ -image ca n b e adapted to pro duce admits a connected v ariatio n that is neither an N ∗ -image nor an H ∗ -image. The in terest in th is v ariation stems from the authors’ version of Parovi ˇ cenko’s theorem from [99, 9]. That theorem states that every compact Ha usdorff space of weigh t ℵ 1 or less is an N ∗ -image; the Co nt inu um Hypo thesis then implies t hat the N ∗ -images are exactly the compact Hausdorff spaces of weigh t 2 ℵ 0 or less. W e proved in [44, 4] a para llel result for H ∗ and contin ua (connected compac t Hausdorff spaces). Since, by Arkhangel ′ ski ˘ ı’s theorem [11, 1], first-countable co mpa ct spa ces hav e weigh t at most 2 ℵ 0 it follows that under CH first-countable compacta/contin ua ar e N ∗ -images/ H ∗ -images resp ectively . Bell’s graph. A ma jor ingredient in our constructio n is Be ll’s graph, constructed in [22, 2]. It is a graph on the or dinal ω 2 , r epresented by a symmetr ic subset E of ω 2 2 . The crucial pro p erty o f this gr aph is that there is no map ϕ : ω 2 → P ( N ) that represents this graph in the se nse that h α, β i ∈ E if and only if ϕ ( α ) ∩ ϕ ( β ) is infinite. Bell’s graph e xists in any forcing extensio n in whic h ℵ 2 Cohen reals are a dded; for the reader’s convenience w e shall describe the constr uc tio n of E and adapt Bell’s pro of so that it applies to con tin uous maps defined on H ∗ . A first-count able co ntinuum Our starting po in t is a connected v ersion of the Alex a ndroff double of the unit int erv al. W e topolog ize the unit square as follows. (1) a local base at p oints of the for m h x, 0 i co nsists of the sets U ( x, 0 , n ) = ( x − 2 − n , x + 2 − n ) × [0 , 1] \ { x } × [2 − n , 1] (2) a local base at p oints of the for m h x, y i , with y > 0 consists of the sets U ( x, y , n ) = { x } × ( y − 2 − n , y + 2 − n ) W e call the resulting space the connected comb and denote it by C . It is straight- forward to verify that C is compact, Hausdor ff and connec ted; it is first-countable by definition. Date : F riday 30-05-2008 at 14:21:32 (cest) . 2000 Mathematics Subje ct Classific ati on. Primary: 54F15. Secondary: 03E50, 03E65, 54A35, 54D40, 54G20. Key wor ds and phr ases. first-coun table con tin uum, contin uous image, H ∗ , Cohen reals. † Supported by NSF gran t DMS-0554896. 1 2 ALAN DO W AND KLAAS PIETER HAR T F or each x ∈ [0 , 1] a nd p ositive a we define to b e the following cr o ss-shap ed closed subset o f C 2 : D x,a =  { x } × [ a, 1] × C  ∪  C × { x } × [ a, 1]  W e note the following tw o proper ties of the sets D x,a (1) if a < b then D x,b is in the in terior of D x,a , and (2) if x 6 = y then D x,a ∩ D y ,a is the union of tw o squares: { x } × [ a, 1] × { y } × [ a, 1] and { y } × [ a, 1] × { x } × [ a, 1] Now take any ℵ 2 -sized subset of [0 , 1 ] a nd index it (faithfully) a s { x α : α < ω 2 } . W e use this indexing to identify E with the subset { h x α , x β i : h α, β i ∈ E } of the unit squar e. Next w e remo ve from C 2 the following op en set: [ h x,y i / ∈ E   { x } × (0 , 1] × { y } × (0 , 1]  ∪  { y } × (0 , 1] × { x } × (0 , 1]   The resulting compact space we denote by C E . Observe that the intersections D x α ,a ∩ C E represent E in the sense that D x α ,a ∩ D x β ,a ∩ C E is nonempty if a nd only if h α, β i ∈ E . W e write D E x,a = D x,a ∩ C E . W e show that C E is (ar cwise) connected. T o b egin: the square S o f the base line of C is a subset o f C E and homeomorphic to the unit square so tha t it is (arcwise) connected. Let h x, a, y , b i b e a p oint of C E not in S . If, say , a = 0 then { h x, 0 i} × ( { y } × [0 , b ]) is an arc in C E that connects h x, 0 , y , b i to the point h x, 0 , y , 0 i in S . If a, b > 0 then h x, y i ∈ E and the whole s q uare { x } × [0 , 1] × { y } × [0 , 1] is in C E and it provides us with an arc in C E from h x, a, y , b i to h x, 0 , y , 0 i . W e find that C E is a first-co unt able contin uum. It remains to show tha t it is not an H ∗ -image. Assume h : H ∗ → C E is a contin uous sur jection and consider, for each α , the sets D E x α , 3 4 and D E x α , 1 2 . Using standard pr op erties of β H , see [77; 7, P rop osition 3.2], w e find for ea ch α a sequence  ( a α,n , b α,n ) : n ∈ N  of op en interv als with rational endp oints, and with b α,n < a α,n +1 for all n , such that h ← [ D E x α , 3 4 ] ⊆ Ex O α ∩ H ∗ ⊆ h ← [ D E x α , 1 2 ], where O α = S n ( a α,n , b α,n ). Because the in tersections of the sets D E x α ,a represent E the int ersections of the O α will do this as w e ll: the conditions ‘ O α ∩ O β is un bounded’ and ‘ h α, β i ∈ E ’ are equiv alen t. In the next subsection we show that fo r (many) h α, β i this equiv alence do es not hold and that therefore C E is not a con tin uous image of H ∗ . Note also that o ur co ntin uum is not a n N ∗ -image either: if g : N ∗ → C E were contin uous and onto we could use clopen subsets of N ∗ and their representing infinite subsets of N to contradict the unrepresentabilit y prop erty of E . Destro ying the equiv alence. W e follow the argument from [22, 2] and w e rely on Kunen’s b o ok [88; 8, Chapter VII] for basic facts on forcing. W e let L = {h α, β i ∈ ω 2 2 : α ≤ β } and we force with the par tial o r der Fn( L . 2) o f finite partial functions with domain in L and r a nge in { 0 , 1 } . If G is a generic filter on Fn( L, 2) then w e let E = {h α, β i : S G ( α, β ) = 1 or S G ( β , α ) = 1 } . T o show that E is as requir ed we ta ke a nice name ˙ F for a function from ω 2 to ( Q 2 ) ω that repr esents a choice of op en sets α 7→ O α as in ab ove in tha t F ( α ) =  h a α,n , b α,n i : n ∈ ω  for all α . As a nice name ˙ F is a subset of ω 2 × ω × Q 2 × Fn( L, 2 ), where for each point h α, n, a, b i the set { p : h α, n, a, b , p i ∈ ˙ F } is a maximal antic ha in in the set of conditions tha t forces the n th term of ˙ F ( α ) to be h a, b i . A FIRST-COUNT ABLE NON-REMAINDER OF H 3 F or each α we le t I α be the set of ordinals that occ ur in the doma ins o f the conditions that app ear as a fifth co ordinate in the elemen ts o f ˙ F with first coo r- dinate α . The sets I α are countable, by the ccc o f Fn ( L, 2 ). W e ma y therefore apply the F r e e -Set Lemma, see [66; 6, Corollar y 44.2 ], a nd find a subs e t A of ω 2 of cardinality ℵ 2 such that α / ∈ I β and β / ∈ I α whenever α, β ∈ A a nd α 6 = β . Let p ∈ Fn( L, 2) b e arbitrar y and take α and β in A with α < β and such that α > η whenever η o ccurs in p . Consider the condition q = p ∪  h α, β , 1 i  . If q forces O α ∩ O β to be bo unded in [0 , ∞ ) then we are done: q forces that the equiv a le nce fails at h α, β i . If q do e s not force the intersection to b e bounded w e c a n extend q to a condition r that forces O α ∩ O β to b e unbounded. W e define an automorphism h of Fn( L, 2) by c hanging the v alue of the conditions only at h α, β i : from 0 to 1 and vice v ersa. The condition p as well as the names ˙ x α and ˙ x β are in v ariant under h . It follows that h ( r ) extends p and h ( r )  S ˙ G ( α, β ) = 0 and O α ∩ O β is unbounded so aga in the equiv alence is forced to fail at h α, β i . R emark. T he arg umen t a b ove go es thro ugh almost verbatim to show that Bell’s graph can a lso be obtained adding ℵ 2 random reals. When forcing with the random real algebra o ne needs only co nsider condtio ns that belong to the σ -alg ebra gener- ated b y the clop en sets of the pro duct { 0 , 1 } L ; these all hav e countable supp orts so that, again by the ccc, one ca n define the sets I α as b efore. The rest of the argument r e mains virtually unc ha nged. References [1] ]cite.MR02516951A. V. 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MR 27#719 Dep ar tment of Ma thema tics, UNC-Charlotte, 9201 University City Bl vd., Char- lotte, NC 28223-000 1 E-mail addr e ss : adow@uncc.edu URL : http:/ /www.mat h.uncc.e du/~adow F acul ty of Electrical Engineering, Ma thematic s a nd Computer Science, TU Delf t, Postbus 5 031, 260 0 GA Delft, the Netherlands E-mail addr e ss : k.p.hart@tudelf t.nl URL : http:/ /fa.its. tudelft. nl/~hart