math.GN 2012-04-24 2

GO-spaces and Noetherian spectra

The Noetherian type of a space is the least k for which the space has a k^op-like base, i.e., a base in which no element has k-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For exam…

Authors: David Milovich

GO-SP A CES AND NOETHERIAN SPECTRA DA VI D MILOVICH Abstract. Th e Noetherian t ype of a space is the least κ f or which the space has a κ op -like base, i.e. , a base in which no elemen t has κ -many s upersets. W e pr ov e some results about Noetherian types of (generalized) ordered spac es and products thereof. F or example: the densit y of a pr o duct of not-too- man y compact linear orders nev er exceeds its No etherian type, with equalit y p ossible only for singular No etherian types; w e prov e a si milar r esult for products of Lindel¨ of GO-spaces. A counta ble pro duct of compact linear orders has an ω op 1 -like base if and only if it i s metrizable, and ev ery metrizable space has an ω op -like base. An i nfinite cardinal κ is the No etherian type of a compact LOTS if and only if κ 6 = ω 1 and κ i s not we akly inaccessible. There is a Lindel¨ of LO TS with No etherian type ω 1 and there consisten tly is a Lindel¨ of LOTS wi th weakly inaccessible N oetherian ty p e. 1. Intr oduction The No ether ia n type of a topo logical spac e is an order- theo retic a nalog of its weigh t. Definition 1.1. Given a ca r dinal κ , de fine a p os et to be κ op - like if no elemen t is below κ -many elemen ts. In the co n text of families of subsets o f a top olog ical space, we will a lw ays im- plicitly order by inclusion. F or example, a descending chain of op en sets of type ω is ω op -like; an ascending chain of op en sets of type ω is ω op 1 -like but not ω op -like. Definition 1.2. Given a space X , • the weight of X , o r w ( X ), is the least κ ≥ ω such that X has a base of siz e at most κ ; • the No etherian typ e of X , or N t ( X ), is the least κ ≥ ω such that X has a base that is κ op -like. Equiv alen tly , N t ( X ) is the le a st κ ≥ ω s uc h that X has a base B such that T A has empt y interior for all A ∈ [ B ] κ . No etherian t yp e w as int ro duced by Peregudov [10]. Preceding this in tro duction are several paper s by Peregudov, ˇ Sapirovski ˘ ı and Malykhin [5, 8, 9, 1 1] ab out the top ological pr op erties N t ( · ) = ω and N t ( · ) ≤ ω 1 . More recently , the author has extensively in vestigated the No etherian type of β N \ N [7] and the No etherian t yp es of homogeneous compacta and dyadic compacta [6]. (See E ngelking [1], Juh´ asz [3], and Kunen [4] for all undefined terms.) A surprising res ult from [6 ] is that no dyadic c o mpactum has No etherian t ype ω 1 . In other words, g iven an ω op 1 -like base of a dyadic compactum X , one ca n construct 2000 Mathematics Subject Classific ation. Pri mary 54F05; Secondary 54A25, 03E04. Some of these r esults are from the author’s thesis, which was partially supp orted b y an N SF graduate fell o wship. 1 2 DA VID MILO VICH an ω op -like base of X . This result do es not gener alize to all compacta. In that same pap er, it was shown how to co ns truct a c o mpactum with Noether ian t yp e κ , for any infinite car dina l κ . It is still a n op en problem whether an y infinite car dinals other than ω 1 are excluded from the sp ectrum of No etherian t yp es of dyadic compa cta, although it was sho wn that there a re dy adic compacta with No ether ia n ω and dyadic compacta with No etherian type κ + , for every infinite cardinal κ with unco un table cofinality . Question 1 .3 . If κ is a singular cardinal with co finalit y ω , then is ther e a dyadic compactum with No e therian t yp e κ + ? Is there a dyadic compac tum with weakly inaccessible Noether ian t yp e? The a bove t wo questions are t ypical of the “s up=max” problems of set-theoretic top ology . See Juh´ asz [3] for a systematic study of these problems. Though the ab ov e tw o questio ns remain o pe n problems, we can now answer the corresp onding ques tions for co mpa ct linea r o rders. The spec tr um o f No ether- ian types of linearly ordered compa cta includes ω , ex cludes ω 1 , includes all singular cardinals, includes κ + for all unco un table car dinals κ , and excludes all w eak inaces- sibles. In the pro cess of pr oving this claim, we will prov e a gener al technical lemma which says r oughly that if X is a pro duct of not-to o-ma ny µ -compact GO-spaces for some fixed c a rdinal µ , then d ( X ) ≤ N t ( X ) and in most cases d ( X ) < N t ( X ). Definition 1.4. • A spa c e X is κ -compact if κ is a ca rdinal and every op en c over of X has a sub c ov e r of size less than κ . • A GO -sp ac e , or gener alized o r dered space, is a subspa c e o f a linearly o r- dered to polo gical spa c e . Equiv alently , a GO-s pa ce is a linear order with a top ology that has a base consisting only of conv ex se ts . • The density d ( X ) of a space X is the least infinite cardinal κ s uc h that X has a dense subset of size at most κ . It is natural to ask what happ ens to the spectrum of No etheria n types of compa ct linear orders if we ge ntly relax the assumption o f compactness. It turns o ut that there are Lindel¨ of linear o rders with No e therian type ω 1 , and, le s s e xpec tedly , that it is co nsistent (relative to existence of an inaccess ible car dinal) that there is a Lindel¨ of linear or der with weakly inacces sible No etherian t yp e. Howev er, it is not consistent for a Lindel¨ of GO -space to have strongly inacessible No etherian type. W e a lso consider the relationship b etw ee n metrizability a nd No etherian type, fo cusing on GO-spa ces. Every metric space has No etheria n type ω . F or a Lindel¨ of GO-space X , X is metr izable if a nd only if N t ( X ) = ω if a nd only if N t ( X ) = ω 1 and X is separable. F or a countable pro duct X of compact linea r orders, X is metrizable if and only if N t ( X ) = ω if and only if N t ( X ) = ω 1 . (Note that e very Lindel¨ of metric space is separable, and every compact GO-space is a compact linear order.) 2. Smal l densities and large Noetherian types Definition 2.1. The π - weight π ( X ) o f a space X is the least infinite cardinal κ such that a space has π -base of size at most κ ; Prop ositio n 2.2. [10] If X is a sp ac e and π ( X ) < cf κ ≤ κ ≤ w ( X ) , then N t ( X ) > κ . GO-SP ACES AND NOETHERIAN SPECTRA 3 Pr o of. Supp ose A is a ba se of X a nd B is π -base of X of size at most π ( X ). W e then hav e |A| ≥ κ ; hence , there exist U ∈ [ A ] κ and V ∈ B such tha t V ⊆ T U . Hence, ther e exists W ∈ A such that W ⊆ V ⊆ T U ; hence, A is not κ op -like.  Note that if X is a pro duct of at most d ( X )-many GO- s paces, then π ( X ) = d ( X ) is witnessed b y the following construction. F or a ny D (top olo g ically) dense in X and of minimal size, collect all the finitely suppor ted pro ducts of top olog ically ope n int erv als with endp oints from the unio n of {± ∞} and the set of all co o rdinates of po in ts from D . T rivially , N t ( X ) ≤ w ( X ) + for a ll space s X . The next example shows that this upper b ound is attained. Example 2.3. [6] The double-arrow spac e , defined as ((0 , 1] × { 0 } ) ∪ ([0 , 1) × { 1 } ) ordered lexicogr aphically , has π -weight ω and weigh t 2 ℵ 0 . By Prop ositio n 2.2, it has No e therian t yp e  2 ℵ 0  + . 3. Lindel ¨ of GO-sp aces Theorem 3.1. Every metric sp ac e has an ω op -like b ase. Pr o of. Le t X b e a metric space. F or ea ch n < ω , let A n be a lo cally finite op en refinement of the (op en) balls of radius 2 − n in X . Set A = S n<ω A n \ { ∅ } . The set A is a base o f X b e c ause if p ∈ X and n < ω , then ther e exis ts U ∈ A n +1 such that p ∈ U a nd U is contained in the ball of r adius 2 − n with center p . Let us show that A is ω op -like. Supp ose that m < ω , U ∈ A , V ∈ A m , a nd U ( V . There then exist p ∈ U and ǫ 0 > ǫ 1 > 0 suc h that the ǫ 0 -ball with center p is contained in U and the ǫ 1 -ball with center p intersects o nly finitely many elements of A n for all n < ω sa tisfying 2 − n > ǫ 0 / 2. I f 2 − m ≤ ǫ 0 / 2, then V is contained in the ǫ 0 -ball with c e n ter p , in con tradiction with U ( V . Hence, 2 − m > ǫ 0 / 2; hence, there are only finitely many p ossibilities fo r m and V given U , for V intersects the ǫ 1 -ball with center p .  Lemma 3. 2. L et X b e a Lindel¨ of GO-sp ac e with op en c over A . The c over A has a c ountable, lo c al ly fi nite r efin ement c onsisting only of c ountable unions of op en c onvex sets. Pr o of. Le t { A n : n < ω } be a countable refinement of A consisting only of op en conv ex sets. F or each n < ω , s et B n = A n \ S m 1 , then choos e U p ∈ A such that p ∈ U p ( V p . Set U = { U p : p ∈ X } . By Lemma 3.2, there exists a countable, lo cally finite refinement B n of U c onsisting only of countable unions of o pen conv ex sets. Since B n is lo cally finite, it has no infinite asce nding chains; hence, we may assume B n is pa ir wise ⊆ -inco mparable bec ause we may shrink B n to its max ima l elements. Let { B n,k : k < ω } = B n . F or each k < ω , set A n,k = U p for some p ∈ X sa tisfying B n,k ⊆ U p . Supp ose m < n and i, j < ω and A m,i = A n,j 6∈ [ X ] 1 . Cho o se p ∈ X such that A n,j = U p ; choose k < ω such that p ∈ B m,k . W e then have B m,i ⊆ A m,i = U p ( V p ⊆ B m,k , in contradiction with the pairwis e ⊆ - inco mparability o f { B m,l : l < ω } . Thu s, { A m,l : l < ω } ∩ { A n,l : l < ω } ⊆ [ X ] 1 for all m < n . By induction, h A n,k i n,k<ω and h B n,k i n,k<ω meet our requirements. Let { X , ≤ , A} ⊆ M ≺ H θ and | M | = ω . Since X is nonsepar a ble, there m ust be a nonempty op en conv ex set W disjoin t from M . Since X is Lindel¨ o f, every isolated p oint of X is in M . Hence, W m ust be infinite. Cho ose { a < c < e } ⊆ W . Cho o se U ∈ A such that U ⊆ ( a, e ). By elementarit y , we may a ssume that hh A n,k , B n,k ii n,k<ω ∈ M for all n, k < ω . F or ea ch n < ω , cho o se i n < ω such that c ∈ B n,i n . T o complete the pro of, it suffices to show that U has infinitely many sup erse ts in A . Fix n < ω . Since c 6∈ M , we cannot hav e A n,i n = { c } ; hence, A n,i n 6 = A m,i m for all m < n . Hence, it suffices to sho w that U ⊆ A n,i n . There exists h I j i j <ω ∈ M such tha t B n,i n = S j <ω I j and each I j is conv ex and o pen. Hence, there exists j < ω such that c ∈ I j . Since U ⊆ ( a, e ) and I j ⊆ B n,i n ⊆ A n,i n , it s uffices to show that ( a, e ) ⊆ I j . Seeking a contradiction, supp ose ( a, e ) 6⊆ I j . Since c ∈ I j , we must have a < b < I j for so me b ∈ X or I j < d < e for so me d ∈ X . By symmetry , we may assume we hav e the for mer. Since X is Lindel¨ of, { p ∈ X : p < I j } ha s a countable cofinal subse t Y . Since I j ∈ M , we may ass ume Y ∈ M . Hence, Y ⊆ M ; hence, there exists y ∈ M such that b ≤ y < I j . Hence, M intersects ( a, e ); hence, M intersects W , which is a bsurd.  Theorem 3.5. L et X b e a Lindel¨ of GO-sp ac e. The fol lowing ar e e quivalent. (1) X is metrizable. (2) X has an ω op -like b ase. (3) X is sep ar able and has an ω op 1 -like b ase. Pr o of. By Theorem 3.1, (1) implies (2 ). By Lemma 3.4, (2) implies (3). Hence , it suffices to show that (3) implies (1). Suppos e X has a coun table dense subse t D a nd GO-SP ACES AND NOETHERIAN SPECTRA 5 an ω op 1 -like base. W e then hav e π ( X ) = ω ; hence, b y Prop os itio n 2.2, w ( X ) = ω ; hence, X is metrizable.  See Exa mple 6 .1 for a no nseparable Lindel¨ of linear order that has No etherian t yp e ω 1 . 4. Small Noetherian types and smaller densities F or co mpact linear ly order e d top olog ical spaces, the theorem at the end of this section streng thens Theorem 3.5. T o pr epare for this theorem, w e first prov e our main tec hnical lemma, which we s tate in very g eneral terms. Lemma 4.1 . Supp ose κ is a r e gular unc ountable c ar dinal, µ is an infinit e c ar dinal, | λ <µ | < κ for al l λ < κ , X is a pr o duct of fewer than κ -many µ -c omp act GO-sp ac es, and N t ( X ) ≤ κ . We then have d ( X ) < κ . Pr o of. Le t X = Q i<ν X i where ν < κ and each X i is a µ -co mpact subspace o f a linearly ordered top olo gical s pace Y i . Seeking a contradiction, supp ose d ( X ) ≥ κ . Let U be a κ op -like base of X , {h Y i , ≤ Y i , X i : i < ν i , U } ∈ M ≺ H θ , | M | < κ , M ∩ κ ∈ κ , and M <µ ⊆ M . (W e can co nstruct M a s the union of an appr opriate elementary chain of length ρ , where ρ is the least regula r car dinal ≥ µ . Such an M is not to o large b ecause ρ < κ , a fact that follows fr om µ ≤ | 2 <µ | < κ and cf ( µ ) < µ ⇒ µ + ≤ | µ cf ( µ ) | < κ ). Since d ( X ) > | M | , there is a finite subpro duct Q i ∈ σ X i of X tha t has a none mpty o p en subset disjoint from M . W e ma y cho ose this op en subset to b e the interior of a set of the for m B = Q i ∈ σ B i where each B i is maximal among the co n vex subsets of X i disjoint fr o m M . Set A i = { p ∈ X i : p < B i } and C i = { p ∈ X i : p > B i } . Since { p : A i < p < C i } = B i , which is no nempty but disjoint from M , we have { A i , C i } 6⊆ M by elementarity . Claim. max { cf ( A i ) , ci( C i ) } ≥ µ for al l i ∈ σ . Pr o of. Seek ing a contradiction, supp o se cf ( A i ) < µ and ci( C i ) < µ . W e then hav e A i ∩ M , C i ∩ M ∈ M . Since A i ∩ M is cofinal in A i , A i = { p : ∃ q ∈ A i ∩ M p ≤ q } ∈ M . Likewise, C i ∈ M , in cont radictio n with the fact that { A i , C i } 6⊆ M .  Therefore, we may ass ume tha t cf ( A i ) ≥ µ fo r all i ∈ σ (by symmetry). Since X is µ -compact, there exists x i = sup Y i ( A i ) = min( B i ) ∈ X i for all i ∈ σ . Claim. Ther e exists y i = sup Y i ( B i ) ∈ ( x i , ∞ ) for al l i ∈ σ , with t he understanding that in this pr o of al l intervals ar e intervals of X i (so y i ∈ X i ). Pr o of. If ci( C i ) ≥ µ , then, by µ -compactness, there exis ts y i = inf Y i ( C i ) ∈ X i . In this case, y i is also max( B i ) beca use y i 6∈ C i . Mo reov er, y i = max( B i ) > min( B i ) = x i bec ause o therwise the interior o f B would b e empty , for x i = sup Y i ( A i ), which is not an isolated po in t in X i . If c i( C i ) < µ , then C i ∈ M , just as in the previous claim’s pro of, so there exists D i ∈ M such that D i is a cofinal subset of { p ∈ X i : p < C i } o f minimal size. In this c a se, D i includes a cofinal subset of B i , so D i 6⊆ M , so | D i | ≥ κ , so µ < κ ≤ | D i | = cf ( B i ), so there exists y i = sup Y i ( B i ) ∈ X i by µ -co mpactness. Also, cf ( B i ) ≥ κ implies sup Y i ( B i ) > min( B i ) = x i . Th us, in any case there exists y i = sup( B i ) ∈ ( x i , ∞ ) for all i ∈ σ .  Let U ∈ U satisfy x i ∈ π i [ U ] ⊆ ( −∞ , y i ) for all i ∈ σ . Since cf ( A i ) ≥ µ > 1 for all i ∈ σ , we then have h x i : i ∈ σ i ∈ Q i ∈ σ W i ⊆ π σ [ U ] where each W i is of the form ( u i , v i ) o r ( u i , v i ] for some u i < x i and v i ≤ y i ; we may ass ume u i ∈ M . Mor eov er, 6 DA VID MILO VICH there exis t p i , q i ∈ X i ∩ M such that u i < p i < q i < x i . Since U is a κ op -like base, it includes few er than κ -many super s ets of T i ∈ σ π − 1 i [( u i , q i )] as members. Since the set of s ups ersets of T i ∈ σ π − 1 i [( u i , q i )] in U is a set in M and a set of size less than κ , it is also a subset o f M . In pa rticular, U ∈ M . Fix an arbitrary i ∈ σ . If cf ( π i [ U ]) < µ , then M would include a cofinal subset of π i [ U ], in contradiction with B i missing M . Therefor e , cf ( π i [ U ]) ≥ µ . Hence, there exists z = sup Y i ( π i [ U ]). By elementarity , z ∈ M , so B i < z , so z = min ( C i ) = sup Y i ( B i ) = y . Beca use of the freedom in how we chose U , it follows that every neighborho o d of x i includes a neighbor ho o d tha t, like π i [ U ], has supremum y i (in Y i ) and has cofinality at lea st µ . Therefore, there is an infinite incr easing sequence of p oints b etw e e n x i and y i that ar e co ntained in every neighborho o d of x i , in contradiction with X i being a subspace of the order ed s pace Y i . Thus, d ( X ) < κ .  Corollary 4.2. Supp ose that X is a pr o duct of at most 2 ℵ 0 -many Lindel¨ of GO- sp ac es s u ch t hat N t ( X ) ≤  2 ℵ 0  + . We then have d ( X ) ≤ 2 ℵ 0 . Corollary 4. 3. Supp ose that κ is a r e gular unc ountable c ar dinal, X is a pr o duct of less than κ -many line arly or der e d c omp acta, and N t ( X ) ≤ κ . We t hen have d ( X ) < κ . The la st co rollary fails for sing ular κ . As we shall s ee in Theor e m 5.1, if λ is an uncountable singular ca rdinal, then N t ( λ + 1 ) = λ , despite the fact that d ( κ + 1) = κ for a ll infinite cardinals κ . Moreover, the s pace ( κ + 1) κ has Noether ian t yp e ω a nd density κ fo r all infinite cardinals κ , so we cannot weak en the ab ov e h yp othesis that X has less than κ -many factors. The equa tio n N t (( κ + 1) κ ) = ω follows fr om a general theorem of Ma lykhin. Theorem 4.4. [5] L et X = Q i ∈ I X i wher e e ach X i has a m inimal op en c over of size t wo (e.g., X i is T 1 ). If sup i ∈ I w ( X i ) ≤ | I | , then N t ( X ) = ω . Pr o of. F or each i ∈ I , let { U i, 0 , U i, 1 } b e a minimal op en c over of X i . Since w ( X ) = sup i ∈ I w ( X i ), we may choos e A to b e a base of X of size at mo st | I | and choose an injection f : A → I . Let B denote the set of a ll nonempty sets o f the form V ∩ π − 1 f ( V )  U f ( V ) ,j  where V ∈ A and j < 2. Since f is injective, every infinite subset of B ha s empt y interior. Hence, B is an ω op -like base of X .  Theorem 4.5. L et X b e a pr o duct of c ountably m any line arly or der e d c omp acta. The fol lowing ar e e quivalent. (1) X is metrizable. (2) X has an ω op -like b ase. (3) X has an ω op 1 -like b ase. (4) X is sep ar able and has an ω op 1 -like b ase. Pr o of. By Theor em 3.1, (1) implies (2), which triv ia lly implies (3). By Coro l- lary 4.3, (3) implies (4). Finally , (4) implies (1) b ecause if X is separ a ble, then π ( X ) = ω , so w ( X ) = ω by Prop osition 2.2.  5. The Noetherian spectrum of the comp act orders Theorem 4.5 implies that no linear ly ordered compactum has Noetherian t ype ω 1 . What is the class of No ether ian types of linearly o r dered compacta? W e shall prov e GO-SP ACES AND NOETHERIAN SPECTRA 7 that a n infinite ca r dinal κ is the No etherian type of a linearly order e d compactum if and only if κ 6 = ω 1 and κ is not w eakly inaccessible. Theorem 5.1. L et κ b e an un c ount able c ar dinal and give κ + 1 the or der top olo gy. If κ is r e gular, then N t ( κ + 1) = κ + ; otherwise, N t ( κ + 1) = κ . Pr o of. Using Corolla ry 4 .3, the low er b ounds o n N t ( κ + 1) ar e easy . W e have d ( κ + 1) ≥ λ for all reg ular λ ≤ κ , s o N t ( κ + 1) > λ for all re gular λ ≤ κ . It follows that N t ( κ + 1) ≥ κ a nd N t ( κ + 1) > cf κ . W e can also pr ove these lower bo unds directly using the Pre ssing Down Lemma. Let A b e a ba se of κ + 1 and let λ b e a reg ula r cardinal ≤ κ . Let us show that A is not λ op -like. F or every limit ordinal α < λ , choo se U α ∈ A such that α = max U α ; choo se η ( α ) < α such that [ η ( α ) , α ] ⊆ U α . By the Pr essing Down Lemma, η is co nstant on a stationar y subset S of λ . Hence, A ∋ { η (min S ) + 1 } ⊆ U α for all α ∈ S ; hence, A is not λ op -like. Once again, it follows that N t ( κ + 1 ) ≥ κ and N t ( κ + 1) > cf κ . T rivially , N t ( κ + 1) ≤ w ( κ + 1) + = κ + . Hence, it s uffices to show that κ + 1 has a κ op -like ba se if κ is singular . Suppose E ∈ [ κ ] <κ is unbounded in κ . Let F be the set of limit po in ts of E in κ + 1. Define B by B = { ( β , α ] : E ∋ β < α ∈ F or sup( E ∩ α ) ≤ β < α ∈ κ \ F } . The s et B is a κ op -like base of κ + 1.  Definition 5.2. Given a po set P with ordering ≤ , let P op denote the set P with ordering ≥ . Theorem 5. 3 . Su pp ose κ is a s ingu lar c ar dinal. Ther e then is a line arly or der e d c omp actum with No etherian t yp e κ + . Pr o of. Set λ = cf κ and X = λ + + 1. Partition the set of limit ordinals in λ + int o λ -many statio nary sets h S α i α<λ . Let h κ α i α<λ be an incr e asing sequence of regular cardinals with supremum κ . F or each α < λ and β ∈ S α , set Y β = ( κ α + 1) op . F or each α ∈ X \ S β <λ S β , set Y α = 1. Set Y = S α ∈ X { α } × Y α ordered lex icographi- cally . W e then hav e N t ( Y ) ≤ w ( Y ) + ≤ | Y | + = κ + . Hence, it suffices to s how that Y has no κ op -like base. Seeking a contradiction, supp ose A is a κ op -like base of Y . F or ea ch α < λ , let U α be the set of all U ∈ A that have a t least κ α -many sup ersets in A . F or all isolated points p of Y , ther e exists α < λ such that { p } 6∈ U α ; whence, p 6∈ S U α . Since h α + 1 , 0 i is iso lated for all α < λ + , there exist β < λ and a set E of successo r ordinals in λ + such that | E | = λ + and ( E × 1 ) ∩ S U β = ∅ . Let C b e the closure of E in λ + . The set C is c lo sed un b ounded; hence, ther e exists γ ∈ C ∩ S β +1 . Set q = h γ , κ β +1 i . W e then hav e q ∈ E × 1; hence, q 6∈ S U β . Since q has coinitiality κ β +1 , any lo cal bas e B at q will co n tain an element U such that U has κ β -many sup e rsets in B . Hence, there exists U ∈ U β such that q ∈ U ; hence, q ∈ S U β , which yields our desired contradiction.  Theorem 5.4. No line arly or der e d c omp actum has we akly inac c essible No etherian typ e. Mor e gener al ly, for every we akly inac c essible κ , pr o duct s of fewer than κ -many line arly or der e d c omp acta do not have No et herian typ e κ . Pr o of. Supp ose κ is weakly inacces sible, X is a pro duct of fewer than κ -many linearly or dered compacta, and N t ( X ) ≤ κ . It suffices to prov e N t ( X ) < κ . By Corollar y 4.3, we hav e d ( X ) < κ ; hence, each fac to r of X has π - weight less 8 DA VID MILO VICH than κ ; hence , π ( X ) < κ . If w ( X ) ≥ κ , then N t ( X ) > κ by Prop osition 2.2, in contradiction with o ur as sumptions ab out X . Hence, w ( X ) < κ ; hence, N t ( X ) ≤ w ( X ) + < κ .  6. The Lindel ¨ of spectr um The sp ectrum of Noetheria n t yp es of Lindel¨ of linear ly ordered top ologica l spa ces trivially includes the sp ectrum o f No etherian types o f compact linear ly order e d top ological s paces. Mor e interestingly , the inclusion is strict, as the next example shows. Example 6. 1. Theorem 4.5 fails for Lindel¨ of linear ly order ed to po logical spa ces. Let X be ( ω 1 × Z ) ∪ ( { ω 1 } × { 0 } ) ordered lexicogra phically . The space X is Lindel¨ of and nonseparable and {{h α, n i} : α < ω 1 and n ∈ Z } ∪ { X \ ( α × Z ) : α < ω 1 } is an ω op 1 -like base of X . Mor eov er, X has no ω op -like base b eca use ev ery lo c a l base at h ω 1 , 0 i includes a descending ω 1 -chain of neighbo r ho o ds. Thu s, N t ( X ) = ω 1 . E asily generalizing this exa mple, if κ is a r egular car dinal and X is ( κ × Z ) ∪ ( { κ } × { 0 } ) ordered lexicogra phically , then X is κ -compact and N t ( X ) = κ . A consequence of Lemma 4.1 is that Lindel¨ of linear ly ordered topo logical spaces cannot hav e strongly inaccessible No etherian type, just as in the compact case. More generally , we hav e the following theorem, whic h is pr ov e d just as Theore m 5.4 was prov ed. Theorem 6.2. Supp ose κ is a we akly inac c essible c ar dinal, | λ <µ | < κ for al l λ < κ , and X is a X is a pr o duct of fewer than κ -many µ - c omp act GO-sp ac es. We then have N t ( X ) 6 = κ . Pr o of. Supp ose that N t ( X ) ≤ κ . Let us show that N t ( X ) < κ . By Le mma 4.1, we hav e d ( X ) < κ ; hence, each factor of X has π -weight less than κ ; hence, π ( X ) < κ . If w ( X ) ≥ κ , then N t ( X ) > κ by P rop osition 2.2, in contradiction with our assumptions abo ut X . Hence, w ( X ) < κ ; hence, N t ( X ) ≤ w ( X ) + < κ .  Corollary 6.3. If κ is stro ngly inac c essible, then the class of No etherian typ es of µ -c omp act GO-sp ac es ex cludes κ if and only if µ < κ . Pr o of. “ If ”: Theorem 6.2. “Only if ”: Exa mple 6.1.  On the o ther hand, it is consistent (re la tive to the consistency of an inaccess i- ble), that some Lindel¨ of linea rly order ed to p olo gical space has w eakly inacces s ible No etherian type. T o show this, we first force 2 ℵ 0 ≥ κ wher e κ is weakly inaccessible (say , by adding κ -many Cohen reals). Next, we construct the desired linear o rder in this fo rcing extension using the following theore m. Theorem 6 .4. If κ is a we ak inac c essible and 2 ℵ 0 ≥ κ , then ther e is a Lindel¨ of line ar or der Z such that N t ( Z ) = κ . Pr o of. Le t B b e a Bernstein subset of X = [0 , 1], i.e. , B includes some po in t in P and misses some p oint in P , for all p erfect P ⊆ X . Let f : B → κ b e s ur jective. F or ea ch x ∈ B , set Y x = ω op + ω f ( x ) + ω , which is L indel¨ of. F or ea ch x ∈ X \ B , s e t Y x = { 0 } . Set Z = S x ∈ X ( { x } × Y x ) ordered lexicog raphically . First, let us s how that Z is Lindel¨ o f. Let U b e an open cov er o f Z . F or every x ∈ X \ B , h x, 0 i has neighborho o ds O x and U x such that U ∋ U x ⊇ O x = S a

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